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Intro to Mechanics of Materials

Starting with a 2D solid material. Let \mathbf{u} be the displacement field of the material. Let \mathbf{\epsilon} be the normal (\epsilon) and shear (\gamma) strains. Let \mathbf{\sigma} be the normal (\sigma) and shear (\tau) stresses.

\mathbf{u} = \begin{pmatrix} u_x \\ u_y \end{pmatrix} \:\:\:\: displacement \mathbf{\epsilon} = \begin{pmatrix} \epsilon_x \\ \epsilon_y \\ \gamma_{xy} \end{pmatrix} = \begin{pmatrix} \frac{\partial u_x}{\partial x} \\ \frac{\partial u_y}{\partial y} \\ \frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x} \end{pmatrix} \:\:\:\: strain \mathbf{\sigma} = \begin{pmatrix} \sigma_x \\ \sigma_y \\ \tau_{xy} \end{pmatrix} \:\:\:\: stress

Let \mathbf{L} be the strain-displacement differential operator for converting \mathbf{u} to \mathbf{\epsilon}.

\mathbf{L(u)}= \begin{bmatrix} \frac{\partial}{\partial x} & 0 \\ 0 & \frac{\partial}{\partial y} \\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} \end{bmatrix} \begin{pmatrix} u_x \\ u_y \end{pmatrix} = \mathbf{\epsilon}

Let \mathbf{E} be the elasticity matrix to convert strain into stress, where E is Young’s modulus, G is the modulus of rigidity, and \nu is Poisson’s ratio. The matrix shown is for the plane stress condition; that is \sigma_z = 0. For convenience, we will apply the substitution E' = E/(1-\nu^2).

\mathbf{\sigma} = \mathbf{E\epsilon} = \begin{bmatrix} E' & E'\nu & 0 \\ E'\nu & E' & 0 \\ 0 & 0 & G \end{bmatrix} \begin{pmatrix} \epsilon_x \\ \epsilon_y \\ \gamma_{xy} \end{pmatrix}

Newton’s second law at any point in the material may be written as follows:

\mathbf{f} = \rho \frac{\partial^2 \mathbf{u}}{\partial t^2}

Where \mathbf{f} is the sum of forces per unit volume and \rho is the density. We are concerned with the stress in the material, but other forces can be added, as needed. Let \mathbf{f}_s be the net force due to stress.

σy τxy σx \mathbf{f}_s= \mathbf{L}^T(\mathbf{\sigma}) = \begin{bmatrix} \frac{\partial}{\partial x} & 0 & \frac{\partial}{\partial y} \\ 0 & \frac{\partial}{\partial y} & \frac{\partial}{\partial x} \end{bmatrix} \begin{pmatrix} \sigma_x \\ \sigma_y \\ \tau_{xy} \end{pmatrix}

Let \mathbf{f}_b be the sum of other forces per unit volume, such as body forces like gravity and magnetism, or reference-frame forces like centrifugal force and coriolis force.

\mathbf{f}_b + \mathbf{L}^T(\mathbf{\sigma}) = \rho \frac{\partial^2 \mathbf{u}}{\partial t^2}