Navier partial differential equations describe the displacement field as a function of body forces and structural properties of the material. Knowing the displacement field, you can calculate the strain and stress fields:
Here, vector u is the displacement, ρ is the mass density, μ is the shear modulus, λ is the Lame modulus of the material, and f is a vector of volume forces. The shear modulus and Lame modulus can be expressed via the Young's (elastic) modulus E and the Poisson's ratio ν:
A typical programmatic workflow for solving a linear elasticity problem includes these steps:
Create a special structural analysis container for a solid (3-D), plane stress, or plane strain model.
Define 2-D or 3-D geometry and mesh it.
Assign structural properties of the material, such as Young's modulus, Poisson's ratio, and mass density.
Specify a damping model and its values for a dynamic problem.
Specify gravitational acceleration as a body load.
Specify boundary loads and constraints.
Specify initial displacement and velocity for a dynamic problem.
Solve the problem and plot results, such as displacement, velocity, acceleration, stress, strain, von Mises stress, principal stress and strain.
For modal analysis problems, use the same steps for creating a model and specifying materials and boundary constraints. In this case, the solver finds natural frequencies and mode shapes of a structure.
For plane stress and plane strain problems, you also can use the PDE Modeler app. The app includes geometry creation and preset modes for applications.
|Assign structural properties of material for structural model|
|Specify damping parameters for transient structural model|
|Specify body load for structural model|
|Specify boundary loads for structural model|
|Specify boundary conditions for structural model|
|Set initial conditions for a transient structural model|
|Solve heat transfer or structural analysis problem|
|Assemble finite element matrices|
|Evaluate stress for dynamic structural analysis problem|
|Evaluate strain for dynamic structural analysis problem|
|Evaluate von Mises stress for dynamic structural analysis problem|
|Evaluate reaction forces on boundary|
|Evaluate principal stress at nodal locations|
|Evaluate principal strain at nodal locations|
|Interpolate displacement at arbitrary spatial locations|
|Interpolate velocity at arbitrary spatial locations for all time steps for transient structural model|
|Interpolate acceleration at arbitrary spatial locations for all time steps for transient structural model|
|Interpolate stress at arbitrary spatial locations|
|Interpolate strain at arbitrary spatial locations|
|Interpolate von Mises stress at arbitrary spatial locations|
|Find structural material properties assigned to geometric region|
|Find damping model assigned to structural dynamics model|
|Find structural boundary conditions and boundary loads assigned to geometric region|
|Find initial displacement and velocity assigned to geometric region|
|Find body load assigned to geometric region|
|StructuralMaterialAssignment Properties||Structural material property assignments|
|StructuralDampingAssignment Properties||Damping assignment for a structural analysis model|
|BodyLoadAssignment Properties||Body load assignments|
|StructuralBC Properties||Boundary condition or boundary load for structural analysis model|
|GeometricStructuralICs Properties||Initial displacement and velocity over a region|
|NodalStructuralICs Properties||Initial displacement and velocity at mesh nodes|
|PDE Modeler||Solve partial differential equations in 2-D regions|
Analyze a 3-D mechanical part under an applied load and determine the maximal deflection.
Perform a 2-D plane-stress elasticity analysis.
Perform modal and transient analysis of a tuning fork.
Use modal analysis results to compute the transient response of a thin 3-D plate under a harmonic load at the center.
Solve a coupled thermo-elasticity problem.
Solve a coupled elasticity-electrostatics problem.
Calculate the deflection of a structural plate acted on by a pressure loading.
Include damping in the transient analysis of a simple cantilever beam.
Analyze the dynamic behavior of a beam clamped at both ends and loaded with a uniform pressure load.
Calculate the vibration modes and frequencies of a 3-D simply supported, square, elastic plate.
Perform coupled electro-mechanical finite element analysis of an electrostatically actuated micro-electro-mechanical (MEMS) device.
Use the PDE Modeler app to compute the von Mises effective stress and displacements for a steel plate clamped along an inset at one corner and pulled along a rounded cut at the opposite corner.
Linear elasticity equations for plane stress, plane strain, and 3-D problems.