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AN OPEN TOOLBOX FOR DAMAGE SIMULATION

Michele D’Ottavio1, Falk K. Wittel2, and Bernd Kroplin3

1,3Institute for Statics and Dynamics of Aerospace Structures, Universitat Stuttgart, Pfaffenwaldring 27,D-70569 Stuttgart, Germany

2Institute for Building Materials, ETH Zurich, Schafmattstr. 6, ETH-Honggerberg HIF, CH-8093 Zurich,Switzerland

Abstract

An open Micro-Mechanisms Toolbox is proposed with the aim of extending the capabili-ties of current platforms for structural design by means of physically sound models for damagemechanisms. Based on the widely accepted ABAQUS Finite Element package, we prototypi-cally implement three different tools which allow a refined failure analysis of short and long fiberreinforced composites. The key is the inclusion at the structural level of dedicated models forspecific damage mechanisms activated at smaller scales. The proposed toolbox should be con-sidered as a first step towards the realization of a consistent simulation tool capable to cope withthe extremely complex phenomena associated to the failure of advanced materials and structures.

Key words: Numerical Failure Analysis; Finite Element Method; Damage Mechanisms; Micro-Mechanics; Representative Volume Elements; Puck’s Theory; Discrete Element Method; Lami-nated Plate Elements; Unified Formulation.

1. INTRODUCTION

Many times structural engineers are faced with failure phenomena that can not be explained on the structuralscale as macro-events. Empirical design rules derived from single case studies seem to provide help at firstsight, nevertheless these ad hoc solutions rapidly exceed their range of validity. Rational explanations can notbe found unless the physical nature of macroscopically observed failure, as a summation of certain interactingmicro-events on finer length scales, is taken into account [1].

Advanced engineering materials are in general strongly heterogeneous and cracks are, hence, faced with theheterogeneities when they grow from the atomistic up to the macroscopic scale [2]. Heterogeneities are forexample pores, voids, aggregates, particles, fibers, accumulations, plies and more, that are all specified by acharacteristic size and associated to a representative length scale [3]. On their representative scale, micro-crackgrowth is trapped for example by deflection from their preferred crack path or cracks tip blunting. Statisticaldescriptions are necessary to account for the scale dependent disorder [4, 5]. Through interactions, crackscan merge and grow to become relevant for higher length scales. This way a hierarchy of length scales canbe identified (see Fig. 1(a)). For a physical interpretation of structural failure, models must capture thishierarchy of length scales [6]. While in principle all model types can be employed for the abstraction of specific

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damage mechanisms, a practically relevant failure description should allow the transfer of system response tothe hierarchically higher level - to bridge the scales [7].

Damage mechanisms are determined by geometry, stiffness and strength ratios of the involved constituents ontheir representative scale [4]. This explains why identical mechanisms can be found in different material systemssuch as steel reinforced concrete and glass fiber reinforced epoxy composites or spruce wood under tension andmetal matrix composites. Thinking in terms of material independent damage mechanisms interacting throughdifferent scales, we propose a framework suitable for a physically based description of failure for a wide classof engineering materials - the Micro-Mechanisms Toolbox (MMT). Our focus is set on short- and long-fiberreinforced materials. The concept is to provide an open, extendable collection of generic mechanism andtheir interaction to give users the possibility to tailor their specific damage scenarios by selecting appropriatemechanisms. The framework for the activation and interaction of the dedicated damage tools is the commercialFinite Element Method (FEM) based software ABAQUS, which is a widely accepted, open analysis code forstructural design.

(a) Internal structure of a woven com-posite laminate with possible genericdamage mechanisms.

(b) FEM simulation procedure and ex-tensions for the damage simulation.

Figure 1.

Our toolbox is integrated into the generic simulation scheme of FEM as depicted in Fig.1(b). For our purposeswe identify three successive steps of the FE simulation process: namely (i) pre-processing, (ii) solving ofgoverning equations and (iii) post-processing. Two different strategies to introduce material damage can befollowed: in a static approach the FE analysis has no influence on the outcome of the constitutive behavior,therefore this can be called material pre-processing and is located at Fig.1(b)(i); contrary, in a dynamicapproach, the material state information of the equilibrium database is interactively defined by the actualfield variables. Therefore it needs to be embedded into the solution procedure (Fig.1(b)(ii)) and can becalled material runtime processing. Dedicated post-processing procedures (Fig.1(b)(iii)) finally allow for thetransformations of the final equilibrium database into properties that are for example comparable to NDTsignals.

In this article, we first present the architecture of the MMT and the way it is embedded into ABAQUS (Sec. 2).

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Three prototypical tools in form of user-friendly plug-ins are subsequently briefly addressed. A plug-in for thematerial pre-processing is proposed in Sec. 3 and one for a material runtime processing is discussed in Sec. 4.A plug-in for a dedicated FE formulation for composite laminates is described in Sec. 5. Some concludingremarks and an outlook close this contribution.

2. ARCHITECTURE OF THE TOOLBOX

The FE package ABAQUS is widely spread throughout the engineering science community, mainly because ofits open philosophy, and it can be regarded as a standard for damage simulation. Users find well documentedinterfaces to include their own material models and elements. Additionally, users can control the whole systemvia the ABAQUS Scripting Interface, which is an extension of the Python object-oriented programminglanguage, giving the user the possibility to include any desired Python module. Thanks to the ABAQUS GUIToolkit, customized applications and plug-ins for the pre-processor can be implemented as Python scriptsas well [8]. With these tools at hand, it is possible to create complex toolboxes like the Micro-MechanismsToolbox as a collection of plug-ins, Python scripts and user subroutines for damage and failure in compositematerials. Plug-ins build the interface to the user, through which all tools can be accessed. Therefore, itis straightforward to increase the functionality of the FE package by adding tools or creating new ways ofcombining pre-existing tools.

Fig. 2 schematically depicts the current architecture of the MMT and highlights the modular structure of theimplemented tools, that interact with the ABAQUS core exploiting separated interfaces. Three prototypicaltools have been implemented, namely the RVE plug-in for the material pre-processing, the ’X’-PUCK plug-infor the material runtime processing and the Unified Formulation Plug-in (UFP) for a dedicated FE formulationfor multilayered plates.

UELABAQUS / Standard

PYTHONinterpreter

Commandline

interface

ABAQUS/CAEkernel

PYTHONscript

GUI

ABAQUS/CAE

UMAT

RVE

input file

outputdatabase

MEDIT

UFP

X-PUCK

Material DB

DEM-tools

FBMTCMSFM

Figure 2. Architecture of the MMT with the three prototypical tools.

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3. THE RVE PLUG-IN

If a specific material volume is subject to damage initiation or growth, its effective material properties change.Within a continuum representation, effective properties of a volume should not depend on its extensions andare called statistically homogeneous [2, 4, 9]. The smallest volume where this condition applies is calledRepresentative Volume Element (RVE) or unit cell. If the micro-structure of a material can be built up by anRVE, the non-linear material behavior can be calculated via numerical experiments on the unit cell (materialpre-processing). This holds as long as homogenization is allowed, namely provided that the conditions (c1)the RVE can be regarded as representative in the micro-scale, and (c2) the RVE can be regarded as a pointin the macro-scale, are satisfied [10].

In order to obtain the complete non-linear material response, a series of numerical tests such as tensile,compressive and shear tests must be conducted on the individual RVE. The number of tests and the way theparameters defining the macroscopic behavior are extracted from the results of these tests (homogenization)depend on the target material model. Our protoypical tool currently refers to the anisotropic elasto-plasticmaterial model with linear kinematic hardening, which is available in the ABAQUS material library [11]. Acollection of parameterized two-dimensional (2D) and three-dimensional (3D) RVEs have been implementedand stored within a repository - an RVE catalog. Additionally, prototypical RVEs with appropriate boundaryconditions and loading cases are available to compute the non-linear material response of a user-defined unitcells. The result of the computations consists of a set of material parameters which is stored in an ABAQUS-compatible Material Data-Base (Material DB in Fig. 2) and can be used in a subsequent structural analysis.

The functionality of the RVE plug-in is enriched by the possiblity of combining the material laws obtainedfrom single unit cells. This allows to increase the complexity of the material representation by means of twodifferent approaches. On the one hand, in cases where the condition (c2) is violated (e.g., short fiber reinforcedplastics with fiber aspect ratio >10), the combined cell approach can still be used to obtain valid results of theconstitutive behavior [12]. On the other hand combinations can be employed to introduce at the macroscalethe effects of various damage mechanisms on the constitutive behavior [13].

In this article, we limit our attention on the theoretical concepts underlying the combination of RVEs for adamage representation, while for a detailed description of the combined cell approach we refer to, e.g., [12].Material properties show disorder over a certain volume that can be built up of many geometrically identicalRVEs. Statistically distributed material properties lead to a statistical break-down of such a disordered system.Within the effective medium theory, this can be represented by first calculating different RVEs like a completelyintact one and as many failed ones as damage mechanisms are to be considered. In a successive step theresulting properties of all RVEs are combined in a constitutive equation using the classic rule of mixture [13, 14]

σcombined = (1− d) · σintact + d · σdamaged. (1)

d is the damage parameter ranging from d = 0 (completely intact) to d = 1 (completely damaged). For eachdamage parameter a separate evolution law has to be defined, which represents to some extent the intrinsicmaterial disorder. Some commonly employed evolution laws d = d(ε) have been implemented in the RVEplug-in and are depicted in Fig. 3.

d

�

1

�f

0

ramp

stepWeibull

Figure 3. Examples of the damage evolution laws implemented into the RVE plug-in.

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4. THE ’X’-PUCK PLUG-IN

In many cases the representation of damage on one material scale is sufficient and the RVE approach provesadequate. However, for certain composite classes, the damage evolution is a complex interplay of geometryand physics, characterized by the interaction of different damage mechanisms on various scales [15]. In thesematerials, single damage mechanisms act on their representative scale, but have the ability to activate orinhibit other mechanisms on the same or other length scales [16]. It is advantageous to describe mechanismsseparately by adequate models and to couple them indirectly, for example by adding the non-linear strainportions to calculate the macroscopic non-linear material behavior [17]. The embedment and interaction onthe laminate scale of single damage mechanisms and their evolution is the key point for failure predictions.

Many failure theories for fiber reinforced composites were proposed, from which the Puck theory appearsto be one of the best available currently, since it captures most features of the experimental results being,however, still simple enough to be applied in engineering design of laminates with optimum strength [18]. Fora detailed discussion on the theory we refer to [19, 20, 21]. In this article, we emphasize that Puck’s theory, asa combination of a micro-mechanical model with a phenomenological one, represents a compromise betweenphysical reality and applicability. For this reason, in our MMT we employ this theory as a framework forthe failure analysis on the structural scale, and enrich it by replacing phenomenological failure and evolutioncriteria by numerical results obtained from specific micro/meso-scale simulations.

Two basically independent fracture criteria are included: one for Fiber Fracture (FF) and one for Inter-FiberFracture (IFF), see Fig. 4(a). Fiber failure in an unidirectional (UD) composite can basically occur undertension or under compression by fiber kinking. This mechanism goes along with a strong energy release andloss of stiffness, especially under tension, since a large amount of energy is stored in the stiff fibers. Fiber failureis described by the maximum normal stress condition, hence for combined stress states, failure is assumed tooccur at the same fiber stress as under pure uniaxial stress σ1.

Inter-Fiber Fracture

shear (mode )B

�12�2

�f =0°

tensile (mode )A

�2�2

�f =0°

compressive (mode )C

�2

�2

�f

Fiber Fracture

compressive

�1

�1

tensile

�1

�1

12

3

(a) Failure modes of single plies

-150 -100 -50 0

-50

0

50

100

-10050

C

AB

B

�2 [MPa]

�1

2[M

Pa]

-Yc Yt

Rzz

A

S

-S

atan p12atan p12+-

(b) IFF envelope for σ1 = 0

Figure 4. Schematic representation of the failure modes defined in Puck’s theory and the failure envelope forthe Inter-Fiber Failure modes.

While FF is considered in a rather simple way, Puck’s theory for IFF is much more advanced, due to thecomplex nature of this damage. One reason is that failure is brittle without any preceding macroscopic plasticdeformation. Therefore, failure criteria adopted from yield criteria (e.g., von Mises or Hill criteria) renderuseless and one should focus on criteria for brittle fracture like Mohr, which states that fracture is exclusivelycreated by stresses acting on a fracture plane, that has to be calculated along with the stresses relative to thatplane. The IFF (σ2, τ21) envelope, defining the upper bound of the stress state, is described by three failuremodes A, B and C (Fig. 4(b)). This is due to the fact that the strength characteristics of the compression

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and tension domain (σ2 < 0 and σ2 > 0) are independent of each other. The plane with the highest effort isidentified as the ’action plane’ by the fracture plane angle θf , and fracture calculations are transformed intothis plane. For judging the significance of IFF one has to be careful to distinguish between the fairly harmless,tolerable modes A, B describing crack formation caused by transverse tensile stresses (σ2 > 0), in contrast tothe ’explosive’ effect that oblique Mode C cracks, caused by transverse compressive stresses (σ2 < 0) due totheir wedge shape (|θf | ≥ 30◦). All these aspects, which take care of the physical reality of the phenomena,are joined within a formulation of the criteria easy to use and which does not require too many empiricalparameters to be determined.

After the definition of failure envelopes, it is a necessary step to define the post-failure material behavior.Again a separate approach has to be used for FF and IFF. Puck proposed to abruptly reduce the stiffnessin fiber direction, once failure has occurred. This procedure is a strong non-linearity and leads to numericaldifficulties. A solution can be found in mitigation procedures [22, 23]. In the case of IFF, the degradationprocedure is more subtle since failed plies are bridged by neighboring plies, so that they still carry a respectableamount of load. However, the reduction of the ply stiffness leads to a load redistribution in the laminate andpossibly to further damage in the other plies. The degradation functions briefly discussed above represent themain tool for damage prediction. Therefore at this point, physical, micro-mechanical models are employedmost efficiently.

(a) Fiber Bundle Model (b) Transverse Crack Model (c) Shear Failure Model

Figure 5. The three DEM-based micro-mechanical models for the simulation of damage evolution.

We use three different micro-mechanical models based on statistical physics and solved by the Discrete ElementMethod (DEM), which allow a detailed analysis of specific constitutive behavior (Fig. 5). A Fiber Bundle Model(FBM) (Fig. 5(a)) is employed for determining the failure evolution of parallel systems under tension, such asfiber bundles [24]. A second model is the Transverse Crack Model (TCM) (Fig. 5(b)), which was designedto monitor the evolution and dynamic crack growth inside the plies and allows for the physical interpretationof the degradation to a damage state [25]. The third model is an FBM-based Shear Failure Model (SFM)(Fig. 5(c)) and has been expressly designed for reproducing the behavior of an interface between two solidblocks [26]. In general, the input parameters required by these methods involve quantities like the geometryof the model, some averaged dynamical parameters and the associated intrinsic disorder (which is defined byWeibull distributions), and the type of interactions (in particular for the FBM-based models). In any case,the output of these models are degradation laws representing the evolution of the mechanism-specific damageparameter (or, equivalently, a stiffness loss) as a function of the applied strain, which can be convenientlyemployed for replacing the experimentally derived degradation curves of Puck’s theory.

Puck’s theory has been implemented as an ABAQUS User Material (UMAT) for both 2D and 3D materialmodels along with a GUI allowing a user-friendly definition of the parameters required by this model. Thethree micro-mechanical DEM-based models are independent FORTRAN codes and are started by an additionalPython plug-in. The computation of the mechanism-specific stiffness reduction is performed before thestructural analysis is started. The resulting damage evolution laws are stored in tabular manner in predefinedfiles, which are accessed by the UMAT subroutine during the subsequent non-linear FEM computation.

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5. THE ’UNIFIED FORMULATION’ PLUG-IN

All runtime material processing tools like that based on Puck’s theory evaluate the failure criteria from thestress state obtained through a generally non-linear FE computation at each iteration step. As a consequence,the capability of the FE to adequately resolve the local stress field plays a fundamental role in failure analyses.In this section, we refer to linear FE analyses of transversely anisotropic composite plates. Laminated panelsare conveniently discretized with 2D elements in order to save computational time and, more important, toease local mesh refinements. Unfortunately, commonly available 2D elements for laminated plates/shells arebased on computationally extremely convenient models which in many cases fall short of meeting the necessaryaccuracy in the stress analysis [27]. The accuracy of 2D formulations for composite laminates depends notonly on the panel geometry (e.g., the thickness ratio), but also on a complex interplay of loading conditions,laminate stacking sequence and mechanical properties of the layer. Therefore, a thorough assessment of theerrors induced by the simplified 2D kinematics can be only made by means of reference solutions obtainedfrom expensive 3D analyses.

The third tool we prototypically implemented in the MMT attempts to close precisely this gap by providing a2D finite plate element formulation particularly dedicated to accurate stress analysis of composite laminates.Instead of a single finite element, a whole family of FEs has been implemented in the ABAQUS User Element(UEL) subroutine by means of the ’Unified Formulation’ (UF) previously introduced by Carrera (see, e.g., [28,29, 30]). The laminated plate theories engendered by the UF range from the most basic classical formulationsup to quasi-3D advanced formulations and, hence, permit an assessment of simplified models entirely withina 2D model space. In this article, we limit our scope to a summary of the main features of the ’UnifiedFormulation’ and refer to the appropriate literature for its detailed description, in particular to [31].

The most interesting feature of the UF consists in the capability of providing a series of hierarchically ordered2D plate elements. This is achieved by using an axiomatic thickness expansion whose polynomial order is afree parameter. Our implementation offers linear up to fourth-order thickness expansion of the variables. If theassumptions are introduced at the multilayered level, so-called Equivalent Single Layer (ESL) formulations areobtained, and the number of degrees of freedom (DOF) is independent from the number of layers constitutingthe laminate (Fig. 6(a)). In contrary, making assumptions for each lamina yields Layer-Wise (LW) or discrete-layer formulations, and the number of DOF depends on the number of layers present in the laminate [32](Fig. 6(b)). Both kinds of laminate descriptions are covered by the implemented UF-based elements.

h/2

-h/2

z

h /2k

-hk/2

zk

0�

k� 0

(a) Equivalent Single Layer (ESL)

h/2

-h/2

z

h /2k

-hk/2

zk

0�

k� 0

(b) Layer-Wise (LW)

Figure 6. Descriptions of a multilayered plate depending on the range of the axiomatic thickness assumptions.

An additional feature of the proposed implementation is the possibility to choose between ’classical’ displacement-based element formulations and ’advanced’ elements based on a partially mixed formulation expressly proposedby Reissner for an accurate stress analysis in composite laminates [33]. The so-called Reissner’s Mixed Varia-tional Theorem (RMVT) allows to make independent assumptions for the displacements and for the transversestress components and, hence, permits the a priori satisfaction of their interlaminar continuity. The additionaltransverse stress unknowns may be retained in the global stiffness matrix or they can be statically condensed

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at the element level [34]. In this case, a ’classical’ formulation with only displacement field components asnodal variables is obtained. In view of a full satisfaction of the so-called ’C0

z -Requirements’ [35] - which statethat, at a bi-material interface, the transverse stress and displacement fields should be continuous, while thethrough-thickness slope of the displacement should be discontinuous - a computationally efficient Zig-Zagfunction can be superimposed to an ESL displacement field [36, 37].

The generality of the UF-based elements is ensured by the use of full 3D constitutive equations at the laminalevel. The most common formulations, like Classical Laminate Theory (CLT) or First-order Shear DeformationTheory (FSDT), may be recovered from an ESL formulation with first-order thickness assumptions by simplypenalizing the transverse stress contributions alone (FSDT), or together with the transverse shear contributions(CLT). The well-known numerical issues associated to 2D element with a 3D constitutive law (thicknesslocking) can be dealt with by means of pragmatic countermeasures as proposed in [38]. The implementedfour-node element has an in-plane approximation with bilinear shape functions. Some remedies for shearlocking have been included in the implementation by resorting to the classical reduced and selectively reducedquadrature techniques for the spurious terms [39].

According to the MMT spirit, the UF-based elements have been implemented into ABAQUS in form of aplug-in, the ’Unified Formulation’ Plug-in (UFP). It consists of the subroutine UEL, a GUI which collects thenecessary inputs (geometry, material, boundary and loading condition, element type), a collection of Pythonscripts generating a complete model file and, finally, a post-processing program for the visualization of someresults. This latter tool has been conceived to work independently by exploiting the freeware Medit program[40]. As an alternative, a back-transformation of the elaborated results into the ABAQUS CAE format maybe performed, thus allowing a subsequent visualization within the CAE.

6. CONCLUSION

As we have addressed in this article, it is rarely possible to perform damage simulations for composites bylooking at only one scale. Thinking in terms of damage mechanisms and not in terms of particular solutions,it becomes possible to extend the FE analysis by a collection of tools to represent the material behavior onrespective scales. This idea has led to the concept of a Micro-Mechanisms Toolbox (MMT).

Having chosen the commercial FE package ABAQUS as a convenient basis platform, we embed tools into thesimulation procedure in form of plug-ins. This way, a user-friendly graphical user interface (GUI) serves as aguide for the users by supporting and limiting them in the selection of suitable tools during the preparationof the complete model. Exemplarily we implemented three different, prototypical plug-ins, namely the RVEplug-in, the ’X’-PUCK plug-in and the ’Unified Formulation’ plug-in (UFP). A detailed description of thetoolbox can be found in [41, 42].

Plug-ins and respective tools are not to be considered as stand-alone implementations of known theories, validfor certain material representation levels. The appeal of this approach is based on the possibility to mergethese different levels by combining the respective tools. To this aim, the MMT has been implemented in amodular way exploiting several different ABAQUS interfaces. This permits on the one hand to modify, extendand refine single tools with own implementations, and on the other hand to let several tools interact withinone simulation. As an example, starting from the material pre-processing with the RVE plug-in, the damageevolution may be enriched by the ’X’-PUCK plug-in, for which the local state is obtained from a higher-orderlaminate theory defined by the UFP.

We are aware of the limitations which characterize the present state of the MMT, further works are in factin progress. All proposed implementations are basically open-source and it is our hope that other researchersavail themselves of the advantage of such an approach and enrich the MMT by their own experience andmodels.

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REFERENCES

1. B. Cotterell: The Past, Present and Future of Fracture Mechanics, Eng. Fract. Mech. 69 (2002), p. 533-553.

2. J. Lemaitre and J.L. Chaboche: Mechanics of Solid Materials, Cambridge University Press (1990).

3. D. Krajcinovic and J. VanMier (Edts.): Damage and Fracture of Disordered Materials, Springer (2000).

4. J. Lemaitre: A Course on Damage Mechanics, 2nd. ed., Springer (1996).

5. H. J. Herrmann and S. Roux (Edts.): Statistical Models for the Fracture of Disordered Media, ElsevierScience Publishers B.V. (1990).

6. D. Krajcinovic: Damage Mechanics, in Mech. Mater. 8 (1989), p. 117–197.

7. D. Krajcinovic: Essential Damage Mechanics - bridging the scales, in O. Allix and F. Hild (Edts.): Conti-nuum Damage Mechanics of Materials and Structures, Elsevier (2002), p. 17–47.

8. ABAQUS Version 6.5: ABAQUS GUI Toolkit User’s Manual (2005).

9. D. Raabe: Computational Materials Science: The Simulation of Materials Microstructures and Properties,Wiley-VCH (1998).

10. D. Gross and T. Seelig: Bruchmechanik mit einer Einfuhrung in die Mikromechanik, Springer-Verlag, 3rded., Berlin (2001).

11. ABAQUS Version 6.5: Analysis User’s Manual, Vol. 3: Materials (2005).

12. M. Dong, S. Schmauder, T. Bidlingmaier and A. Wanner: Prediction of the Mechanical Behavior of ShortFiber Reinforced MMCs by Combined Cell Models, in Comput. Mater. Sci. 9 (1997), p. 121-133.

13. K. Zhu and S. Schmauder: Modellierung des Schadigungsverlaufs in Faserverbundwerkstoffen mit duktilerMatrix, SFB 381 report 2000-2002, Projekt C5 (2002), p. 221-224.

14. E. Jacquet, F. Trivaudey and D. Varchon: Calculation of the Transverse Modulus of a UnidirectionalComposite Material and of the Modulus of an Aggregate: Application of the Rule of Mixtures, in Com-pos. Sci. Tech. 60 (2000), p. 345-350.

15. R. Talreja (Edt.): Damage Mechanics of Composite Materials, Elsevier Science Ltd (1994).

16. O. Mauzac and R. Schirrer: Crack Propagation in Amorphous Brittle Polymers Toughened by RubberParticles: A Micro-Mechanical Study, in D. Baptiste (Edt.): Mechanics and Mechanisms of Damage inComposites and Multi-Materials, ESIS11, Mechanical Engineering Publications, London (1991), p. 81–93.

17. N. Weigel, D. Dinkler, B. Kroplin: Micromechanically Based Continuum Damage Mechanics Material Lawfor Fiber-Reinforced Ceramics, in Comput. Struct. 79 (2001), p. 2277-2286.

18. M. Hinton, A.S. Kaddour and P. Soden: Failure Criteria in Fibre Reinforced Polymer Composites: TheWorld Wide Failure Exercise, Elsevier (2004) or Compos. Sci. Tech. 58 (1998); 62 (2002); 64 (2004).

19. A. Puck: Festigkeitsanalyse von Faser-Matrix-Laminaten: Modelle fur die Praxis, Hanser Verlag 1996.

20. A. Puck and H. Schurmann: Failure Analysis of FRP Laminates by means of Phyiscally Based Phenomeno-logical Models, in Compos. Sci. Tech. 58 (7/1998), p. 1045-1067.

21. A. Puck and H. Schurmann: Failure Analysis of FRP Laminates by means of Phyiscally Based Phenomeno-logical Models, in Compos. Sci. Tech. 62 (2002), p. 1633-1662.

10

22. N.F. Knight, C.C. Rankin and F.A. Brogan: Controlling Progressive Failure Analyses Using ArtificialViscous Damping, in AIAA-2001-1181 (2001), p. 12-22.

23. N.F. Knight, C.C. Ranklin and F.A. Brogan: STAGS Computational Procedure for Progressive failureAnalysis of laminated Composite Structures, in Int. J. Non-Linear Mech. 37 (2002), p. 833-849.

24. S. L. Phoenix and B. J. Beyerlein: Statistical Strength Theory for Fibrous Composite Materials, in A. Kellyand C. Zweben (Edts.) Comprehensive Composite Materials, vol. 1, chap. 19, Pergamon Press (2000), p. 559–640.

25. F.K. Wittel, F. Kun, B.H. Kroplin and H.J. Herrmann: A Study of Transverse Ply Cracking using aDiscrete Element Method, in Comput. Mater. Sci. 28 (2003), p. 608-619.

26. F. Raischel, F. Kun, H.J. Herrmann: A Simple Beam Model for the Shear Failure of Interfaces, in Phys.Rev. E 72, 046126 (2005), cond-mat/0502122.

27. D. H. Robbins Jr and J. N. Reddy: Variable Kinematic Modeling of Laminated Composite Plates, inInt. J. Numer. Meth. Eng. 39 (1996), p. 2283–2317.

28. E. Carrera: Theories and Finite Elements for Multilayered Plates and Shells: A Unified Compact Formu-lation with Numerical Assessment and Benchmarking, in Arch. Comput. Meth. Eng. 10 (2003), p. 215–296.

29. A. Robaldo, E. Carrera and A. Benjeddou: Unified Formulation for Finite Element Thermoelastic Analysisof Multilayered Anisotropic Composite Plates, in J. Therm. Stresses 28 (2005), p. 1031–1065.

30. D. Ballhause, M. D’Ottavio, B. Kroplin and E. Carrera: A Unified Formulation to Assess MultilayeredTheories for Piezoelectric Plates, in Comput. Struct. 83 (2005), p. 1217–1235.

31. E. Carrera and L. Demasi: Classical and Advanced Multilayered Plate Elements based upon PVD andRMVT. Part 1: Derivation of Finite Element Matrices. Part 2: Numerical Implementations, in Int. J. Numer.Meth. Eng 55 (2002), p. 191–231,253–291.

32. J. N. Reddy: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed., CRCPress, Boca Raton, FL (2004).

33. E. Reissner: On a Certain Mixed Variational Theorem and a Proposed Application, in Int. J. Numer.Meth. Eng. 20 (1984), p. 1366–1368.

34. L. Demasi: Treatment of Stress Variables in Advanced Multilayered Plate Elements Based upon Reissner’sMixed Variational Theorem, in Comput. Struct. 84 (2006), p. 1215–1221.

35. E. Carrera: Historical Review of Zig-Zag Theories for Multilayered Plates and Shells, in Appl. Mech. Rev56 (2003), p. 287–308.

36. H. Murakami: Laminated Composite Plate Theory with Improved In-plane Response, in J. Appl. Mech.53 (1986), p. 661–666.

37. E. Carrera: On the Use of Murakami’s Zig-Zag Function in the Modeling of Layered Plates and Shells, inComput. Struct. 82 (2004), p. 541–554.

38. G. M. Kulikov and S. V. Plotnikova: Equivalent Single-Layer and Layerwise Shell Theories and Rigid-BodyMotions - Part I: Foundations, in Mech. Adv. Mater. Struct. 12 (2006), p. 275–283.

39. O. C. Zienkiewicz and R. L. Taylor: The Finite Element Method, Vol. 1: The Basis, Butterworth-Heinemann, 5th ed. (2000).

40. Medit <http://www.ann.jussieu.fr/∼frey/logiciels/medit.html> (2006).

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41. F. K. Wittel, M. D’Ottavio: An Open Toolbox for Damage Simulation, in G. Busse, B. Kroplin, F. K. Wittel(Edts.): Damage and its Evolution in Fiber-Composite Materials: Simulation and Non-Destructive Evalua-tion, ISBN 3-930683-90-3 (2006), p. 475-480.

42. M. D’Ottavio, F. K. Wittel and J. Reiser: Mechanism Based Toolbox: Prototypical Implementation ofTools, in G. Busse, B. Kroplin, F. K. Wittel (Edts.): Damage and its Evolution in Fiber-Composite Materials:Simulation and Non-Destructive Evaluation, ISBN 3-930683-90-3 (2006), p. 481-518.