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15. Need for stress-strain relation

The relationship between stress and strain is crucial for understanding material behavior under external loads. This chapter introduces the stress-strain relation and focuses on formulating the linear stress-strain relationship and its implications in solid mechanics. The importance of additional equations, known as constitutive relations, is emphasized to solve equilibrium equations for deformed bodies.

Sections

Need for stress-strain relation

The section discusses the necessity of establishing a stress-strain relationship to analyze the behavior of a deformed body under external loads.

1 Section Overview

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Linear Stress-Strain Relation

This section discusses the linear relationship between stress and strain in solid mechanics, emphasizing its necessity for solving equilibrium equations.

2 Section Overview

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2.1 Taylor’s expansion

This section explores Taylor's expansion as a method to relate stress to strain in solid mechanics.

2.2 Independent components in tensor C

This section discusses the independent components of the stiffness tensor C and the symmetries associated with stress and strain tensors.

2.2.1 Minor Symmetry

This section discusses minor symmetry within the stiffness tensor in the context of solid mechanics, highlighting its implications for the independence of stress and strain components.

2.2.2 Major Symmetry

This section discusses Major Symmetry in the stiffness tensor, explaining its implications for material constants and energy considerations.

Voigt Notation

Voigt Notation simplifies the representation of stress and strain tensors in solid mechanics by reducing them to six independent components.

3 Section Overview

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Learning Objectives

  • The stress-strain relation is fundamental for predicting material deformation.

  • Lagrangian description is utilized to relate displacement, velocity, and acceleration.

  • The stiffness tensor has major and minor symmetries, reducing the number of independent material constants.

Key Concepts

Stress-Strain Relation

A mathematical formulation that describes the relationship between stress (internal forces) and strain (deformation) in materials.

Lagrangian Description

A method of describing motion where quantities are expressed in terms of the reference configuration of particles.

Stiffness Tensor

A fourth-order tensor that relates stress and strain using a linear approximation, incorporating material properties.

Linear Stress-Strain Relation

An approximation where stress is directly proportional to strain, valid for small deformations.

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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