Overview

Test Case

Diffusion and mass transport effects due to the hydrostatic stress-migration are considered in a rectangular plane of length L and height H. Concentration constraints C1 and C2 are applied to the x = 0 and x = L ends of the plates, respectively. A steady-state analysis is performed to determine the distribution of concentration in the plate due to hydrostatic stress-migration.

Analysis Assumptions and Modeling Notes

Two approaches are used to solve this boundary value problem. The first solution uses a diffusion element type (PLANE238) and applies the hydrostatic-stress driving force V as a transport velocity body load (BF,VELO). The second approach uses a coupled structural-diffusion element type (PLANE223 with KEYOPT(1) = 100001). The UX DOF are constrained at location x = 0 m and the UY DOF are constrained at location y = 0 m and y = 0.1 m. An acceleration ACCELX is applied to the plate to create a stress-gradient in the X-direction. A Migration Model (TB,MIGR) is used to defined the volume of the migrating particle such that the transport velocity V is achieved.

Steady-state diffusion and structural-diffusion analysis is performed to determine the distribution of concentration in the plate. The obtained results are compared against the analytical solution at L/2.

For a 1D static case, the governing equation of diffusion is:

(226–1)

This equation takes the form of a homogeneous second-order linear ordinary differential equation:

(226–2)

The corresponding characteristic polynomial for this equation is:

(226–3)

Considering the real roots,

(226–4)

the generation solution of Equation 226–2 can be written as:

(226–5)

Coefficients and can be determined from the boundary conditions:

(226–6)

Substituting Equation 226–6 into Equation 226–5, we obtain:

(226–7)

Results Comparison

Figure 375: Nodal Concentration Plot at Time = 1 s using PLANE238 Elements

Figure 376: Nodal Concentration Plot at Time = 1 s using PLANE223 Elements