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Continuum Mechanics 4: Strains from Deformation Tensors

Introduction

This article is part 4 in my series on continuum mechanics. The focus is on how to define and calculate different reference and current configuration strains based on the deformation gradient.

Intro: How to Calculate the Polar Decomposition

As I discussed earlier, the deformation gradient can be divided into stretch and rotation tensors: \( \mathbf{F}=\mathbf{R}\mathbf{U} = \mathbf{v}\mathbf{R}. \) Often the deformation gradient is known, and the goal is to calculate the stretch and rotation tensors R, U, v. The right Cauchy Green stretch tensor can be written \(\mathbf{U}^2=\mathbf{F}^\top\mathbf{F}\), from which we get \(\mathbf{U}=\sqrt{\mathbf{F}^\top\mathbf{F}}.\) To solve this we write: $$\mathbf{U}^2=\sum_{i=1}^3 \lambda_i^2 \hat{\mathbf{N}}_i \otimes \hat{\mathbf{N}}_i.$$To calculate the square root we can simply operate with that function on the principal values giving:$$\mathbf{U}=\sum_{i=1}^3 \lambda_i \hat{\mathbf{N}}_i \otimes \hat{\mathbf{N}}_i$$From which we can also calculate:$$\mathbf{R}=\mathbf{F}\mathbf{U}^{-1}$$$$\mathbf{v}=\mathbf{F}\mathbf{R}^{\top}$$That’s how the Polar Decomposition can be calculated!

Strain

1️⃣ The strain cannot depend on rotations, so it cannot depend on the deformation gradient F.
2️⃣ The strain can only depend on the deformation tensors U and v.

Reference Configuration Strains

The most general equation for the reference strain tensor is \(\mathbf{E}=\hat{\mathbf{E}}(\mathbf{U})\). To make this tensor equation easier to understand I will write it in the following spectral form: $$\mathbf{E}=\sum_{i=1}^3 f(\lambda_i) \hat{\mathbf{N}}_i \otimes \hat{\mathbf{N}}_i.$$The function f(.) is arbitrary, and different choices for f(.) will give different strain measures. For the calculated value to be a reasonable strain, the scalar function f(.) has to satisfy the following 3 rules: $$f(1)=0$$$$f'(1)=1$$$$f(\lambda) \text{ has to monotonically increase}$$

Green-Lagrange Strain
One common reference configuration strain is the Green-Lagrange strain.This strain is based on the following equation for f(.):$$f(\lambda_i) = \frac{1}{2} (\lambda_i^2-1)$$Giving the following strain equation:$$\mathbf{E} = \frac{1}{2} \left[ \mathbf{U}^2 – \mathbf{I}\right]$$
Biot Strain
Another common reference strain is the Biot strain. This strain is based on the following equation for f(.):$$f(\lambda_i) = \lambda_i-1$$Giving the following strain equation:$$\mathbf{E} = \mathbf{U} – \mathbf{I}$$

Current Configuration Strains

For solid mechanics FE simulations the strain is typically expressed in the current configuration. In this case the strain tensor in a most general form can be written:$$\mathbf{e}=\hat{\mathbf{e}}(\mathbf{v}).$$This equation is better expressed in its spectral representation:$$\mathbf{e}=\sum_{i=1}^3 f(\lambda_i) \hat{\mathbf{n}}_i \otimes \hat{\mathbf{n}}_i.$$The function f(.) has to satisfy the same 3 rules as for the reference configuration strains:$$f(1)=0$$$$f'(1)=1$$$$f(\lambda) \text{ has to monotonically increase}$$The first rule ensures that the strain is zero if no deformation is applied, the second rule makes the strain measure become equal to the traditional at small strains, and the third rule make sure the strain increases when the deformation increases.

Nominal Strain
The nominal strain is the same as the Engineering Strain, and is commonly used in FE simulations. The strain is based on the following equation for f(.):$$f(\lambda_i) = \lambda_i-1$$Giving the following strain equation:$$\mathbf{e} = \mathbf{v} – \mathbf{I}$$
True Strain
The True strain is also called Logarithmic strain, or Cauchy strain. This strain is based on the following equation for the f(.):$$f(\lambda_i) = \ln(\lambda_i)$$Giving the following strain equation:$$\mathbf{e} = \ln[\mathbf{v}]$$This is my favorite strain measure. Which one is yours?
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