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Why the Deformation Gradient is Important

Introduction

This article is part one of my series on continuum mechanics. The focus of this article is on the kinematics and the deformation gradient. In future parts of this series I will explain how the deformation gradient is used in polymer mechanics theory.

Note:

“The Deformation Gradient rules polymer mechanics theory.”

Large Strain Kinematics

Consider an object that is located at \(\Omega_0\) at time t=0. Specifically, consider a location specified by the vector \(\mathbf{X}\), relative to a fixed coordinate system. At some later time (t), that specific location has moved to \(\mathbf{x}(t)\). To study strains and stresses, it is necessary to introduce the deformation function: \(\mathbf{x} = \mathcal{X}(\mathbf{X},t)\). In this equation \(\mathbf{X}\) is the reference (material) location, and \(\mathbf{x}\) is the current (spatial) location.

Also note the following definitions: Lagrangian formulation = everything is referred back to the initial location; Eulerian formulation = everything is referred to the current location. Now, based on these definitions, we can write the deformation gradient as:

\[\mathbf{F} = \displaystyle\frac{\partial\mathcal{X}(X,t)}{\partial X}\]

\[ F_{ij} = \partial x_i / \partial X_j\]

Example: Simple Shear

In simple shear the deformation is defined by the following equations: \(x_1 = X_1 + \gamma X_2\), \(x_2 = X_2\), and \(x_3 = X_3\). From these equations we can directly obtain the following deformation gradient:

\[\mathbf{F} = \begin{bmatrix} 1 & \gamma & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}\]

Summary

  • The deformation gradient is a non-symmetrical 3×3 matrix.
  • The deformation gradient is super important for polymer mechanics and FE analysis.
  • Stresses and strains are often calculated from the deformation gradient.
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