Hyperelasticity

Hyperelastic constitutive laws are used to model materials with a nonlinear relation between stress and strain based on a strain energy density function. This type of material is typically found in rubber, foam, and biological tissue. Many different hyperelastic material models are available in the Nonlinear Structural Materials Module, but there is also support for users to define their own strain energy density functions. The following hyperelastic models are available:

• Arruda–Boyce
• Blatz–Ko
• Delfino
• Extended tube
• Fung anisotropic
• Gao
• Gent
• Mooney–Rivlin:
  • Two parameters
  • Five parameters
  • Nine parameters
• Murnaghan
• Neo–Hookean
• Ogden
• St. Venant–Kirchhoff
• Storakers
• van der Waals
• Varga
• Yeoh
• Fibers (anisotropic hyperelasticity):
  • Holzapfel–Gasser–Ogden
  • Linear elastic
  • Uniaxial data
  • User-defined anisotropic hyperelasticity
  • Thermal expansion in fibers
• Mullins effect:
  • Ogden–Roxburgh
  • Miehe
• Large strain:
  • Viscoelasticity
  • Viscoplasticity
  • Creep
  • Polymer viscoplasticity
  • Porous plasticity
• Phase-field damage

Porous Plasticity and Pressure-Dependent Plasticity

The modeling of plastic deformation in soils, porous metals, foams, and aggregates has a main difference with respect to traditional metal plasticity: the yield function and plastic potential are not only defined in terms of the deviatoric stress tensor, but also include dependencies on the hydrostatic pressure. The following plasticity models are available:

• Porous plasticity:
  • Shima–Oyane
  • Gurson
  • Gurson–Tvergaard–Needleman (GTN)
  • Fleck–Kuhn–McMeeking (FKM)
  • FKM–GTN
  • Capped Drucker–Prager
• Pressure-dependent plasticity:
  • Drucker–Prager
  • Foam
  • Elliptic
  • Parabolic
  • Cap and cutoff
• Large-strain
• Nonlocal plasticity:
  • Implicit gradient

Shape Memory Alloys

Shape memory alloys (SMAs) refer to materials that can remember their original shape after undergoing large deformations- when heated above a certain temperature. The material models available in the Nonlinear Structural Materials Module provide the necessary settings for the austenite and martensite start and finish temperatures, as well as important phase transformation parameters. Two common SMA models are available: Lagoudas and Souza–Auricchio.

Creep and Viscoplasticity

Creep is an inelastic time-dependent deformation that occurs when a material is subjected to stress (typically much less than the yield stress) at sufficiently high temperatures. In COMSOL Multiphysics^®, there are several creep models that can be combined with each other by adding additional creep nodes. The material models for viscoplasticity are used for rate-dependent inelastic deformations, and such models also undergo creep as part of their behavior. The polymer viscoplasticity models can handle large viscoplastic strains in rubber, polyethylene, and other polymers. The following creep and viscoplasticity models are available:

• Creep:
  • Norton (power law)
  • Norton–Bailey
  • Garofalo (hyperbolic sine)
  • Coble
  • Nabarro–Herring
  • Weertman
  • Large-strain creep
  • User defined
  • Isotropic hardening:
    • Time hardening
    • Strain hardening
    • User defined
  • Thermal effects:
    • Arrhenius
    • User defined
• Viscoplasticity:
  • Anand
  • Anand–Narayan
  • Bingham
  • Chaboche
  • Peric
  • Perzyna
  • Large-strain viscoplasticity
  • User defined
  • Isotropic hardening:
    • Linear
    • Ludwik
    • Swift
    • Voce
    • Hockett–Sherby
    • User defined
  • Kinematic hardening:
    • Linear
    • Armstrong–Frederick
    • Chaboche
• Polymer viscoplasticity:
  • Bergstrom–Boyce
  • Bergstrom–Bischoff
  • Parallel network
  • User defined

Plasticity

Many materials have a distinct elastic regime in which the deformations are recoverable and path independent. When the stresses exceed a certain level, i.e., the yield limit, permanent plastic strains will appear. Elastoplastic material models are common, both when modeling metals and soils. The Nonlinear Structural Materials Module offers functionality for defining the properties of elastoplastic material models with small or large plastic strains, including user-defined yield surfaces and flow rules. The following plasticity models are available:

• von Mises yield criterion
• Tresca yield criterion
• Orthotropic Hill criterion
• Isotropic hardening:
  • Perfectly plastic
  • Linear
  • Ludwik
  • Johnson–Cook
  • Swift
  • Voce
  • Hockett–Sherby
  • Hardening function
  • User defined
• Kinematic hardening:
  • Linear
  • Armstrong–Frederick
  • Chaboche
• Large-strain plasticity
• Nonlocal plasticity:
  • Implicit gradient

Nonlinear Elasticity

As opposed to hyperelastic materials, where the stress–strain relationship becomes significantly nonlinear at moderate to large strains, nonlinear elastic materials present nonlinear stress–strain relationships even at infinitesimal strains. The following nonlinear elasticity models are available:

• Ramberg–Osgood
• Power law
• Uniaxial data
• Shear data
• Bilinear elastic
• Fibers (anisotropy):
  • Thermal expansion in fibers

Additional material models are available with the Geomechanics Module.

Viscoelasticity

Viscoelastic materials have a time-dependent response even if the loading is constant in time. Many polymers and biological tissues exhibit this behavior. Linear viscoelasticity, which is included in the Structural Mechanics Module and the MEMS Module, is a commonly used approximation where the stress depends linearly on the strain and its time derivatives (strain rate). The nonlinear elastic and hyperelastic material models can be extended with viscoelasticity to achieve a nonlinear stress–strain relationship. The following viscoelasticity models are available:

• Small-strain viscoelasticity:¹
  • Burgers
  • Generalized Kelvin–Voigt
  • Generalized Maxwell
  • Kelvin–Voigt
  • Maxwell
  • Standard linear solid
  • Fractional derivatives
  • Volumetric and deviatoric viscoelasticity
• Temperature effects:
  • Williams–Landel–Ferry
  • Arrhenius
  • Tool–Narayanaswamy–Moynihan
  • User defined
• Large-strain viscoelasticity:
  • Generalized Maxwell
  • Kelvin–Voigt
  • Standard linear solid

Damage

The deformation of quasibrittle materials, such as concrete or ceramics, under mechanical loads is characterized by an initial elastic deformation. If a critical level of stress or strain is exceeded, a nonlinear fracture phase will follow the elastic phase. As this critical value is reached, cracks grow and spread until the material fractures. The occurrence and growth of the cracks play an important role in the failure of brittle materials, and there are a number of theories to describe such behavior. The following damage models are available:

• Equivalent strain criterion:
  • Rankine
  • Smooth Rankine
  • Norm of elastic strain tensor
  • User defined
• Phase–field damage
• Regularization:
  • Crack band
  • Implicit gradient
  • Viscous regularization

Parameter Estimation

Nonlinear material models rely on numerous material parameters, each requiring identification for accurate modeling predictions. This necessitates using an extensive dataset of experimental results for parameter estimation. With the Nonlinear Structural Materials Module, it is possible to calibrate both built-in and user-defined material models with experimental data through the application of nonlinear least-squares parameter estimation techniques and efficient gradient-based optimization solvers.