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Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science by Mark F. Horstemeyer

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2.4 CONTINUUM THEORY CONSTITUTIVE EQUATIONS

A complete theoretical discussion of continuum field theory constitutive equations can be found in Malvern [19] among others. The constitutive equation means “to represent the constitution of the material.” When an external stimulus occurs to our country, our constitution determines what our value system is to respond to the external stimulus. When an external stimulus occurs to a metal, its internal constitution determines what the response will be as well. Hence, a constitutive equation is then the mathematical representation of the inner “value” system of the material. This is a foundational point to the thesis of the multiscale modeling idea.

In continuum theory, the governing equations include the conservation laws of mass, momentum, and energy. When counting the number of unknowns in these equations compared with the number of available equations, one finds that there are too many unknowns. As such, more equations are required; these are the constitutive equations. Now some have called these equations constitutive laws; however, in the strictest sense of the scientific method in which a working hypothesis, which started from an observation, moves from a theory to a law, the constitutive relations should not be a law in the same vein as the laws of thermodynamics, gravity, and so on. Each constitutive equation is different from one material to another. Probably the simplest constitutive equation for an ideal elastic solid used for ...

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