eFunda: Stress-Strain Relations of Materials
engineering fundamentals Stress-Strain Relations of Materials
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Hooke was probably the first person that suggested a mathematical expression of the stress-strain relation for a given material.

The most general stress-strain relationship (a.k.a. generalized Hooke's law) within the theory of linear elasticity is that of the materials without any plane of symmetry, i.e., general anisotropic materials or triclinic materials. If there is a plane of symmetry, the material is termed monoclinic. If the number of symmetric planes increases to two, the third orthogonal plane of material symmetry will automatically yield and form a set of principal axes. In this case, the material is known as orthotropic. If there exists a plane in which the mechanical properties are equal in all directions, the material is called transversely isotropic. If there is an infinite number of planes of material symmetry, i.e., the mechanical properties in all directions are the same at a given point, the material is known as isotropic.

Please distinguish 'isotropic' from 'homogeneous.' A material is isotropic when its mechanical properties remain the same in all directions at a given point while they may change from point to point; a material is homogeneous when its mechanical properties may be different along different directions at given point, but this variation is consistant from point to point. For example, consider three common items on a dining table: stainless steel forks, bamboo chopsticks, and swiss cheese. Stainless steel is isotropic and homogeneous. Bamboo chopsticks are homogeneous but not isotropic (they are transversely isotropic, strong along the fiber direction, relatively weak but equal in other directions). Swiss cheese is isotropic but not homogeneous (The air bubbles formed during production left inhomogeneous spots).

Both stress and strain fields are second order tensors. Each component consists of information in two directions: the normal direction of the plane in question and the direction of traction or deformation. There are nine (9) components in each field in a three dimensional space. Since they are symmetric, engineers usually rewrite them from a 3×3 matrix to a vector with six (6) components and arrange the stress-strain relations into a 6×6 matrix to form the generalized Hooke's law. For the 36 components in the stiffness or compliance matrix, not every component is independent to each other and some of them might be zero. This information is summarized in the following table.

  Independent
Constants
Nonzero
On-axis
Nonzero
Off-axis
Nonzero
General
Triclinic
(General Anisotropic)
21 36 36 36
Monoclinic 13 20 36 36
Orthotropic 9 12 20 36
Transversely
Isotropic
5 12 20 36
Isotropic 2 12 12 12

 

A more detailed discussion of stress, strain, and the stress-strain relations of materials can be found in the Mechanics of Materials section.

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