The capability to perform a coupled structural-diffusion analysis exists in the following elements:

These elements support the effects of diffusion strain and hydrostatic stress migration (transport of particles due to a hydrostatic stress gradient) when used in coupled-field analyses having structural and diffusion DOFs.

Constitutive Equations

In a coupled structural-diffusion analysis, the total strain is composed of elastic {ε^el} and diffusion {ε^d} parts, respectively:

(10–74)

where:

{ε} = total strain vector = [ε_x ε_y ε_z ε_xy ε_yz ε_xz]^T= {ε^el} + {ε^di}

{ε^el} = elastic strain vector (output as EPEL)

{ε^di}= diffusion strain vector (output as EPDI)

{σ} = stress vector = [σ_x σ_y σ_z σ_xy σ_yz σ_xz]^T (output as SIG)

ΔC = concentration change = C - C_ref

C = concentration (input/output as CONC);

= normalized concentration (input/output as CONC)

= saturated concentration (input as CSAT on MP command)

C_ref = reference concentration (input as CREF on MP command)

[E] = elastic stiffness matrix (inverse defined in Equation 2–4 as [D]^-1 or input using TB,ANEL command)

{β}= vector of coefficients of diffusion expansion = [β_x β_y β_z 0 0 0]^T (input using BETX, BETY, BETZ on MP command)

In addition to the diffusion strain coupling, the structural and diffusion DOFs are coupled by the hydrostatic stress migration effect:

(10–75)

where:

{J} = diffusion flux density (input/output as DF)

= diffusivity matrix

Dxx, Dyy, and Dzz = diffusivity coefficients in the element's X, Y, and Z directions, respectively (input as DXX, DYY, DZZ on MP command)

T = absolute temperature = Tc + Toff

Tc = current temperature (input/output as TEMP on D or BF commands)

Toff = offset temperature from absolute zero to zero (input on TOFFST command)

Ω/k = atomic volume constant (input as C2 on TBDATA command with TB,MIGR)

Ω = atomic volume

k = Boltzmann constant

= hydrostatic stress

Substituting Equation 10–74 into Equation 10–75, we obtain the following expression for the diffusion flux density {J}:

(10–76)

where:

For more information on Equation 10–75 and the related material constant input, see Migration Model in the Material Reference.

For more information on diffusion analysis, see Diffusion.

Derivation of Structural-Diffusion Matrices

Applying the variational principle to the structural equation (discussed in Derivation of Structural Matrices) and the diffusion equation (Equation 9–6) coupled by the constitutive equations (Equation 10–74 and Equation 10–76), produces the following finite element matrix equation for the structural-diffusion analysis:

(10–77)

where:

[M^u] = element mass matrix (defined by Equation 2–58)

[C^u] = element structural damping matrix (discussed in Damping Matrices)

[K^u] = element stiffness matrix (defined by Equation 2–58)

{u} = nodal displacement vector

{F} = sum of the element nodal force (defined by Equation 2–56) and element pressure (defined by Equation 2–58) vectors

[C^d] = element diffusion damping matrix (defined by Equation 9–9)

[K^d] = element diffusion conductivity matrix (defined by Equation 9–9)

{C} = nodal concentration vector

{R} = nodal diffusion flow rate vector (defined by Equation 9–9)

= element diffusion strain stiffness matrix

= element transport conductivity matrix

= nonlinear part of the element diffusion conductivity matrix

= element conductivity matrix associated with diffusion strain

= nonlinear part of the element conductivity matrix associated with diffusion strain

[B] = strain-displacement matrix (see Equation 2–44)

= concentration gradient (output as CG)

{N} = element shape functions