C16H

Natural Sciences Tripos Part II

MATERIALS SCIENCEMATERIALS SCIENCE

C16: Composite Materials

Prof. T. W. Clyne

Lent Term 2013 14

Name............................. College..........................

Lent Term 2013-14

II

Part II Materials Science: Course C16: Composite Materials - Student Handout C16H1

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Course C16: Composite Materials Synopsis (12 lectures)

Lecture 1 - Overview of Types of Composite System Overview of Composites Usage. Types of Reinforcement and Matrix. Carbon and Glass Fibres. PMCs, MMCs and CMCs. Aligned Fibre Composites, Woven Rovings, Chopped Strand Mat, Laminae and Laminates.

Lecture 2 - Elastic Constants of Long Fibre Composites Recap of Axial and Transverse Youngs Moduli for an Aligned Long Fibre Composite, derived using the Slab Model. Errors for Transverse Loading and Use of Halpin-Tsai Equations. Derivation of Shear Moduli and Poisson Ratios. Number of Elastic Constants for Systems with different Degrees of Symmetry.

Lecture 3 - Elastic Loading of a Lamina Plane Stress Loading of a Uniaxial Lamina and the Kirchoff Assumptions. Off-axis loading of a Lamina. Elastic Constants as a Function of Loading Angle. Tensile-shear Interactions and Lamina Distortions.

Lecture 4 - Elastic Loading of a Laminate A Laminate considered as a Stack of Laminae. Elastic Properties of Laminates as a Function of Loading Angle. Elastic Constants of Some Simple Laminates. Balanced Laminates. Coupling Stresses and Symmetric Laminates.

Lecture 5 - Short Fibre & Particulate Composites Stress Distributions The Shear Lag Model for Stress Transfer. Interfacial Shear Stresses. The Stress Transfer Aspect Ratio. Stress Distributions with Low Reinforcement Aspect Ratios. Numerical Model Predictions. Hydrostatic Stresses and Cavitation.

Lecture 6 - Short Fibre & Particulate Composites Stiffness & Inelastic Behaviour Load Partitioning and Stiffness Prediction for the Shear Lag Model. Fibre Aspect Ratios needed to approach the Long Fibre (Equal Strain) Stiffness. Inelastic Interfacial Phenomena. Interfacial Sliding and Matrix Yielding. Critical Aspect Ratio for Fibre Fracture.

Lecture 7 - The Fibre-Matrix Interface Interfacial Bonding Mechanisms. Measurement of Bond Strength. Pull-out & Push-out Testing. Control of Bond Strength. Silane Coupling Agents. Interfacial Reactions and their Control during Processing.

Lecture 8 - Fracture Strength of Composites Axial Tensile Strength of Long Fibre Composites. Transverse and Shear Strength. Mixed Mode Failure and the Tsai-Hill Criterion. Failure of Laminates. Internal Stresses in Laminates. Failure Sequences. Testing of Tubes in combined Tension and Torsion.

Lecture 9 - Fracture Toughness of Composites Energies absorbed by Crack Deflection and by Fibre Pull-out. Crack Deflection . Toughness of Different Types of Composite. Constraints on Matrix Plasticity in MMCs. Metal Fibre Reinforced Ceramics.

Lecture 10 - Compressive Loading of Fibre Composites Modes of Failure in Compression. Kink Band Formation. The Argon Equation. Prediction of Compressive Strength and the Effect of Fibre Waviness. Failure in Highly Aligned Systems. Possibility of Fibre Crushing Failure.

Lecture 11 - Thermal Expansion of Composites and Thermal Residual Stresses Thermal Expansivity of Long Fibre Composites. Transverse Expansivities. Short Fibre and Particulate Systems. Differential Thermal Contraction Stresses. Thermal Cycling. Thermal Residual Stresses.

Lecture 12 - Surface Coatings as Composite Systems Misfit Strains in Substrate-Coating Systems. Force and Moment Balances. Relationship between Residual Stress Distribution and System Curvature. Curvature Measurement to obtain Stresses in Coatings. Limitations of Stoney Equation. Sources of Misfit Strain. Driving Forces for Interfacial Debonding.


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Booklist D.Hull & T.W.Clyne, "An Introduction to Composite Materials", Cambridge University Press, (1996) [AN10a.86]

Web-based Resources Most of the material associated with the course (handouts, question sheets, examples classes

etc) can be viewed on the web and also downloaded. This includes model answers, which are released after the work concerned should have been completed. In addition to this text-based material, resources produced within the DoITPoMS project are also available. These include libraries of Micrographs and of Teaching and Learning Packages (TLPs). The following TLPs are directly relevant to this course: U Mechanics of Fibre Composites U Bending and Torsion of Beams


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Lecture 1: Overview of Composites & Types of Composite System Stiff, Light, Corrosion-Resistant Structures The Attractions of Composites

Fig.1.1 Data for some engineering materials, in the form of a map of Youngs modulus against

density


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Fibres used in Composite Materials Carbon Fibres

Fig.1.2 Effect of heat treatment temperature on the strength and Youngs modulus of carbon

fibres produced from a PAN precursor


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Glass Fibres

Polymeric Fibres

Fig.1.3 Structures of (a) cellulose & (b) Kevlar (poly paraphenylene terephthalamide) molecules


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Other Reinforcements


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Fibre Distributions and Orientations

Fig.1.4 A fibre laminate (stack of plies), illustrating the nomenclature system


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Lecture 2: Elastic Constants of Long Fibre Composites Use of the Slab Model

Fig.2.1 Schematic illustration of loading geometry and distributions of stress and strain, and

effects on the Youngs moduli and shear moduli, for a uniaxial fibre composite and for the slab model representation


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Halpin-Tsai Expressions

Fig.2.2 Predicted dependence on fibre volume fraction, for the epoxy-glass fibre system, of (a) transverse Youngs modulus and (b) shear moduli of long fibre composites


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Poisson Ratios

Fig.2.3 Schematic representation of the three Poisson ratios of an aligned composite

Fig.2.4 Predicted dependence on fibre volume fraction, for the epoxy-glass fibre system, of the

three Poisson ratios


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Stress, Strain, Stiffness & Compliance Tensors

Fig.2.5 Examples of how 2-D relative displacement components can represent different

combinations of shear and rigid body rotation


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Lecture 3: Elastic Loading of a Lamina Symmetry & Use of Matrix Notation for Matter Tensors

Matrix Notation

Effect of Material Symmetry on the Number of Independent Elastic Constants

Fig.3.1 Indication of the form of the Spq and Cpq matrices (matrix notation for Sijkl and Cijkl

tensors), for materials exhibiting different types of symmetry. All of the matrices are symmetrical about the leading diagonal


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Off-axis Elastic Constants of Laminae Loading Parallel and Normal to Fibre Axis

Loading at Arbitrary Angles to Fibre Axis

Fig.3.2 (a) Relationship between the fibre-related axes in a lamina (1, 2 & 3) and the co-

ordinate system (x, y & z) for an arbitrary in-plane set of applied stresses. (b) Illustration of how such an applied stress state ij (x, y & xy) generates stresses in the fibre-related framework of ij (1, 2 & 12)


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Derivation of Transformed Stress-Strain Relationship For a thin lamina, stresses and strains in the through-thickness (3) direction are neglected, so

that the 3, 4, and 5 components in matrix notation are of no concern. Therefore, when a lamina is loaded parallel or normal to the fibre axis, the strains that interest us are given by

12 12

= S 1 212

=S11 S12 0S12 S22 00 0 S66

1 212

(3.1)

in which, by inspection of the individual equations, it can be seen that

S11 =1E1

S12 =12E1

=21E2

S22 =1E2

S66 =1G12

The first step in establishing the lamina strains for off-axis loading is to find the stresses, referred to the fibre axis (1, 2 and 12), in terms of the applied stress system (x, y and xy). This is done using the equation expressing any second rank tensor with respect to a new coordinate frame

ij =aikajl kl

in which aik is the direction cosine of the (new) i direction referred to the (old) k direction. Obviously, the conversion will work in either direction provided the direction cosines are defined correctly. For example, the normal stress parallel to the fibre direction 11, sometimes written as 1, can be expressed in terms of the applied stresses '11 (= x), '22 (= y) and '12 (= xy)

11 =a11a11 11 +a11a12 12+a12a11 21 +a12a12 22

The angle is that between the fibre axis (1) and the stress axis (x). Referring to the figure, these direction cosines take the values

a11 =cos (= c)a12 =cos 90 ( )=sin (= s)a21 =cos 90 + ( ) = sin (= s)a22 =cos (= c)

Carrying out this operation for all three stresses 1 212

= T x y xy

(3.2)

where

T =c2 s2 2css2 c2 2cscs cs c2 s2( )

The same matrix can be used to transform tensorial strains, so that 1212

= T x y xy


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However, to use engineering strains (xy = 2xy etc), T must be modified (by halving the elements t13 and t23 and doubling elements t31 and t32 of the matrix T ), so as to give

12 12

= T x y xy

(3.3)

in which

T =c2 s2 css2 c2 cs

2cs 2cs c2 s2( )

The procedure is now a progression from the stress-strain relationship when the lamina is loaded along its fibre-related axes to a general one involving a transformed compliance matrix, S , which will depend on . The first step is to write the inverse of Eqn.(3.3), giving the strains

relative to the loading direction (ie the information required), in terms of the strains relative to the fibre direction. This involves using the inverse of the matrix T , written as T 1

x y xy

= T 1 12 12

in which

T 1 =c2 s2 css2 c2 cs2cs 2cs c2 s2( )

Now, the strains relative to the fibre direction can be expressed in terms of the stresses in those directions via the on-axis stress-strain relationship for the lamina, Eqn.(3.1), giving

x y xy

= T 1 S 1 212

Finally, the original transform matrix of Eqn.(3.2) can be used to express these stresses in terms of those being externally applied, to give the result

x y xy

= T 1 S T x y xy

= S x y xy

(3.4)

The elements of S are therefore obtained by concatanation (the equivalent of multiplication) of the matrices T 1 , S and T . The following expressions are obtained


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S11 =S11c4 + S22s4 + 2S12 + S66( )c2s2S12 =S12 c4 + s4( ) + S11 + S22 S66( )c2s2S22 =S11s4 + S22c4 + 2S12 + S66( )c2s2S16 = 2S11 2S12 S66( )c3s 2S22 2S12 S66( )cs3S26 = 2S11 2S12 S66( )cs3 2S22 2S12 S66( )c3sS66 = 4S11 + 4S22 8S12 2S66( )c2s2 +S66 c4 + s4( )

(3.5)

It can be seen that S S as 0.


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Effect of Loading Angle on Stiffness

Fig.3.3 Variation with loading angle of the Youngs modulus Ex and shear modulus Gxy for a

lamina of epoxy-50% glass fibre


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Tensile-Shear Interaction Behaviour

Fig.3.4 Variation with loading angle of the tensile-shear interaction compliance S16, for a

lamina of rubber-5% Al fibre, and photos of 4 specimens (between crossed polars) under axial tension, lined up at the appropriate values of , showing tensile-shear distortions


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Lecture 4: Elastic Loading of a Laminate Obtaining the Elastic Constants of a Laminate

Fig.4.1 Schematic depiction of the loading angle between the x-direction (stress axis) and the

reference direction (N5>A0;05=?;84B;B>B7>F=8BC740=6;4k between the reference direction and the fibre axis of the k th ply (1k direction)


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Stiffness of Laminates

Fig.4.2 Variation with loading angle (between the stress axis and the reference (N

direction) of the Youngs modulus of a single lamina and of two simple laminates, made of epoxy-50% glass fibre. (The equal stress model was used to obtain the transverse Youngs modulus of the lamina, E2.)


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Tensile-Shear Interactions and Balanced Laminates

Fig.4.3 Variation with loading angle (between the stress axis and the reference (N

direction) of the interaction ratio, xyx (ratio of the shear strain xy to the normal strain x) of a single lamina and of three simple laminates, made of epoxy-50% glass fibre. (The equal stress model was used to obtain the transverse Youngs modulus of the lamina, E2.)


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In-plane Stresses within a Loaded Laminate

Fig.4.4 (a) Predicted stresses within one ply of a loaded crossply laminate (epoxy-50%glass)

and (b) a schematic of these stresses for loading parallel to one of the fibre axes


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Coupling Stresses and Symmetric Laminates

Fig.4.5 Elastic distortions of a crossply laminate as a result of (a) uniaxial loading and

(b) heating


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Lecture 5: Short Fibre & Particulate Composites - Stress Distributions The Shear Lag Model for Short Fibre Composites

Displacements of Fibre and Matrix

Fig.5.1 Schematic illustration of the basis of the shear lag model, showing (a) unstressed system,

(b) axial displacements, u, introduced on applying tension parallel to the fibre and (c) variation with radial location of the shear stress and strain in the matrix


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Derivation of Equations The model is based on assuming that the build-up of tensile stress along the length of the fibre

occurs entirely via the shear forces acting on the cylindrical interface. This leads immediately to the basic shear lag equation:

d fdx =

2 ir0

(5.1)

The interfacial shear stress, i, is obtained by considering how the shear stress in this direction varies within the matrix as a function of radial position. This variation is obtained by equating the shear forces on any two neighbouring annuli in the matrix:

2 r1 1 dx=2 r2 2 dxie1 2

=r2r1

= i r0r

The displacement of the matrix in the loading direction, u, is now considered. The shear strain at any point in the matrix can be written both as a variation in this displacement with radial position and in terms of the local shear stress and the shear modulus of the matrix, Gm

= Gm=

i r0r

Gmand =dudr

It follows that an expression can be found for the interfacial shear stress by considering the change in matrix displacement between the interface and some far-field radius, R, where the matrix strain has become effectively uniform (du/dr 0).

duur0uR = i r0Gm

drrr0

R

i =uR ur0( )Gmr0 ln

Rr0

(5.2)

The appropriate value of R is affected by the proximity of neighbouring fibres, and hence by the fibre volume fraction, f. The exact relation depends on the precise distribution of the fibres, but this needn't concern us too much, particularly since R appears in a log term. If an hexagonal array of fibres is assumed, with the distance between the centres of the fibres at their closest approach being 2R, then simple geometry leads to

Rr0

2

=

2 f 3 1f

Substituting for i in the basic shear lag equation now gives d fdx =

2 uR ur0( )Gmr02 12 ln

1f

The displacements uR and ur0 are not known, but their differentials are related to identifiable strains. The differential of ur0 is simply the axial strain in the fibre (assuming perfect interfacial adhesion and neglecting any shear strain in the fibre - which is taken as being much stiffer than the matrix)


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dur0dx = f =

fEf

while the differential of uR, ie the far-field axial strain of the matrix, can be taken as the macroscopic strain of the composite

duRdx 1

Differentiating the expression for the gradient of stress in the fibre and substituting these two relations into the resulting equation, with the shear modulus expressed in terms of Young's modulus and Poisson's ratio [Em=2 Gm (1+m)], leads to

d2 fdx2 =

n2r02

f Ef 1( ) (5.3)

in which n is a dimensionless constant (for a specified composite), given by

n = 2EmEf 1+ m( )ln 1f

(5.4)

This is a second order linear differential equation of a standard form, which has the solution

f =Ef 1 +Bsinhnxr0

+Dcosh nxr0

and, by applying the boundary condition of f = 0 at x = L (the fibre half-length), the constants B and D can be solved to give the final expression for the variation in tensile stress along the length of the fibre

f =Ef 1 1 coshnxr0

sech ns( )

(5.5)

in which s is the aspect ratio of the fibre (=L/r0). From this expression, the variation in interfacial shear stress along the fibre length can also be found, using the basic shear lag equation, by differentiating and multiplying by (-r0/2),

i =Ef n12 sinh

nxr0

sech ns( ) (5.6)

An estimate can now be made of the axial modulus of the composite. This is done by using the Rule of Averages (1 = f f

_ + (1-f) m_ ), with the average matrix stress taken as its Young's

modulus times the composite strain and the average fibre stress obtained by integrating the above expression for f over the length of the fibre. This leads to

E1 =11

= f Ef 1tanh ns( )

ns

+ 1 f( )Em (5.7)


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The Stress Transfer Length (Aspect Ratio)

Fig.5.2 Predicted (shear lag) variations in (a) fibre tensile stress and (b) interfacial shear stress

along the axis of a glass fibre in a polyester-30% glass composite subject to an axial tensile strain of 10-3, for two fibre aspect ratios


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Fibre End Regions - Hydrostatic Stresses and Cavitation

(a) (b) Fig.5.3 Photoelastic (frozen stress) models under applied axial load, showing the stress field

in the matrix around two stiff reinforcements having the same aspect ratio, with (a) cylindrical and (b) ellipsoidal shapes

Fig.5.4 Predicted (finite element) hydrostatic stress fields around sphere and cylinder (s=5) of

SiC in an Al matrix, with an applied axial tensile stress of 100 MPa (and differential thermal contractions stresses corresponding to a temperature drop of 50 K)


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Lecture 6: Short Fibre & Particulate Composites - Stiffness & Inelastic Behaviour Shear Lag Model Predictions for Stiffness

Fig.6.1 Predicted composite/matrix Youngs modulus ratio, as a function of fibre/matrix Youngs

modulus ratio, for aligned short fibre composites with 30% fibre content and fibre aspect ratio (s) values of (a) 30 and (b) 3. Shear lag model predictions are reliable when s is relatively large. For very short fibres, the predictions become inaccurate, due to neglect of the stress transfer across the fibre ends, which is more significant for shorter fibres


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Approach to Rule of Mixtures (Long Fibre) Stiffness

Fig.6.2 A set of four (rubber 5% Al fibre) photoelastic models under axial load, showing how

the stress field and the axial extension change as the aspect ratio and degree of alignment of the fibres are changed


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Interfacial Sliding and Matrix Yielding

Fig.6.3 Plots of the dependence of peak fibre stress, f0, (at the onset of interfacial sliding or

matrix yielding) on the critical shear stress for these phenomena, i*. Plots are shown for different fibre aspect ratios, with n values typical of polymer- and metal-based composites. Also indicated are typical value ranges for fracture of fibres and for matrix yielding and interfacial debonding


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Critical Fibre Aspect Ratio


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Lecture 7: The Fibre-Matrix Interface Bonding Mechanisms and Residual Stresses

Bonding Mechanisms


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Residual Stress Distributions

Fig.7.1 Predicted stress distribution around and within a single fibre, in a polyester-35% glass

long fibre composite, as a result of differential thermal contraction (T drop of 100 K)


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Silane Coupling Agents for Glass Fibres

Fig.7.4 Depiction of the action of silane coupling agents, which are used to generate improved

fibre-matrix bonding for glass fibres in polymeric matrices. The silane reacts with adsorbed water to create a strong bond to the glass surface. The R group is one which can bond strongly to the matrix

Objectives for MMCs and CMCs


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Bond Strength Measurement Single Fibre Pull-out Testing

Fig.7.2 Schematic stress distributions and load-displacement plot during single fibre pull-out

testing. The interfacial shear strength, *, is obtained from the pull-out stress, 0,*


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Single Fibre Push-out Testing

Fig.7.3 Schematic stress distributions and load-displacement plot during the single fibre push-

out test. One difference from the pull-out test is that the Poisson effect causes the fibre to expand (rather than contract), which augments (rather than offsets) the radial compressive stress across the interface due to differential thermal contraction


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Lecture 8: Fracture Strength of Composites Axial, Transverse and Shear Strengths of Long Fibre Composites

Fig.8.1 Schematic depiction of the fracture of a unidirectional long fibre composite at critical

values of (a) axial, (b) transverse and (c) shear stresses Axial Strength

Transverse and Shear Strengths


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Failure Criteria for Laminae subject to In-plane Stresses Maximum Stress Criterion

Mixed Mode Failure and the Tsai-Hill Criterion

Fig.8.2 Single ply failure stresses, as a function of loading angle: (a) maximum stress criterion, for polyester-50%glass (1*=700 MPa, 2*=20 MPa, 12*=50 MPa) and (b) maximum stress and Tsai-Hill criteria, plus experimental data, for epoxy-50%carbon (1*=570 MPa, 2*=32 MPa, 12*=56 MPa)


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Experimental Data for Single Laminae

Fig.8.3 Schematic illustration of how a hoop-wound tube is subjected to simultaneous tension

and torsion in order to investigate failure mechanisms and criteria


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Failure of Laminates Failure Sequences in Laminates

Fig.8.4 Loading of the crossply laminate of Fig.4.4 parallel to one of the fibre directions:

(a) cracking of transverse plies as 2 reaches 2*, (b) onset of cracking parallel to fibres in axial plies as 2 (from inhibition of Poisson contraction) reaches 2* and (c) final failure as 1 in axial plies reaches 1*


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Failure of Laminates under Uniaxial and Biaxial Loading

Fig.8.5 Stresses within an angle-ply laminate of polyester-50%glass fibre, as a function of the ply angle: (a) stresses within one of the plies, as ratios to the applied stress. and (b) applied stress at failure (maximum stress criterion, with 1*=700 MPa, 2*=20 MPa and 12*=50 MPa)


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Fig.8.6 Stresses within an angle-ply laminate of polyester-50%glass fibre, as a function of the

ply angle, when subjected to biaxial loading, with x=2y: (a) stresses within one of the plies, as ratios to the applied x. and (b) applied stress, x, at failure (maximum stress criterion, with 1*=700 MPa, 2*=20 MPa and 12*=50 MPa)


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Lecture 9: Fracture Toughness of Composites Fracture Energies of Reinforcements and Matrices

Crack Deflection at Interfaces Planar Systems

Fig.9.1 Schematic load-displacement plots for 3-point bend testing of monolithic SiC and a SiC

laminate with (weak) graphitic interlayers

Fig.9.2 SEM micrographs showing the layered structures of (a) a mollusc and (b) a SiC laminate


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Energy of Interfacial Debonding in Fibre Composites

Fig.9.3 Schematic representation of the advance of a crack in a direction normal to the fibre

axis, showing interfacial debonding and fibre pull-out processes


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Energy of Fibre Pull-out

Gcpo =Ndx0L0

L

rx02 i* = fr2

r i*L

L33

= fs

2r i*3 (9.1)

Effects of Fibre Flaws and Weibull Modulus

Fig.9.4 Schematic depiction of stress distribution, and associated probability of fracture, along a

fibre bridging a matrix crack, for (a) fixed fibre strength * (m=) and (b) strength which varies along the length of the fibre, due to the presence of flaws (finite m)


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Fracture Energy of a Metal Fibre Reinforced Ceramic Composite

(a) (b) Fig.9.5 Microstructure of a composite (Fiberstone) comprising coarse stainless steel fibres in

a matrix which is predominantly alumina, illustrated by (a) an X-ray tomograph, showing the fibres only, and (b) an optical micrograph of a polished section

There have been many attempts to produce ceramic-matrix composites with high toughness, but with limited success. Probably the most promising approach is to introduce a network of metallic fibres, and this is the basis of a commercial product (Fiberstone see Fig.9.5). The fibres are often about 0.5 mm diameter, although finer fibres can be used. During fracture, fibres bridge the crack and energy is absorbed by both frictional pull-out and plastic deformation - see Fig.9.6. These mechanisms are likely to dominate any other contributions to the work of fracture.

Fig.9.6 Schematic representation of the fracture of Fiberstone, showing: (a) overall fracture

geometry, (b) fibres undergoing debonding, possibly fracture, and then frictional pull-out and (c) fibres undergoing debonding, plastic deformation and then fracture


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The work of fracture can thus be estimated by summing the energy absorbed via both processes, assuming that a fraction g of the fibres bridging the crack plane undergo pull-out and the remainder (1-g) undergo plastic deformation and rupture.

Gcnet = Gcpo +Gcfd (9.2)

An expression for the fibre pull-out work was derived previously (Eqn.(9.1)), but the relationship between N and f depends on fibre orientation distribution, and that treatment referred to aligned fibres. For this type of composite, it can be taken as isotropic (random), in which case N is half that for the aligned case (see EE Underwood, Quantitative Stereology. 1970, Addison-Wesley)

N = f2r2 (9.3)

leading to

Gcpo =gfs2r i*6 (9.4)

where s is here the ratio of , the (average) length of fibre extending beyond the crack plane, to the fibre radius, r.

Fig.9.7 Data from tensile testing of single 304 stainless steel fibres, showing (a) a set of 10

stress-strain curves and (b) the distribution of corresponding work of deformation values The work done during plastic deformation and rupture of fibres can be estimated by assuming

that interfacial debonding extends a distance x0 from the crack plane - see Fig.9.6(c). The energy is obtained by summing the work done on each fibre, as if it had an original length 2x0 and were being subjected to a simple tensile test

Gcfd = (1 g)2x0NUfd = (1 g)2x0f

2r2

Wfdr

2 = (1 g)x0 fWfd (9.5)

where Ufd and Wfd are the work of deformation of the fibre, expressed respectively per unit length (J m-1) and per unit volume (J m-3). The latter is given by the area under the stress-strain curve of the fibre. Some such curves, for the fibres used in Fiberstone, are shown in Fig.9.7, together with corresponding Wfd values. The value of is in this case given by the product of x0 and *, the fibre strain to failure, leading to

Gcfd = (1 g)*

fWfd =

(1 g)srfWfd*

(9.6)


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Use of Eqns.(9.2), (9.4) and (9.6) allows prediction of the composite fracture energy, although it requires measurements or assumptions to be made concerning several parameters. In addition to the single fibre work of deformation, Wfd, and the failure strain, *, estimates are required for the proportion of fibres undergoing pull-out, g, the interfacial frictional sliding stress, i*, and the (average) length of fibre extending beyond the crack plane, , and hence the protrusion aspect ratio, s (= /r) Nevertheless, predictions can be made, based on experimental data or on plausible assumptions, and compared with measured composite fracture energies. An example is shown in Fig.9.8, where it can be seen that, even with the relatively low fibre content (~10-15%) that is normally present, the work of fracture is both predicted and observed to be substantial. The experimental Gc values were obtained by impact (Izod) testing.

Fig.9.8 Comparison between experimental data for the fracture energy of Fibrestone

composites, as a function of fibre volume fraction, and predictions obtained using Eqns.(9.4) amd (9.6), for fine and coarse fibres

The value of s can be estimated from observation of fracture surfaces. However, its difficult to be sure whether particular fibres have predominantly undergone pull-out, rather than plastic deformation and rupture - of course, some fibres could deform plastically and then pull out. In any event, very strong bonding may be undesirable, since this will tend to inhibit both pull-out and plasticity, although very poor bonding may allow fracture to take place without the fibres being significantly involved in the process. An intermediate bond strength is likely to give optimal toughness.

It also worth noting that, for a given fibre protrusion aspect ratio, s (= /r), both pull-out and plastic deformation contributions increase linearly with the absolute scale (fibre diameter). Composites reinforced with coarser fibres therefore tend to be tougher, particularly for this type of composite. Its clear that refining the scale of the microstructure does NOT always give benefits!


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Lecture 10: Compressive Loading of Fibre Composites

Euler Buckling


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Kink Band Formation

Fig.10.1 Optical micrograph of an axial section of a carbon fibre composite after failure under

uniaxial compression, showing a kink band

Fig.10.2 Predicted kinking stress, as a function of misalignment angle, for epoxy-60%carbon

composites, with two different interfacial shear strengths


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Failure by Fibre Crushing in Highly Aligned Systems

(a) (b) Fig.10.3 (a) Fragment of SiC monofilament extracted from a Ti-35%SiC composite after loading

under axial compression and (b) schematic of the crushing process

Fig.10.4 Stresses in Ti-35%SiC monofilament composite (average axial values for fibre, matrix

and composite) as axial strain is increased by external loading. At zero strain, stresses in fibre and matrix are from differential thermal contraction. The matrix yields when the stress in it reaches mY. It is assumed that no matrix work hardening occurs during plastic straining. Failure occurs when the fibre stress reaches the critical value f*

Failure is expected when the fibre stress reaches f*, taken to be a single, fixed value. The composite strength c* can readily be predicted, provided it can be assumed that the matrix yields before composite failure and that matrix work hardening is negligible, since it is then given by

c* = E1ccmY + E1c' c* cmY( ) (10.1) in which the composite moduli before and after matrix yielding are given by


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E1c = fEf + 1 f( )EmE1c' = fEf

Now, the strains at matrix yield and at final failure can be written as

cmY =mY +mT

Em

c* = f* + fT

Ef

Substituting into Eqn.(10.1), and applying the residual stress force balance condition f fT + 1 f( )mT = 0

then leads to c* = f f* + 1 f( )mY

A correction should be applied for the effect of misalignment in reducing the stress parallel to the fibre axis, leading to

c* =f f* + 1 f( )mY

cos20 (10.2)

This predicted strength is independent of the thermal residual stresses (whereas the strain at which failure occurs will depend on them).

Fig.10.5 Experimental strength data, as a function of the initial angle between fibre and loading

axes, during compression of misaligned Ti-35%SiC specimens. Also shown are predicted curves for failure by kink band formation and by fibre crushing, obtained by substitution of the values shown into the kinking equation and Eqn.(10.2) respectively


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Lecture 11: Thermal Expansion of Composites & Thermal Residual Stresses

Thermal Expansivity Data for Reinforcements and Matrices

Fig.11.1 Thermal expansion coefficients for various materials over a range of temperature


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Derivation of Expression for Composite Axial Expansivity

Fig.11.2 Schematic showing thermal expansion in the fibre direction of a long fibre composite,

using the slab model


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Transverse Thermal Expansivities

Fig.11.3 Predicted thermal expansivities of Al-SiC uniaxial fibre composites, as a function of

fibre content


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Thermal Stresses in Composite Systems Magnitudes of Thermal Residual Stresses Stresses in Composites during Thermal Cycling

Fig.11.4 Neutron diffraction data for an Al-5vol%SiC whisker (short fibre) composite, showing

lattice strains (& hence stresses) within matrix & reinforcement during unloaded thermal cycling. (111) reflections were used for both constituents. The gradients shown are calculated values for elastic behaviour, assuming a fibre aspect ratio of 10


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Lecture 12: Surface Coatings as Composite Systems Force and Moment Balances

A Substrate-Deposit System with a Uniform Misfit Strain

Fig.12.1 Schematic depiction of the generation of curvature in a flat bi-material plate, as a result

of the imposition of a uniform misfit strain, . The strain and stress distributions shown are for the case indicated, obtained using Eqns.(12.10) & (12.11)


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Relation between Curvature and Misfit Strain The forces P and P generate an unbalanced moment, given by

M =P h + H2

(12.1)

where h and H are deposit and substrate thicknesses respectively. Since the curvature, , (through-thickness strain gradient) is given by the ratio of moment, M, to beam stiffness,

=M (12.2)

P can be expressed as

P = 2h + H (12.3)

The beam stiffness is given by

= b E yc( )H

h

yc2 dyc = bEd h h2

3 h + 2

+ bEs H

H 23 + H +

2

(12.4)

where , the distance from the neutral axis (yc = 0) to the interface (y = 0) is given (see Appendix on p.64) by

= h2Ed H 2Es

2 hEd + HEs( ) (12.5)

The magnitude of P is found by expressing the misfit strain as the difference between the strains resulting from application of the P forces.

= s d =P

HbEs+

PhbEd

Pb = hEdHEshEd + HEs

(12.6)

Combination of this with Eqs.(12.3)-(12.5) gives a general expression for the curvature, , arising from imposition of a uniform misfit strain,

= 6EdEs h + H( )hH Ed2h4 + 4EdEsh3H + 6EdEsh2H 2 + 4EdEshH 3 + Es2H 4 (12.7)

Note that, for a given deposit/substrate thickness ratio, h/H, the curvature is inversely proportional to the substrate thickness, H. This scale effect is important in practice, since it means that relatively thin substrates are needed if curvatures are to be sufficiently large for accurate measurement. Predicted curvatures, obtained using this equation, are shown in Fig.12.2. Curvatures below about 0.1 m-1 (radius of curvature, R > 10 m) are difficult to measure accurately.


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Biaxial Stresses In practice, there are often in-plane stresses other than those in the x-direction. For an isotropic

in-plane stress state, there is effectively another stress equal to x in a direction at right angles to it (z-direction); this induces a Poisson strain in the x-direction. Assuming isotropic stiffness and negligible through-thickness stress (y = 0), the net strain in the x-direction can be written

xE = x y + z( ) = x 1( ) so that the relation between stress and strain in the x-direction can be expressed

x x

=E1( ) = E ' (12.8)

and this modified form of the Youngs modulus, E, (the biaxial modulus) is usually applicable in expressions referring to substrate/coating systems having an equal biaxial stress state. Stoneys Equation the Thin Coating Limit

A simplified form of Eq.(12.7) applies for coatings much thinner than the substrate (h


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The stress distributions in Fig.12.1, 12.3 and 12.4 were obtained using these equations. The adoption of curvature can effect substantial changes in stress levels and high through-thickness gradients can result. It may be seen from Eqns.(12.10) and (12.11) that (for a given value of h/H), since P is proportional to H and is inversely proportional to H, the stresses (at y=-H, 0 and h) do not depend on H, ie the stress distribution is independent of scale. However, the curvature is not. Substrates must be fairly thin if measurable curvatures are to be generated, although the maximum thickness could be as small as 50 m, or as large as 50 mm, depending on various factors.

Fig.12.2 Predicted curvature, as a function of the fall in temperature, for four different

substrate/deposit combinations

Curvature Measurement Techniques


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Accuracy of the Stoney Equation

Fig.12.3 Predicted dependence on thickness ratio of (a) curvature and (b) stress in deposit

(coating), obtained using Eqns.(12.7), (12.10) and (12.11), and the Stoney equation (Eqn.(12.9).) The Poisson ratios of substrate and deposit were both taken as 0.2

Possible Sources of a Misfit Strain


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Driving Force for Interfacial Debonding


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Appendix Location of the Neutral Axis

Fig.12.4 Location of the Neutral Axis of a Bi-Material Beam

The force balance

b y( )H

h

dy= 0 (12..1)

can be divided into contributions from the two constituents and expressed in terms of the strain

b Ed(y)0

h

dy+b Es(y)H

0

dy= 0 (12..2)

which can then be written in terms of the curvature (through-thickness strain gradient) and the distance from the neutral axis

b Ed y ( )0

h

dy+b Es y ( )H

0

dy= 0 (12..3)

Removing the width, b, and curvature, , which are constant, this gives

Edy22 y

0

h

+Esy22 y

H

0

= 0

Edh22 h

+ Es

H 22 H

= 0

Edh + EsH( ) = 12 Edh2 EsH 2( )

= h2Ed H 2Es

2 hEd + HEs( )

(12..4)


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Property Data (Room Temperature) Fibres

Fibre

Density (Mg m-3)

Axial Modulus E1 (GPa)

Transverse Modulus E2 (GPa)

Shear Modulus G12 (GPa)

Poisson Ratio 12

Axial Strength * (GPa)

Axial CTE

1 ( K-1)

Transverse CTE

2 ( K-1) Glass 2.45 76 76 31 0.22 3.5 5 5 Kevlar 1.47 154 4.2 2.9 0.35 2.8 -4 54

Carbon (HS) 1.75 224 14 14 0.20 2.1 -1 10 Carbon (HM) 1.94 385 6.3 7.7 0.20 1.7 -1 10

Diamond 3.52 1000 1000 415 0.20 4 3 3 Boron 2.64 420 420 170 0.20 4.2 5 5 SiC

(monofilament) 3.2 400 400 170 0.20 3.0 5 5

SiC (whisker)

3.2 550 350 170 0.17 8 4 4

Al2O3 ( continuous)

3.9 385 385 150 0.26 1.4 8 8

Al2O3 ( staple)

3.4 300 300 120 0.26 2.0 8 8

W 19.3 413 413 155 0.33 3.3 5 5

Matrices

Matrix Density

(Mg m-3) Young's Modulus E (GPa)

Shear Modulus G (GPa)

Poisson Ratio

Tensile Strength * (GPa)

Thermal Expansivity ( K-1)

Epoxy 1.25 3.5 1.27 0.38 0.04 58 Polyester 1.38 3.0 1.1 0.37 0.04 150

PEEK 1.30 4 1.4 0.37 0.07 45 Polycarborate 1.15 2.4 0.9 0.33 0.06 70 Polyurethane

Rubber 1.2 0.01 0.003 0.46 0.02 200

Aluminium 2.71 70 26 0.33 0.07 24 Magnesium 1.74 45 7.5 0.33 0.19 26

Titanium 4.51 115 44 0.33 0.24 10 Borosilicate

glass 2.23 64 28 0.21 0.09 3.2


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Question Sheet 1 [Can be attempted after lecture 8: property data on C16H65 can be used if necessary.] 1. Show that the Young's modulus of a composite lamina (having the elastic constants, referred

KFK?


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4. An angle-GCPV\C8D@E8K


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Question Sheet 2 [Can be attempted after lecture 12; property data on C16H65 can be used if necessary.] 1. A strut is in the form of a hollow cylinder with an outside diameter of 25 mm and a bore of

20 mm. It is manufactured from MMC material composed of 70 vol.% SiC monofilaments in a titanium alloy matrix, with the SiC fibres aligned approximately parallel to the axis of the strut. However, the limitations of the manufacturing process are such that fibre misalignments F=LGKF\8I


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3. (a) During formation of a coating on a substrate, its common for a misfit strain, , to be created, representing the difference between the (stress-free) in-plane dimensions of the two constituents. For example, this often arises during deposition and/or subsequent cooling. This misfit creates stresses and stains in the coating (and possibly in the substrate). Show that the relation between the stress and strain in the coating, in any (in-plane) direction, can be expressed =

E1( )

where E is the Youngs modulus and is the Poisson ratio. (b) The curvature, , arising from a misfit strain, , between a coating (deposit) of thickness

h and a substrate of thickness H is given by

= 6EdEs h + H( )hH Ed2h4 + 4EdEsh3H + 6EdEsh2H 2 + 4EdEshH 3 + Es2H 4

where Ed and Es are the Youngs moduli of deposit and substrate. Show that, in the limit of h


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(d) Show that, if the width of the sheet (length along the axis of the cylinder) is 0.5 m, then the beam stiffness ( = EI) of the sheet is 216.7 N m2 and the bending moment that would be needed in order to bring the sheet into contact with the cylindrical former would be 433 N m, assuming that the steel remained elastic. Calculate the required bending moment for the actual case, with the steel undergoing plastic deformation at a yield stress of 300 MPa (but neglecting any work hardening).

{40%} [Steel: Youngs modulus, E = 200 GPa; Rubber: Youngs modulus value more than 4 orders of magnitude lower]

{2011 Tripos} 5. John Harrison, the famous clock-maker credited with developing a time-keeping system

sufficiently reliable to establish longitude at sea, was reportedly the first to create a bi-metallic strip (for compensation of the effects of temperature change), which he did by casting a thin brass layer onto a thin steel sheet. Show that, if both layers have a thickness of 0.1 mm, and the strip is 100 mm long, then the temperature change required to generate a lateral deflection of 1 mm at its end is about 4.6 K, assuming that the system remains elastic.

Sketch the (approximate) through-thickness distributions of stress and strain within the above strip, after it had been heated by 100 K. Give your view as to whether such heating would be likely to cause plastic deformation within the strip, given that the yield stresses of both constituents are expected to be of the order of 100 MPa.

[The curvature, , of a bi-material strip comprising two constituents of equal thickness (h), arising from a misfit strain of between them, is given by

= 12

h E1E2+14 + E2E1

where E1 and E2 are the Youngs moduli of the constituents. The relationship between curvature, , end deflection, y, and length, x, of a bi-material strip may be expressed as

=2sin tan1 y x( )

x2 + y2( )

For steel: Youngs modulus, E = 200 GPa; thermal expansivity, = 13 10-6 K-1 For brass: Youngs modulus, E = 100 GPa; thermal expansivity, = 19 10-6 K-1]

{2012 Tripos}


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Examples Class I [Property data on C16H65 can be used if necessary.] 1. (a) The components of the compliance tensor of an epoxy-glass fibre composite lamina,

referred to the fibre axis direction and the transverse direction, can be written

S =S11 S12 0S12 S22 00 0 S66

=1 / E1 12 / E1 0

21 / E2 1 / E2 00 0 1 /G12

Using information in the Data Book, show that the interaction compliance giving the shear strain arising from a normal stress, when the lamina is loaded at an angle to the fibre axis, is

S16 = 2S11 2S12 S66( )c3s 2S22 2S12 S66( )cs3 in which c = cos and s = sin. (b) Using the following measured values of elastic constants of the composite

E1 = 50GPa, E2 = 5GPa,12 = 0.3,G12 = 10GPa

calculate the shear strain induced in the lamina when a normal tensile stress of 100 MPa is 8GGC@C


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3. On the page Failure of Laminates and the Tsai-Hill criterion, use the facility at the end to create a polyester-50%glass angle-GCPC8D@E8K


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Examples Class II [Property data on C16H65 can be used if necessary.] 1. (a) Outline the function of silane coupling agents, which are sometimes applied to glass

fibres prior to composite manufacture. (b) Show that the contribution to the fracture energy of a directionally-reinforced long fibre composite (loaded parallel to the fibre axis) from fibre pull-out is given by

Gcp =f s2 r i*

3

where f is the fibre volume fraction, s is the fibre pull-out aspect ratio (pull-out length / diameter), r is the fibre radius and i* is the fibre-matrix interfacial shear strength, which may be taken as the shear stress during frictional sliding. (c) Inspection of the fracture surface of an epoxy-50% glass fibre (8 m diameter) composite reveals an approximate distribution of fibre pull-out aspect ratio of: 25% with s~10, 50% with s~20 and 25% with s~30. Estimate the expected contribution from fibre pull-out to the fracture energy of this composite, assuming the interfacial shear strength to be 20 MPa. What characteristic of the fibre determines the average pull-out aspect ratio? {from 2007 Tripos}

2. (a) A small aircraft is being designed and a choice must be made between an aluminium alloy

and a composite for the fuselage material. The fuselage, which will approximate to a cylinder of diameter of 2 m, is expected to experience internal pressures up to 0.6 atm (0.06 MPa) above that of the surrounding atmosphere, axial bending moments of up to 500 kN m and torques of up to 600 kN m. The composite fuselage would be produced by filament-winding 8KV\KFK?


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4. A diamond coating of 1 m thickness is deposited by CVD onto a 1 mm thick titanium JL9JKI8K

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