progressive damage and nonlinear analysis of pultruded composite structures

Progressive damage and nonlinear 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

Received 5 May 2002; accepted 16 September 2002

Abstract

This study combines a simple damage modeling approach with micromechanical models for the progressive damage analysis of pultruded

composite materials and structures. Two micromodels are used to generate the nonlinear effective response of a pultruded composite system

made up from two alternating layers reinforced with roving and continuous filaments mat (CFM). The layers have E-glass fiber and vinylester

matrix constituents. The proposed constitutive and damage framework is integrated within a finite element (FE) code for a general nonlinear

analysis of pultruded composite structures using layered shell or plate elements. The micromechanical models are implemented at the

through-thickness Gaussian integration points of the pultruded cross-section. A layer-wise damage analysis approach is proposed. The Tsai–

Wu failure criterion is calibrated separately for the CFM and roving layers using ultimate stress values from off-axis pultruded coupons under

uniaxial loading. Once a failure is detected in one of the layers, the micromodel of that layer is no longer used. Instead, an elastic degrading

material model is activated for the failed layer to simulate the post-ultimate response. Damage variables for in-plane modes of failure are

considered in the effective anisotropic strain energy density of the layer. The degraded secant stiffness is used in the FE analysis. Examples of

progressive damage analysis are carried out for notched plates under compression and tension, and a single-bolted connection under tension.

Good agreement is shown when comparing the experimental results and the FE models that incorporate the combined micromechanical and

damage models.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: C. Micromechanics; C. Finite element analysis; C. Damage mechanics; E. Pultrusion

1. Introduction

The process of damage development and failure in

composite materials is very complicated. The effective

properties of the composite usually depend on the average

stresses or strains in the phases. However, an analytical

micromechanical damage analysis should properly take into

account the detailed microstructure, the spatial deformation

fields, existing defects, criteria for microfailure and its

evolution, and the way different defects and modes of failure

interact as loading progresses. Modeling these microme-

chanical aspects is difficult, if not impossible.

Theories for failure prediction were early developed in

anisotropic natural materials such as wood. Comprehensive

reviews of failure theories and mechanisms in composites

are also available [15,27,29,30,34]. Maximum strain criteria

were utilized by Petit and Waddoups [25] in their nonlinear

analyses of laminated composites. Tsai [35] used the yield

criterion for orthotropic and ideally plastic material, derived

by Hill [17], as a failure criterion for a unidirectional

lamina. Tsai and Wu [36] proposed a quadratic tensor

polynomial as a failure criterion. Wu [37] used experimental

results to examine the effect of the interaction coefficient

and proposed different approaches to determine it. Hashin

[16] proposed a three-dimensional failure theory for

transversely isotropic materials, in which the appropriate

stress invariants were used to construct separate quadratic

functions for fiber and matrix failure modes. The failure

theory of Hashin uses both information about the material

symmetry and physical considerations that arise from the

different characteristics of the failure mechanisms involved.

Christensen [7] developed a three-dimensional theory for

laminates using restrictions on the effective properties of the

lamina. These restrictions reduce the five independent

1359-8368/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

PII: S1 35 9 -8 36 8 (0 2) 00 1 03 -8

Composites: Part B 34 (2003) 235–250

www.elsevier.com/locate/compositesb

* Corresponding author. Tel.: þ1-404-894-4716; fax: þ1-404-894-0211.

E-mail addresses: [email protected] (R. Haj-Ali), hkilic@

sama.ce.gatech.edu (H. Kilic).

http://www.elsevier.com/locate/compositesb

material properties for transversely isotropic material to

three properties. As a result, the out-of-plane components of

the stiffness matrix of the lamina do not depend on the fiber

orientation when the stiffness matrix is transformed around

the through-thickness axis. The simplified three-dimen-

sional constitutive relation is used in combination with axial

and transverse failure criteria expressed in terms of the

strain invariants to form three-dimensional failure

functions.

The response of polymer fiber composites in the axial

mode can be treated as brittle elastic. On the other hand,

transverse response is dominated by matrix behavior;

damage develops and accumulates prior to ultimate failure.

This type of damage can be attributed to microcracking,

interface or interphase failure, voids, and other effects that

generate ductile nonlinear behavior of the matrix phase.

This provides a rational for progressive damage analysis of

laminated structures. In the progressive damage modeling,

the evolution of the state of damage in the material and its

associated effective stress–strain response, as a result of

continued applied loading, are determined. Failure criteria

for single laminae are often used in the analysis of

composite structures. After first-ply failure is detected at a

point within the structure, the analysis is then continued

using simplistic ply-discount methods. The principal theme

of these methods can be summarized as follows: if a failure

is detected in the fiber direction, the lamina axial stiffness is

reduced to zero, while if a failure is detected in the

transverse direction, the lamina transverse stiffness and the

axial shear stiffness are reduced to zero [28]. In addition,

different approaches have been proposed regarding stress

unloading as a result of fiber or matrix failure. Hahn and

Tsai [8] studied the behavior of cross-ply laminates under

different loading and unloading stress paths, after initial

failure. Several studies of progressive failure analysis using

layered plate and shell models have been reported [5,6,23].

Other studies involving three-dimensional finite element

(FE) analysis have been limited to specialized cases and

specific loading conditions [21,26,32]. Haj-Ali and Peck-

nold [9–11] studied progressive damage in a unidirectional

lamina. The damage formulation is done at the microlevel

by adding interphase/interface type sub-cells between the

fiber and matrix in the unit-cell (UC). Two main damage

modes are identified: the fiber or axial mode, and the matrix

or transverse mode. These two modes are further divided

into tensile and compressive modes. The interface model

satisfies the traction continuity between the fiber and the

matrix in the transverse direction. It functions as a ‘fuse’ in

the model and it accounts for the direct transverse damage

mode in the post-failure regime. The properties of the

interface are chosen to be different in tension and

compression, which allows modeling the different response

of the composite in transverse tension and compression.

Axial damage mode is modeled through the fiber material. A

microbuckling criterion is used in axial compression failure,

and a constant stress failure criterion is used in tension.

Once failure occurs, a strain softening scheme is used to

unload the axial stress in the fiber.

A number of studies has been conducted on the failure

and progressive damage analysis of pultruded composite

structures. These studies include experimental and analyti-

cal work to investigate and model the damage accumulation

in pultruded structural components. Barbero and Trovillion

[4] carried out several tests to observe the post-critical

behavior of pultruded FRP composite columns. The focus

was on the stiffening of the system in the post-critical range.

The Southwell’s method was modified by Tomblin and

Barbero [33] to account for the imperfections and local

buckling. The buckling load was obtained by using a

quadratic expansion of the load to approximate the post-

critical path. The importance of the modified Southwell’s

method is its ability to account for material imperfections.

The damage accumulation was investigated by re-loading

the tested columns, during which the mode shape was

observed as identical to the mode shape of the first test,

however, the deflections were larger at small loads. This

indicates that the material was softer upon re-loading and

that permanent damage occurred previously. Experimental

and analytical investigations on pultruded FRP box beams

under three-point bending were carried out by Mottram

[22]. The objective was to evaluate the simplified design

analysis proposed earlier by Johnson [19]. Failure analyses

were performed using thin-walled beam theory. Five

different failure modes were evaluated. These were

compression face buckling, shear buckling inside the wall,

material failures in compression, tension, and shear. The

material was modeled using linear elastic material proper-

ties. The design and analysis equations failed when the

spans of the beams were short. The analysis gave good

predictions after modifying the original formula to account

for thick walls. Bank and Yin [3] investigated the post-

buckling regime of pultruded I-beams, focusing on the web-

flange junction failure. FE 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 FE code. Their 3D model included eight-node solid

elements with orthotropic elastic material properties.

Analytical results showed that the transverse tensile stress

was the dominating failure mode at the junction. Palmer

et al. [24] simulated and tested for the progressive tearing

failure of pultruded composite box beams. An out-of-plane

tearing damage mode was observed in beams subjected to

three point bending tests. The quasi-static beam tests were

modeled by using an explicit dynamic FE program which

gave out consistent results with the test data.

This study integrates a simple damage modeling

approach with micromechanical models for the progressive

damage analysis of pultruded composite materials and

structures. Two micromodels are applied to a pultruded

composite material system made up from two alternating

layers reinforced with roving and continuous filaments mat

H. Kilic, R. Haj-Ali / Composites: Part B 34 (2003) 235–250236

(CFM). The proposed constitutive and damage framework

is integrated within a FE code for general nonlinear analysis

of pultruded composite structures using layered shell or

plate elements. The first part of this paper describes the

pultruded composite material system used in this study. The

micromechanical models for the roving and CFM layers are

described in Section 3. A layer-wise damage analysis

approach is then proposed for the progressive damage

analysis. Experimental and structural analyses are described

in the second part of the study. Examples of progressive

damage analysis are carried out for notched plates under

compression and tension, and for a single-bolted connection

under tension.

2. Pultruded composite material system

The studied pultruded composite material system con-

sists of vinylester resin reinforced with two alternating

layers of unidirectional E-glass roving and CFM. The fibers

in the CFM are relatively long, swirl, and randomly oriented

in the plane. The roving layers consist of unidirectional fiber

bundles that run in the pultrusion direction along the entire

span of the member. The CFM and roving layers can have

different thicknesses through the cross-section, depending

on the level of reinforcement and number of mats used. The

average fiber volume fractions (FVFs) were determined by

Haj-Ali and Kilic [13], using burn-out tests, as 0.407 for

the roving and 0.305 for the CFM. These are average FVF

values within each layer. These are calculated by assuming

a uniform thickness of all CFM or roving layers. The

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

the roving and CFM volumes, is 0.34. The cross-section

thickness of the tested plate was 0.5 in. with 0.172 in. roving

layers and 0.328 in. CFM layers. There are an apparent

number of void systems spread inside the pultruded section.

Haj-Ali and Kilic [13] showed that the E-glass/vinylester

pultruded material system has lower initial off-axis moduli

in tension than the corresponding compressive moduli. In

addition, the nonlinear response of the pultruded material is

softer in tension than that in compression.

3. Micromechanical framework for pultruded

composites

A combined micromechanical and structural framework

was proposed by Haj-Ali et al. [12,14] for the general

nonlinear analysis of pultruded FRP composites, as shown

in Fig. 1. The material subroutine (UMAT) of the FE code

ABAQUS [1] is used for implementing the framework. This

FE code is extensively used throughout this study for the

failure and progressive damage analysis of pultruded

composite structures. Nonlinear 3D micromechanical

models for the different layers, roving and CFM, of the

pultruded section are used to generate the effective response

at various locations through the thickness of the cross-

section.

Fig. 1. A framework for progressive damage and nonlinear analysis of pultruded composite structures.

H. Kilic, R. Haj-Ali / Composites: Part B 34 (2003) 235–250 237

The proposed structural modeling approach depicted in

Fig. 1 includes layered shell or plate elements that are used

in the pultruded structural model. Each CFM or roving layer

through the thickness of the cross-section, i.e. through the

cross-section of a given element, is explicitly assigned an

odd number of Gaussian integration points. The effective

material response at each of these integration points is

generated using the appropriate roving or CFM micromodel.

Since these are general 3D micromodels, an added

constraint is used to enforce a plane stress state in each

layer.

As mentioned, two independent 3D micromechanical

material models are used for the roving and CFM layers.

The 3D nonlinear micromechanical model for the roving

layers was developed by Haj-Ali et al. [10,12,14]. It is

based on a rectangular UC model with four sub-cells. This

model can be used for a unidirectional fiber reinforced

material. The roving layer is idealized as a periodic

unidirectional 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 average stresses and strains in the

sub-cells. These micromechanical relations are similar to

those generated from the general method of cells that was

proposed by Aboudi [2]. The proposed micromodel

employs mechanics-of-materials type considerations with-

out the need for the detailed displacement-based poly-

nomial expansions.

A simplified phenomenological model was proposed for

the CFM medium by Haj-Ali et al. [12,14]. This model is

constructed using weighted responses 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 distributed in the plane of the mat. The proposed

CFM micromodel generates the effective nonlinear 3D

response from average responses of the two unidirectional

layers. The overall in-plane stress response is generated by

averaging the in-plane responses of the two layers with

equal in-plane strains. The relative thickness in each of the

two layers is determined based on the FVF in the CFM

medium. The average out-of-plane response is generated

using traction continuity between the two layers. The CFM

effective medium represents an in-plane isotropic model.

The micromechanical relations for the roving and CFM

micromodels are reviewed in Appendix A. The in situ

properties of the proposed models are calibrated by using

simple coupon tests. The fiber and matrix constituents are

calibrated in the elastic range from known or assumed

properties of the matrix, fiber, FVFs of the roving and CFM

layers, and their relative thicknesses. The calibrated fiber

and matrix properties are shown in Table 1. The fiber is

assumed to be linear elastic and transversely isotropic. The

nonlinear material response of the matrix is achieved using

the J2 deformation theory along with the Ramberg–Osgood

(R–O) uniaxial stress–strain representation. It is important

to note that the matrix constituent is defined herein as a

collective medium that surrounds the E-glass reinforcement.

This medium may include additives in the vinylester

polymeric matrix, such as clay particles or glass micro-

spheres. Therefore, in the proposed micromodels, the

mechanical in situ properties attributed to the matrix are

overall effective properties of this isotropic medium. V-

notch shear tests are used to calibrate the nonlinear matrix

behavior. The tested off-axis coupons in tension usually

show lower stiffness and more nonlinear response than the

compression coupons, Haj-Ali and Kilic [13]. Therefore, the

matrix R–O parameters are re-calibrated to account for

the additional softening in tension by using the transverse

tension coupon test results.

4. Failure criteria for CFM and roving layers

A layer-wise failure and damage analysis is proposed.

Failure criteria and stiffness degradation are independently

calculated for the CFM and roving layers. The Tsai–Wu

failure criterion is used and calibrated for the two layers.

The criterion represents a general quadratic failure surface

in the stress space. It can be expressed as

Fisi þ Fijsisj ¼ 1 i; j ¼ 1;…; 6 ð1Þ

where Fi and Fij are constant coefficients that depend on

ultimate stress values. Only the in-plane modes of failure are

considered in this study. The Tsai–Wu failure for a plane-

stress state is given by

Fðs11;s22; t12Þ ¼ F1s11 þ F11s211 þ F2s22 þ F22s

222

þ 2F12s11s22 þ F66t212

¼ 1 ð2Þ

Table 1

Elastic properties and matrix nonlinear Ramberg–Osgood (R–O)

parameters FVF in: roving layers ¼ 0.407, CFM layers ¼ 0.305

E (1000 ksi) n b n t0 (ksi)

Fiber (E-glass) 10.5 0.25 – – –

Matrix(vinylester

+ fillers) (tension)

0.730 0.30 4 6.0 2.0

Matrix(vinylester

+ fillers) (compression)

0.730 0.30 1 4.0 7.0

Table 2

Ultimate stress values for E-glass/vinylester plate and the corresponding

stresses in the layers


where

F1 ¼1

sT11

21

sC11

; F11 ¼1

sT11s

C11

;

F66 ¼1

t212

; F2 ¼1

sT22

21

sC22

;

F22 ¼1

sT22s

C22

; F12 ¼ 21

2

ffiffiffiffiffiffiffiffiffiF11F22

pð3Þ

The stresses in Eq. (3) are the ultimate stresses usually

obtained from the following five coupon tests: uniaxial

tension and compression ðsT11;s

C11Þ; transverse tension and

compression ðsT22;s

C22Þ; and V-notch shear test (t12).

Because the Tsai–Wu failure criterion is considered

separately for both the roving and CFM layers, the in situ

stress values must be estimated for each layer. These are

determined by a stress analysis with the micromodels and

calculating the stress states in each layer that correspond to

the experimentally obtained ultimate stresses of the

pultruded composite. The ultimate stress values used for

the failure analysis of each layer, and the corresponding

experimental results for the pultruded plate, are shown in

Table 2. Each experimental value in the table is the average

of 6–10 coupon tests. The experimental data were reported

by Haj-Ali and Kilic [13].

The first step is the transverse calibration of the two

failure criteria, i.e. to find the transverse ultimate stress

values used for both the roving and CFM layers. These

Fig. 2. Schematic illustration of the combined nonlinear and damage modeling approach using micromechanical and ED material models.

Fig. 3. Predicted stress failure envelopes of off-axis pultruded composite plate in compression.


values are determined from the corresponding overall

ultimate stresses in both transverse compression and tension

modes. The transverse modes of failure are sudden and

dynamic in nature. This allows the assumption that a

simultaneous transverse failure occurs in both the CFM and

roving layers. Therefore, the ultimate transverse stresses in

both layers are determined when the overall values match

the experimental data, as shown in arrows marked by (1)

and (2) in Table 2. Calibration of the uniaxial ultimate

stresses of the layers is performed afterwards. The CFM

Fig. 4. Predicted stress failure envelopes of off-axis pultruded composite plate in tension.

Fig. 5. Off-axis notched plate specimens and FE model.


axial ultimate stress values are the same as those in the

transverse direction because the CFM layer is an in-plane

isotropic medium. This calibration step is referred as (3) and

(4) in Table 2. However, the uniaxial ultimate stresses of the

roving layers are those that correspond to the applied overall

ultimate values, steps (5) and (6) in Table 2. Finally, the

ultimate shear stress values of both layers are those that

simultaneously correspond to the applied overall ultimate

shear, step (7) in Table 2.

A fiber or matrix types of failure modes cannot be

directly determined from the Tsai–Wu criterion for the

roving. The terms in the failure criteria, Eq. (2), can be used

to infer the dominant mode at failure. This study employs

the scheme used by Kim et al. [20]. In this scheme, if the sum

of the first five terms in Eq. (2) is greater than the last term,

then the failure mode is either an axial (fiber) or a transverse

(matrix) mode. If this condition is reversed, then the mode

of failure is an axial-shear failure. The sum of the axial

stress terms is then compared with the transverse stress

terms to determine if the mode of failure is a fiber or a

matrix mode.

5. Progressive damage analysis approach

The first step in the damage analysis is to establish a

suitable failure criterion for failure initiation in the layers.

Degradation of the composite material is performed based

on the type of failure mode that has been detected. The

discount method has been frequently used to degrade the

stiffness of a failed layer in laminated composites [18,21,23,

26,31]. For example, if a uniaxial fiber failure mode has

been detected in a layer under a plane-stress state, then the

effective elastic modulus in the fiber direction, and the in-

plane Poisson’s ratio are set to a small number. Another

approach is to discount these terms gradually following a

damage evolution using continuum damage mechanics

framework. Progressive damage modeling is very important

Fig. 6. Progressive damage analysis of notched pultruded plate under compression with off-axis angle u ¼ 608:


because it allows for the simulation of degraded structural

responses, especially when the structure can continue

resisting additional applied loads despite failure at limited

locations. Progressive damage can provide answers to how

the structure reaches its ultimate load. Load redistribution

and damage interactions allow accurate predictions of the

overall and localized deformations.

Failure theories are proposed only at the homogenized

level in each layer. The micromechanical models are

used for a perfect nonlinear medium. Therefore, these

micromodels are not capable of representing post-

ultimate damage responses. Furthermore, the microme-

chanical formulations are not valid after a first failure has

occurred. The modeling strategy presented herein decou-

ples the nonlinear micromechanical models from the

post-ultimate progressive damage. The latter is carried

out separately using the homogenized representation of

damage.

Fig. 2 shows a schematic structural and material response

framework representing the combined micromechanical and

damage modeling approach. The area OAA0 represents the

nonlinear response and the stored strain energy density at

the point where initial failure is detected. The constitutive

micromechanical model is not used after this stage. Instead,

a new fictitious material is defined where its initial strain

energy and secant stiffness are the same as the effective

micromechanical model at point A. The elastic-degrading

(ED) material model is now active and is using an energy-

based formulation to define the new stress state and its

secant stiffness matrix.

Next, an energy based damage model is derived. The

strain energy density function for the overall anisotropic

pultruded composite material is expressed as:

W ¼ 12sij1ij ¼

12ðs11111 þ s22122 þ s33133 þ s12g12

þ s13g13 þ s23g23Þ ð4Þ

Assuming that, damage is limited only to in-plane

modes, damage variables are introduced in the energy

terms that include the corresponding in-plane stress

components. The strain energy function is re-written

Fig. 7. Progressive damage analysis of notched pultruded plate under compression with off-axis angle u ¼ 908:


as:

W ¼ 12½ð1 2 l1Þs11111 þ ð1 2 l2Þs22122 þ s33133

þ ð1 2 l4Þs12g12 þ s13g13 þ s23g23�

0 # l1;l2;l4 # 1:0 ð5Þ

The variables l1, l2, and l4 represent the damage in

axial (fiber), transverse (matrix), and shear modes,

respectively. Evolution of the damage variables is now

needed. A simple evolution function is attached to each

damage variable. In this explicit approach, the damage

variables depend linearly on the direct strain in each

mode of failure. The damage variable reaches its

maximum value, 1, at m1 f where 1 f is the strain when

the failure is first detected. The coefficient m is estimated

empirically by carrying out different analyses of failed

coupons. A more sophisticated approach will be to form

damage potential functions that can be used to relate the

rate of damage growth. The current approach does not

include damage potentials in the formulation mainly for

the lack of sufficient test data that can be used to

calibrate the in situ damage growth. In this sense,

the proposed damage modeling is phenomenological.

However, the formulation is general and can allow

adding damage potentials instead of the specified damage

functions.

The secant stress–strain relations for an orthotropic

material is written as:

s11

s22

s33

t12

t13

t23

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

¼

C11

C12 C22 Symmetric

C13 C23 C33

0 0 0 C44

0 0 0 0 C55

0 0 0 0 0 C66

26666666666664

37777777777775

111

122

133

g12

g13

g23

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

ð6Þ

Fig. 8. Progressive damage analysis of notched pultruded plate under tension with off-axis angle u ¼ 608:


Eq. (6) is now substituted in Eq. (5) to yield:

W ¼ 12½ð12l1ÞðC11111þC12122þC13133Þ111

þð12l2ÞðC12111þC22122þC23133Þ122

þðC13111þC23122þC33133Þ133

þð12l4ÞðC44l212þC55g

213þC66g

223Þ� ð7Þ

The total secant stress–strain relation for the damaged

material is derived from:

sij¼›W

›1ij

��li

ð8Þ

The result is a total stress–strain relation for the damaged

material in the form:

s11

s22

s33

t12

t13

t23

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

¼

ð12l1ÞC11

12l1þl2

2

� �C12 ð12l2ÞC22 Symmetric

12l1

2

� �C13 12l2

2

� �C23C33

0 0 0 ð12l4ÞC44

0 0 0 0 C55

0 0 0 0 0 C66

2666666666666666664

3777777777777777775

�

111

122

133

g12

g13

g23

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

ð9Þ

The damage model and the micromechanical models are

implemented in the ABAQUS FE code [1]. The micro-

models are used to calculate the nonlinear effective

response for each Gaussian integration point at any

through-thickness layer. Different in-plane strain incre-

ments are given as an input at the different Gaussian

points. The written material subroutine is required to

update the effective stress for each material point. The

nonlinear micromechanical response is calculated if no

previous failure mode has been detected. The failure

criteria are evaluated for the CFM and roving layers once

the micromechanical stress update step is completed. In

the case, where a failure criterion is active, the current

state of deformation is used to define an equivalent

homogenized anisotropic material and the micromodels

are not used in future stress update increments. Instead,

the ED material model is used with the previously stored

strain energy and stress state, as described schematically

in Fig. 2.

6. Ultimate stress predictions for off-axis pultruded

coupons

The calibrated failure criteria and the proposed

damage approach are used in the analysis of off-axis

coupons under plane-stress. The nonlinear micromodels

are used for the CFM and roving layers. Net-section

ultimate failure is defined once the monotonically

increasing load is not sustained, which usually occurs

after the CFM and roving fail. The predicted and

experimentally obtained ultimate stress values for off-

axis tests are shown in Figs. 3 and 4. The test data

represents the maximum and minimum ultimate stress

values from 1/2 and 1/4 in. thick coupons. Each test was

repeated 3–5 times. Detailed description of the testing

procedures along with geometry of the different coupons

can also be found in Haj-Ali and Kilic [13]. The

experimental ultimate stress values from the 0 and 908

coupons are used to calibrate the two failure criteria. The

solid line is the prediction of the proposed damage

analysis approach when the Tsai–Wu criterion is used for

both roving and CFM layers. The overall ultimate stress

predictions are in good agreement with the experimental

results. The predicted compression failure values for the

158 are relatively higher than experimental values

presented in Fig. 3. This may be due to fiber

microbuckling that can contribute to compressive strength

reduction. This difference is not as evident as in the

tension results of Fig. 4.

7. Progressive damage analysis of pultruded notched

plates

Off-axis tests for notched pultruded coupons were

performed to compare the prediction of the nonlinear

damage modeling approach. Pultruded off-axis plates were

cut and notched with a circular hole. The geometries of

the plates are described in Fig. 5. The plates were tested

for both compression and tension loading. The tests were

carried out for coupons with off-axis angles: 0, 15, 30, 45,

60, and 908. Damage analyses are carried out using 8-

node layered shell elements. The total number of layers is

9, with 5 CFM and 4 roving. Three integration points are

used through the thickness for each layer. A convergence

study was done to select the appropriate mesh size. The

CFM and roving layers are explicitly modeled using their

nonlinear micromodels prior to failure. The nonlinear

response of the shell cross-section is integrated during the

analysis from all through-thickness integration points. The

Tsai–Wu failure criterion is used for both the CFM and


roving layers. The calibrated coefficients are the same as

those used in Section 6.

Figs. 6 and 7 show the results of the off-axis notched-

plates under compression for two representative cases (off-

axis angles of 60 and 908). These results include

experimental remote load versus the remote displacement

and the predicted FE results using the micromodels and the

ED damage model. The ultimate load is the last point on the

experimental load–deflection curve. The predicted ultimate

loads, from FE models, are very close to the notched coupon

tests under compression.

Next, a damage index, F, is introduced as the maximum

value of the Tsai–Wu failure criterion that an integration

point has experienced during all loading states. Damage

index contours are used to show the progression of damage

in the structure. When F is larger than 1 at an integration

point, it means that the Tsai–Wu failure criterion has been

exceeded and the damage variables, li, are active. There-

fore, the CFM or roving micromodel is not being used;

instead the material response at this integration point is

being generated by the ED material model. The damage

index contours are plotted separately for the CFM and

roving layers at different load steps or increments. Damage

initiation starts at a relatively high load magnitude. It starts

at 0.87 of the failure load for the 608 off-axis plate and 0.82

of the failure load for the 908 case. The brittle nature of

failure is also expressed in the fact that there are a small

number of failed elements in the contour plots prior to

ultimate state.

Figs. 8 and 9 show the experimental and the FE analysis

results of two representative cases: off-axis 60 and 908

plates under tension. In these tension cases, the experimen-

tal curves are plotted with applied load versus a displace-

ment obtained from an extensometer. The extensometer has

a gage length of 2 in.; it was placed on the coupon as shown

in Fig. 5. The extensometer could not be used in the full

duration of the test. The brittle and sudden nature of damage

violently detaches the extensometer from the coupon once

the ultimate failure is reached and the coupon starts to break.

Therefore, it was decided to try to briefly hold the tests at

Fig. 9. Progressive damage analysis of notched pultruded plate under tension with off-axis angle u ¼ 908:


about 80% of the estimated ultimate load and remove the

extensometer before proceeding. The test data for the

extensometer was recorded until its removal. A dashed

horizontal line in Figs. 8 and 9 is used to indicate the

maximum ultimate load at which the notched plate failed.

Damage initiation first starts in the CFM layers. It starts at

relatively lower load values than those in the compression

case. The overall prediction of the model is good especially

in the low displacement range. The predicted ultimate

failure loads are very close to the experimentally obtained

values.

8. Progressive damage analysis of pultruded FRP bolted

connections

This section describes the results of damage analyses

performed to simulate the response of two previous tests on

bolted pultruded composites by Steffen et al. [38,39].

E-glass/vinylester pultruded composite coupons with

6.4 mm thickness were used for the testing. These coupons

include 0 and 908 roving orientations measured with respect

to applied tension load. A simple pin-loaded bolted

connection was tested. The objective of this section is to

use the developed micromechanical and damage modeling

capabilities in order to analyze these tests. To this end, plane

stress models are used with 8-node reduced integration

elements. Fig. 10 shows the FE mesh and geometry of tested

plate. Due to the large stiffness ratio between the steel pins

and the pultruded material (at least 10), the pin is modeled

Fig. 10. FE model of a single pin-loaded connection.

Fig. 11. Progressive damage response of bolted plates.


with rigid elements. Contact is explicitly modeled between

the pin and the plate. A displacement-control loading up to

failure is carried out in the FE analysis.

Fig. 11 shows the load displacement response from the

models and the experimental results up to a displacement

range of 0.04 in. (0.1 cm). Convergence is not achieved

after a significant damage is accumulated, especially

beyond the first ultimate load (around 0.04 in.). This can

be attributed to large deformations and the severe contact

conditions after crushing the material under the heavy

pressure from the pin. It is not possible to include or capture

all of these modes and the FE analysis was terminated due to

convergence.

Figs. 12 and 13 show contours for the Tsai–Wu damage

index, F, at different increments for the 0 and 908 plates,

respectively. In the case of 08 plate, damage first starts in the

roving layers at a load level of 2.46 kips. Significant damage

in the CFM starts at a load level of 2.85 kips. In the case of

908 plate, damage starts in the roving layers at 1.06 kips and

then starts to accumulate in the CFM layers at 1.66 kips. The

damage index contours for the 908 plate show that the CFM

layers are more damaged.

9. Conclusions

A layer-wise damage modeling approach is integrated

with a micromechanical models to form a frame work for the

progressive damage analysis of pultruded composite struc-

tures. The modeling approach is applied to a pultruded

composite system made up from two alternating layers

reinforced with roving and CFM. Separate failure criteria and

ED models are considered for the CFM and roving layers.

The proposed damage modeling strategy employs the

nonlinear micromodels for the layers prior to reaching a

first failure. Once a failure is detected, the micromodels are

no longer used. Instead, a homogenized anisotropic ED

model is used at the failed material point. Progressive

damage analyses are carried out for notched plates under

compression and tension and a single-bolted connection

under tension. Contours of the attained maximum value of

the Tsai–Wu criterion are plotted for the CFM and roving to

indicate the damage progression. Good agreement is shown

when comparing the experimental results and FE models that

incorporate the combined micromechanical and damage

models.

Fig. 12. Damage progression in a single pin-loaded connection with u ¼ 08:


Acknowledgements

This work was supported by NSF through the Civil and

Mechanical Systems (CMS) Division under grant number

9876080.

Appendix A. Micromechanical models for the roving

and CFM layers

The nonlinear 3D micromodels for the CFM and roving

layers are briefly derived in this Appendix. The complete

formulation can be found in Haj-Ali et al. [12,14]. A

rectangular UC model used for a unidirectional fiber

reinforced material is used for the roving layer. The traction

and displacement continuity equations are approximated in

terms of average stresses and strains.

The long fibers are aligned in the x1-direction, while the

x2–x3 are the transverse directions as shown in Fig. 1. The UC

is divided into four sub-cells due to symmetry. The traction

and displacement continuity relations between the sub-cells

are approximated in terms of the appropriate components

of the average stress (s (a ), a ¼ 1, 2, 3, 4) and strain

(1 (a ), a ¼ 1, 2, 3, 4) vectors in the four sub-cells. The

overall average stress and strain vectors for the UC are

denoted by ð �sÞ and ð �1Þ; respectively. The notations for the

stress and strain vectors are:

{sðaÞi }T ¼ {s11;s22;s33; t12; t13; t23}ðaÞ

{1ðaÞi }T ¼ {111; 122; 133;g12; g13;g23}ðaÞ ðA1Þ

i ¼ 1;…; 6; a ¼ 1;…; 4

where (a ) denotes the sub-cell number in the UC and (i )

denotes the stress or strain component. The total volume of

the UC is equal to one. The volumes of the four sub-cells are:

v1 ¼ hb; v2 ¼ ð1 2 hÞb;

v3 ¼ hð1 2 bÞ; v4 ¼ ð1 2 hÞð1 2 bÞðA2Þ

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

the longitudinal relations are:

1ð1Þ1 ¼ 1

ð2Þ1 ¼ 1

ð3Þ1 ¼ 1

ð4Þ1 ¼ �1

ðRÞ1

v1sð1Þ1 þ v2s

ð2Þ1 þ v3s

ð3Þ1 þ v4s

ð4Þ1 ¼ �s

ðRÞ1

ðA3Þ

Consideration of the interfaces with normals in the x2-

direction, yields the traction continuity conditions for

Fig. 13. Damage progression in a single pin-loaded connection with u ¼ 908:


in-plane stress components (22) and (12), respectively. The

corresponding strain compatibility conditions follow from

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

and (4), respectively. This allows writing the traction and

compatibility relations for the transverse stress and strain

components (22) and for axial shear (12). The continuity

relations between the sub-cells are:

sð1Þ2 ¼ s

ð2Þ2 ; s

ð3Þ2 ¼ s

ð4Þ2

v1

v1 þ v2

1ð1Þ2 þ

v2

v1 þ v2

1ð2Þ2 ¼ h1ð1Þ2 þ ð1 2 hÞ1ð2Þ2 ¼ �1

ðRÞ2

v3

v3 þ v4

1ð3Þ2 þ

v4

v3 þ v4

1ð4Þ2 ¼ h1ð3Þ2 þ ð1 2 hÞ1ð4Þ2 ¼ �1

ðRÞ2

ðA4Þ

For the in-plane shear, the relations are:

sð1Þ4 ¼ s

ð2Þ4 ; s

ð3Þ4 ¼ s

ð4Þ4

v1

v1 þ v2

1ð1Þ4 þ

v2

v1 þ v2

1ð2Þ4 ¼ h1ð1Þ4 þ ð1 2 hÞ1ð2Þ4 ¼ �1

ðRÞ4

v3

v3 þ v4

1ð3Þ4 þ

v4

v3 þ v4

1ð4Þ4 ¼ h1ð3Þ4 þ ð1 2 hÞ1ð4Þ4 ¼ �1

ðRÞ4

ðA5Þ

Consideration of the interfaces with normals in the x3-

direction, yields the traction continuity conditions for

out-of-plane stress components (33) and (13), respect-

ively. These relations are expressed as:

sð1Þ3 ¼ s

ð3Þ3 ; s

ð2Þ3 ¼ s

ð4Þ3 ðA6Þ

v1

v1 þ v3

1ð1Þ3 þ

v3

v1 þ v3

1ð3Þ3 ¼ b1ð1Þ3 þ ð1 2 bÞ1ð3Þ3 ¼ �1

ðRÞ3

v2

v2 þ v4

1ð2Þ3 þ

v4

v2 þ v4

1ð4Þ3 ¼ b1ð2Þ3 þ ð1 2 bÞ1ð4Þ3 ¼ �1

ðRÞ3

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

are:

sð1Þ5 ¼ s

ð3Þ5 ; s

ð2Þ5 ¼ s

ð4Þ5 ðA7Þ

v1

v1 þ v3

1ð1Þ5 þ

v3

v1 þ v3

1ð3Þ5 ¼ b1ð1Þ5 þ ð1 2 bÞ1ð3Þ5 ¼ �1

ðRÞ5

v2

v2 þ v4

1ð2Þ5 þ

v4

v2 þ v4

1ð4Þ5 ¼ b1ð2Þ5 þ ð1 2 bÞ1ð4Þ5 ¼ �1

ðRÞ5

In the transverse shear mode, the traction continuity at

all the interfaces between the sub-cells must be satisfied.

The continuity equations for the transverse shear are:

sð1Þ6 ¼ s

ð2Þ6 ¼ s

ð3Þ6 ¼ s

ð4Þ6

v11ð1Þ6 þ v21

ð2Þ6 þ v31

ð3Þ6 þ v41

ð4Þ6 ¼ �1

ðRÞ6

ðA8Þ

Eqs. (A2)–(A8) define the needed micromechanical

relations between the stresses and the strains in the

sub-cells and the overall average stresses and strains of

the roving. These relations are used in incremental (rate)

form because the constitutive relations in the matrix sub-

cells are nonlinear.

A simplified phenomenological model is proposed for

the CFM medium using weighted responses of a uni-

directional layer in both axial and transverse type modes.

The CFM layer is a medium where resin is reinforced with

several mats of relatively long swirl filaments. The fibers are

randomly distributed in the plane of the mat. The proposed

CFM micromodel generates the overall effective nonlinear

3D response from average responses of the two uni-

directional layers with axial and transverse fiber orien-

tations. The overall in-plane average stress response is

generated by averaging the in-plane stress responses of the

two layers while the in-plane strain is the same in all sub-

cells. The FVF in the CFM is used to define the relative

thicknesses of the two layers. The overall out-of-plane

response is generated using traction continuity between the

two layers. The CFM effective medium should be

represented with in-plane isotropic model. The current

model does satisfy this requirement when the fiber is

isotropic. In the case where the fiber is orthotropic, the

resulting effective properties should be integrated and

averaged in the radial direction.

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

also convenient to divide the sub-cells into two parts. The

matrix-mode layer (part-A) is composed of sub-cells (1) and

(2), while the fiber-mode layer (part-B) is composed of sub-

cells (3) and (4). The relative thickness of each layer is

defined using the FVF. The out-of-plane direction is

represented by the x3-axis. The formulation of the CFM

can be presented in terms of average stresses and strains in

sub-cells A and B as intermediate variables. Therefore, these

two parts or sub-cells can be considered in the CFM

formulation as two independent layers. The FVFs within

the two parts are the same and provide the relations:

V1

V1 þ V2

¼ h ¼ vfC

V4

V3 þ V4

¼ j ¼ vfCðA9Þ

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

ment continuity, between parts A and B, are expressed by

�sðCÞO ¼ s

ðAÞO ¼ s

ðBÞO

�1ðCÞi ¼ 1

ðAÞi ¼ 1

ðBÞi

ðA10Þ

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:

�sðCÞi ¼

1

VðVAs

ðAÞi þ VBs

ðBÞi Þ

�1ðCÞO ¼

1

VðVA1

ðAÞO þ VB1

ðBÞO Þ

ðA11Þ

Within the matrix-mode layer (part-A), the following

relations for all stress and strain components should be

satisfied:

�sðAÞ ¼ sð1Þ ¼ sð2Þ ðA12Þ


�1ðAÞ ¼

1

VA

ðV11ð1Þ þ V21

ð2ÞÞ

The corresponding equations for the fiber-mode layer (part-

B) are:

�sðBÞO ¼ s

ð3ÞO ¼ s

ð4ÞO

�1ðBÞi ¼ 1

ð3Þi ¼ 1

ð4Þi

�sðBÞi ¼

1

VB

ðV3sð3Þi þ V4s

ð4Þi Þ

�1ðBÞO ¼

1

VB

ðV31ð3ÞO þ V41

ð4ÞO Þ

ðA13Þ

Eqs. (A9)–(A13) define the 3D micromechanical relations

between the average stresses and strains in the fiber and

matrix sub-cells of the CFM layer. These relations are used in

an incremental form because the constitutive relations in the

two matrix sub-cells are nonlinear.

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