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This theory library includes wider research formulations. Check the released model directory for selectable models, required inputs, limits and exercises. The event-driven RVE branch is not enabled in hosted Workbench.

Theories / Micromechanics / Aligned discontinuous fibers

Semi-analytical aligned discontinuous-composite theory

Reference formulation — not enabled in hosted Workbench.

This Revision 8 chapter documents a separate research formulation. Its event-driven breaks, nonlinear shear, RVE series coupling and stochastic failure are not selectable in hosted Workbench. Cox shear-lag elastic homogenization is available, not this failure model. Released models and connections ↗

1. Theory basis and role in CDS

Henry and Pimenta model aligned short-fiber material across specimen, RVE, fiber, neighboring-fiber interaction, and interaction-segment scales. Random fiber ends create nonuniform overlaps, matrix shear transfers load, fiber strength is stochastic, and final failure occurs when a local cluster becomes unstable.

The reference implementation associates this formulation with aligned-discontinuous material type 1 / criterion 6; this is not a selectable hosted Workbench path. Its specimen response supplies longitudinal ply stiffness and tensile strength to the common orthotropic ply property definition. Laminate ABD assembly and laminate-scale progressive degradation remain separate downstream operations.

2. Paper-to-CDS traceability

Henry–Pimenta mechanismRevision 8 statusCDS implementation
Random longitudinal fiber-end locationsImplementedEach RVE draws one end location per fiber over the selected fiber length.
Square n × n aligned-fiber RVEImplementedThe user controls fibers per row and the number of independent RVEs.
Four-nearest-neighbor topologyImplementedHorizontal and vertical interactions are rebuilt from fiber ends and inserted breaks.
Generic nonlinear matrix constitutive lawImplementedAn optional piecewise-linear τ(γ) table is accepted; otherwise CDS builds an elastic-yield-friction law.
Broken and shear-lag interaction segmentsImplementedSame-fiber endpoints create broken segments; different-fiber endpoints create nonlinear shear-lag segments.
Length- and field-scaled Weibull strengthImplementedOrdered thresholds use fiber/reference length, Weibull scale and shape, and stress-field correction.
Event-driven break insertion and rebuildingImplementedThe governing break coordinate is inserted, its interaction is deactivated, and the network is rebuilt at the same strain.
Debonding and frictional pull-out toughnessImplementedResistance is integrated over realized overlaps from mode-II toughness and residual friction.
Dugdale critical-cluster instabilityImplementedSquare clusters are screened with the nonlinear energy-release expression and physical cluster radius.
Specimen curve from RVEs in seriesImplementedRVE strains are interpolated at common stress and averaged; the first RVE cutoff governs.

3. Interaction-segment mechanics

3.1 Discontinuity reconstruction

For each horizontal and vertical neighbor pair, CDS sorts both fiber ends and all inserted break coordinates. Consecutive discontinuities define the current interaction segments. If both endpoints belong to the same fiber, the segment is broken; if they belong to different fibers, load crosses the matrix through shear lag.

Eq. ADF-01Broken-segment relation
σBrI = (Ef/2) εBrI
Equation detailsExplanation · variables · model connection · reference

Relates interaction stress to strain in a segment bounded by discontinuities on the same fiber.

VariablesσBrIBroken-segment interaction stressPaEfFiber Young’s modulusPaεBrIBroken-segment interaction straindimensionless

Model connectionSupplies the broken-segment response used in the length-weighted interaction strain.

Theory basisHenry and Pimenta (2017), reference formulation

3.2 Nonlinear shear lag

With fiber half-thickness T = φf/4 and effective matrix gap tm, the fiber stress difference follows:

Eq. ADF-02Shear-lag differential equation and parameter
d²Δσ/dx² = ±λ²Δσ,   λ = [2|Gm|/(TtmEf)]1/2
Equation detailsExplanation · variables · model connection · reference

Describes how the stress difference between neighboring fibers varies along an interaction segment. The active matrix-law branch determines the sign and stiffness used.

VariablesΔσStress difference between neighboring fibersPaxPosition along the fibermλShear-transfer parameter1/mGmMatrix shear modulus for the active constitutive branchPaTFiber half-thickness, equal to one quarter of the fiber diametermtmEffective matrix gap between fibersmEfFiber Young’s modulusPa

Model connectionProvides the nonlinear shear-lag segment response before segments are combined in series.

Theory basisHenry and Pimenta (2017), reference formulation

The active secant/tangent behavior comes from the selected piecewise-linear matrix law. The interaction stress is common to its segments, and their strain contributions are length-weighted in series. The weakest segment limits the interaction.

Eq. ADF-03Series interaction strain
εRVE(σI) = (1/lf) Σs Δls εsI(σI)
Equation detailsExplanation · variables · model connection · reference

Averages segment strains by their lengths at a common interaction stress.

VariablesεRVERepresentative-volume straindimensionlessσICommon interaction stressPalfFiber lengthmΔlsLength of segment smεsIStrain of segment s at the common interaction stressdimensionlesssSegment indexdimensionless

Model connectionConnects individual broken and shear-lag segments to the representative-volume response.

Theory basisHenry and Pimenta (2017), reference formulation

4. Progressive fiber-break events

Each fiber receives ordered Weibull thresholds. Length and stress-field scaling follow:

Eq. ADF-04Length- and field-scaled Weibull law
F(σf) = 1 − exp[−ClCσ(σf/σ0)m],   Cl = lf/(4lr)
Equation detailsExplanation · variables · model connection · reference

Expresses the probability of fiber failure with corrections for fiber length and the nonuniform stress field.

VariablesFCumulative probability of fiber failuredimensionlessσfFiber stressPaσ0Weibull reference strength scalePamWeibull shape parameterdimensionlessClFiber-length scaling factordimensionlessCσStress-field scaling factordimensionlesslfFiber lengthmlrWeibull reference lengthm

Model connectionGenerates the ordered fiber-strength thresholds that trigger break insertion and network rebuilding.

Theory basisHenry and Pimenta (2017), reference formulation

Four adjacent interactions generate the fiber peak. When that peak exceeds the current threshold, CDS inserts a break at the governing neighboring discontinuity at the end of the longest segment, advances the ordered threshold, deactivates the responsible interaction, and rebuilds the network without advancing applied strain. Events repeat until stable.

5. RVE, specimen, and cluster failure

Eq. ADF-05RVE stress
σRVE = Vf(ΣσVI + ΣσHI)/[2n(n−1)] + Vmσm
Equation detailsExplanation · variables · model connection · reference

Combines the horizontal and vertical interaction stresses with the matrix contribution to obtain the representative-volume stress.

VariablesσRVERepresentative-volume average axial stressPaVfFiber volume fractiondimensionlessVmMatrix volume fractiondimensionlessσVIVertical-neighbor interaction stressPaσHIHorizontal-neighbor interaction stressPanFibers per row in the square arraycountσmMatrix axial stressPa

Model connectionProduces each representative volume’s stress–strain curve for specimen assembly.

Theory basisHenry and Pimenta (2017), reference formulation

Independent RVEs are placed in series. At a common stress, their interpolated strains are averaged; the earliest RVE cutoff limits the specimen.

Eq. ADF-06Series-coupled specimen strain
εspec(σ) = (1/NRVE) Σr εr(σ)
Equation detailsExplanation · variables · model connection · reference

Averages representative-volume strains evaluated at the same axial stress.

VariablesεspecSpecimen axial straindimensionlessσCommon axial stressPaNRVENumber of representative volumescountεrAxial strain in representative volume rdimensionlessrRepresentative-volume indexdimensionless

Model connectionBuilds the specimen response from volumes in series; the earliest volume cutoff limits the specimen.

Theory basisHenry and Pimenta (2017), reference formulation

Cluster termination uses the nonlinear Dugdale energy release rate and the paper's cluster-radius scaling:

Eq. ADF-07Nonlinear critical-cluster energy
JNL = (32/π³)(Xg²/Eg)a ln[sec(πσ/(2Xg))],   a = naφf/√(2Vf)
Equation detailsExplanation · variables · model connection · reference

Evaluates nonlinear energy release for a damaged cluster and relates its physical radius to the fiber array.

VariablesJNLNonlinear energy release rateJ/m²XgStrength parameter of the cluster modelPaEgElastic modulus used in the cluster modelPaaPhysical cluster radiusmσApplied axial stressPanaCluster-size parameterdimensionlessφfFiber diametermVfFiber volume fractiondimensionless

Model connectionThe cluster energy release is compared with calibrated fracture resistance to determine instability.

Theory basisHenry and Pimenta (2017), reference formulation

Eq. ADF-08Debonding and frictional pull-out resistance
gdeb = (2Vf/φf)GIIcΔl,   gpo = (Vf/φf)τμΔl²
Equation detailsExplanation · variables · model connection · reference

Evaluates the debonding and frictional pull-out contributions for a realized overlap length.

VariablesgdebDebonding resistance per unit areaJ/m²gpoFrictional pull-out resistance per unit areaJ/m²VfFiber volume fractiondimensionlessφfFiber diametermGIIcMode-II interfacial fracture toughnessJ/m²ΔlRealized overlap lengthmτμResidual frictional shear stressPa

Model connectionThese resistance contributions enter the critical-cluster instability assessment.

Theory basisHenry and Pimenta (2017), reference formulation

6. Connected workflow

  1. ConstituentsFiber elastic/Weibull data and matrix elastic, strength, toughness, friction, and optional τ(γ) data
  2. Stochastic geometryFiber length/diameter, volume fraction, n × n RVE, independent realizations, and seed
  3. Interaction networkEnds + breaks → broken/shear-lag segments → nonlinear segment series response
  4. Fiber eventsLocal peaks → ordered Weibull threshold → break insertion → network rebuild at the same strain
  5. RVE and specimenInteraction average → complete RVE curves → common-stress series coupling
  6. Cluster and ply property connectionDugdale cutoff → E1/Xt → 3D ply properties → laminate and structural analysis

7. What to supply and what to examine

Study quantityPhysical meaning
Fiber variability and interactionsSpecify fiber-strength statistics, representative-volume size, flaw assumptions and calibrated fracture resistance.
Matrix shear behaviorSupply measured shear strain and shear stress in Pa. Strain values must be nonnegative and strictly increasing.
Specimen responseSeries-coupled specimen curves.
RVE responseRVE curves, break counts, active-interaction counts, and cutoff state.
Fiber-break eventsEvery progressive fiber-break event and governing interaction.
Critical clustersCritical cluster location, size, JNL, and resistance.
Matrix shear lawResolved matrix shear-law points and piecewise tangent.

8. Calibration and validation boundary

  • Calibrate the matrix shear law, Weibull scale/shape/reference length, stress-field correction, interface limit, mode-II toughness, and friction stress to the intended material state.
  • RVE row count, RVE count, strain resolution, cluster limit, and flaw opportunities require convergence studies.
  • Static and analytical release checks do not replace a complete model run or coupon validation.
  • Use measured ply overrides when a qualified dataset should supersede a predicted property.

9. Primary reference

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