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Solve static linear elasticity problems for 3-D solids, as well as 2-D
plane stress and plane strain problems

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.

$$\left(\lambda +\mu \right)\nabla \left(\nabla \cdot u\right)+\mu {\nabla}^{2}u+f=0$$

Here, vector **u** is the displacement,
*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*.

$$\mu =\frac{E}{2\left(1+\nu \right)},\text{\hspace{1em}}\lambda =\frac{\nu E}{\left(1+\nu \right)\left(1-2\nu \right)},\text{\hspace{1em}}f=\left(\begin{array}{l}{f}_{x}\\ {f}_{y}\end{array}\right)$$

A typical programmatic workflow for solving a linear elasticity problem includes the following 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 body loads.

Specify boundary loads and constraints.

Solve the problem and plot results, such as displacement, stress, strain, von Mises stress, principal stress and strain.

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.

`createpde` | Create a model |

`structuralProperties` | Assign structural properties of a material for a structural model |

`structuralBodyLoad` | Specify body load for a structural model |

`structuralBoundaryLoad` | Specify boundary loads for a structural model |

`structuralBC` | Specify boundary conditions for a structural model |

`solve` | Solve heat transfer or structural analysis problem |

`interpolateDisplacement` | Interpolate displacement at arbitrary spatial locations |

`interpolateStress` | Interpolate stress at arbitrary spatial locations |

`interpolateStrain` | Interpolate strain at arbitrary spatial locations |

`interpolateVonMisesStress` | Interpolate von Mises stress at arbitrary spatial locations |

`evaluateReaction` | Evaluate reaction forces on boundary |

`evaluatePrincipalStress` | Evaluate principal stress at nodal locations |

`evaluatePrincipalStrain` | Evaluate principal strain at nodal locations |

`pdeplot` | Plot solution or mesh for 2-D geometry |

`pdeplot3D` | Plot solution or surface mesh for 3-D geometry |

`findStructuralProperties` | Find structural material properties assigned to geometric region |

`findStructuralBC` | Find structural boundary conditions and boundary loads assigned to geometric region |

`findBodyLoad` | Find body load assigned to geometric region |

`StructuralModel` | Structural model object |

`StaticStructuralResults` | Static structural solution and its derived quantities |

StructuralMaterialAssignment Properties | Structural material property assignments |

BodyLoadAssignment Properties | Body load assignments |

StructuralBC Properties | Boundary condition or boundary load for structural analysis model |

PDE Modeler | Solve partial differential equations in 2-D regions |

**Deflection Analysis of a Bracket**

Analyze a 3-D mechanical part under an applied load and determine the maximal deflection.

**Stress Concentration in Plate with Circular Hole**

Perform a 2-D plane-stress elasticity analysis.

**Deflection of a Piezoelectric Actuator**

Solve a coupled elasticity-electrostatics problem.

**Clamped, Square Isotropic Plate With a Uniform Pressure Load**

Calculate the deflection of a structural plate acted on by a pressure loading.

**Dynamics of a Damped Cantilever Beam**

Include damping in the transient analysis of a simple cantilever beam.

**Dynamic Analysis of a Clamped 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.

**von Mises Effective Stress and Displacements**

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.

Mathematical definition and discussion of the elliptic equation

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