Biomedical Engineering Reference
In-Depth Information
6.3
Constitutive Equations
Figure 6.3 shows the emplacement of the constitutive equation in the framework
of solid mechanical modeling of a material particle. The constitutive equation
combines the kinematic and equilibrium equations by indicating the dependence
of strains on stresses.
According to the level of description, the constitutive equations are distinguished
as microscopic, stochastic, and macroscopic [8]. Microscopic constitutive equations
describe the material behavior based on variables (e.g., dislocation density) taken
from materials physics. A transfer of the microscopic material behavior on the
macroscopic level has not been successful till now. Stochastic constitutive equations
also describe the material behavior on the micro level. For that purpose, proba-
bilistic processes are applied. The transfer on the macroscopic level is due to its
possible mathematical structure. Macroscopic constitutive equations are also called
phenomenological models
. Their field of application is the continuum mechanical
modeling of components and structures. The material is idealized as a homoge-
neous continuum, which constitution is described by a few variables such as strain,
temperature, and occasionally by additional so-called internal variables. In the case
of multiphase or inhomogeneous materials, such as cellular or fiber-reinforced ma-
terials, this assumption does not hold in the first place. By averaging the different
material properties, an approximate homogeneity can be assumed, cf. Figure 6.4.
External forces
Displacements
Continuum mechanical
modeling
Equilibrium
Kinematic
Constitutive
equation
Stresses
Measure for
loading
Strains
Measure for
deformation
Figure 6.3
Solid mechanical modeling of engineering materials.
Homogenization
Multiphase material
Equivalent material
Figure 6.4
Averaging the materials properties to obtain an equivalent material.
