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Progressive Failure Analysis of Composite StructuresUsing a Constitutive Material Model (USERMAT)Developed and Implemented in ANSYS ©

Elisa Pietropaoli

Received: 5 May 2011 /Accepted: 29 June 2011# Springer Science+Business Media B.V. 2011

Abstract Composites are materials characterized by complex failure phenomena that onsetand interact. Several approaches are available in literature to predict the behaviour of compositestructures taking into account failure modes. However, most of them require the knowledge ofexperimental parameters, which generally are not provided by composite suppliers. Progressivefailure techniques represent a valuable alternative to these methodologies because they rely onfailure criteria and ply-discount techniques often based on the choice of a single degradationfactor whose value is chosen by the analyst as small enough to prevent convergence problems infinite element analyses. The aim of this work is to analyze the behaviour of a compositestructure taking into account the damage onset and evolution. The analysis is performed byusing a constitutive material model (USERMAT) developed and implemented in the Finiteelement software ANSYS©. The accuracy of the procedure proposed is assessed by comparingnumerical results and experimental data taken from literature.

Keywords Composites . Finite element analysis . Damage . Progressive failure

1 Introduction

Composites are materials constituted by two components (fiber and matrix) whose stiffnessand strength-to-failure are extremely different. The presence of these constituents, mixed atmacroscopic level, leads to high performance of composites but makes these materialsinherently anisotropic and characterized by many failure modes such as matrix breakageand fibre failure. Failures that develops within a layer are defined “intralaminar damage”,however when dealing with multi-directional laminates (obtained by stacking unidirectionalply of composite materials with different orientations) also other kinds of damage can

Appl Compos MaterDOI 10.1007/s10443-011-9220-0

E. Pietropaoli (*)CIRA, Italian Aerospace Research Center, via Maiorise, 81043 Capua, CE, Italye-mail: [email protected]

Elisa. Pietropaolie-mail: [email protected]

develop such as debonding between adjacent layers (intralaminar failure or delamination)and failure transverse to the laminate plane (translaminar failures). These failure modes canonset and grow at different scales, interact and coalesce. A comprehensive methodology forthe analysis of composite structures should take into account of all the damage modes andof their interactions especially when the safety is of main concern such for aircraftapplications. In past years, the methodologies available were not deemed as effective andaccurate enough to be used in the design of composite structures for aerospace applicationsand a huge degree of conservatism was still applied to the design phase thus preventing theperformance of composites to be fully exploited. Nowadays, many efforts are beingdedicated to the development of robust and accurate methods and as first achievement,commercial finite element codes begin to allow analyses of composites to be performedtaking into account both intralaminar and interlaminar damage.

When dealing with intralaminar damage, the onset of failure in an individual ply (firstply failure) generally does not lead the structure to collapse [1] and this condition may notoccur until the failure has spread to multiple plies (final failure) (Fig. 1).

The different approaches available for analysing this phenomenology can be classifiedfollowing [2] as damage mechanics or progressive ply damage. Damage mechanic approachesare formulated using physically based equations to represent the damage onset and evolutiontaking into account the microstructure of the composite material [3–5]. Progressive plydamage methodologies rely on the combined use of failure criteria for the identification of thedamage [6–8] and on ply-discount techniques [9–17] for the simulation of the progressivedecrease of the structural stiffness and strength. These ply-discount techniques are generallynot-physically based and the undamaged properties are instantaneously reduced through aconstant degradation factor whose value is not equal to zero but it is selected as small enoughto prevent convergence problems in finite element analyses.

Even thought damage mechanics methodologies seem to be the most appropriate way totake into account of the physics of the damage in composites [18, 19], these approachesrequire the knowledge of many experimental parameters, which generally are not providedby composite suppliers. An extensive and expensive material characterization is requiredthat in most cases is not feasible. Thus, progressive failure techniques still represent avaluable alternative either when the objective is to design damage tolerant structure or toevaluate the residual strength of a composite structure [20–24].

App

lied

load

F

First ply Failure

End-shortening

Final failure (collapse)

δ

δ

Failure

FPF

LPF

Fig. 1 Progressive damage of acomposite laminate

Appl Compos Mater

The aim of this paper is to analyse the behaviour of a composite structures subjectedto static loads taking into account onset and progression of damage. These analyses willbe performed by using a progressive failure technique implemented as user materialsubroutine in the finite element code ANSYS ©. The accuracy of the numerical resultswill be assessed through the comparison between numerical and reference experimentaldata taken from literature.

2 Progressive Failure Methodology

Composite materials behave differently under tension and compression loads, therefore it isnecessary to define five dissimilar strength parameters shown in Fig. 2 [25].

Failure criteria for composite laminates are mainly analytical approximations orcurve fittings of experimental results. Most failure criteria for composite material (Tsai-Hill,Tsai-Wu and Hoffman criteria) have been thought as an extension of the VonMises criterion to aquadratic criterion [8] and allow the failure of a whole ply to be checked. The Hashin’s [6]criteria differs slightly from the latter and they allow to distinguish between failure in tensionand in compression and to take into account a three dimensional state of stress (Table 1).

It is worth noting that in Table 1 two different shear strengths are defined respectively inthe 12 plane (axial failure shear S12) and in 23 plane (transverse failure shear).

The criterion in Eq. 1 has been chosen, among those available in literature to check forthe onset of delaminations.

Delamination onsets33

Zt

� �2

þ s23

S23

� �2

þ s13

S13

� �2

� 1 ð1Þ

Degradation of plies that have partially or completely failed is a critical phase for theprediction of the ultimate failure. As plies fail progressively, strain energy of all plies mustbe redistributed and this induces a modification of the stress within the material.Degradation rules or ply discount techniques, such as those proposed in Table 2, can beused to reduce the stiffness of damaged plies in order to reflect the presence of matrixcracks, fiber failure and of the other damage modes.

1

2

XT: Longitudinal tensile strenght

σ1>0

Xc: Longitudinal compressive strenght

σ1<0

Yt: Transverse tensile strenght

σ2>0 σ2<0

Yc: Transverse compressive strenght

σ6<0

S: In plane strenght

Fig. 2 Strength parameters for composite laminates referred to the material principal axes [25]

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Different degradation rules are available in literature [9–17]. However, most of them arebased on the introduction of a degradation factor (k in Table 2) that allows the stiffness ofdamaged plies to be reduced to a percentage of their own value in the undamaged state.Unfortunately, there is no correlation between the physics of damage mechanisms and thedegradation factor, whose value is always selected by chance.

By using these degradation rules, it is possible to follow the progression of the damage andto determine the strength beyond the first ply failure bymeans of progressive failure procedures.

The basic steps of these procedures may be synthesised as: determination of the stressesdistribution on a ply by ply basis (material axes), application of failure criteria for each ply,degradation of the material properties to take into account the typical post-damage stiffnessreduction of the material.

Failure mode Material propertiesdegradation rules

Matrix failure(tension)

syy > 0Ey ¼ kEy;Gxy ¼ kGxy

Matrix failure(compression)

syy < 0

Fiber failure(tension)

sxx > 0All the properties arereduced.

Fiber failure(compressione)

sxx < 0

Fiber matrix failure(shear-out)

sxx > 0 Ey ¼ kEy; vxy ¼ kvxy;vyz ¼ kvyz

Delaminationonset Ez ¼ kEz

Table 2 Ply discount technique(k is a degradation factor).Properties indicated without abar are those of theundamaged ply

Table 1 Hashin’s failure criteria

Matrix tensile failure

s22 þ s33 > 0 s22þs33Yt

� �2þ s2

12þs213

S212þ s2

23�s22s33

S223� 1

Matrix compressive failure

s22 þ s33 < 0 1YC

YC2S23

� �2� 1

� �s22 þ s33ð Þ þ 1

4S223s22 þ s33ð Þ2 þ 1

S223s223 � s22s33

� þ 1

S212s212 þ s2

13

� � 1

Fibre tensile failure

s11 > 0 s11Xt

� �2þ 1

S212s212 þ s2

13

� � 1

Fibre compressive failure

s11 < 0 s11Xc

� �� 1

Fiber-matrix shear failures11Xt

� �2þ s12

S12

� �2� 1

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3 User Material Subroutine for Progressive Failure Analysis (Intralaminar Damage)

A user material subroutine (usermat [26]) in ANSYS allows a constitutive material modelto be defined. This subroutine is called at each material integration point of the elementsduring the solution phase. ANSYS passes in stresses, strains, and state variable values at thebeginning of the time (or load) increment and strain increment at the current increment,then updates the stresses and state variables, according to the values defined by the usermat,at the end of the time (or load) increment [26].

A progressive failure technique has been implemented in ANSYS as usermat. Thisroutine is based on failure criteria in Table 1 and the instantaneous degradation rules inTable 2. The stiffness of the composite is provided as material Jacobian matrix defined as@Δs ij=@Δ"ij where Δs ij is the stress increment, and Δ"ijis the strain increment within a

W

L

STACKING SEQUENCE: [-45°/90°/45°/0°]St(b)

(a)

L 118mmW 38mm d 6.35mmt 1.1mm

Longitudinal tensile modulus 148 GPa Longitudinal tensile strength

2000 MPa

Transverse tensile modulus 9.5 GPa Transverse tensile strength 50 MPa Poisson ratio 12 and 13 0.3 Longitudinal compressive

strength 1500 MPa

Poisson ratio 23 0.49 Transverse compressive strength

150 MPa

In plane shear modulus 12-13 4.5 GPa In plane shear strength 100 MPa In plane shear modulus 2-3 3.17GPa

d

Fig. 3 Geometry and material properties description

Table 3 Advantages and disadvantages of using a usermat

User material subroutine (Fortran) Post-processing routine (APDL)

Time for the analysis Only one processor should be used. More than one processor can be used.

The solution can be found without stopand restart the analysis.

The analysis must be performed assuccession of analyses (stop, incrementof the load, restart).

Applicability Whichever structure without requiringany modifications.

If the mesh is very fine the storage ofinformation is a time consuming processand it requires a lot of memory space.

Post-processing Straightforward because state variablescan be plotted as element or nodalresults.

Time consuming.

User friendliness Once implemented the usermat, the userhas a custom-executable for ANSYStherefore, he/she can be unknown ofdetails of the implementation.

The user must know how the post-processing routine is written.

Compile of the code Required. Not required.

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load step of a non-linear incremental analysis. Each component of the stiffness matrix hasbeen defined equal to the coefficient Cij of the composite material considered asorthotropic. These coefficients Cij are computed as in Ref.[27] starting from the values ofthe properties E1,E2,E3,n12,n13,n23,G12,G23,G13 . It is worth noting that the values of theseproperties are initially those defined by the user for the undamaged material. Then, they areupdated following the rules reported in Table 2 and in agreement with the values of the“state variables”, which have been used to store information on the damage state, and that,are themselves updated based on the outcomes of failure criteria. The transformation of thestiffness in the global coordinate system, according to the laminate staking sequence

00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

5

10

15

20

25

End shortening (mm)

For

ce (K

N)

#MS01

#MS02

Experimental results

DEVIATION FROM LINEARITY

INITIAL STIFFNESS

Fig. 5 Load versus end shorteninggraph. Numerical andExperimental [28] results

#MSO1 #MSO2

#MSO2

#MSO1

Fig. 4 Mesh for the models #MSO1 (208 elements) and #MSO2 (800 elements)

Appl Compos Mater

defined for each element (as SECTION [26]), is performed automatically by ANSYSoutside of the usermat.

Often progressive failure procedures have been implemented as post-processing routinesespecially when using ANSYS [20–22]. This is principally because it is quite easy toimplement post-processing routines in ANSYS by means of the Parametric DesignLanguage (APDL), which allows stopping and restarting analyses, storing data as matricesand obtaining directly data from the model and from the result file. However, the storage offailure data in the post-processing phase becomes infeasible when a huge number ofelements are used in the FE model. Furthermore, many operations are performed “out-of-core” as input/output computations thus preventing the analysis to be time effective. Indeed,these difficulties can be easily overcome by means of a USERMAT: the Progressive failureanalysis is performed “in-core” within the solution phase and the post-processing phase(visualization of the damage state) is straightforward because failure data are stored aselement or nodal results. The Table 3 summarizes the main advantages and disadvantagesassociated to the different implementations.

4 Analysis of a Specimen with a Hole Subjected to Static Compressive Load

The structural response of a specimen with a hole subjected to compressive loads has beenanalysed by using the progressive failure methodology presented in Section 3. Geometryand material properties are described in Fig. 3. The specimen is fully constrained at hisedges and out of plane displacements are prevented. The load is applied along the specimenmajor axis.

Two different meshes have been built (#MSO1 and #MSO2 in Fig. 4). In both cases,solid elements (20 node hexahedral layered) have been used and a degradation factor equalto1e-08 has been chosen for the progressive failure analysis.

Numerical results obtained by using the developed procedure for the specimendescribed in Fig. 3 have been compared with experimental results taken from literature

MN

MX

020

4060

80100

ANSYS 12.0.1

MN

MX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=21SUB =6TIME=21MT (NOAVG)LAYR=8DMX =1.102SMX =100

(a) (b) (c)

ELEMENT SOLUTION

STEP=23SUB =4TIME=22.575MT (NOAVG)LAYR=8DMX =1.203SMX =100

Fig. 6 a Percentage of damage at 21 KN (#MS02), b C-Scan before final failure [28], c Percentage ofdamage at 22.6 KN (#MS02)

Table 4 Comparison between the numerical results

Model identifier First Failure Load Deviation from linearity Final Failure Load

#MS01 8.2KN 19KN 26.13 KN

#MS02 7.6 KN 19 KN 22.59 KN

Appl Compos Mater

[28]. Really, the specimen proposed by Suemasu et al in Ref. [28] have the samedimensions and material but has different thickness (2.2 mm instead of 1.1 mm) andstacking sequence ([(45/0/-45/90)2]S instead of [(45/0/-45/90)]S) with respect to the onesimulated numerically. However, numerical analyses of the Suemasu’s specimen gave asresults a stiffness much higher that that measured experimentally. Chua Hui Eng [29]proposed to use the specimen in Fig. 3 for the comparison with the Suemasu’s one andeffectively, even thought this choice seems to be rude approximation of the experiments,the results are very encouraging. Indeed, it is possible to recognize in Fig. 5 a perfectcorrespondence between the numerical results obtained by using ANSYS and experi-mental data in terms of stiffness of the structure (measured as slope of the load versus endshortening graph). Furthermore, the load at which the force versus end-shortening curvedeviates from linearity (19KN) is well simulated. Finally, by increasing the mesh densitya good prediction of the final collapse load can be obtained as shown in Fig. 5 andTable 4.

In addition, the failure maps obtained numerically have been compared with a C-Scantaken from Ref. 28 as shown in Fig. 6. The position and the extension of the damage zoneare in good agreement as well as the ply splitting shown in Fig. 7 which has been obtainednumerically as highly distorted elements. It is worth noting that the percentage of damagehas been computed considering an element as completely failed when in all its plies afailure mode is detected.

The percentage of delaminated plies computed taking into account the criterion in Eq. 1is shown in Fig. 8. Also in this case, the damage develops near the hole and propagates inthe direction orthogonal to the applied load.

MNMX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=23SUB =4TIME=22.575MT (NOAVG)LAYR=8DMX =1.203SMX =84.375

Fig. 8 Percentage of delaminatedplies at 22.6KN (#MS02)

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=23SUB =4TIME=22.575MT (NOAVG)LAYR=8DMX =1.203SMX =100

PLY SPLITTING HIGHLY DISTORTED

ELEMENTS

(a) (b) (c)

Fig. 7 a Microscope picture at 22.7KN [28] b Deformed shape with distorted elements at 22.6KN (#MS02),c Percentage of damage at 22.6 KN (#MS02)

Appl Compos Mater

5 Stiffened Panel with an Embedded Delamination

The structural response of a stiffened panel has been analysed taking into account bothintralaminar (fiber/matrix) and interlaminar (delamination) damage. Indeed, the usermatdefined in Section 3 has been used for intralaminar damage whereas a procedure based on

1250

1300

1350

1400

1450

1500

1550

1600

1650

1700

1750

-4600 -4100 -3600 -3100 -2600

Applied Strain

Deb

on

ded

Are

a (m

m^2

)

EXPERIMENTAL RESULTS

NUMERICAL RESULTS

-1

-0.5

0

0.5

1

1.5

2

-5000 -4000 -3000 -2000 -1000 0

Applied Strain (με )

Ou

t of p

lan

e d

isp

lace

men

t Uz

(mm

)

EXPERIMENTAL RESULTS

Numerical results (delamination only)

Numerical results

-900

-800

-700

-600

-500

-400

-300

-200

-100

0-6000 -5000 -4000 -3000 -2000 -1000 0

Applied Strain (με )

Lo

ad F

(KN

)

EXPERIMENTAL RESULTS

Numerical results (delamination only)

Numerical results

(a) (b)

(c)

Fig. 10 Comparison between numerical and experimental results [31]

R0

bc

bfs bl

W

L

xc

Yc

SKIN: [+45°/-45°/0°/90°]3S

STRINGER 1 - 2 - 3 : [+45°/-45°/0°3/90°/0°3/-45°/+45°]

hw

tf

tw

tc

(b)

(a)

12

3

Bl 11mm Tb 3.33mm Bf 55mm Tf 1.16mm Bc 48.2mm Tc 2.80mm S 94mm Tw 2.47mm R0 20mm Hw 45mm W 375mm XC 113mm L 446mm YC 205mm

Longitudinal tensile modulus 140 GPa Longitudinal tensile strength

2100 MPa

Transverse tensile modulus 10.5 GPa Transverse tensile strength 70 MPa Poisson ratio 12 and 13 0.3 Longitudinal compressive

strength 1650 MPa

Poisson ratio 23 0.51 Transverse compressive strength

240 MPa

In plane shear modulus 12-13 5.2 GPa In plane shear strength 105 MPa In plane shear modulus 2-3 3.48GPa Critical ERR – Mode I 260 J/m2

Critical ERR – Mode II 950 J/m2

Critical ERR – Mode III 1200 J/m2

Fig. 9 Geometry and material properties of the stiffened panel [31]

Appl Compos Mater

the combined use of the Virtual Crack Closure Technique and of a failure release approachhas been adopted for simulating the delamination growth [30].

The geometry of the panel and the material properties as well as the stacking sequencefor the components of the panel, are described in Fig. 9.

Displacement controlled non-linear analyses have been performed by using a constantload increment equal to 0.07 mm considering both intralaminar and interlaminar damage(label “Numerical results” in Fig. 10) and interlaminar damage only (label “Numericalresults (delamination only) in Fig. 10). The global (−2700 με) and the local buckling load(−1100 με) are very close to the experimental results (global and local buckling loadrespectively equal to −2850 με and −1250 με). The delamination growth rate (defined asthe ratio debonded area/applied strain) obtained by using ANSYS is in a very closeagreement with the experimental results (Fig. 10-a). No significant differences can beappreciated in Fig. 10-c between the results of the two numerical analyses whereas for an

MNMX

X

YZ

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=26SUB =1TIME=26MT (NOAVG)MIDDLEDMX =4.18SMX =66.667

AAMN

MX

X

YZ

-3.439-2.606

-1.772-.938888

-.105535.727818

1.5612.395

3.2284.061

ANSYS 12.0.1NODAL SOLUTION

STEP=26SUB =1TIME=26UZ (AVG)RSYS=0DMX =4.169SMN =-3.439SMX =4.061

MNMX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=26SUB =1TIME=26MT (NOAVG)MIDDLEDMX =4.18SMX =66.667

a b c

Fig. 12 a deformed shape and out of plane displacement contours at −4000 με. b and c percentage ofdamage for the most external ply of the delaminated area

MNMX

020

4060

80100

ANSYS 12.0.1

MN

MX

020

4060

80100

ANSYS 12.0.1

MNMX

020

4060

80100

ANSYS 12.0.1

XMNM

020

4060

80100

ANSYS 12.0.1

I

II IV

III

DELAMINATION

THICK SUBLAMINATE

THIN SUBLAMINATE

DELAMINATION

THICK SUBLAMINATE

THIN SUBLAMINATE

DELAMINATION

THICK SUBLAMINATE

THIN SUBLAMINATE

DELAMINATION

THICK SUBLAMINATE

THIN SUBLAMINATE

I

III

II

IV

Fig. 11 Failure maps at −4000 με for the most external plies (plies of the thinnest sublaminate). The positionof a ply along the thickness is sketched at the left hand side of each map

Appl Compos Mater

applied compressive strain greater than −3000 με the results of the two numerical analysesare not overlapped as shown in Fig. 10-b.

This difference can be explained by analysing the damage state within the structure. Asshown in Figs. 11 and 12 only the plies of the thinnest sublaminate of the delaminated areaare damaged. Since the stringers are undamaged, the global behaviour of the panel is notaffected by the onset and evolution of intralaminar (matrix/fiber) failure. On the contrary,the out of plane displacement (graph b in Fig. 10) measured locally in the middle of thedelaminated area (Point A in Fig. 12) decreased due to the ply failure.

The failure mode detected in the most external ply (Fig. 13) is matrix failure, whereasthere are both fibre and matrix failure modes in the ply of the thinnest sub-laminate nearestto the delamination plane (Fig. 14).

6 Conclusions

A constitutive model for composite materials has been defined and implemented asusermat in ANSYS. This model includes failure criteria and a ply discount techniquefor taking into account the onset and progression of intralaminar damage. By using thisusermat, two composite structures (a specimen with a hole and a stiffened panel) havebeen analysed and results have been compared against experimental results taken fromliterature. A good agreement between the numerical results and the experimental databoth in terms of global structural behaviour and failure loads has been obtained. Alimited sensitivity analysis has been performed on the specimen with a hole showingthat the use of progressive failure approaches can be dependent on the mesh density(and on the load increment too). Therefore dedicated algorithms should by used toreduce this dependency when the objective is to use these procedures for predicting andnot only for simulating the structural behaviour.

FIBER MATRIX SHEAR FAILURE

MATRIX FAILURE FIBER FAILURE

MNMX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=26SUB =1TIME=26MT (NOAVG)MIDDLELAYR=1DMX =4.18SMX =45.833

MN

MX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=26SUB =1TIME=26MT (NOAVG)MIDDLELAYR=1DMX =4.18SMX =66.667

MNMX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=26SUB =1TIME=26MT (NOAVG)MIDDLELAYR=1DMX =4.18SMX =50

IV

DELAMINATION

THICK SUBLAMINATE

THIN SUBLAMINATE

IV

Fig. 14 Failure maps at −4000 με for the +90 ply (IV)

MNMX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=26SUB =1TIME=26MT (NOAVG)MIDDLELAYR=1DMX =4.18SMX =16.667

MNMX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=26SUB =1TIME=26MT (NOAVG)MIDDLELAYR=1DMX =4.18SMX =33.333

MNMX

020

4060

80100

ANSYS 12.0.1ELEMENT SOLUTION

STEP=26SUB =1TIME=26MT (NOAVG)MIDDLELAYR=1DMX =4.18SMX =16.667

FIBER MATRIX SHEAR FAILURE

MATRIX FAILURE FIBER FAILURE

DELAMINATION

THICK SUBLAMINATE

THIN SUBLAMINATE

I

Fig. 13 Failure maps at −4000 με for the +45 ply (I)

Appl Compos Mater

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