SolidMechanicsPDEComponent
SolidMechanicsPDEComponent[vars,pars]
yields solid mechanics PDE terms with variables vars and parameters pars.
Details




- SolidMechanicsPDEComponent returns a differential operator for solid mechanics analysis.
- SolidMechanicsPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:
- SolidMechanicsPDEComponent models the resulting displacement of a body subject to applied loads and constraints.
- SolidMechanicsPDEComponent creates PDE components for stationary, time-dependent, parametric, frequency-response and eigenmode analysis.
- SolidMechanicsPDEComponent models solid mechanics phenomena with dependent variables displacement
,
and
in units of meters
, independent variables
in units of
, time variable
in units of seconds
and angular frequency
in units of radians per second.
- SolidMechanicsPDEComponent creates PDE components in two and three space dimensions.
- Stationary variables vars are vars={{u[x1,…,xn],v[x1,…,xn],…},{x1,…,xn}}.
- Time-dependent or eigenmode variables vars are vars={{u[t,x1,…,xn],v[t,x1,…,xn],…},t,{x1,…,xn}}.
- Frequency-dependent variables vars are vars={{u[x1,…,xn],v[x1,…,xn],…},ω,{x1,…,xn}}.
- The equations for different analysis types SolidMechanicsPDEComponent generates depend on the form of vars.
- The stationary equilibrium equations of the solid mechanics PDE SolidMechanicsPDEComponent with mass density
, stress tensor
, strain tensor
, displacement vector
and body load vectors
or
is based on:
- The time-dependent equilibrium equation of the solid mechanics model SolidMechanicsPDEComponent is based on:
- The eigenfrequency equation of the solid mechanics model SolidMechanicsPDEComponent with eigenvalues
is based on:
- The frequency response equation of the solid mechanics model SolidMechanicsPDEComponent with angular frequency
is based on:
- The units of the solid mechanics model terms are a force density in
.
- The following parameters pars can be given:
-
parameter default symbol "AnalysisType" Automatic none "BodyLoad" 0 , body force density in
"BodyLoadValue" 0 , body acceleration in
"MassDensity" - , density in
"Material" - none "SolidMechanicsMaterialModel" "LinearElasticIsotropic" none "SolidMechanicsModelForm" "Solid" none - When the "AnalysisType" is Automatic, then the model generated depends on the form of vars.
- For eigenfrequency analysis, "AnalysisType" needs to be set to "Eigenmode" and time-dependent variables tvars need to be used.
- If a "Material" is specified, the material constants are extracted from the material data; otherwise, relevant material parameters need to be specified.
- For isotropic material, any two moduli can be used:
-
parameter name "BulkModulus" "LameParameter" "PoissonRatio" "PWaveModulus" "ShearModulus" "YoungModulus" - The parameter "SolidMechanicsModelForm" can be "Solid", "PlaneStress" or "PlaneStrain".
- For the "PlaneStress" and the "PlaneStress" models, a "Thickness" parameter needs to be defined.
- The following material models are available:
-
material model name usage "LinearElasticIsotropic" small deformation "LinearElasticOrthotropic" small deformation "LinearElasticAnsiotropic" small deformation "StVernantKirchhoffIsotropic" large deformation, educational, nonlinear "NeoHookeanIsotropic" large deformation, nonlinear "YeohIsotropic" large deformation, nonlinear - The default material model is a linear elastic isotropic material model.
- The kinematic equation uses an infinitesimal, small deformation strain measure:
- For nonlinear material laws, the kinematic equation uses the Green-Lagrange strain measure based on the deformation gradient
, where
is the identity matrix:
- The constitutive equation for the linear elastic material models with elasticity matrix
,
an initial stress,
an initial strain and
a thermal strain is given by:
- The normal strain components
and the shear strain components
use Voigt notation with the following order:
- The normal stress components
and the shear stress components
use Voigt notation with the following order:
- The "PlaneStrain" model assumes 0 strain in the
direction for
.
- The "PlaneStress" model assumes 0 stresses in the
direction for
.
- "RegionSymmetry""Axisymmetric" uses a truncated cylindrical coordinate system and assumes a displacement of
in the
direction and
. Note that
.
- The constitutive equation for the linear elastic isotropic material with Young's modulus
and Poisson ratio
is given by:
- A thermal strain
can be added with the coefficient thermal expansion
in units
, thermal strain temperature
in
and thermal strain reference temperature
in
:
- The following subparameters can be given for the "LinearElasticIsotropic" material model:
-
parameter default symbol "InitialStrain" 0 , initial strain
"InitialStress" 0 , initial stress in
"PoissonRatio" Automatic , Poisson ratio
"ThermalExpansion" 0 , coefficient of thermal expansion in
"ThermalStrainTemperature" 0 , temperature in
"ThermalStrainReferenceTemperature" 0 , temperature in
"YoungModulus" Automatic , Young modulus in
- The constitutive equation for the linear elastic orthotropic material model with compliance matrix
is given by:
- The elasticity matrix is the inverse of compliance matrix
.
- The linear elastic orthotropic compliance matrix
with shear modulus
is given by:
- For the linear elastic orthotropic material model, the coefficient of thermal expansion
depends on direction:
- The following subparameters can be given for the "LinearElasticOrthotropic" material model:
-
parameter default symbol "InitialStrain" 0 , initial strain
"InitialStress" 0 , initial stress in
"PoissonRatio" - ,
,
,
,
,
Poisson ratios
"ShearModulus" - ,
,
shear moduli in
"ThermalExpansion" 0 ,
,
, coefficient of thermal expansion in
"ThermalStrainTemperature" 0 , temperature in
"ThermalStrainReferenceTemperature" 0 , temperature in
"YoungModulus" - ,
,
Young moduli in
- Poisson ratio, shear modulus and Young's modulus are specified as formal indexed variables.
- For the anisotropic material model, the full elasticity matrix
needs to be specified:
- Alternatively, the compliance matrix
can be specified.
- For the linear elastic anisotropic material model, the coefficient of thermal expansion
depends on direction:
- The following subparameters can be given for the "LinearElasticAnisotropic" material model:
-
parameter default symbol "ComplianceMatrix" - , compliance matrix
"ElasticityMatrix" - , elasticity matrix,
"InitialStrain" 0 , initial strain
"InitialStress" 0 , initial stress in
"ThermalExpansion" 0 ,
,
,
,
,
, coefficient of thermal expansion in
"ThermalStrainTemperature" 0 , temperature in
"ThermalStrainReferenceTemperature" 0 , temperature in
- The "StVenantKirchhoffIsotropic" material model is based on the energy density function
, where
is the first Lamé constant,
is the second constant, and
is the Green–Lagrange strain.
- The "NeoHookeanIsotropic" material model is based on the energy density function
, where
is the first Lamé constant,
is the second constant,
is the right Cauchy–Green tensor, and
is the deformation gradient.
- The "YeohIsotropic" material model is based on the energy density function
, where the
are model constants,
,
is the first invariant, and
with
the deformation gradient.
is a model parameter set to
.
- SolidMechanicsPDEComponent uses "SIBase" units. The geometry has to be in the same units as the PDE.
- If the SolidMechanicsPDEComponent depends on parameters
that are specified in the association pars as …,keypi…,pivi,…], the parameters
are replaced with
.


















Examples
open allclose allBasic Examples (3)
Scope (15)
Activate a solid mechanics PDE model:
Define a stationary solid mechanics model with a particular material:
Define a model with material values specified:
Stationary Analysis (2)
Stationary Plane Stress Analysis (3)
Compute the displacement of a rectangular steel plate held fixed at the left and with a forced displacement on the right. Set up the region, variables and parameters:
Compute the displacement of a rectangular steel plate held fixed at the left and with a pressure applied at the right end. Set up the region, variables and parameters:
Compute the displacement of a rectangular steel plate held fixed at the bottom and with pressures applied at the remaining sides. Set up the region, variables and parameters:
Verify that the normal strain in the direction is about 0:
Visualize the normal strain in the direction:
Verify that the normal strain in the direction is about 0:
Time-Dependent Analysis (3)
Define a time-dependent solid mechanics model for a particular material:
Simulate a time-dependent force on a beam. Set up a region, variables and a material:
Create a time dependent force at the right end of the beam:
Set up zero initial conditions and the initial velocity condition:
Solve the time-dependent PDE and monitor the progress:
Visualize the -displacement over time at a query point:
Simulate a time-dependent force on a cross section of a spoon considering a Rayleigh damping model. Set up a region, variables and a material:
Set up the PDE and solve while monitoring progress:
Eigenmode Analysis (2)
Text
Wolfram Research (2021), SolidMechanicsPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/SolidMechanicsPDEComponent.html (updated 2023).
CMS
Wolfram Language. 2021. "SolidMechanicsPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/SolidMechanicsPDEComponent.html.
APA
Wolfram Language. (2021). SolidMechanicsPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SolidMechanicsPDEComponent.html