Aspects of implementing constant traction boundary conditions in computational homogenization via semi-Dirichlet boundary conditions

Abstract

In the past decades computational homogenization has proven to be a powerful strategy to compute the overall response of continua. Central to computational homogenization is the Hill–Mandel condition. The Hill–Mandel condition is fulfilled via imposing displacement boundary conditions (DBC), periodic boundary conditions (PBC) or traction boundary conditions (TBC) collectively referred to as canonical boundary conditions. While DBC and PBC are widely implemented, TBC remains poorly understood, with a few exceptions. The main issue with TBC is the singularity of the stiffness matrix due to rigid body motions. The objective of this manuscript is to propose a generic strategy to implement TBC in the context of computational homogenization at finite strains. To eliminate rigid body motions, we introduce the concept of semi-Dirichlet boundary conditions. Semi-Dirichlet boundary conditions are non-homogeneous Dirichlet-type constraints that simultaneously satisfy the Neumann-type conditions. A key feature of the proposed methodology is its applicability for both strain-driven as well as stress-driven homogenization. The performance of the proposed scheme is demonstrated via a series of numerical examples.

Appendix: Implementation of TBC via the Lagrange multiplier method

This methodology is essentially based on solving the incremental minimization problem of homogenization

$$\begin{aligned} {}^\text {{M}}\!{W}({}^\text {{M}}\!{\varvec{F}})= & {} \inf _{\mathbf {d}} {}^\text {{M}}\!{W}(\mathbf {d}) \qquad \text {with} \nonumber \\ {}^\text {{M}}\!{W}(\mathbf {d})= & {} \displaystyle \frac{1}{\mathscr {V}_\text {0}}\int _{\mathcal {B}_\text {0}} W(\varvec{F};\varvec{X}) \, \text{ d }V\, , \end{aligned}$$

(15)

with $\mathbf {d}$ being the unknown global vector of deformations which minimizes the average incremental energy of the micro-structure for a given macro deformation gradient. In order to impose the constraint $\langle {\varvec{F}} \rangle - {}^\text {{M}}\!{\varvec{F}} \overset{!}{=} \varvec{0}$, the Lagrange multiplier method is used which yields the Lagrangian

$$\begin{aligned} {{L}}(\mathbf {d}, \varvec{\lambda }; {}^\text {{M}}\!{\varvec{F}}) = {}^\text {{M}}\!{W}(\mathbf {d}) - \varvec{\lambda }: [\langle {\varvec{F}} \rangle - {}^\text {{M}}\!{\varvec{F}}]\, , \end{aligned}$$

(16)

with $\varvec{\lambda }$ being the Lagrange multiplier. It can be readily verified that

$$\begin{aligned} \displaystyle \frac{\partial {{{L}}(\mathbf {d}, \varvec{\lambda }; {}^\text {{M}}\!{\varvec{F}})}}{\partial {}^\text {{M}}\!{\varvec{F}}} = {}^\text {{M}}\!{\varvec{P}} \qquad \text {with} \qquad {}^\text {{M}}\!{\varvec{P}} = \varvec{\lambda }\,. \end{aligned}$$

(17)

Next, the derivatives of the Lagrangian functional with respect to its variables are set to zero and the following system of equations is obtained.

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{\partial {{{L}}(\mathbf {d}, {}^\text {{M}}\!{\varvec{P}};{}^\text {{M}}\!{\varvec{F}})}}{\partial \mathbf {d}} = \varvec{0}~~ \Rightarrow ~~ \underbrace{\displaystyle \frac{\partial {{}^\text {{M}}\!{W}(\mathbf {d})}}{\partial {\mathbf {d}}}}_{\mathbf {R}_{\varvec{\text {int}}}} - \underbrace{{}^\text {{M}}\!{\varvec{P}}: \displaystyle \frac{\partial {\langle {\varvec{F}} \rangle }}{\partial {{\mathbf {d}}}}}_{\mathbf {R}_{\varvec{\text {ext}}}} = \varvec{0}\,, \\ \displaystyle \frac{\partial {{{L}}(\varvec{\mathbf {d}}, {}^\text {{M}}\!{\varvec{P}}; {}^\text {{M}}\!{\varvec{F}})}}{\partial {}^\text {{M}}\!{\varvec{P}}} = \varvec{0}~~ \Rightarrow ~~ {}^\text {{M}}\!{\varvec{F}}-\langle {\varvec{F}} \rangle = \varvec{0}\,. \end{array}\right. } \end{aligned}$$

(18)

Linearization of this system of equations yields

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbf {R}_{\varvec{int}}}(\mathbf {d}_{i+1}) - {\mathbf {R}_{\varvec{ext}}}({}^\text {{M}}\!{\varvec{P}}_{i+1}) = {\mathbf {R}_{\varvec{int}}}(\mathbf {d}_{i}) - {\mathbf {R}_{\varvec{ext}}}({}^\text {{M}}\!{\varvec{P}}_{i}) \\ \quad +\displaystyle \frac{\partial {\mathbf {R}_{\varvec{int}}}}{\partial {\mathbf {d}}}\mid _{i} \cdot \, \Delta \mathbf {d}_{i} - \displaystyle \frac{\partial \mathbf {R}_{\varvec{ext}}}{\partial {}^\text {{M}}\!{\varvec{P}}}\mid _{i} \, : \Delta {}^\text {{M}}\!{\varvec{P}}_{i} \overset{!}{=} \varvec{0}\,, \\ {}^\text {{M}}\!{\varvec{F}}-\langle {\varvec{F}} \rangle (\mathbf {d}_{i+1}) = {}^\text {{M}}\!{\varvec{F}}-\langle {\varvec{F}} \rangle (\mathbf {d}_{i}) -\displaystyle \frac{\partial \langle {\varvec{F}} \rangle }{\partial {\mathbf {d}}}\mid _{i} \cdot \, \Delta \mathbf {d}_{i} \overset{!}{=} \varvec{0}\,. \end{array}\right. } \end{aligned}$$

(19)

where $\mathbf {K} = \partial {\mathbf {R}_{\varvec{\text {int}}}}/\partial {\mathbf {d}}$ is the assembled tangent stiffness matrix. In order to represent the system of Eq. (19) in matrix format, all tensor products should be reformulates as matrix multiplications. To do so, we first derive the nodal equivalents of the global matrices $\partial \mathbf {R}_{\varvec{\text {ext}}}/\partial {}^\text {{M}}\!{\varvec{P}}$ and $\partial \langle {\varvec{F}} \rangle /\partial {\mathbf {d}}$ as

$$\begin{aligned}&\displaystyle \frac{\partial }{\partial {}^\text {{M}}\!{\varvec{P}}}({}^\text {{M}}\!{\varvec{P}}:\displaystyle \frac{\partial \langle {\varvec{F}} \rangle }{\partial {\varvec{\varphi }^{J}}}) = \varvec{I} \otimes \langle {\text{ Grad }{N^{J}}} \rangle \qquad \text {and} \nonumber \\&\displaystyle \frac{\partial \langle {\varvec{F}} \rangle }{\partial {\varvec{\varphi }^{J}}} = \varvec{I} \, \overline{\otimes }\,\langle {\text{ Grad }{N^{J}}} \rangle \, , \end{aligned}$$

(20)

respectively. The non-standard tensor product $\overline{\otimes }$ of a second-order tensor $\varvec{A}$ and a vector $\varvec{b}$ is the third-order tensor $\mathbbm {D} = \varvec{A} \, \overline{\otimes }\, \varvec{b}$ with components ${D}_{ijk} = {A}_{ik} {b}_{j}$. Next, the double contraction of the third-order tensor (20)$_{1}$ with $\Delta {}^\text {{M}}\!{\varvec{P}}$ and the dot product of the third-order tensor (20)$_{2}$ with nodal $\Delta \varphi ^{J}$ in matrix format are derived as

$$\begin{aligned}&\varvec{I} \otimes \langle {\text{ Grad }{N^{J}}} \rangle : \Delta {}^\text {{M}}\!{\varvec{P}} = \Delta {}^\text {{M}}\!{\varvec{P}} \cdot \langle {\text{ Grad }N^{J}} \rangle \nonumber \\&\quad \equiv \left[ \begin{array}{llll} \langle {\text{ Grad }N^{J}} \rangle _{x} &{} \langle {\text{ Grad }N^{J}} \rangle _{y} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \langle {\text{ Grad }N^{J}} \rangle _{x} &{}\quad \langle {\text{ Grad }N^{J}} \rangle _{y} \\ \end{array}\right] \nonumber \\&\qquad \qquad \begin{bmatrix} {}^\text {{M}}\!{P}_{xx}\\ {}^\text {{M}}\!{P}_{xy}\\ {}^\text {{M}}\!{P}_{yx}\\ {}^\text {{M}}\!{P}_{yy} \end{bmatrix}\, , \end{aligned}$$

(21)

$$\begin{aligned}&\varvec{I} \, \overline{\otimes }\, \langle {\text{ Grad }{N^{J}}} \rangle \cdot \Delta \varvec{\varphi }^{J} = \Delta \varvec{\varphi }^{J} \otimes \langle {\text{ Grad }N^{J}} \rangle \nonumber \\&\quad \equiv \left[ \begin{array}{ll} \langle {\text{ Grad }N^{J}} \rangle _{x} &{}\quad 0 \\ \langle {\text{ Grad }N^{J}} \rangle _{y} &{}\quad 0 \\ 0 &{}\quad \langle {\text{ Grad }N^{J}} \rangle _{x} \\ 0 &{}\quad \langle {\text{ Grad }N^{J}} \rangle _{y} \end{array}\right] \begin{bmatrix} \Delta \varphi ^{J}_{x} \\ \Delta \varphi ^{J}_{y} \end{bmatrix}, \end{aligned}$$

(22)

respectively. Hence, the final matrix format representation of the system of Eq. (19) the reads

where n denotes the total number of nodes and $\text{ Grad }N$ is represented as $\nabla N$ for the sake of space. This system turns out to be not singular and can be solved without further modifications.