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# Theory

The finite element method is basically concerned with the determination of field variables. The most important ones are the stress and strain fields. As basic measure of strain in CalculiX the Lagrangian strain tensor E is used for elastic media, the Eulerian strain tensor e is used for deformation plasticity and the deviatoric elastic left Cauchy-Green tensor is used for incremental plasticity. The Lagrangian strain satisfies ([8]):

| (1)

where are the displacement components in the material frame of reference and repeated indices imply summation over the appropriate range. In a linear analysis, this reduces to the familiar form:

| (2)

The Eulerian strain satisfies ([8]):

| (3)

where are the displacements components in the spatial frame of reference.

Finally, the deviatoric elastic left Cauchy-Green tensor is defined by ([22]):

| (4)

where is the elastic Jacobian and is the elastic deformation gradient. The above formulas apply for Carthesian coordinate systems.

The stress measure consistent with the Lagrangian strain is the second Piola-Kirchhoff stress S. This stress, which is internally used in CalculiX for all applications (the so-called total Lagrangian approach, see [6]), can be transformed into the first Piola-Kirchhoff stress P (the so-called engineering stress, a nonsymmetric tensor) and into the Cauchy stress t (true stress). All CalculiX input (e.g. distributed loading) and output is in terms of true stress. In a tensile test on a specimen with length L the three stress measures are related by:

| (5)

where is the engineering strain defined by

| (6)

Subsections

- Element Types
  - Eight-node brick element (C3D8)
  - Eight-node brick element with reduced integration (C3D8R)
  - Twenty-node brick element (C3D20)
  - Twenty-node brick element with reduced integration (C3D20R)
  - Four-node tetrahedral element (C3D4)
  - Ten-node tetrahedral element (C3D10)
  - Six-node wedge element (C3D6)
  - Fifteen-node wedge element (C3D15)
  - Eight node shell element (S8 and S8R)
  - Eight node plane stress element (CPS8 and CPS8R)
  - Eight node plane strain element (CPE8 and CPE8R)
  - Eight node axisymmetric element (CAX8 and CAX8R)
  - Three node beam element (B32 and B32R)
- Boundary conditions
  - Single point constraints (SPC)
  - Multiple point constraints (MPC)
- Materials
  - Linear elastic materials
  - Hyperelastic and hyperfoam materials
  - Deformation plasticity
  - Incremental (visco)plasticity
  - User materials
- Types of analysis
  - Static analysis
  - Frequency analysis
  - Buckling analysis
  - Modal dynamic analysis
  - Direct integration dynamic analysis
- Loading
  - Point loads
  - Facial distributed loading
  - Centrifugal distributed loading
  - Gravity distributed loading
  - Temperature loading
  - Initial(residual) stresses

Next: Element Types Up: CalculiX CrunchiX USER'S MANUAL Previous: Simple example problem Contents