By Alfred North Whitehead
This distinct little booklet bargains a brisk creation to a sequence of mathematical strategies, a historical past in their improvement, and a concise precis of the way state-of-the-art reader could use them
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Extra info for An Introduction to Mathematics
The precise role of the plastic strain in the internal variable theory will become clear shortly; in particular, it will be seen that it would be premature to assume the plastic strain to be one of the internal variables, since the additive decomposition of the strain, and the existence of the plastic strain, follow as consequences of the thermodynamics of internal variables. 4). Whether or not internal variables are required will depend on the particular features that one would wish to incorporate in the theory.
4 Isotropic Elasticity It is often the case that materials possess preferred directions or symmetries. For example, timber can be regarded as an orthotropic material, in the sense that it possesses particular constitutive properties along the grain and at right angles to the grain of the wood. The greatest degree of symmetry is possessed by a material that has no preferred directions; that is, say, its response to a force is independent of its orientation. This property is known as isotropy, and a material with such a property is called isotropic.
We also have subsequent elastic ranges, such as the interval (−σ1 , σ1 ), that are reached only as a result of plastic deformation having taken place. It is the feature of irreversibility that sets an elastoplastic material apart from an elastic one; the nonlinear behavior described before is not a feature peculiar to plastic materials, since nonlinearly elastic behavior is possible, and indeed common. But the feature of irreversibility implies that we no longer have a one-to-one relationship between stress and strain.
An Introduction to Mathematics by Alfred North Whitehead