Solid mechanics

Strong formulation

The strong form of deformable solids is given by the Cauchy momentum equation

$$\begin{equation} \nabla \cdot \boldsymbol{\sigma} + \boldsymbol{f} = \rho \boldsymbol{\ddot{u}}, \end{equation}$$

where

• $\boldsymbol{\sigma}$ is the stress tensor
• $\boldsymbol{f}$ is the volumetric body force vector
• $\rho$ is the mass density
• $\boldsymbol{u}$ is the displacement field vector
• $\boldsymbol{\ddot{u}}$ is the acceleration vector

Constitutive equation

Generalized Hooke’s law defines the relation between the stress and strain components

$$\boldsymbol{\sigma} = \boldsymbol{C} \boldsymbol{\varepsilon},$$

where $\boldsymbol{C}$ is a fourth-order elasticity tensor and $\boldsymbol{\varepsilon}$ is the strain tensor.

Strain tensor

• Small displacement theory: $\qquad$

$$\boldsymbol{\varepsilon} = \dfrac{1}{2} \left( \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T \right)$$

• Large displacement theory: $\qquad$

$$\boldsymbol{\varepsilon} = \dfrac{1}{2} \left( \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T + (\nabla \boldsymbol{u})^T \nabla \boldsymbol{u} \right)$$

Voigt notation

In Voigt notation, the stress and strain tensors are represented as vectors. Consequently, the constitutive relation takes the form

$$\boldsymbol{\sigma}_v = \boldsymbol{H} \boldsymbol{\varepsilon}_v,$$

where $\boldsymbol{H}$ is the elasticity matrix which is a symmetric $6 \times 6$ matrix.

Weak formulation

The weak form of solid mechanics is obtained by multiplying the partial differential equation in $(1)$ with the test function of displacement field $\boldsymbol{u}^\prime$ and integrating over the whole domain $\Omega$.

$$\int_{\Omega}\; -\rho \boldsymbol{\ddot{u}} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; (\nabla \cdot \boldsymbol{\sigma}) \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; \boldsymbol{f} \cdot \boldsymbol{u}^\prime \; d\Omega = 0$$

Applying Leibniz rule on the divergence term, we get

$$\int_{\Omega}\; -\rho \boldsymbol{\ddot{u}} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; -\boldsymbol{\sigma} \colon \nabla \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; \nabla \cdot (\boldsymbol{\sigma} \cdot \boldsymbol{u}^\prime) \; d\Omega + \int_{\Omega}\; \boldsymbol{f} \cdot \boldsymbol{u}^\prime \; d\Omega = 0.$$

Applying Divergence theorem on the divergence term, we get

$$\int_{\Omega}\; -\rho \boldsymbol{\ddot{u}} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; -\boldsymbol{\sigma} \colon \nabla \boldsymbol{u}^\prime \; d\Omega + \int_{\Gamma}\; \boldsymbol{u}^\prime \cdot (\boldsymbol{\sigma} \cdot \boldsymbol{n}) \; d\Gamma + \int_{\Omega}\; \boldsymbol{f} \cdot \boldsymbol{u}^\prime \; d\Omega = 0.$$

Since the stress tensor is symmetric, the following relation holds.

$$\begin{align*} \boldsymbol{\sigma} \colon \nabla \boldsymbol{u}^\prime &= \boldsymbol{\sigma} \colon (\nabla {\boldsymbol{u}^\prime})^T \\ &= \boldsymbol{\sigma} \colon \dfrac{1}{2} \left((\nabla {\boldsymbol{u}^\prime})^T + (\nabla {\boldsymbol{u}^\prime})^T \right) \\ &= \boldsymbol{\sigma} \colon \dfrac{1}{2} \left(\nabla {\boldsymbol{u}^\prime} + (\nabla {\boldsymbol{u}^\prime})^T \right) \\ &= \boldsymbol{\sigma} \colon \boldsymbol{\varepsilon}^\prime \end{align*}$$

Substituting in the above relation $\boldsymbol{\sigma} \colon \nabla \boldsymbol{u}^\prime = \boldsymbol{\sigma} \colon \boldsymbol{\varepsilon}^\prime$ and following the definition of traction vector $\boldsymbol{t} = \boldsymbol{\sigma \cdot n}$, we get the final form of our weak formulation:

$$\int_{\Omega}\; -\rho \boldsymbol{\ddot{u}} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Omega}\; -\boldsymbol{\sigma} \colon \boldsymbol{\varepsilon}^\prime \; d\Omega + \int_{\Omega}\; \boldsymbol{f} \cdot \boldsymbol{u}^\prime \; d\Omega + \int_{\Gamma}\; \boldsymbol{t} \cdot \boldsymbol{u}^\prime \; d\Gamma = 0$$

Available Interactions

Load

Applies an external mechanical load to a selected region by prescribing a force vector $\boldsymbol{f}$, resulting in deformation and stress.

Constraint

Fixes all components of the displacement field $\boldsymbol{u}$. Use this interaction to fix displacements at a node, boundary, or within a region of the solid. Use Clamp interaction to fix displacement values to zero.

How to use:

You can use either the expression editor or the matrix editor.

With the expression editor, you must provide a 3×2 matrix for a 3D problem (2×2 for 2D, etc.):

The first column contains 0 or 1, indicating whether a displacement constraint is applied in that dimension.

The second column contains the corresponding displacement values.

Example:

[0, 0; 1, 42; 0, 0]

In the matrix editor, you can enable a dimension (e.g., y) and disable the others (x and z) using checkboxes. This is equivalent to entering 0 or 1in the first column in expression style. Then, you can enter the displacement value 42 for the enabled row corresponding to the y-dimension.

Both styles are equivalent and apply a 3D displacement constraint of $u_y = 42$ in the y-direction, while the other dimensions remain unconstrained.

Unit: Displacement values in meters (m)

Clamp

Clamp is a special case of a Constraint where the displacement is fixed to zero. This boundary condition will constraint all components of the displacement vector $\boldsymbol{u}$ in the targeted region to zero displacement: $u_x = 0, u_y = 0, u_z = 0$

Lump U/F

Applies a lumped displacement $\boldsymbol{U}$ or force $\boldsymbol{F}$ through a boundary of some surface, replacing detailed field distribution with an equivalent lumped value.

How to use:

Specify the target curve. The curve needs to be connected loop. From Actuation mode, select either displacement, force or circuit coupling.

Example:

You can use either the matrix editor or the expression editor to define lumped displacement or force values.

In the matrix editor, you can enable a dimension (e.g., y) and disable the others (x and z) using checkboxes. This is equivalent to entering 0 or 1in the first column in expression style. Then, you can enter the displacement value 0.01 for the enabled row corresponding to the y-dimension.

With the expression editor, you must provide a 3×2 matrix for a 3D problem (2×2 for 2D, etc.):

The first column contains 0 or 1, indicating whether a constraint is applied in that dimension.

The second column contains the corresponding values.

[0, 0; 1, 0.01; 0, 0]

Both styles are equivalent and apply a 3D lumped displacement constraint of $u_y = 0.01$ in the y-direction, while the other dimensions remain unconstrained.

Unit: Displacement values in meters (m) or force in Newtons (N)

Pressure

Prestress

Periodicity

Imposes periodic boundary conditions for the vector field $\boldsymbol{u}$. Reduces computational domain size for symmetric problems.

Example:

Periodicity has same principles, despite to which physics module it belongs to. See how periodicity is used in magnetism as a reference.

Periodicity in electric motors

Symmetry

Proportional damping

Geometric nonlinearity

Contact

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Couplings to Other Physics

This formulation supports the following couplings:

Thermal expansion (Heat solid or Heat fluid)

Piezoelectricity (Electrostatics)

Electric force (Current flow, Electromagnetic waves, Electrostatics or Magnetism H)

Magnetic force (Magnetism A, Magnetism 𝜑 or Magnetism H)

Large displacement (Mesh deformation)

Use cases

• Solid mechanics: Cantilever beam
• Solid mechanics: Backbone curve of a clamped-clamped beam
• Solid mechanics: Combdrive eigenmodes
• Solid mechanics: Combdrive static deflection
• Solid mechanics - Electrostatics coupling in microspeaker