Constitutive models

Seven models of soil and rock, each written down and each checked.

A finite element result is only as good as the stress–strain law at its integration points. KATAI 2D ships seven: from linear elasticity to a double-hardening soil model with small-strain stiffness, a creep model for soft clays, and the generalised Hoek–Brown criterion for rock. This page states what each one computes, which parameters it takes, and the closed-form or published result it has been verified against in release 0.9.0.

ModelWhat it capturesTypical usePython
Linear ElasticHooke’s law; stiffness may increase with depthStiff inclusions, rock at working stress, first estimateslinear_elastic
Mohr–CoulombElastic–perfectly plastic; friction, cohesion, dilatancy, tension cut-offLimit loads, factors of safety, first-pass deformationmohr_coulomb
Hardening SoilStress-dependent stiffness; shear and compression hardening; distinct unloading stiffnessExcavations, foundations, sands and stiff clayshardening_soil
HS smallHardening Soil plus stiffness decay with shear strain from G0Excavation-induced settlements, soil–structure interaction, dynamicshs_small
Soft SoilLogarithmic compression; pre-consolidation memoryNormally consolidated clay, silt, peat in primary compressionsoft_soil
Soft Soil CreepSoft Soil plus time-dependent (secondary) compressionEmbankments on soft ground, long-term settlementsoft_soil_creep
Hoek–BrownCurved strength envelope of a rock mass from σci, mi, GSI, DTunnels, rock foundations, rock slopes in deformationhoek_brown

Conventions on this page: stresses are effective unless stated, and the equations are written compression-positive, as in the soil mechanics literature. The program itself uses the continuum mechanics convention (tension positive) and fixed units — kN, m, day.

Model 1

Linear Elastic

Isotropic Hooke’s law with two constants. It has no strength and no stress dependence, which makes it the right model for parts of the system that are not expected to yield — a concrete block, intact rock at working stress — and the wrong one for soil near failure.

σ′=De ε,G=E2(1+ν),K=E3(1−2ν),Eoed=E (1−ν)(1+ν)(1−2ν)\boldsymbol{\sigma}' = \mathbf{D}^e\,\boldsymbol{\varepsilon}, \qquad G = \frac{E}{2(1+\nu)}, \quad K = \frac{E}{3(1-2\nu)}, \quad E_{\mathrm{oed}} = \frac{E\,(1-\nu)}{(1+\nu)(1-2\nu)}Isotropic linear elasticity and the three moduli derived from E and ν.

Young’s modulus may increase linearly with depth below a reference level, which is the Gibson soil of case KV-FND-011 (settlement within −3.4% of the published finite-layer value):

E(y)=Eref+Einc (yref−y)E(y) = E_{\mathrm{ref}} + E_{\mathrm{inc}}\,(y_{\mathrm{ref}} - y)Stiffness increasing linearly with depth below the reference level.

Verified by

  • Boussinesq strip load, stress at depth: +0.9% … +1.4% (KV-FND-008).
  • Flamant line load: +0.0% … +1.3% (KV-FND-009).
  • Rigid strip footing against Giroud (1972): +1.1% (KV-FND-012).
  • Staged excavation heave against 1D elastic unloading: +0.0% (KV-EXC-001).

Model 2

Mohr–Coulomb

Linear elasticity bounded by the Mohr–Coulomb failure surface: elastic–perfectly plastic, with a non-associated flow rule so that the dilatancy angle can be smaller than the friction angle. Five parameters, all with direct physical meaning, and exact collapse loads to test against — which is why it is the model the limit-load benchmarks are run with.

f=12 (σ1′−σ3′)−12 (σ1′+σ3′)sin⁡φ′−c′cos⁡φ′≤0f = \tfrac{1}{2}\,(\sigma'_1 - \sigma'_3) - \tfrac{1}{2}\,(\sigma'_1 + \sigma'_3)\sin\varphi' - c'\cos\varphi' \le 0Mohr–Coulomb yield function in principal effective stresses, σ′₁ ≥ σ′₂ ≥ σ′₃ compression-positive. The full surface is the six-sided cone obtained by permuting the stresses.
g=12 (σ1′−σ3′)−12 (σ1′+σ3′)sin⁡ψ,ε˙p=λ˙ ∂g∂σ′g = \tfrac{1}{2}\,(\sigma'_1 - \sigma'_3) - \tfrac{1}{2}\,(\sigma'_1 + \sigma'_3)\sin\psi, \qquad \dot{\boldsymbol{\varepsilon}}^{p} = \dot{\lambda}\,\frac{\partial g}{\partial \boldsymbol{\sigma}'}Plastic potential: the same function with the dilatancy angle ψ in place of φ′. ψ = φ′ is associated flow.

Tension cut-off

The Mohr–Coulomb cone admits tensile stresses of up to c′ cot φ′, which soil cannot carry. Three further yield functions limit each principal stress to a tensile strength σt, zero by default, with associated flow:

ft,i=− σi′−σt  ≤  0,i=1,2,3f_{t,i} = -\,\sigma'_i - \sigma_t \;\le\; 0, \qquad i = 1,2,3Tension cut-off: no principal stress may be more tensile than σ_t. The same cut-off acts in the Hardening Soil and Soft Soil families (case KV-CST-011).

Dilatancy cut-off

A dense sand does not dilate for ever. When the void ratio reaches its critical value the mobilised dilatancy is set to zero and further shearing is isochoric:

1+e=(1+einit) exp⁡(εv),ψm=0  for  e≥emax⁡1 + e = (1 + e_{\mathrm{init}})\,\exp(\varepsilon_v), \qquad \psi_m = 0 \ \text{ for } \ e \ge e_{\max}Dilatancy cut-off (volumetric strain expansion-positive). Verified in case KV-CST-004: dilation stops within one increment of ln((1+e_max)/(1+e_init)) and the flow is then isochoric to round-off.
ParameterMeaningUnitPython
E, νYoung’s modulus, Poisson’s ratiokN/m², –E, nu
c′, φ′Effective cohesion and friction angle (or su with φ = 0 for undrained total-stress analysis)kN/m², °c, phi
ψDilatancy angle°psi
γ, γsatUnit weight above and below the phreatic levelkN/m³gamma, gamma_sat
kx, kyPermeabilitym/dayk or kx, ky

Cohesion and stiffness may also increase linearly with depth — the profile of the Davis & Booker problem.

Verified by

  • Prandtl strip footing, Nc = 2 + π: +0.6% (KV-FND-010).
  • Cox (1962) circular footing, axisymmetric, associated flow: +3.7% (KV-FND-013).
  • Davis & Booker (1973), strength increasing with depth: +1.4% (KV-FND-014).
  • Griffiths & Lane (1999) slope by strength reduction: 1.384 against 1.380 (KV-SLP-002).

Recorded limit. The failure bound is applied as a one-sided limit on the major principal stress, which ratchets slightly when the failure deviator moves within an increment: a drained triaxial test stalls 0.24% below its plateau. The deviation is measured and published in the changelog rather than tuned away.

Model 3

Hardening Soil

Real soil is not linear before it fails. Its stiffness grows with confining stress, it is several times stiffer in unloading than in first loading, and a drained triaxial test traces a hyperbola rather than two straight lines. The Hardening Soil model (Schanz, Vermeer & Bonnier 1999) reproduces all three with an elastoplastic formulation: a shear hardening surface that produces the hyperbola, and a cap that produces plastic compression in oedometric and isotropic loading.

ε1=1Ei q1−q/qa,Ei=2 E502−Rf,qa=qfRf\varepsilon_1 = \frac{1}{E_i}\,\frac{q}{1 - q/q_a}, \qquad E_i = \frac{2\,E_{50}}{2 - R_f}, \qquad q_a = \frac{q_f}{R_f}The hyperbolic relation between axial strain and deviatoric stress in drained triaxial loading (after Kondner 1963; Duncan & Chang 1970).
qf=(c′cot⁡φ′+σ3′) 2sin⁡φ′1−sin⁡φ′q_f = \bigl(c'\cot\varphi' + \sigma'_3\bigr)\,\frac{2\sin\varphi'}{1 - \sin\varphi'}The failure deviator is the Mohr–Coulomb one; R_f = q_f / q_a is 0.9 by default.
Drained triaxial compression in the Hardening Soil model, evaluated from the equations at left: c′ = 0, φ′ = 35°, σ′₃ = 100 kPa, E₅₀ = 30 MPa, R_f = 0.9. Failure at q_f = 269 kPa is reached at 4.9% strain.
0 80 160 240 320 0 1 2 3 4 5 6 axial strain ε₁ [%] deviatoric stress q [kPa]
Hardening Soil response asymptote q_a = 299 kPa secant E50 through q_f / 2

Stress-dependent stiffness

All three stiffnesses follow a power law in the stress level, with exponent m — about 0.5 for sands, approaching 1 for soft clays:

E50=E50ref(c′cos⁡φ′+σ3′sin⁡φ′c′cos⁡φ′+prefsin⁡φ′)m,Eur=Eurref(c′cos⁡φ′+σ3′sin⁡φ′c′cos⁡φ′+prefsin⁡φ′)mE_{50} = E_{50}^{\mathrm{ref}}\left(\frac{c'\cos\varphi' + \sigma'_3\sin\varphi'}{c'\cos\varphi' + p^{\mathrm{ref}}\sin\varphi'}\right)^{m}, \quad E_{ur} = E_{ur}^{\mathrm{ref}}\left(\frac{c'\cos\varphi' + \sigma'_3\sin\varphi'}{c'\cos\varphi' + p^{\mathrm{ref}}\sin\varphi'}\right)^{m}Secant stiffness in primary triaxial loading, and the unloading–reloading stiffness.
Eoed=Eoedref(c′cos⁡φ′+σ1′sin⁡φ′c′cos⁡φ′+prefsin⁡φ′)mE_{\mathrm{oed}} = E_{\mathrm{oed}}^{\mathrm{ref}}\left(\frac{c'\cos\varphi' + \sigma'_1\sin\varphi'}{c'\cos\varphi' + p^{\mathrm{ref}}\sin\varphi'}\right)^{m}Tangent stiffness in primary oedometer loading, a function of the major principal stress.

Two hardening mechanisms

Shear hardening is driven by plastic shear strain and moves the shear yield surface towards the Mohr–Coulomb failure line; the mobilised dilatancy follows Rowe’s stress–dilatancy theory. Compression hardening moves an elliptical cap along the hydrostatic axis with the pre-consolidation pressure pp:

fc=q~ 2α2+p′ 2−pp 2,sin⁡ψm=sin⁡φm−sin⁡φcv1−sin⁡φm sin⁡φcvf_c = \frac{\tilde{q}^{\,2}}{\alpha^{2}} + p'^{\,2} - p_p^{\,2}, \qquad \sin\psi_m = \frac{\sin\varphi_m - \sin\varphi_{cv}}{1 - \sin\varphi_m\,\sin\varphi_{cv}}The cap yield surface, and Rowe’s (1962) mobilised dilatancy angle. α is derived internally so that one-dimensional compression reproduces K₀ⁿᶜ and E_oed.
ParameterMeaningUnitPython
E50refSecant stiffness in a drained triaxial test at prefkN/m²E50
EoedrefTangent stiffness in primary oedometer loading at prefkN/m²Eoed
EurrefUnloading–reloading stiffness at prefkN/m²Eur
mPower of the stress dependency–m (0.5)
c′, φ′, ψStrength and dilatancy, as in Mohr–CoulombkN/m², °c, phi, psi

Verified by

  • Oedometer column loaded from 50 to 200 kPa against the integrated Eoed law: −0.93% from a checked-in file, and −0.95% for the same law integrated at a stress point with no finite elements at all (KV-CST-002).
  • Independence of the load path: 10 to 160 load increments move the answer by 0.020%, and a mesh of 85 against 1105 nodes by 5.6×10−16 (KV-NUM-009).
  • Tension cut-off acting in the Hardening Soil family (KV-CST-011).
  • Independence of the stopping tolerance, from the file and per phase (KV-NUM-007, KV-NUM-008).

What the record covers, and what it does not yet. In release 0.9.0 the Hardening Soil family is verified in oedometric compression, in elastic unloading and through its numerical controls. Undrained triaxial stress paths — extension in particular — are not yet pinned to a published laboratory curve in this release; that comparison is being built for the next one. If your design question turns on undrained strength mobilised by this model, cross-check it with an Undrained (B) analysis, where the strength is an input.

Model 4

HS small — Hardening Soil with small-strain stiffness

At the strains that matter behind a retaining wall or under a raft — 10−5 to 10−3 — soil is far stiffer than a conventional laboratory test suggests. Ignoring that puts settlement troughs too wide and too deep, and wall deflections too large. HS small adds two parameters to Hardening Soil: the very-small-strain shear modulus G0, and the shear strain γ0.7 at which the secant modulus has fallen to about 70% of it.

GsG0=11+a γ/γ0.7,a=0.385\frac{G_s}{G_0} = \frac{1}{1 + a\,\gamma/\gamma_{0.7}}, \qquad a = 0.385Secant shear modulus against shear strain: the modified Hardin–Drnevich relation (Santos & Correia 2001).
Gt=G0(1+a γ/γ0.7)2  ≥  Gur=Eur2 (1+νur)G_t = \frac{G_0}{\bigl(1 + a\,\gamma/\gamma_{0.7}\bigr)^{2}} \;\ge\; G_{ur} = \frac{E_{ur}}{2\,(1+\nu_{ur})}Tangent shear modulus, bounded below by the unloading–reloading stiffness of the underlying Hardening Soil model, where plasticity takes over.

On unloading and reloading, Masing’s rule doubles the threshold strain, γ0.7 → 2γ0.7. Below the phase-transformation line the model uses the dilatancy of Li & Dafalias (2000), which allows contraction in loose states.

Stiffness reduction with shear strain in HS small, evaluated from the equations at left for γ₀.₇ = 1.5×10⁻⁴ — the material of verification case KV-CST-008, in which E₀ = 5 E_ur.
0 0.25 0.5 0.75 1 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² shear strain γ [–] G / G₀ [–]
secant G_s / G₀ tangent G_t / G₀, cut off at G_ur / G₀ = 0.2
ParameterMeaningUnitPython
G0refShear modulus at very small strain, at prefkN/m²G0
γ0.7Shear strain at which Gs = 0.722 G0–gamma07
… and every Hardening Soil parameter above.

Verified by

  • Heave of an excavated floor against the inverted secant law, sampled at three points of the degradation curve (1, 2 and 4 m of excavation): −0.14% / −0.30% / −0.72% (KV-CST-008). Riding the virgin backbone instead of Masing’s would read +5.7% / +12.9% / +34.8% — the case discriminates between the two.
  • With G0 = Gur the overlay is redundant and the run falls back onto plain Hardening Soil bit for bit.
  • The “reset small strain” phase option restores the fresh state to +0.026% of a run that never had a history (KV-CST-012).

Recorded limit of release 0.9.0. The strain history is accumulated as a monotone scalar and load reversals are not detected automatically. An unloading that follows a loading phase therefore does not recover small-strain stiffness on its own; the phase option reset_small_strain exists so that the engineer can say where a history begins. A tensor history with reversal detection is in development.

Model 5

Soft Soil

Normally consolidated clays, clayey silts and peat compress in proportion to the logarithm of stress. The Soft Soil model is built on that observation, in the tradition of the Cam-Clay family: a logarithmic compression law, an elliptical cap that remembers the pre-consolidation stress, and Mohr–Coulomb failure.

εv−εv0=λ∗ ln⁡p′p0′(primary loading),εve−εve0=κ∗ ln⁡p′p0′(unloading / reloading)\varepsilon_v - \varepsilon_v^{0} = \lambda^{*}\,\ln\frac{p'}{p'_0} \quad\text{(primary loading)}, \qquad \varepsilon_v^{e} - \varepsilon_v^{e0} = \kappa^{*}\,\ln\frac{p'}{p'_0} \quad\text{(unloading / reloading)}Logarithmic compression with the modified compression index λ* and the modified swelling index κ*.
λ∗=Cc2.3 (1+e),κ∗≈2 Cs2.3 (1+e),Kur=p′κ∗\lambda^{*} = \frac{C_c}{2.3\,(1+e)}, \qquad \kappa^{*} \approx \frac{2\,C_s}{2.3\,(1+e)}, \qquad K_{ur} = \frac{p'}{\kappa^{*}}Relation to the one-dimensional compression and swelling indices, and the stress-dependent elastic bulk modulus.

The slope M of the cap is not an input: it is derived from the normally consolidated earth pressure coefficient so that one-dimensional primary compression reaches that K0nc.

Verified by

  • A confined column walked through the same stress range three times — 50 → 200 kPa primary, back to 50, up to 200 again: −0.01% on primary loading, −1.01% and −1.63% on the two elastic legs; the settlement of the second loading is a tenth of the first, which is the pre-consolidation memory stated as a number (KV-CST-009).
  • The lateral stress ratio reached in primary loading: K0nc = 1 − sin 25° = 0.577, measured +2.11%.

A detail worth knowing. In one-dimensional unloading the lateral stress falls much less than the vertical, so κ* has no exact relation to the oedometer swelling index. A naive oracle, κ* ln(σv/σv0), is off by a factor of 2.2 on this case. The verification writes the closed form for the path the soil actually walks — part of verifying a model is making sure the reference is right.

Model 6

Soft Soil Creep

Soft clay keeps settling after the excess pore pressure has gone. Soft Soil Creep (Vermeer & Neher 1999) makes all inelastic strain time-dependent: the creep rate is governed by how close the current stress is to the pre-consolidation stress, which itself grows as the soil creeps — so the rate decays with the logarithm of time.

ε˙vc=μ∗τ(peqppeq)β,β=λ∗−κ∗μ∗,τ=1 day\dot{\varepsilon}_v^{c} = \frac{\mu^{*}}{\tau}\left(\frac{p^{\mathrm{eq}}}{p_p^{\mathrm{eq}}}\right)^{\beta}, \qquad \beta = \frac{\lambda^{*} - \kappa^{*}}{\mu^{*}}, \qquad \tau = 1\ \text{day}Volumetric creep rate. τ is one day because the 24-hour stage of the standard oedometer test defines the normal-consolidation line.
εvc(t)=μ∗ ln⁡ ⁣(1+tτ),μ∗=Cα2.3 (1+e)\varepsilon_v^{c}(t) = \mu^{*}\,\ln\!\left(1 + \frac{t}{\tau}\right), \qquad \mu^{*} = \frac{C_\alpha}{2.3\,(1+e)}On the normal-consolidation line the rate law integrates exactly to the classical logarithmic creep law; μ* is the modified creep index.

Verified by

  • Ground under its own weight left for 100 days with no change of load: settlement +1.81% from μ* ln(1 + t/τ) H, and +5.93% / +3.14% / +1.64% at 1, 10 and 1000 days — falling with duration, as it must, since the idealised law omits the elastic part (KV-CST-010).
  • The differential witness: the same file with plain Soft Soil — every feature of this model except creep — moves exactly 0.000 m over the same hundred days.

Use with care at OCR = 1. With β of the order of 16, a stress state above the pre-consolidation stress produces very high initial creep rates. Seed the initial state with the K0 procedure and a realistic over-consolidation, as the verification case does.

Model 7

Hoek–Brown

The generalised Hoek–Brown criterion (Hoek, Carranza-Torres & Corkum 2002) describes the strength of a jointed rock mass as a curve whose shape is set by the intact rock and by the quality of the mass. It is entered the way it is measured: the uniaxial compressive strength of intact core σci, the petrographic constant mi, the Geological Strength Index and the disturbance factor D.

σ1′=σ3′+σci(mb σ3′σci+s)a\sigma'_1 = \sigma'_3 + \sigma_{ci}\left(m_b\,\frac{\sigma'_3}{\sigma_{ci}} + s\right)^{a}The generalised Hoek–Brown failure criterion.
mb=mi eGSI−10028−14D,s=eGSI−1009−3D,a=12+16(e−GSI/15−e−20/3)m_b = m_i\,e^{\frac{\mathrm{GSI}-100}{28-14D}}, \qquad s = e^{\frac{\mathrm{GSI}-100}{9-3D}}, \qquad a = \frac{1}{2} + \frac{1}{6}\left(e^{-\mathrm{GSI}/15} - e^{-20/3}\right)Rock-mass constants.
σc=σci sa,σt=s σcimb\sigma_c = \sigma_{ci}\,s^{a}, \qquad \sigma_t = \frac{s\,\sigma_{ci}}{m_b}Uniaxial compressive and (biaxial) tensile strength of the rock mass.
The envelope of the rock mass of verification case KV-CST-016 — σ_ci = 50 MPa, m_i = 7, GSI = 48, D = 0, giving m_b = 1.093, s = 0.0031, a = 0.507 — evaluated from the equations at left. A linear envelope matched at both ends of this range is up to 4.0 MPa below it in between, and above it outside.
0 9 18 27 36 0 2 4 6 8 10 minor principal stress σ′₃ [MPa] major principal stress at failure σ′₁ [MPa]
Hoek–Brown envelope a straight line through its values at 0 and 10 MPa
ParameterMeaningUnitPython
E, νRock-mass modulus (not the intact core’s) and Poisson’s ratiokN/m², –E, nu
σciUniaxial compressive strength of the intact rockkN/m²sigma_ci
miIntact rock parameter (about 4 for claystone, 33 for granite)–mi
GSIGeological Strength Index, 0 … 100–gsi
DDisturbance factor, 0 undisturbed … 1 heavily blasted–D
ψ, σψDilatancy at zero confinement, and the confining stress at which it has died out°, kN/m²psi, sigma_psi

Verified by

  • A circular tunnel under 28 MPa hydrostatic stress, unloaded in stages, against the closed-form elasto-plastic solution: radial stress within 0.59% inside the plastic annulus and 0.72% outside it (KV-CST-016).
  • A triaxial specimen run from a project file: the plateau lands on the envelope to −0.0000% at two cell pressures (KV-CST-015).
  • At the material point: constants, strengths and envelope reproduced exactly, and the limit mb → 0 converging on this engine’s Mohr–Coulomb return in proportion to mb (KV-CST-014).

More on the tunnel case, and on the four analyses the engine refuses for rock, on the rock and tunnels page.

Drainage

Drained, undrained (A), (B), (C) and non-porous

The drainage type decides how pore water takes part in the response. It is a property of the material set, it can be overridden per phase (“ignore undrained behaviour”), and the choice between the undrained variants is a choice of which quantity you trust more — effective strength parameters, or a measured undrained shear strength.

TypeStiffnessStrengthExcess pore pressureUse when
Drainedeffective E′, ν′effective c′, φ′noneSands and gravels; long-term behaviour of clays
Undrained (A)effective E′, ν′effective c′, φ′computedShort-term clay behaviour followed by consolidation; su is a result
Undrained (B)effective E′, ν′undrained su, φ = 0computedShort-term stability where su is measured and must be honoured
Undrained (C)undrained Eu, νuundrained su, φ = 0not computedTotal-stress analysis; no consolidation afterwards
Non-poroustotaltotalnone, no pore pressure at allConcrete, intact rock, structural clusters

In an undrained effective-stress analysis the pore water adds a bulk stiffness to the skeleton:

Kwn=3 (νu−ν′)(1−2νu)(1+ν′) K′,B=Kw/nK′+Kw/n\frac{K_w}{n} = \frac{3\,(\nu_u - \nu')}{(1 - 2\nu_u)(1 + \nu')}\,K', \qquad B = \frac{K_w/n}{K' + K_w/n}Equivalent bulk stiffness of the pore fluid from the undrained Poisson’s ratio (0.495 by default), and Skempton’s B. Either ν_u or B may be given per material.

Verified by

  • Undrained (A) confined column, settlement and effective stress against the closed form: +0.0% on both (KV-CST-001).
  • A two-layer column with νu = 0.497 below and B = 0.90 above: +0.00% on four quantities (KV-CST-006).
  • Undrained (A) against Undrained (C) for the same soil: the ratio of their settlements is 2(1 + νu)/3 exactly, so they converge as νu → 0.5 — 0.33% apart at 0.495, 0.0067% at 0.4999 (KV-CST-007).
  • “Ignore undrained behaviour” against the same material declared drained: bit-identical (KV-CST-005).

Guidance

Choosing a model

A model should be as simple as the question allows and no simpler. In practice:

  • A collapse load or a factor of safety depends on strength, not on the stiffness law. Mohr–Coulomb gives it with five parameters and exact solutions to check against.
  • Wall deflections, settlements behind an excavation, heave at formation level depend on unloading stiffness and its stress dependence: Hardening Soil, and HS small where movements at small strain govern.
  • Settlement of soft ground under fill is logarithmic compression and time: Soft Soil for the primary part, Soft Soil Creep when secondary compression matters.
  • A rock mass around an opening needs a curved envelope: Hoek–Brown.
  • Everything that will not yield — a concrete base, a stiff stratum far from the works — can be Linear Elastic or non-porous, and the run is faster for it.

Whichever you choose, the engine tells you when a parameter combination is physically unusual, when the integration of the model ran out of room at a stress point, and which convergence criteria the run actually met. Those diagnostics are described on the features page; the numerical treatment of the models — return mapping, substepping under an error tolerance, the consistent tangent — on the theory page.

Every model on this page can be run against its own benchmark.

KATAI 2D is free under Apache‑2.0, installs in one line and ships the input files of the benchmarks it is verified against. The fastest way to judge it is to rerun one.

Install KATAI 2D See the verification record Source on GitHub