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## 1.8. Nonlinear Elasticity

### 1.8.1. Stress strain relations

For a linear elastic material, there is always a constant relationship between the applied stress and the resulting strain, regardless the magnitude of the stress and the strain. The stress strain relation (Eq. (1.91)) for such a material is therefore a straight line, as shown in Fig. 1.19a. The elastic modulus corresponding to this stress strain pair is the slope of the curve.

Fig. 1.19: Stress strain relations for: a) linear elastic material. b) perfectly elastic material. c) elastic material, with hysteresis. d) material suffering permanent deformation.

Any material not obeying a linear stress strain relation is said to behave nonlinearly. For a nonlinear elastic material, the stress strain relation may be written as

Remembering that ? and ? generally are tensors, it is clear that nonlinear elasticity may be very complicated mathematically.

Further, we can not ignore the higher order terms in the strain tensor, which were neglected in the derivation on page 15. The full expression for the strain tensor, replacing Eq. (1.74), is (see e.g. Landau and Lifshitz, 1986):

Nonlinear behaviour may have various causes, and appear in many different ways. Fig. 1.19b shows one example. This material has a nonlinear stress strain relation, since the ratio of stress to strain is not the same for all stresses. The relation is, however, identical for the loading and unloading process. Such materials are said to be perfectly elastic.

For nonlinear stress strain relations the elastic modulus is no longer uniquely defined, not even for a specific...

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Deflectometers and Extensometers
Deflectometers and extensometers are used to indicate the deformation of material while it is subjected to stress.
Strain Gauges
Strain gauges are measuring elements that convert force, pressure, tension, etc., into an electrical signal.
Metal Sheet
Metal sheet is metal or alloy stock supplied or available in the form of sheet or foil. It has a thickness between 0.006" and 0.250", and a width of 24" (609.6 mm) or more.
Metal Strip
Metal strip is narrow, thin stock that is usually 3/16 in. (4.76 mm) or less in thickness and under 24 in. (609.6 mm) in width. Metal strips are formed to precise thicknesses and/or width requirements.
Nondestructive Testing (NDT) Material Testers
Non-destructive testing (NDT) material testers are devices, machines and equipment that are used to determine residual stress, alloy type, hardness, microstructure, elasticity, heat treatment and other material conditions without permanently altering or destroying the material being examined.

##### Topics of Interest

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