Distinguish infinitesimal strain, finite-rotation measures and the work-conjugate stresses required by each formulation.
01 Model and scope
Kinematics describes deformation. A constitutive law relates the chosen strain and state to a compatible stress. Corotation, total-Lagrangian kinematics and J2 plasticity therefore describe different parts of a solver route.
Route assumptions and calculation checks
- Large rigid rotation is not objective under infinitesimal strain.
- A kinematic description does not select a constitutive law.
- Match the stress measure, integration domain and consistent tangent.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- Deformation gradient and its determinant.
- Infinitesimal and Green–Lagrange strain.
- Cauchy, first Piola and second Piola stress.
- Rotation and right stretch in polar decomposition.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Identify the infinitesimal approximation
Infinitesimal strain drops products of displacement gradients. It vanishes for infinitesimal rigid rotations, but does not represent arbitrary finite rotation objectively.
2 · Retain finite-rotation kinematics
The deformation gradient maps reference line elements to the current configuration. Green–Lagrange strain retains quadratic displacement-gradient terms and vanishes under a proper rigid rotation.
3 · Pair stress with the chosen description
First Piola stress is conjugate to the deformation gradient in reference-volume virtual work. Second Piola stress pairs with Green–Lagrange strain. Push-forward gives Cauchy stress when J is positive.
4 · Distinguish a composed route
A corotational route removes rotation before a bounded local material update. Its local strain bound, state transport and force tangent remain part of that named implementation. It is not interchangeable with multiplicative finite-strain plasticity.
Consult the corotational J2 and total-Lagrangian StVK sheets for their actual implementation and evidence limits.
04 Algorithm and implementation
Follow the route sheets below to see how this shared mathematics is implemented.
05 Verification and limitations
FOUNDATION · distinguish the selected kinematics
The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.