Stress and strain

Stress and strain#

Strain measures#

Deformation gradient tensor#

The deformation gradient tensor, \(\mathbf{F}\), is a crucial concept in finite deformation (large deformation) continuum mechanics, as it forms the basis for deriving all measures of deformation. It provides a linear mapping between an initial configuration, \(d\mathbf{X}\), to an deformed configuration, \(d\mathbf{x}\), without assuming that displacements or rotations are small. It is expressed as:

\[ \mathbf{F} := \frac{\partial \boldsymbol{\phi}}{\partial \mathbf{X}} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}}, \quad \text{or in component form,} \quad F_{ij} = \frac{\partial x_i}{\partial X_j}. \]

Velocity gradient tensor#

The velocity gradient tensor, \(\mathbf{L}\), quantifies the spatial rate of change of the velocity field, \(\mathbf{v}\), within the material. It captures the local rate of stretching, shearing, and rotation. \(\mathbf{L}\) is defined as the gradient of the velocity vector, and its components are given by:

\[ \dot{\mathbf{F}} = \frac{\partial}{\partial t}\left(\frac{\partial \phi(\mathbf{X},t)}{\partial \mathbf{X}}\right) = \frac{\partial \mathbf{v}}{\partial \mathbf{X}} = \frac{\partial \mathbf{v}}{\partial \mathbf{x}} \cdot \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \mathbf{L} \cdot \mathbf{F}, \quad \Rightarrow \mathbf{L} = \dot{\mathbf{F}} \cdot \mathbf{F}^{-1} \]

\[\begin{split} \bf{L}=\nabla v =\left[ \begin{matrix} \frac{\partial v_x}{\partial x} & \frac{\partial v_x}{\partial y} & \frac{\partial v_x}{\partial z} \\\\ \frac{\partial v_y}{\partial x} & \frac{\partial v_y}{\partial y} & \frac{\partial v_y}{\partial z} \\\\ \frac{\partial v_z}{\partial x} & \frac{\partial v_z}{\partial y} & \frac{\partial v_z}{\partial z} \end{matrix} \right] \end{split}\]

Numerical integration of deformation gradient#

To numerically compute \( \mathbf{F} \) over a discrete time step \( \Delta t \), we use the forward Euler approximation for \( \dot{\mathbf{F}} \):

\[ \mathbf{F}_p^{t+\Delta t} = \mathbf{F}_p^t + \dot{\mathbf{F}} \cdot \Delta t \]

Substituting \( \dot{\mathbf{F}} = \mathbf{L}_p \cdot \mathbf{F}_p^t \):

\[ \mathbf{F}_p^{t+\Delta t} = \mathbf{F}_p^t + \left( \mathbf{L}_p \cdot \mathbf{F}_p^t \right) \cdot \Delta t \]

Factoring \( \mathbf{F}_p^t \):

\[ \mathbf{F}_p^{t+\Delta t} = \mathbf{F}_p^t \cdot \left( \mathbf{I} + \mathbf{L}_p \cdot \Delta t \right) \]

Rate of deformation tensor (strain rate tensor)#

The rate of deformation tensor, \(\mathbf{D}\), also known as the strain rate tensor, represents the symmetric part of the velocity gradient tensor, \(\mathbf{L}\). It describes the rate of change of strain in the material (rate of stretching and shearing). Mathematically, \(\mathbf{D}\) is given by:

\[\begin{split} \mathbf{D} = \frac{1}{2}(\mathbf{L} + \mathbf{L}^T) = \left[ \begin{matrix} \frac{\partial v_x}{\partial x} & \frac{1}{2}(\frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x}) & \frac{1}{2}(\frac{\partial v_x}{\partial z} + \frac{\partial v_z}{\partial x}) \\ \frac{1}{2}(\frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x}) & \frac{\partial v_y}{\partial y} & \frac{1}{2}(\frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y}) \\ \frac{1}{2}(\frac{\partial v_x}{\partial z} + \frac{\partial v_z}{\partial x}) & \frac{1}{2}(\frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y}) & \frac{\partial v_z}{\partial z} \end{matrix} \right] \end{split}\]

As a 6x1 the rate of deformation tensor \(\mathbf{D} = \left[\frac{dv_x}{dx}, \frac{dv_y}{dy}, \frac{dv_z}{dz}, \frac{1}{2}(\frac{dv_x}{dy} + \frac{dv_y}{dx}), \frac{1}{2}(\frac{dv_y}{dz} + \frac{dv_z}{dy}), \frac{1}{2}(\frac{dv_x}{dz} + \frac{dv_z}{dx})\right]^T\)