More generally, in physics, a constitutive equation is a relation between two physical quantities (often tensors) that is specific to a material or substance, and does not follow directly from physical law. It is combined with other equations that do represent physical laws to solve some physical problem, like the flow of a fluid in a pipe, or the response of a crystal to an electric field.
Some constitutive equations are simply phenomenological; others are derived from first principles. A constitutive equation frequently has a parameter taken to be a constant of proportionality in ideal systems.
Therefore, whenever we develop a constitutiveequation to model a tissue, we need to balance the need to accurately model tissue behavior under the range of loading with the need to have a constitutiveequation that is simple enough to in a numerical model and to experimentally measure all the constants in the constitutiveequation.
Therefore, constitutiveequations consist of two major components: constants that must be fit to experimental data and measures of deformation, which may include small or finite deformation as well as the rate of deformation.
The use of linear elastic constitutiveequations is typically restricted for use with bone tissue, since bone tissue is the only tissue that consistently operates in the small strain regime and exhibits a linear relationship between stress and strain.
For the description of history dependent material behavior of metals in addition to the partial differential equations derived from the balance laws of moment and mass constitutiveequations expressing the dependence of the stress from the deformation history are used.
In the case of metals, the formulation of these constitutiveequations is based on the use of internal variables, and the constitutiveequations consist of a system of nonlinear ordinary differential equations, called evolution equations, for these internal variables.
It was also shown that for many constitutiveequations, which are not of monotone type, the interior variables can be transformed so that the transformed variables satisfy equations of monotone type, which means that the boundary value problem for the transformed constitutiveequations can be solved.
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