Micromorphic Theory

This section describes a small portion of micromorphic theory as relevant to understanding the inputs and outputs of the Micromorphic Filter and Tardigarde-MOOSE. Further discussion is provided in Detailed Micromorphic Theory.

Homogenization via Micromorphic Filter

The Micromorphic Filter calculates a variety of homogenized, macroscale quantities using volume and surface integrals of DNS data over selected micro-averaing domains. The following equations (Eq. (1)) define the macroscale density, Cauchy stress, force, acceleration, couple stress, symmetric micro stress, body force couple, and micro spin inertia. Further details of the Micromorphic Filter and micromorphic quantities are provided by Miller 2021 [1], Miller et al. 2022 [2], and a variety of other resources.

(1)\[ \begin{align}\begin{aligned}\rho dv &\stackrel{\text{def}}{=} \int_{da}\rho^{\left(\alpha\right)}\,{dv^{\left(\alpha\right)}}\\\sigma_{ij} n_i da &\stackrel{\text{def}}{=} \int_{da} \sigma_{ij}^{\left(\alpha\right)} n_{i}^{\left(\alpha\right)} \,{da^{\left(\alpha\right)} }\\\rho f_{i} dv &\stackrel{\text{def}}{=} \int_{dv} \rho^{\left(\alpha\right)} f_{i}^{\left(\alpha\right)} \,{dv^{\left(\alpha\right)} }\\\rho a_{i} dv &\stackrel{\text{def}}{=} \int_{dv} \rho^{\left(\alpha\right)} a_{i}^{\left(\alpha\right)} \,{dv^{\left(\alpha\right)} }\\m_{ijk} n_i da &\stackrel{\text{def}}{=} \int_{da} \sigma_{ij}^{\left(\alpha\right)} \xi_k n_{i}^{\left(\alpha\right)} \,{da^{\left(\alpha\right)} }\\s_{ij} dv &\stackrel{\text{def}}{=} \int_{dv} \sigma_{ij}^{\left(\alpha\right)} \,{dv^{\left(\alpha\right)} }\\\rho l_{ij} dv &\stackrel{\text{def}}{=} \int_{dv} \rho^{\left(\alpha\right)} f_{i}^{\left(\alpha\right)} \xi_j\,{dv^{\left(\alpha\right)} }\\\rho \omega_{ij} dv &\stackrel{\text{def}}{=} \int_{dv} \rho^{\left(\alpha\right)} \ddot{\xi_i} \xi_j \,{dv^{\left(\alpha\right)}}\end{aligned}\end{align} \]

The Micromorphic Filter also determines the macroscale deformation gradient, \(F_{iI}\), the micro-deformation tensor, \(\chi_{iJ}\), and the gradient of the micro-deformation tensor, \(\chi_{iJ,K}\). With these terms, deformation measures may be determined including the Green-Lagrange strain (\(E_{ij}\)), Euler-Almansi strain (\(e_{ij}\)), micro-strain (\(\mathcal{E}_{IJ}\)), and micro-deformation gradient (\(\Gamma_{IJK}\)) shown in equation (2).

(2)\[ \begin{align}\begin{aligned}E_{IJ} &= \frac{1}{2} \left( F_{iI} F_{iJ} - \delta_{IJ}\right)\\e_{ij} &= F_{Ii}^{-1} E_{IJ} F_{Jj}^{-1}\\\mathcal{E}_{IJ} &= F_{iI} \chi_{iJ} - \delta_{IJ}\\\Gamma_{IJK} &= F_{iI} \chi_{iJ,K}\end{aligned}\end{align} \]

Tardigrade-MOOSE

The relevant balance equations (in the current configuration) to describe a determinant system with 12 unknowns may be defined for the balance of linear momentum and the balance of the first moment of momentum.

(3)\[ \begin{align}\begin{aligned} \sigma_{lk,l} + \rho \left( f_k - a_k \right) &= 0\\ \sigma_{mk} - s_{mk} + m_{lkm,l} + \rho \left( l_{mk} - \omega_{mk}\right) &= 0\end{aligned}\end{align} \]

Tardigrade-MOOSE solves these equations to find the unknown displacements, \(\mathbf{u}\), and micro-displacements, \(\mathbf{\Phi}\), where \(\mathbf{\chi} = \mathbf{I} + \mathbf{\Phi}\).

Micromorphic Constitutive Models

Micromorphic Linear Elasticity

The model of Micromorphic Linear elasticity intoduced by Eringen and Suhubi [3] is defined in the reference configuration for the second Piola Kirchoff stress, symmetric micro stress, and couple stress as follows:

(4)\[ \begin{align}\begin{aligned}S_{KL} &= \left(\lambda^* + \tau^*\right) E_{MM} \delta_{KL} + 2\left(\mu^* + \sigma^*\right) E_{KL} + \eta^* \mathcal{E}_{MM} \delta_{KL} + \kappa^* \mathcal{E}_{KL} + \nu^* \mathcal{E}_{LK}\\\Sigma_{KL} &= \left(\lambda^* + 2\tau^*\right) E_{MM} \delta_{KL} + 2\left(\mu^* + 2\sigma^*\right) E_{KL} + \left(2\eta^* - \tau^*\right) \mathcal{E}_{MM} \delta_{KL}\\ &+ \left(\nu^* + \kappa^* - \sigma\right) \left(\mathcal{E}_{KL} + \mathcal{E}_{LK}\right)\\M_{KLM} &= \tau_1^* \left(\delta_{LM}\Gamma_{KPP} + \delta_{MK} \Gamma_{PPL}\right) + \tau_2^* \left(\delta_{LM}\Gamma_{NKN} + \delta_{LK} \Gamma_{PPM}\right)\\ &+ \tau_3^* \delta_{LM} \Gamma_{NNK} + \tau_4^* \delta_{MK} \Gamma_{LPP} + \tau_5^* \left(\delta_{LK}\Gamma_{MPP} + \delta_{MK} \Gamma_{NLN}\right)\\ &+ \tau_6^* \delta_{LK} \Gamma_{NMN} + \tau_7^* \Gamma_{LMK} + \tau_8^* \left(\Gamma_{KLM} \Gamma_{MKL}\right) + \tau_9^* \Gamma_{LKM}\\ &+ \tau_{10}^* \Gamma_{MLK} + \tau_{11}^* \Gamma_{KML}\end{aligned}\end{align} \]

If parameters \(\tau^*\), \(\sigma^*\), \(\eta^*\), \(\kappa^*\), \(\nu^*\), and \(\tau_1^*\) through \(\tau_{11}^*\) are set to zero (with the exception of \(\tau_7^*\) which must be non-zero to satisfy positive definiteness of the free energy function [4]), then the constitutive equations reduce as follows.

(5)\[ \begin{align}\begin{aligned}S_{KL} &= \lambda^* E_{MM} \delta_{KL} + 2\mu^* E_{KL}\\\Sigma_{KL} &= \lambda^* E_{MM} \delta_{KL} + 2\mu^* E_{KL}\\M_{KLM} &= \tau_7^* \Gamma_{LMK}\end{aligned}\end{align} \]

Here the second Piola-Kirchhoff and symmetric micro-stresses are equivalent and classical Saint Venant Kirchhoff elasticity is recovered. The Saint Venant-Kirchhoff model is an extension of classical linear elasticity to the geometrically non-linear regime (e.g. large deformations) [5]. For this special case, the micromorphic material parameters \(\lambda^*\) and \(\mu^*\) are equivalent to the classical Lam'e parameters.

Another simplification can be made in which the components of the higher order stress are set to zero (again with the exception that \(\tau_7\) must be non-zero). After setting \(\tau_1\) to \(\tau_6\) and \(\tau_8\) to \(\tau_{11}\) to zero, the simplified equations become:

(6)\[ \begin{align}\begin{aligned}S_{KL} &= \left(\lambda^* + \tau^*\right) E_{MM} \delta_{KL} + 2\left(\mu^* + \sigma^*\right) E_{KL} + \eta^* \mathcal{E}_{MM} \delta_{KL} + \kappa^* \mathcal{E}_{KL} + \nu^* \mathcal{E}_{LK}\\\Sigma_{KL} &= \left(\lambda^* + 2\tau^*\right) E_{MM} \delta_{KL} + 2\left(\mu^* + 2\sigma^*\right) E_{KL} + \left(2\eta^* - \tau^*\right) \mathcal{E}_{MM} \delta_{KL}\\ &+ \left(\nu^* + \kappa^* - \sigma\right) \left(\mathcal{E}_{KL} + \mathcal{E}_{LK}\right)\\M_{KLM} &= + \tau_7^* \Gamma_{LMK}\end{aligned}\end{align} \]

Here \(\tau_7\) may be either set to a fixed value or determined through a calibration process.

The reference configuration stresses may be pushed forward to the current configuration with:

(7)\[ \begin{align}\begin{aligned}\sigma_{ij} &= \frac{1}{J} F_{iI} S_{IJ} F_{jJ}\\s_{ij} &= \frac{1}{J} F_{iI} \Sigma_{IJ} F_{jJ}\\m_{ijk} &= \frac{1}{J} F_{iI} \chi_{kK} M_{IJK} F_{jJ}\end{aligned}\end{align} \]

where \(J\) is the Jacobian of deformation defined as the determinant of the deformation gradient \(\mathbf{F}\).

Note

All elastic material properties are denoted with a star notation \(\left(\right)^*\) to help differentiate from other terms

Constraints on Elastic Parameters

The parameters of the micromorphic linear elastic model of Eringen and Suhubi [3] may not be selected arbitrarily and may depend on each other. Smith 1968 [4] proposed restrictions on the 18 parameters to maintain positive definiteness of the quadratic strain energy function (shown in equation (59). The constraints on second Piola-Kirchhoff and symmetric micro-stresses are written as:

(8)\[ \begin{align}\begin{aligned}\lambda &> 0\\\kappa + \nu &> 2\sigma\\\left(\kappa + \nu - 2\sigma\right) \mu &> 2\sigma^2\\3\lambda + 2\mu &> 0\\\kappa + \nu +3\eta &> 3\tau + 2\sigma\\\left(\kappa + \nu + 2\eta - 3\tau - 2\sigma\right)\left(3\lambda + 2\mu\right) &> \left(3\tau + 2\sigma\right)^2\\\kappa - \nu &> 0\\4\mu \left(\kappa + \nu - 2\sigma\right) &> 2\sigma\end{aligned}\end{align} \]

For the higher order stress parameters, a matrix form may be utilized. Define the matrix \(\mathbf{T}\) as:

(9)\[\begin{split}\mathbf{T} &= \begin{bmatrix} \tau_1 + \tau_2 + 3\tau_3 + \tau_7 + \tau_{10} & 3\tau_1 + \tau_4 + 3\tau_5 + \tau_8 + \tau_{11} & 3\tau_2 + \tau_5 + \tau_6 + \tau_8 + \tau_9 \\ 3\tau_1 + \tau_2 + \tau_3 + \tau_8 + \tau_{11} & \tau_1 + 3\tau_4 + \tau_5 + \tau_7 + \tau_9 & \tau_2 + 3\tau_5 + \tau_6 + \tau_8 + \tau_{10} \\ \tau_1 + 3\tau_2 + \tau_3 + \tau_8 + \tau_9 & \tau_1 + \tau_4 + 3\tau_5 + \tau_8 + \tau_{10} & \tau_2 + \tau_5 + 3\tau_6 + \tau_7 + \tau_{11}. \\ \end{bmatrix}\end{split}\]

The constraints on terms \(\tau_1\) to \(\tau_{11}\) may then be expressed as:

(10)\[ \begin{align}\begin{aligned}\tau_7 + 2\tau_8 &> |\tau_9 + \tau_{10} + \tau_{11}|\\\tau_7 - \tau_8 &> \frac{1}{\sqrt{2}} |\left(\tau_9 - \tau_{10}\right)^2 + \left(\tau_{10} - \tau_{11}\right)^2 + \left(\tau_{11} - \tau_9\right)^2|^{\frac{1}{2}}\\tr\left(\mathbf{T}\right) &> 0\\tr\left(co\left(\mathbf{T}\right)\right) &> 0\\det\left(\mathbf{T}\right) &> 0\end{aligned}\end{align} \]

where \(tr\left(\cdot\right)\) is the trace, \(co\left(\mathbf{\cdot}\right)\) is the cofactor, and \(det\left(\cdot\right)\) is the determinant of a matrix.

Micromorphic Elasto-Plasticity

Note

Discussion coming soon!