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First-ply failure criteria

Physical formulation · assumptions, essential equations, full derivation, algorithm, code and verification references.

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Physical formulationINDICATOR ONLY · no progressive damage update

Maximum stress/strain, Tsai–Hill, and Tsai–Wu indices are evaluated at declared ply positions and material axes.

01 Model and scope

These are first-ply indicators evaluated at prescribed through-thickness positions in each ply's material axes. They compare recovered stresses or strains with allowables, or evaluate a failure surface. Crossing FI=1 flags a criterion-specific first-ply exceedance; QF Solver does not degrade stiffness or propagate the failure.

Route assumptions and calculation checks

  1. Indices use recovered ply stress/strain in axes 1–2 at lower, middle, and upper positions, with tension/compression allowables selected by sign.
  2. Tsai–Wu's default f12*=−0.5 is an assumption; replace it with traceable biaxial test data for engineering use.
  3. Crossing FI=1 reports a first-ply indicator. The solver does not reduce stiffness or model crack growth, progressive damage, fatigue, impact, or delamination.

Useful foundations and prerequisites

Strain, stress and rotation measuresUnits, signs and error measures

02 Essential equations and symbols

X={Xt,σ1≥0Xc,σ1<0,Y={Yt,σ2≥0Yc,σ2<0X=\begin{cases}X_t,&\sigma_1\ge0\\X_c,&\sigma_1<0\end{cases},\quad Y=\begin{cases}Y_t,&\sigma_2\ge0\\Y_c,&\sigma_2<0\end{cases}FIMS=max⁡(max⁡(σ1,0)Xt,max⁡(−σ1,0)Xc,max⁡(σ2,0)Yt,max⁡(−σ2,0)Yc,∣τ12∣S12)FI_{MS}=\max\left(\frac{\max(\sigma_1,0)}{X_t},\frac{\max(-\sigma_1,0)}{X_c},\frac{\max(\sigma_2,0)}{Y_t},\frac{\max(-\sigma_2,0)}{Y_c},\frac{|\tau_{12}|}{S_{12}}\right)FITH=(σ1X)2−σ1σ2X2+(σ2Y)2+(τ12S12)2FI_{TH}=\left(\frac{\sigma_1}{X}\right)^2-\frac{\sigma_1\sigma_2}{X^2}+\left(\frac{\sigma_2}{Y}\right)^2+\left(\frac{\tau_{12}}{S_{12}}\right)^2FITW=F1σ1+F2σ2+F11σ12+F22σ22+2F12σ1σ2+F66τ122FI_{TW}=F_1\sigma_1+F_2\sigma_2+F_{11}\sigma_1^2+F_{22}\sigma_2^2+2F_{12}\sigma_1\sigma_2+F_{66}\tau_{12}^2FI≤1:inside selected surface;FI>1:first-ply exceedance indicatorFI\le1:\quad\text{inside selected surface};\quad FI>1:\quad\text{first-ply exceedance indicator}

Definitions and conventions

σ1,σ2,τ12\sigma{}_{1}, \sigma{}_{2}, \tau{}_{12}
Ply stresses in material axes; τ₁₂ is the in-plane tensor shear stress.
Xt, Xc, Yt, Yc
Positive magnitudes of tensile and compressive strengths along fibre axis 1 and transverse axis 2.
S12S_{12}
Positive in-plane shear strength magnitude.
FI
Criterion-specific failure index; FI≤1 is below that criterion boundary.
RF, MS
Reserve factor under proportional load scaling and margin of safety RF−1.
σ₂σ₁statecriterion surface · RF scales the load directionRecover ply state → index → reserve factorMaximum stress/strain use component-wise limitsTsai–Hill and Tsai–Wu include interactionFI > 1 flags first-ply exceedance only
Material-axis stress is evaluated against a chosen failure surfaceThe index indicates criterion exceedance for the supplied allowables and load direction. It does not start a damage evolution or predict delamination.
6 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

The chain is: recover shell strain at a ply position, rotate to axes 1–2, apply the selected maximum-stress/strain, Tsai–Hill, or Tsai–Wu formula, then report index and reserve factor. The chosen strength data and Tsai–Wu interaction coefficient are assumptions unless supported by traceable material tests.

1 · Recover a local ply state before checking failure

A composite criterion must use ply material axes, not global shell stress. For each ply, QF Solver recovers the lower, middle, and upper through-thickness states from ε⁰+zκ, rotates strain to axes 1–2, and computes ply stress with that ply's reduced stiffness. The governing position can differ for each criterion.

εkshell(z)=ε0+zκ\boldsymbol{\varepsilon}_{k}^{shell}(z)=\boldsymbol{\varepsilon}^{0}+z\boldsymbol{\kappa}εkmat=Tε(θk)εkshell,σkmat=Qkεkmat\boldsymbol{\varepsilon}_{k}^{mat}=\mathcal T_{\varepsilon}(\theta_k)\boldsymbol{\varepsilon}_{k}^{shell},\qquad\boldsymbol{\sigma}_{k}^{mat}=\mathbf Q_k\boldsymbol{\varepsilon}_{k}^{mat}{σ1,σ2,τ12}k,z,z∈{lower,middle,upper}\{\sigma_1,\sigma_2,\tau_{12}\}_{k,z},\qquad z\in\{lower,middle,upper\}

Results depend on stacking order, local shell axes, angle convention, stress-recovery location, and interface-side convention. These inputs belong in the audit trail.

2 · Maximum stress and maximum strain

The maximum-stress criterion computes separate utilization ratios for fibre tension/compression, transverse tension/compression, and in-plane shear, then takes the largest. Maximum strain applies the same structure to material-axis engineering strains and strain allowables. Neither criterion models interaction between the components.

r1={σ1/Xt,σ1≥0−σ1/Xc,σ1<0,r2={σ2/Yt,σ2≥0−σ2/Yc,σ2<0,r6=∣τ12∣/S12r_1=\begin{cases}\sigma_1/X_t,&\sigma_1\ge0\\-\sigma_1/X_c,&\sigma_1<0\end{cases},\quad r_2=\begin{cases}\sigma_2/Y_t,&\sigma_2\ge0\\-\sigma_2/Y_c,&\sigma_2<0\end{cases},\quad r_6=|\tau_{12}|/S_{12}FIMS=max⁡(r1,r2,r6),RFMS=1/FIMSFI_{MS}=\max(r_1,r_2,r_6),\qquad RF_{MS}=1/FI_{MS}FIME=max⁡(ε1e1t/c,ε2e2t/c,∣γ12∣g12)FI_{ME}=\max\left(\frac{\varepsilon_1}{e_{1t/c}},\frac{\varepsilon_2}{e_{2t/c}},\frac{|\gamma_{12}|}{g_{12}}\right)

For strain, γ12 is engineering shear strain. Tensile and compressive allowables are selected by the sign of the normal component.

3 · Tsai–Hill: a quadratic interaction surface

Tsai–Hill combines normal and shear stress terms into one quadratic index. QF Solver follows the documented Azzi–Tsai convention and chooses X and Y tensile or compressive magnitudes according to the signs of σ₁ and σ₂. This choice makes the surface piecewise dependent on the stress quadrant.

FITH=(σ1X)2−σ1σ2X2+(σ2Y)2+(τ12S12)2FI_{TH}=\left(\frac{\sigma_1}{X}\right)^2-\frac{\sigma_1\sigma_2}{X^2}+\left(\frac{\sigma_2}{Y}\right)^2+\left(\frac{\tau_{12}}{S_{12}}\right)^2X={Xt,σ1≥0Xc,σ1<0,Y={Yt,σ2≥0Yc,σ2<0X=\begin{cases}X_t,&\sigma_1\ge0\\X_c,&\sigma_1<0\end{cases},\qquad Y=\begin{cases}Y_t,&\sigma_2\ge0\\Y_c,&\sigma_2<0\end{cases}RFTH=1FITHfor proportional loading and FITH>0RF_{TH}=\frac1{\sqrt{FI_{TH}}}\quad\text{for proportional loading and }FI_{TH}>0

The interaction term couples σ₁ and σ₂ but no progressive damage is applied after crossing FI=1.

4 · Tsai–Wu: include linear tension/compression asymmetry

Tsai–Wu adds linear terms so tensile and compressive strengths can differ. Its quadratic interaction coefficient F12 cannot be inferred uniquely from four uniaxial allowables; it needs biaxial test data or an explicitly labeled assumption. The current default f12* is −0.5.

FITW=F1σ1+F2σ2+F11σ12+F22σ22+2F12σ1σ2+F66τ122FI_{TW}=F_1\sigma_1+F_2\sigma_2+F_{11}\sigma_1^2+F_{22}\sigma_2^2+2F_{12}\sigma_1\sigma_2+F_{66}\tau_{12}^2F1=1Xt−1Xc,F11=1XtXc,F2=1Yt−1Yc,F22=1YtYc,F66=1S122F_1=\frac1{X_t}-\frac1{X_c},\quad F_{11}=\frac1{X_tX_c},\quad F_2=\frac1{Y_t}-\frac1{Y_c},\quad F_{22}=\frac1{Y_tY_c},\quad F_{66}=\frac1{S_{12}^2}F12=f12∗F11F22,−1<f12∗<1,f12∗=−0.5  documented defaultF_{12}=f_{12}^*\sqrt{F_{11}F_{22}},\qquad -1<f_{12}^*<1,\qquad f_{12}^*=-0.5\;\text{documented default}

Because FI_TW contains linear and quadratic terms, its reserve factor is not generally 1/√FI at the current point.

5 · Interpret reserve factor and margin

For proportional loading, scale all current stress components by RF until the selected index reaches one. For Tsai–Hill this gives 1/√FI. For Tsai–Wu, solve the positive root of the load-scaling quadratic. The implementation reports index, reserve factor, margin, pass flag, and contributing components.

FITW(RF σ)=L RF+Q RF2=1FI_{TW}(RF\,\boldsymbol{\sigma})=L\,RF+Q\,RF^2=1RFTW=min⁡{r>0:Qr2+Lr−1=0}RF_{TW}=\min\{r>0:Qr^2+Lr-1=0\}MS=RF−1MS=RF-1

The reserve factor is meaningful only for the assumed proportional loading direction and current ply state. Non-proportional load paths can change which ply and criterion governs.

6 · What this output cannot conclude

FI≤1 means only that the supplied state lies inside the chosen mathematical criterion using the supplied allowables. FI>1 predicts first-ply criterion exceedance. It does not trigger stiffness reduction, crack growth, fibre kinking, matrix cracking, delamination, fatigue, impact damage, or redistribution to neighboring plies.

FI≤1  ⟹  inside the selected criterion surfaceFI\le1\;\Longrightarrow\;\text{inside the selected criterion surface}FI>1  ⟹  predicted first-ply criterion exceedanceFI>1\;\Longrightarrow\;\text{predicted first-ply criterion exceedance}FI>1  ⟹̸  progressive failure simulationFI>1\;\not\Longrightarrow\;\text{progressive failure simulation}

QF Solver validates that strengths are finite and positive and constrains f12* to (−1,1); those checks do not verify that a material data sheet or test campaign supplied the values.

04 Algorithm and implementation

Source excerpt · v0.2.11

The evaluator computes criterion-specific indices and reserve factors in material axes. Values are indicators; no constitutive damage update follows.

Code
f1 = 1.0 / strengths.Xt - 1.0 / strengths.Xc
f2 = 1.0 / strengths.Yt - 1.0 / strengths.Yc
f11 = 1.0 / (strengths.Xt * strengths.Xc)
f22 = 1.0 / (strengths.Yt * strengths.Yc)
f66 = 1.0 / strengths.S12**2
f12 = strengths.f12_star * sqrt(f11 * f22)
linear = f1 * sigma1 + f2 * sigma2
quadratic = f11*sigma1**2 + f22*sigma2**2 + 2*f12*sigma1*sigma2 + f66*tau12**2
index = linear + quadratic
reserve = _positive_tsai_wu_root(linear, quadratic)
Read the complete versioned implementation →
Failure implementation

05 Verification and limitations

INDICATOR ONLY · no progressive damage update

Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.

v0.2.11 verification index
Capability matrix →Recorded cases and original sources →Run a maintained example →

06 Sources and related sheets

Criteria and definitionsFailure implementationAzzi–Tsai referenceTsai–Wu referencev0.2.11 verification index

Continue the reasoning

Isotropic linear elasticity3D orthotropic elasticitySmall-strain J2 plasticityTotal-Lagrangian St. Venant–KirchhoffClassical laminate resultants
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