Characteristic progressive damage modes in a plain weave textile composite under multiaxial loads

Abstract

A finite element-based model was developed to predict progressive damage evolution within a plain weave textile composite subjected to various combinations of in-plane tension and shear. Cracking in the tows, matrix, and interfaces was accounted for through cohesive zone modeling. Shear damage in the tows was accounted for through a continuum damage model. The damage behavior in the tows was stochastic in nature with properties determined from prior investigations of composite microstructures that included randomness in fiber positions. The predicted progressive damage evolution was found to qualitatively match well with experimental observations performed on similar material systems. The effect of temperature change, which modifies the thermally induced stresses in the tows as well as the apparent strength of the tows (due to changes in thermally induced microstresses at the fiber–matrix scale) was examined. Finally, the progressive failure responses under different loadings were compared to identify common characteristic behaviors. The effect of these characteristic behaviors on the textile’s effective response was investigated along with approaches to incorporate the behaviors into a structural scale progressive failure model.

Keywords

Textile composite damage progressive failure cohesive zone continuum damage multiscale

Introduction

Developing an understanding of damage development in composites requires a careful study and characterization of their behavior at a variety of scales. Tests conducted on composite plies yield stress and strain information that can be used in micro-scale models to determine constituent properties through the solution of an inverse problem.¹ By using these properties at the micro-scale, where fibers and matrix are accounted for discretely, it is possible to examine the interaction between thermally and mechanically induced stress fields to predict how they will interact to cause failure of a given constituent.² Information from this micro-scale characterization can then be utilized to define the moduli, strength, and fracture properties for a homogenized transversely isotropic composite material at larger scales. Such analyses are useful for a detailed examination of stresses and damage development in laminate and textile composites. Herein, this larger scale will be referred to as the textile scale, although it is commonly referred to as the “meso-scale” in other work.

It has been previously noted for tape laminates that damage tends to take on characteristic forms.³ In the present work, damage development in a plain weave textile is simulated under a variety of loads to determine whether a similar simplified characterization of the damage state is possible. Provided a simplified characterization of damage development in the textile is possible (which previous studies⁴ suggested may be the case), this characterization can be utilized at the larger scale of composite structures to predict the occurrence of damage in the composite as well as predict the effect of that damage on the overall structural response. Such predictive capability holds great utility in engineering practice. The strategy used herein is similar to the work of Ladevéze and Lubineau,⁵ but with a focus on textile composites rather than tape laminates.

As stated, the present investigation utilizes a textile-scale model to predict the onset and development of damage for a variety of in-plane loadings. The prediction of damage in textiles has been undertaken by a number of researchers using various approaches over the years. Most of the early work in this area used a continuum damage approach in which properties are degraded in elements or at integration points where stresses reach a sufficient level to violate some local failure criteria. Examples include work by Blackketter et al.,⁶ Whitcomb and Srirengan,⁷ Choi and Tamma,⁸ Guagliano and Riva,⁹ Tang and Whitcomb,¹⁰ Zako et al.,¹¹ and Lomov et al.¹² These works utilize a variety of approaches to predict how properties should be degraded when damage occurs, ranging from simple intuition to the results of more comprehensive investigations.^13,14 It was noted by Lomov et al.¹² that accurately predicting the direction of damage evolution could be problematic, which was shown by Gorbatikh et al.¹⁵ to be a consequence of representing a crack as a damaged volume of material. Ivanov et al.¹⁶ addressed this issue by degrading large regions in damaged tows based on the investigations of Ladevéze and Lubineau⁵ for tape laminates. More recently, Hsu and Cheng¹⁷ undertook a study in which they used cohesive behavior to model failure in the composite, although their model did not include any mechanism to account for intra-tow cracking.

Most work to model progressive failure in textiles has focused on a single or small group of load cases. Identifying characteristic behaviors depends on examination of a broader variety of loadings. Karkkainen and Sankar¹⁸ and Karkkainen et al.^19,20 undertook studies along these lines with the goal of characterizing the failure initiation envelope. More recently, McLendon and Whitcomb⁴ examined failure initiation in a textile under various loadings with the explicit goal of identifying characteristic behaviors, finding that a limited number of behaviors were predicted for thousands of different multiaxial loads.

The current study expands and improves upon previous work in several ways. First, progressive damage evolution will be predicted in the textile under various combinations of temperature and in-plane tensile and shear loads in order to identify characteristic behaviors. Furthermore, the fidelity of the textile-scale model is improved by accounting for matrix cracking, progressive failure of the tows under shear, and interfacial failures discretely through the combined use of cohesive zone and continuum damage models. Although such models are not new, they have not been previously combined in this manner for simulating damage at the textile scale. An additional enhancement is that failure in the tows is based on the characteristic behavior observed from micro-scale investigations performed on periodic RVEs of the fiber–matrix that possess random fiber positions,^2,21 making this a multiscale approach.

It will be shown that for the loadings examined, characteristic behaviors are apparent. These characteristic behaviors form the basis of a structural-scale continuum damage model described by McLendon.²¹ In that work, four damage state variables are used to track the evolution of the different characteristic damage modes. These damage parameters are devised based on the damage distribution in the textile-scale analysis considered in the current work, i.e. they constitute a reduced-order representation of the overall damage state in the textile. By tracking the evolution of each reduced-order damage state variable under various multiaxial loadings using textile-scale models, it is possible to characterize each variable’s evolution as a function of load history. Furthermore, by tracking the evolution of the effective stiffness using textile-scale analysis, the effect each state variable has on the textile’s response can be characterized. Combining these characterizations permits the prediction of damage evolution and effective response of textile structures experiencing general multiaxial in-plane load histories via a continuum damage model based on a limited collection of textile RVE analyses performed a priori. Such an approach avoids the computational complexity and expense of trying to perform separate textile RVE analyses for every load history which exists in the overall structure.

The current work is intended to present the foundation of a framework for characterizing damage evolution in a textile composite under general multiaxial loading.²¹ Therefore, a detailed comparison to experiments is beyond the scope of the current work. However, the predicted damage evolution is qualitatively similar to damage observed by other authors²² in comparable textile material systems.

Progressive failure model of textile unit cell

This section describes the models used to approximate the progressive failure behavior of a textile unit cell under thermomechanical loading. Before experiencing damage, the material response is assumed to be linear elastic. Cohesive zones are used to approximate the presence of discrete cracks within various regions of the unit cell, and a continuum damage model is used to approximate the effect of diffuse cracking occurring in the tows under longitudinal shear loading.

Elastic tow properties

The transversely isotropic tow properties for the progressive failure textile model were obtained using a fiber–matrix micro-scale model with randomly positioned fibers assuming a fiber volume fraction v_f (within a tow) of 60%. This procedure is described in greater detail in McLendon and Whitcomb.²³ These properties are given in Table 1. The constituent properties used to calculate these effective tow properties were obtained from experimental tape laminate properties using the procedure outlined by Ballard et al.¹

Table 1.

Predicted IM7/8552 properties for v_f = 60%, used in tows for progressive failure of textile.

Modulus	Value
$E_{11}$	167 GPa
$E_{22}$ , $E_{33}$	11.6 GPa
$ν_{12}$ , $ν_{13}$	0.30
$G_{12}$ , $G_{12}$	5.23 GPa
$G_{23}$	3.87 GPa

Damage models

Damage within the textile unit cell is accounted for in two ways. Larger-scale discrete failures, such as cracking in the neat matrix pockets and interfacial failure between adjacent tows and between tows and the neat matrix pocket, are accounted for through the use of interfacial elements with opening governed by a cohesive zone model. These interfacial elements are inserted a priori into the textile unit cell at likely locations of failure. Damage in the tows themselves is accounted for through a combination of cohesive zone and continuum damage models. Fiber failure is not accounted for; the textile is considered to have failed completely at the onset of fiber failure, which is predicted based on the axial stress in the tows.

Previous studies of fiber/matrix models reveal that under transverse normal loading, the fiber–matrix material exhibits brittle behavior with damage localizing into a single crack that grows in an unstable manner. Therefore, cohesive elements are used to account for matrix cracking under transverse normal load. The strength of these cohesive elements is determined from a distribution of strengths obtained using multiple realizations of fiber–matrix microstructures.

Under longitudinal shear load, micromechanics models showed that the fiber/matrix exhibited a diffuse field of ductile matrix failures in the regions between fibers. As loading was progressively increased to the maximum shear stress, these failures began to coalesce into a single larger damage feature. After this coalescing occurred, further shear strain resulted in a decreasing stress.

The initial damage features under longitudinal shear are small and diffuse when considered at the scale of the textile unit cell. Therefore, their homogenized effect at the textile scale is accounted for using continuum damage mechanics. The stress–strain behavior for this continuum damage mechanics model is defined in such a manner to ensure that eventually, tangential traction across an interfacial element in the tow will exceed the allowable value at a stress level corresponding to the coalescing of diffuse damage into discrete damage features, leading to cohesive opening once the shear stresses in the tows reach the maximum shear stress predicted from the fiber/matrix analyses.

The details of these damage models are described in the following subsections.

Continuum damage model for shear damage in the tows

This section describes the continuum damage model that is applied to the tows to account for diffuse shear damage. Previous investigations into the evolution of damage in a tow’s microstructure² revealed the following progression of damage development under longitudinal shear loading. Ductile failure initiates in the matrix between fibers aligned with the loading direction at a low shear stress level (relative to the maximum shear stress). As longitudinal shear loading is increased, the composite material experiences a gradual increase of this distributed damage around fibers. This leads to a gradual reduction in the effective stiffness of the composite. Eventually, the effective stress in the composite reaches a plateau as additional damage development localizes along a single band running across the microstructure.

Accounting for the distributed damage in the tows and the associated gradual reduction in the material stiffness is accomplished through the use of a continuum damage mechanics model. The continuum damage mechanics approach is very well suited for this type of failure. Its original conception by Kachanov²⁴ is based on the premise that a reduction in the material’s modulus is caused by the formation of a large number of very small defects in the material, which is the case for the initial stage of damage development under longitudinal shear. In the current continuum damage model, the degraded shear moduli $G_{12}^{dam}$ and $G_{13}^{dam}$ at a given location in the tows are obtained from the original shear moduli using equation (1).

G_{12}^{dam} = (1 - d_{12}) G_{12} G_{13}^{dam} = (1 - d_{13}) G_{13}

(1)

where

d_{12}

and

d_{13}

are damage parameters which increase monotonically from zero to a maximum value of one. Their evolution ensures that under monotonically increasing load, the stress–strain relationship for longitudinal shear in the tows follows the path shown by the dashed line in Figure 1, which is comparable to the shear stress–strain behavior observed using a variety of fiber–matrix micro-scale analyses with different realizations of the random fiber locations.² At each load increment of the finite element analysis, an iterative process is used to increase the damage parameter at any location where stress exceeds this dashed line until no shear stress in the tows exceeds this boundary. Unloading follows a linear path back to the origin.

This stress–strain relationship is a piecewise function with three segments. The first segment which accounts for linear elastic deformation is simply linear with no evolution of shear damage.

The intermediate stage, labeled the damaging stage, has a stress–strain relationship following a cubic polynomial $σ_{12} (γ_{12})$ given in equation (2) with coefficient values that are defined based on the conditions of equation (3), which ensure value and slope continuity of the stress under increasing load.

σ_{12} (γ_{12}) = A γ_{12}^{3} + B γ_{12}^{2} + C γ_{12} + D

(2)

σ_{12} (γ_{12}^{init}) = G_{12} γ_{12}^{init} σ_{12} (γ_{12}^{max}) = S_{12}^{tow} \frac{\partial σ_{12}}{\partial γ_{12}} |_{γ_{12}^{init}} = G_{12} \frac{\partial σ_{12}}{\partial γ_{12}} |_{γ_{12}^{max}} = G_{12}^{loc}

(3)

S_{12}^{tow}

is the strength of the tow,

γ_{12}^{init}

is the shear strain at which degradation of the tow begins to occur, and

G_{12}^{loc}

is the tangent shear modulus at the point that localization of the damage begins to occur as described in greater detail later. This “damaging” stage captures the gradual loss of stiffness in the material as damage grows in the tows, but before localization of failure starts to occur. A cubic polynomial was chosen due to the relative ease of calculating the polynomial coefficients as compared to other nonlinear functions that were considered.

The last stage is the localization stage, which begins once the maximum stress observed from the micromechanics models, $S_{12}^{tow}$ , has been reached. In this stage, the stress in the tows continues to gradually increase with shear strain with a constant tangent modulus $G_{12}^{loc}$ that is a small specified fraction of the initial shear modulus. This continuing increase in the stress ensures that as loading increases, the tangential tractions in adjacent cohesive elements increase to a level sufficient to cause cohesive opening. It was noted in the fiber/matrix micromechanics model that once the maximum stress was reached, localization of the damage began to occur. Continuum damage mechanics is not well suited for accounting for this localization (see the investigation by Gorbatikh et al.¹⁵ regarding problems associated with the use of continuum damage to account for the presence of discrete cracks under shear loading). Therefore, it is desirable that the cohesive zones begin to open to account for this localization, since the localization of failure ultimately represents the formation of a discrete crack running along a tow.

This damage model is applied to both of the longitudinal shear moduli in the tows, although for in-plane loading, most damage occurs for the 1–2 component. The parameters used to define the model used in the current study are provided in Table 2.

Table 2.

Parameters for shear damage model in tows.

Property	Value
$G_{12} = G_{13}$	5.23 GPa
$S_{12}^{tow}$	90 MPa
$σ_{12}^{init}$	20 MPa
$γ_{12}^{max}$	0.037
$G_{12}^{loc} / G_{12}$	0.05

Cohesive opening model for tows

Discrete matrix crack opening in the tows is accounted for through the use of cohesive zone interfacial elements inserted along the tow length (Figure 2(b) and (c)). Two distinct types of discrete crack opening were noted for the tows based on the micromechanics analysis. The first, associated with transverse tension, is brittle crack opening resulting from the onset and unstable growth of brittle failure in the matrix. The second, associated with longitudinal shear, is the gradual localization of ductile shear damage (which is initially diffusely distributed within the microstructure) into a band of failed material. These phenomena occur when the normal or shear stress in the tows reach a critical value.

The cohesive zone elements in the tows are able to account for both of these failure mechanisms. When matrix cracking occurs in the tows, the region of the tow surrounding the newly opened matrix crack is unloaded. Further loading leads to additional cracks developing in different regions of the tow cross section. This leads to multiple cracking in the tows observed in Karahan.²² Eventually, the tows will become saturated with matrix cracks. Although the density of cohesive elements inserted into the tows in the current study is primarily determined by the size of model that could be analyzed, it affords a crack density which is comparable to that observed experimentally for similar composite systems.²²

Cohesive zones provide a means of predicting both crack initiation and opening. Initiation of cracking is governed by the maximum traction that the cohesive zone is capable of sustaining. Crack opening is governed through a dissipative traction-separation law such that the energy required to fully open a crack matches the critical fracture energy obtained through crack growth experiments. Various cohesive zone formulations differ in terms of the assumed shape of the traction-separation curve as well as the criteria for determining the maximum traction and critical fracture energy under mixed-mode loading. The formulation of Turon et al.,²⁵ with a few modifications, forms the basis for the current study.

The most significant modification was made to the criterion for obtaining the critical strain energy under mixed-mode loading. This modification applies to the intra-tow cohesive zones only. Turon et al.²⁵ use a power-law fit to delamination tests of unidirectional specimens under various mode mixes. However, it was found that the mode I strain energy release rate from such tests was not appropriate for modeling matrix cracking in a lamina or tow under transverse tension. Using this energy tended to cause gradual crack opening in all the cohesive elements placed in a tow or cross-ply laminate. This is not in agreement with the behavior typically observed in composites experiencing matrix cracking, wherein cracks of increasing density undergo sudden, brittle opening. This difference in behavior may be due to the different directions of crack growth relative to the fiber direction, but a detailed study of this issue has not yet been performed. Realistic matrix cracking behavior was obtained in cross-ply laminates and textile configurations by reducing $G_{Ic}$ , the pure mode I critical strain energy release rate to a value 1/10th of the $G_{Ic}$ from mode I delamination. The exact degree to which the energy was decreased is not thought to be of critical importance to the model behavior provided that brittle behavior is obtained under mode I. However, because there are not mixed-mode tests available, the power-law mixed-mode critical strain energy equation used by Turon et al.²⁵ is not appropriate for use, as it has the potential of yielding strain energies with mode I components that exceed $G_{Ic}$ . Therefore, the critical strain energy under mixed-mode loading in the tows is governed by the following relation

\frac{G_{I}}{G_{Ic}} + \frac{G_{s}}{G_{IIc}} = 1

(4)

G_I is the mode I strain energy and G_s is the total strain energy associated with shear separation. This leads to the following expression for G_c, the critical strain energy release rate under mixed-mode loading.

G_{c} = \frac{G_{Ic}}{1 + B (\frac{G_{Ic}}{G_{IIc}} - 1)}

(5)

where B is the fraction of the current total strain energy associated with shear opening.

In addition to this modification, a variation was made to limit the number of possible values which the damage variable d of the cohesive zone could attain, and a solution approach was adopted which ensured that this value could only ratchet upwards. Specific details of these modification are described in greater detail by McLendon.²¹

The critical strain energy release rate for mode II opening of the cohesive zone (under longitudinal shear) is set to the same value as that obtained from mode II delamination testing of IM7/8552 laminates.²⁵ Delamination growth between two 0° plies under pure mode II loading bears a strong similarity to the failure of a tow under longitudinal shear in terms of the stresses at the crack tip/delamination front and the direction of crack growth.

The final aspect of the cohesive zone to be defined is the maximum traction. Typically, these values would be obtained from experiments on unidirectional laminates under transverse normal and shear load. However, it can be expected that within a tow, this cohesive strength will be subject to variability because of the randomness which exists in the smaller-scale microstructure. Also, as was demonstrated in previous micromechanics analysis, this strength exhibits temperature dependence due to the thermally induced stresses which exist in the fiber–matrix microstructure, particularly for failure under transverse normal loading. The strength variability and temperature dependence from the fiber–matrix scale analysis are incorporated into the textile-scale analysis by specifying a different strength at each quadrature point in the intra-tow cohesive elements at the textile scale. These strengths are specified at the beginning of the progressive failure analysis and stay constant throughout the analysis. Each strength value is obtained at random using a Weibull distribution that is characterized by fitting the strengths from progressive failure analyses performed on hundreds of fiber/matrix microstructural realizations at the same temperature as the tow. The probability density function is defined as

where k is the shape parameter and λ is the scale parameter. The parameters obtained from micromechanics for use in the current study are given in Table 3. Note that a temperature decrease of 160℃ from cure temperature corresponds to room temperature.

Table 3.

Weibull parameters used to define strength in intra-tow cohesive zones.

$Δ T$ from cure (℃)	Normal traction			Shear traction
$Δ T$ from cure (℃)	k	λ (MPa)	5th % (MPa)	k	λ (MPa)	5th % (MPa)
−160	23.96	68.38	60.40	134.73	92.52	90.50
−10	17.24	53.61	45.12	141.10	94.82	92.84

The other cohesive zone properties that are applied to the intra-tow cohesive zones are given in Table 4. An investigation was performed into the effect of using a variable strength value rather than a uniform strength value in the textile by examining the evolution of tow cracking under uniaxial loading when using a uniform strength (which represented the fifth percentile strength from the distribution) as well as four different seedings of the random strength. In all cases for a given temperature, the different seedings of random strength values all provided extremely similar predictions for the overall evolution of tow cracking, although the particular locations of cohesive zone opening varied. When the temperature of the composite was close to cure temperature (i.e. when thermally induced stresses were low), the random strength distribution and uniform strength models predicted similar behaviors in terms of the extent of tow cracking at a given mechanical load. However, when the temperature was reduced to room temperature, the cases which were seeded with random strength values from the Weibull distribution predicted the onset of tow cracking at lower loads than the uniform strength case.

Table 4.

Intra-tow cohesive zone properties.

Property	Value
K	10¹⁷N/m
$G_{Ic}$	20.94 N/m
$G_{IIc}$	1047 N/m

Cohesive opening model for interfaces and the neat matrix pocket

Interfacial elements are also utilized to model failure of the interfaces between the tows (Figure 2(e)) and between the tows and the neat matrix pocket (Figure 2(f)), as well as cracking within the neat matrix pocket (Figure 2(d)). The opening of these elements is governed by the cohesive zone law of Turon et al.²⁵ and use the power-law mixed-mode criterion utilized by Turon et al. The values utilized for the cohesive zone models are provided in Table 5.

Table 5.

Properties for inter-tow, inter-tow-matrix, and intra-matrix cohesive zones.

	$G_{Ic}$ (N/m)	$G_{IIc}$ (N/m)	η	$τ_{n}^{0}$ (MPa)	$τ_{t}^{0}$ (MPa)
Inter-tow	200	1000	1.45	60	90
Inter-tow-matrix	200	1000	1.45	60	90
Intra-matrix	680	1000	1.45	120	180

The inter-tow properties are the same as those used by Hallett et al.,²⁶ which were curve-fit to match experiments performed on IM7/8552 laminates. It is thought that the interface between tows will be reasonably similar to the interface between two adjacent lamina. There was no experimental data available that seemed appropriate for modeling the interface between a tow and the neat matrix pocket, so for the current set of analyses, this interface is modeled using the same properties as the inter-tow interfaces. Tensile strength data as well as $G_{Ic}$ are provided by the manufacturer for 8552 epoxy,²⁷ but shear failure strength and $G_{IIc}$ are not. Values for mode II were chosen to be higher than for mode I by a proportion which is similar to other epoxy resin systems that have been experimentally characterized.²⁸

Progressive failure response under in-plane loadings

This section examines the predicted behavior of the textile under various types of in-plane loading. For each loading, the damage is presented in figures such as Figure 3. In these figures, cohesive elements with quadrature points that have undergone degradation and partial opening, but which have not failed completely are shaded light gray. Cohesive elements which contain quadrature points that have experienced complete failure and have no remaining stiffness are shaded dark gray. Cohesive elements that have not experienced any degradation are left transparent. Each of these analyses was stress controlled. Loading was increased until (1) the model reached an instability under increasing stress (i.e. the effective stress in the model reached its maximum) or (2) until the onset of fiber failure was predicted due to the tow’s axial strength being exceeded, as this is considered a critical type of failure. Fiber failure onset is predicted to occur when more than 0.5% of the tow volume in the textile exceeds 2.6 GPa, the axial tensile strength in a unidirectional IM7/8552 laminate. It is assumed that this strength corresponds to the axial strength of a tow. This value is not considered to be temperature dependent. The current study is primarily focused on the behavior of the textile before fiber failure occurs; therefore, accounting for the temperature dependence of the tow’s axial strength was not deemed to be important for the current investigation.

Failure under uniaxial load

The first loading case that was examined was x-direction uniaxial loading. The predicted progressive failure behaviors for loading at room temperature (with $Δ T$ = −160℃ applied to account for the temperature differential from cure) and at near-cure temperature ( $Δ T$ = −10℃) are now described.

At room temperature, as uniaxial loading was applied to the textile, the first type of damage that was predicted to occur was crack opening in the y-direction tows. The damage in the y-direction tows is shown in Figures 3 and 4. These tows run perpendicular to the applied load, and therefore they experienced large transverse normal stress. The onset of opening began at an applied $σ_{xx}$ of 90 MPa (Figure 3(a)). The tow cracks started to completely open at about $σ_{xx}$ = 100 MPa (Figure 3(b)). As seen in Figure 4, the fraction of the y-tow cohesive zones which experienced complete opening continued to increase until all y-tow cohesive zones were opened at an applied load of $σ_{xx}$ = 400 MPa.

In addition to failure in the y-direction tows, the x-direction tows were also predicted to undergo cracking as shown in Figure 5. Although these tows are aligned with the applied loading, they experience transverse tensile stress due to the thermal load and Poisson contraction as mechanical loading is increased. As shown in Figure 5(c), this stress becomes sufficient to cause complete opening in limited regions of the axial tows at an applied load of $σ_{xx}$ = 391.5 MPa. When fiber failure begins to initiate in the axial tows at an applied uniaxial stress of $σ_{xx}$ = 625.5 MPa, a little more than 35% of the cohesive zones in the axial tows have experienced complete opening, as shown in Figure 5(d).

As the tows undergo matrix cracking, the cohesive elements between the tows (Figure 6) begin to undergo degradation. The degradation in these regions tends to coincide with the edges of opened cracks in the tows as illustrated in Figure 7. This is expected for the inter-tow interfaces, as this interface forms a barrier to further extension of matrix cracks in the tows. As a result, these cracks will tend to turn and run along interfaces between tows, as observed for 2 × 2 carbon-fiber twills by Karahan.²² Similar behavior has been widely observed in tape laminates, where delaminations tend to initiate where matrix cracks interact with inter-ply interfaces. The area of the interface that has experienced degradation increases with increasing load. By the onset of fiber failure, some regions of the interface have experienced complete opening as shown in Figure 6(d) in regions where matrix cracks in the x and y direction tows interact.

Crack opening also occurs in the neat matrix pocket (Figure 8) shortly after tow cracks start to open. As seen in Figure 8(b), the cohesive zones which open are predominantly adjacent to cracks running in the y-direction tows (Figure 3(d)). This is because when a matrix crack running through a tow encounters the neat matrix pocket, the matrix pocket does not pose a barrier to continued crack growth (as opposed to a perpendicular tow). However, the strain energy required to open cracks in the more ductile neat matrix pocket is considerably higher than that needed to open cracks in the brittle fibers. Therefore, cohesive zone failure in the matrix region occurs more gradually than in the tows, with most cohesive elements experiencing partial opening. This pattern of intra-tow matrix cracks extending into the matrix pocket is in good agreement with the microscopic observations of Karahan.²² Once sufficient load has been applied to cause the onset of fiber failure, complete crack opening has occurred in several regions of the neat matrix pocket as seen in Figure 8(d).

Failure of the interface between the tows and the neat matrix pocket (Figure 9) begins to occur at higher load levels than for other cohesive interfaces. Degradation tends to occur in regions where partially opened cracks in the matrix pocket encounter tows which run perpendicular to the crack direction (Figure 9(b) and (c)). When the crack running through the matrix pocket encounters a perpendicular tow, it cannot grow into the tow, and therefore it will turn and follow the tow/matrix pocket interface. Once loading is increased to a level sufficient to cause fiber failure, a large region of this interface has experienced partial degradation, and limited regions have experienced complete failure (Figure 9(d)). These regions coincide with cracks that have completely opened in the neat matrix pocket.

In addition to cohesive zone opening, the accumulation of shear damage was tracked. Due to the low waviness ratio for the textile and the fact that the tows are aligned orthogonally with the applied uniaxial loads, shear stresses remained low, and only a very small amount of shear damage occurs.

As damage evolved in in the textile, the effective compliance of the textile was also tracked. This was accomplished by performing six different “virtual” uniaxial stress–strain tests on the unit cell at each load step. The damage state was held constant for these tests. A single component of unit-magnitude stress loading was applied in the volume-average sense for each test, meaning that the resulting volume-average strains made up the rows of the effective compliance matrix for the textile. Effective orthotropic engineering moduli were obtained from the compliance matrix, which exhibited negligible normal-shear and shear–shear coupling as damage evolved. The evolution of the in-plane moduli is given in Figure 10. An examination of these moduli reveals that there is a drop of roughly 5% in $E_{xx}$ and nearly 20% in $G_{xy}$ associated with the saturation of cracking in the y-direction tows, although $E_{yy}$ remains relatively unchanged (as expected, since cracks running along the y direction do not sever the load path of y-direction normal loading). As the applied stress is further increased, there is a gradual reduction in all three moduli as degradation increases in the matrix pocket as well as on the interfaces. A jump in crack opening in the x-direction tows at $σ_{xx}$ = 530 MPa leads to an additional drop in $G_{xy}$ .

Progressive failure under uniaxial loading was also examined near-cure temperature, with an applied temperature change of $Δ T$ = −10℃. At this temperature, the thermally induced stresses in the textile (and within the tow microstructure) are much lower than for the room-temperature case. The overall behavior for this temperature was similar to that at room temperature, so only the differences noted for this temperature will be highlighted for the sake of brevity. The initiation of critical fiber failure at this temperature was predicted to occur at an applied stress of $σ_{xx} = 688.5$ MPa, a 10% increase relative to the room-temperature case. Like the room-temperature model, y-tow cracking was the predominant failure mechanism under this load, but as seen in Figure 11, cohesive zones begin experiencing complete failure at an applied load of about $σ_{xx} = 220$ MPa (as opposed to $σ_{xx} = 100$ MPa for the room-temperature case). In short, thermally induced strains from cooling the composite after cure tend to result in damage at lower loads (i.e. they have a weakening effect on the textile). However, it was noted in the previous micromechanics study that at the micro-scale, the thermally induced stresses from cooling the composite after cure tend to increase the transverse stress required to cause matrix cracking in a tow. This is accounted for in the current model through the use of different strength distributions for the intra-tow cohesive zones at different temperatures. From this, it can be inferred that the thermally induced stresses at the textile scale have a greater effect than the thermally induced stresses at the micro-scale in regards to the applied load required to cause matrix cracking in the tows of the textile.

Cracking in the x-direction tow occurred at the near-cure temperature as well, although the extent of cracking was much lower at fiber failure than it was for the room-temperature case. A lesser extent of x tow cracking was expected since thermally induced stresses contribute significantly to the transverse tensile stresses in the x-direction tows. Degradation of the inter-tow interface and within the neat matrix pocket followed the same trend as was seen at room temperature. Like the room-temperature case, these forms of damage occurred only after y-tow cracking had initiated. Interestingly, the failure of the tow-matrix interface followed a somewhat different trend than was observed at room temperature. For the near-cure temperatures and uniaxial loading, degradation of the tow-matrix interface tended to coincide with locations where tow cracks met the matrix pocket (as seen in Figure 12), rather than coinciding with locations where crack running through the matrix pocket met perpendicular tows (Figure 9(c)). This was somewhat unexpected since this interface does not pose a barrier to crack growth (the tow crack is free to grow into the matrix and therefore has little reason to turn and grow along the interface). A definitive reason for the different behavior was not determined from more detailed study, but careful examination of the stress state in the textile showed that the room-temperature case experienced much higher compressive tractions across this interface, which will tend to inhibit cracking of the tow-matrix interface and promote crack growth from the tow straight into the pocket of neat matrix.

The evolution of the in-plane moduli for uniaxial loading near-cure temperature, shown in Figure 13, bore a strong resemblance to that seen at room temperature (Figure 10) in the sense of the amount of stiffness loss that occurred when a particular type of damage (i.e. y tow cracking, x tow cracking, etc.) was predicted to occur in the textile. The overall level of property degradation was less than for the room-temperature case, particularly for $G_{xy}$ , but this is to be expected due to the fact that less damage had occurred in the near-cure model than the room-temperature model at the applied stress causing fiber failure.

Failure under equal biaxial load

Progressive failure under in-plane equal biaxial tension shares many characteristics with unidirectional loading. As expected and shown in Figure 14, for textiles at room temperature and near-cure temperature, cracking was predicted to initiate in both tows at approximately the same load. The onset of cracking occurred at a considerably lower stress for the room-temperature model (Figure 14(a)) than for the model near-cure temperature (Figure 14(b)). For both temperatures, cohesive element degradation occurred on the inter-tow interface, in the neat matrix pocket, and on the inter-tow-matrix interface immediately after crack opening began in the tows. The models at both temperatures predicted the onset of fiber failure at nearly the same applied loading: $σ_{xx} = σ_{yy} = 733.5$ MPa for room temperature and $σ_{xx} = σ_{yy} = 742.5$ MPa near-cure temperature. Like uniaxial loading, shear damage in the tows under biaxial loading was negligible.

The evolution of in-plane properties, shown in Figure 15, provides further insight into how different damage mechanisms affect the resulting stiffness of the textile unit cell. Both $E_{xx}$ and $E_{yy}$ experience a 5–6% drop when tow cracking occurs. Furthermore, the total drop in $G_{xy}$ is approximately 35%, which suggests that the effect of tow cracking on the shear modulus is roughly additive (that is, the reduction in $G_{xy}$ resulting from cracks in both tow directions is roughly twice the reduction for tow cracking in just one direction).

Failure under in-plane shear load

Under in-plane shear loading, damage evolves in a characteristically different way than under in-plane normal loading. While shear damage was not a major occurrence under in-plane normal loading, it dominates the behavior under in-plane shear. Figure 16 shows the evolution of shear damage and tow cracking. Figure 16(a) shows the evolution of the volume average of $d_{12}$ taken over the volume of the tows. For this loading, the value of $d_{13}$ in the tows remains negligible compared to $d_{12}$ . As shear loading is initially applied, shear damage starts to develop in the regions where the tow edges cross in the unit cell, labeled A in Figure 16(c), along the edges of the tow, and in regions in close proximity to the edges of crossing tows. Continued loading leads to an increase in shear damage as well as the onset of degradation in the intra-tow cohesive zones in regions of elevated shear damage (and elevated shear stress). With further increases in loading, cohesive zone degradation becomes more widespread, and complete decohesion occurs in a limited number of locations. Additionally, at elevated loads, the shear damage in the tows continues to increase, particularly in regions where it was originally lower, labeled B in Figure 16(e). In Figure 16(e), bands of tow material with lower shear damage are apparent where tow crack opening and interfacial failure have shielded regions of the tow from elevated shear stresses.

As was observed under normal in-plane loads, partial failure of the cohesive zones on the interfaces and in the neat matrix pocket started as a result of the development of adjacent tow cracks. However, the total energy dissipated by degradation in the interfacial and neat matrix pocket cohesive zones under shear before textile failure was approximately 1/7th of that predicted for uniaxial and biaxial loading. Furthermore, the overall effect of this damage evolution on the stiffness of the textile was less than what was observed for unit cells with in-plane normal loading.

As the shear load increased and the cohesive elements in the tows, matrix pockets, and interfaces failed, the load path for carrying shear stress was severed, resulting in an instability under stress-controlled loading. This was considered to be the final failure of the textile under shear. The overall stress–strain response of the textile is shown in Figure 17. It can be noted from the overall response that the gradual development of shear failure in the tows leads to a nonlinear shear response of the textile under shear loading. This is also seen in the evolution of the effective in-plane moduli, shown in Figure 18. There is a large decrease in the in-plane shear modulus associated with the development of shear damage in the tows. There is a slight decrease in the normal moduli resulting from partial crack opening in the tows as well as the opening of cohesive zones elsewhere in the unit cell.

Characteristic progressive failure behaviors

Based on observations of the behaviors under a variety of loadings, it was determined that the damage evolution and effective response of the textile could be characterized using four different modes: matrix cracking in the x-tows and y-tows, shear damage, and the combination of neat matrix pocket cracking and interfacial failure. The following sections describe in a subjective manner the modes of damage, how the volume-average stresses in the textile affect the evolution of these characteristic modes, and how each mode affects the evolution of the effective stiffness predicted for the textile. Finally, a brief description is given for how these characteristic damage modes can be used to develop a damage model to be applied to macro-scale analyses of textiles.

Tow matrix cracking

In-plane normal loading was always observed to cause the development of brittle cracking in the tows running transverse to the applied load. This tow cracking typically developed rapidly within the tow at significantly lower loading than that required to cause fiber failure in the axial tows. It was observed that multiaxial normal loads caused this type of damage to occur at even lower stresses. Additionally, thermally induced stresses from cooling after cure tend to cause a much earlier onset of tow cracking under tensile loading. Tow cracking also occurred under shear loading after the tows had accumulated considerable shear damage. Crack opening under shear was much more gradual than under normal load, with lots of partial opening.

Tow cracking resulted in a characteristic reduction in the Young's modulus for the direction perpendicular to the cracked tow of about 5%. Also, cracking in one tow direction resulted in a drop in the in-plane shear modulus $G_{xy}$ of almost 20%, while cracking in both tows caused a drop of approximately 35%. A convenient way of tracking the evolution of this failure mode was to track the total area of opened (completely failed) cohesive elements in a given tow direction. In McLendon²¹ and in representations of tow cracking in the present work, this quantity has been normalized by the total area of all cohesive zones in the given tow. However, this may not be the most appropriate way to develop the metric. Normalizing by an area that is characteristic of the textile geometry, such as the tow cross-section, would likely be a better approach that is less likely to introduce mesh dependence.

Shear damage

Under shear loading, the predominant mode of damage was the development of distributed shear damage within the tows. Shear damage primarily consisted of $d_{12}$ ; $d_{13}$ was not observed to have a major effect on the response of the textile. It was observed that this type of damage depended only on the volume-average in-plane shear stress. A convenient way of tracking the evolution of the shear damage parameter $d_{12}$ in the tows was to take the volume average of the damage parameter over the tow volumes. Figure 19 shows that this metric correlates very strongly with the applied shear stress for a broad array of multiaxial loadings and has very little dependence on the applied normal stress. The onset of $d_{12}$ shear damage occurred at fairly low shear stresses relative to the shear strength of the textile under pure shear loading. Worsening shear damage led to a steady reduction in $G_{xy}$ for the textile as loading was increased.

Degradation in interfacial and matrix cohesive zones

The final characteristic type of failure that was observed to occur was the degradation and occasional failure of cohesive zones in the neat matrix pocket and on the interfaces between tows and between the tows and neat matrix pocket. This failure mode was strongly linked to the development of tow cracks, and almost always developed as an extension of those cracks into adjacent parts of the textile unit cell. This type of failure can be characterized for the composite by taking the total amount of strain energy that has been dissipated over all the cohesive elements in these regions of the textile, since it was noted that the degree to which degraded cohesive zones were damaged had an impact on the predicted stiffness reduction in the textile (i.e. degraded cohesive zones that had released more strain energy resulted in larger reductions in the predicted effective moduli). This type of failure was observed to result in a gradual reduction in both in-plane Young's moduli as well as the in-plane shear modulus.

Towards a macro-scale damage model

These modes form the basis of a macro-scale damage model for the textile described in McLendon.²¹ A full description of this model is beyond the scope of this work, but the basic principles of the model are presented to emphasize the utility of these characteristic damage modes.

Each characteristic damage mode is expressed as a damage state variable whose value depends upon the accumulated local damage in the textile unit cell. Textile-scale analyses are performed for a variety of multiaxial loadings. For every loading, the evolution of each damage state variable is tracked along with the effective multiaxial stress–strain response of the textile. By examining the results from different analyses, the sensitivity of the textile’s effective moduli to each damage mode can be determined (e.g. how much the x-direction Young’s modulus decreases for a given increase in y-direction tow cracking). Additionally, the various damage state evolution histories are used to build up a database that permits prediction of the damage state for an arbitrary multiaxial load via a table lookup. This damage evolution database and sensitivity data permit the damage state and effective response of the textile to be predicted using a limited number of a priori simulations rather than requiring a concurrent textile-scale analysis for every different stress history that exists in a structure. It is demonstrated in McLendon²¹ that this approach shows fairly good accuracy when compared to full textile-scale analyses for proportional loadings.

Conclusions

A textile-scale model was developed to predict progressive damage growth in a textile composite. This model used cohesive zone elements and a shear continuum damage model to account for the development of various types of local damage. The properties of the cohesive zone and continuum damage models governing intra-tow behavior were based upon a priori investigations using fiber–matrix microstructural models that included randomness in the fiber positions. Qualitatively, the evolution of damage matched behavior that has been observed in experiments performed on other textile systems. Normal loading tended to cause matrix cracking in the tows running perpendicular to the load direction. These cracks subsequently grew into the neat matrix pocket and turned to follow interfaces with crossing tows as the load was increased further. Under shear load, nonlinear behavior of the textile was recovered as a result of accumulation of diffuse shear damage in the tows followed by the opening of discrete cracks in the tows, matrix pockets, and on the interfaces within the textile. The textile-scale model was subjected to several different in-plane loadings at different temperatures. It was noted that in general, the thermally induced stresses resulting from temperature decrease after cure tend to cause earlier onset and more pronounced development of damage in the textile, even though the temperature decrease results in a transverse strength increase of the tows due to micro-scale residual stresses. It was further noted that the examined in-plane normal and shear loads tended to cause characteristic types of damage. By tracking the effective stiffness of the textiles along with the damage evolution, it was noted that each damage mode’s effect on the textile’s stiffness was very consistent across the different multiaxial in-plane loadings that were examined. Therefore, these modes of damage are deemed likely to form a useful basis for the development of a larger-scale textile damage model which can track the evolution of the damage state in a textile, along with the resulting evolution of its effective response, based on the stress history without the need for performing a textile-scale analysis for every multiaxial load that exists at different points in a structure.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Multidisciplinary University Research Initiative grant FA9550-09-1-0686 from the Air Force Office of Scientific Research to Texas A&M University with David Stargel as the program manager.

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