Product: ABAQUS/Standard
ABAQUS/Standard provides for the modeling of the transient or steady-state diffusion of one material through another, such as the diffusion of hydrogen through a metal (Crank (1956), deGroot and Mazur (1962)). The governing equations are an extension of Fick's equations, to allow for nonuniform solubility of the diffusing substance in the base material.
The basic solution variable (used as the degree of freedom at the nodes of the mesh) is the “normalized concentration” (often referred to as the “activity” of the diffusing material), 
, where c is the mass concentration of the diffusing material and s is its solubility in the base material. This means that when the mesh includes dissimilar materials that share nodes, the normalized concentration is continuous across the interface between the different materials. Since 
is the square root of the partial pressure of the diffusing phase, the partial pressure is the same on both sides of the interface; Sievert's law is assumed to hold at the interface.
The diffusion problem is defined from the requirement of mass conservation for the diffusing phase:
where V is any volume whose surface is S,
is the outward normal to S, 
is the flux of concentration of the diffusing phase, and 
is the flux of concentration leaving S.Using the divergence theorem,
The equivalent weak form is
is an arbitrary, suitably continuous, scalar field.This statement can be rewritten as
The diffusion is assumed to be driven by the gradient of a chemical potential, which gives the general behavior
where
is the diffusivity; 
is the solubility; 
is the “Soret effect” factor, providing diffusion because of the temperature gradient; 
is the temperature; 
is the absolute zero on the temperature scale used; 
is the pressure stress factor, providing diffusion driven by the gradient of the equivalent pressure stress, 
; and 
are any predefined field variables.An example of a particular form of this constitutive model is the assumption made for hydrogen diffusion in a metal:
, defined as 
is a fixed datum, R is the universal gas constant, and 
is the partial molar volume of hydrogen in the solid solution. This form is similar to that used by Sofronis and McMeeking (1989) and results in a constitutive expression of the form 
and 
must be calculated from the equations 
Changing variables (
) and introducing the constitutive assumption of Equation 2.13.1–3 into Equation 2.13.1–2 yields
Equilibrium in a finite element model is approximated by a finite set of equations through the introduction of appropriate interpolation functions. Discretized quantities are indicated by uppercase superscripts (for example, 
). The summation convention is adopted for the superscripts. These represent nodal variables, with nodes shared between adjacent elements and appropriate interpolation chosen to provide adequate continuity of the assumed variation. The interpolation is based on material coordinates 
, 
, 2, 3.
The virtual normalized concentration field is interpolated by
are interpolation functions. Then, the discretized equations are written as Time integration in transient problems utilizes the backward Euler method (the modified Crank-Nicholson operator). Adopting the convention that any quantity not explicitly associated with a point in time is taken at 
, we can drop the subscript and write the integrated equations as
The Jacobian contribution from the conservation equation is obtained from the variation of Equation 2.13.1–6 with respect to at time 
. This yields
, we obtain 
Inspecting the above equation, we observe that the Jacobian becomes unsymmetric whenever the diffusivity, 
; the temperature-driven diffusion coefficient, ; or the pressure-driven diffusion coefficient, , is defined as a function of concentration.
