Generalized Hooke’s Law

Linear Elastic Materials

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Hooke’s Law 3D - 1 stress state

• Let’s look at this belt drive to formulate 3D strain tensor.

• The belt experiences a tension state in only one direction.

• Using Hooke’s law, the relation between the stress and strain in the X direction is given by

• Due to Poisson’s effect the material will experience deformation in the other two principal directions.

• There are no shear forces, so the shear strains are zero.

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Hooke’s Law 3D - 2 stress states

• Look at the structure of the trampoline.

• It has multiple springs connected to a frame and center fabric material.

• It has two tensile stress states acting along two principal directions.

• The strains along these directions are given by:

• The strain in the Z direction due to Poisson’s effect is:

• Shear strains are zero.

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Hooke’s Law 3D - 3 stress states

• Let’s look at a scuba diver who experiences hydrostatic pressure from all sides.

• The diver experiences stress from all three principal directions.

• Strains in principal directions

• No shear strains

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Hooke’s Law 3D - 3 stress states (cont.)

• The sum of three principal strains is simply the volumetric strain.

• Simplifying this equation leads to

• This follows Hooke’s law!

• Relation between the bulk modulus and Young’s modulus

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Engineering Shear Strains

• Let’s review shear strain in a cube that shears in the XY plane.

• Due to symmetry,

• Engineering shear strain (γ) is often reported as shear strain by most commercial FE solvers.

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Hooke’s Law 3D

• A more general case such as brake pad assembly.

• Brake pads experience both normal and shear stresses.

• All the strain components:

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Strain Vector in Tensor Form

• In a full 3D case the strain tensor can be written in this form.

• It is in full agreement with the linear form proposed by Hooke’s law.

• The constant tensor is called the Compliance tensor.

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Stress Vector in Tensor Form (cont.)

• Due to its linear nature, the same equation can be expressed in the form of stress vector.

• It still follows the Hooke’s law and the tensor of constants is called the Stiffness matrix.

• This is what Hooke’s law looks like in a 3D isotropic body.

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Anisotropic Materials

• In reality, not all materials are isotropic; some have significant directional behavior.

• In such cases the Young’s modulus and Poisson’s ratio change according to the principal direction.

• Examples include various materials such as wood, fibrous tissues, composites, rolled metal specimens, etc.

• Hooke’s law is still applicable, but the equations need to account for the directional material properties.

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Orthotropic Materials

• Orthotropic materials are special cases in which the material has symmetry about the three orthogonal planes.

• The material has different material properties in each direction.

• The stiffness matrix for such a material takes this form: