The homogenization of heterogeneous materials with the use of multi-scale technique is a common method today [12, 22, 33, 35–37]. The main idea of this method, also called global-local analysis, bases on the direct calculation of the local macroscopic constitutive response from the corresponding microstructure boundary value problem. The foundation of the computational homogenization method is consistent with the other concepts of homogenization techniques. It can be described as a classical four-steps homogenization scheme proposed by Suquet [33]. In the first step, a microstructural RVE is defined. Its constitutive behaviour is assumed to be known. In the second step, microscopic boundary conditions from the macroscopic input variables are formulated and applied to the RVE (so-called macro-to-micro transition). The third step is the calculation of the homogenized macroscopic characteristics based on the analysis of the microstructural RVE (so called micro-to-macro transition). In the last step, the relation between the macroscopic characteristics and inputs is calculated.
The computational homogenization technique has a few crucial advantages. First of all, it does not require any explicitly formulated constitutive equations on a macro-scale level. The macroscopic constitutive behaviour is directly obtained from the corresponding micro-scale boundary value problem analysis [35]. The second advantage is the possibility of taking into consideration the evolution of the microstructure on the macroscopic level analysis and coupling this evolution with results of the macro-scale analysis. Last but not least, there is the advantage that this method can be still effectively used even if the requirement of RVE is not exactly fulfilled for all micro-scale levels analyses.
Basics
The basic scheme of the computational homogenization procedure is shown in Fig. 2. The macroscopic deformation gradient tensor \({{\varvec{F}}}_{M}\) is calculated for each integration point of the macro-scale finite element model (called further macro-model). Next, it is used to define the boundary conditions of the micro-scale model (called further micro-model) which corresponds to the particular integration points. This micro-model defines the microstructure boundary value problem and can be understood in the sense of the RVE. The macroscopic stress tensor \({{\varvec{P}}}_{M}\) is calculated based on the results from the micro-model as averaged stress field over the volume. The stress–deformation relationship at the macro-scale level (macro-model element) is defined as a local macroscopic stiffness tangent matrix. This equivalent matrix is derived directly from the microstructural stiffness. We assume that micro-model is linear, the strains are small and the bone material is elastic. Taking into account the above assumptions, the tangent stiffness matrix can be easily calculated.
In the first step of the computational homogenization procedure a microstructural RVE has to be defined. In the approach the micro-model describes physical and geometrical properties of the microstructure and simultaneously defines RVE. The schematic description of RVE is shown in Fig. 3. The dimensions of RVE should be large enough to reproduce the microstructure heterogeneity in a proper way, on the other hand, they should be small enough to allow for reasonable times of calculations. Because of the non-continuous character of cancellous bones which contain a mixture of trabeculae and voids, it is impossible to define the appropriate corresponding RVE for each element of macro-model. The three cases of micro-model configuration can be distinguished: completely filled with bone material, completely empty and partially filled. The first two cases can be clearly treated as RVE while there is no certainty that the requirements of appropriate RVEs are met for the last case. However, we can still estimate the “homogenized” anisotropic behavior of the material on macro-scale level. Generally, the problem on the micro-scale level can be considered as a standard problem of the continuum solid mechanics. The field of deformation of the micro-model in a point described with the initial position vector \(\vec {X}\) and the actual position vector is \(\vec {x}\) defined by the microstructural deformation gradient tensor:
$$\begin{aligned} F_m =\nabla _{\mathrm{0m}} \vec {x}, \end{aligned}$$
(1)
where the gradient operator \(\nabla _{\mathrm{0m}}\) is taken with respect to the reference microstructural configuration. The equilibrium conditions have to be fulfilled for the RVE:
$$\begin{aligned} \nabla _{\mathrm{0m}} P_m =\vec {0}, \end{aligned}$$
(2)
where the stress tensor \({{\varvec{P}}}_{m}\) is defined as the first Piola–Kirchhoff tensor for the reference domain \(V_{0}\).
In the second step, microscopic boundary conditions from the macroscopic input variables are formulated and applied to the RVE. This macro-to-micro transition is done by enforcement of the macroscopic gradient deformation tensor \({{\varvec{F}}}_{M}\) on the microstructural RVE. This averaging procedure can be used in a few different ways under different assumptions and criteria. There is a general requirement that they should fulfil the assumption of the so-called averaging theorems [18]. The averaging theorems assume that the coupling between micro- and macro-levels is done in an energetically consistent way. In the presented approach the macroscopic input variables are transferred to the microstructural RVE via the boundary conditions (Taylor/Voigt assumption). However, these assumptions are not fulfilled accurately if the appropriate RVE doesn’t exist. The displacement boundary conditions are prescribed in such a way that the position vector \(\vec {x}_{m}\) of a point on the deformed micro-model boundary \(\varGamma \) is defined as:
$$\begin{aligned} \vec {x}_{m} =F_{M} \vec {X}_{m}, \end{aligned}$$
(3)
where \(\vec {X}_{m}\) is the position vector of a point on undeformed micro-model boundary \(\varGamma _{0}\). This condition introduces a mapping of the RVE boundaries as a linear relationship.
The first averaging relation for the kinematic quantities is defined by the macroscopic gradient deformation tensor \({{\varvec{F}}}_{M}\) which is the volume average of the microstructural gradient deformation tensor \({{\varvec{F}}}_{m}\):
$$\begin{aligned} F_{M} =\frac{1}{V_{0}}\int \limits _{V_{0}} {F_{m}} dV_{0} =\frac{1}{V_{0}}\int \limits _{\varGamma _{0}} {\vec {x}} \vec {N}d\varGamma _{0} \,. \end{aligned}$$
(4)
Next, the averaging of stresses for the first Piola–Kirchhoff stress tensor can be defined in a similar way:
$$\begin{aligned} P_{M} =\frac{1}{V_{0}}\int \limits _{V_{0}} {P_{m}} dV_{0}\,. \end{aligned}$$
(5)
the macroscopic first Piola–Kirchhoff stress tensor \({{\varvec{P}}}_{M}\) can be written in the microstructural quantities with the use of the following expression:
$$\begin{aligned} P_{m} =\left( {\nabla _{\mathrm{0m}} \cdot P_{m}} \right) \vec {X}+P_{m} \cdot \left( {\nabla _{\mathrm{0m}} \vec {X}} \right) , \end{aligned}$$
(6)
If we take into account equilibrium condition (Eq. 2) and because of:
$$\begin{aligned} \nabla _{\mathrm{0m}} \vec {X}=I, \end{aligned}$$
(7)
the stress tensor \({{\varvec{P}}}_{M}\) (Eq. 6) can be transformed into:
$$\begin{aligned} P_{m} =\nabla _{\mathrm{0m}} \cdot \left( {P_{m} \vec {X}}\right) \,. \end{aligned}$$
(8)
Now, we can substitute (Eq. 8) into (Eq. 5):
$$\begin{aligned} P_{M} =\frac{1}{V_{0}}\int \limits _{V_{0}} {\nabla _{\mathrm{0m}}} \cdot \left( {P_{m} \vec {X}} \right) dV_{0} \end{aligned}$$
(9)
and obtain the following expression after using the Gauss–Ostrogradsky theorem and the first Piola–Kirchhoff stress vector definition \(\vec {p}=\vec {N}\cdot P_{m}\) :
$$\begin{aligned} P_{M} =\frac{1}{V_{0}}\int \limits _{\varGamma _{0}} {\vec {N}} \cdot P_{m} \vec {X}\,d\varGamma _{0} =\frac{1}{V_{0}}\int \limits _{\varGamma _{0}} {\vec {p}} \vec {X}d\,\varGamma _{0}\,. \end{aligned}$$
(10)
The obtained Eq. (10) can be directly used to calculate averaging stresses on macro-scale level based on results of microstructural analysis.
Finally, the internal work of the microstructure is averaged as well. The average over the volume energy in the micro-model and macro-model should be the same:
$$\begin{aligned} \delta {W}_{0M} =\delta W_{0m} . \end{aligned}$$
(11)
This assumption is known as the Hill–Mandel condition [18].
The third step of computational homogenization procedure is micro-to-macro transition. In this step the homogenized macroscopic characteristics are calculated based on the micro-models analysis. It has been assumed that the micro-model is linear – the strains and deformations are small and the bone material is elastic. The stress–deformation relationship at the macro-scale level (macro-model element) is defined as a local equivalent stiffness matrix. This matrix is derived directly from the microstructural stiffness matrix. For each micro-model all the “internal” degrees of freedom \(\vec {u}_{I}\) are eliminated except those which correspond to the degrees of freedom of nodes of the macro-model elements. These nodes are called transition nodes \(\vec {u}_{T}\). Thus, on the macro-level the micro-models are represented by the stiffness (equivalent stiffness matrix) of these retained degrees of freedom. This equivalent stiffness matrix can be derived based on virtual work equation and Hill–Mandel condition (Eq. 11). The virtual work contribution of the selected element of the macro-model can be expressed as a virtual work of micro-model which corresponds to them. It can be written as follow:
$$\begin{aligned} \delta W_{0M}^{Elem}&= \left[ {\delta \vec {u}_T \delta \vec {u}_I} \right] \left( {\left[ { \begin{array}{c} \vec {P}_T \\ \vec {P}_I \\ \end{array}} \right] } \right. \nonumber \\&\left. -\left[ {{ \begin{array}{cc} {K_{mTT}} &{} \quad {K_{mTI}} \\ {K_{mIT}} &{} \quad {K_{mII}} \\ \end{array}}}\right] \left[ {\begin{array}{c} \vec {u}_T \\ \vec {u}_I \\ \end{array}}\right] \right) , \end{aligned}$$
(12)
where \(\vec {P}_{T}\) and \(\vec {P}_{I}\) are nodal forces which are applied to the micro-model based on the macro-scale response. They can be understood as macroscopic input variables. The stiffness matrix of the microstructure is defined as:
$$\begin{aligned} K_m =\left[ {{ \begin{array}{cc} {K_{mTT}} &{} \quad {K_{mTI}} \\ {K_{mIT}} &{} \quad {K_{mII}} \\ \end{array}}} \right] \, . \end{aligned}$$
(13)
Because the micro-model has to be in internal equilibrium and the internal degrees of freedom are eliminated on macro-level, the equation conjugated to \(\delta \vec {u}_{I}\) in the contribution of the virtual work given above (Eq. 12) has to fulfill:
$$\begin{aligned} \vec {P}_I -K_{mIT} \vec {u}_T -K_{mII} \vec {u}_I =0 \end{aligned}$$
(14)
These equations can be rewritten to express \(\vec {u}_{I}\) in the following format:
$$\begin{aligned} \vec {u}_{I} =K_{mII^{-1}} \left( {\vec {P}_I -K_{mIT} \vec {u}_T} \right) \end{aligned}$$
(15)
Now, the virtual work of micro-model can be written with the use of the transition constituents and (23) as follows:
$$\begin{aligned}&\vec {u}_T \left( {\left( {\vec {P}_T \!-\!K_{mIT} K_{II}^{-1} \vec {P}_I} \right) \!-\!\left( \! {K_{mTT} \!-\!K_{mTI} K_{mII}^{-1} K_{mIT}} \right) \vec {u}_T} \!\right) \nonumber \\&\delta W_{0M}^{Elem} =\delta , \end{aligned}$$
(16)
Finally, the equivalent stiffness matrix can be defined, based on the above equation, in the form:
$$\begin{aligned} K_{M} =K_{mTT} -K_{mTI} K_{mII}^{-1} K_{mIT}\,. \end{aligned}$$
(17)
In the last step the relation between the macroscopic characteristics and inputs is calculated. It has been done by using the finite element macro-model. Both, macro- and micro-models are discussed in the next section.
Models
In the presented approach the finite element method is used to realize computational homogenization procedure on macro- and micro-levels. Both macro- and micro-models are created in Abaqus/Standard code. In this section the macro- and micro-models are described.
In the discussed method there are no restrictions for the macro-model geometry. The discrete geometry can be general and complex as it is needed. However, for each element of macro-model an individual micro-model has to be assigned (Fig. 4). Therefore, the one to one relation causes that the size of macro-model element determines the size of the micro-model in a direct way. Each node of macro-model is by definition the transition node (Fig. 4).
Each micro-model represents the part of the microstructure of cancellous bone (Fig. 4). It simultaneously defines microstructure boundary value problem. From the viewpoint of homogenization procedure, the crucial parts of the micro-model are its boundaries. The micro-model boundaries are the same for all micro-models – hexagonal prisms (Fig. 4). The size and shape of these prisms are described directly by the size and shape of the associated macro-model element. The six faces of prisms are defined explicitly in the micro-model as a membrane-like finite element (MFE) with zero stiffness. The linear Lagrange polynomial functions are used to interpolate the field of displacement in these elements [41]. The six MFEs create the virtual box, the eight vertices of which coincidence with transition nodes (Fig. 4). The macro-to-micro transition is done with the use of the fully prescribed displacement boundary conditions (Eq. 3). The displacements of all nodes on the boundary are defined as:
$$\begin{aligned} \vec {u}_{B} =\left( {F_M -I} \right) \cdot \vec {X}_B, \quad B=1,\ldots , N_B, \end{aligned}$$
(18)
where \(N_{B}\) is the number of boundary nodes. These boundary conditions (Eq. 18) are applied to the model by static condensation using the additional weight functions which describe spatial location of boundary nodes relative to the transition nodes. These functions couple each boundary node on a particular face of a virtual box with three closest vertices of this face [6]. The degrees of freedom of the coupled nodes are eliminated from the micro-model system of equations.