March 2011 

NASA/TM–2011-217079 

Literature Reviews on Modeling Internal 

Geometry of Textile Composites and 

Rate-Independent Continuum Damage 

Su-Yuen Hsu 

Langley Research Center, Hampton, Virginia 
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March 2011 

NASA/TM–2011-217079 

Literature Reviews on Modeling Internal 

Geometry of Textile Composites and 

Rate-Independent Continuum Damage 

Su-Yuen Hsu 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1

Preface 

Textile composite materials have good potential for constructing composite structures 

where the effects of three-dimensional stresses are critical or geometric complexity is a 

manufacturing concern. There is a recent interest in advancing competence within LaRC for 

modeling the degradation of mechanical properties of textile composites. In an initial effort, two 

critical areas are identified to pursue: 

 Construction of internal geometry of textile composites 

 Rate-independent continuum damage mechanics 

If a micromechanical approach is chosen for modeling the degradation of properties, 

consistent definition of the volumes taken by the fiber tows inside a unit cell of a textile 

composite are needed before a micromechanical analysis of the material can start. On the other 

hand, textile composites usually exhibit degradation of mechanical properties before macroscopic 

cracks appear. This phenomenon can be dealt with by applying continuum damage mechanics at 

either the macroscopic level or the microscopic level. A complete micromechanical modeling of 

the degradation of textile composites additionally requires techniques that resolve discrete cracks, 

e.g., cohesive finite elements, which have been investigated and implemented by researchers at 

LaRC, among others. 

This memorandum documents reviews on the above two subjects. Various reviewed 

approaches are categorized, their assumptions, methods, and progress are briefed, and then 

critiques are presented. Each review ends with recommended research. 
2

A Review on Approaches to Constructing Internal Geometry of Textile Composites 

1. Introduction 

The feasibility of finite element damage analysis of unit cells of textile composite 

materials is debilitated by the geometric complexity of the internal material interface. The 

internal geometry of complex fabrics generated by the existing modeling methods in conjunction 

with all of the input geometric data (e.g., weave type and tow spacing) determined “prior to 

fabrication” is either substantially idealized or very expensive to obtain. It is tempting to create 

internal geometry that is suitable for the geometry-based meshing paradigm to create a finite 

element mesh that is conformal with the material interface. (Such a mesh is referred to as a 

“conformal mesh” below.) Up to the present, this undertaking is only possible by substantial 

idealization and even assuming substantial idealization, the task is usually arduous. Most finite 

element modeling of unit cells of fabric composites in the literature falls into this category. The 

difficulty arises from the contact among the fiber tows, which is ubiquitous in actual textile 

composites, and the complexity and large aspect ratio of the volumes taken by the matrix. 

Therefore, one may attempt to develop a special meshing algorithm to obtain a conformal mesh, 

e.g., the method proposed by Kim and Swan (2003b), to convert a voxel-based mesh to a 

conformal mesh. On the other hand, one may attempt to develop generalized finite element 

approaches that use a non-conformal mesh, e.g., the voxel-based mesh refinement (Kim and 

Swan, 2003a), the heterogeneous finite element analysis (Wang, 2003), the structured extended 

finite element method (Belytschko, Parimi, et al., 2003), the mesh superposition method (Nakai, 

Kurashiki, and Zako 2007), and the independent mesh method (Iarve, 2008). 

Whatever roadmap for achieving a finite element solution for the unit cell of a textile 

composite is taken, the first step is to build a geometric model that provides consistent definitions 

of the volumes taken by the fiber tows inside the unit cell. The present review first summarizes 

approaches for creating the internal geometry of textiles that do not involve simulation of the 

mechanical interaction among the fiber tows, and then presents approaches that involve 

simulation of the interaction. 

2. Creating Internal Geometry 

2.1. CAD-based Internal Geometry 

The internal geometry of fabrics has been generated without simulating the mechanical 

interaction among the interlacing fiber tows. Therefore, the internal geometry is essentially 

generated by CAD. Research emphasis in this area has been placed on building automated CAD 

procedures and special CAD libraries to improve efficiency in modeling that is based on the 

paradigm of geometry-based conformal mesh generation (e.g., Wentorf, Collar, et al., 1999). 

Since the mechanical interaction among the fiber tows is not simulated, some input geometric 

data have to be determined empirically or experimentally. 

Highly idealized geometric models in which each fiber tow has a constant elliptical or 

lenticular cross section have been employed in most finite element analyses of unit cells of fabric 

composites (e.g., Thom, 1999; Carvelli and Poggi, 2001; Whitcomb and Tang, 2001; Goyal and 

Whitcomb, 2006). Due to the strong idealizing assumption on geometry, these models require the 

smallest number of input geometric data as compared with other CAD-based geometric models. 
3

Hivet and Boisse (2005) developed an automated CAD procedure to create parametric solid 

models for two-dimensional (2-D) weaves (plain weave, twill, and satin). In the procedure, CAD 

solids, representing the yarns, are generated by sweeping along the yarn trajectories and blending 

the cross sections of the yarns. Several assumptions are made in order to generate yarn-to-yarn 

areal contact, without causing interpenetration, and other experimentally observed geometric 

characteristics. The cross sections of the yarns are represented as four-sided planes. Each side 

can be either straight or curved (parabola or arc). A pair of opposite sides are not in contact with 

any other yarn and this pair may degenerate to two points if a two-sided cross section is desired. 

The cross sections of a yarn are enveloped by a constant rectangle specified for that yarn. The 

shape of a contact side of a yarn is identical to the shape of the trajectory of the other overlapping 

yarn in the contact zone and vice versa. The trajectory in a contact-free zone is assumed to be 

straight in view of the small bending stiffness. In addition, if a yarn consecutively crosses over m 

yarns on the same side, the contact sides of the 2nd, 3rd..., (m-1)th yarns are straight in the contact 

zones. Asymmetry of a cross section about the vertical plane, which is orthogonal to the fabric 

plane and passes through the trajectory of the yarn, is allowed if symmetry cannot be inferred 

from the weave pattern. It is shown that the model requires three and seven input geometric data 

for balanced 2-D fabrics and unbalanced 2-D fabrics, respectively. 

In the work of Crookston, Ruijter, et al. (2007), each tow volume is defined with its 

centerline and several cross sections taken at different locations along the centerline. The 

centerline and cross sections are defined with splines. Both contact and interpenetration among 

the fiber tows are avoided by automatic adjustment of the local cross sections. Accordingly, the 

local fiber volume fractions are adjusted as well in order to obtain a consistent total amount of 

reinforcement. It should be noted that without taking into account the mechanical interaction 

among the fiber tows, there is always excessive freedom in adjusting a cross section. Crookston, 

Ruijter, et al. (2007) pointed out difficulties in automating CAD processes for creation of the 

internal geometry that includes contact among the fiber tows and is suitable for geometry-based 

conformal mesh generation. It should be noted that if contact is allowed, topology very probably 

changes from one tow to another and from one set of values of textile design parameters to 

another. In addition, creating geometry with complicated topology is prone to numerical 

problems. An example is creating intricate matrix volumes by performing boolean operations 

that involve contacting tows in a unit cell. 

2.2. Internal Geometry Created by Simulating Mechanical Interaction among Fiber Tows 

 Lomov, Huysmans, et al. (2001) and Verpoest and Lomov (2005) presented a method, 

which has been implemented in a textile geometry modeling program (WiseTex), to determine the 

centerlines and the cross sections of the yarns that are affected by the mechanical interaction. 

The cross section of a yarn is assumed to be either elliptical or lenticular. The cross-sectional 

height and width are experimentally determined nonlinear functions of the compressive 

transverse force acting on the yarn. Bending and longitudinal shear deformation of the fiber tows 

are assumed to be elastic. It should be noted that in reality, elasticity never dominates shear 

deformation. In the method, each tow is divided into “crimp intervals,” each of which represents 

a section of the tow between interlacing sites. Each crimp interval is assigned a deflection trial 

function with one unknown, “crimp height,” representing the deflection at one end of the crimp 

interval relative to the other end. The trial function is a geometrically-nonlinear approximate 

solution for a beam subjected to shear loads and the no-rotation condition at the ends of the beam 

and is nonlinear in terms of the crimp height. The elastic bending energy stored in a crimp 

interval can be derived from the deflection function in conjunction with the experimentally 
4

measured curvature-dependent bending rigidity, so that the bending energy is a nonlinear function 

of the crimp height. According to the geometrically-linear-theory, the bending energy stored in a 

crimp interval is equal to one half of the product of the shear load and the crimp height. By 

assuming that this relation holds for the energy derived from the geometrically-nonlinear 

deflection, the shear load on a crimp interval is expressed as a nonlinear function of the crimp 

height. Furthermore, the compressive transverse force exerted on a crimp interval by its 

interlacing tows is approximated by the shear load, so the cross-sectional height of a crimp 

interval becomes a nonlinear function of the shear load. Linear kinematic constraints in terms of 

the cross-sectional heights and crimp heights are set up to ensure deflection continuity for each 

tow and point-contact between interlacing tows. Essentially, these constraints can be reduced to 

nonlinear equations in terms of the crimp heights. Numerical iteration is performed between 

minimizing the total bending energy and satisfying the constraints in order to solve for the 

unknown crimp heights, from which all the other variables can be determined. Once the local 

cross sections are determined, the complete yarn geometry can be reconstructed by spline 

approximation. A very short computational time for a complicated unit cell has been reported. In 

the work of Lomov and Verpoest (2006), the above method is applied to simulate deformation of 

the yarns of dry fabrics subject to tension. Friction among the interlacing tows is incorporated 

into the method to simulate the response to shear loading. Such characterization is important to 

prediction of fabric draping. As discussed by Lomov, Ivanov, et al. (2007), the simplified yarn 

deformation and contact in the method may impair fidelity and result in interpenetration among 

fiber tows. Conceivably, the problem is serious for fabrics where some fiber tows are locally 

sandwiched by others or have substantially nonsymmetrical cross sections, e.g., dense fabrics, 

three-dimensional (3-D) fabrics, and fabrics with non-orthogonal yarns. An ad hoc finite-element 

based mesh deformation procedure for eliminating the interpenetration was presented by Lomov, 

Ivanov, et al. (2007). In addition to the ad hoc nature, the procedure has a limitation that it results 

in no contact between the fiber tows. 

 Zhou, Sun, and Wang (2004) developed the so-called multi-chain digital element 

approach to simulate textile fabrication (such as weaving and braiding) and forming in order to 

predict the internal geometry of textiles. A digital element is a one-dimensional (1-D) rod finite 

element (having only the axial stiffness) and a chain is a serial assembly of digital elements. In a 

multi-chain digital element model, each fiber yarn is modeled as a bundle of chains and node-tonode contact elements are dynamically added or removed to simulate frictional contact between 

chains. Thus, a textile fabrication or deformation process, being treated as a geometrically 

nonlinear static contact problem, can be analyzed by using an implicit finite element code. The 

analysis yields the movement of the chains relative to one another, which determines the final 

yarn geometry. Note that the number of chains used for a fiber yarn defines the upper bound for 

the cross-sectional aspect ratio of the deformed yarn. Step-by-step simulations of a 2-D weaving 

process and of a 3-D braiding process were demonstrated by Zhou, Sun, and Wang (2004) and 

simulation of vacuum bag compression of randomly nested layers of a 2-D braided preform was 

demonstrated by Zhou, Mollenhauer, and Iarve (2008). Excellent correlation between the internal 

geometry that is predicted by the 2-D weaving and the 3-D braiding simulations, and the 

experimentally observed geometry is shown in the work of Miao, Zhou, et al. (2008). Their 

numerical study also addresses a major concern of the computational time required for the 

simulation using the multi-chain digital element approach. It is shown that the predicted 

geometry for the numerical examples obtained by using about 20 chains per fiber yarn has quality 

similar to that obtained by using more chains. It is also shown that the time-consuming step-bystep simulation of the fabric processes in the examples is not necessary to obtain the final internal 

geometry. A much shortened simulation that starts with a specified interlacing pattern (topology) 

and fiber yarns having circular cross sections and omits the effect of the chain-to-chain friction 

yields nearly the same result. This finding suggests that the final internal geometry is primarily 
5

determined by the interlacing topology, the initial cross-sectional areas of the yarns, and the 

boundary conditions. Detailed information about the model size and computational time is not 

provided in the paper. In a continuum mechanics analysis of a textile composite, each tow is 

modeled as a solid entity of a unidirectionally continuous fiber reinforced composite. Note that a 

bundle of deformed chains, modeling a deformed yarn, is not a solid; therefore, the bundle cannot 

be used directly to create a solid mesh for the deformed yarn. Zhou, Mollenhauer, and Iarve 

(2008) presented a procedure to construct a solid approximating a bundle of deformed chains so 

that a solid finite element mesh can be created subsequently. First, cross sections of a chain 

bundle are taken at various locations along the yarn path, including its two ends, and each cross 

section is approximated by a polygon. Then, the lateral surface of a yarn can be represented by a 

set of triangular facets which are obtained by connecting the vertices of the cross-sectional 

polygons. Finally, a solid can be defined with the lateral surface and appropriate boundaries of 

the unit cell. It should be noted that the resulting solid geometry has to be adjusted in order to 

eliminate interpenetration or to ensure well-defined contact among the fiber yarns. 

 Durville (2005) modeled a fiber with special 1-D large-strain beam finite elements 

connected in series. By assigning nine degrees of freedom to each node, the kinematic 

formulation of the beam element allows not only the axial strain, bending strains, and transverse 

shear strains but also the cross-sectional planar stains. If a bundle of fibers, instead of a single 

fiber, is modeled as a series of the beam elements, then a yarn can be modeled as a bundle of 

beam element series. Under the circumstances, the incorporation of the cross-sectional planar 

stains should conduce to a reduction in the number of beam element series needed to model the 

yarn. A “symmetric” contact-point search algorithm that uses the normal direction to an 

“intermediate geometry” as the search direction, instead of traditionally the normal direction to 

either one of two contact candidates, is employed in order to more accurately capture contact 

between two contact candidate intervals of high-curvature fiber-like structures. The intermediate 

geometry is defined as the geometric average of the contact candidate intervals. A desired 

number of contact elements per unit length of the intermediate geometry are created between the 

identified contact points on a “fiber” and those on the other contacting “fiber,” not necessarily 

between nodes. In general, Durville’s contact algorithm is an improvement from the multi-chain 

digital element approach. Durville (2007) ran an implicit static finite element code on a cluster of 

six processors to predict the internal geometry of a relaxed plain weave and its internal 

deformation in response to external shearing, bending, and twisting. In the work, each yarn is 

modeled with 28 series of the beam elements and the internal geometry is obtained without 

following the weaving kinematics. The CPU time spent on an analysis is reported to be a couple 

of days. 

3. A Recommended Research Approach to Constructing Internal Geometry of Textile 

Composites Suitable for Damage Analysis 

 CAD-based internal geometry is adequate for predicting the effective properties of textile 

composites. Highly-idealized CAD-based geometric models are not suitable for the analysis of 

damage evolution, which is sensitive to local geometric details. For some 2-D fabrics, with 

relatively simple internal geometry, more realistic CAD-based models in conjunction with all of 

the input geometric data determined prior to fabrication may be sufficient for damage analysis. 

For fabrics with more complicated internal architecture, sophisticated CAD-based modeling is 

prone to robustness problems and involves input geometric data that have to be determined 

experimentally. Suitable internal geometry for damage analysis can be obtained by modeling the 

tows as bundles of series of 1-D finite elements in a contact analysis. However, the numerical 

implementation is too time-consuming for practical implementation, even if the step-by-step 
6

textile process is omitted. A potential solution to this problem recommended herein is to model 

the tows, arranged according to the desired interlacing topology, with 3-D finite elements in a 

surface-to-surface contact analysis. A transversely isotropic elastoplastic material model is used 

for the tows and a pressure-dependent yield strength is used to simulate the effect of fiber-to-fiber 

friction. As suggested by Miao, Zhou, et al. (2008), each fiber tow may be assigned a constant 

cross section when building the initial geometry. It is recommended herein that suitable initial 

cross sections are chosen to avoid interpenetration and subsequently the cross-sectional areas are 

increased in a controlled manner during the simulation of contact. The controlled increase in 

cross-sectional area is achieved by using a fictitious coefficient of transverse thermal expansion 

for the tows. The proposed approach should have another time-saving advantage over the 

approaches using bundles of series of 1-D finite elements because solids are directly obtained 

from the proposed simulation. 

References 

Belytschko, T., Parimi, C., Moës, N., Sukumar, N., Usui, S., 2003. Structured Extended Finite 

Element Methods for Solids Defined by Implicit Surfaces. International Journal for 

Numerical Methods in Engineering 56, 609-635. 

Carvelli, V., Poggi, C., 2001. A Homogenization Procedure for the Numerical Analysis of Woven 

Fabric Composites. Composites: Part A 32, 1425-1432. 

Crookston, J.J., Ruijter, W., Long, A.C., Warrior, N.A., Jones, I.A., 2007. A Comparison of 

Mechanical Property Prediction Techniques Using Conformal Tetrahedra and Voxel-Based 

Finite Element Meshes for Textile Composite Unit Cells. Proceedings, Finite Element 

Modeling of Textiles and Textile Composites, Saint-Petersburg, Russia. 

Durville, D., 2005. Numerical Simulation of Entangled Materials Mechanical Properties. Journal 

of Materials Science 40, 5941-5948. 

Durville, D., 2007. Finite Element Simulation of Textile Materials at Mesoscopic Scale. 

Proceedings, Finite Element Modeling of Textiles and Textile Composites, Saint-Petersburg, 

Russia. 

Goyal, D., Whitcomb, J.D., 2006. Analysis of Stress Concentrations in 2 X 2 Braided Composites. 

Journal of Composite Materials 40, 533-546. 

Hivet, G., Boisse, P., 2005. Consistent 3D Geometrical Model of Fabric Elementary Cell, 

Application to a Meshing Preprocessor for 3D Finite Element Analysis. Finite Elements in 

Analysis and Design 42, 25-49. 

Iarve, E., Zhou, E.G., Whitney, T.J., 2008. Independent Mesh Method Based Prediction of Local 

and Volume Average Fields in Textile Composites. AIAA 2008-1869. 

Kim, H.J., Swan, C.C., 2003a. Voxel-based Meshing and Unit-cell Analysis of Textile 

Composites. International Journal for Numerical Methods in Engineering 56, 977–1006. 

Kim, H.J., Swan, C.C., 2003b. Algorithms for Automated Meshing and Unit Cell Analysis of 

Periodic Composites with Hierarchical Tri-quadratic Tetrahedral Elements 58, 1683–1711. 

Lomov, S.V., Huysmans, G., Luo, Y., Parnas, R.S., Prodromou, A., Verpoest, I., Phelan, F.R., 

2001. Textile Composites: Modelling Strategies. Composites: Part A 32, 1379-1394. 

Lomov, S.V., Verpoest, I., 2006. Model of Shear of Woven Fabric and Parametric Description of 

Shear Resistance of Glass Woven Reinforcements. Composites Science and Technology 66, 

919–933. 

Lomov, S.V., Ivanov, D.S., Verpoest, I., Zako, M., Kurashiki, T., Nakai, H., Hirosawa, S., 2007. 

Meso-FE Modelling of Textile Composites: Road Map, Data Flow, and Algorithms. 

Composites Science and Technology 67, 1870–1891. 

Miao, Y., Zhou, E., Wang, Y., Cheeseman, B.A., 2008. Mechanics of Textile Composites Microgeometry. Composites Science and Technology 68, 1671–1678. 
7

Nakai, H., Kurashiki, T., Zako, M., 2007. A Modeling Technique of Textile Composites with 

Individual Models by the Mesh Superposition Method. Proceedings, Finite Element 

Modeling of Textiles and Textile Composites, Saint-Petersburg, Russia. 

Thom, H., 1999. Finite Element Modeling of Plain Weave Composites. Journal of Composite 

Materials 33, 1491-1510. 

Verpoest, I., Lomov, S.V., 2005. Virtual Textile Composites Software WiseTex. Composites 

Science and Technology 65, 2563–2574. 

Wang, Y., 2003. Micro-stress and Failure Analysis of Textile Composites. AFRL-SR-AR-TR-03-

0365. 

Wentorf, R., Collar, R., Shephard, M.S., Fish, J., 1999. Automated Modeling for Complex 

Woven Mesostructures. Computer Methods in Applied Mechanics and Engineering 172, 273-

291. 

Whitcomb, J., Tang, X., 2001. Effective Moduli of Woven Composites. Journal of Composite 

Materials 35, 2127-2144. 

Zhou, G., Sun, X., Wang, Y., 2004. Multi-chain Digital Element Analysis in Textile Mechanics. 

Composites Science and Technology 64, 239–244. 

Zhou, E.G., Mollenhauer, D., Iarve, E., 2008. Micro-geometric Modeling of Textile Preforms 

with Vacuum Bag Compression: An Application of Multi-chain Digital Element Technique. 

AIAA 2008-1868. 
8

A Review on Rate-Independent Continuum Damage Mechanics 

1. Continuum Damage Mechanics Applied to Four Categories of Material Response 

There have been noticeable advancements in continuum damage mechanics (CDM) over 

the past two decades. Rate-independent CDM theories are intended to describe progressive loss 

of stiffness of a rate-independent material as a result of the loading-induced initiation, growth, 

and coalescence of distributed cracks “inside” the material until the discrete nature of dominating 

cracks cannot be ignored for a structural mechanical problem in hand. 

CDM theories have been developed to model the effects of anisotropic damage on the 

elastic response of materials with initial isotropy (e.g., Chow and Wang, 1987; Valanis, 1990; 

Tang, Jiang, et al., 2002), the elastoplastic response of materials with initial isotropy (e.g., Ju, 

1989; Chow and Chen, 1992; Wu and Nanakorn, 1999; Voyiadjis and Park, 1999), the elastic 

response of continuous-fiber reinforced composites (e.g., Talreja, 1991; Sørensen and Talreja, 

1993; Maire and Chaboche, 1997; Chaboche and Maire, 2001; Varna, Joffe, et al., 2001; 

Williams and Vaziri, 2001; Talreja, 2005; Hassan, 2005; Camanho, Maimí, et al., 2007), and the 

elastoplastic response of continuous-fiber reinforced composites (e.g., Voyiadjis and Deliktas, 

2000; Maa and Cheng, 2002). 

The present review covers the above four types of damaged material response, and 

focuses only on the rate-independent material behavior. 

2. Modeling Effects of Damage 

Many CDM models incorporate the effects of damage by using the concept of effective 

stress, which is usually related to the nominal (macroscopic) stress through a single fourth-order 

damage tensor. For the case of isotropic damage, the fourth-order damage tensor reduces to a 

single scalar damage variable. Other models do not employ the concept of effective stress, but 

instead directly adjust the elastic moduli by using a set of scalar damage variables, or define 

damage in terms of micromechanical quantities. 

2.1. Modeling Effects of Damage by Using the Concept of Effective Stress 

A majority of CDM models either explicitly or implicitly employ the concept of effective 

stress, which is representative of the internal stress state of a damaged material. Damage causes 

reduction in the load-carrying cross-sectional area of material. The traction acting on a reduced 

cross section is calculated from the effective stress tensor by using Cauchy’s formula. It is 

usually assumed that the effective stress is a linear tensorial function of the nominal stress. 

Referring to a Cartesian coordinate system, this relation is written as follows: 

 ij  Mijkl kl

 ij  ijkl kl
9

where  ij is the effective stress tensor,  kl is the nominal stress, and Mijkl is the damage effect 

tensor, and  ijkl is the integrity tensor. The integrity tensor can be rewritten as 

ijkl  ik jl  il jk  Dijkl ( ) 2

1     

where Dijkl is called the fourth-order damage tensor. 

2.1.1. The Role of the Fourth-Order Damage Tensor 

Some researchers directly used the fourth-order damage tensor, Dijkl , as a model variable 

(e.g., Tang, Jiang, et al., 2002; Ju, 1989; Williams and Vaziri, 2001), although its microscopic 

geometric meaning is not clear in general. A special case was given by Tang, Jiang, et al. (2002), 

where Dijkl is related to volume fractions of orthotropically distributed needle-shaped microvoids. 

Many researchers (e.g., Chow and Wang, 1987; Valanis, 1990; Chow and Chen, 1992; 

Wu and Nanakorn, 1999; Voyiadjis and Park, 1999; Maire and Chaboche, 1997; Chaboche and 

Maire, 2001; Voyiadjis and Deliktas, 2000) employed heuristic construction of the fourth-order 

damage effect tensor (or damage tensor) from a symmetric second-order damage tensor which 

characterizes reduction in load-carrying cross-sectional area of material. Although the 

construction of the fourth-order damage effect tensor in these approaches is heuristic and it 

involves ad hoc symmetrization in order to ensure the symmetry of the effective stress, the 

second-order damage tensor is widely used because of its clear microscopic geometric meaning, 

which was described by Voyiadjis and Deliktas (2000) by constructing a fictitious undamaged 

configuration through coordinate transformation. In the work of Voyiadjis and Deliktas (2000), it 

is shown that the fictitious configuration has a reduced volume due to the elimination of damage. 

Jirasek (2007) pointed out that the formulations based on the symmetric second-order tensor are 

limited to orthotropic damage whose principal directions may change along the loading history. 

2.1.2. Relating Damage to Mechanical Behavior 

The elastic stiffness of a damaged material can be expressed as a function of the fourthorder damage tensor or its degenerates by adopting either the hypothesis of strain equivalence or 

the hypothesis of elastic strain energy equivalence. 

2.1.2.1. The Hypothesis of Strain Equivalence 

The effective-stress based CDM models developed by Tang, Jiang, et al. (2002), Ju 

(1989), Maire and Chaboche (1997), Chaboche and Maire (2001), and Williams and Vaziri (2001) 

are constructed with the aid of the hypothesis of strain equivalence proposed by Lemaitre and 

Chaboche (1978). The hypothesis states that the nominal stress applied to a damaged material 

produces the same strain as the effective stress applied to the fictitious undamaged material. This 

approach leads to a non-symmetric elastic stiffness tensor as follows: 

0 0 0 Cijkl  ijmnCmnkl  Cijkl  DijmnCmnkl
10

where Cijkl and 0 Cijmn are the damaged elastic stiffness and undamaged elastic stiffness, 

respectively. Therefore, ad hoc treatment has to be performed on Cijkl to obtain the symmetry 

(Maire and Chaboche, 1997; Chaboche and Maire, 2001; Williams and Vaziri, 2001). 

Conceivably, an alternate approach is to directly use symmetric Cijkl as the damage variable 

(Tang, Jiang, et al., 2002; Ju, 1989). It is worth noting that the hypothesis of strain equivalence 

leads to reductions, in the axial elastic moduli, proportional to the respective axial damages when 

the principal directions of Dijkl are aligned with the material principal axes. 

2.1.2.2. The Hypothesis of Elastic Strain Energy Equivalence

The effective-stress based CDM models in the works of Chow and Wang (1987), Valanis 

(1990), Chow and Chen (1992), Wu and Nanakorn (1999), Voyiadjis and Park (1999), and 

Voyiadjis and Deliktas (2000) are constructed with the aid of the hypothesis of elastic strain 

energy equivalence proposed by Cordebois and Sidoroff (1979). The hypothesis states that the 

density of strain energy stored in a damaged material subject to the nominal stress is equal to the 

density of strain energy stored in the fictitious undamaged material subject to the effective stress. 

This approach yields a symmetric elastic stiffness tensor. It is noted that the afore-mentioned 

volume reduction (Voyiadjis and Deliktas, 2000) does not enter the calculation of stain energy 

density and an anomaly may ensue. For example, consider uniaxial damage of a “material” 

consisting of a bundle of parallel elastic fibers subject to the nominal tensile stress  . Let E

denote the Young’s modulus of the intact material and d denote the fraction of broken fibers due 

to the applied load. Consequently the effective stress is  /1 d and the strain energy density of 

the undamaged material is 2 2  / 2E(1 d ) . Using the hypothesis of strain energy equivalence 

yields the Young’s modulus of the damaged material being 2 E(1 d ) , which is less than the 

expected value E(1 d ) . The latter coincides with the prediction by using the hypothesis of 

strain equivalence. It should be noted that the distinctive factor 1 d is exactly the volume 

fraction of the undamaged material. 

2.2. Modeling Effects of Damage without the Concept of Effective Stress

Some CDM models directly characterize engineering elastic parameters without referring 

to the concept of effective stress (e.g., Camanho, Maimí, et al., 2007; Maa and Cheng, 2002). 

The CDM models in the papers of Camanho, Maimí, et al. (2007) and Maa and Cheng (2002) for 

unidirectional composite laminas subject to plane stress assume that the original principal axes of 

material are retained after damage, only the in-plane axial elastic moduli are reduced, and the 

reductions are proportional to the respective axial damages. Scalar damage variables can be 

defined solely for the purpose to facilitate formulation of the reductions. 

It is possible to define damage variables in terms of micromechanical quantities. In the 

works of Talreja (1990, 1991), Sørensen and Talreja (1993), Varna, Joffe, et al. (2001), Talreja 

(2006), and Singh and Talreja (2008), strain-like second-order tensors are defined in terms of 

micromechanical quantities (such as crack density, crack-face displacement discontinuity per unit 

of applied macro-strain) for damage modes of matrix cracking normal to fibers, fiber-matrix 

interfacial debonding, and interfacial slip in unidirectional ceramic matrix composites, and 

through-the-ply-thickness transverse cracking in polymeric matrix composite laminas. In the 
11

work of Hassan (2005), three micromechanical scalar damage variables are adopted to represent 

the fractions of ineffective broken fibers due to longitudinal loading, cracked matrix due to 

transverse loading, and fiber-matrix interfacial debonding due to longitudinal shear, respectively, 

in unidirectional AS4/PEEK composite laminae. A simple rule-of-mixtures micromechanical 

analysis is employed in the work of Hassan (2005) to derive relations between engineering elastic 

parameters and the scalar damage variables. The results show that only the in-plane axial elastic 

moduli are reduced and the reductions are nearly proportional to the respective damage variables. 

3. Modeling Evolution of Damage

With the elastic stiffness expressed as a function of damage variables, CDM models 

characterize the reduction in the elastic stiffness by tracking the evolution of the damage variables 

as a result of a loading history. To this end, damage evolution equations must be proposed in any 

CDM theory. 

3.1. Modeling Evolution of Damage within the Framework of Thermodynamics 

Damage variables used for a macroscopic description of a damage process are indeed 

internal (or hidden) variables. To describe more complex material behavior, additional internal 

variables may be required. Like any other constitutive theory with internal variables, CDM 

models with internal variables must be formulated in a way such that the second law of 

thermodynamics is satisfied by any admissible change in the material state. Therefore, for such 

CDM models, the damage evolution equations must be derived within the framework of 

thermodynamics. 

3.1.1. Clausius-Duhem Inequality and Internal Variables 

With selected damage variables, CDM constitutive equations are usually derived from or 

devised to satisfy the Clausius-Duhem inequality. The formulation starts with postulating a 

functional form for the Helmholtz free energy or Gibbs free energy. If the material is elastic, a 

free energy function can be readily obtained with a known relation between the elastic stiffness 

(or compliance) and damage variables (e.g., Valanis, 1990; Tang, Jiang, et al., 2002; Maire and 

Carboche, 1997; Chaboche and Maire, 2001; Williams and Vaziri, 2001; Hassan, 2005; Camanho, 

Maimí, et al., 2007). If the material exhibits plastic behavior, the free energy also depends on 

additional internal plasticity variables that complement characterization of the plastic state, and 

may further depend on other internal damage variables that complement characterization of the 

damage state (e.g., Ju, 1989; Chow and Chen, 1992; Wu and Nanakorn, 1999; Voyiadjis and 

Deliktas, 2000; Maa and Cheng, 2002). For this case, the construction of a free energy function 

may be guided by the choice of the hypothesis of strain equivalence or strain energy equivalence 

(e.g., Ju, 1989; Chow and Chen, 1992; and Wu and Nanakorn, 1999). Having an assumed free 

energy function, one may proceed to identify the thermodynamic force conjugate to a damage or 

plasticity variable as the derivative of the free energy with respective to that variable. 

3.1.2. Derivation of Damage Evolution Equations 

Once the free energy, internal variables, and their conjugates are established, damage 

evolution equations are usually derived from either the hypothesis of maximum dissipation or the 
12

so-called linear thermodynamic law. Both arguments are sufficient conditions for the ClausiusDuhem inequality. 

3.1.2.1. Hypothesis of Maximum Dissipation 

A majority of CDM models require a damage potential (damage criterion) as a function 

of the thermodynamic forces conjugate to the damage variables and if plasticity is involved, a 

plasticity potential (yield criterion) as a function of the thermodynamic forces conjugate to the 

plasticity variables. Then, the hypothesis of maximum dissipation under the constraint of a 

damage criterion and additionally under the constraint of a yield criterion, if plasticity is involved, 

may be invoked to obtain “flow rules,” which state that increments of model variables are 

proportional to gradients of the potentials with Lagrange multipliers as the proportionality factors. 

Finally, evolution equations are derived from combining the “flow rules” and consistency 

conditions, which are inferred from the damage and yield criteria (e.g., Tang, Jiang, et al., 2002; 

Ju, 1989; Maire and Carboche, 1997; Chaboche and Maire, 2001; Voyiadjis and Deliktas, 2000). 

The CDM models proposed by Williams and Vaziri (2001), Hassan (2005), Camanho, Maimí, et 

al., (2007), and Maa and Cheng (2002) for unidirectional composite laminas only take into 

account degradation of the in-plane axial elastic moduli. Therefore, for these cases, the above 

thermodynamics-based methodology merely leads to a condition that the damage variables are 

monotonically increasing along the loading history, and optional use of the thermodynamic force 

conjugates for formulating the damage criterion (or yield criterion) and evolution equations. 

3.1.2.2. Linear Thermodynamic Law

Instead of the constrained maximization of dissipation, the linear thermodynamic law that 

the rate of change of a thermodynamic variable is proportional to the conjugate thermodynamic 

force can be used to devise evolution equations (e.g., Valanis, 1990; Chow and Chen, 1992; Wu 

and Nanakorn, 1999). From a theoretical standpoint, this concise approach does not require a 

damage criterion or yield criterion. A suitable equation governing evolution of a model variable 

can be contrived by judiciously implementing the linear thermodynamic law so that the 

dissipation due to an increment of the variable is positive definite. 

3.2. Modeling Evolution of Directly Measurable Damage Variables

A polynomial expansion of the free energy is used in the works of Talreja (1990, 1991), 

where damage variables are defined in terms of micromechanical quantities. As dictated by 

classical linear elasticity theory, the polynomial expansion is quadratic in strain components. On 

the other hand, the researchers’ prior experience suggests that a linear expansion in the damage 

variables as well as products of some paired damage variables (accounting for coupled damage 

mechanism such as fiber-matrix interfacial slip in conjunction with matrix cracking) should be 

adequate. The damage-stress-strain relationship with indeterminate coefficients for the 

polynomial terms is derived by differentiating the free energy with respective to strains. In the 

subsequent works of Sørensen and Talreja (1993), Varna, Joffe, et al. (2001), Talreja (2006), and 

Singh and Talreja (2008), a procedure involving experimental measurement of micromechanical 

quantities (e.g., crack density) or a combination of experimental measurement and 

micromechanical analysis (e.g., crack-opening displacement) is used to evaluate the coefficients 

and evolution of damage variables for some simple loading conditions. Interestingly, the 
13

procedure takes advantage of the micromechanical root of the damage variables and thus does not 

resort to the second law of thermodynamics in the macroscopic form. 

4. A Recommended Research Approach to Modeling Degradation of Elastic Properties 

It is well known that effective elastic properties of a continuum with periodically 

distributed micro-cracks can be computed with good accuracy by using a micromechanical model 

containing a small number of degrees of freedom. This suggests that the concept of effective 

stress in CDM could be generalized and incorporated into the common stress averaging scheme 

of micromechanics. Therefore, one might characterize damage with internal volumes (say, 

“ineffective volumes”) where stress is reduced due to degradation, and then proceed with 

micromechanical formulation, which is particularly useful to model inhomogeneous materials. 

The tensorial representation of damage is recommended not only because of its operational 

conciseness but also its ability to capture desired physical meaning. It should be noted that an 

ineffective volume has attributes such as size, shape, and orientation. The thermodynamic 

framework has proven to be a successful tool for modeling mechanical behavior of continuums. 

From a perspective of developing constitutive models for the purpose of computational structural 

analysis, the thermodynamic approach is necessary as complicated interaction among microdamages cannot be realistically or efficiently simulated merely by micromechanical analysis. 

References 

Camanho, P.P., Maimí, P., Dávila, C.G., 2007. Prediction of size effects in notched laminates 

using continuum damage mechanics. Composites Science and Technology 67, 2715-2727. 

Chaboche, J.L., Maire, J.F., 2001. New progree in micromechanics-based CDM models and their 

application to CMCs. Composites Science and Technology 61, 2239-2246. 

Chow, C.L., Wang, J., 1987. An anisotropic theory of elasticity for continuum damage mechanics. 

International Journal of Fracture 33, 3-16. 

Chow, C.L., Chen, X.F., 1992. An anisotropic model of damage mechanics based on endochronic 

theory of plasticity. International Journal of Fracture 55, 115-130. 

Cordebois, J.P., Sidoroff, F., 1979, Damage-induced elastic anisotropy. In Boehler, J.P. (Ed.), 

Comportement Mécanique des Solides Anisotropes, Martinus Nijhoff Publisher, 19-22. 

Hassan, N.M., 2005. Damage development in static and dynamic deformation of fiber-reinforced 

composite plates. Ph.D. dissertation, Virginia Polytechnic Institute and State University, 

Blacksburg, Virginia. 

Jirásek, M., 2007. Modeling of localized inelastic deformation. Short course lecture notes, Czech 

Technical University, Prague. 

Ju, J.W., 1989. On energy-based coupled elastoplastic damage theories: constitutive modeling 

and computational aspects. International Journal of Solids and Structures 25, 803-833. 

Lemaître, J., Chaboche, J.L., 1978. Aspect phénoménologique de la rupture par endommagement. 

Journal de Mécanique Appliquée 2, 317-365. 

Maa, R.H., Cheng, J.H., 2002. A CDM-based failure model for predicting strength of notched 

composite laminates. Composites: Part B 33, 479-489. 

Maire, J.F., Chaboche, J.L., 1997. A new formulation of continuum damage mechanics (CDM) 

for composite materials. Aerospace Science and Technology 4, 247-257. 

Tang, X.S., Jiang, C.P., Zheng, J.L., 2002. Anisotropic elastic constitutive relations for damaged 

materials by application of irreversible thermodynamics. Theoretical and Applied Fracture 

Mechanics 38, 211-220. 
14

Valanis, K.C., 1990. A theory of damage in brittle materials. Engineering Fracture Mechanics 36, 

403-416. 

Voyiadjis, G.Z., Park, T., 1999. Kinematics description of damage for finite strain plasticity. 

Journal of Engineering Science 37, 803-830. 

Wu, H.C., Nanakorn, C.K., 1999. A constitutive framework of plastically deformed damaged 

continuum and a formulation using the endochronic concept. International Journal of Solids 

and Structures 36, 5057-5087. 

Sørensen, B.F., Talreja, R., 1993. Analysis of damage in a ceramic matrix composite. 

International Journal of Damage Mechanics 2, 246-271. 

Talreja, R., 1990. Internal variable damage mechanics of composite materials. In Boehler, J.P. 

(Ed.), Yielding, Damage, and Failure of Anisotropic Solids, Mechanical Engineering 

Publications, London, 509–533. 

Talreja, R., 1991. Continuum modeling of damage in ceramic matrix composites. Mechanics of 

Materials 12, 165-180. 

Talreja, R., 2006. Damage analysis for structural integrity and durability of composite materials. 

Fatigue and Fracture of Engineering Materials and Structures 29, 481-506. 

Singh, C.V., Talreja, R, 2008. Analysis of multiple off-axis ply cracks in composite laminates. 

International Journal of Solids and Structures 45, 4574-4589. 

Varna, J., Joffe, R., Talreja, R., 2001. A synergistic damage-mechanics analysis of transverse 

cracking in [θ/904]S laminates. Composites Science and Technology 61, 657-665. 

Voyiadjis, G.Z., Deliktas, B., 2000. A coupled anisotropic damage model for the inelastic 

response of composite materials. Computational Methods in Applied Mechanics and 

Engineering 183, 159-199. 

Williams, K.V., Vaziri, R., 2001. Application of a damage mechanics model for predicting the 

impact response of composite materials. Computers and Structures 79, 997-1011. 
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Literature Reviews on Modeling Internal Geometry of Textile Composites 

and Rate-Independent Continuum Damage 

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Hsu, Su-Yuen

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14. ABSTRACT

Textile composite materials have good potential for constructing composite structures where the effects of three-dimensional 

stresses are critical or geometric complexity is a manufacturing concern. There is a recent interest in advancing competence 

within Langley Research Center for modeling the degradation of mechanical properties of textile composites. In an initial 

effort, two critical areas are identified to pursue: (1) Construction of internal geometry of textile composites, and (2) 

Rate-independent continuum damage mechanics. This report documents reviews on the two subjects. Various reviewed 

approaches are categorized, their assumptions, methods, and progress are briefed, and then critiques are presented. Each 

review ends with recommended research.

15. SUBJECT TERMS

Textile composite; Continuum damage mechanics; Geometry; Micromechanics

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