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Derivation of relationships

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The most general linear relationship between two symmetric tensors and which is homogeneous (i.e. independent of direction) is:

Where A and B are constants, and is the Kroneker delta tensor. When is the stress and is the strain, this is an expression of Hooke's law, also known as the constitutive equations of linear elasticity.

Bulk modulus

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Consider a cube of linearly elastic material with each side having length L. A purely compressive force consists of a force directed normal to each face, causing each face to move a distance . The stress and strain may be written as:

Thus:

and the constitutive equations become:

The incompressibility or bulk modulus K is defined as

where . Taking the derivative, it follows that , thus:

and it is clear that the constant A is simply the bulk modulus K.

Shear modulus

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For a pure shear force applied only to the z-face of the cube:

Since only off-diagonal elements are non-zero, the constitutive equations become:

The shear modulus is defined as

where and thus:

and it is clear that the constant B is simply twice the shear modulus G. The constitutive equations may now be written:

Young's modulus and Poisson's ratio

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Figure 1: A cube with sides of length L of an isotropic linearly elastic material subject to tension along the x axis, with a Poisson's ratio of 0.5. The green cube is unstressed, the red is expanded in the x direction by due to tension, and contracted in the y and z directions by .

By the definition of Poisson's ratio, if a positive force is applied only to the x-faces of the cube, they will move a distance of , and the other faces will move a distance of . The stress and strain are written:

Thus:

The constitutive equations become:

Young's modulus is defined as:

thus:

and

These two equations may be solved for any one of the variables in terms of the other two, yielding the relationships:

Lame's first parameter and the p-wave modulus

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If equal and opposite forces are applied to the x faces of the cube, and a force is applied to the other faces such that these other faces do not move, then:

Thus:

The constitutive equations become:

and

The p-wave modulus is defined as:

and Lame's first parameter is defined as:

Thus:

These two equations, along with the relationships derived above may be used to express any two elastic moduli in terms of any other two.


Volumetric change

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The relative change of volume ΔV/V due to the stretch of the material can be calculated using a simplified formula:

which, for small deformations reduces to:

where

is material volume
is change in material volume
is original length, before stretch
is the change of length along the direction of compression:

Note that for an incompressible material, which implies that . For a material which does not have any transverse expansion or contraction, the volume change will be simply , which implies that .