3.2.4.5. Conservation of Solid Momentum

Aria currently solves the quasistatic form of the solid momentum equations. Furthermore, the solid stress is treated as a linear elastic material. In this limit, the Cauchy momentum equation is given by

(3.24) $\div\T = \vector{0}$

where $\T$ is the solid stress tensor. We construct the G/FEM residual form of (3.21) by contracting with the unit coordinate vector in the $k$ -direction, $\e_k$ , multiplying by the weight function $\phi^i$ and integrating over the volume. Using the vector identity $(\div\T)\bcdot\e_k\phi^i = \div(\T\bcdot\e_k\phi^i) - \T^t:\grad(\e_k\phi^i)$ and integrating by parts gives

(3.25) $\symRes_{m,k}^i = \int\limits_\Vol \T^t:\grad\left(\e_k\phi^i\right)\dV = 0$

Here, the surface contribution, $\int_\Surf \n\bcdot\T\bcdot\e_k\phi^i\dS$ , has been dropped because Aria currently only supports Dirichlet and natural (homogeneous) boundary conditions for the solid equation.

In Aria, each term in (3.25) is specified separately as identified in equation (3.26).

(3.26) $\symRes_{m,k}^i = \underbrace{\int\limits_\Vol \T^t:\grad\left(\e_k\phi^i\right)\dV}_\mathrm{DIFF} = 0$

Currently, Aria does not support direct specification of the more popular stress-strain parameterization that utilizes Young’s modulus $E$ , Poisson’s ratio $\nu$ and coefficient of thermal expansion $\alpha$ (note, the shear modulus $G = \mu$ ). The relationship between these two parameterizations is summarized here for convenience.

$2 \mu & = \frac{E}{\left(1+\nu\right)} \\ \lambda & = \frac{\nu E}{\left(1+\nu\right)\left(1-2\nu\right)} \;\; = \;\; 2 \mu \frac{\nu}{\left(1-2\nu\right)} \\ \beta & = \frac{\alpha E}{\left(1-2\nu\right)} \;\; = \;\; \alpha \left( 3 \lambda + 2 \mu \right)$

When the solid momentum equation is solved, the physical coordinates of the region, used to calculate integration quantities, is deformed by the displacement vector. The current configuration, $\mathbf{x}$ , is calculated from the reference configuration, $\mathbf{X}$ , by

$\mathbf{x} = \mathbf{X} + \mathbf{d},$

where $\mathbf{d}$ is the displacement field. The displacement field is chosen automatically by the code, but may be overridden by setting a solution option. These displaced, physical coordinates are used for integration of all of the equations.

Additionally, the code supports the integration of the solid momentum equations in the reference configuration, rather than in the current configuration. This is activated through the use of the REF_DIFF equation term rather than the DIFF term. When this is used, the Cauchy stress tensor is rotated into the Second Piola-Kirchhoff stress tensor. Integration quantities are also calculated in the reference configuration. Care should be taken to ensure that the constitutive model used also changes its strain measure to one which is work conjugate with the Second Piola-Kirchhoff stress tensor.

If both the MESH and SOLID equations are defined in the same region, then the TALE mesh motion algorithm is activated automatically. Here, the reference frame for the SOLID equation is changed to the TALE reference frame, which is defined as

$\underline{\mathbf{x}}_r = \underline{\mathbf{X}} + \underline{\mathbf{d}}_m - \underline{\mathbf{d}}_s$

where $\underline{\mathbf{x}}_r$ is the new TALE reference frame, $\underline{\mathbf{d}}_m$ is MESH_DISPLACEMENTS, and $\underline{\mathbf{d}}_s$ is SOLID_DISPLACEMENTS. This affects the calculation of strain measures, such as $\underline{\underline{\mathrm{F}}}$ , and integration quantities, such as $\mathrm{det}(J)$ , among others. More details of the TALE method will be available soon in a pending report.