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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 ([15]):

| (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 ([15]):

| (3)

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

Finally, the deviatoric elastic left Cauchy-Green tensor is defined by ([46]):

| (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 [8]), 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)
  - Six-node shell element (S6)
  - Eight-node shell element (S8 and S8R)
  - Six-node plane stress element (CPS6)
  - Eight-node plane stress element (CPS8 and CPS8R)
  - Six-node plane strain element (CPE6)
  - Eight-node plane strain element (CPE8 and CPE8R)
  - Six-node axisymmetric element (CAX6)
  - Eight-node axisymmetric element (CAX8 and CAX8R)
  - Three-node beam element (B32 and B32R)
  - Three-node fluid element (D)
  - Two-node unidirectional gap element (GAPUNI)
  - Two-node 3-dimensional dashpot (DASHPOTA)
  - Two-node 3-dimensional spring (SPRINGA)
- Fluid Section Types: Gases
  - Orifice
  - Bleed Tapping
  - Preswirl Nozzle
  - Straight and Stepped Labyrinth
  - Characteristic
  - Carbon Seal
  - Gas Pipe
  - Restrictor, Long Orifice
  - Restrictor, Enlargement
  - Restrictor, Contraction
  - Restrictor, Bend
  - Restrictor, Wall Orifice
  - Restrictor, Entrance
  - Restrictor, Exit
  - Restrictor, User
  - Branch, Joint
  - Branch, Split
  - Vortex
  - Möhring
  - Change of the absolute into a relative system and vice versa
- Fluid Section Types: Liquids
  - Pipe, Manning
  - Pipe, White-Colebrook
  - Pipe, Sudden Enlargement
  - Pipe, Sudden Contraction
  - Pipe, Entrance
  - Pipe, Diaphragm
  - Pipe, Bend
  - Pipe, Gate Valve
  - Pump
- Boundary conditions
  - Single point constraints (SPC)
  - Multiple point constraints (MPC)
  - Contact
- 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
  - Steady state dynamics
  - Direct integration dynamic analysis
  - Heat transfer
  - Acoustics
  - Shallow water motion
  - Hydrodynamic lubrication
  - Irrotational incompressible inviscid flow
  - Electrostatics
  - Stationary groundwater flow
  - Diffusion mass transfer in a stationary medium
  - Aerodynamic Networks
  - Hydraulic Networks
- Convergence criteria
- Loading
  - Point loads
  - Facial distributed loading
  - Centrifugal distributed loading
  - Gravity distributed loading
  - Temperature loading in a mechanical analysis
  - Initial(residual) stresses
  - Concentrated heat flux
  - Distrubuted heat flux
  - Convective heat flux
  - Radiative heat flux
- Zienkiewicz-Zhu error estimator

Next: Element Types Up: CalculiX CrunchiX USER'S MANUAL Previous: Hydraulic pipe system Contents