Continuum Modeling
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Continuum Modeling Lee Davison
16
MRS BULLETIN/FEBRUARY 1988
Continuum Modeling
sity, temperature and local rate of deformation, but not on the deformation itself. The behavior of solids is more complicated. Measurements are made using gauges fixed to the material and the theoretical concepts, both microscopic and macroscopic, refer to the deformation from a reference configuration, not simply the current rate of deformation at a given point. Accordingly, discussion of the deformation of solids requires use of two coordinate systems. A material or Lagrangian system is used to identify the individual material points in the reference configuration, and a spatial or Eulerian system locates these points after the body has been deformed. (The simplicity of the fluid description comes from the fact that we need not distinguish one particle from another and so use only the spatial coordinates.) The behavior of solids is simplified when attention is restricted to small deformations, because we need not distinguish between the two coordinate systems. We take advantage of this fact to streamline the discussion of this article, despite the fact that large deformations, with their associated nonlinear effects, are issues of prime importance to much of the current work in constitutive modeling. We consider material bodies in a cartesian spatial coordinate system in which the coordinates of the place x are denoted x„i = 1,2,3. The velocity of the material point at this place is represented by its components Vj(x,t). The gradient of this velocity field is important to the discussion of viscous behavior and enters the theory through the
is determined by) the forces. Forces of elastic origin depend on strains, while forces of viscous origin are associated with the deformation rate. Since coordinate frames are chosen quite arbitrarily, it is necessary that the constitutive equations be restricted to forms that are invariant to this choice. Imposition of this restriction leads to models consistent with observations such as the one that tension in an elongated rubber band depends on its elongation but not its position or orientation in space. The field equations of classical continuum mechanics derive from the principles of balance of mass, momentum, and energy. In the class of theories considered, the principle of balance of moment of momentum is satisfied automatically by the requirement that the stress tensor be symmetric, i.e., that f,y = tji, a condition built into the constitutive equations. The remaining conditions, representing conservation of mass and balance of momentum and energy respectively, take the forms 0 =
dp(x,t) t , .. + {pvk).k at
P»i = hi pe - tjjdjj + qu
where p is the mass density, e is the internal energy density, and q is the heat flux vector, and where the superimposed dots denote time differentiation at fixed material points (summation over repeated indices is presumed).
Constitutive Modeling The starting point for discussing the If the material is subjected to a small mechanical behavior of contin
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