Constituents to effective ply properties

Table of Contents

1 Preface

2 Abstract

A compact elastic micromechanics framework is described for predicting the effective orthotropic stiffness of continuous fiber, discontinuous fiber, woven, filler-reinforced, and general multiphase composite materials. The model permits selection among four main homogenization branches: rule of mixtures, Halpin–Tsai, Mori–Tanaka engineering approximation, and a self-consistent engineering approximation. Constituent properties are supplied through Workbench vector inputs, while microstructure descriptors are supplied through scalar flags and scalar parameters. Fiber architecture effects are represented using fiber volume fraction, aspect ratio, and Tsu-Wei Chou axial and planar architecture parameters, denoted $F_{a}$ and $F_{p}$. The model outputs effective Young’s moduli, shear moduli, Poisson ratios, and related quantities for use in subsequent laminate and structural studies.

3 Nomenclature

4 1. Model Scope

The solver model predicts effective properties of composite materials from constituent properties and architecture descriptors. The main elastic outputs are the orthotropic engineering constants

These properties are used to generate Abaqus material definitions. Choose from the following stiffness models:

Choose the material architecture independently of the stiffness model:

This separation allows, for example, a discontinuous-fiber material to be evaluated using a simple rule-of-mixtures approximation or a more interaction-aware self-consistent approximation.

5 2. Constituent Property Inputs

Specify the properties of each constituent. The fiber property vector contains anisotropic stiffnesses, strengths, density, and thermophysical values. In Volume I only the elastic terms are required:

The matrix vector contains isotropic elastic properties,

and the filler vector contains particle properties,

The matrix shear modulus is computed from isotropic elasticity:

Similarly, for an isotropic particle phase,

6 3. Constituent Stiffness and Compliance

where $r$ denotes the phase. For an isotropic phase, the stiffness tensor is

with

For an orthotropic fiber, the compliance matrix in material coordinates is conventionally written as

The reciprocal Poisson ratios satisfy

The stiffness matrix is obtained by inversion:

7 4. Volume Fractions and Phase Normalization

The model normalizes phase volume fractions so that

where $V_{v}$ is the optional void volume fraction. If the filler is treated implicitly, the matrix is first modified and the explicit particle phase is set to zero. If the filler is treated explicitly, the global particle fraction is estimated from the filler fraction inside the resin phase:

The remaining matrix fraction is then

This normalization is important because all mixture and mean-field equations assume that the phase fractions sum to unity.

8 5. Implicit Filled Matrix Model

Before fiber/matrix homogenization, a filler can be used to modify the matrix modulus. The current script uses a Halpin–Tsai style spherical-particle correction. The effective filled matrix modulus is

where

The filled matrix shear modulus is

with

A consistent effective Poisson ratio is recovered by

This option is computationally efficient and appropriate when filler is treated as a matrix modifier rather than as a separate interacting phase.

9 6. Rule of Mixtures

The rule-of-mixtures model is the simplest model and is retained as a bounding and verification option. In the fiber direction, the iso-strain assumption gives

The phase-averaged stress is

Substituting phase constitutive laws yields the Voigt axial modulus

For transverse loading, the inverse or Reuss estimate is used:

Similarly,

The shear moduli are estimated using inverse mixtures:

and

The major Poisson ratios are computed by direct mixture:

The same form is used for $\nu_{13}$ and $\nu_{23}$. The rule-of-mixtures option is not intended to capture inclusion interaction, but it provides a transparent reference case and useful numerical sanity check.

10 7. Halpin–Tsai Model

The Halpin–Tsai model provides a semi-empirical correction for reinforcement geometry and phase contrast. For a generic engineering modulus $M$, the Halpin–Tsai form is

where

For transverse Young’s modulus, the code uses the common circular-fiber estimate

so that

with

The third-direction modulus is treated analogously:

The shear moduli are estimated using

where

The code uses $\xi_{G} \approx 1$ as a default engineering estimate. The same equation form is applied to $G_{13}$ and $G_{23}$. Halpin–Tsai is more physically representative than rule of mixtures for transverse and shear properties, but the parameter $\xi$ is empirical and should be calibrated when reliable data are available.

11 8. Mori–Tanaka Mean-Field Theory

The classical Mori–Tanaka method is based on an inclusion phase embedded in a matrix reference medium. In compact tensor notation, the inclusion strain concentration tensor is

where $\mathbf{S}$ is the Eshelby tensor. The Mori–Tanaka stiffness is commonly written as

For multiple inclusion phases, a normalized mean-field form may be written as

where $\Phi_{MT}$ is an interaction factor depending on $V_{f}$ and stiffness contrast. The implemented form is an engineering approximation to the interaction behavior of a mean-field model:

The transverse and shear terms are based on Halpin–Tsai values with a small aspect-ratio-dependent correction. The aspect ratio factor is

A representative transverse correction is

The Mori–Tanaka option should therefore be interpreted as a computationally robust engineering approximation to mean-field behavior rather than a complete fourth-order tensor Mori–Tanaka implementation.

12 9. Self-Consistent Field Theory and Implemented Approximation

In the full self-consistent theory, each phase is embedded in the unknown effective medium instead of in the matrix. The governing consistency equation is

The phase concentration tensor is

A fixed-point update form is

with

The transverse moduli are blended from Halpin–Tsai and a stiffness-interaction term:

The shear moduli use analogous blended expressions:

and similarly for $G_{13}^{SC}$. This form is designed to remain stable in Workbench while providing a stronger interaction correction than the pure Halpin–Tsai model.

13 10. Continuous Fiber Composite Model

For continuous fibers, the length efficiency is set to unity:

The axial orientation factor is forced close to the aligned limit:

For continuous aligned fibers, the axial modulus approaches the Voigt expression

However, the transverse and shear properties are still computed from the selected micromechanics branch. This distinction is important: even when the axial response is mixture-like, the transverse stiffness and shear stiffness remain controlled by matrix constraint, fiber packing, and phase interaction. The continuous fiber branch is appropriate for unidirectional prepreg laminae, pultruded rods, filament-wound laminae, and tow-level properties in woven fabrics.

14 11. Discontinuous Fiber Composite Model

Model scope. This section describes the compact elastic architecture-efficiency path. The unified solver's criterion-6 tensile branch implements the complete Henry and Pimenta (2017) event-driven workflow: reconstructed broken/shear-lag interaction segments, a generic nonlinear matrix shear law, ordered Weibull breaks, same-strain network rebuilding, RVE series coupling, pull-out resistance, and nonlinear critical-cluster termination. See the aligned discontinuous-fiber theory and traceability page.

In short-fiber composites, a fiber does not generally reach the full far-field axial stress because stress must build up through matrix-to-fiber shear transfer. The model uses a compact length efficiency factor

where

This expression has the correct limits:

and

The axial contribution of short fibers is then reduced by length and orientation factors:

The implemented code uses the Chou axial factor $F_{a}$ as the default orientation efficiency:

The transverse moduli are modified by a planar/random architecture correction. A representative engineering correction is

with a similar expression for $E_{3}$. This correction reduces directional stiffness when fibers are less aligned and redistributes stiffness into planar directions as the random component increases.

15 12. Tsu-Wei Chou Architecture Factors

The model uses compact Chou-style architecture parameters to describe orientation effects without requiring the Workbench user to enter a full orientation tensor. The axial parameter $F_{a}$ represents the fourth-order projection of fiber direction onto the primary material axis:

The planar or random contribution is represented as

or as an independently supplied parameter when calibrated from data. The axial stiffness contribution is governed primarily by $F_{a}$:

A compact architecture-modified stiffness expression is

For a perfectly aligned material,

For planar random orientation, $F_{a}$ is smaller and $F_{p}$ is larger. For three-dimensional random orientation, the axial fourth-order projection is lower still. The $F_{a},F_{p}$ representation is less general than a full second- and fourth-order orientation tensor, but it is much more compact and well suited for Workbench front-panel entry.

16 13. Orthotropic Engineering Constant Extraction

After the selected micromechanics branch and architecture corrections are applied, the model assembles effective engineering constants. The major Poisson ratios are estimated by direct mixture:

The effective compliance matrix is then assembled as

The stiffness matrix used for anisotropic Abaqus export is

When the positive-definite projection flag is enabled, the model applies lower bounds to moduli and regularizes ill-conditioned matrices to prevent nonphysical Abaqus material definitions.

17 14. Workflow Summary

1. Parse material and architecture flags.
2. Construct matrix, fiber, filler, and void phase properties.
3. Apply implicit filled-matrix correction if required.
4. Normalize all phase volume fractions.
5. Choose the micromechanics model.
6. Choose the material architecture.
7. Compute baseline elastic constants.
8. Apply discontinuous-fiber, Chou, or woven architecture corrections.
9. Compute Poisson ratios and assemble the compliance matrix.
10. Pass the dense property state through the thermophysical, void, damage, and failure branches documented in Volumes II and III.
11. Apply the global constituent-limit recovery check to the final property state.
12. Rebuild compliance, stiffness, finite-element properties, and material exports from the recovered state.
13. Store results in Workbench scalar, vector, matrix, and selected-string outputs.

18 15. Constituent-Limit Recovery

Every homogenization branch is required to recover the exact active constituent at a pure-phase endpoint. This is an implementation verification condition as well as a physical consistency condition. If phase $r$ is dominant within the selected tolerance, the final mechanical and thermophysical property vector is replaced by that phase vector:

The recovered vector includes the elastic constants, density, specific heat, directional conductivity, and directional CTE. Thus $V_f\rightarrow 1$ returns the fiber properties exactly; matrix- and filler-dominated limits recover their corresponding phase definitions. The void endpoint uses the defined void state.

Recovery is deliberately applied after homogenization, architecture, thermophysical, void, and damage/failure operations. Every dependent quantity is then regenerated: orthotropic and anisotropic stiffness and compliance matrices, effective properties, scalar diagnostics, Workbench FE matrices, and the selected material export. This ordering prevents a later correction from overwriting the endpoint state or an export from retaining stale pre-recovery values. See and .

19 16. Applicability and Limitations

The current implementation is most appropriate for preliminary design, material screening, embedded Workbench workflows, and automated Abaqus material-card generation. It is not intended to replace detailed finite-element RVE homogenization when exact tow geometry, interphase behavior, nonlinear matrix response, or progressive damage evolution must be resolved explicitly.

20 References

Advani, S. G., and Tucker, C. L. III (1987). The use of tensors to describe and predict fiber orientation in short fiber composites. Journal of Rheology, 31, 751–784.

Benveniste, Y. (1987). A new approach to the application of Mori–Tanaka’s theory in composite materials. Mechanics of Materials, 6, 147–157.

Budiansky, B. (1965). On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids, 13, 223–227.

Chou, T.-W., Nomura, S., and Taya, M. (1980). A self-consistent approach to the elastic stiffness of short-fiber composites. Journal of Composite Materials, 14, 178–188.

Christensen, R. M. (1979). Mechanics of Composite Materials. Wiley.

Cox, H. L. (1952). The elasticity and strength of paper and other fibrous materials. British Journal of Applied Physics, 3, 72–79.

Henry, J., and Pimenta, S. (2017). Semi-analytical simulation of aligned discontinuous composites. Composites Science and Technology, 144, 230–244. doi:10.1016/j.compscitech.2017.01.027.

Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London A, 241, 376–396.

Folgar, F., and Tucker, C. L. III (1984). Orientation behavior of fibers in concentrated suspensions. Journal of Reinforced Plastics and Composites, 3, 98–119.

Halpin, J. C., and Kardos, J. L. (1976). The Halpin–Tsai equations: A review. Polymer Engineering and Science, 16, 344–352.

Hill, R. (1965). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13, 213–222.

Jones, R. M. (1999). Mechanics of Composite Materials, 2nd ed. Taylor & Francis.

Mori, T., and Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21, 571–574.

Mura, T. (1987). Micromechanics of Defects in Solids, 2nd ed. Martinus Nijhoff.

Nemat-Nasser, S., and Hori, M. (1999). Micromechanics: Overall Properties of Heterogeneous Materials, 2nd ed. Elsevier.

Tandon, G. P., and Weng, G. J. (1984). The effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites. Polymer Composites, 5, 327–333.

Volume II - Section 6: Thermal Conductivity Micromechanics

6.1 Purpose and Scope

Figure V-II-S6-1. Thermal-conductivity homogenization workflow used by the Workbench micromechanics implementation.

6.2 Local Heat-Conduction Law

The local thermal field is assumed to satisfy Fourier’s law in each phase,

where $q_{i}^{(r)}$ is the phase heat flux vector, $k_{ij}^{(r)}$ is the phase thermal conductivity tensor, and $T^{(r)}$ is the local temperature field. For an effective homogeneous composite, the corresponding macroscopic relation is

For an orthotropic material aligned with the material principal axes, the conductivity tensor is diagonal,

The solver implementation outputs the three principal components $k_{1}$, $k_{2}$, and $k_{3}$ because these are the values required by Abaqus for an orthotropic conductivity card.

6.3 Rule-of-Mixtures Conductivity Limits

For heat flow parallel to continuous fibers or aligned high-conductivity inclusions, the Voigt-type expression is used,

For heat flow normal to layered or fiber-like phase paths, the Reuss-type series expression is

For a fiber-matrix-filler-void system, the implemented parallel idealization is

The transverse or through-thickness conductivity may be estimated as

and

In the numerical implementation, the void conductivity $k_{v}$ is assigned a small positive value rather than zero so that division by zero is avoided.

Figure V-II-S6-2. Parallel and series conductivity idealizations used as upper and lower engineering estimates.

6.4 Filler-Modified Matrix Conductivity

If filler is treated implicitly as a matrix modifier, the filled matrix conductivity is estimated using a Maxwell-type expression for dispersed particles,

Here $k_{m}^{*}$ is the effective conductivity of the filled resin, $k_{m}$ is the neat matrix conductivity, $k_{p}$ is the particulate filler conductivity, and $V_{p}$ is the filler volume fraction within the resin phase. This expression is consistent with the dilute spherical-inclusion limit and provides a stable engineering model for filled polymer matrices.

If filler is treated explicitly, the filler enters the multiphase conductivity equations through its own global volume fraction.

6.5 Mean-Field Conductivity Approximation

The thermal-conductivity analogue of a mean-field inclusion model can be expressed using a thermal concentration tensor $\mathbf{B}_{r}$,

For an inclusion phase embedded in a matrix reference medium,

where $\mathbf{P}_{r}$ is a thermal polarization tensor that depends on inclusion geometry and reference conductivity. The current Workbench-safe script does not perform a full tensor integration for $\mathbf{P}_{r}$; instead, it uses robust engineering approximations based on the parallel, series, filled-matrix, and architecture-corrected expressions.

6.6 Self-Consistent Conductivity Statement

The full self-consistent conductivity problem would embed each phase in the unknown effective medium and solve

The corresponding fixed-point update is

6.7 Textile Architecture Effects

For woven fabrics, conductivity is redistributed by the warp and weft tow architecture. Let $w_{warp}$ and $w_{weft}$ denote the normalized warp and weft fractions. If $k_{tow}$ denotes the tow-axis conductivity and $k_{t}$ denotes a transverse tow conductivity, the in-plane fabric conductivities are approximated as

where $f_{c}$ is the architecture-adjusted crimp factor. Through-thickness conductivity is reduced by the same architecture penalty in the implemented model,

where $\epsilon_{k}$ is a small positive numerical floor. The same architecture framework supports plain weave, twill, satin, basket weave, leno weave, non-crimp stitched fabrics, and triaxial braids.

Figure V-II-S6-3. Textile architecture effects on in-plane thermal conductivity.

6.10 Verification Checks

The following checks are recommended for the thermal-conductivity implementation.

1. If $V_{f} = 0$ and no filler is present, the model must recover $k_{m}$.
2. If $V_{f} \rightarrow 1$ for aligned continuous fibers, $k_{1}$ must approach $k_{f1}$.
3. If the void volume fraction increases, $k_{2}$ and $k_{3}$ must decrease.
4. For a balanced fabric, $k_{1}$ and $k_{2}$ should approach each other.
5. For an unbalanced fabric, $k_{1}/k_{2}$ should track the warp/weft weighting.

6.11 References

Chamis, C. C. (1983). Simplified composite micromechanics equations for hygral, thermal and mechanical properties. SAMPE Quarterly, 15, 14-23.

Mura, T. (1987). Micromechanics of Defects in Solids. Martinus Nijhoff.

Benveniste, Y. (1987). A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials, 6, 147-157.

Dvorak, G. J. (2013). Micromechanics of Composite Materials. Springer.

Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer.

Jones, R. M. (1999). Mechanics of Composite Materials. Taylor & Francis.

Volume II - Section 7: Effective Thermal Expansion and Thermoelastic Micromechanics

7.1 Purpose and Scope

This section documents the coefficient of thermal expansion (CTE) models used by the Workbench micromechanics code. The thermal expansion module converts constituent thermal expansion data for fiber, matrix, filler, void, tow, and fabric phases into effective orthotropic engineering properties. The final outputs are the three material-axis thermal expansion coefficients (_1), (_2), and (_3), which are exported to both effective properties and Abaqus material cards.

The implementation is intentionally consistent with the mechanical micromechanics framework. The same volume fractions, fabric architecture flags, crimp factors, warp/weft weighting factors, and filler/void corrections used for stiffness prediction are also used to assemble thermoelastic material properties.

Figure V-II-S7-1. Multiscale CTE prediction workflow.

7.2 Thermal Strain and Thermoelastic Constitutive Law

For a homogeneous orthotropic material subjected to a temperature increment ΔT, the free thermal strain vector in engineering notation is written as

where

The thermoelastic stress-strain relation is

where C is the effective stiffness matrix. This form is the basis for the Abaqus *EXPANSION, TYPE=ORTHO export generated by the solver script.

Figure V-II-S7-2. Thermal expansion of an orthotropic composite coupon.

7.3 Phase-Averaged Expansion in Composite Constituents

For a composite with fiber, matrix, filler, and void phases, the unconstrained volumetric expansion estimate is

The phase-volume constraint is

Because voids do not carry significant thermal stress, the model normally sets (_v=0) for the void contribution and accounts for voids primarily through dilution of load-carrying and heat-carrying phases.

7.4 Stiffness-Weighted Longitudinal CTE

For continuous fiber composites, axial expansion is constrained by the stiffness of the phases in the fiber direction. The longitudinal CTE estimate used by the model is therefore stiffness weighted:

This expression captures the fact that high-modulus fibers dominate axial expansion even when the matrix has a much larger unconstrained thermal expansion coefficient.

7.5 Transverse and Through-Thickness CTE Estimates

The current engineering implementation uses mixture-style transverse and through-thickness expansion estimates:

7.6 Filled-Matrix CTE

When the filler is treated implicitly as part of the resin-rich phase, the filled matrix CTE is estimated by phase averaging:

where (V_p^{res}) is the filler volume fraction inside the resin phase. This filled-matrix value is then used as the matrix phase input for subsequent fiber and fabric homogenization.

7.7 Fabric Architecture Effects on CTE

For textile composites, warp and weft directions exchange the roles of axial and transverse tow properties. The model uses architecture-weighted CTE assembly:

where (w_{warp}) and (w_{weft}) are normalized fabric architecture fractions satisfying

Crimp and architecture effects modify the mechanical constraints and thermal directions indirectly through the same factors used in stiffness assembly.

Figure V-II-S7-3. Warp/weft CTE directions in a woven RVE.

7.8 Architecture Library and CTE Interpretation

The fabric architecture flag controls the interpretation of the warp/weft CTE assembly. Plain weaves usually exhibit larger crimp and stronger interlacing effects. Satin weaves usually have lower interlacing and reduced tow waviness. Non-crimp fabrics preserve near-straight tow paths and therefore maintain stronger directional anisotropy. Triaxial braids introduce bias-angle effects that couple the two in-plane expansion directions.

The implemented fabric architecture set includes:

• plain weave,
• 2 by 2 twill,
• 4-harness satin,
• 5-harness satin,
• 8-harness satin,
• basket weave,
• leno weave,
• non-crimp biaxial stitched fabric,
• triaxial braid.

7.11 Representative CTE Trend

The stiffness-weighted expression causes the longitudinal CTE of carbon/epoxy composites to decrease rapidly with increasing fiber volume fraction because carbon fibers have low or negative axial CTE and high axial stiffness.

Figure V-II-S7-4. Example CTE trend versus fiber volume fraction.

7.12 Assumptions and Limitations

The current CTE implementation is an engineering micromechanics approximation. It is appropriate for rapid material-property prediction, Workbench integration, and finite-element preprocessing. It does not solve the full thermoelastic inclusion boundary-value problem for arbitrary ellipsoidal inclusions. For high-fidelity thermoelastic homogenization, the full fourth-order concentration tensors from Mori-Tanaka or self-consistent theory should be coupled directly to the phase eigenstrain tensors.

The implemented model assumes:

• small-strain linear thermoelasticity,
• temperature-independent constituent properties unless externally updated,
• orthotropic effective material axes,
• architecture correction factors shared with the mechanical model,
• voids contribute negligibly to stiffness and thermal expansion constraint.

7.13 References

1. Jones, R. M. Mechanics of Composite Materials. Taylor & Francis, 1999.
2. Chou, T. W. Microstructural Design of Fiber Composites. Cambridge University Press, 1992.
3. Mura, T. Micromechanics of Defects in Solids. Martinus Nijhoff, 1987.
4. Dvorak, G. J. Micromechanics of Composite Materials. Springer, 2013.
5. Schapery, R. A. Thermal expansion coefficients of composite materials based on energy principles. Journal of Composite Materials, 1968.
6. Rosen, B. W. Thermoelastic behavior of fiber composites. In Fiber Composite Materials, ASM, 1965.
7. Hashin, Z. Analysis of composite materials - A survey. Journal of Applied Mechanics, 1983.
8. Ishikawa, T. and Chou, T. W. Stiffness and strength behavior of woven fabric composites. Journal of Materials Science, 1982.
9. Naik, N. K. Woven Fabric Composites. Technomic, 1994.

Volume II – Textile and Thermophysical Micromechanics

Section 1 – Introduction and Multiscale Framework

1.1 Purpose and Scope

Volume II documents the textile, hybrid, and thermophysical extensions of the Workbench micromechanics solver model. Volume I established the elastic micromechanics foundation for continuous and discontinuous fiber composites. The present volume extends that foundation to woven, braided, stitched, and non-crimp fabric architectures, and to the effective thermophysical properties needed for coupled thermal and structural finite-element simulations.

The model treats the composite as a hierarchy of homogenization problems. Constituent fibers, polymer matrix, particulate filler, and void phases are first assembled into local material descriptions. These local descriptions are then homogenized into tow properties, textile representative volume element properties, and finally orthotropic lamina properties suitable for Abaqus material-card export.

The Volume II model branches predict the following quantities:

• effective orthotropic elastic constants influenced by fabric architecture;
• effective density;
• effective specific heat capacity;
• effective thermal conductivity in the material directions;
• effective coefficients of thermal expansion in the material directions;
• architecture factors for weave, braid, non-crimp, and stitched reinforcement systems;
• effective properties and Abaqus thermophysical material-card entries.

The goal is not to replace detailed finite-element textile unit-cell analysis. Instead, the intent is to provide a robust, deterministic, Workbench-safe engineering model that captures the dominant effects of tow homogenization, warp–weft balance, crimp, interlacing, fillers, voids, and orthotropic thermal response.

Figure V-II-S1-1. Multiscale textile and thermophysical homogenization workflow.

1.2 Multiscale Material Hierarchy

The model is organized around the assumption that each length scale may be represented by an effective property vector. The general homogenization mapping is written as

where $\mathbf{P}^{(r)}$ is the property vector for phase $r$, $V_{r}$ is its volume fraction, $N$ is the number of active phases, and $\mathbf{\chi}$ denotes geometry or architecture descriptors such as weave type, crimp amplitude, warp/weft balance, stitch knockdown, binder fraction, or braid angle.

For the textile and thermophysical model, the phase-property vector is defined as

where $\mathbf{C}$ is the stiffness tensor, $\rho$ is density, $\mathbf{\alpha}$ is the coefficient of thermal expansion tensor, $\mathbf{k}$ is the thermal conductivity tensor, $C_{p}$ is specific heat capacity, and $\mathbf{S}$ represents strength allowables when damage and failure calculations are enabled.

Figure V-II-S1-2. Phase-property representation used by the textile and thermophysical model.

1.3 Phase Volume Fractions

The phase fractions satisfy the conservation constraint

For the fiber–matrix–filler–void representation used in the solver implementation,

where $V_{f}$ is fiber volume fraction, $V_{m}$ is matrix volume fraction, $V_{p}$ is particulate filler volume fraction, and $V_{v}$ is void volume fraction.

When filler is treated as an implicit modifier of the resin phase, the model first constructs a filled-matrix property set and then performs fiber/tow/fabric homogenization using that modified matrix. When filler is treated explicitly, the filler remains a distinct phase in the phase list used for the homogenization calculation.

1.4 Textile Architecture as a Reduced-Order Descriptor

A detailed textile unit cell may require tow cross-section shape, nesting, local waviness, inter-tow resin pockets, contact surfaces, and through-thickness binder geometry. The implemented Workbench model reduces this complexity to a set of architecture descriptors that can be specified through scalar inputs:

where $I_{fabric}$ is the fabric architecture identifier, $F_{warp}$ and $F_{weft}$ are warp and weft weighting fractions, $k_{c}$ is a crimp factor, $k_{i}$ is an interlacing factor, $k_{s}$ is an in-plane shear factor, $k_{stitch}$ is a stitching knockdown factor, $V_{b}$ is binder or through-thickness reinforcement fraction, and $\theta_{b}$ is a bias or braid angle.

The architecture library in the current solver implementation includes plain weave, 2 by 2 twill, four-harness satin, five-harness satin, eight-harness satin, basket weave, leno weave, non-crimp biaxial stitched fabric, and triaxial braid. These options are implemented as engineering corrections rather than as a finite-element textile meshing scheme.

1.5 General Textile Homogenization Operator

The textile-level homogenization problem may be written in the compact form

In the engineering implementation, this relationship is evaluated through warp–weft weighted tow properties modified by architecture factors. The same conceptual form is also used for conductivity and thermal expansion,

and

This unified notation is useful because all material properties ultimately feed the same finite-element material description.

1.7 Volume II Output Philosophy

The final orthotropic property vector assembled by the Volume II branches is

The effective thermophysical properties are exported to Abaqus using the material-card blocks

1.8 Assumptions and Limitations

The Volume II formulation is a reduced-order engineering homogenization model. The principal assumptions are:

1. tow properties can be represented by homogenized orthotropic engineering constants;
2. textile architecture can be represented using warp/weft weights and architecture correction factors;
3. crimp and interlacing are treated as stiffness and shear knockdown factors rather than explicit curved tow finite elements;
4. filler and void phases are represented through effective phase-volume terms;
5. thermophysical properties are homogenized through volume averaging, series/parallel conductivity estimates, and architecture corrections;
6. all outputs are deterministic scalar, vector, matrix, or string quantities suitable for Workbench and Abaqus export.

These assumptions make the method much faster than detailed textile finite-element unit-cell analysis, but they also limit its ability to capture local tow contact, resin pocket stress concentration, inter-tow delamination, nesting effects, local permeability, and nonlinear compaction.

1.9 Section Summary

Section 1 defines the multiscale and software framework for Volume II. The following sections document the individual textile and thermophysical model components: textile representative volume elements, fabric architecture factors, warp–weft averaging, crimp correction, filled matrix micromechanics, void effects, density, specific heat, thermal conductivity, thermal expansion, and final orthotropic property assembly.

References

Advani, S. G., and Tucker, C. L. III (1987). The use of tensors to describe and predict fiber orientation in short fiber composites. Journal of Rheology, 31, 751–784.

Chou, T.-W. (1992). Microstructural Design of Fiber Composites. Cambridge University Press.

Dvorak, G. J. (2013). Micromechanics of Composite Materials. Springer.

Ishikawa, T., and Chou, T.-W. (1982). Stiffness and strength behavior of woven fabric composites. Journal of Materials Science, 17, 3211–3220.

Jones, R. M. (1999). Mechanics of Composite Materials, 2nd ed. Taylor & Francis.

Mura, T. (1987). Micromechanics of Defects in Solids, 2nd ed. Martinus Nijhoff.

Naik, N. K. (1994). Woven Fabric Composites. Technomic.

Whitcomb, J. D. (2001). Three-dimensional stress analysis of plain weave composites. Composite Materials: Fatigue and Fracture, ASTM STP.

2. Fabric Micromechanics Theory

2.1 Scope and purpose

This section documents the fabric micromechanics branch of the Workbench solver composite-material model. The fabric branch extends the elastic micromechanics framework of Volume I from unidirectional and discontinuous reinforcement to textile architectures. The current implementation supports common woven, stitched, non-crimp, and braided fabrics using a reduced-order representative-volume-element (RVE) framework. The goal is not to resolve every tow cross-section with a mesoscopic finite-element mesh. Instead, the model provides a fast engineering homogenization suitable for material screening, Workbench embedding, and automatic finite-element material-card generation.

For a woven or textile composite, choose the fabric family that represents the reinforcement. Architecture-specific factors then modify tow-dominated stiffness, in-plane shear response, thermophysical properties, and through-thickness reinforcement contributions. This section describes the theory behind those factors and explains how the model maps textile architecture into final orthotropic engineering constants.

Figure V-II-S2-1. Fabric micromechanics workflow implemented in the Workbench solver model.

2.2 Textile composite scale hierarchy

The fabric formulation assumes three nested scales. At the constituent scale, fiber, matrix, filler, and void properties are specified independently. At the tow scale, fibers and matrix are homogenized into an equivalent orthotropic tow. At the textile scale, warp, weft, bias, stitch, and binder contributions are assembled into an effective fabric lamina.

The homogenization chain is represented as

where $\mathbf{P}_{f}$, $\mathbf{P}_{m}$, and $\mathbf{P}_{p}$ are the fiber, matrix, and particle property sets, respectively; $V_{f}$, $V_{m}$, $V_{p}$, and $V_{v}$ are fiber, matrix, particle, and void volume fractions; $\mathcal{H}_{tow}$ is the selected tow homogenization model; and $\mathcal{H}_{fabric}$ is the textile-level homogenization operator.

The final fabric property vector is

2.3 Representative volume element definition

A textile RVE is the smallest repeating domain that preserves the relevant fabric architecture. The RVE contains a warp tow family, a weft tow family, matrix-rich pockets, and optional binder or stitching reinforcement. For non-crimp fabrics and triaxial braids, the RVE additionally contains bias tow families.

The phase volume-fraction constraint is

At the fabric level, the tow fractions satisfy

For a two-family woven fabric without explicit bias or binder phases,

Balanced woven fabrics use

whereas unbalanced fabrics use architecture- or user-specified values. In the solver implementation, this weighting is controlled by fabric_warp_bias_fraction, woven_warp_volume_fraction, and woven_weft_volume_fraction.

2.4 Supported fabric architecture library

The following fabric architectures are supported.

Figure V-II-S2-2. Common textile architectures represented in the fabric branch.

2.5 Tow homogenization

The tow is treated as a unidirectional composite sub-material. The selected constituent-level micromechanics model produces tow properties from the fiber and matrix properties. In high-level operator form,

where $V_{f,tow}$ is the fiber volume fraction inside the tow.

The axial tow modulus used in the implemented engineering approximation is

A transverse tow modulus may be estimated using a Halpin-Tsai-like expression,

with

The same tow-level concept is applied to shear properties,

These tow-level properties become the input to the fabric-level homogenization step.

2.6 Warp-weft stiffness assembly

The fabric stiffness is assembled from warp and weft tow contributions. For a balanced woven architecture, the warp and weft fractions are equal. For an unbalanced fabric, the weighting is user-defined or architecture-derived.

The normalized tow-family fractions are

The implemented in-plane Young’s moduli are represented by

where $f_{c}$ is the architecture-adjusted crimp factor and $f_{b}$ is the balance factor. The through-thickness modulus is treated as a tow-transverse-dominated quantity modified by interlacing and optional binder reinforcement,

where $f_{i}$ is the interlacing penalty factor.

2.7 Crimp, interlacing, balance, and shear correction factors

Fabric architecture influences stiffness through geometric tow waviness and the number of over-under interlacings. The model uses reduced-order scalar correction factors rather than explicit meso-scale finite-element tow geometry.

The crimp factor is modeled as a decreasing function of the tow waviness amplitude ratio,

where $A_{c}$ is tow waviness amplitude, $L_{c}$ is crimp wavelength, and $a_{c}$ is an architecture-dependent constant. Plain weaves use a larger effective $a_{c}$ because they interlace more frequently, while satin weaves use smaller values because they have longer floats and lower interlacing density.

Figure V-II-S2-3. Crimp and interlacing correction concept used by the fabric branch.

The balance factor is defined from warp and weft fractions,

This factor approaches unity for balanced fabrics and decreases as the architecture becomes strongly warp- or weft-dominated.

The interlacing penalty is represented by

where $I_{a}$ is the architecture interlacing factor and $b_{i}$ is a calibration constant. The in-plane shear correction is expressed as

where $f_{s}$ is the architecture shear factor and ${\bar{G}}_{12,tow}$ is the averaged tow-level shear modulus.

2.8 Architecture-specific interpretation

Plain weave has the highest interlacing density among the common balanced fabrics. It therefore receives the largest crimp and interlacing penalties. Twill fabrics reduce the interlacing frequency and improve drape, so the stiffness knockdown is less severe. Satin fabrics introduce longer tow floats, which reduce crimp and preserve tow axial stiffness but may reduce fabric stability during handling. Basket weaves group tows and reduce crossing density; leno weaves lock yarns together and improve dimensional stability but increase local yarn distortion. Non-crimp fabrics preserve tow straightness and therefore use very low crimp penalties, but stitching effects are included through a stitch knockdown factor. Triaxial braids include a bias-angle contribution that redistributes stiffness and increases in-plane shear coupling.

The architecture-dependent stiffness correction can be summarized as