# SolidMechanicsPDEComponent

SolidMechanicsPDEComponent[vars,pars]

yields solid mechanics PDE terms with variables vars and parameters pars.

# Details

• 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 "MaterialModel" "LinearElasticIsotropic" none "ModelForm" "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 three-dimensional isotropic material, any two moduli can be used:
•  parameter name "BulkModulus" "LameParameter" "PoissonRatio" "ShearModulus" "YoungModulus"
• The parameter "ModelForm" can be "Solid", "PlaneStress" or "PlaneStrain".
• For the model form "PlaneStress", 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 defromation, 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 .
• 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 CauchyGreen tensor, and is the deformation gradient.
• 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

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## Basic Examples(3)

Define a solid mechanics PDE model:

Define a symbolic solid mechanics PDE model:

Define a symbolic time-dependent solid mechanics PDE model:

## 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)

Compute the deflection of a spoon held fixed at the end and with a force applied at the top. Set up variables and parameters:

Set up the PDE and the geometry:

Visualize the displacement:

Define a symbolic solid mechanics PDE model that considers thermal expansion:

### 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:

Solve the equations:

Visualize the displacement:

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:

Solve the equations:

Visualize the displacement:

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:

Solve the equations:

Visualize the displacement:

Compute the strain:

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:

Visualize the normal strain in the direction:

Find the shear strain:

Visualize the normal strain in the - direction:

### Stationary Plane Strain Analysis(1)

Define a 2D plane strain PDE model:

### 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:

Set up the PDE model:

Create a time dependent force at the right end of the beam:

Fix the beam on the left:

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:

Visualize the -displacement over time at a query point:

Visualize the displacement:

### Eigenmode Analysis(2)

Define an eigenmode solid mechanics model for a particular material:

Compute the eigenmodes of an iron bracket. Set up the region, variables and a material:

Compute the eigenvalue and modes:

Visualize the seventh to tenth eigenmode:

### Neo-Hookean Material Model(1)

Set up the variables and the model parameters for a neo-Hookean hyperelastic material model:

Create the PDE model with boundary conditions:

Show the actual displacement:

Compute the strain:

Compute the stress:

Wolfram Research (2021), SolidMechanicsPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/SolidMechanicsPDEComponent.html (updated 2022).

#### Text

Wolfram Research (2021), SolidMechanicsPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/SolidMechanicsPDEComponent.html (updated 2022).

#### CMS

Wolfram Language. 2021. "SolidMechanicsPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. 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

#### BibTeX

@misc{reference.wolfram_2022_solidmechanicspdecomponent, author="Wolfram Research", title="{SolidMechanicsPDEComponent}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/SolidMechanicsPDEComponent.html}", note=[Accessed: 21-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_solidmechanicspdecomponent, organization={Wolfram Research}, title={SolidMechanicsPDEComponent}, year={2022}, url={https://reference.wolfram.com/language/ref/SolidMechanicsPDEComponent.html}, note=[Accessed: 21-March-2023 ]}