Mechanics of fracture

VOLUME 6

Cracks in composite materials

Mechanics of fracture

edited by GEORGE C. SIR

VOLUME 1

Methods of analysis and solutions of crack problems

VOLUME 2

Three-dimensional crack problems

VOLUME 3

Plates and shells with cracks

VOLUME 4

Elastodynamic crack problems

VOLUME 5

Stress analysis of notch problems

VOLUME 6

Cracks in composite materials

Mechanics of fracture

Cracks in composite materials

A compilation of stress solutions for composite systems with cracks

G. C. SIH Lehigh University Bethlehem, Pennsylvania USA

and

E. P. CHEN

Sandia Laboratories Albuquerque, New Mexico USA

II 1981 MARTINUS NIJHOFF PUBLISHERS

THE HAGUE / BOSTON / LONDON

6

Distributors:

for the United States and Canada Kluwer Boston, Inc. 190 Old Derby Street Hingham, MA 02043 USA

for all other countries Kluwer Academic Publishers Group Distribution Center P.O. Box 322 3300 AH Dordrecht The Netherlands

Library of Congress Catalog Card Number: 81-50355 ISBN-13: 978-94-009-8342-7 e-ISBN-13: 978-94-009-8340-3 001: 10.1007/978-94-009-8340-3 ISBN 90-247-3006-6 (series)

Copyright © 1981 by Martinus Nijhoff Publishers bv, The Hague. Softcover reprint of the hardcover 1st edition 1981 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, photocopying, recording, or otherwise, without the prior written permission of the publisher, Martinus Nijhoff Publishers bv, P.O. Box 566, 2501 CN The Hague, The Netherlands.

Contents

Editor's preface

Introductory chapter Failure of composites as predicted by the strain energy

density theory G. C. Sih

Chapter 1 Cracks in materials possessing homogeneous anisotropy

1.1 Introduction 1 1.2 Anisotropic elasticity 1 1.3 Plane and anti-plane problems of cracks in anisotropic

materials 9 1.4 Penny-shaped crack in transversely isotropic bodies of infinite

extent 22 1.5 A finite width orthotropic body with a central crack 33 1.6 Through crack in an orthotropic layer of finite height 45 1.7 Transversely isotropic cylinder containing a penny-shaped

crack 55 1.8 Transversely isotropic layer containing a penny-shaped crack 66 1.9 Bending of anisotropic plates with cracks 76 1.10 Generalized plane deformation of aniostropic materials with

cracks 87 1.11 Concluding remarks 97 1.12 Appendix: Method of solution for solving dual integral

equations 99 References 100

Chapter 2 Nonhomogeneous materials with cracks

2.1 Introduction 102

IX

2.2 Shear modulus varying a direction normal to the plane crack 102 2.3 Interaction of axial inhomogeneity with a penny-shaped crack 109

VI Contents

2.4 Appendix: Stress field and coefficient for a plane crack in nonhomogeneous solid 115

References 116

Chapter 3 Interface cracks in bimaterial systems

3.1 Introduction 117 3.2 Straight line cracks between two dissimilar media 118 3.3 Bimaterial solid debonded over a penny-shaped region 136 3.4 Cracks normal to bimaterial interface 146 3.5 Bending of bimaterial plates with cracks at and normal to

interface 154 3.6 Appendix: Crack tip stress field and stress intensity factor

solutions 166 A3.1 Oscillatory stresses in plane extension 167 A3.2 Oscillatory stresses in plate bending 168 A3.3 Concentrated forces at an arbitrary point 171 A3.4 A partially bonded cylinder in an infinite medium 172 A3.5 Bending of bimaterial plates debonded along a straight

line 173 A3.6 Partially de bonded circular insert in a bent plate 175

References 176

Chapter 4 Composite material with a cracked layer and fiber

4.1 Introduction 178 4.2 A cracked layer sandwiched between two half-planes 179 4.3 Layered composite with a crack normal to interface 198 4.4 Penny-shaped crack parallel to interface of layered composite 218 4.5 Embedded cylinder with a crack normal to the interface 233 4.6 Cracks in composites with orthotropic layers 255 References 275

Chapter 5 Dynamic response of dissimilar materials with cracks

5.1 Introduction 277 5.2 Parallel crack in a sandwiched layer 278 5.3 Sandwiched layer with a crack normal to interface 304

Contents VII

5.4 Embedded penny-snaped crack parallel to composite interface 328 5.5 Cracked cylindrical fiber embedded in a matrix 351 5.6 Anti-plane shear of interface rectangular cracks III layered

orthotropic dissimilar materials 375 5.7 Orthotropic layered composite debonded over a penny­

shaped region subjected to sudden shear 386 5.8 Diffraction of time-harmonic waves by interface cracks in

dissimilar media 397 5.9 Moving cracks in layered media of dissimilar materials 421 Appendix: Inverse Laplace transform of dynamic stress intensity factor 437 References 439

Chapter 6 Plane extension and bending of laminate composite

plates with cracks: static and dynamic loading

6.1 Introduction 441 6.2 A brief review of existing laminate plate theories 443 6.3 A laminate plate theory with boundary layer 461 6.4 Sudden extension of a cracked laminate 489 6.5 Bending theory of laminated plate: static and dynamic 502 References 527

Author's Index 531

Subject Index 535

Editor's preface

Composites offer great promise as light weight and strong materials for high performance structures. One of the major advantages of these materials as compared with metals is the basic way in which heterogeneity resist crack extension. In a fiber/matrix composite system, the fibers tend to cause cracks to form at closer spacing and delay the formation of a large crack. The enhancement of local failure such as fiber breaking, matrix cracking and interface debonding further reduces the energy level which might have otherwise reached the point of catastrophic failure.

Even though substantial tests have been made on composite materials, little has been gained in the understanding and development of a predic­tive procedure for composite failure. There are fundamental difficulties associated with incorporating the nonhomogeneous and anisotropic prop­erties of the composite into the continuum mechanics analysis. Additional uncertainties arise from voids and defects that are introduced in the composite during manufacturing. Even a small quantity of mechanical imperfections can cause a marked influence on the composite strength. Moreover, the interface properties between the fibers and matrix or bonded laminae can also affect the load transmission characteristics significantly. It would be impossible to establish predictive procedures for composite failure unless realistic guidelines could be developed to control the manufacturing quality of composite systems. (1 A discipline concerned specifically with the failure of material by crack initiation and propagation has been known as fracture mechanics. It has had much success in characterizing the fracture behavior of polycrystals by scaling the size of the macrocrack and specimens such that the microstructure of the material can be overshadowed and the assumption of material homogeneity and isotropy can be justified. Whether this same assumption could be carried over to explain failure of the fiber/matrix composite system depends to a great extent on how the specimen size is scaled relative to fiber spacing and size of the critical crack or damage zone corresponding to catastrophic fracture. It is apparent that the failure modes considered in ordinary composite specimens are observed at the

x Editor's preface

scale level where heterogeneity and anisotropy of the material cannot be ignored. The presence of many small cracks and/or a large crack will further complicate the analytical development of predictive techniques.

In view of the foregoing remarks, there remains a wide gap between analysis and standard test procedures. There is still lacking the capability of estimating failure loads expected in service from strengths determined on small-scale laboratory specimens. It is not the intention of this Volume to resolve the general problem of composite failure but rather to analyze composite systems with crack-like defects in terms of the stress intensity factor parameters so commonly used in the linear theory of fracture mechanics. The basic stress and/or displacement solutions are presented for a variety of composite systems which serve as the basis for developing predictive procedures. The failure of relatively simple composite struc­tures is examined by the strain energy density criterion in the Introduc­tory Chapter. This criterion, unlike the conventional G 1c or k1c approach, is particularly suited for treating cracks or defects that spread in a nonself-similar fashion. Mixed mode crack extension will generally occur in composites as the rule rather than the exception. The free surfaces created by broken fibers, cracked matrix and debonded interfaces will lie in planes most likely not normal to the direction of the applied load.

Two analytical models are proposed for predicting the failure of unidirectional fiber-reinforced composites subjected to a load that is oriented at an arbitrary angle with the crack plane. The first model assumes that the composite is an ideal homogenous anisotropic body and the second assumes that failure takes place in a layer of matrix material. Brittle fracture is assumed to occur when the strain energy stored in an element ahead of the crack reaches a critical value, Sc Results are presented for E-glass fiber-reinforced plastics, stainless steel fiber rein­forced aluminum and graphite fiber epoxy materials and compared with available experimental data. The degree of material nonhomogeneity, which depends on the fiber volume fraction and the difference of elastic properties of the fiber and matrix, affects the state of affairs in the region ahead of the crack tip. This nonhomogeneity is found to be more significant in composites with plastic resin matrix than with metal. The matrix cracking model agrees well with the experimental results on pre cracked Scotchply 1002 sheets provided that the load direction is not too closely aligned with the fiber orientation.

The strain energy density criterion applies equally well to the dynamic fracture of unidirectional fibrous composites. Under impact, waves are reflected and refracted from the fibers and create a complex pattern. By

Editor's preface XI

assuming that matrix cracking is the dominant mode of failure, the maximum intensification of the local strain energy density is found to occur very quickly when the waves traveling from one end of the crack tip reach the other. The critical dynamic stress level is determined as a functi0n of the geometry and material parameters of the composite. When Smin exceeds Sc, the crack begins to propagate with a velocity that depends on the degree of nonhomogeneity in the composite system. This dependence is discussed for a crack running in the matrix material.

Two failure modes that commonly arise in angle-ply laminates made of layered unidirectional composites are presented. They are known as thru-lamina and interlaminar cracking. The latter has also been referred to as delamination. Results are given for a balanced four-layered lami­nate. Possible delamination is assumed to coincide with the locations of minimum strain energy density. Such a location indeed occurs in a small region confined to the free edges.

The theory of anisotropic elasticity is briefly reviewed in Chapter One in order to define the material constants for systems with different elastic symmetry. Special emphases are given to the determination of the stresses and displacements near a plane of discontinuity or a crack defined as two neighboring surfaces free of tractions. The boundaries of the cracks are selected to conform with the preferred directions in the material.

Rectangular and circular cracks in materials with rectilinear and polar anisotropy are treated. A system is said to be orthotropic when the crack plane coincides with one of the principal axes of rectilinear anisotropy. A number of crack boundary value problems involving in-plane extension, out-of-plane shear and bending are solved. When a material possesses only one axis of elastic symmetry that is directed normally through the center of a penny-shaped crack, the governing equations correspond to those for a transversely isotropic body and are simplified considerably.

It is customary to express the local stresses in terms of the radial distance and angle measured from the right hand side of the crack tip. Although the material anisotropy does not affect the crack tip stress singularity, the inverse square root of the radial distance, the angular distribution is disturbed by the material constants. For convenience, the stress intensity factors for cracks in anisotropic bodies are defined so that when the bodies are infinite in extent and subjected to self-equilibrating loads, they are the same as those for the corresponding isotropic bodies. The anisotropic elastic constants will enter into the stress intensity factor expression in situations where the crack interacts with a specimen bound­ary or when the applied load is non self-equilibrating.

XII Editor's preface

Previously unpublished results on the generalized plane deformation of anisotropic materials with cracks is also discussed in Chapter One. This formulation differs from the plane strain solution in that all six indepen­dent stress components prevail even though the loading is two­dimensional. The stress intensity factors for a concentrated force oriented in an arbitrary direction and applied in one of the crack surfaces are obtained. This permits the generation of other solutions involving a variety of tractions applied to the crack surfaces.

Chapter Two considers cracks in a nonhomogeneous material whose elastic constants may vary with the space coordinates. Since the influence of the Poisson's ratio is known to be less significant, only the variation in the modulus of elasticity will be considered. Stress solutions are obtained for the rectangular and circular crack in a material whose shear modulus varies as a continuous function of the space variable directed normal to the crack plane. The functional relationship of the crack tip stresses is entirely different from that for a homogeneous material. The order of the stress singularity depends on the degree of material nonhomogeneity. It follows that the coefficient of the singular stress field can no longer be interpreted as the same kind of stress intensity factor as in the linear theory of fracture mechanics.

A class of problems dealing with cracks at the dividing line of two homogeneous dissimilar materials is considered in Chapter Three. The idealized condition of perfect bonding* and traction-free surfaces sepa­rated by the crack point at the interface give rise to a stress singularity that is oscillatory in character. This oscillation leads to interpenetration of portions of the crack surface in the immediate vicinity of the singular point and is incomprehensible to the ordinary mind. Such a localized region must be excluded from the analysis. A typical feature of dissimilar material problems with interface cracks is that at least two stress intensity factors will be nonzero even if the loading is symmetric or skew­symmetric with respect to the plane of the crack. For cracks along straight and curved interfaces owing to extensional and bending loads, the methods of eigenfunction expansion and complex function theory may be coupled to solve a variety of boundary value problems. Solutions are also presented for other crack configurations involving a debonded circular region and a crack normal to the interface.

Stress and strain analysis of layered composites are prerequisites to

* Perfect bonding implies that stresses and displacements are assumed to be continuous across the interface.

Editor's preface XIII

determining the mechanical behavior of laminates. The objective of Chapter Four is to investigate the effect of a crack in the fiber or matrix of a unidirectional composite. No attempt is made to analyze the layered system as a plate specimen. This will be addressed in Chapter Six. Instead, attention will be focused on the redistribution of the stresses near the crack. The material remote from the crack is assumed to possess the average properties of the fiber and matrix. The simplified model consists of a cracked layer sandwiched between two isotropic or anisotropic homogeneous materials. Matrix cracking and fiber breaking are simulated by having the crack aligned parallel or normal to the interface. Defects are taken to be in the form of a rectangular or circular plane of discontinuity. Numerical results are displayed graphically to illustrate how the stress intensity factors are affected by the nonhomogeneous nature of the layered composite.

Chapter Five is devoted to the dynamic response of cracked layered composites of infinite extent. Laplace transformation on the time variable is made such that the stress intensity factor can be extracted from the asymptotic solution of the stresses in the transformed plane. This proce­dure requires only the Laplace inversion of the stress intensity factor which fluctuates with time. Regardless of whether the crack is directed parallel or normal to an interface, the intensity of the local stresses rise to a peak and then oscillate in time with decreasing amplitude. The mag­nitude and location of the peak depends on the arrival of the travelling waves from one end of the crack tip to the other. This in turn is dictated by the crack length to layer height ratio and difference in the elastic moduli of the adjoining layers. The way with which material nonhomogeneity affects initiation and/or arrest characteristics of cracks in composites is investigated by solving the problem of a moving crack in a layered medium. The velocity of the crack may be increased or decreased depending on the construction of the composite.

Static and dynamic laminated plate theories are developed in Chapter Six. These theories are approximate in that they assume the stresses can be represented by functions in terms of the space variables in product form. Unlike the classical plate theories, the thickness-dependence is not assumed arbitrarily but determined by imposing the condition of plane strain on the asymptotic stress field. The basic configuration is that of a three-layered plate containing a through crack, although the general formulation applies to any number of layers. The stress intensity factors associated with the in-plane stresses are found to be different from those for the transverse normal stress. They are, however, related through the

XIV Editor's preface

plane strain condition. The laminated plate theory solution does not apply in regions close to a free surface or interface where high stress gradients are likely to occur. Such characteristics are constructed by introducing the concept of boundary layer. The thickness of this layer will, in general, depend on the elastic and geometric parameters of the laminate.

Dynamic stress intensity factors for cracked laminates under the action of sudden stretching and bending are obtained. The complex system of governing equations are solved by isolating the singular portion solution from the numerical analysis. The local stresses tend to fluctuate with time. Their intensity reaches a peak and then acquire an oscillatory character with decreasing magnitude. These results are helpful for understanding the way with which composite laminates behave under impact.

I am indebted to those individuals who were associated with many of the composite material research programs at Lehigh University. Their contributions are gratefully acknowledged. Finally, the typing of this volume would never have been completed without the continued service and patience of my secretaries Mrs. Barbara DeLazaro and Mrs. Con­stance Weaver.

Lehigh University Bethlehem, Pennsylvania

October, 1980

G. C. SIH

G. C. Sih

Introductory chapter:

Failure of composites as predicted by the strain energy density theory

I. Preliminary remarks

If composite materials are to be used effectively in high performance structural applications, it is necessary to develop accurate reliable predic­tive methods for describing their failure behavior. The various failure modes observed at the laboratory scale level are fiber breakage, matrix separation and interface debonding. Should the load be increased indefin­itely, the existing defects will grow while new ones may be created and damage of the composite will eventually reach catastrophic proportions. The progressive failure process in a composite is extremely complex and it would be impossible to analytically describe each successive stage in detail. This would not only be impractical but also would shed little light on engineering applications. On the other hand, methods of analysis procedures are needed to bridge the gap between composite testing and design. Only in this way can reliable estimates of the allowable stresses expected in service be made from strength and fracture toughness data obtained from small-scale laboratory tests. This capability still does not generally exist because of the complexity of cracks interacting with material inhomogeneity and anisotropy, effects that cannot be ignored in fiber-reinforced composites. To treat this subject, the discipline of frac­ture mechanics,* which is concerned with failure by crack initiation and propagation, is a natural tool to use for this purpose.

* The term fracture mechanics acquires a much broader meaning in this communication than the conventional G Ie or k,c concept. It addresses the initiation, stable growth and/or unstable propagation of cracks.

XVI

(a) Global Homogeneous

Anisotropic Continuum

Introductory Chapter

(b) Global Nonhomogeneous

Isotropic Continuum

Figure 1. Scaling of the continuum element with reference to crack size and microstructure of the composite.

Prior to the application of fracture mechanics, however, a few remarks on scaling the continuum element relative to defect size and microstruc­ture of the composite is in order. Loosely speaking, there are two schools of thought in the literature. The first assumes a global homogeneous anisotropic* continuum such that crack extension takes place in an idealized material with the gross combined properties of the constituents. This approach excludes detailed consideration of local damages such as broken fibers, cracks in matrix, etc. The second is to consider a global nonhomogeneous anisotropic medium where the crack tip with a known radius of curvature can exist only in one phase of the composite material

* The homogeneous anisotropic assumption has also been applied to continuum models using the concept of "micromechanical" elements whose mechanical properties are deter­mined from laboratory specimens that are several orders of magnitude larger in size. The degree to which the mechanical behavior of composites is dependent on damage due to the size variance between the specimen and continuum element is not at all clear. Homogeneity at the global scale level may not hold in the local region where material is being damaged.

II Strain energy density criterion XVII

at a given instance. The tip must either lie in the fiber, matrix or at the interface between two media. Figures l(a) and l(b) illustrate the scaling of the continuum element with reference to the crack size and microstruc­ture of the composite. The validity of these approaches depends of course on how the specimen dimensions are scaled with respect to the allowable strcss level and size of the cracks or damage zone. The reason is that failure modes are dictated by the rate and amount of energy dissipated which depends on specimen size and the kind and rate of loading. The sequential order of discrete failure modes can also greatly influence the life of composite laminates. This is known as the stacking sequence effect.

The experimental procedure for measuring the effective moduli of a composite should be distinguished from that of strength determination. The former assumes the generic stress/strain relations and geometries to remain unchanged under load and material damage is kept to a minimum or is assumed to be negligible. Strength prediction implies the monitoring of progressive failure under increasing load up to the final stage of fracture. It is the specimen size and loading rate that determine whether analysis should account for progressive material damage.*

II. Strain energy density criterion

The main concern in fracture analysis consists in characterizing the action or phenomenon that causes a structural member to fail by slow and/or rapid crack extension. This necessitates the selection of a quantity that can most consistently relate the observed fracture mode with such factors as material properties, type of loading, shape of member, etc. The strain energy density function, d Wid V, as proposed by Beltrami, Haigh and Ludwik at the beginning of the 20th century possesses the unique feature that it can be expressed in terms of strains or stresses:

d W 1 + v [2 2 2 V ( )2] d V = 2E 0" 1 + 0"2 + 0" 3 - 1 + v 0"1 + 0"2 + 0"3 (1)

in which 0"1, 0"2 and 0"3 are the principal stresses, v is Poisson's ratio and E is Young's modulus. In the linear theory of elasticity, dW/dV may be regarded as the sum of two energy components, (dW/dV)d and

* This is analogous to the typical phenomenon of metal fracture which is always preceded by slow crack growth.

XVIII Introductory Chapter

(dW/dV)v' The former

(2)

can be identified with shape change and is, in fact, the Huber, von Mises and Hencky yield condition while the latter

(3)

is associated with volume change and condition for fracture. The value of d Wid V at fracture, say (d Wid V)C' can be evaluated from a simple tension test by calculating the area under the true stress/true strain curve as

(4)

where ec is the critical strain. Both Matthaes [1] and Gillemot [2] have discussed the association of (d Wid V)c with the failure of tensile bar specimens. Experimental values of (d Wid V)c were also reported in [4].

The way with which (dW/dV)c is related to the condition of crack instability was established by Sih [3,4] through the critical strain energy density factor

(5)

with rc being the radius of a core region surrounding the crack tip.* The size of this region is characteristic of the material [5,6]. An important advancement of the strain energy density criterion [4] is that the energy absorbed in changing volume and shape should be considered simultane­ously when searching for possible sites of failure. A possible means of weighing (dW/dV)d in equation (2) and (dW/dV)v in equation (3) is to consider the stationary values of the total strain energy density function with respect to a set of polar coordinates measured from the crack front. The locations of (dW/dV)rnax correspond to yielding and (dW/dV)m;n to fracture or crack extension. The critical value of equation (1), (dW/dVL

* In ductile fracture [8], rc represents the last ligament of slow crack growth just prior to the onset of rapid fracture.

II Strain energy density criterion XIX

is assumed to remain constant along the crack front, a condition that has been used to determine crack profiles in elastic [7] and elastic-plastic materials [8].

Direction of crack inItiatIOn. Failure modes in composite materials are more complex than those in more conventional materials. Because of nonhomogeneity and anisotropy, self-similar crack growth is not likely to occur in composites even in simple tension. The classical concept of energy release rate Ole may no longer be easily applied. A more convenient approach is to apply the strain energy density criterion. For the onset of rapid fracture, this criterion may be stated as follows:

(1) Crack initiation takes place in a direction determined by the relative minimum of the strain energy density factor, i.e.,

as = 0(a2s>0) at 0 = 00 ao ao 2 ' (6)

(2) Rapid crack growth occurs when the minimum strain energy density factor reaches a critical value:

(7)

The Se -criterion is particularly useful for analyzing crack systems involv­ing complex loadings where the direction of crack initiation is not known a priori. The parameter Sc is, in fact, directly related to K 1c by the relation*

S =(l+v)(l-2v) K c 27TE 1e

(8)

This implies that the Se values for metal alloys can be found in the ASTM standard fracture toughness testing procedures. An alternate way of evaluating Se has been presented in [10, 11].

Since the material in the immediate vicinity of the crack front is highly stressed and its behavior is not exactly known, the continuum mechanics analysis must necessarily be restricted to regions outside of a small zone surrounding the crack edge. For three-dimensional cracks, the shape of this core may be conveniently taken as the interior of all spheres like that

* The stress intensity factor K, and k, are related by K 1 = hk, .

xx

Crock Plane

z

y

Spherical

Core Region

(a) Three Dimensions

x

Introductory Chapter

Circular Core Region

Crock

( b) Two Dimensions

Figure 2. Core region surrounding the crack front.

of Figure 2(a), the three-dimensional crack, Figure 2(a), and circular for the two-dimensional crack, Figure 2(b). From the linear theory of iso­tropic and homogeneous elasticity, the strain energy density function is found to possess a singularity of the order* l/r near the crack front and can be expressed in terms of three stress intensity factors kj (j = 1, 2, 3):

dW S -=-+ ... dV r

(9a)

where

(9)

The coefficients a ij (i, j = 1, 2, 3) are given by

16M cos c{Ja ll = (3 - 4v cos 0)(1 + cos 0) (10a)

16M cos c{Ja12 = 2 sin O(cos 0 -1 +2v) (lOb)

16M cos c{Ja22 = 4(1- v )(1- cos 0) + (3 cos 0 -1)(1 + cos 0) (lOc)

16M cos c{Ja33 = 4 (10d)

* The order of strain energy density singularity remains the same for finite deformation [9].

II Strain energy density criterion XXI

where /-L is the shear modulus of elasticity. The direction of crack initiation and condition of fracture are then determined by application of equations (6) and (7). For two-dimensional crack problems, the element with d V = r dr dO . dz is always situated in a place normal to the crack edge such that the angle <p can be set to zero in equations (10).

Incremental crack growth. In order to achieve a predictive capability for composite fracture, the sequence of failure events, whereby a crack extends stably under an increasing load up to rapid growth, must be determined from the composite configurations and applied loads. The rate at which the crack grows incrementally can be associated with the amount of energy absorbed and subsequently released by each element ahead of the advancing crack. Suppose that the (dW/dV)c value for a material is known, then the condition

(11)

prevails for stable crack growth as well as the onset of catastrophic fracture, equation (5).

If the fracture process due to increasing load is unstable, then each increment of crack growth will increase monotonically, i.e.,

(12)

The corresponding strain energy density factors will also increase accord­ingly

(13)

such that the ratio Sj/rj (j = 1, 2, ... ,n) is a constant as assumed by equation (11). The last ligament of growth rc corresponds to instability as illustrated in Figure 3. A stable fracture process corresponds to decreas­ing increments of crack growth:

(14)

where ro represents the last increment of growth before crack arrest. The strain energy density factors SI, S2, ... , So also decrease according to

XXII

Crack

a

Radius Of Core Region

Introductory Chapter

Onset Of Rapid

Crack ~ Growth t

Figure 3. Incremental stable crack growth leading up to rapid fracture.

(15)

The factor

(16)

is below the critical value Sc In a composite structure, a combination of the conditions described by

equations (12) and (14) can exist. That is the increments of crack growth may either increase or decrease depending on several factors such as the prevailing mechanical constraint, the material properties, nature of load­ing, etc.

III. Failure of fiber, matrix or interface

As mentioned earlier, real failure modes in fiber-reinforced composite systems are very complex as they may involve fiber breakage, matrix cracking, fiber/matrix debonding, and laminae delamination during a period of gradually increasing load. Although a precise quantitative assessment of all the failure modes is beyond the scope of this communi­cation, reasonably accurate predictions of the allowable stress can be made if anyone of the aforementioned failure modes tends to dominate at the onset of rapid fracture. The local damage done prior to global

III Failure of fiber, matrix or interface XXIII

instability simply alters the stiffness and/or available energy to cause final fracture. Therefore, it is useful to have analytical representation of the individual failure modes and to understand how the physical properties of the constituents affect the fracture behavior of composite systems.

Critical strain energy density function for fiber and matrix. Needless to say, data from fracture tests on isolated fibers and on matrix materials are prerequisite to the development of predictive procedures. Standard sim­ple tension tests as shown in Figures 4(a) and 4(b) may be performed to obtain the values of (dW/dV)f and (dW/dV)m at fracture in accordance with equation (4). The specimen size and loading rate should be selected such that the influence of preexisting defects will not have to be consi­dered in the strength calculation. Special consideration must also be given to the dependency of d Wid V at fracture on specimen size. The way in which (dW/dV)f and (dW/dV)m are utilized in estimating the fracture resistance of larger size composite members, in which existing defects or cracks must be accounted for, will be examined by application of the strain energy density criterion.

Crack initiation in fiber, matrix or interface. The commonly observed failure modes of fiber breaking and/or matrix cracking are generally determined by the combination of loading, location of defect and material nonhomogeneity. The fracture mechanics viewpoint is that all failure initiates from a preexisting defect. Figures 5(a) and 5(b) illustrate three

CTm

o EO m

(a) Fiber Specimen (b) Matrix Specimen

Figure 4. Measurements of the strain energy density function at fracture.

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III Failure of fiber, matrix or interface xxv

sites of possible failure initiating from a crack of length 2a situated near the interface. The first possibility is matrix cracking if the loading angle {3 is sufficiently large since the shear modulus of the fiber t-tf is much greater than that of the matrix t-tm' As (3 is decreased towards zero, more load is transmitted to the fiber and there is more likelihood of fiber breakage. Failure within the interface is the third possibility.

Assuming that the defect does not grow along the interface, the cohesive failure model in Figure 5(a) may be adopted. The upper half­plane is occupied by the matrix material with elastic properties t-tm, Vm

and the lower half-plane by the fiber with elastic properties t-tf' vf' For this configuration, asymmetry prevails in both loading and material, and hence two types of stres~ intensity factors kl and k2 are present [12]

"-

kl = (T~[cos (£ log 2a) + 2£ sin (£ log 2a)]/cosh ('1T£)

k2 = -(T~[sin (£ log 2a) - 2£ cos (£ log 2a)]/cosh ('1T£)

in which £ is a bi-material constant given by

£ =~ log [(Km +-.l)j(Kf +~)] 2'1T t-tm t-tf t-tf t-tm

(17a)

(17b)

(18)

The relation Km = 3 - 4vm holds for both the matrix and fiber material under the condition of plane strain. Once kl and k2 are known, the strain energy density factor S may be computed from equation (9) with k3 = 0 and the following coefficients for a;j (i, j = 1, 2):

16t-tmal1 = (Km -1)e-2(7T-e)E{1 + cos [0 +2£ log (r/a)]}

+ e-2 (7T-e)£(cos 0 + 2£ sin 0)2 + e2(7T-e)£ - 2(cos 0 + 2£ sin 0)

x cos [0 +2£ log (r/a)]

16t-tma12 = -(Km _1)e-2Cn--e)£ sin [0 + 2£ log (r/a)]

+ 2(cos 0 + 2£ sin 0) sin [0 + 2£ log (r/a)]

16/-Lma22 = (Km -1)e-2 (7T-e)£{1-cos [0 +2£ log (r/a)]}

(19a)

(19b)

+e-2 (7T-e)£(cos 0+2£ sin 0?+e2 (1T-e)£+2(cos 0+2£ sin 0)

x cos [0 + 2 £log (r/a)] (19c)

Substituting equations (19) into (9), the resulting expression S may then be minimized with respect to O. This yields the direction of crack initiation, 00, as indicated in Figure 5(a). Matrix breaking occurs when

XXVI Introductory Chapter

Smin reaches Sc which can be found from the product rm (dWjdV)m with rm being the size of the core region* for the matrix material.

The foregoing treatment also applies to fiber breaking simply by interchanging the subscripts m and f in equations (18) and (19). The value Sc for the fiber becomes rf (d Wid V)f in which case rf must be evaluated for the fiber material.

Should the dissimilar material properties of the fiber, matrix and interface and loading direction be combined to enhance crack spreading along the interface, then a third layer of material with thickness 2h may be added to the model as shown in Figure 5(b). Modeling of the properties of this interface layer, however, is open to uncertainty. Sih and Moyer [14] have shown that the difference between an interface with gradual modulus variation and constant average modulus can be signific­ant depending on the boundary conditions specified remotely away from the interface. The results suggest that even though the interface may be thin in comparison with the dimensions of the two joining materials, it can exert appreciable influence on the gross mechanical behavior of the composite. A possible way of circumventing this problem is to optimize the properties of the constituents with the local stress/strain condition such that the interface will not be overly sensitive to the response of the composite.

If the original defect, say a crack, is located either in the matrix or fiber, then the material surrounding the crack may be regarded as homogeneous and equation (9) may be used. Stress intensity factor solutions for a variety of configurations and loadings involving matrix or fiber weakened by defects can be found in several Chapters of this Volume.

IV. Unidirectional fiber/matrix composite

The unidirectional fibrous composite is the simplest two-phase material constructed by embedding a system of parallel fibers in a matrix. Two basic continuum mechanics approaches will be considered:

(1) a crack propagating in a layer of isotropic matrix material sandwiched between two anisotropic solids, and

(2) a crack extending in a homogeneous and anisotropic material

* The size of the core region around a crack tip for a given material has received much attention [6, 13]. It can be interpreted as the zone outside of which failure is assumed to initiate.

IV Unidirectional fiber/matrix composite XXVII

having the gross mechanical properties of the fiber and matrix or compo­site. Critical stresses for these two models will be obtained and compared with available experimental data.

Matrix cracking model. Experimental work [15] on Scotchply 1002 (trade name by the Minnesota Mining and Manufacturing Company) has shown that unidirectional composites fail by crack propagation between the fibers starting from an initial defect such as voids or air bubbles in the composite. On the basis of this observed failure mode, the analytical model in Figure 6(a) is chosen with a crack of length 2a in a layer of matrix material having elastic properties Em, v'" or I1-m' Vm. The material surrounding this layer of thickness 2h is taken to be orthotropic with the elastic properties E 1, E 2 , V12, V21 and 11-12. The subscripts 1 and 2 designate the directions parallel and perpendicular to the fiber as illus­trated in Figure 6(a). On the average, the spacing 2h between the fibers may be estimated from the fiber volume fraction Vf by assuming that all the fibers having radius R, are uniformly distributed throughout the composite. The resulting expression is

2h = R[vi1T/Vf -2] (20)

For the configuration shown in Figure 6(a), the applied stress (J makes an angle (3 with the crack plane and the intensity of the crack tip stress field is described by the following kj (j = 1, 2) factors:

kl = <P(I)(Jv!a sin2 (3

k2 = P(I)(Jv!a sin (3 cos (3

(21a)

(21b)

in which the functions <p(~) and P(~) depend on the elastic constants and geometric parameters of the composite. They can be computed numeri­cally from a system of Fredholm integral equations [16]. The gross elastic properties for the Scotchply 1002 composite with a fiber volume fraction of 56.5% can be obtained from equations (AI) in Appendix A and they are given by

El = 6.13 X 106 psi,

E2 = 1.42 X 106 psi,

11-12 = 0.53

V12 =0.27

V23 = 0.34

(22a)

(22b)

(22c)

(a)

Ma

trix

C

rack

ing

cr

E2 VE

l

~

'Y"

_ X

E2 VE

l

cr

(b)

Cra

ck i

n H

om

og

en

eo

us

an

d

An

iso

tro

pic

M

ed

ium

Fig

ure

6.

Mod

elin

g of

uni

dire

ctio

nal

fibr

ous

com

posi

tes

unde

r an

gle

load

ing.

><: ~ .....

. .....

.

~ 2 l:l

.. ~ 0- ~ 9 s:l

"=l 1?

....

IV Unidirectional fiber/matrix composite XXIX

where v12/E1 = V21/E2' The matrix is made of epoxy resin with Em = 4.5 X 105 psi and Vm = 0.35 and fibers with E f = Efl = Ef2 = 10.5 X 106 psi and vf = 0.20. When the layer thickness is taken to be very small in comparison with the crack length 2a, the values of ct>(O :md 1Jr(g) evaluated at g = 1 are

ct>(1) =0.31, 1Jr(1)=0.16 (23)

In general, both ct>(1) and 1Jr(1) decrease with Vf which means that fiber reinforcement tends to lower the intensification of stresses near the crack tip.

Before application of the Sc -criterion * to the unidirectional fibrous composite, it is essential to emphasize the failure mode of Scotchply 1002. First of all, the final fracture is known to occur in a brittle fashion in that there exists a distinct point of instability on the load versus deflection curve. Most of the stored energy is assumed to be released at the moment of incipient fracture where the crack begins to propagate. The remaining energy dissipated prior to global instability is sufficiently small so that the specimen geometry and stiffness are not affected appreciably as the critical load point is approached. To be recognized also is that failure initiates at an angle (Jo with reference to the crack plane although the final crack path may follow along the direction of the fiber. The values of (Jo for each (3 may be found by first computing S in equation (9b) with the aid of equations (21) and (23), and then applying the conditions specified by equation (6). The fracture angles (Jo are found to be negative as listed in Table I together with the corresponding values of Smin/a2a obtained from

Smin = a 2 a(0.096a l1 sin4 (3 + 0.050a 12 sin3 (3 cos (3 + 0.26a22 cos4 (3) (24)

It should be reiterated that the foregoing model assumes that energy is released only in the matrix. Incipient fracture is assumed to occur when Smin reaches Sc for the epoxy resin. Hence, only a knowledge of Sc for the matrix material and a valid stress analysis that accounts for a reduction in load transmitted to the matrix owing to fiber reinforcement are needed. When local nonhomogeneity of the composite is considered, it is no

* Stable crack growth is assumed to be governed by the condition of Cd Wid V)c = const as indicated in equation (11).

xxx Introductory Chapter

TABLE I Fracture angles and normalized strain energy density factors for the matrix cracking model (Scotchply 1002 with a 56.5% fiber volume fraction)

{3 -00 Smin/a2a (3 -00 Smin/u2a

5.00 77.700 .00012 50.00 35.400 .01841 10.00 71.400 .00052 55.00 31.600 .02257 15.00 65.600 .00123 60.00 27.700 .02688 20.00 60.300 .00231 65.00 23.600 .03112 25.00 55.500 .00380 70.00 19.300 .03508 30.00 51.000 .00575 75.00 14.800 .03849 35.00 46.900 .00819 80.00 10.000 .04113 40.00 42.900 .01113 85.00 5.100 .04280 45.00 39.100 .01456 90.00 0.000 .04337

longer meaningful to discuss the global fracture toughness of the compo­site system since all the free surfaces are assumed to be created in the matrix, and none in the fibers nor the fiber-to-matrix interfaces.

Homogeneous anisotropic model. Inquiries have been made into the possi­bility of defining a fracture toughness value for the entire composite system in much the same way as for a single-phase homogeneous material such as metal. Sih et al. [17] first derived an energy release rate expres­sion for a line crack in a solid possessing rectilinear anisotropy. Conceptu­ally speaking, it is not clear how the creation of free surfaces in fracturing fibers, matrix and fiber-matrix interfaces can be combined in such a way as to use an average release of energy through a segment of equivalent homogeneous and anisotropic material at incipient fracture. The validity of this idealization will obviously depend on how the discrete nature of the composite affect the failure modes and will be assessed quantitatively by using the Sc-criterion.

Referring to Figure 6(b), a crack of length 2a is situated in a homogeneous anisotropic medium with the principal axes of elastic sym­metry denoted by 1 and 2. A uniform tensile stress of magnitude a is applied in the y-direction which makes an angle {3 with the crack plane. The stresses aI, a2 and T]2 in the neighborhood of the crack tip are given by [17]

(2Sa)

IV Unidirectional fiber/matrix composite XXXI

(2Sb)

(2Sc)

where Re [ ] denotes the real part of the complex function in the bracket. The stress intensity factors* kl and k2 take the same forms as those shown in equations (21) with <P(1) = '1'(1) = 1. The complex vari­ables Zl and Z2 are:

Z2 = (cos (J + S2 sin (J)1/2 (26)

The polar coordinates rand (J are measured from the right hand side crack tip. The angular distribution of the stresses is clearly affected by the anisotropy of the material through the parameters s 1 and S2 which are roots of the fourth order characteristic equation**

(27)

The constants bij correspond to those appearing in the strain-stress relationship:

(28)

For the condition of plane strain, bij can be related to Sij as follows:

bu = Sl1 S 33 - Si3

b12 = b21 = S12S33 - S13 S 23

(29a) S33 S33

b22 = S22S33 - S~3

b16 =b61 = S16 S 33 - S13 S 36

(29b) S33 S33

b66 = S66 S 33 - S~6

b26 = b62 = S26 S 33 - S23 S 36

(29c) S33 S33

* The stress intensity factors [17J are defined so that for cracks in an infinite medium under self-equilibrating loads, they are independent of the elastic constants and the same as those for the isotropic material. ** By an energy consideration, it can be shown that the roots of equation (27) are either complex or pure imaginary and cannot be real. Thus, the four roots separate into two sets of distinct complex conjugates. The parameters s, and S2 correspond to those with positive imaginary parts.

XXXII Introductory Chapter

For unidirectional composites, there exist only three principal directions that are mutually perpendicular to one another and only nine indepen­dent elastic constants are nonzero. Since S44 and S55 do not appear in equations (29), only seven of the nine quantities of Sij need to be identified with the engineering constants Eb Eb etc. They are:

1 S22 = E 2 '

1 S66=­

iJ.12

(30a)

(30b)

(30c)

(30d)

As in the case of isotropic elasticity, the stresses in equations (25) may be inserted into the strain energy density function

(31)

to yield an expression involving l/r the coefficient of which is the strain energy density factor

(32)

The quantities Aij (i, j = 1, 2) are complicated functions of the anisotropic elastic constants and depend on the angle e through Zl and Z2' They are different from the aij (i, j = 1,2) in equations (10) for an isotropic material.

All =±[bllA 2+ b22c2+ b66E2+2b12AC+2b16AE +2b26CE] (33a)

A12 =±[b11AB + b22CD+ b66EF+ b12(AD + BC)+b16(AF+ BE) + b26(CF + DE)] (33b)

A22 = ±[bll B2 + b22D2 + b66F2 + 2b 12BD + 2b16BF + 2b26DF] (33c)

IV Unidirectional fiber/matrix composite XXXIII

The following contractions have been made:

A =Re [~(S2_~)], S1 - S2 Z2 Z1

(34a)

[ 1 (S1 S2)] C=Re -- --- , S1-S2 Z2 Z1 [ 1 (1 1 )] D=Re -- ---

S1 - S2 Z2 Z1 (34b)

[ S1S2 (1 1 )] E=Re -- ---S1 - S2 Z1 Z2 '

[ 1 (SI S2)] F=Re -- ---S1-S2 Z1 Z2 (34c)

Suppose that the unidirectional composite is made by embedding Modulite II graphite fibers in an epoxy resin matrix. The elastic proper­ties of these two constituents are

E f = 35 X 106 psi,

Em = 0.5 X 106 psi,

(35a)

(35b)

The fibers are assumed to be isotropic so that E f = Ef , = Efo and the fiber volume fracture for the Modulite II 5206 composite is approximately 0.65. The gross mechanical properties of the composite may be calculated from equations (A.1) in Appendix A. The results are

El = 22.92 X 106 psi,

/-L12 = 0.8 X 106 psi,

E2 = 2.22 X 106 psi

V12 = 0.34, V23 = 0.43

(36a)

(36b)

The constants SI1> S12, ... ,S66 can then be obtained from equations (30) in unit of (psi)-1:

Sl1 = 0.04 X 10-6 ,

S22 = 0.45 X 10-6,

S12 = S13 = -0.02 X 10-6

S23 = -0.19 X 10-6 , S66 = 1.25 X 10-6

(37a)

(37b)

Consequently, b11 , b12 , ••• ,b66 follow from equations (29) having the same units as Sij:

bll = 0.04 X 10-6,

b22 = 0.37 X 10-6 ,

b12 = -0.02 X 10-6

b66 = 1.25 X 10-6

(38a)

(38b)

where b16 = b26 = O. Once the coefficients bij in equation (27) are known,

XXXIV Introductory Chapter

TABLE II Fracture angles and normalized strain energy density factors for the homogeneous anisotropic model (Modulite II 5206 graphite-epoxy composite)

(3 -eo Smin/(T2a (3 -eo Sminl(T2a

5.00 75.200 .00041 50.00 40.900 .06078 10.00 69.500 .00182 55.00 38.500 .07176 15.00 64.500 .00447 60.00 36.300 .08232 20.00 60.100 .00853 65.00 34.400 .09206 25.00 56.100 .01411 70.00 32.800 .10062 30.00 52.500 .02119 75.00 31.700 .10774 35.00 49.300 .02967 80.00 31.200 .11321 40.00 46.200 0.3932 85.00 31.400 .11682 45.00 43.500 0.4982 90.00 32.400 .11839

it is found that

Sl = 0.636i, S2 = 5.261i (39)

With k1 and k2 in equation (32) being the same as an inclined tensile crack in an isotropic material, the S factor is obtained:

Differentiating equation (40) with respect to e and setting the result equal to zero, the values of eo corresponding to Smin are found and tabulated in Table II for {3 increasing from 5° to 90° in increments of 5°. Although the negative fracture angle eo decreases as {3 is increased, the normalized strain energy density factor Smin/u-2 a tends to increase reaching a max­imum at {3 = 90°. Unlike the isotropic material solution, the fracture angle eo is different from zero even when the applied stress is normal to the crack plane. In addition to eo = - 32.4° shown in Table I, eo = + 32.4° is also a possible solution which implies the tendency for the crack to bifurcate when the modulus of elasticity in the direction parallel to the crack is larger than that normal to it, i.e., E J > E 2 .

v. Comparison of results obtained from amso­tropic and matrix cracking model

Two possible analytical models have been presented in this section for describing the brittle fracture of unidirectional fiber/matrix composites

V Comparison of results xxxv

under angle loading. Results obtained from these two models will now be compared theoretically for the Modulite II 5206 graphite-epoxy compo­site. Experimental data on the fracture of E-glass fiber reinforced Scotch­ply 1002 composite will also be discussed in connection with the theoreti­cal prediction.

Modulite II 5206 graphite-epoxy. The mechanical properties for this material are given in equations (35) and (36). For the matrix cracking model, <P(1) and '1'(1) in equations (21) take the values

<P(1) = 0.22, '1'(1) = 0.07 (41)

The strain energy density factor becomes

S = (T2 a(0.0484a ll sin4 {3 + 0.0308a 12 sin3 {3

cos {3 + 0.0049a22 sin2 (3 cos2 (3) (42)

where aij (i, j = 1, 2) are given by equations (10) with <I> = 0 and P = Pm'

Making use of the conditions in equations (6) and (42), the fracture angles 80 and Smin/a2a are obtained (Table III). By assuming that fracture occurs when Smin = So the ratio ((Tc)f3/((TJf3~'rr/2 is obtained and plotted as a function of {3 in Figure 7 by using equation (42) for the matrix cracking model and equation (40) for the homogeneous anisotropic model. The two curves are normalized at {3 = 90° and their deviations begin to increase significantly when {3 is reduced. The discrepancy can be attri­buted to the difference in the values for 80 and Smin/(T2a given in Tables II and III. Another point to be noted is that the matrix crack propagation model predicts a fracture angle 80 = 0 at {3 = 90° as opposed to 80 =!= 0 for the anisotropic model.

E-glass epoxy resin composite. A comparison of the results presented in Table I for Scotchply 1002 will now be made with those for the homogeneous anisotropic model. Referring to the elastic constants in equations (22) and (29), the quantities b ij can be computed. Equation (27) may then be solved to yield

Sl =0.611i, S2 = 3.222i (43)

The strain energy density factor S takes the same form as equation (40) except the quantities Aij are obtained from equations (33) and (43). The

XXXVI Introductory Chapter

TABLE III Fracture angles and normalized strain energy density factors for the matrix cracking model (Modulite II 5206 graphite-epoxy composite)

(3 -eo

5.00 76.600 10.00 67.900 15.00 60.500 20.00 54.100 25.00 48.600 30.00 43.700 35.00 39.300 40.00 35.300 45.()0 31.400

7.0

(/) (/)

6.0 ~ Ui 0 '-' . .;:::

5,0 . .: u '0 Q) N

0 4.0 E 0 Z

I

N 3.0

~ ~CQ

b" :::::::: 2.0 ~CQ

b"

1,0

0

Smin/cr 2 a

.00002

.00010

.00024

.00046

.00081

.00219

.00195

.00279

.00382

(3

50.00 55.00 60.00 65.00 70.00 75.00 80.00 85.00 90.00

Modulite IT 5206 Graphite-Epoxy

-eo

27.700 24.200 20.600 17.200 13.700 10.300

6.900 3.400 0.000

Matrix Model

Anisotropic Model

f3 - Crack Angle

Smin/cr 2 a

.00502

.00635

.00775

.00915

.01045

.01158

.01245

.01300

.01318

Figure 7. Normalized critical stress versus crack angle (3 for Modulite II 5206 graphite epoxy.

V Comparison of results XXXVII

TABLE IV Fracture angles and normalized strain energy density factors for the homogeneous anisotropic model (Scotchply 1002 with a 56.5% fiber volume fraction)

(3 -(Jo Smin/a.2a (3 -(Jo Smin/(T2a

5.00 75.200 .00093 50.00 44.100 .11599 10.00 70.600 .00403 55.00 41.700 .13528 15.00 66.400 .00968 60.00 39.500 .15358 20.00 62.500 .01807 65.00 37.600 .17026 25.00 58.900 .02927 70.00 36.000 .18485 30.00 55.500 .04313 75.00 35.000 .19697 35.00 52.400 .05929 80.00 34.600 .20634 40.00 49.500 .07727 85.00 35.200 .21264 45.00 46.700 .09641 90.00 37.200 .21546

first condition of the strain energy density criterion given by equation (7) leads to the results in Table IV.

As {3 is varied, Tables I and IV show that 80 and Srn;n/er2a differ significantly for the two models. A graphical comparison of the critical stress as a function of {3 with experimental data [18] are displayed in Figures 8 and 9 for the critical half crack lengths of 0.578" and 0.785", respectively. All the curves are normalized with respect to the value of ere at {3 = 90°. For 45°.s {3.s 90°, the theoretical curves (solid for the matrix model and dotted for the anisotropic model) agreed reasonably well with the experimental data. The discrepancies become more noticeable as the angle {3 is reduced. This is not surprising, since at low crack angles, more and more fibers tend to be aligned with the tensile axis and to be broken and/or debonded from the matrix. These energy dissipative mechanisms were not included in the analysis. The solid curves for the matrix cracking model in Figures 8 and 9 appear to be in better agreement with the experimental data [18].

Similar results are reported in Appendix B for E-glass fiber epoxy resin composite with fiber volume fractions of 0.1, 0.2, 0.5 and 0.6. Graphical display of the normalized critical stress (ere)f3/(ere)f3=7T/2 versus crack angle {3 are also provided for both the matrix cracking and homogeneous anisotropic model.

Metal composites. Since the degree of material nonhomogeneity depends on the fiber volume fraction and the difference of elastic properties of the fiber and matrix, there should be less variations between the analytical predictions of the two models when both the fiber and matrix are made of

XXXVIII Introductory Chapter

12.0 ---Theoretico 1 Curve (Anisotropic Model)

10.0 - Theoretica I Curve

(Matrix Model)

A Experimental Data 'in I ac~ 0.578 in ..>< 8.0 I <n I <n I ~ .... \

(J) \ 6.0

0 \ tJ

\ .... 8 \

\ I 4.0 \ bU \

\

" "-2.0

o 60· 90·

/3 - Crack Angle

Figure 8. Critical stress versus crack angle for Scotchply 1002 with ac = 0.578 in.

metal. Results on aluminum matrix reinforced by stainless steel fibers indeed bear out this expectation. Refer to the calculated values of 00 and Smin/(J"2a in Tables C.I to C.UI and the critical stress versus crack angle curves in Figures C.1 to C.} of Appendix C.

VI. Angle-ply laminates

A system made of two or more layers of fiber-reinforced composite is referred to as a laminate. The fibers in each layer may be oriented in different directions to give different strengths and stiffnesses in the various directions. Figure 10(a) shows a four-layer laminate with each layer being considered as a plate element made of unidirectional fibrous composites. The plates are stacked and bonded together with various angles between the fibers in one plate to the next indicated by the angles

VI Angle-ply laminates

14.0

12.0

10.0

. iii 8.0 -'" Vl Vl

~ iii 6.0

" u -8 4.0

bU

2.0

o

...

- - - Theoretico I Curve (AnisotropIc Model)

- Theoretical Curve

30·

(Matrix Model)

... Experimental Data ac~ 0.785 in

60·

f3 - Crack Angle

XXXIX

90·

Figure 9. Critical stress versus crack angle for Scotchply 1002 with ac = 0.785 in.

±(3 (angle-ply) as illustrated in Figures 11. Upon loading, the entire assembly acts as a single unit in that the load will be transferred from layer to layer through the interfaces. Moreover, the laminate is said to be balanced when the layers are stacked symmetrically with reference to the mid-plane of the composite plate, Figures 10 and 1l.

It was observed in [19] that balanced laminates, when subjected to uniaxial loading, failed by a combination of thru-Iamina and interlaminar (delamination) cracking. These two modes of failure tend to trade off with one another as the fiber angles are varied. For the Scotchply 1002E glass-epoxy composite, a large amount of delamination occurred at low fiber angles corresponding to 15°~I(3I~30° while thru-Iamina cracking dominated at the higher fiber angles. Thru-Iamina cracking is obviously more damaging as it leads to the eventual fracture of the composite into

*y-j

--2

bj

.17

' <1t

~

----

I ho

-r

--L

y

.j r Lay

" 1(

+1

---L

-

Lh

l

La

yer

/ L

aye

r 2

(-)

Adh

esiv

e L

ayer

S""

, L

aye

r 3

(-)

Laye

r

Laye

r 4

(+)

La

yer

4(+

)

(a)

(b)

Fig

ure

10.

Fou

r-la

yer

bala

nced

lam

inat

e: (

a) i

deal

ized

mod

el,

(b)

wit

h ad

hesi

ve l

ayer

.

>:::

t""'

~

~ $:l..

>::

('

) is

~ 9 .§ ~

....

VI Angle-ply laminates XLI

y 2

(a) Layer I or 4 ( + )

y 2

-{3

(b) Layer 2 or 3

y

( c) Layers I and 2 or 3 and 4

Figure 11. Fiber orientations: (a) layer 1 or 4 (+), (b) layer 2 or 3 (-), (c) layers 1 and 2 or 3 and 4.

XLII Introductory Chapter

separate pieces. The two aforementioned failure modes always interact with each other near the free edges and their individual contributions cannot be easily accounted for.

Stress distribution. The interlaminar cracking between the plies, or de­lamination, at low fiber angles has been observed [19] to take place within a thin layer of the fibrous material. Instead of taking this layer as an idealized interface of zero thickness* as in Figure 10(a), an additional layer of material representing the adhesive between the plies is intro­duced. Delamination is then assumed to occur by breaking the adhesive layer or the neighboring material. Figure 10(b) shows the analytical model of a four-layer laminate containing the additional adhesive layers with thickness ho which is small in comparison with the plate element with thickness hI' Each plate element is assumed to possess three mutually perpendicular directions of material symmetry and will be described by the homogeneous, orthotropic theory of elasticity. The adhesive layers are assumed to be isotropic in nature.

Puppo and Evensen [21] have provided a solution to the problem in Figure 8(b) where the laminate is 2b in width and 2a* in length and is extended uniformly by an amount ±u* along the x-axis. The in-plane stresses in the anisotropic layers labelled by the subscripts 1 and 2 in Figure 12 are given by**

(Ux)l = (UJ2 = ux, (Uy)1 = (uy)z = 0,

(Txy)1 =-(Txyh= Txy

in which

(44a)

(44b)

(45a)

(45b)

* By definition, delamination cannot take place at the idealized interface where perfect bonding is assumed by requiring the stresses and displacements to be continuous. The choice of cohesive or adhesive failure will depend on the composite system under consideration [20]. ** Although the classical lamination theory in [22] and [23] gives a reasonably good estimate of the stress solution in the interior region of the laminate, it does not account for the effect of interlaminar shears close to the edges of the composite plate. In general, interlaminar normal stress can also come into play depending upon the stacking sequence of the layers. Refer to Pipes and Pagano [24] for a discussion on this subject.

An

iso

tro

pic

¥~ (

CTX

) I

Laye

r I

">.

(Txy

) ,

(<Ty

) ,

Tzy

z JLx

Ad~;yS~~e--~TZX

L--

--A

nis

otr

op

ic

'" -T

zx

Lay

er

2 (<

T X}2

(<T Y

}2

Fig

ure

12.

Str

ess

com

po

nen

ts o

n i

ndiv

idua

l la

yers

.

~

;::

a« ~

I ~ ~

S ;:1 9'

~ ~

en e ::1

XLIV Introductory Chapter

The interlaminar shear stresses in the adhesive layers are given by

(46)

where

(47)

with /-t * being the shear modulus of the adhesive. The quantities Qij are the stiffness coefficients referred to the axes x and y in Figures 11. They are identified with the generalized plane stress-strain· relationship as follows:

(48)

Since each of the plate elements is assumed to be in a state of generalized plane stress, this model does not account for interlaminar normal stress. The results in equations (45) can also be expressed in terms of the applied uniaxial tensile stress (T in the direction of the displacement u *, i.e.,

(49)

Hence, the quantity u*/a* in equations (45) and ·(46) can be eliminated and (T", 'T"y, 'T"z can be expressed directly in terms of (T.

Of special interest is the interlaminar shear stress, equation (46), developed in the adhesive layer. For illustrating this, equation (46) is calculated numerically using the elastic constants for Scotchply 1002 glass-epoxy laminate and the results are plotted in Figure 13. Note that the shear stress 'Txz increases with the fiber angle {3 and reaches a maximum at {3 = 35°. From there on, T"z decreases and becomes negative. The cases when {3 = 0° and 90° correspond to the unidirectional compo­sites with fiber parallel and perpendicular to the loading axis.

It should be mentioned that the existing lamination theories [21]

VI Angle-ply laminates

(J) (J) Q) "-(j) "-o Q) .c Cf)

"­o .5: E o

-.::: Q) :s "0 Q)

.!::! o E

1.6

1.2

0.8

0.4

o

" -0.4 :z

y •

Angle ±f3

Figure 13. Interlaminar shear stress versus fiber orientation.

XLV

through [24] do not account for pre-existing mechanical imperfections which normally act as stress raisers that lead to crack extension. Sih and Hilton [25] have used a variational principle and have developed an approximate three-dimensional isotropic lamination theory. The theory has been applied to solve the problem of a crack through each of the laminae with boundary layers constructed near the free surfaces and interfaces. The problem of a through crack in anisotropic laminates can also be solved in a similar manner.

Thru-Iaminar cracking. For laminates with fiber angles 45°~ if3i ~ 90°, the dominant fracture mode is that of cracking through the individual laminae with little or no delamination. Failure of the balanced four-layer Scotchply 1002 is attributed to crack propagation initiated from crack­like imperfections inherent in the cured laminate. It is assumed that the

XLVI Introductory Chapter

entire laminate fractures when anyone lamina reaches the point of unstable rapid crack propagation. Thus, the Sc -criterion of fracture will be applied to a single lamina as in the case of the unidirectional compo­sites.

Since the crack is generally aligned along the fibers making an angle with the tensile axis, the initial direction of crack extension will not be collinear with the main crack. In this case, both stress-intensity factors are present and they are

(50)

which correspond to the problem of a single crack in a laminae with 56.5% E-glass fiber volume fraction. The stresses fr2 and T12 are referred to the lamina principal axes as shown in Figures 11 and they can be obtained from

[frl] [ cos2 {3, fr 2 = sin2 {3, T12 -sin {3 cos {3,

sin2 {3, cos2 {3,

sin {3 cos {3,

sin 2{3 ] [frx] -sin 2{3 fry

cos 2{3 Txy

in which fr2 and T12 may be alternatively written as

fr2 = frx sin2 {3 + fry cos2 {3 - TXY sin 2{3

Tn = (fry - frx) sin {3 cos {3 + Txy cos 2{3

(51)

(52a)

(52b)

The stresses frx and Txy are given by equations (45) with fry = o. Using equations (50) and (52), the strain-energy-density factor S in equation (9b) may be computed:

(53)

Now, differentiating S in equation (53) with respect to e and setting the result equal to zero, a relation between the direction of crack initiation, measured by the fracture angle eo, and the fiber angle {3 is obtained. The results, presented graphically in Figure 14, can be inserted into equation (53) to yield Smin.

A comparison between theory and experiment will now be made. In [19], balanced four-layer Scotchply laminate specimens 10 in. long and 0.05 in. thick (0.01 in. per ply) with fiber orientations of ± 15°, ±30°,

VI Angle-ply laminates XLVII

CT

120°

r-~x A-BO <l> t 0'1 CT c: <!

<l> 60° ..... :J -() 0 .....

LL

ch° I

30°

Angle f3

Figure 14. Direction of crack initiation as a function of fiber angle for angle-ply laminates.

±45°, ±60°, and ± 75° were tested by loading in the zero direction. The tensile loads at incipient fracture were measured and they are given as round dots in Figure 15 plotted as a function of the fiber angle (3. The half crack length in these specimens is estimated to be approximately 0.03 in. The theoretical results represented by the solid curve are obtained from equation (53) by holding Sc at (3 = 90° to be constant. Note that the agreement is good for 30° ~ 1(31 ~ 90°. At (3 = ±15°, the theoretical predic­tion is somewhat higher than the experimental result. This is because a substantial amount of delamination has occurred at this fiber orientation which is not accounted for the the thm-Iaminar crack model.

XLVIII Introductory Chapter

120.0

100.0

~ CT

If) 80.0 .><: /I

If) Half Crack Length':::::: 0.03

If)

Q)

• Experimental Data "-

en 60.0 0 (.) .... .... () 40.0 I bU

20.0

o Angle ±f3

Figure 15. Critical stress variations with fiber orientation.

Delamination. At low fiber orientations 15°,,; i{3i ,,; 30°, the specimens exhibited interlaminar cracking (i.e., cracking between the "+" and "-" layers in Figures 10) prior to complete thm-Iaminar failure. The region of interlaminar cracking or delamination is triangular in shape as illustrated in Figure 16(a). This scissoring effect was not evident in the high angle fiber orientations. Experiments for angles below {3 = ± 15° are not availa­ble. It is anticipated that the failure mode tends to that of the unidirec­tional composite for small values of {3.

Txz

0- ~

~TXY

o-x

J:7

y~

cP

~ x

0-

(0 )

F

igur

e 16

. In

terl

amin

ar c

rack

ing:

(a)

pla

ne v

iew

, (b

) co

re r

egio

n.

( b

) ~

L Introductory Chapter

The possible sites of delamination will be predicted by application of the Sc -theory without assuming an initial flaw size. Referring to Figure 16(b), an interfacial core region of radius ro is centered around a point where delamination may start. Consider the energy stored in a differential volume d V situated on the boundary of the core region as

(54)

The in-plane strains in equation (54) are given by

1-- - ----ex = ~ [( 0 22 0 66 - 0~6)O"x +(0160 26 - 0 12 066)O"Y

+ (0120 26 - 0 160n)Txy] (55a)

1-- -- --_ ey = Ll [( 0 160 26 - 0 12 066)O"x + (011 0 66 - Oi6)O"y

+ (012 0 16 - 0 11 (26)'TXY ] (55b)

1-- -- ----'Yxy = Ll [( 0 120 26 - 016022)O"x +( 0 12 0 16 - 0 11 Q26)O"y

+ (011 022 - Oi2)Txy] (SSe)

in which d stands for

The shearing strain 'Yxz can be expressed in terms of 'Txz by calculating the shear modulus /.L * of the adhesive layer while the last two terms in equation (54) vanish because 'Tyz = O"Z = 0 for the present model. Inserting equation (55) into (54) and simplifying the result, the strain-energy­density factor becomes *

S = (~~}o = ;~ [(022 0 66 - 0~6)0"~+(011 0 22 - Oi2)T~y

+2(0120 26 - 0 160n)O"x'TXY + :* 'T~z ] (57)

* Although the exact shape of the initial defect which initiates delamination within the core region is not known. it is assumed that it reflects a llr singularity to the energy-density field outside the core, i.e., d Wid V = Sir for r --' roo

VI Angle-ply laminates LI

The stiffness coefficients Oij referred to the x and y axes can be related to the Oij referred to the principal axes 1 and 2 through the angle {3 as follows:

0 11 = 011 cos4 (3 + 2( 0 12 + 2066) sin2 {3 cos2 {3 + 0 22 sin4 {3

0 12 = (0 11 + 0 22 - 4066) sin2 {3 cos2 {3 + 012(sin4 (3 + cos4 (3)

(58a)

(58b)

0 16 = (0 11 - 0 12 - 2066) sin {3 cos3 {3 + (012 - 0 22 + 2066) sin3 {3 cos {3 (58c)

022 = 0 11 sin4 (3 + 2( 0 12 + 2066) sin2 {3 cos2 {3 + 0 22 cos4 {3 (58d)

0 26 = (0 11 - 0 12 - 2066) sin3 {3 cos {3 + (012 - 0 22 + 2066) sin {3 cos3 {3 (58e)

0 66 = (0 11 + 0 22 - 2012 - 2066) sin2 {3 cos2 {3 + 066(sin4 (3 + cos4 (3) (58f)

From equations (45) to (46), it is seen that the stresses in equation (46) depend only on the variable y. For the delamination problem, the Sc -theory has the following interpretation:

(1) Interlaminar cracking occurs at a location determined by the relative minimum value of the strain-energy-density factor S, i.e.,

as (a2 S ) -=0 -2>0 at y = Yo ay ay

(59)

(2) Delamination occurs when the minimum strain-energy-density fac­tor S reaches a critical value:

Smin = Sc for y = Yo (60)

Figure 17 gives a plot of the normalized S-value in equation (57) versus the distance y, the path along which delamination may occur. Indeed, the strain-energy-density factor possesses a minimum value for each value of {3. Failure is assumed to occur outside the core region with an average radius of '0 = 0.02 in. Experimental data on delamination is lacking. It must be cautioned that the measured values of the critical load or (Yc for {3 = ±15° and 300 in [19] at which delamination was observed also include the effect of thru-Iaminar cracking. More refined experiments must be designed such that the quantitative effects of thru-Iaminar and interlami­nar cracking may be separated.

LII

N

b ~ "-en 0-

Introductory Chapter

Core Region ~

/3=650 II ----~/3~=-8~0~0--------41~1

1.5

/3 = 40°

1.0

/3 = 30°

/3 = 20°

/3 = 10°

0.6 0.8 1.0

Y/b

Figure 17. Variations of normalized S -value along the y -direction.

VII. Impact and cracking of unidirectional composites

When a fibrous composite is loaded suddenly, waves are reflected and refracted through the fibers and matrix in a complex pattern. Additional disturbances result from defects or cracks that are inherent in the compo­site. Because of the complexity of wave scattering in the presence of material nonhomogeneity and cracks, only a few simple cases of dynamic

VII Impact and cracking of unidirectional composites LIII

failure analyses have been treated [26-28]. To be considered in this section are the impact response of a unidirectional fibrous composite with an initial stationary crack and a running crack in the matrix.

Impact loading. Without loss in generality, the unidirectional fibrous composite with a crack in the matrix will be modeled by a layer of cracked material with elastic properties ILl, VI, PI sandwiched between two media with properties IL2' V2, P2' The number of layers surrounding the cracked layer is sufficiently large such that the average shear modulus ILz. Poisson's ratio V2 and mass density P2 can be used. Refer to Figure 18. The composite is subjected to an impact load oriented in a direction that makes an angle (3 with the crack plane. The crack surfaces experi­ence both normal and shear impact with magnitude ao and To, respec­tively. This gives rise to both stress intensity factors kl(t) and k2(t), which

[I

- _I h I

1

/

• 1 I

---~

y _/ = == -== ------=- - - -- -- -- - - -

/-I-2,7/2,P2 • f3

/ )= ~ /-1-1·7/ 1' PI X I

j,-/~a-j I I

A ~==-=------=

/-I-2,7/2,P2

------

I I t

- - - ---

----

Figure 18. Unidirectional composite under normal and shear impact.

LIV Introductory Chapter

in the dynamic case will vary as functions of time. The functional relationship of the dynamic stresses in terms of the local polar coordinates rand (J as indicated in Figure 18 is the same as that obtained from the static equations of elasticity. Suppose that the shear modulus 1L2 is much larger than ILl such that 1L2/1L1 = 10.0 and that the same mass density and Poisson's ratio are used for the matrix and the surrounding composite material, i.e., PI = P2 and VI = V2 = 0.29. Figures 19 and 20 give the numerical values of the normalized dynamic stress intensity factors k1 (t)/(J"oFa and k2(t)/ToFa as functions of the dimensionless time variable c21 t/a where C21 = (lLl/Pl)1/2 is the shear wave velocity in the matrix material. All the curves tend to first rise to a peak and then oscillate with decreasing amplitude. These peaks decrease in magnitude as the crack

1.2

1.0 a/h= 0.5

0.8

~ o

~ 0.6

0.4

0.2

o 2.0

2.0

4.0

C 2 I t /a

1-'-2/1-'-1 = 10.0

l/I = l/2= 0.29

PI =P2

6.0 8.0 10.0

Figure 19. Normalized k,(t) factor as a function of time for a crack in matrix layer.

VII Impact and cracking of unidirectional composites

1.2

1.0

0.8

o

~ 0.6 -(\J

""

0.4

0.2

o

o/h = 0.5

f-L2/f-L I = 1 0

vI = v2 = 0.29

PI =P2

2.0 4.0 6.0 8.0

LV

10.0

Figure 20. Normalized k 2 (t) factor as a function of time for a crack in matrix layer.

length 2a is increased relative to the height 2h of the matrix layer. A second peak is observed on the k 1(t)/O'oJ(i curve for small time t when a/h = 2.0 corresponding to a long crack in a narrow layer. This local disturbance corresponds to the arrival of the reflected waves from the interfaces that are in close proximity of the crack tip. In what follows, only the case of a/h = 1.0 will be discussed.

Once k 1(t) and kz(t) are known, the dynamic strain energy density factor S can be calculated* from equation (9). In the dynamic problem,

* Since the qualitative feature of the stress field around a stationary crack for both static and dynamic loadings are the same, the coefficients au are the same as those shown by equations (10). The only difference lies in the stress intensity factors.

LVI Introductory Chapter

a 15° 30° 45° 60° 75° 90° {3

Figure 21. Fracture angle as a function of crack angle at different time.

1-'-2/1-'-, = 10.0, a/h = 1.0

v,=v2 =0.29, P,=P2

75°h------------------------------30°

---------------------------~~----45° 60° ,----------------______ ~~ ___

60°

o

Figure 22. Fracture angle as a function of time for different crack angle.

VII Impact and cracking of unidirectional composites LVII

the fracture angle -00 can no longer be uniquely determined by the crack angle {3 alone. Figure 21 gives a plot of -00 versus {3 and shows that the fracture angle tends to increase slightly for a given crack angle if the crack initiation occurs at the early stage of loading. This is better illustrated in Figure 22 which shows the variations for -00 with C21t/a for angles of {3 = 15°, 30°, ... , 90°. Note that the fracture angle is time dependent only for C21t/a < 1.0. The minimum strain energy density factor Smin may now be obtained by inserting the values -00 from Figure 22 into the expres­sion for S. The normalized strain energy density factor 161L1Smin/(T2a is plotted against C21t/a for different crack angle {3 as shown in Figure 23. The quantities (To and 'To in Figures 19 and 20 are related to (T as

(To = (T sin2 {3, 'To = (T sin {3 cos {3 (61)

All the curves in Figure 23 possess a maximum at a time t which depends on {3. This maximum first increases with {3 from 30° to approximately 60° and then decreases as {3 approaches 90°.

In order to find the critical stress at which fracture occurs for a fixed crack position {3, the elapsed time at which Srnin first reaches Sc must be known. To this end, it would be simpler to compare the ratio of the dynamic and static failure stress denoted by (Trn and (T" respectively. Let

Figure 23. Variations of normalized minimum strain energy density factor with time for different (3.

LVIII

5.0

4.0

'" 3.0 b "-bE

2.0

1.0

o

fL2/fLi = 10.0, a/h = 1.0

1/1= 1/2= 0.29, PI =P2

1.0 2.0 3.0 C21 t 10

Introductory Chapter

300~ __ _

4.0 5.0

Figure 24. Ratio of dynamic and static failure stress versus time for different (3.

16~lSc = 4(1- 2v1 )u;a be given such that the same Sc governs unstable fracture regardless of whether the load is static or dynamic. Then the dynamic failure stress may be estimated from

(62)

The ratio um/us plotted as a function of cZlt/a is shown in Figure 24. The lowest failure stress U m occurs at C 21 t/ a = 1.5 and (3 = 60°. It is seen that the most vulnerable position of the crack is not at (3 = 90° but when the crack is tilted at an angle to the direction of impact. The difference will depend on the material and geometric parameters describing the uni­directional composite.

Crack propagation. When the stress U m exceeds the critical level, failure of the composite can take place by a process of crack propagation which involves the cracking of the matrix at relatively high velocities. The process of fast fracture in a nonhomogeneous material is little understood because of the multitude of parameters that can affect crack propagation.

VII Impact and cracking of unidirectional composites LIX

Analytical modeling of composite failure by crack propagation is of great interest, since information on how the combination of fiber and matrix can influence the initiation and arrest of cracks is scarce and will be most useful for determining the resistance of fibrous composites to running cracks.

Referring to Figure 18, if the crack in the matrix layer begins to propagate ata velocity v under a uniform applied stress (T on the surrounding material for {3 = 90°, then the crack tip stress field when referenced to a set of local polar coordinates rand e will differ from that of the stationary crack. For a moving crack, the local stresses in a rectangular cartesian coordinate system are [29]

_kI(v)[ 2) 2 2 (Tx - 1- (1 +'TI2 (2'T11 + 1-'TI2)f('TII)-4'T1I'TI2f('TI2)]/m('TII, 'TI2)

v2r

(Ty = k~(v) [4'T1(f12f( 'TI2) - (1 + 'TI~?f( 'TII)]/m( 'TIl, 'TI2) v2r

kiev) 2 TXY= ~ 2'T1I(1+'TI2)[g('TII)-g('TI2)]/m(Th,'TI2)

v2r

in which m('TII, 'TI2) stands for

The functions f( 'TIJ and g( 'TIJ for j = 1, 2 are defined as

f2( 'TI) =! sec e[(1 + 'TI; tan2 8)-1/2 + (1 + 'TI; tan2 8)-1]

g2( 'TIJ = !sec e[(l + 'TI; tan2 8)-1/2 - (1 + 'TI; tan2 e)-I]

where

(63a)

(63b)

(63c)

(64)

(65a)

(65b)

(66)

while the dilatational and shear wave speeds designated by Cll and C211

respectively, are given by

_ [AI +2I-LI] _ [I-LI]I/2 Cll - ,C21-

PI Pl (67)

LX Introductory Chapter

Since the crack is assumed to run straight, only the stress intensity factor k 1(v) appears in equations (63) and it takes the form

(68)

If the material properties at a finite distance, h, away from the crack are different from those in which the crack is propagating, then the function F in equation (68) must depend on additional constants that describe the changes of the local environment, i.e.,

F =F(~ ~ 1-1-2 V2 P2) , , , , h C21 1-1-1 V1 PI

(69)

Equation (69) shows the multiplicity of parameters that can influence the stress intensity factor kl(v). Their individual contributions can be studied analytically to gain a better understanding of the interaction of crack propagation with material nonhomogeneity.

The strain energy density criterion applies equally well to moving cracks without any major alteration except for the physical interpretation of the results. Substituting equations (63) into the strain energy density function expression for dW/dV, S is obtained since S = r(dW/dV):

- (1- 2Vl)(2'TJi + 1- 'TJ~)(1 + 'TJ~)]f( 'TJl) + 32'TJi'TJ~f( 'TJ2)

- 16'TJl 'TJz(1 + 'TJi)(l + 'TJ~)f( 'TJl)f( 'TJ2)

+ 8'TJi(1 + 'Y)~)2[g( 'TJ1) - g( 'TJ2)]2} (70)

In equation (70), the Poisson's ratio and the mass density of the matrix layer 1 and surrounding material 2 have been assumed equal, i.e., PI = P2

and VI = V2 = 0.25. In [29], it was shown that the ratio SviSd for a moving crack remains

nearly constant in the low crack velocity range and starts to decrease rapidly for v in the range of 0.4C21 and 0.5C21 . The small volume change

VII Impact and cracking of unidirectional composites LXI

TABLE V Bifurcation predicted by the strain energy density criterion for VI = 0.25

vlc2I 0.1 0.2 0.3 0.4 0.5 0.6 0.8

00 0° 0° 0° 0° ±15.1° ±48.6° ±65.2°

8f.L1S~;n 1.008 1.036 1.090 1.185 1.361 1.631 3.892

(T2a

in an element moving ahead of the crack tends to suggest crack bifurca­tion* so that energy can be dissipated more quickly by material separa­tion on both sides of the main crack. More specifically, Sv > Sd at the locations of the minimum values of S or Smin and can be determined from as/ao = 0 for 0 = 00 , For VI = 0.25, values of Smin were found at both sides of the crack corresponding to 00 = ± 15.10 and occurred at a crack velocity of approximately O.5e21 . Table V shows the values of 8f.LISmin/a.2a and 00 for different crack velocities v. Bifurcation at the higher crack vel­ocities is rare but may occur in materials that are relatively free of obstacles and hence it is less likely for the crack to encounter an obstacle and change direction. Consequently, it can be driven to a higher speed. This limitation also applies to crack propagation in a composite since the crack can only propagate in one phase of the composite at a given time. The occurrence of bifurcation, however, will be affected by the nonhomogeneous nature of the composite material.

Now that the preliminary discussion on crack propagation in the homogeneous material has been completed, the effect of material nonhomogeneity will be analyzed. The minimum S or Smin in equation (70) for different values of a/ h, v/ C21 and f.L2/ f.LI is calculated for a moving crack. When f.L2/ f.LI > 1, the crack is in the matrix material that is relatively soft in comparison with the surrounding material located at a distance h from each side of the crack.

Figure 25 shows a plot of 8f.LISmin/a2a versus a/h for f.L2/f.LI = 5.0 and V/C21 = 0.1,0.2, ... ,0.5. For short cracks or small ratios of a/h assuming that h is fixed, the crack velocities are not as sensitive to changes in the

* The mathematical model of crack bifurcation should be interpreted with caution when compared to experiments where crack branching or curving is highly sensitive to the presence of obstacles such as inclusions or voids within the material.

LXII

0 N

b "-c 'E en

::l.. co

0.8

0.7

0.6

0.5

0.4

0.3

T o 0.5

1-'-2/1-'-1 = 5.0

vl=v2 =0.25

I I

Introductory Chapter

""""i;;;;;~;;::::..._0.4 0.3 0.2 0.1

I I 1.0 1.5 2.0

o/h

2.5 3.0

Figure 25. Normalized mllllmum strain energy density factor versus a/h for f.L2/f.LI = 5.0.

VII Impact and cracking of unidirectional composites

o N

b "­

c: E

(f)

:[ co

14

\ 1.3 \

1.2

1.1

0.9

0.8

\ \ \ \ \ \ \

\ "­

"'-......... __ v/c 21 = 0.5 ---

LXIII

Figure 26. Normalized minimum strain energy density factor versus a/h for f.L2/f.LI =2.0.

applied stress or (J as compared with the longer cracks where a small change in (J can lead to larger variations of v. In general, Smin tends to decrease with increasing crack length. If the degree of nonhomogeneity is not as severe, i.e., for IJdlkl = 2.0, then Figure 26 suggests that the variations in (J for a given material or Smin = const seem to alter the crack velocities by approximately the same amount regardless of the crack size.

The influence of the relative stiffness or 1k2/ Ikl ratio on crack length is exhibited in Figure 27 in which 81k 1Smin/(J2a = 0.9 is taken. As the crack travels at v = 0.3C21 , there is a tendency for it to increase its length should

LXIV Introductory Chapter

1.0 8tL i Smin = 0.9

0"2 a

0.8

fL2 ItL 1 = 2.0 .s:::.

0.6 "-c

0.4

0.2

o

Figure 27. Crack length as a function of velocity for 81J..1Sm;n/u2a = 0.9.

the ratio f.L2/(L1 be suddenly altered, say decreased from 5.0 to 2.0. The crack length is observed to increase from a = 0.35h to a = 0.57h, i.e., by a factor of 1.63.

As long as the crack is propagating in the material with properties (L" VI' PI' the crack front stress field will be governed by equations (63) and the velocities corresponding to bifurcation will be the same as those shown in Table V. However, the stored energy in elements ahead of the crack will be affected by (L2/f.Ll and a/h. By letting (L2/(Ll = 2.0 and

TABLE VI Influence of material nonhomogeneity on Sm;n for 11-2/11-1 = 2.0 and V/C21 = 0.5

a/h 0.1 0.2 0.4 0.5 1.0 2.0 3.0

1.345 1.301 1.172 1.108 0.901 0.775 0.736

VIII Concluding remarks LXV

V/C 21 = 0.5, Table VI shows that 8/-tISmin/a2a tends to decrease with increasing a/ h.

VIII. Concluding remarks

The strain energy density criterion has been applied to analyze the fracture behavior of unidirectional fiber-reinforced composites. Matrix cracking initiated from an inherent defect in the form of a crack is assumed to be the dominant mode of fracture. Depending on the fiber volume fraction and the material properties of the constituents, the composite system was modeled either by invoking the assumption of homogeneous anisotropy or nonhomogeneous isotropy. The former as­sumes fracture to occur in an idealized material with the combined anisotropic elastic properties of the fiber and matrix while the latter assumes cracking to take place in the isotropic matrix located in between the fibers. The agreement between theory and experiment for the E-glass epoxy resin with 56.5% fiber volume fraction was good for fiber orienta­tions from 30° to 90° with respect to the direction of loading. As the fiber direction approaches the loading axis, there is the tendency for more fibers to break, a condition which was not included in the analysis. Theoretical predictions of Modulite II 5206 graphite, E-glass epoxy with different fiber volume fractions and stainless steel-aluminum composites were also presented.

The impact response of unidirectional composites is treated by assum­ing fracture to initiate from a finite crack in the matrix. Load transmission through the composite takes place in the form of wave propagation. The reflective waves from the composite interfaces and one crack tip to the other cause intensification of the local stress field. The results can be expressed in terms of the critical dynamic stress to initiate fracture. Unlike the behavior of a homogeneous material, the lowest critical stress corresponds to impact loading directed at an angle with respect to the crack plane rather than normal to it.

The failure of angle-ply laminates made of layers of unidirectional fiber-reinforced material is also considered. Thru-laminar and interlami­nar (or delamination) failure modes are discussed separately. Delamina­tion becomes significant when the orientation of the angle-ply with respect to the applied load is reduced. Failure of the laminate is assumed to coincide with unstable thru-Iaminar cracking of one of the laminae. The simultaneous occurrence of delamination and thru-cracking has yet to be investigated analytically.

LXVI Introductory Chapter

IX. Appendix A: Calculation of gross mechani­cal properties of unidirectional composites

The gross mechanical properties of fiber-reinforced unidirectional compo­sites may be calculated from the following formulas [16]

(Ala)

(Alb)

(A1c)

(A1d)

(Ale)

in which 1, 2 and 3 correspond to the principal axes of elastic symmetry. The axes 1 and 2 are in the plane of the composite sheet specimen while 3 is directed through the thickness. The symbols in equations (A1) stand for

Vf = fiber volume fraction Vm = matrix volume fraction = 1- Vf

Efl = fiber longitudinal Young's modulus Ef2 = fiber transverse Young's modulus Em = matrix Young's modulus

Vf = fiber Poisson's ratio Pm = matrix Poisson's ratio /-Lf = fiber shear modulus

/-Lm = matrix shear modulus

X. Appendix B: E-glass epoxy reSin composites

This section provides additional information on the fracture angle 00, normalized strain energy density factor Sminla.2a and critical stress «(jJ(3/«(jJ(3~"IT/2 for the unidirectional E-glass epoxy resin composite mod­elled by cracking in the isotropic matrix and anisotropic homogeneous medium. Four different fiber volume fractions are considered. They are

X Appendix B: E-glass epoxy resin composites LXVII

Vf = 0.1, 0.2, 0.5 and 0.6. The elastic properties of the E-glass fiber are

Ef = Efl = Efz = 10.50 X 106 psi

/-Lf = 4.40 x 106 psi, vf = 0.20

The elastic properties of the epoxy resin are given by

Em = 0.50 X 106 psi

/-Lm = 0.19x 106 psi, Vm =0.35

(B.la)

(B.lb)

(B.2a)

(B.2b)

Fiber volume fraction 0.1. The gross mechanical response of the compo­site will be affected by the fiber volume fraction. For Vf = 0.1, equations (A. 1) in Appendix A may be used to obtain

EI = 1.50 X 106 psi,

E2 = 0.62 X 106 psi,

!LIZ = 0.27 X 106 psi

V12 = 0.34

Vn = 0.45

This leads to the characteristic roots

Sl =0.634i, S2 = 2.361i

(B.3a)

(B.3b)

(B.3c)

(B.4)

that satisfy equation (27). The functions <1>(1) and pel) in the stress intensity factor equations (21) for the matrix cracking model are

<1>(1) = 0.71, pel) =0.50 (B.5)

The final results for the anisotropic homogeneous and matrix cracking model are given in Tables B.I and B.II. A plot of the normalized critical stress versus crack angle given in Figure B.1. shows that there is little difference between the two models.

Fiber volume fraction 0.2. As before, the gross elastic properties of the composite for Vf = 0.2 are found as

EI = 2.50 X 106 psi,

E z = 0.72 X 106 psi,

/-L12 = 0.27 X 106 psi

VIZ = 0.32

V23 = 0.44

(B.6a)

(B.6b)

(B.6c)

LXVIII Introductory Chapter

TABLE B.I Fracture angle and normalized strain energy density factor for homogeneous anisotropic model (E-glass epoxy resin with VI = 0.2)

(3 -80 Srnin/a-2a (3 -80 Srnin/a-2a

5.00 76.500 .00272 50.00 45.700 .27568 10.00 72.400 .01149 55.00 42.900 .31609 15.00 68.500 .02690 60.00 40.200 .35359 20.00 64.800 .04903 65.00 37.600 .38738 25.00 61.300 .07752 70.00 35.200 .41704 30.00 57.900 .11154 75.00 33.200 .44249 35.00 54.700 .14987 80.00 31.700 .46386 40.00 51.600 .19107 85.00 31.100 .48127 45.00 48.600 .23353 90.00 32.400 .49431

The roots SI and S2 take the values

SI =0.6l5i, (B.7)

For the matrix cracking model, it is found that

<P(l) = 0.65, pel) = 0.38 (B.8)

Refer to Tables B.lII and B.IV for the results on eo and Sminlu2a. The curves in Figure B.2 reveals a slightly larger deviation between the two models when the fiber volume fraction is increased from 0.1 to 0.2.

TABLE B.II Fracture angle and normalized strain energy density factor for matrix cracking model (E-glass epoxy resin with VI = 0.2)

(3 -80 Smin/rr2a (3 -lJo Srninla2a

5.00 79.500 .00104 50.00 42.400 .10167 10.00 74.800 .00431 55.00 38.400 .11889 15.00 70.300 .00993 60.00 34.300 .13602 20.00 65.900 .01785 65.00 29.900 .15252 25.00 61.700 .02793 70.00 25.000 .16773 30.00 57.700 .03994 75.00 19.600 .18089 35.00 53.800 .06867 80.00 13.600 .19117 40.00 50.000 .08480 85.00 7.000 .19776 45.00 46.200 .08480 90.00 0.000 .20003

X Appendix B: E-glass epoxy resin composites LXIX

70 E - G lass Epoxy Composite

Vf ;: 0.1 (/) (/) 6.0 (I) .... -(f) c ()

:-E 5.0 Matrix Model .... u "C Anisotropic Model (I)

.!:::! 0 4.0 E .... 0 z

(\J 3.0 ~ " ~CQ

bU

::::::. 2.0 ~CQ

bU

1.0

o f3 - Crack Angle

Figure B.1. Normalized (Ie versus (3 curve for E-glass epoxy composite ("f = 0.1).

Fiber volume fraction. 0.6. The elastic constants for the composite with a tends to increase the modulus of elasticity for the composite:

El = 5.50 X 106 psi,

E2 = 1.33 X 106 psi,

!L12 = 0.50 X 106 psi

V12 = 0.28

V23 = 0.36

(B.9a)

(B.9b)

(B.9c)

LXX Introductory Chapter

TABLE B.III Fracture angle and normalized strain energy density factor for homogeneous anisotropic model (E-glass epoxy resin with Vf ~ 0:2)

(3 -lJo Smin/a2a (3 -lJo Smin/a2a

5.00 75.800 .00199 50.00 44.700 .22672 10.00 71.400 .00853 55.00 42.200 .26254 15.00 67.300 .02024 60.00 39.800 .29622 20.00 63.S00 .03740 65.00 37.600 .32679 25.00 59.900 .05993 70.00 35.700 .35358 30.00 56.500 .08739 75.00 34.300 .37614 35.00 53.400 .11897 80.00 33.600 .39421 40.00 50.300 .15358 85.00 33.900 .40744 45.00 47.500 .18996 90.00 35.800 .41511

Equations (B.9) can then be used to obtain

Si = 0.620i, S2 = 3.140i (B.10)

The required values of .p(1) and 1]1(1) for kl and k2 are

.p(1) = 0.37, 1]1(1)=0.19 (B.ll)

The necessary information in Tables B.V and B.VI can be found by making use of equations (B.10) and (B.ll). Figure B.3 shows that as the crack angle (3 decreases, the difference between the critical stresses predicted by the two curves can differ by an appreciable amount.

TABLE B.IV Fracture angle and normalized strain energy density factor for matrix cracking model (E-glass epoxy resin with Vf ~ 0.2)

(3 -IJD Smin/a2a (3 -lJo Smin/a2a

5.00 78.600 .00062 50.00 38.500 .07609 10.00 73.100 .00260 55.00 34.700 .09148 15.00 67.900 .00609 60.00 30.600 .10720 20.00 63.000 .01118 65.00 26.400 .12260 25.00 58.400 .01793 70.00 . 21.800 .13690 30.00 54.100 .02638 75.00 16.900 .14925 35.00 50.100 .03651 80.00 11.500 .15883 40.00 46.200 .04829 85.00 5.900 .16491 45.00 42.300 .06157 90.00 0.000 .16700

X Appendix B: E-glass epoxy resin composites LXXI

70 E - Glass Epoxy Composite

Vf = 0.2 en en 6.0 a.> ...

U5 c Matrix Model u

:-E 5.0 ... 0 Anisotropic Model -c a.> !::! c 4.0 E ... 0 z

(\J 3.0 j;: II

~CQ

~

" 2.0 ~CQ

~

1.0

o 30° 60°

f3 - Crack Angle

Figure B.2. Normalized (Fe versus {3 curve for E-glass epoxy composite (Vr =

0.2).

Fiber volume fraction 0.6. The elastic constants for the composite with a 60% fiber volume fraction are

E1 = 6.50 X 106 psi,

E2 = 1.70 X 106 psi,

11-12 = 0.64

V 12 =0.26

V23 = 0.33

based upon which S1 and S2 are calculated:

S1 =0.627i, S2 = 3.009i

(B.12a)

(B.12b)

(B.12c)

(B. 13)

LXXII Introductory Chapter

TABLE B.Y Fracture angle and strain energy density factor for homogeneous anisotropic model (E-glass epoxy resin with VI = 0.5)

(3 -00 Smin/a.2a (3 -00 Sminla2 a

5.00 74.900 .00102 50.00 44.300 .12845 10.00 70.400 .00442 55.00 41.900 .14995 15.00 66.200 .01062 60.00 39.800 .17037 20.00 62.400 .01987 65.00 38.000 .18899 25.00 58.800 .03222 70.00 36.600 .20524 30.00 55.500 .04753 75.00 35.600 .21866 35.00 52.400 .06544 80.00 35.400 .22891 40.00 49.500 .08538 85.00 36.200 .23559 45.00 46.800 .10665 90.00 38.400 .23820

The kl and k2 factors in equations (21) are determined as

<P(1) = 0.29, 11'(1) =0.16 CB.14)

Tables B.VII and B.VIIl summarize, respectively, the results for the aniso­tropic and matrix model. Comparing the curves in Figures B.3 and B.4, there appears to be negligible differences in the critical stresses for the two models when Vf is increased from 0.5 to 0.6.

TABLE B.VI Fracture angle and strain energy density factor for matrix cracking model (E-glass epoxy resin

with VI = 0.5)

(3 -00 Smit/(Y2 a (3 -00 Smin/a2a

5.00 77.900 .00016 50.00 35.900 .02293 10.00 71.700 .00067 55.00 32.100 .02801 15.00 66.100 .00159 60.00 28.200 .03327 20.00 60.800 .00296 65.00 24.100 .03844 25.00 56.000 .00485 70.00 19.800 .04326 30.00 51.600 .00730 75.00 15.200 .04741 35.00 47.500 .01034 80.00 10.300 .05063 40.00 43.500 .01399 85.00 5.200 .05266 45.00 39.700 .01821 90.00 0.00 .05336

LXXIII

70 E - Glass Epoxy Composite

Vf = 0.5 IJl IJl 6.0 Q) .... (jj

0 u

'';:::: 5.0 Matrix Model .;: 0

-0 Anisotropic Model Q) N

"0 4.0 E .... 0 z

C\I 3.0 l:: " ~Cl:l.

be..>

" 2.0 ~Cl:l.

be..>

1.0

o 30° 60° /3 - Crack Angle

Figure B.3. Normalized fy, versus {3 curve for E-glass epoxy composite (Vf =

0.5).

TABLE B.VII Fracture angle and normalized strain energy density factor for homogeneous anisotropic model (E-glass epoxy resin with V f = 0.6)

i3 -110 Smin/a2a i3 -110 Smin/u2a

5.00 74.800 .00083 50.00 44.500 .10380 10.00 70.300 .00360 55.00 42.200 .12110 15.00 66.200 .00863 60.00 40.100 .13752 20.00 62.400 .01613 65.00 38.300 .15247 25.00 58.900 .02614 70.00 36.900 .16551 30.00 55.600 .03RS2 75.00 36.000 .17625 35.00 52.600 .05299 80.00 35.900 .18441 40.00 49.700 .06908 85.00 36.800 .18967 45.00 47.000 .08623 90.00 39.200 .19161

LXXIV Introductory Chapter TABLE B.VIII Fracture angle and normalized strain energy density factor for matrix cracking model (E-glass epoxy resin with Vf = 0.6)

(3 -00 Smin/(]"2a (3 -00 Smin/(]"2a

5.00 78.000 .00010 50.00 36.600 .01476 10.00 72.100 .00045 55.00 32.700 .01796 15.00 66.500 .00106 60.00 28.800 .02127 20.00 61.300 .00197 65.00 24.600 .02453 25.00 56.600 .00320 70.00 20.200 .02755 30.00 52.200 .00479 75.00 15.500 .03017 35.00 48.100 .00675 80.00 10.600 .03219 40.00 44.290 .00908 85.00 5.400 .03347 45.00 40.300 .01177 90.00 0.000 .03391

7.0

E - Glass Epoxy Composite

If) Vf = 0.6 If) 6-0 Q) ... +-(f)

c (J

+- 5.0 Matrix Model .~

u "0 Anisotropic Model Q)

.!::! c 4.0 E ... 0 Z

I C\.I 3.0 ~ " ~~

bO ........ ~~

2.0

bO

1.0

o f3 - Crack Angle

Figure B.4. Normalized a c versus (3 curve for E-glass epoxy composite ("f =

0.6).

XI Appendix C: Stainless steel-aluminum composite LXXV

XI. Appendix C: Stainless steel-aluminum com­posite

Consider an aluminum matrix whose elastic properties are given by

Em = 10.50 X 106 psi

/-Lm = 3.94 X 106 psi, Vm =0.33

Stainless steel fibers with elastic properties

/-Lf = 11.54 X 106 psi, Vf = 0.30

(C.1a)

(C.1b)

(C.2a)

(C.2b)

are embedded in the aluminum matrix to form a unidirectional compo­site. To be considered are two different fiber volume fractions, namely Vf = 0.1 and 0.4.

Fiber volume fraction 0.1. Application of the formulas in equations (A. 1) yields the following anisotropic constants for the stainless steel-aluminum composite with Vf = 0.1:

El = 12.45 x 10 psi,

E2 = 11.56 X 106 psi,

/-L12 = 4.35 X 106 psi

V12 =0.33

V23 = 0.35

The parameters Sl and S2 are

Sl = 0.836i, S2 = 1.225i

For the matrix cracking model, it is necessary to know

4>(1) = 0.90, '1"(1) = 0.88

(C.3a)

(C.3b)

(C.3c)

(C.4)

(C.5)

for finding the crack tip stress intensity factors. Without going into details, -eo and S"'in/u2a for both of the models can be found with the aid of equations (C.4) and (C.5). Refer to Tables C.l and c.rr. It can be seen from Figure C.1 that both models give nearly the same results.

Fiber volume fraction 0.4. Let the fiber volume function of the stainless steel-aluminum composite be increased to 0.4. This will result in a

LXXVI Introductory Chapter

TABLE C.I Fracture angle and normalized strain energy density factor for homogeneous anisotropic model.~ (stainless steel-aluminum with Vr = 0.1)

(3 -80 Smin/a-2a (3

5.00 79.900 .00018 50.00 10.00 76.500 .00075 55.00 15.00 73.100 .00169 60.00 20.00 69.700 .00297 65.00 25.00 66.400 .00453 70.00 30.00 63.000 .00627 75.00 35.00 59.600 .00809 80.00 40.00 56.200 .00991 85.00 45.00 52.700 .01164 90.00

corresponding increase of the elastic moduli:

El = 18.30 X 106 psi,

E2 = 15.51 X 106 psi,

i-L12 = 5.87 X 106 psi

V12 = 0.32

V23 = 0.36

which may be used to render

Sl =0.777i, S2 = 1.365i

TABLE c.n

-80

49.100 45.300 41.300 37.000 32.100 26.300 19.300 10.500

0.000

Smin/a- 2a

.01322

.01463

.01586

.01691

.01781

.01858

.01921

.01966

.01983

(C.6a)

(C.6b)

(C.6c)

(C.7)

Fracture angle and normalized strain energy density factor for matrix cracking model (stainless steel-aluminum with VI = 0.1)

(3 ·-110 Smin/a-2a (3 -110 Smin/a-2a

5.00 80.000 .00016 50.00 48.800 .01149 10.00 76.600 .00065 55.00 45.000 .01275 15.00 73.100 .00146 60.00 40.900 .01385 20.00 69.700 .00257 65.00 36.400 .01481 25.00 66.300 .00392 70.00 31.200 .01564 30.00 62.900 .00542 75.00 25.300 .01634 35.00 59.500 .00700 80.00 18.100 .01690 40.00 56.000 .00858 85.00 9.600 .01728 45.00 52.500 .01009 90.00 0.000 .01742

XI Appendix C: Stainless steel-aluminum composite LXXVII

22 Stainless Steel-Aluminum Composite

(f)

Vf = 0.1 (f)

~ 2.0 Cf)

0 u

"- 1.8 u Matrix Model

-0 (]) N

0

E 1.6 "-0 z

(\J

"- 14 f::: " Anisotropic ~CQ

bU Model :::::: 1.2 ~CQ

bU

1.0

lL-____ ~ ______ ~ ______ ~_ o o

o 30 60 90

f3 - Crack Angle

Figure c.1. Normalized (To versus {3 curve for steel-aluminum composite (VI =

0.1).

The quantities <P(1) and -P(1) take the values

<P(1) = 0.67, -pel) = 0.63 (C.S)

which are found to decrease as the fiber volume fraction is increased from 0.1 to 0.4. Tables c.nI and C.IV give the calculated values of the fracture

LXXVIII Introductory Chapter

TABLE c.m Fracture angle and normalized strain energy density factor for homogeneous anisotropic model (stainless steel-aluminum with V f = 0.4)

(3 -00 Sm;n/a-2a (3 -00 Sm;n/a-2a

5.00 79.300 .00013 50.00 48.600 .01010 10.00 77.900 .00055 55.00 44.900 .01123 15.00 72.400 .00125 60.00 41.100 .01222 20.00 69.000 .00220 65.00 36.900 .01308 25.00 65.600 .00337 70.00 32.400 .01383 30.00 62.300 .00468 75.00 27.100 .01447 35.00 58.900 .00608 80.00 20.500 .01501 40.00 55.500 .00748 85.00 11.800 .01541 45.00 52.100 .00884 90.00 0.000 .01557

angle and normalized strain energy density factor, respectively, for the two models. The difference is again found to be small. This is also shown graphically in Figure C.2.

Equations (C.3) and (C.6) show that the elastic moduli in the principal directions of material symmetry do not vary appreciably. The prevailing nonhomogeneity has a negligible influence on the way with which the metal composite is modeled. This observation can also be seen from the curves in Figures C.l and C.2.

TABLE C.IV Fracture angle and normalized strain energy density factor for matrix cracking model (stainless steel-aluminum with V f = 0.4)

(3 -00 Sm;,,/a- 2 a (3 -00 Smin/cr2 a

5.00 79.900 .00008 50.00 48.100 .00613 10.00 76.300 .00034 55.00 44.300 .00684 15.00 72.800 .00076 60.00 40.200 .00748 20.00 69.300 .00134 65.00 35.700 .00805 25.00 65.800 .00204 70.00 30.600 .00855 30.00 62.400 .00284 75.00 24.600 .00897 35.00 58.900 .00368 80.00 17.600 .00930 40.00 55.400 .00453 85.00 9.250 .00953 45.00 51.800 .00536 90.00 0.000 .00961

XII References

2.2

(j) (j)

OJ 2.0-'--if)

0 u <-'-- 1.8 u -0 OJ ~ 0 E 1.6 '--0 Z

N

~ 14 "

~(Q

bU

'-... 1.2 ~(Q

0'

1.0

o

Anisotropic Model

Stainless Steel-Aluminum

Composite

MatrIX Model

f3 - Crack Angle

LXXIX

Figure C.2. Normalized (Te versus (3 curve for steel-aluminum composite (Vt =

0.4).

XII. References

[1] Matthaes, K, Betrachtugen zur theorie der werkstoff-festigkeit, Zeitschrift fiir Metallkunde, Vol. 43, pp. 11-19 and 90-95, (1952).

[2] Gillemot, L. F., "Criterion of crack initiation and spreading", International Journal of Engineering Fracture Mechanics, Vol. 8, pp. 239-253, (1976).

[3] Sih, G. c., A special theory of crack propagation, Methods of analysis and solutions of crack problems, Mechanics of Fracture I, edited by G. C. Sih, Noordhoff International Publishing, Leyden, pp. 21-45, (1973).

LXXX Introductory Chapter

[4] Sih, G. c., A three-dimensional strain energy density factor theory of crack propaga­tion, Three dimensional crack problems, Mechanics of Fracture II, edited by G. C. Sih, Noordhoff International Publishing, Leyden, pp. XV-LIII, (1975).

[5] Sih, G. c., Strain energy density and surface layer energy for blunted cracks or notches, Stress analysis of notch problems, Mechanics of Fracture V, edited by G. C. Sih, Noordhoff International Publishing, Alphen aan den Rijn, pp. XIII-CX, (1978).

[6] Theocaris, P. S., Experimental determination of the core region in mixed-mode fracture, Mixed Mode Crack Propagation, edited by G. C. Sih and P. S. Theocaris, Sijthoff and Noordhoff International Publishers, Alphen aan den Rijn, pp. 21-36, (1981).

[7] Sih, G. C. and Kiefer, B. V., Stable growth of surface cracks, Journal of the Engineering Mechanics Division, ASCE, Vol. 106, No. EM2, pp. 245-253, (1980).

[8] Sih, G. C. and Kiefer, B. V., Nonlinear response of solids due to crack growth and plastic deformation, Nonlinear and Dynamic Fracture Mechanics, edited by N. Perrone and S. N. Atluri, The American Society of Mechanical Engineers, AMD-Vol. 35, pp. 135-156, (1979).

[9] Stephenson, R A., The equilibrium field near the tip of a crack for finite plane strain of incompressible elastic materials, California Institute of Technology, Report No. 44, (1980).

[10] Ivanova, V. S. and Ginsztler, J., The examination of fracture toughness with the aid of Gillemot's absorbed specific energy till fracture, Absorbed specific energy and/or strain energy density criterion, edited by G. C. Sih, E. Czoboly and F. Gillemot, Sijthoff and Noordhoff International Publishers, Alphen aan den Rijn, pp. 101-106, (1981).

[11] Wnuk, M. P. and Mazur, A., Experimental verification of the strain energy density criterion for the stable phase of tensile fracture, Absorbed specific energy and/or strain energy density criterion, edited by G. C. Sih, E. Czoboly and F. Gillemot, Sijthoff and Noordhoff International Publishers, Alphen aan den Rijn, pp. 297-308, (1981).

[12] Rice, J. Rand Sih, G. c., Plane problems of cracks in dissimilar media, Journal of Applied Mechanics, Vol. 32, pp. 418-423, (1965).

[13] Kipp, M. E. and Sih, G. c., The strain energy density failure criterion applicd to notched elastic solids, International Journal of Solids and Structures, Vol. 2, pp. 153-173, (1975).

[14] Sih, G. C. and Moyer, Jr., E. T., Influence of interface on composite failure, Proceedings of Advanced Composites, Technology Conferences, El Segundo, California, pp. 283-312, (1979).

[15] Lauraitis, K., Tensile strength of off-axis unidirectional composites, University of Illinois TAM Report No. 344, (1971).

[16] Sih, G. C. and Chen, E. P., Fracture analysis of unidirectional composites, Journal of Composite Materials, Vol. 7, pp. 230-244, (1973).

[17] Sih, G. c., Paris, P. C. and Irwin, G. R, On cracks in rectilinearly anisotropic bodies, International Journal of Fracture Mechanics, Vol. 1, pp. 189-203, (1965).

[18] Wu, E. M. and Reuter, R c., Crack extension in fiber-glass'reinforced plastics, University of Illinois TAM Report No. 275, (1965).

[19J Lauraitis, K., Failure modes and strength of angle-ply laminates, University of Illinois TAM Report No. 345, (1971).

[20J Sih, G. c., Fracture mechanics of adhesive joints, Journal of Polymer Engineering and Science, Vol. 20, No. 14, PP. 977-981, (1980).

l21J Puppo, A. H. and Evensen, H. A., Interlaminar shear in laminated composites under generalized plane stress, Journal of Composite Materials. Vol. 4, pp. 204-220, (1970).

XII References LXXXI

[22] Reissner, E. and Stavsky, Y., Bending and stretching of certain types of heterogeneous aeolotropic elastic plates, Journal of Applied Mechanics, Vol. 28, pp. 402-408, (1961).

[23] Stavsky, Y., Bending and stretching of laminated aeolotropic plates, Proceedings American Society of Civil Engineering, Journal of the Engineering Mechanics Division, Vol. 87, pp. 31-56, (1961).

[24] Pipes, R. B. and Pagano, N. J., Interlaminar stresses in composite laminates under uniform axial extension, Journal of Composite Materials, Vol. 4, pp. 538-548, (1970).

[25] Sih, G. C. and Hilton, P. D., Three-dimensional analysis of laminar composites with through cracks, American Society of Testing Materials, STP 539, pp. 3-35, (1975).

[26] Sih, G. C. and Chen, E. P., Normal and shear impact of layered composite with a crack: dynamic stress intensification, Journal of Applied Mechanics, Vol. 47, pp. 351-358, (1980).

[27] Sih, G. C. and Chen, E. P., Effect of material nonhomogeneity on crack propagation characteristics, International Journal of Engineering Fracture Mechanics, Vol. 13, No.3, pp. 431-438, (1980).

[28] Sih, G. C., Influence of interface modeling on composite failure, Proceedings of the Third International Sympo,~ium on Continuum Models of Discrete Systems, University of Waterloo, (in press).

[29] Sih, G. c., Dynamic crack problems: strain energy density fracture theory, Elas­todynamic Crack Problems, Mechanics of Fracture IV, edited by G. C. Sih, Noordhoff International Publishing, Leyden, pp. XVII-XLVII, (1977).