Total-Lagrangian St. Venant–Kirchhoff

Composed formulationAUDITED GO_WITH_LIMITATIONS · maturity not promoted

A finite-kinematic route using reference coordinates and an StVK law for selected static serial solid cases.

01 Model and scope

The total-Lagrangian formulation integrates forces over the reference configuration. Green–Lagrange strain is objective under finite rigid rotation, and the StVK law applies linear isotropic elasticity to that strain. This combination has finite-kinematic terms but is only a simple verification material; it is not physically reliable for every large-strain response.

Read the kinematics, constitutive law and solution lifecycle together. A component's evidence does not qualify their combination.

Route assumptions and calculation checks

Green–Lagrange strain retains quadratic displacement-gradient terms and is zero for a pure finite rigid rotation.
StVK is linear in E but does not become a generally reliable material law at large strain merely because the kinematics are finite.
The audited evidence is selected static serial TET4/HEX8, GO_WITH_LIMITATIONS, with no maturity promotion; contact, coupled J2, dynamics, MPI/PETSc, high-order, follower-load, and reduced/hourglass routes are excluded.

Useful foundations and prerequisites

Strain, stress and rotation measures Units, signs and error measures

02 Essential equations and symbols

\mathbf F=\mathbf I+\nabla_X\mathbf u,\quad J=\det\mathbf F>0

Definitions and conventions

$X, x = X+u$: Reference and current coordinates.
$F = \partial{}x/\partial{}X$: Deformation gradient; J=det(F) must remain positive.
$E = \frac{1}{2}(F^TF-I)$: Green–Lagrange strain measured against the reference configuration.
$S$: Second Piola–Kirchhoff stress, work-conjugate to E.
$P = FS$: First Piola–Kirchhoff stress, used with reference gradients.

Reference configuration and finite-rotation strainThe strain measure is objective for rigid rotation, but StVK remains a simple linear-in-E material. Finite kinematics alone do not validate arbitrary large-strain response.

5 detailed steps

03 Step by step derivation

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

In the v0.2.11 TET4 kernel, reference gradients form F, then E, second Piola stress S, first Piola stress P=FS, and internal force. A consistent tangent contains both material and geometric contributions. The separate static TET4/HEX8 audit remained GO_WITH_LIMITATIONS and did not promote maturity.

1 · Keep the reference configuration fixed

Total-Lagrangian integration stores the original coordinates X and reference volume Ω₀ throughout the solve. The deformation gradient maps reference directions to current directions. This choice makes the virtual-work integral independent of repeatedly rebuilding the integration domain in the deformed configuration.

\mathbf x(\mathbf X)=\mathbf X+\mathbf u(\mathbf X)

The TET4 implementation rejects an inverted or singular current deformation as well as invalid reference geometry.

2 · Use Green–Lagrange strain for finite-rotation kinematics

The term FᵀF removes a pure rigid rotation because RᵀR=I. Expanding E shows both the linear symmetric-gradient term and a quadratic gradient term. That quadratic term is discarded by infinitesimal elasticity but retained here.

\mathbf E=\frac12\left(\mathbf F^T\mathbf F-\mathbf I\right)

Objectivity of this strain measure does not make an arbitrary constitutive law objective by itself; stress measures and work conjugacy must also match.

3 · Apply the St. Venant–Kirchhoff material law

StVK is the finite-kinematic counterpart of a linear isotropic law written in Green–Lagrange strain. It returns second Piola stress S. The law remains linear in E, even though E is nonlinear in displacement; the global force–displacement relation is consequently nonlinear.

\mathbf S=\lambda\operatorname{tr}(\mathbf E)\mathbf I+2\mu\mathbf E

Here E_Y denotes Young's modulus; bold E denotes Green–Lagrange strain. StVK can predict poor or nonphysical behavior at large strain despite the exact finite-rotation kinematics.

4 · Push stress forward and assemble reference virtual work

Second Piola stress is work-conjugate to Green–Lagrange strain. The first Piola stress P=FS enters the reference-volume force integral. The consistent tangent separates a material contribution, which changes S with E, from a geometric contribution due to the current F and stress.

\mathbf P=\mathbf F\mathbf S

For the affine TET4 kernel, reference gradients and volume are constant. Higher-order or distorted element routes require their own integration and tangent evidence.

5 · Read the audit boundary

The source verification summary reports a bounded static serial campaign using selected TET4 and HEX8 StVK cases and records GO_WITH_LIMITATIONS; its final audit leaves maturity unchanged. The envelope excludes contact, coupled J2, dynamics, MPI/PETSc, higher-order elements, follower loads, and reduced/hourglass HEX8 variants.

\text{audited route: selected static serial TET4 / HEX8 cases}

A small geometric-nonlinear smoke example is deliberately WARNING. Treat it as an execution demonstration, not proof of convergence, physical validation, or qualification outside the recorded route.

04 Algorithm and implementation

Source excerpt · v0.2.11

The TET4 kernel evaluates Green–Lagrange strain and StVK second Piola stress, maps it to first Piola stress, and integrates force over the reference volume.

deformation = self.deformation_gradient(coords, displacement)
strain = self.green_lagrange_strain(deformation)
stress = self.second_piola_stress(strain)
first_piola = deformation @ stress
internal = volume * np.einsum('iJ,aJ->ai', first_piola, gradients).reshape(12)
tangent = self._consistent_tangent(deformation, stress, gradients, volume)
return internal, symmetrize(tangent)

Read the complete versioned implementation →

v0.2.11 TET4 total-Lagrangian kernel