# Generalize Hooks Law

## Table of Contents

## Generalize Hooks Law (1678)

- Any combination of stresses can be utilized to anticipate the deformations that a given material would experience by using the generalized Hooke’s Law.
- Hooks law is applicable when deformation is small.

The linear relationship between stress and strain applies to 0< \sigma < \sigma_{yield}

Another finding of the expanded Hooke’s Law is that strain can exist in the absence of stress. For example, if the member is experiencing a load in the y-direction (which in turn causes stress in the y-direction), Hooke’s Law shows that strain in the x-direction does not equal to zero.

This is because as the material is being pulled outward by the y-plane, the material in the x-plane moves inward to fill in the space once occupied, just like pulling an elastic band apart causes it to grow thinner. In this situation, the x-plane does not have any external force acting on it but they experience a change in length. Therefore, it is valid to say that strain exists without stress in the x-plane.

- Anisotropic materials are defined as having varying characteristics in different directions.
- If there are axes of symmetry in 3 perpendicular directions, the material is called ORTHOTROPIC material.
- If an orthotropic material has two or three two-fold axes of rotational symmetry that are mutually orthogonal, each of the axis’ directions will generally affect the material’s mechanical properties. As a result, orthotropic materials are anisotropic, meaning that the direction of measurement affects their characteristics. In contrast, the qualities of an isotropic material remain constant in all directions.
- An example of an orthotropic material with three mutually perpendicular axes is wood, in which the properties (such as strength and stiffness) along its grain and in each of the two perpendicular directions are different. Hankinson’s formula offers a way to measure the variation in strength in several directions. Another example is a metal that has been rolled to form a sheet; the properties in the rolling direction and each of the two transverse directions will be different due to the anisotropic structure that develops during rolling.

Generalize Hooks Law (Anisotropic Form)

- \sigma_{ij} = E_{ijkl} \: \epsilon_{kl} where, i,j,k,l=1,2,3

- Total unknown Constant – 81 (normal strain will depend on shear strain also.)
- When we have stress symmetry \sigma_{ij} = \sigma_{ji} , total unknown Constant =6×9=54
- When we have strain symmetry with stress symmetry ( \sigma_{ij} = E_{ij} \: \epsilon_{j} ) total unknown constant=6×6=36

- According to Cauchy, the six components of stress and the six components of strain have a linear relationship, generalizing Hooke’s law to three-dimensional elastic systems. The six components of stress and strain are arranged into column vectors in the matrix representation of the stress-strain relationship, which is

- Now if C_{ij} = C_{ji} that material is known as Anisotropic or Aelotropic material, Total constant \frac{36-6}{2}+6=21
- For Anisotropic material having one plane of symmetry (Mono Clinic) Total Constant=13
- For Anisotropic material having two planes of symmetry (Orthotropic) Total Constant=9
- For orthotropic material, the total number of constants will be 9 (3E,3\mu,3G) (normal strain will not depend on shear strain)

- For Transversely Isotropic material (same properties in one plane) Total Constant=5
- For Isotropic material Total Constant=2 generally orthotropic, linear-elastic materials (wood, laminated plastics, cold rolled steels, reinforced concrete, various composite materials)

## Method of Superposition

The superposition principle essentially asserts that, for a linear elastic structure, the aggregate effect of several loads operating concurrently is equivalent to the algebraic sum of the effects of each load operating separately.

### Conditions

There are two conditions that a structure must satisfy for the principle to be valid. Structures that do satisfy these two conditions are referred to as linear elastic Structures. The two conditions are as follows:

- The equations of equilibrium must be based on the undeformed shape of the structure: If the size of the deformations occurring to the structure is small enough to be considered negligible, it can be determined that the undeformed shape of the structure can be used as a basis for the equilibrium equations
- The material used in the structure must be Linearly Elastic: For a material to be considered this way, Its stress-strain relationship must relate to Hooke’s Law of elasticity, referring to the stiffness of the materials
- the presence of the deflection does not alter the action of the applied load. An example method of a section or free body diagram is an example of a method of superposition.

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