Failure initiation damage and durability

Volume III — Damage, Failure, and Software Implementation

Section 1 — Damage and Failure Framework

Executive Summary

This section introduces the damage, failure, and implementation layer of the Workbench micromechanics master model. Volumes I and II define the elastic and thermophysical homogenization framework used to compute effective orthotropic properties for continuous fiber, discontinuous fiber, woven textile, filled matrix, and hybrid composite architectures. Volume III extends the same framework into damage degradation, failure-index evaluation, software data flow, interfaces, and automatic Abaqus material-card export.

The purpose of Section 1 is to define the top-level mathematical and software structure for Volume III. Later sections describe the individual damage models, Tsai-Wu criterion, Hashin criterion, Workbench material properties, and Abaqus export options in greater detail.

1.1 Scope of Volume III

The present volume documents four major capability groups implemented in the solver micromechanics script:

1. Continuum damage mechanics (CDM) for stiffness degradation.
2. Failure initiation models, including Tsai-Wu and Hashin failure indices.
3. composite model, using scalar and vector arrays rather than nested structures or user-defined solver functions.
4. Abaqus material-card generation, including engineering constants, transversely isotropic, orthotropic, and fully anisotropic material definitions.

The damage and failure layer is applied after the micromechanics homogenization layer. Therefore, the model can be viewed as a sequential operator chain:

where $\mathcal{H}$ is the homogenization operator, $\mathcal{D}$ is the damage operator, $\mathcal{F}$ is the failure-evaluation operator, and $\mathcal{E}$ is the export operator used to assemble Workbench and Abaqus outputs.

1.2 Position of Damage and Failure in the Multiscale Workflow

The model first predicts the undamaged effective composite properties. Damage is then applied to the effective stiffness, not to the constituent stiffnesses. This implementation is computationally efficient and appropriate when damage variables are interpreted as homogenized lamina-level state variables.

The undamaged effective stiffness is obtained from the selected micromechanics model:

where $\mathbf{C}^{0}$ is the undamaged effective stiffness tensor. The damaged stiffness is then written as

where $\mathbf{D}$ is the damage degradation operator. This double-sided form preserves major symmetry of the stiffness matrix and is consistent with common engineering implementations of continuum damage mechanics.

A failure model then evaluates whether the current stress state exceeds an initiation criterion:

where $FI$ is the failure index, $\mathbf{\sigma}$ is the stress vector, and $\mathbf{X}$ denotes the set of strength allowables.

1.3 Damage Variables Used in the solver Model

The implemented damage model uses three scalar damage variables:

where $d_{f}$ is the fiber-direction damage variable, $d_{m}$ is the matrix/transverse damage variable, and $d_{s}$ is the shear damage variable. Each variable is bounded as

The corresponding damage operator in Voigt notation is

The use of a single matrix damage variable for both transverse directions is consistent with the engineering assumption that transverse normal damage is matrix dominated. The shear damage variable is applied to all three shear modes in the simplified composite model.

1.4 Engineering-Constant Form of Damage Degradation

and

where the superscript $0$ denotes the undamaged value and $d$ denotes the damaged value. These equations summarize the stiffness-degradation logic implemented in the current Workbench-safe solver script.

1.5 Failure Indices and Failure Initiation Logic

The solver script provides two failure-index options: Tsai-Wu and Hashin. The Tsai-Wu failure model is an interactive polynomial criterion for anisotropic materials. In compact form,

Failure initiation is predicted when

The Hashin model separates fiber and matrix failure modes. The governing Hashin failure index is evaluated as the maximum of the active mode indices:

where $ft$, $fc$, $mt$, and $mc$ denote fiber tension, fiber compression, matrix tension, and matrix compression, respectively.

The failure module is optional. It can be disabled when strength data are unavailable or when the model is used only for property prediction and Abaqus material-card generation.

1.7 Damage and Failure Settings

Table 1 summarizes the damage, failure and export settings.

Table 1. Damage, failure and export settings.

The damage variables are interpreted as prescribed state variables. The current script does not automatically evolve damage with load history; rather, it provides the constitutive degradation framework into which an external progressive-damage update law can be inserted later.

1.9 Figure Placeholders

Figure 1. Damage and failure position in the full micromechanics workflow.
[Insert flowchart showing homogenization, damage degradation, failure evaluation, property evaluation, and Abaqus export.]

Figure 2. Continuum damage operator acting on the undamaged effective stiffness matrix.
[Insert schematic showing $\mathbf{C}^{0}$, $\mathbf{D}$, and $\mathbf{C}^{d}$.]

1.10 References

1. Lemaitre, J. (1996). A Course on Damage Mechanics, 2nd ed. Springer.
2. Ladevèze, P. (1983). A damage computational method for composite structures. Computers & Structures.
3. Tsai, S. W., and Wu, E. M. (1971). A general theory of strength for anisotropic materials. Journal of Composite Materials, 5, 58-80.
4. Hashin, Z. (1980). Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics, 47, 329-334.
5. Matzenmiller, A., Lubliner, J., and Taylor, R. L. (1995). A constitutive model for anisotropic damage in fiber-composites. Mechanics of Materials, 20, 125-152.
6. Abaqus Documentation. Material definitions: engineering constants, orthotropic elasticity, anisotropic elasticity, expansion, conductivity, and specific heat.
7. Jones, R. M. (1999). Mechanics of Composite Materials, 2nd ed. Taylor & Francis.

1.11 Section Summary

Subsequent sections of Volume III document each damage and failure model and then describe the Workbench and Abaqus implementation details in a software-reference format.

Volume III — Section 2

Continuum Damage Mechanics and Stiffness Degradation

2.1 Purpose and Scope

The CDM module is not a full progressive failure solver by itself. Instead, it applies user-supplied damage state variables to the effective engineering constants and exports both undamaged and damaged properties. This structure allows Workbench, Abaqus, test correlation scripts, or a future progressive damage law to update the damage variables externally while keeping the micromechanics model stable and transparent.

In the current implementation, the damage variables are:

2.2 Undamaged Effective Stiffness State

Before damage is applied, the micromechanics branch produces an undamaged orthotropic material state:

The corresponding compliance matrix in engineering Voigt notation is:

The undamaged stiffness matrix is recovered from:

This undamaged state is saved in the solver implementation before degradation so that the output vector can report both pre-damage and post-damage material properties.

2.3 Scalar Damage Variables

The model uses scalar damage variables bounded between zero and unity:

Here, $d_{f}$ denotes fiber-direction damage, $d_{m}$ denotes matrix/transverse damage, and $d_{s}$ denotes shear damage. The limiting values have the following interpretation:

The solver code clips damage values to remain below unity to avoid exactly singular stiffness matrices. This is essential for Workbench and Abaqus workflows, where a zero stiffness can produce ill-conditioned matrices or unstable material cards.

2.4 Damage Operator in Engineering Notation

The implemented damage operator is diagonal in engineering notation:

The first component corresponds to the material 1-direction, which is typically the fiber-dominated direction. The second and third components correspond to transverse matrix-dominated normal responses. The final three components correspond to shear responses.

A double-sided damage operation gives the damaged stiffness:

The double-sided form preserves the major symmetry of the stiffness matrix:

2.5 Engineering-Constant Damage Laws Used in the Code

The implemented stiffness degradation laws are:

The squared form is intentionally conservative relative to a linear knockdown law and reflects the double-sided degradation operator. It also maintains continuity of stiffness as damage evolves.

The Poisson ratios are not explicitly degraded in the current script:

This is a practical engineering assumption. In a future extension, Poisson ratios may be recomputed from a fully degraded compliance matrix or linked to damage-mode-specific evolution laws.

2.6 Positive-Definite Stiffness Protection

For numerical robustness, the model applies lower bounds to damaged moduli when the positive-definite projection flag is active. The intent is to ensure:

At the engineering constant level, the stabilized form may be written as:

where $E_{min}$ and $G_{min}$ are small positive stiffness floors. This protection prevents singular compliance matrices during Abaqus export and protects the Workbench execution path from divide-by-zero errors.

2.7 Damage and Thermophysical Properties

The current implementation applies CDM only to mechanical stiffness. Effective density, thermal conductivity, coefficient of thermal expansion, and specific heat are computed independently of the damage variables:

This is appropriate for a first-order engineering material card where damage represents stiffness degradation rather than physical mass loss, crack-surface heat-transfer changes, or evolving thermal expansion. If a coupled thermomechanical damage model is required, thermal conductivity and CTE may be degraded by additional variables or functions of crack density.

2.8 Relationship Between Damage and Failure Criteria

Damage and failure are separate modules in the solver script. Damage modifies the stiffness used to define the exported material response, while failure indices evaluate whether a stress state satisfies a failure criterion. The conceptual workflow is:

Undamaged Micromechanics
        ↓
Effective Engineering Constants
        ↓
Apply Damage Variables
        ↓
Damaged Engineering Constants
        ↓
Evaluate Failure Indices
        ↓
Export Workbench and Abaqus Outputs

The current code uses placeholder stress components unless a future Workbench interface extension supplies applied stress or strain states. The damage module therefore provides stiffness degradation capability independent of stress-driven damage evolution.

2.11 Abaqus Material-Card Implications

When Abaqus export is enabled, the engineering constants written to the material card are the damaged values:

For *ELASTIC, TYPE=ENGINEERING CONSTANTS, this produces:

*ELASTIC, TYPE=ENGINEERING CONSTANTS
E1d, E2d, E3d, NU12, NU13, NU23, G12d, G13d, G23d

The Abaqus card is therefore consistent with the damaged stiffness state used by the solver model. The original undamaged properties remain available in the scalar output vector for diagnostics.

2.12 Implementation Notes

1. Store undamaged effective stiffness values.
2. Clip damage variables to admissible bounds.
3. Apply squared degradation factors to $E_{i}$ and $G_{ij}$.
4. Apply stiffness floors if positive-definite protection is active.
5. Export damaged properties to subsequent analysis.
6. Use damaged constants in the Abaqus material card.

This structure is robust for Workbench because it avoids nested functions, dynamic objects, and nonlinear iterations inside the damage module.

2.13 Figure Placeholders

Figure 2.1. CDM workflow in the micromechanics model.
[Insert schematic showing undamaged micromechanics → damage variables → damaged stiffness → Workbench/Abaqus outputs.]

Figure 2.2. Physical interpretation of $d_{f}$, $d_{m}$, and $d_{s}$.
[Insert schematic showing axial fiber damage, transverse matrix cracking, and shear degradation.]

Figure 2.3. Stiffness degradation curves.
[Insert plot of $E_{1}^{d}/E_{1}^{0}$, $E_{2}^{d}/E_{2}^{0}$, and $G_{12}^{d}/G_{12}^{0}$ versus damage variable.]

2.14 References

1. Lemaitre, J. (1996). A Course on Damage Mechanics. Springer.
2. Kachanov, L. M. (1958). Time of the rupture process under creep conditions. Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk.
3. Ladevèze, P. (1983). A damage computational method for composite structures. Computers & Structures.
4. Matzenmiller, A., Lubliner, J., and Taylor, R. L. (1995). A constitutive model for anisotropic damage in fiber-composites. Mechanics of Materials.
5. Hashin, Z. (1980). Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics.
6. Tsai, S. W., and Wu, E. M. (1971). A general theory of strength for anisotropic materials. Journal of Composite Materials.
7. Talreja, R. (1985). Transverse cracking and stiffness reduction in composite laminates. Journal of Composite Materials.
8. Barbero, E. J. (2018). Finite Element Analysis of Composite Materials Using Abaqus. CRC Press.

Volume III - Section 3

Composite Failure Criteria

3.1 Purpose of the Failure Module

The current implementation supports two classical composite failure formulations:

1. Tsai-Wu interactive failure criterion.
2. Hashin mode-separated failure criterion.

The purpose of the failure module is not to replace a full progressive damage analysis, but to provide a consistent first-ply or material-point-level failure indicator using the same material property properties carried into the analysis.

Figure 3.1. Failure evaluation workflow from constituent strengths and material stresses to scalar failure indices.

3.2 Stress Components Used by the Failure Model

The failure model is evaluated in the local material coordinate system. The local stress vector is written as

where the subscripts 1, 2, and 3 denote the principal orthotropic material directions. Direction 1 is normally associated with the dominant fiber or tow direction, while directions 2 and 3 are matrix-dominated transverse directions.

In the present solver script, the failure stress state is retained as a placeholder stress vector unless external stress inputs are added through Workbench. This design permits the same code framework to be coupled later to structural analysis outputs, coupon-level stress states, or ply-level laminate analysis.

3.3 Strength Inputs

The fiber material property input vector carries the primary tensile, compressive, and shear allowables used by both Tsai-Wu and Hashin models. The strength set is

where

• $X_{t}$ and $X_{c}$ are longitudinal tensile and compressive strengths,
• $Y_{t}$ and $Y_{c}$ are transverse tensile and compressive strengths,
• $Z_{t}$ and $Z_{c}$ are third-direction tensile and compressive strengths,
• $S_{12}$, $S_{13}$, and $S_{23}$ are shear strengths.

The solver input mapping is summarized in Table 3.1.

Table 3.1. Strength input mapping used by the current solver implementation.

3.4 Tsai-Wu Failure Criterion

The Tsai-Wu criterion is an interactive polynomial failure theory for anisotropic composite materials. It combines linear stress terms, quadratic stress terms, and interaction terms into a scalar failure index. Failure is predicted when the index reaches unity.

The general form is

with failure initiation defined by

For the three-dimensional orthotropic stress state used in the solver model, the expanded form is

The linear coefficients distinguish tensile and compressive strengths:

The direct quadratic coefficients are

and the shear coefficients are

When biaxial interaction data are unavailable, the implementation uses the common engineering approximation

The Tsai-Wu index is useful when a smooth scalar interaction surface is desired. It is especially convenient for optimization, design-space exploration, and quick screening of material systems.

Figure 3.2. Schematic Tsai-Wu interaction surface in normal stress space.

3.5 Hashin Failure Criterion

Hashin failure theory separates failure into physically interpretable fiber and matrix modes. This is useful for composite materials because fiber-dominated and matrix-dominated failures have different consequences for stiffness degradation, residual strength, and progressive damage modeling.

The solver model evaluates four Hashin-type mode indices:

1. Fiber tension.
2. Fiber compression.
3. Matrix tension.
4. Matrix compression.

The governing Hashin failure index is the maximum of the active mode indices:

Failure initiation is predicted when

3.6 Fiber Tension Mode

For positive longitudinal stress, the fiber tension index is

This mode represents tensile rupture or tensile-driven degradation of the fiber-dominated direction. In a progressive model, this index would normally be associated with the fiber damage variable $d_{f}$.

3.7 Fiber Compression Mode

For negative longitudinal stress, the fiber compression index is implemented as

3.8 Matrix Tension Mode

The matrix tension mode is evaluated when the transverse normal stress sum is tensile:

For $\sigma_{T} \geq 0$, the matrix tension failure index is

Matrix tension is associated with transverse cracking, microcrack initiation, and matrix-dominated stiffness loss.

3.9 Matrix Compression Mode

For $\sigma_{T} < 0$, the matrix compression index is implemented as

This form accounts for the interaction between transverse compression and shear loading, which is often critical in matrix-dominated composite failure.

Figure 3.3. Hashin failure mode map showing fiber tension, fiber compression, matrix tension, and matrix compression regions.

3.10 Failure Model Selection

Choose the failure criterion appropriate to the material and the available strength data.

Table 3.2. Failure model selection logic.

This flag allows Workbench to activate or deactivate failure calculations without changing the solver script.

3.11 Failure Outputs

The primary scalar outputs from this section are the Tsai-Wu and Hashin failure indices.

Table 3.3. Failure scalar outputs.

Model outputs use a consistent metadata convention for downstream reporting.

3.12 Relationship to Damage Variables

The failure criteria compute initiation indices. The continuum damage variables described in Section 2 degrade stiffness:

A future progressive failure extension may connect the failure indices to damage update laws of the form

where $N$ may represent cycle count for fatigue extensions, $T$ may represent temperature, and $\Delta t$ is the analysis increment.

The present solver implementation does not automatically evolve damage variables from the failure indices. Instead, damage variables are prescribed through Workbench scalar inputs, while the failure indices provide diagnostic measures.

3.13 Implementation Workflow

The failure calculation sequence is summarized as follows:

Read strength allowables
        |
        v
Read or define local stress state
        |
        v
Compute Tsai-Wu coefficients
        |
        v
Evaluate Tsai-Wu failure index
        |
        v
Evaluate Hashin mode indices
        |
        v
Return scalar failure outputs

This workflow is intentionally simple and deterministic for Workbench execution.

3.14 Applicability and Limitations

The Tsai-Wu model provides a smooth scalar interaction criterion but does not distinguish physical failure modes. Hashin theory provides physically interpretable failure modes but remains an initiation criterion unless coupled to damage evolution laws. Neither criterion alone determines post-failure stiffness evolution. In the current model, stiffness degradation is controlled separately through continuum damage variables.

The failure indices should therefore be interpreted as material-point-level initiation measures and design screening metrics rather than complete progressive failure simulations.

References

1. Tsai, S. W., and Wu, E. M. (1971). A general theory of strength for anisotropic materials. Journal of Composite Materials, 5, 58-80.
2. Hashin, Z. (1980). Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics, 47, 329-334.
3. Hashin, Z., and Rotem, A. (1973). A fatigue failure criterion for fiber reinforced materials. Journal of Composite Materials, 7, 448-464.
4. Daniel, I. M., and Ishai, O. (2006). Engineering Mechanics of Composite Materials, 2nd ed. Oxford University Press.
5. Jones, R. M. (1999). Mechanics of Composite Materials, 2nd ed. Taylor & Francis.
6. Puck, A., and Schurmann, H. (1998). Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology, 58, 1045-1067.
7. Camanho, P. P., and Matthews, F. L. (1999). A progressive damage model for mechanically fastened joints in composite laminates. Journal of Composite Materials, 33, 2248-2280.

Volume III – Section 4

Damage and Failure Evaluation Workflow

4.1 Purpose and Scope

The damage and failure workflow is intentionally modular. The effective elastic and thermophysical property predictions can be executed without activating damage or failure. When damage variables are prescribed, the stiffness tensor or engineering constants are degraded before output. When strength allowables and a stress state are available, failure indices are evaluated using the selected criterion. The solver implementation currently supports a prescribed-damage workflow and a failure-index workflow; it does not automatically evolve damage in time unless an external Workbench or solver-level update law supplies new damage variables.

The key workflow outputs are:

• damaged orthotropic engineering constants,
• undamaged reference properties,
• prescribed damage variables,
• Tsai-Wu failure index,
• Hashin modal failure indices,
• maximum governing failure index,
• Abaqus material card data derived from the damaged properties.

4.2 Position of the Damage/Failure Module in the Full Model

The damage and failure module acts after micromechanical homogenization and architecture correction. The full material-property workflow may be summarized as:

Constituent inputs
      -> phase properties
      -> filled matrix / hybrid phase assembly
      -> micromechanics model selection
      -> architecture correction
      -> effective undamaged properties
      -> CDM stiffness degradation
      -> failure index evaluation
      -> effective properties
      -> Abaqus material card string

The model first computes undamaged engineering constants

After damage degradation, the final output property vector becomes

In the current engineering implementation, the Poisson ratios are retained from the effective undamaged material state unless a later extension explicitly evolves damage-dependent coupling terms.

4.3 Damage State Vector

The prescribed damage state is represented by three scalar variables:

where $d_{f}$ is the fiber-direction damage variable, $d_{m}$ is the matrix-normal damage variable, and $d_{s}$ is the shear damage variable. Each variable is constrained to the physically meaningful interval

In the solver script, numerical clipping is applied to avoid exactly singular stiffness values:

This rule is not a constitutive law; it is a numerical protection used to maintain finite output properties for Workbench and Abaqus export.

4.4 Engineering-Constant Degradation Workflow

The transverse moduli are degraded by the matrix-normal damage variable:

The shear moduli are degraded by the shear damage variable:

The quadratic degradation form is numerically smooth and ensures monotonic stiffness reduction. It also mirrors the double-sided tensor degradation form introduced in Section 2:

The direct engineering-constant workflow is preferred in the Workbench script because it avoids user-defined solver functions and avoids repeated tensor assembly for simple engineering studies.

4.5 Positive-Definite Stiffness Protection

After degradation, each modulus is bounded below by a positive floor value:

The floor is not intended to represent real residual material stiffness. It prevents singular compliance matrices when the output is converted into a full stiffness matrix for anisotropic Abaqus export.

The compliance matrix used for orthotropic reconstruction is

The corresponding stiffness matrix is obtained by inversion:

4.6 Failure-Model Selection

Choose the active failure model:

The selected failure model is evaluated after the damaged properties have been assembled. The failure model does not modify the stiffness by itself. Instead, it reports failure indices that may be used by Workbench, Abaqus, or an external design loop.

4.7 Stress State for Failure Evaluation

The general three-dimensional stress vector is

In the current solver implementation, the stress variables are initialized as placeholders unless supplied by a future extension. This design keeps the material-property calculator independent of a structural stress solver. In a coupled workflow, the stress state would be supplied by:

1. a laminate analysis tool,
2. a finite-element postprocessor,
3. a Workbench stress module,
4. a load-envelope evaluation routine,
5. or a progressive damage loop.

The failure index function can therefore be interpreted as a local material-point assessment operating on the stress vector of Eq. (4.17).

4.8 Tsai-Wu Workflow

When Tsai-Wu is active, the solver computes the scalar failure index

Failure initiation is indicated when

The workflow is:

Read strengths
      -> compute Tsai-Wu coefficients
      -> read stress vector
      -> evaluate quadratic interaction index
      -> report FI_TsaiWu

A design margin may be calculated as

A positive margin indicates that the evaluated stress state is below the Tsai-Wu failure surface.

4.9 Hashin Workflow

When Hashin is active, the model evaluates separate modal indices:

where $ft$ denotes fiber tension, $fc$ denotes fiber compression, $mt$ denotes matrix tension, and $mc$ denotes matrix compression. Failure initiation is indicated when

The workflow is:

Read strengths
      -> identify stress sign
      -> evaluate fiber mode
      -> evaluate matrix mode
      -> select maximum modal index
      -> report FI_Hashin

Unlike Tsai-Wu, Hashin provides a failure-mode classification. This makes it useful for diagnostic interpretation and progressive damage modeling.

4.10 Governing Failure Status Logic

The implemented failure-status logic is summarized by a governing index:

when both criteria are active. A binary failure flag can be defined as

The current solver file outputs the indices directly. Workbench can apply Eq. (4.24) externally if a binary pass/fail output is desired.

4.11 Recommended Progressive-Damage Extension

Although the current model uses prescribed damage variables, the same framework can be extended to progressive damage by updating damage as a function of failure index. A simple rate-independent update form is

where

and the Macaulay bracket is

This extension is not required for the present Workbench material-property model, but it provides a clear path for coupling the micromechanics module to nonlinear laminate or finite-element analyses.

4.13 Figure Placeholders

Figure 4.1. Damage and failure workflow.
[Insert flowchart showing homogenization -> damage degradation -> failure index evaluation -> reported properties -> Abaqus material card.]

Failure-model selection. Compare Tsai-Wu and Hashin predictions using the same stress state and strength data.

Figure 4.3. Progressive damage extension.
[Insert schematic showing failure index exceeding unity, damage update, stiffness degradation, and re-evaluation.]

4.14 Implementation Notes

The damage and failure workflow is intentionally kept separate from the homogenization model. This allows the same effective property solver to be used for purely elastic material characterization, damaged-property estimation, or failure-index screening. This separation is also important for Workbench deployment because it avoids state-dependent solver memory unless the user intentionally implements a progressive damage loop outside the script.

1. Damage variables are scalar inputs, not hidden internal state variables.
2. Failure criteria report indices but do not automatically degrade stiffness.
3. Abaqus export uses the final degraded engineering constants.

These rules make the material model deterministic, repeatable, and suitable for automated design studies.

4.15 References

Hashin, Z. (1980). Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics, 47, 329–334.

Ladevèze, P. (1983). A damage mechanics approach for composite materials. Journal of Engineering Materials and Technology, 105, 280–286.

Lemaitre, J. (1996). A Course on Damage Mechanics. Springer.

Matzenmiller, A., Lubliner, J., and Taylor, R. L. (1995). A constitutive model for anisotropic damage in fiber-composites. Mechanics of Materials, 20, 125–152.

Tsai, S. W., and Wu, E. M. (1971). A general theory of strength for anisotropic materials. Journal of Composite Materials, 5, 58–80.