3.1.3.1.2. FiniteStrainContinuum

The FiniteStrainContinuum element is a displacement-based finite element for 2D and 3D large deformation analysis in continuum mechanics. This element formulation accounts for geometric nonlinearity through finite strain kinematics, making it suitable for problems involving large displacements, large rotations, and moderate to large strains.

The element supports both Total Lagrangian (TL) and Updated Lagrangian (UL) formulations, automatically adapts to 2D or 3D analysis based on the problem dimension (rank), and can be combined with various material models including hyperelastic and inelastic constitutive laws.

3.1.3.1.2.1. Overview

Element type: FiniteStrainContinuum

The element implements:

  • Total Lagrangian (TL) formulation: Reference configuration-based approach using the initial geometry throughout the analysis

  • Updated Lagrangian (UL) formulation: Current configuration-based approach where the reference is updated at each increment

  • 2D formulation: Plane stress or plane strain analysis with triangular or quadrilateral elements (displacement DOFs: u, v)

  • 3D formulation: Full 3D analysis with tetrahedral or hexahedral elements (displacement DOFs: u, v, w)

  • Geometric nonlinearity: Full consideration of large displacement and rotation effects through geometric stiffness contributions

  • Material nonlinearity: Compatible with hyperelastic, plastic, and other nonlinear constitutive models

The element computes the tangent stiffness matrix including both material and geometric contributions, internal force vector, and mass matrix for dynamic analysis. The formulation includes strain-displacement relations appropriate for finite deformations.

3.1.3.1.2.2. Parameters

3.1.3.1.2.2.1. Mandatory parameters

type

Must be set to "FiniteStrainContinuum"

material

A material block defining the constitutive behavior for finite strains. The material block must include a type parameter specifying the material model.

Common material types include:

  • "PlaneStress": For 2D plane stress analysis (can handle finite strains)

  • "PlaneStrain": For 2D plane strain analysis

  • "Isotropic": For 3D isotropic elasticity

  • "NeoHookean": Hyperelastic material model for rubber-like behavior

  • Other finite strain material models as available

Within the material block, material-specific parameters must be provided:

  • E: Young’s modulus

  • nu: Poisson’s ratio

  • Additional parameters depending on the material type

3.1.3.1.2.2.2. Optional parameters

method

Specifies the Lagrangian formulation to use. Options are:

  • "TL": Total Lagrangian formulation (default)

  • "UL": Updated Lagrangian formulation

If not specified, the Total Lagrangian formulation is used.

3.1.3.1.2.3. Examples

3.1.3.1.2.3.1. Example 1: 2D Nonlinear Beam (Total Lagrangian)

A cantilever beam subjected to large displacements using the default Total Lagrangian formulation:

ContElem =
{
  type = "FiniteStrainContinuum";

  material =
  {
    type = "PlaneStress";
    E    = 1.e6;
    nu   = 0.25;
  };
};

This configuration is used with a nonlinear solver in: examples/ch03/NewtonRaphson.pro

3.1.3.1.2.3.2. Example 2: 3D Finite Strain Analysis

A 3D structure undergoing large deformations:

ContElem =
{
  type = "FiniteStrainContinuum";

  material =
  {
    type = "Isotropic";
    E    = 1.e6;
    nu   = 0.25;
  };
};

This configuration is used in: examples/ch03/NewtonRaphson3D.pro

3.1.3.1.2.3.3. Example 3: Dynamic Analysis with Finite Strains

The element can be used for explicit dynamic analysis with finite strain effects:

ContElem =
{
  type = "FiniteStrainContinuum";

  material =
  {
    type = "PlaneStress";
    E    = 1.e6;
    nu   = 0.25;
    rho  = 1000.;
  };
};

This configuration is used in dynamic stress wave propagation: examples/ch05/StressWave_20x20.pro

3.1.3.1.2.4. Additional Examples

The FiniteStrainContinuum element is used in various examples demonstrating large deformation behavior:

  • Nonlinear beam analysis: examples/ch03/cantilever8.pro, examples/ch03/cantilever8PrescribedDisp.pro

  • Newton-Raphson examples: examples/ch03/NewtonRaphson.pro, examples/ch03/NewtonRaphson3D.pro

  • Dynamic analysis: examples/ch05/StressWave_20x20.pro

  • Contact mechanics: examples/contact/contact_test.pro, examples/contact/contact_test02.pro

  • Buckling analysis: examples/solver/dissipatedEnergySolver/delam_buckling100.pro, examples/solver/dissipatedEnergySolver/delam_buckling200.pro

3.1.3.1.2.5. Solver Requirements

The FiniteStrainContinuum element requires a nonlinear solver due to the geometric nonlinearity. Use solvers such as:

  • NonlinearSolver: Standard Newton-Raphson solution procedure

  • RiksSolver: Arc-length method for tracing equilibrium paths

  • DissipatedEnergySolver: For problems with softening behavior

  • ExplicitSolver: For explicit dynamic analysis

  • smallstraincontinuum - Small strain continuum element

  • materials - Available material models

  • solvers - Nonlinear solution procedures

  • tutorial1 - Introduction to PyFEM input files