Elastic-degrading analysis of pultruded composite structures

Elastic-degrading analysis of pultruded composite structures

Hakan Kilic, Rami Haj-Ali *

Department of Structural Engineering and Mechanics, School of Civil and Environmental Engineering, Georgia Institute of Technology,

Atlanta, GA 30332-0355, USA

Abstract

This study presents a combined micromechanical and structural modeling approach for the elastic-degrading (ED) analysis of

pultruded composite materials and structures. Pultruded composites with continuous filament mats and roving layers are considered

in this study. The overall effective response of the pultruded composite material is predicted using 3D micromechanical models for

the layers that span the thickness. These micromodels account for the nonlinear response in their matrix while the fiber constituent is

assumed to be linear elastic and transversely isotropic. The nonlinear material response of the matrix is achieved using an isotropic

ED model that better captures the effective nonlinear behavior of the pultruded material when compared with using a J2-plasticity

type model. A structural framework analysis is generated by integrating the micromechanical models within 3D or layered-shell

finite element (FE) models. The result is a general global–local modeling approach for the nonlinear ED analysis of pultruded

structures. Off-axis pultruded coupons were tested to examine the prediction capability of the micromodels for the nonlinear effective

material behavior. A structural verification is also carried out by using detailed FE models of off-axis pultruded plates with a central

hole, and four-point bending tests of pultruded coupons. The proposed micromodels combined with FE are able to effectively

predict the nonlinear material and structural responses.

� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Micro-mechanics; Finite element analysis; Nonlinear; Elastic-degrading analysis; Pultrusion

1. Introduction

Pultruded structural members, made from fiber-

reinforced polymers (FRP), are currently used in a

number of civil and infrastructural applications. These

composites are long and have relatively thick-walled

cross-sectional profiles similar to those found in stan-

dard steel members. Roving and continuous filamentmats (CFM) with E-glass fibers are often used as rein-

forcements in pultruded composites. Their high strength

and stiffness to weight ratios, resistance to corrosion,

high resistance to fatigue, light-weight, and easiness in

fabrication have been increasing the use of pultruded

composites in various civil engineering applications.

Pultruded structural components may exhibit significant

nonlinear material response under various multi-axialloading conditions. This nonlinear response is due to the

relatively small fiber volume fraction (FVF), soft poly-

meric matrix, large thickness that allow the development

of large shear stresses, and the existence of manufac-

turing defects in the form of microcracks and voids. The

nonlinear response is particularly important near loaded

fastener holes, edges, and cutouts, that tend to amplify

the nonlinearity due to stress concentrations. The re-

sponse at these stress concentration areas can have

crucial influence over the structural mode and magni-tude of failure. Thus, there exists a need, in the analysis

of pultruded structures, for a comprehensive nonlinear

material model that can predict the overall behavior

under multi-axial loading conditions.

Different nonlinear constitutive models have been ap-

plied to laminated composites under plane-stress condi-

tions, e.g. Hahn and Tsai [4], and Hashin et al. [10]. In

fact, multi-axial nonlinear models for pultruded com-posites have been limited. This can be attributed to the

fact that pultruded composite structural members are

designed to carry predominant axial loading and the

unidirectional roving is the main reinforcement that

is often used. As a result, constant axial properties

and linear material response are usually assumed and

combined with beam or plate theories to perform a

*Corresponding author. Tel.: +1-404-894-4716; fax: +1-404-894-

0211.

E-mail addresses: [email protected] (H. Kilic), rami.haj-

[email protected] (R. Haj-Ali).

0263-8223/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

PII: S0263-8223 (02 )00296-9

Composite Structures 60 (2003) 43–55

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mail to: [email protected]

structural analysis. Luciano and Barbero [11] proposed

models that can predict the overall initial stiffness from

micromechanical theories for each composite system

(layer) that forms the cross-section of a pultrudedmember. Classical lamination theory and mechanics of

laminated beams were used to predict the overall axial

stiffness. Haj-Ali and Kilic [6,8,9] proposed a 3D mi-

cromechanics-based framework for linear and nonlinear

analysis of pultruded composite materials and structures.

These material and structural frameworks integrate dif-

ferent nonlinear 3D micromechanical models for each

of the reinforcement layers that form the pultrudedcross-section. Each micromodel recognizes the lin-

ear or nonlinear response of the fiber and matrix con-

stituents. Haj-Ali and Kilic [8] also applied the nonlinear

homogeneous orthotropic (plane-stress) macromodels of

Hahn and Tsai [4], and Hashin et al. [10] to predict the

nonlinear multi-axial response of E-glass/vinylester pul-

truded composite material. Finite element analysis of

pultruded composite structures have been performedmostly using linear orthotropic and homogeneous ma-

terial properties. Bank and Yin [2] investigated the post-

buckling regime of pultruded I-beams, focusing on the

web–flange junction failure. A finite element analysis

with a node separation technique was performed to

simulate the local separation of the flange from the web,

following the local buckling of the flange. The analysis

was performed by using a nonlinear implicit finite ele-ment (FE) code. Their 3D model included eight-node

solid elements with orthotropic elastic material proper-

ties. Lui et al. [12] performed analyses to predict the be-

havior of pultruded fiber reinforced polymer (PFRP)

portal frames. A finite element model was developed

using composite shell elements, and considering semi-

rigid behavior of beam-to-column frame connections.

Linear elastic material properties were assumed. Brooksand Turvey [3] performed lateral buckling tests on pul-

truded I-section cantilevers and used FE models with

4-node shell elements. Smith et al. [14,15] performed an

experimental and numerical research on GFRP connec-

tions. Detailed shell FE analysis was used, and a con-

densation method was proposed for the effective

moment–rotation response of the connection while using

the experimentally obtained anisotropic material prop-erties for the section.

This study presents an isotropic elastic degrading

(ED) constitutive model for the matrix constituents used

within 3D micromodels for pultruded composites. The

new ED material model allows for modeling the com-

posite softening as a result of damage in the matrix. This

modeling approach is applied to pultruded composites-

with CFM and roving layers. In the first part of thispaper, the micromechanical and structural frameworks

for the layers are introduced. The constitutive modeling

framework is integrated into a general 2D and 3D FE

code. The calibration process of the micromodels and

the ED constitutive material model for the matrix are

discussed. The experimental part of this study includes

off-axis coupon tests that are used to examine the ability

of the micromodels to predict the nonlinear in-planemulti-axial response. Structural verification tests follow

and the micromodels are integrated with nonlinear 3D

FE to simulate the response of off-axis pultruded plates

with a circular hole and for pultruded beams under four-

point bending loading.

2. A combined micromechanical and structural framework

A combined micromechanical and structural frame-

work was proposed by Haj-Ali and Kilic [6,8,9] for the

general nonlinear analysis of laminated and pultruded

FRP composites. This framework is modified in thisstudy in order to include a nonlinear ED model for the

matrix constituents. The proposed framework is illus-

trated in Fig. 1 and applied for the ED analysis of

pultruded composites. Two separate 3D micromechan-

ical material models are used for the roving and CFM

layers. The 3D nonlinear micromechanical model for

the roving layers was developed by Haj-Ali and Kilic

[5,6,8,9]. It is based on a rectangular unit-cell (UC)model with four sub-cells. This model is used in this

study to idealize the roving layer as a periodic medium

with arrays of fibers having a square section. The roving

micromodel is derived by writing approximate traction

and displacement continuity relations in terms of aver-

age stresses and strains in the four sub-cells. A simplified

phenomenological micromodel was proposed by Haj-Ali

and Kilic [6,8,9] for the CFM medium using weightedresponses of a unidirectional layer in both axial and

transverse type modes. The CFM layer is a medium

where resin is reinforced with several mats of relatively

long and swirl filaments. The fibers are randomly dis-

tributed in the plane of the mat. The proposed CFM

micromodel generates the overall effective nonlinear 3D

response from the average responses of the two unidi-

rectional layers with axial and transverse fiber orienta-tions. The FVF in the CFM is used to define the relative

thicknesses of the two layers. The overall in-plane ef-

fective stress response is composed from the average

stress components of the two layers, while their in-plane

strains are equal. The overall out-of-plane response is

generated using traction continuity between the two

layers. The micromechanical relations for the roving and

CFM micromodels are reviewed in Appendix A.The upper level of the framework in Fig. 1 depicts an

FE structural model using 3D continuum or layered-

shell type elements. In the case where 3D continuum

elements are used at the structural level, a sublaminate

model exists at each Gaussian material point in order to

generate an effective nonlinear continuum response for a

periodic layered medium with alternating roving and

44 H. Kilic, R. Haj-Ali / Composite Structures 60 (2003) 43–55

CFM layers. The sublaminate model considers the me-

dium as a perfectly bonded layered system. This modelgenerates a through-thickness equivalent response for

each material point (Gauss point). The equivalence be-

tween the actual layered response and a homogeneous

continuum response is defined using the 3D lamination

theory. A representative volume element (RVE) in the

form of a two-layer repeated stacking sequence is used

to illustrate the sublaminate model. A hybrid vector is

prescribed over this RVE; it is composed of in-planestrain and out-of-plane stress components. The nonlin-

ear responses of the CFM and roving layers are taken

from their 3D micromodels. The micromechanical re-

lations for the sublaminate model can also be found in

Appendix A.

In the case where layered-shell elements are used to

model the pultruded structure, the sublaminate model is

not needed. Instead, each CFM or roving layer throughthe thickness of the cross-section, i.e. through the cross-

section of the element, is explicitly assigned one or more

Gaussian integration points. The response at each of

these integration points is generated from the appro-

priate roving or the CFM micromodel with a plane-

stress constraint imposed on its 3D formulation.

The proposed 3D micromechanical models along

with the matrix ED model are implemented numericallyalong with an efficient stress-update algorithm suited for

a general FE code. The trial incremental stresses and

strains typically violate the nonlinear constitutive rela-

tions in the sub-cells. The stress-update algorithms are

integrated at all nested levels of the proposed micro-

mechanical framework in order to satisfy the actual

stress–strain relations as well as the traction and com-patibility constraints for each nested micromodel. Each

UC and the sublaminate model have different compati-

bility and traction equations that need to be satisfied

through a separate correction scheme.

The material subroutine (UMAT) of the ABAQUS [1]

finite element code is used to implement the proposed

nonlinear material and structural frameworks for the

analysis of pultruded structures. This subroutine oper-ates at each integration point and its task is to update the

stresses and stiffness for a given strain increment. History

variables are used to define and store the states of de-

formation in all nested levels (layers, matrix, and fiber).

3. Isotropic elastic-degrading (ED) constitutive model

This section deals with the formulation of an isotro-

pic ED model for the matrix. This model employs the

Richard–Abbott (R–A) [13] formula to represent the

uniaxial nonlinear matrix response. The advantage of

this type of a model over the J2-plasticity is in its ability

to soften the matrix behavior while the overall response

is still monotonic due to the fiber constituents.

The stress is decomposed into elastic and inelasticparts as

rij ¼ reij � rI

ij ¼ ðseij � sIijÞ þ ðrem � rI

mÞdij ð1Þ

where sij is the deviatoric stress and rm is the meanstress.

Fig. 1. ED analysis framework.

H. Kilic, R. Haj-Ali / Composite Structures 60 (2003) 43–55 45

The linear and the inelastic parts of the total stress

tensors are expressed, respectively, as

reij ¼ 2G0eij þ K0ekkdij ¼ 2G0eij þ 3K0emdij ð2Þ

rIij ¼ 2ksG0eij þ kmK0ekkdij ¼ 2ksG0eij þ 3kmK0emdij ð3Þ06 ks6 1; 06 km6 1 ð4ÞTwo material damage variables are introduced in the

ED formulation. The first is ks that is associated with thedeviatoric strain energy. The second is km that is derivedin terms of mean strain em and reflects hydrostatic

energy.

The damage variables are taken as functions of strain

invariants

ks ¼ ksðJ 02Þ; km ¼ kmðemÞ; J 0

2 ¼1

2eijeij;

eij ¼ eij �1

3ekkdij ð5Þ

The R–A nonlinear formula is used to represent uniaxial

response in both modes. The uniaxial shear stress is

expressed in terms of engineering shear strain as

s ¼ G1c

1þ G1cs0

�� n� �ð1=nÞ þ Gpc ð6Þ

and the mean stress is written in terms of mean strain as

rm ¼ 3K1em

1þ 3K1emrom

�� m� �ð1=mÞ þ 3Kpem ð7Þ

where

G1 � G0 � Gp; n; s0K1 � K0 � Kp; m; ro

m

ð8Þ

are linear and nonlinear material parameters.

Specializing Eqs. (2), (3), and (5) and comparing with

Eqs. (6) and (7), the material damage variables are de-rived as

ks ¼ 1

�� Gp

G0

�1

0B@ � 1

1þ G1ces0

�� n� �ð1=nÞ

1CA ð9Þ

km ¼ 1

�� Kp

K0

�1

0B@ � 1

1þ 3K1emrom

�� m� �ð1=mÞ

1CA ð10Þ

where ce is the effective shear stress.

ce �ffiffiffiffiffiffiffiffiffiffiffiffi2eijeij

p; ce �

ffiffiffiffiffiffiffi4J 02

pð11Þ

The material parameters in Eqs. (12) and (13) are foundafter the calibration of the model by using the experi-

mental data. The model used in this study does not

consider damage due to hydrostatic pressure. Therefore,

km is equal to 0 and only three parameters in Eq. (12) areneeded for each mode

n;Gp

G0;G0s0

ð12Þ

m;Kp

K0;K0rom

ð13Þ

4. Calibration of the proposed micromodels

Numerical constitutive calculations are performed for

the fiber and matrix at the lower level of the proposed

modeling framework. Therefore, their in situ properties

are required for the two micromodels. The two matrix

constituents in the roving and CFM sub-cells are as-

sumed to have the same elastic and nonlinear parame-

ters. The fibers in the roving and the CFM are both

made from E-glass material. The matrix medium con-sists of vinylester resin mixed with small clay particles

and voids or microcracks. The proposed nonlinear iso-

tropic ED model is used to model the effective behavior

of the matrix medium. The FVFs in the unidirectional

roving and the CFM were determined in this study by a

series of burn-out tests. The average FVFs were found

to be 0.407 for the roving and 0.305 for the CFM. These

are average FVF values calculated by assuming a uni-form thickness of all CFM or roving layers. The CFM

and roving layers can have different thicknesses through

the cross-section, depending on the level of reinforce-

ment and number of mats used. The relative average

thickness of the CFM layers is (0.328/0.5), while the

relative thickness for the roving is (0.172/0.5). The

combined average FVF in the pultruded material, i.e. in

both the roving and CFM volumes, is 0.34.The micromodels are calibrated in the elastic range

from known or assumed in-situ properties of the matrix

and the fiber, relative thicknesses of the roving and

CFM, and the FVFs in these layers. Once the FVFs and

the relative thickness of the roving and CFM are known,

the linear elastic calibration is initiated for the properties

of the fiber and matrix constituents. Coupon tests are

used to calibrate the proposed micromodels in the linearelastic range. Fibers are considered as linear, elastic, and

transversely isotropic materials and the E-glass fiber

elastic properties are taken from the literature. During

Table 1

Elastic properties and nonlinear Richard–Abbott parameters. Fiber

volume fractions: roving layers¼ 0.407, CFM layers¼ 0.305. Matrix isvinylester with clay additives and calibrated both from tension (þ) andcompression ()) tests

Tension (þ) Compression ())

E (1000 ksi) v Gp=G0 n G0=s0

E-glass fiber 10.5 0.25

Matrix

(vinylester

resin ðþÞ

0.730 0.30 )0.2 2.0 72.0

þ clay addi-tives) ())

0.730 0.30 )0.5 2.0 18.0


the calibration, an attempt is made to match some of the

overall measured effective properties of the pultruded

material by changing the matrix properties. Table 1 in-cludes the calibrated elastic properties of the fiber and

matrix. Table 2 lists the predicted 3D effective elastic

properties of the pultruded material and the corre-

sponding experimental results for the in-plane proper-

ties. The predicted stiffness values all cannot be verified

mainly due to complicated testing process for out-of-

plane properties.

The V-notch shear test is used to calibrate the non-linear ED model for the matrix, such that the overall

predicted behavior matches the experimental stress–

strain curve. Fig. 2 illustrates the pronounced nonlinear

axial-shear stress–strain response of the pultruded ma-

terial. The R–A stress–strain formula is calibrated for

the matrix by varying its parameters until the overall

effective response of the pultruded material is matched

with the V-notch test results. The axial-shear nonlinearproperties of the matrix model are listed in Table 1. The

solid line in Fig. 2 shows the overall axial-shear response

using these calibration parameters. The Ramberg–

Osgood (R–O) uniaxial curve is also used, along with

the J2-plasticity to model the nonlinear behavior of the

matrix. The dashed line in Fig. 2 shows the overall axial

shear response generated from the micromodels with J2-

plasticity for the matrix. The ED model better captures

the overall axial shear response for strains larger than

1.5%.

Haj-Ali and Kilic [7] performed coupon testing todetermine both the linear and nonlinear behavior of a

pultruded E-glass/vinylester composite material system.

They showed that the material system has a lower initial

elastic modulus in tension than the corresponding

compressive modulus. Also, the nonlinear response of

the pultruded material is softer in tension than the be-

havior in compression. This is attributed to the ‘‘active’’

voids, microcracks in the tension mode of loading. TheED model is calibrated from the axial-shear parameters

for compression mode of loading. Therefore, the matrix

R–A parameters are re-calibrated to account for the

additional softening in the tension mode by stress–strain

curves for transverse coupons under tension (Fig. 3).

The elastic properties of the matrix constituent are not

changed. The calibrated R–A parameters in tension are

listed in Table 1. The ED and the J2-plasticity modelsare both calibrated for this case. In the latter one, the

calibrated transverse tension response curve has a con-

stant stiffness after 0.8% strain, as seen in Fig. 3. This is

because the matrix has completely softened and the

overall stiffness is predominantly influenced from the

linear fiber response. Therefore, the J2-plasticity model

does not show more softening beyond this point; It is

sufficient to describe the nonlinear response only in

Table 2

Experimental and predicted micromechanical effective elastic properties of E-glass/vinylester pultruded composite with CFM and roving layers. Units

are in (1000 ksi)

Tension tests (+) Compression tests ())

E1 E2 E3 (1000 ksi) G12 G13 G23 v12 v13 v23

Experimental ()) 2.800 1.838 0.645 0.260

Experimental ðþÞ 2.633 1.486 0.330

Micromodel 2.825 1.783 1.287 0.662 0.434 0.411 0.275 0.295 0.316

Fig. 2. Axial-shear stress–strain behavior of E-glass/vinylester composite used to calibrate the matrix nonlinear properties.


compression, as reported in Haj-Ali and Kilic [6,8,9].

Because the nonlinear response in tension is more pro-

nounced, the ED model provides a better calibration

that accounts for this softening response in tension.The calibrated R–A and R–O shear stress–strain

curves for the matrix are shown in Fig. 4. The solid lines

represent the R–A curves and the dashed ones are for

the R–O. The R–A curves allow more softening for the

matrix behavior, which can capture the additional

nonlinear monotonic response of the pultruded com-

posite material under tension. The fact that the matrix

R–A curves have negative slope does not influence theoverall positive tangent stiffness of the composite due to

the undamaged fiber medium. Loading redistribution is

achieved in the constitutive frameworks by using secant

stiffness with unloading paths to the origin without ac-

cumulation of plastic strains.

5. Off-axis stress–strain response in tension

The nonlinear prediction capability of the proposed

material modeling framework is first examined for uni-form states of deformations. Off-axis coupon tests sub-

ject to axial tension are used to validate the prediction of

the micromodels. These coupons were cut from 1/4 and

1/2 in. thick plates with different off-axis angles for the

roving. Six orientations were used (0�, 15�, 30�, 45�, 60�and 90�). Both of the plates had the same FVF. The testswere repeated 3–5 times and were performed in dis-

placement control mode in order to reach the ultimateload and complete breakage of the coupon. The off-axis

coupon geometry and the testing details, such as loading

rate and geometry size, were described in Haj-Ali and

Kilic [7]. Strain gages were used to monitor the axial

strains at the center of these coupons. The predicted

Fig. 4. Calibrated matrix shear responses from V-notch and transverse tension tests.

Fig. 3. Transverse tensile response of pultruded E-glass/vinylester composite coupons used to calibrate the matrix nonlinear properties.


tensile responses, from the ED and J2-plasticity models

are compared with the experimental results. Figs. 5 and

6 show the predicted stress–strain curves with the ex-

perimental results for two representative off-axis orien-

tations. The EDmodel predictions are in good agreement

with the experimental results for both cases. The ED

model better captures the tensile nonlinear response ofthe material than the J2-plasticity model.

6. Nonlinear analysis of pultruded plates with a circular

hole

The proposed nonlinear micromodels are used to

predict the nonlinear response under multi-axial stress–

strain states. To this end, a series of tests were per-

formed with off-axis pultruded plates with a circular

hole and subject to uniaxial tension. The geometry of

the plates is shown in Fig. 7. Four strain gages were

attached on the plates. The first strain gage is placed

remotely such that it is not affected by the hole nor by

the end-clamping of the grips. The other three strain

gages are placed to obtain different strain data at loca-tions near the stress concentration region. The off-axis

angles used in these plates are: 0�, 15�, 30�, 45�, 60�, and90�. Tests were run under displacement control with aloading rate of 0.002 in./min.

FE models with the previously calibrated micro-

models are used to simulate the tests with the off-axis

plates. The top-left quadrant of the plate is modeled

in the FE analysis, as shown in Fig. 7. A quartermodel with height of 3.5 in. is used for the model which

Fig. 6. Predicted off-axis tensile stress–strain responses using ED and J2-plasticity models––II.

Fig. 5. Predicted off-axis tensile stress–strain responses using ED and J2-plasticity models––I.


consists of 664 3D 20-node brick elements with 3961

nodes. One element is used through the thickness due to

the plane-stress conditions.

Figs. 8–13 show the FE predictions compared with

the experimental results for all off-axis angles. Predicted

FE results are compared to the experimental data and

plotted as four curves in the form of remote stress versusthe strains measured from the four strain gages. The last

point in each experimental curve is the ultimate state of

loading. Beyond this point, the response is not mono-

tonic and a softening process starts with a brittle type

dynamic failure. The fourth strain gage (G4) is the

closest to the edge of the hole. Therefore, a consistent

softer response is shown for the G4 curves in all off-axis

tests. As expected, the level of nonlinear response variesfor the different plates increasing with the increased off-

axis angle. Overall, good predictions are shown by the

FE models compared with the experimental results. This

confirms the ability of the proposed micromodels to

predict a nonlinear multi-axial state of deformation with

stress concentration due to the circular holes.

7. Nonlinear analysis of pultruded beams under four-point

bending

Four-point bending tests were performed in order to

investigate the prediction capability of the proposed

micromodels for small pultruded beams under bending.

The dimensions of the specimen are shown in Fig. 14.Six specimens were cut from 0.5 in. thick E-glass/vinyl-

ester pultruded composite plate. Three specimens have

the roving fibers aligned along the x-direction, while theother three have their roving in the y-direction. A dis-placement control mode is used to apply the loading.

The FE model consists of a quarter of the plate due to

the symmetry of geometry and loading conditions. The

FE model includes 315 20-node brick elements with re-duced integration. One element is used in the y-direction

Fig. 8. Predicted remote average stress versus strains from four gages mounted on a pultruded plate under tension––I.

Fig. 7. Geometry of a pultruded plate with a circular hole and its FE

model.


Fig. 10. Predicted remote average stress versus strains from four gages mounted on a pultruded plate under tension––III.

Fig. 9. Predicted remote average stress versus strains from four gages mounted on a pultruded plate under tension––II.

Fig. 11. Predicted remote average stress versus strains from four gages mounted on a pultruded plate under tension––IV.


Fig. 13. Predicted remote average stress versus strains from four gages mounted on a pultruded plate under tension––VI.

Fig. 12. Predicted remote average stress versus strains from four gages mounted on a pultruded plate under tension––V.

Fig. 14. Geometry of a pultruded beam under four-point bending loading and its FE model.


assuming a state of plane deformation, and neglecting

the edge effects on out-of-plane (XZ) motion. The com-parison of experimental and the FE results is shown in

Fig. 15. The test data show the mid-span displacement

versus the total applied load. The nonlinear load–

deflection predictions from the micromechanical models

are in good agreement with the test results for both

orientations.

8. Conclusions

A combined micromechanical and structural model-

ing approach for the elastic-degrading (ED) analysis of

pultruded composite materials and structures is pre-

sented. The overall effective response of the pultruded

composite material is generated from 3D microme-

chanical models used for the different layers that span

the thickness. These micromodels can recognize thenonlinear response in their matrix and fiber constituents.

The fiber is assumed to be linear elastic and transversely

isotropic. The nonlinear material response of the matrix

is achieved using an isotropic ED model. The ability of

the micromodels to predict the elastic stiffness and the

nonlinear multi-axial stress–strain response is examined

with different E-glass/vinylester off-axis coupon tests.

The proposed micromodels are able to predict the multi-axial nonlinear behavior of these coupons. The ED

model for the matrix can capture the additional non-

linear behavior of the pultruded material in monotonic

tension when compared with using a J2-plasticity model.

Verification tests were also performed for off-axis plates

in tension. FE models are used to simulate these tests.

Good predictions are demonstrated by the FE models

with the ED micromodels. Finally, pultruded beams are

tested under four-point bending and used to verify thegeneral prediction capability of the proposed ED mi-

cromodels. Good results are obtained by the FE struc-

tural component models.

Acknowledgement

This work was supported by NSF under grant num-ber 9876080.

Appendix A

A.1. Sublaminate model: equivalent continuum for roving

and CFM layers

The following derivation is for a sublaminate model

with two layers, the roving and CFM. The stress and

strain vectors for each layer are transformed to material

coordinates, and are partitioned into in-plane and out-

of-plane components

�rri ¼ f r11 r22 s12 g; �rro ¼ f r33 s13 s23 g; ðA:1Þ�eei ¼ f e11 e22 c12 g; �eeo ¼ f e33 c13 c23 g ðA:2Þwhere the overbars indicate homogenized sublaminate

quantities.

The in-plane strains and out-of-plane stresses are

assumed to be the same throughout the sublaminate:

�rro ¼ rðCÞo ¼ rðRÞ

o ðA:3Þ

�ee ¼ eðCÞi ¼ eðRÞi ðA:4Þwhere ðCÞ and ðRÞ denote CFM and roving layers, re-

spectively.

Fig. 15. Predicted FE for load–deflection results of four-point bending.


The complementary stresses and strains may differ in

the roving and CFM layers. The homogenized in-plane

stresses and out-of-plane strains are taken as weighted

averages, using the relative CFM and roving thickness,tC and tR, respectively, as

�rri ¼1

ðtC þ tRÞðtCrðCÞ

i þ tRrðRÞi Þ ðA:5Þ

�eeo ¼1

ðtC þ tRÞðtCeðCÞo þ tReðRÞo Þ ðA:6Þ

Eqs. (A.3)–(A.6) completely characterize the sublami-nate model.

A.2. Micromechanical model for the roving layer

The UC is divided into four sub-cells due to sym-

metry. The traction and displacement continuity rela-

tions between the sub-cells are approximated in terms of

the appropriate components of the average stress ðrðaÞ,

a ¼ 1; 2; 3; 4Þ and strain ðeðaÞ, a ¼ 1; 2; 3; 4Þ vectors in thesub-cells. The long fibers are aligned in the x1 direction.The other cross-section directions are referred to as the

transverse directions. The overall average stress and

strain vectors for the UC are denoted by ð�rrÞ and ð�eeÞ,respectively.

The notations for the stress and strain vectors, de-

fined in this section, are

frðaÞi gT ¼ f r11; r22; r33; s12; s13; s23 gðaÞ

feðaÞi gT ¼ f e11; e22; e33; c12; c13; c23 gðaÞ

i ¼ 1; . . . ; 6 a ¼ 1; . . . ; 4 ðA:7Þ

where ðaÞ denotes the sub-cell number in the UC and (i)denotes the stress or strain component or the mode-i.The total volume of the UC is taken to be equal to one.

The volumes of the four sub-cells are

v1 ¼ hb; v2 ¼ ð1� hÞb; v3 ¼ hð1� bÞ;v4 ¼ ð1� hÞð1� bÞ ðA:8Þ

The axial strains are the same in all the sub-cells.

Therefore, the longitudinal relations (mode-1) are

eð1Þ1 ¼ eð2Þ1 ¼ eð3Þ1 ¼ eð4Þ1 ¼ �eeðRÞ1

v1rð1Þ1 þ v2r

ð2Þ1 þ v3r

ð3Þ1 þ v4r

ð4Þ1 ¼ �rrðRÞ

1

ðA:9Þ

Considering the interfaces with normals in the x2 di-rection, the corresponding strain compatibility condi-

tions for the modes 2 and 4 follow from separatelyconsidering sub-cells (1) and (2), and sub-cells (3) and

(4), respectively. These relations are used to express

traction and compatibility relations for the transverse

stress and strain components (22), mode-2, and for axial

shear (12), mode-4. For the case of direct transverse

mode-2, i.e. components (22), the continuity relations

between the sub-cells are:

rð1Þ2 ¼ rð2Þ

2

rð3Þ2 ¼ rð4Þ

2

v1v1 þ v2

eð1Þ2 þ v2v1 þ v2

eð2Þ2 ¼ �eeðRÞ2

v3v3 þ v4



ðA:10Þ

For the in-plane shear (mode-4), the relations are

rð1Þ4 ¼ rð2Þ

4

rð3Þ4 ¼ rð4Þ

4

v1v1 þ v2



v3v3 þ v4



ðA:11Þ

Considering the interfaces with normals in the x3 di-rection, the corresponding strain compatibility condi-

tions for the modes 3 and 5 follow from separately

considering sub-cells (1) and (3), and sub-cells (2) and

(4), respectively. These relations are expressed for thedirect stress component 33 (mode-3), as

rð1Þ3 ¼ rð3Þ

3

rð2Þ3 ¼ rð4Þ

3

v1v1 þ v3



v2v2 þ v4



ðA:12Þ

For the out-of-plane shear component 13 (mode-5),

the relations are

rð1Þ5 ¼ rð3Þ

5

rð2Þ5 ¼ rð4Þ

5

v1v1 þ v3



v2v2 þ v4



ðA:13Þ

Finally, in the transverse shear mode, mode-6, the

traction continuity at all the interfaces between the sub-

cells must be satisfied. Since this relation is satisfied

using the sub-cells� average stress, the traction continuityand compatibility equations for the transverse shear are

rð1Þ6 ¼ rð2Þ

6 ¼ rð3Þ6 ¼ rð4Þ

6

v1eð1Þ6 þ v2e

ð2Þ6 þ v3e

ð3Þ6 þ v4e

ð4Þ6 ¼ �eeðRÞ6

ðA:14Þ

Eqs. (A.8)–(A.14) completely define the micromechani-

cal relations between the stresses and the strains in the

sub-cells and the overall average stresses and strains ofthe roving.

A.3. Micromechanical model for the CFM layer

The CFM UC model is a collection of four sub-cells.

The matrix-mode layer (part-A) is composed of sub-cells

(1) and (2), while the fiber-mode layer (part-B) is com-

posed of sub-cells (3) and (4). The relative thickness of


each layer is defined using the FVF. The formulation of

the CFM can be presented in terms of average stresses

and strains in parts A and B which can be considered in

the CFM formulation as two independent layers. Thefiber volume fractions within the two parts are the same

and provide the relations:

V1V1 þ V2

¼ h ¼ vfCV4

V3 þ V4¼ n ¼ vfC ðA:15Þ

The out-of-plane traction continuity and interface dis-

placement continuity, between parts A and B, are ex-

pressed by

�rrðCÞo ¼ rðAÞ

o ¼ rðBÞo

�eeðCÞi ¼ eðAÞi ¼ eðBÞi

ðA:16Þ

where a CFM quantity is denoted by a (C) superscript

and an overbar is used to denote an averaged variable.The homogenized in-plane stresses and out-of-plane

strains are taken as weighted averages, using the FVF in

the CFM, as

�rrðCÞi ¼ 1

VðVArðAÞ

i þ VBrðBÞi Þ

�eeðCÞo ¼ 1

VðVAeðAÞo þ VBeðBÞo Þ

ðA:17Þ

Within the matrix-mode layer (part-A), the followingrelations for all stress and strain components should be

satisfied

�rrðAÞ ¼ rð1Þ ¼ rð2Þ

�eeðAÞ ¼ 1

VAV1eð1Þ�

þ V2eð2Þ� ðA:18Þ

The corresponding equations for the fiber-mode layer(part-B) are

�rrðBÞo ¼ rð3Þ

o ¼ rð4Þo

�eeðBÞi ¼ eð3Þi ¼ eð4Þi

�rrðBÞi ¼ 1

VBV3r

ð3Þi

�þ V4r

ð4Þi

�

�eeðBÞo ¼ 1

VBV3eð3Þo

�þ V4eð4Þo

�ðA:19Þ

Eqs. (A.15)–(A.19) define the 3D micromechanical

relations between the average stresses and strains in the

fiber and matrix sub-cells of the CFM layer.

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Elastic-degrading analysis of pultruded composite structures