A laminate combines independently defined plies through their thicknesses, orientations and constitutive properties. Classical laminate theory represents in-plane strain as a mid-plane strain plus a term linear in the thickness coordinate. Integrating transformed ply stiffness produces the membrane, coupling and bending blocks.
The A block relates membrane strain to membrane force, B represents extension–bending coupling and D relates curvature to bending moment. Symmetry of a correctly assembled layup eliminates B to numerical precision; balance alone does not guarantee symmetry. The stacking order and ply thickness must remain explicit.
Thermal expansion, moisture expansion and any enabled process shrinkage are free strains. Integrate their equivalent force and moment resultants with the same ply definitions used for stiffness. Recover elastic strain by subtracting free strain before calculating stress. A free laminate can curve while its net force and moment remain zero; zero applied load does not imply zero ply stress.
4.1 Ply transformations and laminate constitutive response governing relation
Equation details — explanation, variables and reference
Applied force and moment equal the constitutive response minus the enabled free-strain resultants. Move the free-strain vector to the load side when solving for strain and curvature.
N: membrane force per width (N/m); M: moment per width (N); A, B, D: stiffness blocks (N/m, N, N·m); ε⁰: mid-plane strain; κ: curvature (1/m); starred resultants: integrated free-strain contributions.
Theory basis4.2 Laminate Analysis
Theories / Laminates
A unified constituent-to-process-to-laminate reference. Generate or measure the three-dimensional property definition independently for every ply, solve optional heat, moisture, cure, and crystallization histories, assemble the multi-material laminate, and progress its selected failure models under mechanical and process loading.
Theory map
Models, equations, figures, and cross-referencesIndependent data for every ply
Resolve automatic, constituent-generated, or directly measured 3D properties independently for every physical ply.
Open material routing →Hygrothermal CLT and ABD
Transform ply stiffness, integrate A/B/D matrices, add environmental resultants, and recover through-thickness fields.
Open ABD theory →Failure, degradation, and redistribution
Evaluate selected criteria independently by ply, update damaged properties, and resolve equilibrium at the same load.
Open progressive theory →Properties and delamination
Assemble effective laminate properties and cohesive response for structural assessment.
Open delamination theory →Carpet plots, envelopes and optimization
Review the equations, fixed inputs, search construction and limits of the interactive Workbench. Distinguish these screening tools from saved coupled analyses.
Open Workbench methods →Complete reference
One continuous sequenceFramework & Scope
→02Ply Property Definition
→03Material Routing
→04Failure Criteria
→05Hygrothermal CLT & ABD
→06Transverse Traction & z-Response
→07Progressive Failure
→08Damage Redistribution
→09Effective Properties
→10Cohesive Delamination
→11Verification & Validation
→12Assumptions & Limitations
→13Theory References
→4.3 5. Hygro-Thermo-Mechanical CLT and ABD
5. Hygro-Thermo-Mechanical CLT and ABD
The framework retains Classical Lamination Theory and extends its generalized equilibrium to include per-ply temperature and moisture fields and the in-plane Poisson response induced by prescribed σz, all evaluated with current damaged stiffness.
Meaning, notation and theory source
Integrates transformed damaged ply stiffness through thickness to form A, B, and D.
Uses Q from Eq. 05.3.02-02 after material routing and damage updates from Eq. 05.3.08-01.
Definitions and derivation: 5. Hygro-Thermo-Mechanical CLT and ABD
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Environmental fields and free strain
Meaning, notation and theory source
Defines temperature and moisture change independently for every ply relative to its reference state.
These resolved ply fields generate free strain in Eq. 05.3.05-03 and support later gradient-generation inputs.
Definitions and derivation: Environmental fields and free strain
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
Meaning, notation and theory source
Adds thermal and moisture expansion in the local ply constitutive relation.
Uses α and β from Eq. 05.3.02-01 and the fields from Eq. 05.3.05-02.
Definitions and derivation: Environmental fields and free strain
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Meaning, notation and theory source
Integrates environmentally induced ply stress equivalents into laminate loads.
Must be recomputed with the current Q̄d after every damage event before Eq. 05.3.05-05.
Definitions and derivation: Environmental fields and free strain
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Damaged generalized equilibrium
Meaning, notation and theory source
Solves membrane strains and curvatures from mechanical, thermal, moisture, and σz-Poisson equivalent resultants.
Uses A, B, and D from Eq. 05.3.05-01; its response is recovered at three nodes per ply.
Definitions and derivation: Damaged generalized equilibrium
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
At each ply surface, ε(z)=ε⁰+zκ and σk(z)=Q̄dk[ε(z)−εfk(z)]. Both total and mechanical strain are retained. Failure criteria use mechanical strain and the solver checks the top and bottom of every ply.
4.4 9. Effective Laminate and Transport Properties
9. Effective Laminate and Transport Properties
The full inverse ABD matrix is used so apparent membrane properties preserve extension–bending coupling in unsymmetric laminates.
Meaning, notation and theory source
Inverts the complete ABD matrix so unsymmetric extension–bending coupling is retained.
ABD is assembled by Eq. 05.3.05-01.
Definitions and derivation: 9. Effective Laminate and Transport Properties
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Meaning, notation and theory source
Extracts laminate moduli and Poisson ratios from the zero-moment compliance terms.
Uses the upper-left compliance terms of Eq. 05.3.09-01 and total thickness h.
Definitions and derivation: 9. Effective Laminate and Transport Properties
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Effective z-direction elastic and expansion properties
Meaning, notation and theory source
Uses a unit uniform σz virtual load and the thickness-averaged recovered εz to calculate Ez.
The virtual response uses the same Poisson-coupled transverse constitutive recovery as the r11 load solution.
Definitions and derivation: Effective z-direction elastic and expansion properties
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
Meaning, notation and theory source
Thickness-averages rotated local G13/G23 compliances and inverts the result to obtain Gxz, Gyz, and xz–yz coupling.
Retains ply orientation and multi-material transverse shear behavior.
Definitions and derivation: Effective z-direction elastic and expansion properties
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
The diagonal terms of K̄sh provide Gxz and Gyz; its off-diagonal term is the xz–yz shear-coupling stiffness. Unit temperature and moisture load cases produce αz, βz, in-plane CTE/CME values, and free-warpage curvatures while retaining Poisson coupling.
Density, heat capacity, conductivity, and diffusivity
Meaning, notation and theory source
Thickness-averages density and mass-weights laminate specific heat.
Uses per-ply physical properties from Eq. 05.3.02-01.
Definitions and derivation: Density, heat capacity, conductivity, and diffusivity
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Meaning, notation and theory source
Uses parallel in-plane and k3-based series through-thickness conductivity, then computes diffusivity.
Uses ρL and cp,L from the preceding physical-property relation.
Definitions and derivation: Density, heat capacity, conductivity, and diffusivity
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
In-plane conductivity uses a thickness average; through-thickness conductivity uses a series model. These are pristine homogenized properties: the current model does not degrade conductivity or heat capacity with cracking and does not solve transient transport.
4.5 2. Common Ply Property Definition
2. Common Ply Property Definition
Every ply supplies the same constitutive property definition, allowing different material types and failure theories to share the laminate equilibrium solver.
Meaning, notation and theory source
Collects the mechanical, strength, expansion, density, and transport properties required by every ply model.
Feeds the reduced-stiffness definition in Eq. 05.3.02-02 and all later failure and environmental equations.
Definitions and derivation: 2. Common Ply Property Definition
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
| Family | Required quantities | Purpose |
|---|---|---|
| Elastic | E1, E2, E3; G12, G13, G23; ν12, ν13, ν23 | Damaged ABD response and layerwise transverse recovery |
| Strength | Xt, Xc, Yt, Yc, Zt, Zc; S12, S13, S23 | Sign-dependent in-plane and transverse failure evaluation |
| Strain allowables | ε1t/c, ε2t/c, ε3t/c; γ12, γ13, γ23 | Maximum Strain and supplemental transverse failure |
| Expansion | α1, α2, α3; β1, β2, β3; ply ΔT and ΔC | Thermal and moisture contributions at each ply |
| Physical | ρ, k1, k2, k3, cp | Laminate density, conductivity, heat capacity, and diffusivity |
Plane-stress reduced stiffness
Meaning, notation and theory source
Defines the orthotropic constitutive matrix in local ply axes under plane stress.
Its coefficients are expanded in Eq. 05.3.02-03 and transformed before ABD integration in Eq. 05.3.05-01.
Definitions and derivation: Plane-stress reduced stiffness
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Meaning, notation and theory source
Develops Q from E1, E2, G12, and the reciprocal Poisson ratios.
Derived from the elastic variables introduced in Eq. 05.3.02-01.
Definitions and derivation: Plane-stress reduced stiffness
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Strengths and strain allowables must be positive magnitudes. The solver chooses the tensile or compressive value from the sign of the recovered local component.
4.6 3. Micromechanics and Material Routing
3. Micromechanics and Material Routing
Release r15 adds a source router before the established material/failure router. Each ply first resolves where its pristine properties come from, then resolves which progressive criterion is physically eligible.
Three property sources per ply
Four architectures and failure eligibility
| ID | Architecture | Resolved family | Failure selection |
|---|---|---|---|
| 1 | Continuous UD | Deterministic/continuous | Criteria 1–5 |
| 2 | Aligned discontinuous/short fiber | Aligned discontinuous | Criterion 6 forced |
| 3 | Woven, NCF, braid | Deterministic/continuous | Criteria 1–5 |
| 4 | General multiphase | Deterministic/continuous | Criteria 1–5 |
A hybrid laminate may combine generated UD, fabric, short-fiber, multiphase, and direct measured plies. Every ply retains its own environment, criterion, and irreversible damage state during the common damaged-ABD solution.
Phase fractions and reduced-order homogenization
Meaning, notation and theory source
Normalizes fiber, matrix, filler, and void fractions before the per-ply property generator is evaluated.
The normalized phase state supplies every homogenization branch and is returned with the generated micromechanics properties.
Definitions and derivation: Phase fractions and reduced-order homogenization
CDS laminate theory reference. Consult the released model directory for current Workbench methods and scope.
Halpin, J. C., and Tsai, S. W. (1976). Environmental Factors in Composite Materials Design. U.S. Air Force Materials Laboratory.
Meaning, notation and theory source
Provides longitudinal isostrain and transverse/series engineering bounds for the generated ply definition.
The resolved value enters the full 3D orthotropic compliance and stiffness pair before laminate transformation.
Definitions and derivation: Phase fractions and reduced-order homogenization
Halpin, J. C., and Tsai, S. W. (1976). Environmental Factors in Composite Materials Design. U.S. Air Force Materials Laboratory.
Meaning, notation and theory source
Uses constituent contrast and a geometry parameter to estimate transverse, shear, filler, or architecture-sensitive properties.
Model ID 2 selects this branch independently for every source-mode-1 ply.
Definitions and derivation: Phase fractions and reduced-order homogenization
Halpin, J. C., and Tsai, S. W. (1976). Environmental Factors in Composite Materials Design. U.S. Air Force Materials Laboratory.
Model IDs select ROM/Reuss, Halpin–Tsai, approximate Mori–Tanaka, or approximate self-consistent engineering estimates. Fabric controls modify warp, weft, bias, crimp, stitch, binder, and interlacing contributions. All generated branches return the same 3D ply definition and positive-definite 6×6 stiffness/compliance pair.
Expansion, density, and transport
Meaning, notation and theory source
Conserves phase mass while returning the density and mass-weighted heat capacity used by laminate effective-property outputs.
Generated values enter the common ply definition and later thickness/mass averages.
Definitions and derivation: Expansion, density, and transport
Meaning, notation and theory source
Uses stiffness-weighted longitudinal CTE and directional parallel/series conductivity relations for the pristine generated ply.
These coefficients drive per-ply environmental resultants and laminate transport properties.
Definitions and derivation: Expansion, density, and transport
Halpin, J. C., and Tsai, S. W. (1976). Environmental Factors in Composite Materials Design. U.S. Air Force Materials Laboratory.
Aligned-discontinuous transfer and strength
Meaning, notation and theory source
Relates overlap length and matrix transfer stiffness to the fraction of fiber stiffness effectively developed.
ηL modifies the aligned-discontinuous ply response routed to criterion 6.
Definitions and derivation: Aligned-discontinuous transfer and strength
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Meaning, notation and theory source
Samples ordered stochastic fiber strength thresholds at the selected fiber and reference lengths.
Each current threshold drives event-based break insertion, interaction rebuilding, and critical-cluster evaluation in criterion 6.
Definitions and derivation: Aligned-discontinuous transfer and strength
Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18, 293–297.
A General-Purpose Hygro-Thermo-Mechanical Progressive-Failure Framework for Continuous and Aligned-Discontinuous Composite Laminates (20 August 2026), source paper supplied for this theory update.
Only the aligned-discontinuous architecture enters the stochastic break-neighborhood and critical-cluster solver. The embedded micromechanics engine generates pristine properties; progressive damage remains solely in the laminate solver, preventing double degradation.
4.7 6. Uniform Transverse Traction and z-Response
6. Uniform Transverse Traction and z-Response
Release r11 adds a reduced-order layerwise section solution for spatially uniform σz, τxz, and τyz. These directions may be static or progressive, contribute to ply failure, and are reported at three nodes per active ply.
Nine-direction load matrix
| Rows | Directions | Column 1 | Column 2 | Column 3 |
|---|---|---|---|---|
| 1–3 | x, y, xy membrane | Static stress or resultant | Progressive flag | + tension / − compression |
| 4–6 | x, y, xy bending/twisting | Static surface stress or moment resultant | Progressive flag | Signed ramp direction |
| 7–9 | σz, τxz, τyz | Static traction in Pa | Progressive flag | Signed ramp direction |
When a row is progressive, its static value is ignored and the shared final progressive magnitude is applied with the sign in column 3. Multiple flagged directions ramp together. Static directions remain superposed throughout the ramp. Rows 7–9 are always entered in Pa, independent of whether rows 1–6 use stress-based or N/M resultant-based loading.
Local traction transformation
Meaning, notation and theory source
Rotates prescribed global xz and yz shear tractions into each ply’s local 1–3 and 2–3 axes.
The local tractions drive transverse strain recovery and progressive modes 9–12.
Definitions and derivation: Local traction transformation
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
Transverse strain superposition
Meaning, notation and theory source
Superposes Poisson coupling, direct σ3/E3 compliance, thermal expansion, and moisture expansion for each ply.
Its mechanical contribution drives modes 7–8 and its total value is integrated to recover w(z).
Definitions and derivation: Transverse strain superposition
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
Meaning, notation and theory source
Recovers local 1–3 and 2–3 engineering shear strains from the current damaged shear moduli.
Criterion 2 compares these strains with γ13 and γ23; other criteria use S13 and S23 stress envelopes.
Definitions and derivation: Transverse strain superposition
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
For each ply, σz also creates in-plane Poisson eigenstrains −ν13σz/E1 and −ν23σz/E2. Their transformed equivalent N and M resultants are included before the damaged ABD solve, so membrane, bending, transverse traction, thermal, and moisture effects remain mechanically coupled.
Compatible displacement recovery
Meaning, notation and theory source
Integrates through-thickness strain from the laminate bottom so displacement remains continuous across ply interfaces.
The reconstructed w field is exported with compatible u and v at three nodes per ply.
Definitions and derivation: Compatible displacement recovery
ADC General-Purpose Progressive-Failure Model — r11: Uniform 3D Traction and z-Property Extension (22 August 2026), release 2026082201.
Integrating εz from the laminate bottom produces a continuous through-thickness displacement. The output stores compatible u, v, and w at three recovery nodes per active ply.
4.8 11. Inputs and Model Selection
11. Inputs and Model Selection
Input data define both the laminate mechanics and the theory used by each ply. Values must describe the same material state, units, conditioning, and temperature range.
| Input group | Required content |
|---|---|
| Fiber database | 3D elastic, strength/Weibull, expansion, conductivity, heat capacity, and moisture data |
| Matrix database | elastic, strength, density, thermal, moisture, toughness, and friction data |
| Filler database | elastic, density, aspect ratio, expansion, conductivity, heat capacity, and moisture data |
| Per-ply source routing | Material or solved-lamina selection, fiber architecture, model choice, volume fractions and void content |
| Laminate | Ply count, thicknesses, orientations, stacking order, reference plane, three recovery nodes per active ply |
| Loading | six membrane/bending directions plus uniform σz, τxz, τyz; static/progressive flag and signed ramp direction |
| Ply routing | Material family and compatible failure model |
| Mechanical | Elastic constants, strengths, and strain allowables |
| Environmental | Per-ply ΔT and ΔC with α1–α3 and β1–β3, ready for later gradient generation |
| Physical | Density, k1, k2, k3, and specific heat |
| Process boundaries | Independent top and bottom boundary schedules with time, temperature, moisture, boundary types, and coefficient/custom-flux values |
| Ply kinetics | Ply definitions for inert, thermoset, or semicrystalline thermoplastic state, diffusion, heat, and shrinkage |
| Short fiber | Vf, fiber length/diameter, overlap geometry, interface shear, Weibull parameters, RVE controls |
| Cohesive | Penalty stiffnesses, strengths, fracture energies, BK exponent, and interface separation history |
For each ply, choose its material or solved lamina, thickness and angle. Supply the elastic properties, strength allowables and environmental properties needed by the selected analysis. Missing data must not be interpreted as a pass.
4.9 12. Outputs and Event Histories
12. Outputs and Event Histories
Outputs separate global laminate response, ply-surface state, damage history, effective properties, and optional interface response so the progressive sequence can be audited.
| Output family | Contents |
|---|---|
| Laminate state | Pristine/current A, B, D and ABD; mid-plane strains; curvatures; mechanical and environmental resultants |
| Layerwise fields | Three nodes per ply; global/local/principal in-plane fields and global/local transverse fields |
| Contributions | Total, mechanical, thermal, and moisture stress and strain histories reported independently and by superposition |
| Progressive events | Step, load factor, ply, surface, mode, FI, stress vector, z-coordinate |
| Retention | Directional stiffness ratios, terminal condition, and collapse/conditioning flags |
| Effective properties | Ex, Ey, Gxy, Ez, Gxz, Gyz, transverse Poisson ratios, CTE/CME, density, cp, conductivities, diffusivities, and transverse shear coupling |
| Delamination | Interface separations, tractions, mode mix, cohesive damage, effective bending feedback |
Read stress, strain, deformation, damage and retained stiffness together. Total strain describes deformation; mechanical strain drives strain-based failure.
Progressive visualization
The optional progressive-damage MP4 synchronizes ply damage, stress redistribution, retained stiffness, the load–strain path, full-text failure callouts, and Figure 6 principal stress/strain histories. A second optional MP4 progressively reveals the six Figure 4 global through-thickness stress and strain panels. Every failure-event increment is forced into the frame list so same-load redistribution events are not skipped.
Chapter review
Thermal expansion, moisture expansion and any enabled process shrinkage are free strains. Integrate their equivalent force and moment resultants with the same ply definitions used for stiffness. Recover elastic strain by subtracting free strain before calculating stress. A free laminate can curve while its net force and moment remain zero; zero applied load does not imply zero ply stress.
References and source sections
References are retained with the formulations they support. Software instructions describe implementation scope; a cited source does not establish independent validation of a CDS calculation.
- Nettles, A. T. (1994). Basic Mechanics of Laminated Composite Plates. NASA Reference Publication 1351.
- Cox, H. L. (1952). The elasticity and strength of paper and other fibrous materials. British Journal of Applied Physics, 3, 72–79.
- Pimenta, S., and Robinson, P. (2014). An analytical shear-lag model for composites with brick-and-mortar architecture considering non-linear matrix response and failure. Composites Science and Technology, 104, 111–124.
- Henry, J., and Pimenta, S. (2017). Semi-analytical simulation of aligned discontinuous composites. Composites Science and Technology, 144, 230–244.