HOMOGENIZATION#
Description#
This object class provides the way of introducing linear and nonlinear mechanical constitutive equations into the standard phase field approach. In the region where both phases coexist, the local behaviour in both phases is interpolated in order to replace an heterogeneous medium by an equivalent homogeneous one [MS2].
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DESCRIPTION |
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uniform strain field among the phases |
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interpolation scheme |
VoigtIt is a Voigt model, which is also referred to as the uniform strain model. Its basic assumptions are that the strain field is uniform among the phases. One distinct set of constitutive equations is attributed to each individual phase k at any material point. Each phase at a material point then possesses its own stress/strain tensor \((\ten \sigma_1,\ten \varepsilon_1)\) and \((\ten \sigma_2,\ten \varepsilon_2)\). The overall strain and stress quantities \(\ten\Sigma,\ten E\) at this material point must then be averaged or interpolated from the values attributed to each phase, using the well-known results of homogenization theory. Following a naive representation depicted in the figure below, each material point, i.e. \(\cal V\), within a diffuse interface can be seen as a local mixture of the two abutting phases \(1\) and \(2\) with proportions fixing \({\cal V}_1\) and \({\cal V}_2\) given by complementary functions of \(\phi\). It must be emphasized that this representation involves the presence of fields \(\Psi_1\) and \(\Psi_2\) in phases \(2\) and \(1\) respectively, which has no incidence on the bulk of those phases.
Fig. 87 Schematic illustration of the underlying material representative volume element \({\cal V} = \left\{{\cal V}_1 \cup {\cal V}_2,\,\, {\cal V}_1 \cap {\cal V}_2 = \emptyset \right\}\) at each material point of a diffuse interface.#
The local energy stored in the effective homogeneous elastic material is expressed in terms of the average value of the local elastic energy with respect to both phases weighted by their volume fractions:
(625)#\[f_{e}(\phi, c, \ten{\varepsilon}) = \frac{1}{2}\,(\ten E - \ten \varepsilon_1):\tenf C:(\ten E - \ten \varepsilon_1) = h_u(\phi)\,f_{e1} + (1-h_u(\phi))f_{e2}\]where the elastic energy densities of \(1\) and \(2\) phases can be expressed as:
(626)#\[\begin{split}\left\{ \begin{array}{l} f_{e1} = \displaystyle{\frac{1}{2}\,(\ten E - \ten \varepsilon^\star_1):\tenf C_1 : (\ten E - \ten \varepsilon^\star_1)}\\\\ f_{e2} = \displaystyle{\frac{1}{2}\,(\ten E - \ten \varepsilon^\star_2) :\tenf C_2 : (\ten E - \ten \varepsilon^\star_2)} \end{array} \right.\end{split}\]Using Voigt’s model, we assume a uniform total strain at any point in the diffuse interface between elastoplastically inhomogeneous phases. The effective stress is expressed in terms of the local stress average with respect to both phases weighted by the volume fractions:
(627)#\[ \ten\Sigma = h_u(\phi)\, \ten \sigma_1 + (1 - h_u(\phi))\,\ten \sigma_2 = \tenf C : (\ten E - \ten \varepsilon_1)\qquad {\rm and} \qquad \ten E = \ten \varepsilon_1 = \ten \varepsilon_2\]where the effective elasticity tensor \(\tenf C\) is obtained from the mixture rule of the elasticity matrix for both phases and the effective Eigenstrain \(\ten{E}^\star\) vary continuously between their respective values in the bulk phases as follows:
(628)#\[\begin{split}\begin{array}{l} \tenf C = h_u(\phi)\, \tenf C_1 + (1 - h_u(\phi)) \tenf C_2\\\\ \ten \varepsilon_1=\tenf C^{-1}:(h_u(\phi)\,\tenf C_1:\ten \varepsilon^\star_1(c)+(1-h_u(\phi))\tenf C_2:\ten \varepsilon^\star_2(c)) \end{array}\end{split}\]KhachaturyanIt is an interpolation scheme, where the material behaviour is described by a unified set of constitutive equations that explicitly depend on the concentration or the phase variable. Each material parameter is usually interpolated between the limit values known for each phase. Linear mixture interpolations are adopted respectively for eigenstrain and elasticity moduli tensor:
(629)#\[\ten E^\star = h_u(\phi)\, \ten \varepsilon^\star_1 + (1 - h_u(\phi))\, \ten \varepsilon^\star_2\quad,\quad \tenf C = h_u(\phi)\,\tenf C_1 + (1-h_u(\phi))\,\tenf C_2\]Hooke’s law relates the strain tensor to the stress tensor by the following expression:
(630)#\[\begin{split}\begin{aligned} \ten\Sigma & = \tenf C : (\ten E - \ten E^\star)\nonumber\\ & = (h_u(\phi)\,\tenf C_1(c) + (1-h_u(\phi))\,\tenf C_2(c)) : (\ten E - h_u(\phi)\, \ten \varepsilon^\star_1 - (1 - h_u(\phi))\, \ten \varepsilon^\star_2) \end{aligned}\end{split}\]Contrary to the previous Voigt homogenization scheme, the elastic energy of the effective homogeneous material is no longer the average of energy densities of both phases. It is indeed not possible to distinguish an explicit form for the elastic energy densities in each phase. The elastic energy is then postulated as:
(631)#\[f_{e}(\phi, c, \ten{\varepsilon}) = (\ten E - h_u(\phi)\, \ten \varepsilon^\star_1(c) - (1 - h_u(\phi))\, \ten \varepsilon^\star_2(c)) : \tenf C : (\ten E - h_u(\phi)\, \ten \varepsilon^\star_1(c) - (1 - h_u(\phi))\, \ten \varepsilon^\star_2(c))\]