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:
  • parameterdefaultsymbol
    "AnalysisType"Automaticnone
    "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 nameusage
    "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:
  • parameterdefaultsymbol
    "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:
  • parameterdefaultsymbol
    "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:
  • parameterdefaultsymbol
    "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: 03-July-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_solidmechanicspdecomponent, organization={Wolfram Research}, title={SolidMechanicsPDEComponent}, year={2022}, url={https://reference.wolfram.com/language/ref/SolidMechanicsPDEComponent.html}, note=[Accessed: 03-July-2022 ]}