II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

global state variables
models are consistent.
(e.g., stress and strain) are dealt with on a global level which ensures that all constitutive
Internal state variables (e.g., backstress) see no rotations whatsoever.
The drawback to work]ng in the unrotated reference frame is that we must accurately determine the rotation
tensor, R, which is not a straightforward numerical calculation. We present an incremental, algebraic algorithm to
accomplish this task in Section 3.4.
2.3 Fundamental Equations
The equilibrium equation for the body is
div T+pb=O
(2.3.1)
where p is the mass density
per unit volume and b is a specific body force vector.
We seek the solution to Equation (2.3.1) subject to the boundary conditions
u = f(t) on Su (2.3.2)
where Su represents the portion of the boundary on which kinematic quantities are specified (displacement). In
addition to satisfying the kinematic boundary conditions given by Equation (2.3.2), we must satisfy the traction
boundary conditions
Ton=s(t)on S~ (2.3.3)
where ST represents the portion of the boundary on which tractions are specified. The boundary of the body is given
by the union of Su and ST, and we note that for a valid mechanics problem Su and ST have a null intersection.
The jump conditions at all contact discontinuities must satis& the relation
(T+- T-). n= Oon Sc (2.3.4)
where Sc represents the contact surface intersection and the subscripts “+” and “-” denote different sides of the
contact
surface.
To utilize dynamic relaxation as a solution strategy for quasktatics problems,
equilibrium equations into equations of motion by adding an acceleration term. Thus,
divT + pb = pu
we must first convert the
(2.3.5)
where u is the acceleration of the material point. Now, all that remains is to introduce the concept of mesh
homogenization and artificial darnping as well as integrate forward in time horn initial conditions until the transient
dynamic response has damped out to the static result with equilibrium satisfied. Further description of the
implementation of the dynamic relaxation method will be discussed in a later section (Section 3.7).
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