A method of two-scale analysis with micro-macro decoupling scheme: application to hyperelastic composite materials

Abstract

The aim of this study is to propose a strategy for performing nonlinear two-scale analysis for composite materials with periodic microstructures (unit cells), based on the assumption that a functional form of the macroscopic constitutive equation is available. In order to solve the two-scale boundary value problems (BVP) derived within the framework of the homogenization theory, we employ a class of the micro-macro decoupling scheme, in which a series of numerical material tests (NMTs) is conducted on the unit cell model to obtain the data used for the identification of the material parameters in the assumed constitutive model. For the NMTs with arbitrary patterns of macro-scale loading, we propose an extended system of the governing equations for the micro-scale BVP, which is equipped with the external material points or, in the FEM, control nodes. Taking an anisotropic hyperelastic constitutive model for fiber-reinforced composites as an example of the assumed macroscopic material behavior, we introduce a tensor-based method of parameter identification with the ‘measured’ data in the NMTs. Once the macro-scale material behavior is successfully fitted with the identified parameters, the macro-scale analysis can be performed, and, as may be necessary, the macro-scale deformation history at any point in the macro-structure can be applied to the unit cell to evaluate the actual micro-scale response.

Appendices

Relationship between the macro- and micro-scale traction vectors

In this appendix, we derive relationship (11), which is critical to the numerical material testing with a general-purpose FEM software. For the sake of simplicity, a rectangular parallelepipe-shaped unit cell is considered, with its boundary surfaces assumed to be perpendicular to one of the axes of the micro-scale coordinate system so that only the $J$-th component of the side vector $\varvec{L}^{[J]}$ is a non-zero value $L_{J}~(J=1, 2, 3)$. In other words, the $J$-th basis vector $\varvec{E}^{[J]}$ coincides with the outward unit normal vector of the side vector $\varvec{L}^{[J]}$ of the unit cell of $L_{1} \times L_{2} \times L_{3}$. It is also assumed that the basis vectors $\varvec{E}^{[1]}, \varvec{E}^{[2]}, \varvec{E}^{[3]}$ are common to the micro- and macro-scale coordinate systems, $O\text{- }Y_1 Y_2 Y_3$ and $O\text{- }X_1 X_2 X_3$.

As indicated in Eq. (6), the macroscopic 1st Piola–Kirchhoff (PK) stress $\tilde{\varvec{P}}$ is the volumetric average of the corresponding microscopic stress ${\varvec{P}}$ over the domain of the initial configuration $\mathcal{Y _0}$ of a unit cell. This homogenization formula for the stress can be transformed to

$$\begin{aligned} \tilde{\varvec{P}}&=\frac{1}{|\mathcal Y _0|} \int _\mathcal{Y _0}{ \left[ \nabla _Y \cdot \left( {{\varvec{P}} \otimes {\varvec{Y}}} \right) - \left( {\nabla _Y \cdot {\varvec{P}}} \right) \otimes {\varvec{Y}} \right] } dY \nonumber \\&=\frac{1}{|\mathcal Y _0|}\int _\mathcal{Y _0} {\nabla _Y \cdot \left( {{\varvec{P}} \otimes {\varvec{Y}}} \right) } dY \end{aligned}$$

(37)

where we have used the micro-scale self-equilibrium equation (3) along with $\nabla _Y \cdot \left( {{\varvec{P}} \otimes {\varvec{Y}}} \right) = \left( {\nabla _Y \cdot {\varvec{P}}} \right) \otimes {\varvec{Y}} + {\varvec{P}}.$ The application of the Gauss divergence theorem yields the following relationship:

$$\begin{aligned} \tilde{\varvec{P}} =\frac{1}{|\mathcal Y _0|}\int _{\partial \mathcal Y _0} {\left( \varvec{P} \otimes \varvec{Y} \right) \cdot \varvec{N} } ds =\frac{1}{|\mathcal Y _0|}\int _{\partial \mathcal Y _0} {{\varvec{T}}^{(\varvec{N})} \otimes {\varvec{Y}}} ds \end{aligned}$$

(38)

where ${\varvec{T}}^{(\varvec{N})}:= \varvec{P} \cdot \varvec{N}$ is the microscopic Piola traction vector, wtth $\varvec{N}$ being an arbitrary outward unit normal vector at the boundary surface $\partial \mathcal Y _0$. Note here that, due to the assumption on the geometry of the unit cell, the outward unit normal vector $\varvec{N}$ at the unit cell boundairs $\partial \mathcal Y _0^{[J]}$ coincides with $\varvec{E}^{[J]}$.

Since the same basis vectors are used for the micro- and macro-scale coordinate systems, the macro-scale Piola traction vector on the macroscopic surface, which is parallel to the boundary surface $\partial \mathcal Y _0^{[J]}$ with the outward unit normal vector $\varvec{E}^{[J]}$ can be can be written as the area average of the corresponding micro-scale Piola traction vector on $\partial \mathcal Y _0^{[J]}$. For example, the macro-scale Piola traction vector $\tilde{\varvec{T}}^{({\varvec{E}}^{[1]} )}$ associated with $\partial \mathcal Y _0^{[1]}$ and $\varvec{E}^{[1]}$, whose compnents are $\{1, ~0,~0\}^\mathrm{T}$, can be expressed as follows:

$$\begin{aligned} \tilde{\varvec{T}}^{({\varvec{E}}^{[1]} )}&= \tilde{\varvec{P}}\cdot \varvec{E}^{[1]} \nonumber \\&= \left( {\frac{1}{{\left| \mathcal Y _0 \right| }}\int _{\partial \mathcal Y _0} {{\varvec{T}}^{({\varvec{N}})} \otimes {\varvec{Y}}dS} } \right) \cdot {\varvec{E}}^{[1]} = \frac{1}{{\left| \mathcal Y _0 \right| }}\int _{\partial \mathcal Y _0} {{\varvec{T}}^{({\varvec{N}})} Y_1 dS} \nonumber \\&= \frac{1}{{\left| \mathcal Y _0 \right| }}\left( {\int _{\partial \mathcal Y _0^{[1]} } {{\varvec{T}}^{({\varvec{E}}^{[1]} )} Y_1 dS} + \int _{\partial \mathcal Y _0^{[{ - 1}]} } {{\varvec{T}}^{({\varvec{E}}^{[-1]} )} Y_1 S} } \right. \nonumber \\&\quad \quad + \int _{\partial \mathcal Y _0^{[2]} } {{\varvec{T}}^{({\varvec{E}}^{[2]} )} Y_1 dS} + \int _{\partial \mathcal Y _0^{[- 2]} } {{\varvec{T}}^{({\varvec{E}}^{[ - 2]} )} Y_1 dS} \nonumber \\&\quad \quad \left. + \int _{\partial \mathcal Y _0^{[3]} } {{\varvec{T}}^{({\varvec{E}}^{[3]} )} Y_1 dS} + \int _{\partial \mathcal Y _0^{[ - 3]} } {{\varvec{T}}^{({\varvec{E}}^{[ - 3]} )} Y_1 dS} \right) \end{aligned}$$

(39)

where Eq. (38) has been utilized. Then, due to the anti-periodicity of the Piola traction vector (4), we apply ${\varvec{T}}^{({\varvec{E}}^{[-J]} )}=-{\varvec{T}}^{({\varvec{E}}^{[J]} )}$ along with $\left. Y_1 \right| _{\partial \mathcal Y _0^{[J]}} = \left. Y_1 \right| _{\partial \mathcal Y _0^{[-J]} } ~ (J \ne 1)$ to (39) to have

$$\begin{aligned} \tilde{\varvec{T}}^{({\varvec{E}}^{[1]} )} = \frac{\left. Y_1 \right| _{\partial \mathcal Y _0^{[1]} } - \left. Y_1 \right| _{\partial \mathcal Y _0^{[-1]} }}{{{\left| \mathcal Y _0 \right| }}} \int _{\partial \mathcal Y _0^{[1]} } {{\varvec{T}}^{({\varvec{E}}^{[1]} )} dS}. \end{aligned}$$

(40)

Using the equivalent expressions $L_1\!=\!{\left. Y_1 \right| _{\partial \mathcal Y _0^{[1]} } \!-\! \left. Y_1 \right| _{\partial \mathcal Y _0^{[-1]} }}, \left| \mathcal Y _0 \right| \!=\! L_1 L_2 L_3$ and $\left| {\partial \mathcal Y _0^{[1]} } \right| \!=\! L_2 L_3$, we arrive at the following relationship:

$$\begin{aligned} \tilde{\varvec{T}}^{({\varvec{E}}^{[1]} )} =\tilde{\varvec{P}} \cdot \varvec{E}^{[1]} = \frac{1}{|\partial \mathcal Y _0^{[1]}|} \int _{\partial \mathcal Y _0^{[1]}} { \varvec{T}^{({\varvec{E}}^{[1]} )} } ds \end{aligned}$$

(41)

Since the same expression can be obtained for the traction vectors on the other two boundary surfaces, Eq. (11) has been proven.

Anisotropic hyperelastic model with fabric vectors

We consider one class of anisotropic hyperelastic constitutive models, whose functional form is expressed by means of the invariants of the right Cauchy-Green (CG) deformation tensor $\varvec{C}$ along with the so-called “fabric” vector indicating the direction of reinforcements such as fibers. Although the application for the macroscopic BVP is assumed, we do not distinguish the micro- and macro-scale variables below.

The elastic energy functional of the employed anisotropic hyperelastic model is given as

$$\begin{aligned} W = W_{\mathrm{vol}} \left( {J} \right) +W_{\mathrm{iso}} \left( \bar{\varvec{C}}; {\varvec{A}}, {\varvec{B}} \right) \end{aligned}$$

(42)

where $\varvec{A}$ and $\varvec{B}$ are the two distinct directions of fiber alignment in the reference configuration and can be referred to as the fibric vectors. Here, $W_{\mathrm{vol}}(J)$ is the energy function of the Jacobian $J:=\det \varvec{F}$ associated with the volumetric deformation. $W_{\mathrm{iso}} \left( \bar{\varvec{C}}; {\varvec{A}}, {\varvec{B}} \right)$ is the isochoric component of the energy functional by means of the deviatoric part of the right CG deformation tensor, which is defined as

$$\begin{aligned} \bar{\varvec{C}} = \bar{\varvec{F}}^\mathrm{{T}} \bar{\varvec{F}} = J^{ - 2/3} {\varvec{F}}^\mathrm{{T}} {\varvec{F}} = I_3^{ - 1/3} {\varvec{C}} \end{aligned}$$

(43)

with $\bar{\varvec{F}} = J^{ - 1/3} {\varvec{F}}$ and $I_3 =\det \varvec{C}= J^2$.

The 2nd Piola–Kirchhoff (PK) stress can be obtained by differentiating the energy function (42) with respect to the right CG deformation tensor $\varvec{C}$ as follows:

$$\begin{aligned} {\varvec{S}} = 2\frac{{\partial W}}{{\partial {\varvec{C}}}} = 2\frac{{\partial W_{\mathrm{{vol}}} }}{{\partial {\varvec{C}}}} + 2\frac{{\partial W_{\mathrm{{iso}}} }}{{\partial {\varvec{C}}}} = {\varvec{S}}_{\mathrm{{vol}}} + {\varvec{S}}_{\mathrm{{iso}}} \end{aligned}$$

(44)

Here, we have defined the volumetric and isochoric components of ${\varvec{S}}, {\varvec{S}}_{\mathrm{{vol}}}$ and ${\varvec{S}}_{\mathrm{{iso}}}$, are respectively expressed as

$$\begin{aligned} {\varvec{S}}_{\mathrm{{vol}}}&= 2\frac{{\partial W_{\mathrm{{vol}}} }}{{\partial {\varvec{C}}}} = J\frac{{\partial W_{\mathrm{{vol}}} }}{{\partial J}}{\varvec{C}}^{ - 1}\end{aligned}$$

(45)

$$\begin{aligned} {\varvec{S}}_{\mathrm{{iso}}}&= I_3^{ - 1/3} \mathbb Q :{\varvec{\bar{S}}} \end{aligned}$$

(46)

where

$$\begin{aligned} \mathbb Q&= {\varvec{I}} - \frac{1}{3}{\varvec{C}}^{ - 1} \otimes {\varvec{C}}\end{aligned}$$

(47)

$$\begin{aligned} \bar{\varvec{S}}&= 2\frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{\varvec{C}}}} \nonumber \\&= \bar{\gamma }_1 {\varvec{1}} + \bar{\gamma }_2 \bar{\varvec{C}} + \bar{\gamma }_4 \left( {{\varvec{A}} \otimes {\varvec{A}}} \right) + \bar{\gamma }_5 \left( {{\varvec{A}} \otimes \bar{\varvec{C}} \varvec{A} + \bar{\varvec{CA}} \otimes {\varvec{A}}} \right) \nonumber \\&\quad + \bar{\gamma }_6 \left( {{\varvec{B}} \otimes {\varvec{B}}} \right) + \bar{\gamma }_7 \left( {{\varvec{B}} \otimes \bar{\varvec{CB}} + \bar{\varvec{C}}\bar{\varvec{B}} \otimes {\varvec{B}}} \right) \nonumber \\&\quad + \bar{\gamma }_8 \left( {{\varvec{A}} \cdot {\varvec{B}}} \right) \left( {{\varvec{A}} \otimes {\varvec{B}}} \right) \end{aligned}$$

(48)

along with

$$\begin{aligned} \left. \begin{array}{l} \bar{\gamma }_1 = 2\left( {{\displaystyle \frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{I}_1 }}} + \bar{I}_1 {\displaystyle \frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{I}_2 }}}} \right) , \\ \bar{\gamma }_2 = - 2{\displaystyle \frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{I}_2 }}}, \quad \bar{\gamma }_4 = 2{\displaystyle \frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{I}_4 }}}, \quad \bar{\gamma }_5 = 2{\displaystyle \frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{I}_5 }}}, \\ \bar{\gamma }_6 = 2{\displaystyle \frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{I}_6 }}}, \quad \ \ \ \bar{\gamma }_7 = 2{\displaystyle \frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{I}_7 }}}, \quad \bar{\gamma }_8 = 2{\displaystyle \frac{{\partial W_{\mathrm{{iso}}} }}{{\partial \bar{I}_8 }}} \\ \end{array} \right\} \end{aligned}$$

(49)

Here, $\mathbf{1}$ and $\varvec{I}$ are the 2nd-order identity tensor and the 4th-order symmetric identify tensor, respectively. Information about the derivation of these formulae is found in [40].

One of the examples of this class is typified by Kaliske et al. [38, 39], which seems to be reasonable within the present framework of two-scale coupling analysis with the micro-macro decoupling scheme. We employ their model in this study and provide its concrete functional form below. The volumetric and isochoric parts of the energy functional in [39] by are respectively given as

$$\begin{aligned} W_{\mathrm{{vol}}}&= \frac{1}{D}\left( {J - 1} \right) ^2 \end{aligned}$$

(50)

$$\begin{aligned} W_{\mathrm{iso}}&= W_{\mathrm{{iso}}} (\bar{I}_1 ,\bar{I}_2 ,\bar{I}_4 ,\bar{I}_5 ,\bar{I}_6 ,\bar{I}_7 ,\bar{I}_8; a_i, b_j, c_k ,d_l, e_m, f_n, g_o; \varvec{A},~\varvec{B}) \nonumber \\&= \sum \limits _{i = 1}^3 {a_i \left( {\bar{I}_1 - 3} \right) ^i } + \sum \limits _{j = 1}^3 {b_j \left( {\bar{I}_2 - 3} \right) ^j } \nonumber \\&\quad + \sum \limits _{k = 2}^6 {c_k \left( {\bar{I}_4 - 1} \right) ^k } + \sum \limits _{l = 2}^6 {d_l \left( {\bar{I}_5 - 1} \right) ^l } + \sum \limits _{m = 2}^6 {e_m \left( {\bar{I}_6 - 1} \right) ^m } \nonumber \\&\quad + \sum \limits _{n = 2}^6 {f_n \left( {\bar{I}_7 - 1} \right) ^n } + \sum \limits _{o = 2}^6 {g_o \left( {\bar{I}_8 - \varsigma } \right) ^o } \end{aligned}$$

(51)

Here, $D, a_i, b_j, c_k, d_l, e_m, f_n, g_o$ are scalar-valued material parameters, and $\bar{I}_1, \bar{I}_2, \bar{I}_4, \bar{I}_5, \bar{I}_6, \bar{I}_7, \bar{I}_8$ are the invariant of $\bar{\varvec{C}}$ defined as follows:

$$\begin{aligned} \left. \begin{array}{ll} \bar{I}_1 = \mathrm{tr }\bar{\varvec{C}}, &{} \bar{I}_2 = {\displaystyle \frac{1}{2}} \left( {\mathrm{{tr}}^2 \bar{\varvec{C}} - \mathrm{tr }\bar{\varvec{C}}^2 } \right) , \\ \bar{I}_4 = {\varvec{A}} \cdot \bar{\varvec{C}}\varvec{A}, &{} \bar{I}_5 = {\varvec{A}} \cdot \bar{\varvec{C}}^2 {\varvec{A}}, \quad \bar{I}_6 = {\varvec{B}} \cdot \bar{\varvec{C}}\varvec{B}, \\ \bar{I}_7 = {\varvec{B}} \cdot \bar{\varvec{C}}^2 {\varvec{B}}, &{} \bar{I}_8 = \left( {{\varvec{A}} \cdot {\varvec{B}}} \right) {\varvec{A}} \cdot \bar{\varvec{C}}\varvec{B} \\ \end{array} \right\} \end{aligned}$$

(52)

where $\varsigma = \left( {{\varvec{A}} \cdot {\varvec{B}}} \right) ^2$.

This model is capable of representing a certain class of orthotropic hyperelastic behavior with the fibric vectors $\varvec{A}$ and $\varvec{B}$ as input data. If, for example, we assume $\varvec{B}=\mathbf{0}$, then the resulting energy functional can be used for a class of transversely isotropic materials introduced in [38].