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Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

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Abstract

We rigorously determine the scale-independent short range elastic parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure. This is done using both classical periodic homogenization and a new procedure involving the concept of apparent material stiffness of a unit-cell under affine Dirichlet boundary conditions and Neumann’s principle on the overall representation of anisotropy. We explain our idea of “maximal” stiffness of the unit-cell and use state of the art first order numerical homogenization methods to obtain the needed parameters for a given tetragonal unit-cell. These results are used in the accompanying paper (d’Agostino et al. in J. Elast. 2019. Accepted in this volume) to describe the wave propagation including band-gaps in the same tetragonal metamaterial.

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Notes

  1. As is well-known, under affine loading, the response of a large periodic structure is periodic up to a vanishing boundary layer.

  2. For the micromorphic model, postulate d) implies a direct interpretation of what the new degrees of freedom (the non symmetric micro-distortion \(P\in \mathbb{R}^{3\times 3}\)) is. While we do not discard such a direct micro-macro relation, we rather believe that any simple relation will fall short of the truth for the relaxed micromorphic model.

  3. For the presentation we have chosen throughout the simplest representation of the curvature energy – a one constant isotropic format.

  4. Despite the name micromorphic model.

  5. The situation is different when one considers homogenization towards a second gradient continuum (or micromorphic approximations thereof) where there is no independent kinematical field. More precisely, the case \(\widehat{\mathbb{C}}_{\textrm{e}}\gg 1\), \(L_{c}\ll 1\) would be consistent with determining \(P\) as some average of the micro-displacements over a unit-cell.

  6. Equation (4)1 and (4)2 together imply \(\textrm{Div} ( \mathbb{C}_{\textrm{micro}}\,\textrm{sym}\,P ) =f\). Constraining \(\textrm{sym}\,P=\textrm{sym}\,\nabla u\) gives \(\textrm{Div} ( \mathbb{C}_{\textrm{micro}}\,\textrm{sym}\,\nabla u ) =f\).

  7. The curvature expression in the Eringen-Mindlin-model or gradient elasticity model would typically include a sixth-order tensor [5], in contrast to the relaxed micromorphic model, which only needs a fourth-order tensor.

  8. Equation (7)1 and (7)2 together imply that \(\textrm{Div} [ \mathbb{C}_{\textrm{micro}}\,\textrm{sym}\,P-\mu \,L_{c}^{2}\Delta P ] =f\). Here \(\mathbb{C}_{\textrm{micro}}\) is invariably coupled to the characteristic length \(L_{c}\). Constraining \(P=\nabla u\) gives \(\textrm{Div} [ \mathbb{C}_{\textrm{micro}}\,\textrm{sym}\,\nabla u-\mu \,L_{c}^{2}\Delta \nabla u ] =f\). This is the fourth-order equilibrium equation of the second gradient formulation (8).

  9. The relaxed micromorphic model cannot be obtained as penalty formulation of gradient elasticity and in a 1-D setting it reduces to linear elasticity with stiffness \(\mathbb{C}_{\textrm{macro}}\).

  10. Letting \(L_{c}\rightarrow 0\) in (7) leads to the algebraic side condition \(\widehat{\mathbb{C}}_{\textrm{e}}\,(\nabla u-P)=\mathbb{C}_{\textrm{micro}}\,\textrm{sym}\,P\). Due to the more general format of \(\widehat{\mathbb{C}}_{\textrm{e}}\) as compared to \(\mathbb{C}_{\textrm{e}}\) and \(\mathbb{C}_{\textrm{c}}\) in (13), it is not possible to analytically solve for \(\textrm{sym}\,P\) and no transparent formula connecting \(\widehat{\mathbb{C}}_{\textrm{e}}\) and \(\mathbb{C}_{\textrm{micro}}\) to \(\mathbb{C}_{\textrm{macro}}\) like (15) results. The formally scale-independent material parameters of the classical Eringen-Mindlin-model are \(\widehat{\mathbb{C}}_{\textrm{e}}\) and \(\mathbb{C}_{\textrm{micro}}\) and the scale-independent parameters of \(W_{\textrm{GE}}\) are \(\mathbb{C}_{\textrm{micro}}=\mathbb{C}_{\textrm{macro}}\). For the Cosserat model, the respective scale-independent stiffness is \(\mathbb{C}_{\textrm{e}}=\mathbb{C}_{\textrm{macro}}\). However, considering (footnote 8) \(\textrm{Div} [ \mathbb{C}_{\textrm{micro}}\,\textrm{sym}\,P-\mu \,L_{c}^{2}\Delta P ] =f\), it is not strictly possible to say that \(\mathbb{C}_{\textrm{micro}}\) is scale-independent in the Eringen-Mindlin model. The identification of \(\mathbb{C}_{\textrm{micro}}\) (and therefore also \(\widehat{\mathbb{C}}_{\textrm{e}}\)) in the Eringen-Mindlin model may be length-scale dependent after all.

  11. It is indeed well known in the field of homogenization techniques (see, e.g., [20, 68]) that the homogenization of a unit-cell on which one imposes periodic boundary conditions mimics the behavior of a very large specimen of the associated equivalent Cauchy continuum. Usually, homogenization techniques only provide a direct transition from the micro to the macro-scale without considering the intermediate (transition) scale in which all relevant microstructure-related phenomena are manifest. Some attempts to introduce a transition scale via the homogenization towards a micromorphic continuum are made in [39, 79], even if it is clear that a definitive answer is far from being provided (see [39, 79] and references cited there). Our relaxed micromorphic model naturally provides the bridge between the micro and macro behavior of the considered homogenized material with the simple and transparent tensor homogenization formulas (15).

  12. In this way, artificial boundary layer effects are avoided.

  13. And not any of the ambiguous extended versions for generalized continua [26, 27, 29, 40].

  14. Since

    $$\begin{aligned} & \frac{1}{2} \bigl\langle \mathbb{C}_{\textrm{KUBC}}^{V} \,\overline{E}, \overline{E} \bigr\rangle \bigl\vert V (x ) \bigr\vert \\ &\quad =\inf \biggl\{ \int _{\xi \in V (x )}\frac{1}{2} \bigl\langle \mathbb{C} (\xi ) \bigl(\textrm{sym}\nabla _{\xi }v (\xi )+ \overline{E} \bigr),\textrm{sym}\nabla _{\xi }v (\xi )+ \overline{E} \bigr\rangle \,d\xi \: |\:v\in C_{0}^{\infty } \bigl(V \! (x ),\mathbb{R}^{3} \bigr) \biggr\} \\ & \quad v\equiv 0\;(\textrm{constant strain assumption: Taylor/Voigt}) \\ &\quad \leq \int _{V}\frac{1}{2} \bigl\langle \mathbb{C}(\xi ) \overline{E}, \overline{E} \bigr\rangle \,d\xi = \frac{1}{2} \biggl\langle \overline{E}, \int _{V} \mathbb{C}(\xi )\,d\xi \,\overline{E} \biggr\rangle = \frac{1}{2} \vert V \vert \biggl\langle \overline{E}, \frac{1}{ \vert V \vert } \int _{V} \mathbb{C}(\xi )\,d\xi \,\overline{E} \biggr\rangle = \frac{1}{2} \vert V \vert \langle \overline{E},\mathbb{C}_{ \textrm{Voigt}} \,\overline{E} \rangle \end{aligned}$$
    (31)

    it is clear that \(\langle \mathbb{C}_{\textrm{KUBC}}^{V}\, \overline{E}, \,\overline{E} \rangle \leq \langle \mathbb{C}_{\textrm{Voigt}} \,\overline{E},\overline{E} \rangle \) for all applied loadings \(\overline{E}\in \textrm{Sym}\left(3\right)\). On the other hand, it is natural to require as well \(\langle \mathbb{C}_{\textrm{micro}}\,\overline{E}, \, \overline{E} \rangle \leq \langle \mathbb{C}_{\textrm{Voigt}} \,\overline{E},\overline{E} \rangle \), where equality will be obtained if and only if the material on the micro-scale is homogeneous, i.e., \(\mathbb{C}(\xi )=\textrm{const}\).

  15. An equivalent, more algorithmic procedure to determine \(\mathbb{C}_{\textrm{KUBC}}^{V}\) is obtained as follows. Consider again (37)

    $$ \langle \overline{\sigma },\overline{\varepsilon } \rangle = \frac{1}{ \vert V \vert } \int _{ V} \bigl\langle \sigma (\xi ), \varepsilon (\xi ) \bigr\rangle \,d\xi = \frac{1}{ \vert V \vert } \int _{V} \bigl\langle \mathbb{C}(\xi )\,\varepsilon (\xi ), \varepsilon (\xi ) \bigr\rangle \,d \xi ,\quad \textrm{Div}\,\sigma \! (\xi )=0, \quad \sigma ^{\,T}\!(\xi )=\sigma (\xi ), $$
    (38)

    and \(\widetilde{v}=\overline{\varepsilon }\cdot \xi \) at the boundary. Let us define the corresponding linear solution operator of the linear elastic problem at the micro-scale \(\mathscr{L}(\xi )\cdot \overline{ \varepsilon }=\varepsilon (\xi )\), (“localization tensor”) and insert this back into (37). This gives

  16. Here, \(\mu \,L^{2}_{c} \langle \mathbb{L}\,\textrm{Curl}\,P,\textrm{Curl}\,P \rangle \) would represent the most general quadratic anisotropic curvature energy in the relaxed micromorphic model, where \(\mathbb{L}\) is a fourth-order tensor mapping non-symmetric second-order tensors to non-symmetric second-order tensors.

  17. \(\mathbb{C}_{\textrm{micro}}\) could be isotropic nevertheless, since isotropy is a subclass of the tetragonal symmetry.

  18. Considering the Voigt upper bound \(\mathbb{C}_{\textrm{Voigt}}:= \frac{1}{ \vert V \vert }\int _{V}\mathbb{C}(\xi )\,d\xi \) as representing the maximal microscopic stiffness is not useful for two reasons: First, \(\mathbb{C}_{\,\textrm{Voigt}}\) will be isotropic and lose the information of the geometry of the microstructure. Second, the actual deformation in any unit-cell will never exhibit constant strain.

  19. Here, \(x\) is the macro space variable of the continuum, while \(\xi \) is the micro-variable spanning inside the unit-cell.

  20. For a discussion of the non-uniqueness of the unit-cell, see [74].

  21. N.B. the value of \(\lambda _{\textrm{micro}}\) is correctly 5.270 and not 5.981 because of the definition of \(\widehat{\lambda }\) in (47).

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Acknowledgements

Patrizio Neff thanks Samuel Forest (Ecole des Mines, Paris), Geralf Hütter (TU Freiberg) and Jörg Schröder (University of Duisburg-Essen) for helpful discussions. The authors are also indebted to Lev Truskinovsky (ESPCI, Paris) for pertinent remarks which helped improve the paper.

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Authors and Affiliations

  1. Head of Chair for Nonlinear Analysis and Modelling, Fakultät für Mathematik, Universität Duisburg-Essen, Mathematik-Carrée, Thea-Leymann-Straße 9, 45127, Essen, Germany

    Patrizio Neff

  2. Universität Siegen, Institut für Mechanik, Heisenberg-group, Paul-Bonatz-Straße 9-11, 57076, Siegen, Germany

    Bernhard Eidel

  3. GEOMAS, INSA-Lyon, 20 avenue Albert Einstein, 69621, Villeurbanne, France

    Marco Valerio d’Agostino & Angela Madeo

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  1. Patrizio Neff
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  2. Bernhard Eidel
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  3. Marco Valerio d’Agostino
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  4. Angela Madeo
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Correspondence to Patrizio Neff.

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Dedicated to Prof. Dr. Dr. h. c. Hans-Dieter Alber on the occasion of his 70. birthday with great esteem

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Neff, P., Eidel, B., d’Agostino, M.V. et al. Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization. J Elast 139, 269–298 (2020). https://doi.org/10.1007/s10659-019-09752-w

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