SolidMechanicsPDEComponent

✖
`SolidMechanicsPDEComponent`

Updatedshow changeshide changes[Experimental]

✖

SolidMechanicsPDEComponent[vars,pars]

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

Details

SolidMechanicsPDEComponent returns partial differential equations 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 displacements , and in units of meters [] as dependent variables, as 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,…,x_n],v[x₁,…,x_n],…},{x₁,…,x_n}}.
Time-dependent or eigenmode variables vars are vars={{u[t,x₁,…,x_n],v[t,x₁,…,x_n],…},t,{x₁,…,x_n}}.
Frequency-dependent variables vars are vars={{u[x₁,…,x_n],v[x₁,…,x_n],…},ω,{x₁,…,x_n}}.
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 [ $TemplateBox[{InterpretationBox[, 1], {"kg", , "/", , {"m", ^, 3}}, kilograms per meter cubed, {{(, "Kilograms", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ], stress tensor [], strain tensor , displacement vector [] and body load vectors [ $TemplateBox[{InterpretationBox[, 1], {"N", , "/", , {"m", ^, 3}}, newtons per meter cubed, {{(, "Newtons", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ] or [ $TemplateBox[{InterpretationBox[, 1], {"m", , "/", , {"s", ^, 2}}, meters per second squared, {{(, "Meters", )}, /, {(, {"Seconds", ^, 2}, )}}}, QuantityTF]$ ] are 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 [ $TemplateBox[{InterpretationBox[, 1], {"N", , "/", , {"m", ^, 3}}, newtons per meter cubed, {{(, "Newtons", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ].
The following parameters pars can be given:

parameter	default	symbol
"AnalysisType"	Automatic	none
"BodyLoad"	0	, body force density [ $TemplateBox[{InterpretationBox[, 1], {"N", , "/", , {"m", ^, 3}}, newtons per meter cubed, {{(, "Newtons", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ]
"BodyLoadValue"	0	, body acceleration [ $TemplateBox[{InterpretationBox[, 1], {"m", , "/", , {"s", ^, 2}}, meters per second squared, {{(, "Meters", )}, /, {(, {"Seconds", ^, 2}, )}}}, QuantityTF]$ ]
"MassDensity"	-	, density [ $TemplateBox[{InterpretationBox[, 1], {"kg", , "/", , {"m", ^, 3}}, kilograms per meter cubed, {{(, "Kilograms", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ]
"Material"	-	none
"MaterialSymmetry"	"Isotropic"	none
"SolidMechanicsMaterialModel"	"LinearElastic"	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 linear elastic isotropic material, any two moduli can be used:
parameter name

"BulkModulus"

"LameParameter"

"PoissonRatio"

"PWaveModulus"

"ShearModulus"

"YoungModulus"
The parameter "SolidMechanicsModelForm" can be "Solid", "PlaneStress", "PlaneStrain", "ExtendedPlaneStress" or "ExtendedPlaneStrain".
For the "PlaneStress", the "PlaneStrain", the "ExtendedPlaneStress" or "ExtendedPlaneStrain" models, a "Thickness" parameter needs to be defined.
The default material model is a linear elastic isotropic material model.
The following material symmetries for linear elastic small deformation models are available:
material symmetry name

"Isotropic"

"Orthotropic"

"TransverselyIsotropic"

"Anisotropic"
The following isotropic compressible hyperelastic large deformation material models are available:
material model name

"StVenantKirchhoff"

"NeoHookean"
The following nearly isotropic incompressible hyperelastic large deformation material models are available:
material model name

"ArrudaBoyce"

"Gent"

"MooneyRivlin"

"NeoHookean"

"Yeoh"
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 of [], thermal strain temperature [] and thermal strain reference temperature []:

The following subparameters can be given for the "LinearElastic" "Isotropic" material model:

parameter	default	symbol
"InitialStrain"	0	, initial strain
"InitialStress"	0	, initial stress []
"PoissonRatio"	Automatic	, Poisson ratio
"ThermalExpansion"	0	, coefficient of thermal expansion []
"ThermalStrainTemperature"	0	, temperature []
"ThermalStrainReferenceTemperature"	0	, temperature []
"YoungModulus"	Automatic	, Young modulus []

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 "LinearElastic" "Orthotropic" material model:

parameter	default	symbol
"InitialStrain"	0	, initial strain
"InitialStress"	0	, initial stress []
"PoissonRatio"	-	, , , , , Poisson ratios
"ShearModulus"	-	, , shear moduli []
"ThermalExpansion"	0	, , , coefficient of thermal expansion []
"ThermalStrainTemperature"	0	, temperature []
"ThermalStrainReferenceTemperature"	0	, temperature []
"YoungModulus"	-	, , Young moduli []

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 "LinearElastic" "Anisotropic" material model:

parameter	default	symbol
"ComplianceMatrix"	-	, compliance matrix
"ElasticityMatrix"	-	, elasticity matrix,
"InitialStrain"	0	, initial strain
"InitialStress"	0	, initial stress []
"ThermalExpansion"	0	, , , , ,, coefficient of thermal expansion []
"ThermalStrainTemperature"	0	, temperature []
"ThermalStrainReferenceTemperature"	0	, temperature []

Compressible and nearly incompressible hyperelastic material models are available. Nearly incompressible models are the default, when available.
The "ArrudaBoyce" material model is based on the energy density function , where is the number of polymer chains in a network, the number of segments in a single chain, the Boltzmann constant, the absolute temperature, a chain stretch and the Langevin function.
The "Gent" material model is based on the energy density function , where is the first strain invariant, the shear modulus and the limiting value.
The "NeoHookean" material model is based on the energy density function , where is the first Lamé constant, the second constant, the right Cauchy–Green tensor and the deformation gradient.
The compressible "StVenantKirchhoff" material model is based on the energy density function , where is the first Lamé constant, the second constant and the Green–Lagrange strain.
The "MooneyRivlin" material model is based on the energy density function , where are material coefficients and and are the first and second isochoric strain invariants.
The "Yeoh" material model is based on the energy density function , where the are model constants, is the first invariant.
When available, the "Compressibility" of a model can be specified as either "NearlyIncompressibile" or "Compressibile".
Near incompressibility is implemented as a hydrostatic pressure added to the strain energy density function , where is the material's bulk modulus.
Plane strain and plane stress model forms are available for all hyperelastic models.
All hyperelastic material models can make use of a standard reinforcing material model, to model transverse isotropic material, like fiber reinforced materials.
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 …,keyp_i…,p_iv_i,…], the parameters are replaced with .

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Define a solid mechanics PDE model:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-2r8h1y

Define a symbolic solid mechanics PDE model:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-q7jiop

Define a symbolic time-dependent solid mechanics PDE model:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-pwme5h

Scope (22)Survey of the scope of standard use cases

Basic Examples (3)

Activate a solid mechanics PDE model:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-7fjeus

Out[1]=1

Define a stationary solid mechanics model with a particular material:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-spnkan

Define a model with material values specified:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-dlnlar

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:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-me42fu

Set up the PDE and the geometry:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-52jbt6

Visualize the displacement:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-46lc3

Out[5]=5

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

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-uj7ron

Stationary Plane Stress Analysis (4)

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:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-fhf1c5

Solve the equations:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-dom0eo

Out[4]=4

Visualize the displacement:

In[6]:=6

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-xua0ft

Out[6]=6

Compute a plane stress case as an extended model. This allows for the computation of the out-of-plane strain and verifies that the out-of-plane stress is 0. A rectangular steel plate is held fixed at the left and with a forced displacement on the right. Set up the region, variables and parameters. The variables now include all three directions:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-n3hln

Solve the equations with thee semi-dependent variables:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-9scgt4

Out[2]=2

Note that now there are three output variables in the list of displacements. Visualize the displacement for the main variables:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-jbpbkr

Out[3]=3

Compute the strain:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-72dans

Out[4]=4

Note, that the strain is a 3 by 3 array. Visualize the out-of-plane strain:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-cuaa5q

Out[5]=5

Compute the stress from the strain:

In[6]:=6

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-uxc2y4

Out[6]=6

Note, that the stress is a 3 by 3 array. Verify the plane stress condition:

In[7]:=7

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-45ogn

Out[7]=7

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:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-sgb7g1

Solve the equations:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-wi1a45

Out[2]=2

Visualize the displacement:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-yomnx8

Out[3]=3

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:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-e873ml

Solve the equations:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-q5ou39

Out[2]=2

Visualize the displacement:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-3w5wsg

Out[3]=3

Compute the strain:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-gkcijv

Out[4]=4

Verify that the normal strain in the direction is about 0:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-6s310l

Out[5]=5

Visualize the normal strain in the direction:

In[6]:=6

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-v71wd0

Out[6]=6

Verify that the normal strain in the direction is about 0:

In[7]:=7

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-ew8tkg

Out[7]=7

Visualize the normal strain in the direction:

In[8]:=8

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-cwsxx4

Out[8]=8

Find the shear strain:

In[9]:=9

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-df4jm8

Out[9]=9

Visualize the normal strain in the - direction:

In[10]:=10

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-5p4ku2

Out[10]=10

Stationary Plane Strain Analysis (3)

Define a 2D plane strain PDE model:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-c1va06

Out[1]=1

Define an extended plane strain PDE model:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-4xv79c

Out[1]=1

Compute a plane strain case as an extended model. This allows for the computation of the out-of-plane stress and verifies that the out-of-plane strain is 0. Set up the region, variables and parameters. The variables now include all three directions:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-odgagf

Set up the solid mechanics PDE component:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-lmscmx

Out[2]=2

Set up the PDEs:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-rz669l

Solve the PDE:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-molvgw

Out[4]=4

Compute the strains from the displacements:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-8pbvhd

Out[5]=5

Note that the strain is a 3×3 array. Verify the plane strain condition:

In[6]:=6

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-027b51

Out[6]=6

Compute the stress from the strain:

In[7]:=7

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-il12ny

Out[7]=7

Note that the stress is a 3×3 array. Visualize the out-of-plane stress:

In[8]:=8

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-181jjq

Out[8]=8

Time-Dependent Analysis (3)

Define a time-dependent solid mechanics model for a particular material:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-450lar

Simulate a time-dependent force on a beam. Set up a region, variables and a material:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-t8lqqt

Set up the PDE model:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-i8xdo

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

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-t20coc

Fix the beam on the left:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-nluht4

Set up zero initial conditions and the initial velocity condition:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-wv5dwt

Solve the time-dependent PDE and monitor the progress:

In[6]:=6

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-n9v1yy

Visualize the -displacement over time at a query point:

In[7]:=7

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-gfxhhz

Out[7]=7

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:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-cbcojn

Set up the PDE and solve while monitoring progress:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-kif3lg

Out[2]=2

Visualize the -displacement over time at a query point:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-jhjba8

Out[3]=3

Visualize the displacement:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-7klnxf

Out[4]=4

Eigenmode Analysis (2)

Define an eigenmode solid mechanics model for a particular material:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-5d0pcu

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

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-0atj1m

Compute the eigenvalue and modes:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-mrapau

Visualize the seventh to tenth eigenmode:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-gg9a67

Out[3]=3

Hyperelastic Material Model (2)

Set up the variables and the model parameters for a piece of rubber that is held fixed at the left and a pressure is applied on the right. A neo-Hookean hyperelastic material model is used:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-vp861

Set up the geometry:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-y66v6d

Set up the equation and solve for the displacement:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-vi720k

Visualize the deformed body over the original shape of the body:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-otxfds

Out[5]=5

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

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-dug6s7

Create the PDE model with boundary conditions:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-1sk6i6

Show the actual displacement:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-4qdjh2

Out[3]=3

Compute the strain:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-krjyx4

Out[4]=4

Compute the stress:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-9p7agd

Out[5]=5

Fiber Reinforced Material Model (1)

Set up the variables and the model parameters for a fiber-reinforced piece of rubber that is held fixed at the left and a pressure is applied on the right. A neo-Hookean hyperelastic material model is used:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-8dga3y

Set up the geometry:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-noeesy

Visualize the fiber reinforcement in the geometry:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-1wesjg

Out[4]=4

Set up the equation and solve for the displacement:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-0n0zjr

Visualize the deformed body over the original shape of the body:

In[6]:=6

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-gghoup

Out[6]=6

Compute the strain:

In[7]:=7

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-o1c4ro

Out[7]=7

Compute the stress:

In[8]:=8

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-2v03fr

Out[8]=8

Compute the von Mises stress:

In[9]:=9

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-qee0re

Out[9]=9

Visualize the von Mises stress:

In[10]:=10

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-nn66n8

Out[10]=10

Multi-material Models (1)

A compliance matrix needs to be specified as a matrix. This is also true for multi-material models. This example shows how to do that. Specify two compliance matrices:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-iz8a2q

Create a multi-material compliance matrix, where material 1 is to be used for values in the geometry where , and material 2 is used in all other cases:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-3way18

Note that the compliance matrix now is a matrix:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-o74jv7

Out[4]=4

Look at a specific entry:

In[5]:=5

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-he5i5f

Out[5]=5

Set up the parameters to make use of the multi-material compliance matrix:

In[6]:=6

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-jx3jw1

Create the solid mechanics PDE component:

In[7]:=7

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-nq1dcb

Out[7]=7

Non-axis-aligned Material Model (1)

For anisotropic elastic material that is not axis aligned, an orientation matrix can be specified to adjust the material to that. This also works with thermal expansion and is shown here. Variables and parameter are set up:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-jr99lw

Solve the PDE:

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-gzyc9h

Visualize the displacement:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-em28q4

Out[3]=3

Applications (1)Sample problems that can be solved with this function

Geotechnical (1)

When modeling soil in geotechnical applications, the Young modulus can change with the depth of the soil. This example explores a position-dependent Young's modulus. We use a rectangular slab of soild that is 100 meters wide and 100 meters deep:

In[1]:=1

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-kyte7q

Out[1]=1

Next, we set up variables and parameters. At this point, we have a symbolic Young modulus :

In[2]:=2

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-mam35q

Set up the PDE:

In[3]:=3

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-1n8nmk

We have a force in the negative direction acting on part of the top:

In[4]:=4

✖

https://wolfram.com/xid/0cezk7pym0vk09ov8loxe-vb57uw

On the left- and right-hand side we constrain the movement in the direction; in the direction, the soil is able to move freely. At the bottom, the soil can move in the direction but not in the direction. This models a scenario where the soil is "standing" on a harder ground that does not move: