*2.1.2.1. Failure criteria-based approach*

In this method, specific relationships based on either stress or strain components are intro‐ duced for the onset of failure modes. One fashion for the presentation of this method is the use of a single relationship to represent all the failure modes collectively. A case in point is the Tsai-Wu failure criterion [16] which offered a response surface for damage in stress or strain space. This approach is not able to distinguish between individual failure modes. Moreover, it is not meant to consider the degradation mechanism of composites. In order to propose a precise degradation mechanism, the induced failure modes have to be identified individually so that the corresponding stiffness coefficients can be decreased gradually.

There is another presentation type under this approach which offers a distinct relationship for each failure mode. Perhaps the most popular models of this kind are those proposed by Chang-Chang [17] and Hashin [18]. Nevertheless, all of the aforementioned failure criteria have been developed for UD composite materials, rather than woven fabric composites. In these failure criteria, the authors have assumed that failure in the longitudinal and transverse directions correspond to failure in fibers and matrix, respectively. As an example, Table 1 [14] shows the introduced failure criterion for each failure mode by Chang-Chang. For the Hashin failure criteria, β in the first equation is 1. According to this table, the failure modes of UD composites are classified as compression as well as tension in both fibers and matrix (longitudinal and transverse) directions, whereas in woven composite materials, fibers (wefts and warps) are spread in both longitudinal and transverse directions.


**Table 1.** Chang-Chang failure criteria [14].

Another notable point is that in the former type of 'surface' failure criteria (also called failure 'envelope'), some terms of both longitudinal and transverse stresses appear in the correspond‐ ing relationships, meaning that the stress values in both principal directions can accelerate the failure occurrence. On the other hand, in the second category, no terms of transverse (longi‐ tudinal) stresses are observed in the presented relationship corresponding to the failure in longitudinal (transverse) direction, as shown in Table 1. In other words, the latter technique of failure introduction assumes that the transverse stresses do not affect the failure in the longitudinal direction, and vice versa. The validity of such presumption should be challenged particularly for consolidated fabric composites, as there have been recent evidences of influence of coupling between warp and weft yarns in dry and coated fabrics [19–22] as will be further expanded on in section 2.1.3.1 Furthermore, as indicated earlier, both of these modeling types can forecast the initiation of damage and not the damage growth. In fact, after fulfillment of such relationships, the corresponding stiffness coefficients decrease to zero.

Recently, Materials Science Corporation (MSC) presented a model for woven composite materials by generalizing the Hashin failure criteria [14]. They assumed tensile and compres‐ sion in both warp and fill directions, crushing failure, shear failure due to matrix cracking without fiber breakage, and tensile matrix failure in the out of plane direction. However, for failure in the warp and fill directions—the most common and predominant failure mode in woven composites—the relationship was analogous to that of Hashin's. That is, the effect of stress in the second main direction is not seen in the relationship for the first main direction.

#### *2.1.2.2. Plasticity-based approach*

Unidirectional composite materials are predominantly known as brittle materials showing an elastic and linear response until failure. However, woven composite materials can show a nonlinear response even in the early stage of a tensile test in the fibers' direction [23]. Some facts, such as wavy architecture of woven composites, visco-elastic behavior of matrix, and matrix cracking, can be postulated as the reasons of such non-linearity. Although earlier studies [23] showed that matrix cracking in woven composites can partially cause a non-linear tensile behavior, there has not been a full-scale investigation into discriminating all different sources and understand their effects. One of the easiest modeling methods to take the non-linear behavior of woven composites into account is the plasticity approach, in which the non-linear behavior of the composite is equivalently modelled as a plastic material behavior, regardless of the reason of non-linearity. In the first study in this area, Hill introduced a plastic model for anisotropic materials [24]. Vaziri et al. proposed a comprehensive plasticity model for fiber reinforced composite plies based on a rate-independent orthotropic plasticity theory [25]. In this method, a relationship known as flow rule is considered between the effective strain and the effective stress. The coefficients of the flow rule are obtained using several tensile tests. Although this approach is very suitable to prediction loading regimes, it is not as accurate for events containing loading and unloading regimes, such as impact events, especially after the occurrence of some damage. The loading-unloading response of damaged woven composites has not been fully characterized in the literature yet. Each of the aforementioned non-linearity sources in woven composites probably has a different influence on the unloading behavior of these materials. In addition, in woven composites under tensile loading, the first failure mode is matrix cracking, decreasing the stiffness in a non-linear manner and causing a non-linear global response as illustrated in Figure 1 [23]. However, there is a fact known as 'crack saturation' in which crack growth is saturated and stopped. After crack saturation, a linear response is seen until yarn breakage. As a result, in some cases after damage initiation, some linear responses are observed in reality, which cannot be modeled by the plasticity approach. Another point regarding the limitation of the plasticity approach is that the behavior of composite materials in general, and their stress-strain responses in particular, under compres‐ sion are not the same as that of tensile loading. This fact can be interpreted in a way that fiber yarns in composites, which may be comparable to thin beams, are more endangered with compression loading—buckling—rather than tensile loading. As a result, the ultimate compression strength of composite materials is predominantly less than their maximum tensile strength [26]. Also, the non-linearity in tensile loading of textile composites does not necessa‐ rily exist in compression loading. Hence, the plasticity method may not be fully applicable to simulate the behavior of woven composites under real-life loadingssuch as crashes in which some regions of the structure undergo compression. Finally, the effect of stress in one yarn direction upon the failure in the other yarn direction (coupling effect) cannot be considered in the plasticity approach.

#### *2.1.2.3. Continuum damage mechanics-based approach*

**Tensile fiber mode** *ef*

**Compressive fiber mode** *ec*

**Tensile matrix mode** *em*

**Compressive matrix mode** *ed*

**Table 1.** Chang-Chang failure criteria [14].

240 Non-woven Fabrics

*2.1.2.2. Plasticity-based approach*

<sup>2</sup> =( *<sup>σ</sup>aa Xt*

<sup>2</sup> =( *<sup>σ</sup>bb Yt* ) 2 <sup>+</sup>*β*( *<sup>σ</sup>ab Sc* ) 2

Another notable point is that in the former type of 'surface' failure criteria (also called failure 'envelope'), some terms of both longitudinal and transverse stresses appear in the correspond‐ ing relationships, meaning that the stress values in both principal directions can accelerate the failure occurrence. On the other hand, in the second category, no terms of transverse (longi‐ tudinal) stresses are observed in the presented relationship corresponding to the failure in longitudinal (transverse) direction, as shown in Table 1. In other words, the latter technique of failure introduction assumes that the transverse stresses do not affect the failure in the longitudinal direction, and vice versa. The validity of such presumption should be challenged particularly for consolidated fabric composites, as there have been recent evidences of influence of coupling between warp and weft yarns in dry and coated fabrics [19–22] as will be further expanded on in section 2.1.3.1 Furthermore, as indicated earlier, both of these modeling types can forecast the initiation of damage and not the damage growth. In fact, after fulfillment of such relationships, the corresponding stiffness coefficients decrease to zero.

Recently, Materials Science Corporation (MSC) presented a model for woven composite materials by generalizing the Hashin failure criteria [14]. They assumed tensile and compres‐ sion in both warp and fill directions, crushing failure, shear failure due to matrix cracking without fiber breakage, and tensile matrix failure in the out of plane direction. However, for failure in the warp and fill directions—the most common and predominant failure mode in woven composites—the relationship was analogous to that of Hashin's. That is, the effect of stress in the second main direction is not seen in the relationship for the first main direction.

Unidirectional composite materials are predominantly known as brittle materials showing an elastic and linear response until failure. However, woven composite materials can show a nonlinear response even in the early stage of a tensile test in the fibers' direction [23]. Some facts, such as wavy architecture of woven composites, visco-elastic behavior of matrix, and matrix cracking, can be postulated as the reasons of such non-linearity. Although earlier studies [23] showed that matrix cracking in woven composites can partially cause a non-linear tensile behavior, there has not been a full-scale investigation into discriminating all different sources and understand their effects. One of the easiest modeling methods to take the non-linear behavior of woven composites into account is the plasticity approach, in which the non-linear

<sup>2</sup> =( *<sup>σ</sup>bb* 2*Sc* ) 2 + ( *Yc* 2*Sc* ) 2 <sup>−</sup><sup>1</sup> *<sup>σ</sup>bb Yc* + ( *<sup>σ</sup>ab Sc* ) 2

) <sup>+</sup>*<sup>β</sup>* ( *<sup>σ</sup>ab Sc*

<sup>2</sup> =( *<sup>σ</sup>aa Xc* ) 2 ) <sup>−</sup><sup>1</sup> { <sup>≥</sup>0 *failed* <0 *elastic*

<sup>−</sup><sup>1</sup> { <sup>≥</sup>0 *failed* <0 *elastic*

> <sup>−</sup><sup>1</sup> { <sup>≥</sup>0 *failed* <0 *elastic*

<sup>−</sup><sup>1</sup> { <sup>≥</sup>0 *failed* <0 *elastic*

> The continuum damage mechanics covers the initiation as well as propagation of various failure modes, including matrix cracking, delamination, and fiber fracture under tension as well as compression. This approach was introduced by Kachanov [27] and developed for composite materials by Frantziskonis [28]. Basically, the method employs a set of damage variables each of which representing a certain failure mode and varying from zero to one, covering no damage, partial damage, and complete damage states in a material element. Frantziskonis also defined the damage variable, *r*, as the ratio of damaged volume to the total volume. It was assumed that the strain for the damaged and undamaged part is the same, and then the stress was calculated employing the mixed rule.

> In another work under continuum damage mechanics-based approach, Feng et al. introduced a damage model for woven fabric materials [29]. Due to the stress concentration in textile

**Figure 1.** Tensile behavior of a woven composite sample [23].

composites arising from their complex fibrous architectures as proven in the micromechanical approaches using FEM [15], such materials can experience damage in early stages of defor‐ mation. Therefore, Feng et al. employed a stress variation factor (SVF) to correct the stress value by multiplying the nominal stress by the defined SVF. In their study, a woven lamina was assumed as two unidirectional (cross-ply) layers. Hashin failure criteria were utilized to predict the damage occurrence in each ply. For degradation mechanism, the corresponding stiffness coefficients were instantaneously reduced to zero. They compared the numerical and experimental results of a transverse compression test—punch test—on a woven composite specimen. The results were comparable in most cases.

Ladevaze and LeDantec introduced a continuum damage model for UD composite materials [30]. They wrote the strain energy based on the stress, stiffness, and damage variables of materials. Thereafter, driving forces, each of which being associated with a certain damage variable, was obtained by taking the derivative of strain energy with respect to damage variables. In fact, the damage evolution was based on such driving forces. In two subsequent investigations [31, 32], this approach was generalized for woven fabric materials. For instance, Johnson presented a 2D damage model that was able to predict the elastic failure in the warp and weft directions and the elastic-plastic in-plane shear failure for matrix [31]. Johnson used a linear relationship for the damage evolution based on the driving forces. In addition, that work employed a power law function to consider plastic hardening functions for cyclic loads. The required parameters for the model were found using a tensile test. The damage model was implemented in an explicit finite element code in order to compare with the experimental results. In low velocity impact tests, in which there was considerable delamination failure, there was a substantial difference between numerical and experimental results, as shown in Figure 2.

**Figure 2.** Comparison between numerical and experimental impact results [32].

composites arising from their complex fibrous architectures as proven in the micromechanical approaches using FEM [15], such materials can experience damage in early stages of defor‐ mation. Therefore, Feng et al. employed a stress variation factor (SVF) to correct the stress value by multiplying the nominal stress by the defined SVF. In their study, a woven lamina was assumed as two unidirectional (cross-ply) layers. Hashin failure criteria were utilized to predict the damage occurrence in each ply. For degradation mechanism, the corresponding stiffness coefficients were instantaneously reduced to zero. They compared the numerical and experimental results of a transverse compression test—punch test—on a woven composite

Ladevaze and LeDantec introduced a continuum damage model for UD composite materials [30]. They wrote the strain energy based on the stress, stiffness, and damage variables of materials. Thereafter, driving forces, each of which being associated with a certain damage variable, was obtained by taking the derivative of strain energy with respect to damage variables. In fact, the damage evolution was based on such driving forces. In two subsequent investigations [31, 32], this approach was generalized for woven fabric materials. For instance, Johnson presented a 2D damage model that was able to predict the elastic failure in the warp and weft directions and the elastic-plastic in-plane shear failure for matrix [31]. Johnson used a linear relationship for the damage evolution based on the driving forces. In addition, that work employed a power law function to consider plastic hardening functions for cyclic loads.

specimen. The results were comparable in most cases.

**Figure 1.** Tensile behavior of a woven composite sample [23].

242 Non-woven Fabrics

Another research group introducing a damage model for woven composites is Iannucci and co-workers [33–35]. Iannucci et al. developed a progressive damage model for woven glass fiber-epoxy composite laminates and implemented it in a finite element code [33]. Their model was valid for shell elements with plain stress assumption. In addition, the proposed model was applicable to situations where there was no considerable damage, for example delami‐ nation, in the specimen. They postulated two unidirectional layers instead of one woven layer and found the Young's modulus and other mechanical properties required for modeling. Damage parameters varying from 0 to 1 were allocated for two types of failure modes, including matrix cracking and fiber fracture. The employed failure criteria, however, had been originally developed for unidirectional composite laminates. They also eliminated the terms of shear stresses in the employed failure criteria. A significant point of their work was that they defined an advanced model for damage propagation and degradation mechanisms, in that the rate of damage growth was proportional to the extent of damage and the value of corresponding failure criterion. However, one of the drawbacks of the model was that the damage growth rate could increase infinitely. The latter aspect of the model may not be reasonable on account of two facts:


In the follow-up studies [34, 35], these authors introduced new failure modes in their originally developed model. Also, they made some efforts to distinguish between compression and tension failures. The presented damage model was rather accurate when the damage extent was not considerable. On the other hand, when there was a significant damage in the specimen, deviation between the model and experimental results was rather notable.

One of the latest research activities presenting a damage model for woven composites was performed by Cousigne et al. [36]. Employing LS-DYNA, they wrote a subroutine for woven fabric materials assuming a non-linear material behavior. Ramberg-Osgood equation [37] was exploited in different directions, covering longitudinal, transverse, and in-plane shear deformations. Effective macro-mechanical properties were determined based on tensile tests, and maximum stress failure criteria were chosen for damage initiation. Four post-damage models were considered as depicted in Figure 3. The force-displacement curves in Figure 4 demonstrate that the numerical results were not in a full agreement with the experimental data.

**Figure 3.** Four post-damage models for woven fabric materials: (a) linear damage, (b) non-linear damage, (c) constant stress level, (d) step-based material degradation [36].

**Figure 4.** Force–displacement curves of 25J impact on a woven composite specimen [36].

Other researchers have aimed to use the pre-defined material models in commercial finite element programs to predict the macro-behavior of woven composite materials. A case in point is the study on low-velocity impact of thermoplastic woven composite specimens by Brown et al. [38]. They conducted an investigation into the applicability of MAT 162 of LS-DYNA material library—one of the most advanced material models for damage initiation and propagation in composite materials [14]—for the impact modeling of woven composite materials. At first, the parameters of MAT 162 were found using tensile tests on the woven specimens. Thereafter, the obtained parameters were used in impact simulations. Results showed that the identified damage model could not lead to highly accurate predictions against actual impact test data. In other words, the results again showed that the failure behavior of woven composite materials is more complicated than UDs and the original damage models need modifications.

#### *2.1.3. Experimental studies*

damage growth rate could increase infinitely. The latter aspect of the model may not be

**•** The maximum damage growth rate would be the stress wave speed in the material [33].

**•** Because of the crack growth saturation, as discussed before in section 2.1.2.2, the crack propagation cannot continue infinitely; it is restricted between the cells of weave architec‐

In the follow-up studies [34, 35], these authors introduced new failure modes in their originally developed model. Also, they made some efforts to distinguish between compression and tension failures. The presented damage model was rather accurate when the damage extent was not considerable. On the other hand, when there was a significant damage in the specimen,

One of the latest research activities presenting a damage model for woven composites was performed by Cousigne et al. [36]. Employing LS-DYNA, they wrote a subroutine for woven fabric materials assuming a non-linear material behavior. Ramberg-Osgood equation [37] was exploited in different directions, covering longitudinal, transverse, and in-plane shear deformations. Effective macro-mechanical properties were determined based on tensile tests, and maximum stress failure criteria were chosen for damage initiation. Four post-damage models were considered as depicted in Figure 3. The force-displacement curves in Figure 4 demonstrate that the numerical results were not in a full agreement with the experimental

**Figure 3.** Four post-damage models for woven fabric materials: (a) linear damage, (b) non-linear damage, (c) constant

stress level, (d) step-based material degradation [36].

deviation between the model and experimental results was rather notable.

reasonable on account of two facts:

ture.

244 Non-woven Fabrics

data.

There have been several studies in the literature encompassing experimental research on woven composite materials. To illustrate a few, the compression behavior of woven fabric materials was investigated by Song et al. under quasi-static and high strain rate loading, employing a split Hopkinson bar [39]. The obtained results demonstrated that regardless of strain rate, the predominant failure mode was shear mode, while delamination happened only at high strain rates. Another category of most cited studies in this area links to the papers presenting differences between unidirectional and woven fiber reinforced composites. For instance, Evci and Gulgec performed an experimental study to find the difference between the impact behavior of thermoplastic woven and unidirectional laminates with the same fiber/ matrix materials and thickness [40]. They introduced plain weave composites as an outstand‐ ing replacement for UDs in dynamics applications, owing to the fibers woven architecture that confines the damage. In addition, these materials were found to be more sensitive to strain rate, that is, their ultimate strength rises more substantially as compared with UD materials under high-strain rate loadings, mainly due to their higher visco-elasticity. Another article in this area is the study on the delamination modes I and II as well as impact resistance of woven fabric composite materials, performed by Kim and Sham [41]. Their results showed that woven composite materials are superior in comparison to UD composite materials under the above fracture modes and impact resistance criteria.

Mallikarachchi and Pellegrino endeavored to introduce the most elaborated failure criteria for symmetric two-ply woven carbon fiber reinforced plastics, by conducting numerous experi‐ ments [42]. In the first step, they used the Karkkainen relationship for failure occurrence including terms based on three forces and three moments applied to an element. This single equation is similar to the first presentation approach of failure criteria (section 2.1.2.1), which presents a surface for failure onset, and is not able to inform which type of failure arises in the sample. Tension, compression, shear, bending, and twist tests were performed with various lay-up configurations so as to find unknown constants of the utilized relationship/surface of failure. For example, [+45/-45] specimens were subjected to uniaxial testing and bending testing, separately, to induce shear and twisting modes, respectively. After identifying the unknown coefficients, they did several follow-up tests as combinations of five aforementioned loading types, in order to ensure that the presented model is reliable. However, the failure envelop relationship was not valid for some of the combined loading tests. Hence, instead of one equation for all the failure modes, they defined three failure modes, including in-plane, bending, and combined in-plane and bending, based on the three forces and three moments applied to the material element. The results showed that these failure criteria are valid for combined loadings. However, in the above approach, the final failure of woven composite was of interest and there was no identification of the first failure mode—matrix cracking—which decreases the stiffness of the material structure. In other words, such models are very reliable for prediction of catastrophic damage in textile composites but not for first-failure and progressive damage. In addition, although three relationships for failure of in-plane, bending, and combined loadings were considered, they cannot distinguish between individual failure modes under each type of deformation. Finally, the introduced relationships are based on the applied moments and forces, making the failure prediction analytics rather difficult for implementation in user-defined FE subroutines.

#### *2.1.3.1. Effect of loading modes and yarns interaction*

Generally speaking, among different deformation modes, the uniaxial behavior of materials has been investigated most. Woven composite materials are not an exception to this. However, the design of a targeted composite structure should not be based on the results of uniaxial tests only. This is because in sensitive events, such as a crash, structures experience different loading modes *simultaneously*, including tension, compression, and shear, in different directions. As an illustration, in an impact testing on a rectangular composite specimen, the elements in upper and lower parts of the material system may undergo biaxial compression and tension, respectively. Although conventional materials which have the same properties in all the directions should have almost the same properties (e.g., Young's modulus) under uniaxial loading or combined loading, the properties of composite materials that are anisotropic is not necessarily similar under different types of loadings. The account of combined loading is specifically more critical for woven composites, resulting from the geometrical and mechanical interactions between warp and weft yarns. In general, in order to design any composite structure (woven or non-woven) with the highest performance reliability, it would be desired to characterize the mechanical behavior of the material under several combined loadings, rather than individual deformation modes. One common type of combined loadings is biaxial loading in which primarily a test specimen is subjected to equi-biaxial tension. However, providing facilities to apply such multi-axial tests is sometimes expensive and challenging. Moreover, creating a homogenous deformation in the zone of interest in biaxial specimens, normally far away from the loading jig, is crucial so as to obtain reliable characterization results. Furthermore, stress concentration in the specimen resulting from the sample shape in biaxial tests makes their analysis more complicated. In other words, a correct calculation of effective stress and strain either by analytical or numerical approaches should be considered next to the experimental testing. Currently, there is limited standard for biaxial testing of composites [43], whereas there are several standards for testing individual deformation modes, such as ASTM D3039 for the uniaxial tensile testing of composite materials. The biaxial testing of dry woven fabric materials using customized fixtures has substantiated the presence of severe coupling effects between warp and weft directions under different deformation modes [19]. In another recent study regarding a state-of-the-art combined biaxial-shear testing of dry carbon fabrics, Nosrat-Nazemi et al. conducted shear as well as biaxial tension-shear experi‐ ments [20]. Their results clearly demonstrated that there is another nonlinear coupling between the global shear behavior of the woven fabric specimens and the pre-tension applied to the samples.

presenting differences between unidirectional and woven fiber reinforced composites. For instance, Evci and Gulgec performed an experimental study to find the difference between the impact behavior of thermoplastic woven and unidirectional laminates with the same fiber/ matrix materials and thickness [40]. They introduced plain weave composites as an outstand‐ ing replacement for UDs in dynamics applications, owing to the fibers woven architecture that confines the damage. In addition, these materials were found to be more sensitive to strain rate, that is, their ultimate strength rises more substantially as compared with UD materials under high-strain rate loadings, mainly due to their higher visco-elasticity. Another article in this area is the study on the delamination modes I and II as well as impact resistance of woven fabric composite materials, performed by Kim and Sham [41]. Their results showed that woven composite materials are superior in comparison to UD composite materials under the above

Mallikarachchi and Pellegrino endeavored to introduce the most elaborated failure criteria for symmetric two-ply woven carbon fiber reinforced plastics, by conducting numerous experi‐ ments [42]. In the first step, they used the Karkkainen relationship for failure occurrence including terms based on three forces and three moments applied to an element. This single equation is similar to the first presentation approach of failure criteria (section 2.1.2.1), which presents a surface for failure onset, and is not able to inform which type of failure arises in the sample. Tension, compression, shear, bending, and twist tests were performed with various lay-up configurations so as to find unknown constants of the utilized relationship/surface of failure. For example, [+45/-45] specimens were subjected to uniaxial testing and bending testing, separately, to induce shear and twisting modes, respectively. After identifying the unknown coefficients, they did several follow-up tests as combinations of five aforementioned loading types, in order to ensure that the presented model is reliable. However, the failure envelop relationship was not valid for some of the combined loading tests. Hence, instead of one equation for all the failure modes, they defined three failure modes, including in-plane, bending, and combined in-plane and bending, based on the three forces and three moments applied to the material element. The results showed that these failure criteria are valid for combined loadings. However, in the above approach, the final failure of woven composite was of interest and there was no identification of the first failure mode—matrix cracking—which decreases the stiffness of the material structure. In other words, such models are very reliable for prediction of catastrophic damage in textile composites but not for first-failure and progressive damage. In addition, although three relationships for failure of in-plane, bending, and combined loadings were considered, they cannot distinguish between individual failure modes under each type of deformation. Finally, the introduced relationships are based on the applied moments and forces, making the failure prediction analytics rather difficult for

Generally speaking, among different deformation modes, the uniaxial behavior of materials has been investigated most. Woven composite materials are not an exception to this. However, the design of a targeted composite structure should not be based on the results of uniaxial tests

fracture modes and impact resistance criteria.

246 Non-woven Fabrics

implementation in user-defined FE subroutines.

*2.1.3.1. Effect of loading modes and yarns interaction*

The biaxial behavior of coated fabric materials has also shown that these materials behave substantially different by changing the ratio of applied stress in the warp direction to the applied stress in the weft direction. One of the first studies in this area, performed by Reinhardt, showed that the stiffness and ultimate strain of coated fabrics changes with different loading ratios between warp and weft [21]. In another study of coated fabric composites, Galliot and Luchsinger presented a simple model for non-linear unequi-biaxial tensile behavior of PVCcoated polyester fabrics based on experiments. In their model, the Young's modulus of different directions changed by changing the loading ratio [22]. Actually, this change resulted from the significant effect of interaction between warps and wefts. A linear relation between the Young's modulus and the loading ratio was also found in their investigation experimen‐ tally.

By refereeing to the above biaxial tests on dry and coated woven fabrics, the effect of interac‐ tions between warp and weft yarns under different modes has been well evidenced [19–22]. However, the influence of yarns' interaction in fully consolidated woven composites has not been explored in full. In three investigations [44–46], Welsh et al. studied the biaxial behavior of IM7-977-2 carbon-epoxy and E-glass vinyl plain weave composite laminates. Their results showed the existence of biaxial strengthening effect. However, the researchers of World Wide Failure Exercise recommended more experimental studies, as well as more attempts to advance the generalized set-ups of biaxial tests, on consolidated woven composites because of the scarcity of information in this area [3]. Regarding analytical studies on combined loading of consolidated textile composites, Welsh and co-workers [44–46] endeavored to predict the behavior of fabric composites using a multi-continuum theory (MCT). However, there was not sufficient agreement between the experimental and MCT results. In another study, Key et al. utilized multi-continuum theory in which warp and weft yarns as well as the matrix were considered as three model constituents [47]. Comparison between the analytical results and the experimental data obtained by Welsh et al. showed more compatibility because of employing a progressive damage model based on a continuous material property degradation state.

In conclusion, although the interweaved fiber architectures of woven composites bring assets such as out-of-plane impact resistance, ease of molding processes, and better resistance to damage growth, such complex material structures also cause difficulties for their analyses; three of which are in-plane as well as out-of-plane waviness and coupling effects between warp and weft yarns. In the modeling works reviewed above, researchers successfully took the advantage of some simplifications and assumptions in order to consider some of the afore‐ mentioned complexities. However, there is no explicit and practical damage model developed for woven composites yet, as also highlighted in the World Wide Failure Exercise [3]. It is believed that as the first step toward such comprehensive models under different deformation modes and different loading regimes (including loading–unloading and viscoelasticity), the underlying local damage mechanisms and their progression should be further explored under *individual* and *combined* failure modes via advanced experimental/numerical studies to understand their effects on the global stress-strain behavior of woven composites. Some preliminary results in this direction will be presented in Section 3.

#### **2.2. Damage models for non-woven fabric composites**

Several efforts have been put into research to accurately predict the mechanical behavior of non-woven fabric composites. As an example of such predictions, Cox assumed the paper material is composed of perfectly homogeneous plane of non-interactive long straight fibers which are oriented randomly. The material elastic modulus was calculated for small defor‐ mation regimes [48]. Applying an orthotropic model, Backer and Petterson investigated the tensile behavior of non-woven composites [49]. In an orthotropic model, stiffness coefficients in an arbitrary direction for small deformation can be estimated by knowing the stiffness constants in two main (principal) directions. There was a fair compatibility between the obtained initial elastic modulus from analytical and experimental analyses. However, nonwoven composites can be inherently random anisotropic materials due to the non-uniformity of oriented fibers. Non-uniform distribution of fibers was first taken into account in the studies [50–51]. In these works, the authors presumed that a non-woven fabric specimen is made of many layers of fibers with various orientations bonded to each other. However, fiber reorientation was ignored on account of the bonding between fiber layers. Demirci et al. discussed the anisotropy of non-woven fabrics through continuum mechanics, based on the randomness of fiber orientation [52]. Utilizing image processing of data acquired by Scanning Electron Microscopy (SEM) and X-ray tomography, orientation distribution function (ODF) was measured as a representative of orientation randomness of fibers. Then, using the Fourier series, some parameters were defined to show the level of orthotropy. The analytical results which had good agreement with experimental results demonstrated a significant directiondependence of nonwovens. In one of the other latest studies employing continuum mechanics [53], Ridruejo et al. presented a constitutive model for in-plane behavior of non-woven felts that included three parts: fibrous network, fibers, and damage. For modeling the fibers network, they followed Cox's model in which fibers do not have any interaction with each other. However, fiber orientation in large deformation was considered in the model. The damage of bonds and fibers were incorporated in the study in a phenomenological way. However, this method was not able to give a profound insight into the micro-level mechanical behavior of these materials which have more complex fibrous architecture than woven and UD composites. In other words, most of reported analytical works cannot precisely consider actual effects such as fibers reorientation, fibers sliding, and progressive damage propagation.

By refereeing to the above biaxial tests on dry and coated woven fabrics, the effect of interac‐ tions between warp and weft yarns under different modes has been well evidenced [19–22]. However, the influence of yarns' interaction in fully consolidated woven composites has not been explored in full. In three investigations [44–46], Welsh et al. studied the biaxial behavior of IM7-977-2 carbon-epoxy and E-glass vinyl plain weave composite laminates. Their results showed the existence of biaxial strengthening effect. However, the researchers of World Wide Failure Exercise recommended more experimental studies, as well as more attempts to advance the generalized set-ups of biaxial tests, on consolidated woven composites because of the scarcity of information in this area [3]. Regarding analytical studies on combined loading of consolidated textile composites, Welsh and co-workers [44–46] endeavored to predict the behavior of fabric composites using a multi-continuum theory (MCT). However, there was not sufficient agreement between the experimental and MCT results. In another study, Key et al. utilized multi-continuum theory in which warp and weft yarns as well as the matrix were considered as three model constituents [47]. Comparison between the analytical results and the experimental data obtained by Welsh et al. showed more compatibility because of employing a progressive damage model based on a continuous material property degradation

In conclusion, although the interweaved fiber architectures of woven composites bring assets such as out-of-plane impact resistance, ease of molding processes, and better resistance to damage growth, such complex material structures also cause difficulties for their analyses; three of which are in-plane as well as out-of-plane waviness and coupling effects between warp and weft yarns. In the modeling works reviewed above, researchers successfully took the advantage of some simplifications and assumptions in order to consider some of the afore‐ mentioned complexities. However, there is no explicit and practical damage model developed for woven composites yet, as also highlighted in the World Wide Failure Exercise [3]. It is believed that as the first step toward such comprehensive models under different deformation modes and different loading regimes (including loading–unloading and viscoelasticity), the underlying local damage mechanisms and their progression should be further explored under *individual* and *combined* failure modes via advanced experimental/numerical studies to understand their effects on the global stress-strain behavior of woven composites. Some

Several efforts have been put into research to accurately predict the mechanical behavior of non-woven fabric composites. As an example of such predictions, Cox assumed the paper material is composed of perfectly homogeneous plane of non-interactive long straight fibers which are oriented randomly. The material elastic modulus was calculated for small defor‐ mation regimes [48]. Applying an orthotropic model, Backer and Petterson investigated the tensile behavior of non-woven composites [49]. In an orthotropic model, stiffness coefficients in an arbitrary direction for small deformation can be estimated by knowing the stiffness constants in two main (principal) directions. There was a fair compatibility between the obtained initial elastic modulus from analytical and experimental analyses. However, non-

preliminary results in this direction will be presented in Section 3.

**2.2. Damage models for non-woven fabric composites**

state.

248 Non-woven Fabrics

In order to arrive at a better understanding of the mechanical behavior of non-woven fabrics, representative unit cells (RUCs) can be employed based on numerical homogenization. The first study of this kind focusing on the micromechanics of non-woven composites was carried out by Petterson [54]. Straight fibers were modeled based on a specific statistical distribution within a unit cell. Another assumption in the simulation was a rigid bond between fibers. Thereafter, Hearle and Stevenson attempted to improve Peterson' fibrous network model by taking fiber curls into account [55]. The results demonstrated that fiber curl has a significant effect on the initial modulus of non-woven fabrics. The previous models were valid for in plane deformations. Narter et al. presented a 3D micro-mechanical model for prediction of elastic constants of non-woven composites, taking into account fiber elastic modulus, fiber linear density, and fabric bulk density [56]. Another extension of Petterson's work, considering the time-dependence behavior of felt composites, was performed by Kothari and Patel [57]. They investigated the creeping behavior of non-woven fabrics by considering the viscoelastic behavior of fibers. In another work, Silberstein et al. presented a micromechanical model for non-woven polymer fabrics that was able to implement an elasto-plastic behavior of fibers [58]. The obtained initial modulus and yield stress were based on fiber properties, network geometry, and network density. However, as similarly discussed for the RVE micromechanics of woven fabrics (section 2.1.1), this approach has several disadvantages, the main of which being lack of damage localization. Namely, this technique states that when damage initiates in a representative unit cell, it arises in the whole sample. However, this behavior could be contrasting with what is observed in reality where failure begins in a point of material medium, and then propagates to other points of the sample. Moreover, the RUC procedure may not able to fully predict damage propagation of non-woven felts due to the presence of more random‐ ness than, e.g., woven composites. Furthermore, on account of existence of voids, various fiber concentrations, and non-uniform fiber orientations at different points of a non-woven fabric, the homogenization rule may not be fully valid. In other words, the micro behavior of representative volume cells should be very cautiously extended to the macro behavior of nonwoven felts. Therefore, some researchers decided to sacrifice computational time for accom‐ plishing more reliable results by modeling the entire specimen at the micro/meso level instead of a single unit cell. Although this method is computationally expensive and may not be directly applied to large scale structures, small laboratory-scale specimens can be modeled and compared to experiments. In this type of modeling, however, simulation of bonding between fibers is controversial. In some studies, rigid contacts were defined between fibers. The first group of analyses that implemented this assumption was performed by Britton et al. [59–61]. Although they considered rigid contacts in bonds, they took into account bonding breakage. For the determination of bonding failure, they assumed when the force of a fiber exceeds a certain value, the fiber is debonded from the bond point. In another study, Wu and Dzenis considered the elastic behavior of planar fiber networks numerically [62]. In their simulations, slippage of fibers on each other and their angular displacement were overlooked. Constitutive equation was found based on the average of dissipated energy in each fiber. The numerical results were fairly comparable with analytical predictions. To make use of more advanced fiber contact models, Grindstaff and Hansen used springs to model bonds between fibers [63]. In order to determine the stiffness of springs, tensile tests were carried out on one single bond. One of the other advanced contact simulations was performed by Ridruejo et al. [64]. They simulated the behavior of glass fiber non-woven fabrics by explicitly introducing the fiber bundles using a random distribution. The implemented mechanical behavior and geometrical parameters of fibers were measured by removing a single fiber from the fabric and applying tensile test on it. In order to model a realistic contact, tiebreak contact was utilized in which fibers were jointed at their crossovers until bonding failure occurs. Although fibers do not move together after debonding, there was a pre-defined friction between them due to the use of this type of contact, which brought another capability of the model to consider fibers sliding. Experimental results of that study demonstrated that the main and first induced failure mode in non-woven fabrics under tension is bonding failure which propagates and creates a wide band. There was a good agreement between numerical and experimental results. Farukh et al. simulated thermally bonded nonwovens with a new insight into the bonding issue [65]. It has been reported that during the manufacturing process, high temperature and pressure at bond points can cause changes in molecular arrangement of fibers [66]. Additionally, stress concen‐ tration exists in fiber crossover points of nonwovens, similar to woven composites. As a result, the mechanical behavior of fibers in a fibrous network is different between an original fiber before the fabrication process and a fiber removed from the fabricated fabric sample. Hence, they extracted a fiber from the fabric sample in a way that the fiber was jointed to bonds at both ends. Their tensile tests showed that the ultimate strength and strain of fibers within fabrics are less, in comparison with those of unprocessed fibers. The numerical simulation was able to predict the general behavior of non-woven felts under tension with a reasonable accuracy.
