10 Continuum Mechanics Fundamentals

10.1 Displacement and Configuration

In continuum mechanics, we deal with continuous media and their deformation under external forces.

Displacement $\mathbf{u}$: The change in position of a point in the body, defined as $\mathbf{u} = \mathbf{x} - \mathbf{x}_0$, where $\mathbf{x}_0$ is the original position, often written using capital $\mathbf{X}$.

Configuration:

$\mathbf{X}$: The original position of a point in the body before deformation, with $E_i$ being the coordinate system in the reference configuration: $\mathbf{X}=X_iE_i$
$\mathbf{x}$: The current position of the same point after deformation, with $e_i$ being the coordinate system in the current configuration: $\mathbf{x}=x_ie_i$

10.2 Deformation Gradient

The Deformation Gradient $\mathbf{F}$ is a tensor that describes the local deformation of the material. It relates the current configuration to the reference configuration:

\[\mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \begin{bmatrix} \frac{\partial x_1}{\partial X_1} & \frac{\partial x_1}{\partial X_2} & \ldots \\ \frac{\partial x_2}{\partial X_1} & \frac{\partial x_2}{\partial X_2} & \ldots \\ \vdots & \vdots & \ddots \end{bmatrix}\]

10.3 Strain

Strain $\boldsymbol{\varepsilon}$ is a measure of deformation defined as the symmetric part of the deformation gradient:

\[\boldsymbol{\varepsilon} = \frac{1}{2}(\mathbf{F}^T \mathbf{F} - \mathbf{I})\]

where $\mathbf{I}$ is the identity tensor.

Under the assumption of small deformations, the strain can be approximated as:

\[\boldsymbol{\varepsilon} \approx \frac{1}{2}(\nabla \mathbf{u} + \nabla \mathbf{u}^T) = \nabla^s\mathbf{u}\]

where $\nabla \mathbf{u}$ is the gradient of the displacement field and $\nabla^s\mathbf{u}$ is the symmetric gradient.

10.4 Stress and Constitutive Relation

Stress $\boldsymbol{\sigma}$ is a measure of internal forces within the material, defined using a constitutive relation that relates stress to strain. For linear elasticity:

\[\boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon}\]

where $\mathbf{C}$ is the elasticity tensor (material property matrix) and $\boldsymbol{\sigma}$ is the Cauchy stress tensor.

10.5 Tractions

Tractions $\mathbf{t}$ represent the force per unit area acting on a surface, related to stress by:

\[\mathbf{t} = \boldsymbol{\sigma} \cdot \mathbf{n}\]

where $\mathbf{n}$ is the outward normal to the surface.

10.6 Mass Conservation

The principle that mass is conserved in a closed system. In continuum mechanics, this is expressed as:

\[\int_{\Omega} \rho \, dv = \text{constant}= \int_{\Omega} \rho_0 \, dV\Rightarrow \rho J = \rho_0\]

where $\rho$ is the current density, $\rho_0$ is the reference density, and $J = \det(\mathbf{F})$ is the Jacobian determinant of the deformation gradient.

10.7 Balance of Linear Momentum

The balance of linear momentum states:

\[\int_{\Omega} \rho \frac{\partial^2 \mathbf{x}}{\partial t^2} \, dv = \int_{\Omega} \mathbf{f} \,dv +\int_{\partial \Omega} \mathbf{t} \, da\]

This leads to the equation of motion:

\[\text{div}(\boldsymbol{\sigma}) + \mathbf{f} = \rho \ddot{\mathbf{u}}\]

10.8 Balance of Angular Momentum

The balance of angular momentum is expressed as:

\[\frac{d}{dt}\int_{\Omega} \mathbf{x} \times (\rho v)\, dv = \int_{\Omega} \mathbf{x} \times \mathbf{f} \,dv +\int_{\partial \Omega} \mathbf{x} \times \mathbf{t} \, da\]

10.9 The Weak Form Derivation

For quasi-static problems, we assume the acceleration terms are zero, so the balance of linear momentum becomes:

\[\nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = 0\]

with

\[\nabla \cdot \boldsymbol{\sigma} = \left(\frac{\partial \sigma_{ij}}{\partial x_j}\right)\mathbf{e}_i\]

where $\sigma_{ij}$ are the components of the stress tensor.

10.9.1 Multiplication by Test Function and Integration

We multiply by a test function $\mathbf{v}$ and integrate over the domain:

\[\int_{\Omega} (\nabla \cdot \boldsymbol{\sigma} + \mathbf{f}) \cdot \mathbf{v} \, d\Omega = 0\]

10.9.2 Product Rule Application

We use the product rule for the divergence of a tensor contracted with a vector:

\[\nabla \cdot (\boldsymbol{\sigma}\cdot\mathbf{v}) = (\nabla \cdot \boldsymbol{\sigma})\cdot \mathbf{v} + \boldsymbol{\sigma}:\nabla(\mathbf{v})\]

where:

\[\boldsymbol{\sigma}:\nabla(\mathbf{v}) = \sigma_{ij}\frac{\partial v_j}{\partial x_i}\]

Rearranging:

\[(\nabla \cdot \boldsymbol{\sigma}) \cdot \mathbf{v} = \nabla \cdot (\boldsymbol{\sigma}\mathbf{v}) - \boldsymbol{\sigma} : \nabla\mathbf{v}\]

10.9.3 Divergence Theorem Application

Applying the divergence theorem to the integrated equation:

\[\int_{\Omega} \nabla \cdot (\boldsymbol{\sigma}\mathbf{v}) d\Omega - \int_{\Omega} \boldsymbol{\sigma} : \nabla\mathbf{v} d\Omega + \int_{\Omega} \mathbf{f} \cdot \mathbf{v} d\Omega = 0\]

\[\implies \int_{\partial\Omega} (\boldsymbol{\sigma}\mathbf{v}) \cdot \mathbf{n} d\Gamma - \int_{\Omega} \boldsymbol{\sigma} : \nabla\mathbf{v} d\Omega + \int_{\Omega} \mathbf{f} \cdot \mathbf{v} d\Omega = 0\]

The term ($\boldsymbol{\sigma}\boldsymbol{n}$) is the traction vector $\boldsymbol{t}$. The term $\nabla \boldsymbol{v}$ is the gradient of the test function, and its symmetric part is the strain tensor for the test function, $\boldsymbol{\varepsilon}(\boldsymbol{v})$.

10.9.4 Final Weak Form

This derivation yields the weak form of the momentum equation, which forms the foundation for finite element elasticity analysis.

--- title: "Continuum Mechanics Fundamentals" --- ## Displacement and Configuration In continuum mechanics, we deal with continuous media and their deformation under external forces. **Displacement** $\mathbf{u}$: The change in position of a point in the body, defined as $\mathbf{u} = \mathbf{x} - \mathbf{x}_0$, where $\mathbf{x}_0$ is the original position, often written using capital $\mathbf{X}$. **Configuration:** - $\mathbf{X}$: The original position of a point in the body before deformation, with $E_i$ being the coordinate system in the reference configuration: $\mathbf{X}=X_iE_i$ - $\mathbf{x}$: The current position of the same point after deformation, with $e_i$ being the coordinate system in the current configuration: $\mathbf{x}=x_ie_i$ ## Deformation Gradient The **Deformation Gradient** $\mathbf{F}$ is a tensor that describes the local deformation of the material. It relates the current configuration to the reference configuration: $$\mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \begin{bmatrix} \frac{\partial x_1}{\partial X_1} & \frac{\partial x_1}{\partial X_2} & \ldots \\ \frac{\partial x_2}{\partial X_1} & \frac{\partial x_2}{\partial X_2} & \ldots \\ \vdots & \vdots & \ddots \end{bmatrix}$$ ## Strain **Strain** $\boldsymbol{\varepsilon}$ is a measure of deformation defined as the symmetric part of the deformation gradient: $$\boldsymbol{\varepsilon} = \frac{1}{2}(\mathbf{F}^T \mathbf{F} - \mathbf{I})$$ where $\mathbf{I}$ is the identity tensor. Under the assumption of small deformations, the strain can be approximated as: $$\boldsymbol{\varepsilon} \approx \frac{1}{2}(\nabla \mathbf{u} + \nabla \mathbf{u}^T) = \nabla^s\mathbf{u}$$ where $\nabla \mathbf{u}$ is the gradient of the displacement field and $\nabla^s\mathbf{u}$ is the symmetric gradient. ## Stress and Constitutive Relation **Stress** $\boldsymbol{\sigma}$ is a measure of internal forces within the material, defined using a constitutive relation that relates stress to strain. For linear elasticity: $$\boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon}$$ where $\mathbf{C}$ is the elasticity tensor (material property matrix) and $\boldsymbol{\sigma}$ is the Cauchy stress tensor. ## Tractions **Tractions** $\mathbf{t}$ represent the force per unit area acting on a surface, related to stress by: $$\mathbf{t} = \boldsymbol{\sigma} \cdot \mathbf{n}$$ where $\mathbf{n}$ is the outward normal to the surface. ## Mass Conservation The principle that mass is conserved in a closed system. In continuum mechanics, this is expressed as: $$\int_{\Omega} \rho \, dv = \text{constant}= \int_{\Omega} \rho_0 \, dV\Rightarrow \rho J = \rho_0$$ where $\rho$ is the current density, $\rho_0$ is the reference density, and $J = \det(\mathbf{F})$ is the Jacobian determinant of the deformation gradient. ## Balance of Linear Momentum The balance of linear momentum states: $$\int_{\Omega} \rho \frac{\partial^2 \mathbf{x}}{\partial t^2} \, dv = \int_{\Omega} \mathbf{f} \,dv +\int_{\partial \Omega} \mathbf{t} \, da$$ This leads to the equation of motion: $$\text{div}(\boldsymbol{\sigma}) + \mathbf{f} = \rho \ddot{\mathbf{u}}$$ ## Balance of Angular Momentum The balance of angular momentum is expressed as: $$\frac{d}{dt}\int_{\Omega} \mathbf{x} \times (\rho v)\, dv = \int_{\Omega} \mathbf{x} \times \mathbf{f} \,dv +\int_{\partial \Omega} \mathbf{x} \times \mathbf{t} \, da$$ ## The Weak Form Derivation For quasi-static problems, we assume the acceleration terms are zero, so the balance of linear momentum becomes: $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = 0$$ with $$\nabla \cdot \boldsymbol{\sigma} = \left(\frac{\partial \sigma_{ij}}{\partial x_j}\right)\mathbf{e}_i$$ where $\sigma_{ij}$ are the components of the stress tensor. ### Multiplication by Test Function and Integration We multiply by a test function $\mathbf{v}$ and integrate over the domain: $$\int_{\Omega} (\nabla \cdot \boldsymbol{\sigma} + \mathbf{f}) \cdot \mathbf{v} \, d\Omega = 0$$ ### Product Rule Application We use the product rule for the divergence of a tensor contracted with a vector: $$\nabla \cdot (\boldsymbol{\sigma}\cdot\mathbf{v}) = (\nabla \cdot \boldsymbol{\sigma})\cdot \mathbf{v} + \boldsymbol{\sigma}:\nabla(\mathbf{v})$$ where: $$\boldsymbol{\sigma}:\nabla(\mathbf{v}) = \sigma_{ij}\frac{\partial v_j}{\partial x_i}$$ Rearranging: $$(\nabla \cdot \boldsymbol{\sigma}) \cdot \mathbf{v} = \nabla \cdot (\boldsymbol{\sigma}\mathbf{v}) - \boldsymbol{\sigma} : \nabla\mathbf{v}$$ ### Divergence Theorem Application Applying the divergence theorem to the integrated equation: $$\int_{\Omega} \nabla \cdot (\boldsymbol{\sigma}\mathbf{v}) d\Omega - \int_{\Omega} \boldsymbol{\sigma} : \nabla\mathbf{v} d\Omega + \int_{\Omega} \mathbf{f} \cdot \mathbf{v} d\Omega = 0$$ $$\implies \int_{\partial\Omega} (\boldsymbol{\sigma}\mathbf{v}) \cdot \mathbf{n} d\Gamma - \int_{\Omega} \boldsymbol{\sigma} : \nabla\mathbf{v} d\Omega + \int_{\Omega} \mathbf{f} \cdot \mathbf{v} d\Omega = 0$$ The term ($\boldsymbol{\sigma}\boldsymbol{n}$) is the traction vector $\boldsymbol{t}$. The term $\nabla \boldsymbol{v}$ is the gradient of the test function, and its symmetric part is the strain tensor for the test function, $\boldsymbol{\varepsilon}(\boldsymbol{v})$. ### Final Weak Form This derivation yields the weak form of the momentum equation, which forms the foundation for finite element elasticity analysis.