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13. Local Volumetric Strain

The discussion centers on local volumetric strain and the local rotation tensor as essential concepts in continuum mechanics. It explains how volumetric strain is defined through the change in volume of small elements under deformation, and how displacement gradients relate to the strain tensor and local rotation. The chapter highlights the independence of volumetric strain from the choice of line elements and describes the physical significance of infinitesimal rotations in deformable bodies.

Sections

Local Volumetric Strain

This section introduces the concept of local volumetric strain as a measure of volume change in deformable bodies.

1 Section Overview

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1.1 Formula for volumetric strain

This section discusses the concept of local volumetric strain and introduces the formula for calculating it during deformation.

Strain Tensor

This section focuses on local volumetric strain and the strain tensor, essential concepts in understanding deformation in solid mechanics.

2 Section Overview

Start current section content and materials

Local Rotation Tensor

This section discusses the local rotation tensor and its relation to deformation in solid mechanics, particularly focusing on the mapping of reference configurations to deformed states.

3 Section Overview

Start current section content and materials

3.1 Rodrigues’ Rotation Formula

Rodrigues' Rotation Formula provides a mathematical representation for rotations in three-dimensional space, applicable for small and large angles.

3.1.1 An example for rotation

This section discusses local rotation in a body undergoing deformation, specifically focusing on the rotation matrix and its application in analyzing small rotations.

3.2 Extracting the axis and angle of local rotation

This section discusses the concepts of local rotation through the axial vector and the angle of rotation derived from the skew-symmetric part of the displacement gradient tensor.

3.2.1 An example

This section illustrates the concept of local rotation and strain in a coordinate system through a practical example.

Learning Objectives

  • Local volumetric strain quantifies the change in volume per unit volume in a deforming body.

  • The volumetric strain is defined in terms of the original and deformed volumes of small elements.

  • The strain tensor encompasses both longitudinal and shear strains, and its components can be derived from displacement gradients.

Key Concepts

Local Volumetric Strain

A measure of the change in volume per unit volume of a small region within a deformed body.

Strain Tensor

A mathematical representation that includes both the symmetric (strain) and anti-symmetric (rotation) parts of the displacement gradient tensor.

Local Rotation Tensor

Describes the infinitesimal local rotations occurring within the material as it deforms.

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting
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