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# Why the Deformation Gradient is Important

## Introduction

### Next Part

##### 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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