nach oben

Journal of Elasticity

Erschienen in:

16.10.2019

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

verfasst von: Patrizio Neff, Bernhard Eidel, Marco Valerio d’Agostino, Angela Madeo

Erschienen in: Journal of Elasticity | Ausgabe 2/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel Topological Defects and Metric Anomalies as Sources of Incompatibility for Piecewise Smooth Strain Fields

Nächster Artikel Effective Description of Anisotropic Wave Dispersion in Mechanical Band-Gap Metamaterials via the Relaxed Micromorphic Model

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

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

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.

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

Despite the name micromorphic model.

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.

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 $\textrm{Div} ( \mathbb{C}_{\textrm{micro}}\,\textrm{sym}\,\nabla u ) =f$.

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.

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).

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 $\mathbb{C}_{\textrm{macro}}$.

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.

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).

In this way, artificial boundary layer effects are avoided.

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

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., $\mathbb{C}(\xi )=\textrm{const}$.

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

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-019-09752-w/MediaObjects/10659_2019_9752_Equa_HTML.png

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.

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

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.

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

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

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).

Abdulle, A.: Analysis of a heterogeneous multiscale FEM for problems in elasticity. Math. Models Methods Appl. Sci. 16(04), 615–635 (2006) MathSciNetMATH

Aivaliotis, A., Daouadji, A., Barbagallo, G., Tallarico, D., Neff, P., Madeo, A.: Low-and high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface (2018). arXiv preprint. arXiv:1810.12578

Aivaliotis, A., Daouadji, A., Barbagallo, G., Tallarico, D., Neff, P., Madeo, A.: Microstructure-related Stoneley waves and their effect on the scattering properties of a 2d Cauchy/relaxed-micromorphic interface. Wave Motion 90, 99–120 (2019) MathSciNet

Aivaliotis, A., Tallarico, D., Daouadji, A., Neff, P., Madeo, A.: Scattering of finite-size anisotropic metastructures via the relaxed micromorphic model (2019). arXiv preprint. arXiv:1905.12297

Auffray, N., Bouchet, R., Brechet, Y.: Derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behavior. Int. J. Solids Struct. 46(2), 440–454 (2009) MATH

Barbagallo, G., Madeo, A., d’Agostino, M.V., Abreu, R., Ghiba, I.-D., Neff, P.: Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics. Int. J. Solids Struct. 120, 7–30 (2017) MATH

Barbagallo, G., Tallarico, D., d’Agostino, M.V., Aivaliotis, A., Neff, P., Madeo, A.: Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures. Int. J. Solids Struct. 162, 148–163 (2019)

Bauer, S., Neff, P., Pauly, D., Starke, G.: Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions. ESAIM Control Optim. Calc. Var. 22(1), 112–133 (2016) MathSciNetMATH

Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures, vol. 5. North-Holland Publishing Company, Amsterdam (1978) MATH

10.

Biswas, R., Poh, L.H.: A micromorphic computational homogenization framework for heterogeneous materials. J. Mech. Phys. Solids 102, 187–208 (2017) ADSMathSciNet

11.

Boutin, C., Rallu, A., Hans, S.: Large scale modulation of high frequency waves in periodic elastic composites. J. Mech. Phys. Solids 70, 362–381 (2014) ADSMathSciNetMATH

12.

Bouyge, F., Jasiuk, I., Boccara, S., Ostoja-Starzewski, M.: A micromechanically based couple-stress model of an elastic orthotropic two-phase composite. Eur. J. Mech. A, Solids 21(3), 465–481 (2002) MATH

13.

Braides, A.: A handbook of ${\varGamma }$-convergence. Handb. Differ. Equ. 3, 101–213 (2006) MATH

14.

Burgeth, B., Welk, M., Feddern, C., Weickert, J.: Mathematical morphology on tensor data using the Löwner ordering. In: Visualization and Processing of Tensor Fields, pp. 357–368. Springer, Berlin (2006)

15.

Cosserat, E., Cosserat, F.: Théorie des corps déformables (1909). (engl. translation by D. Delphenich, pdf available at http://www.uni-due.de/%7ehm0014/Cosserat_files/Cosserat09_eng.pdf) MATH

16.

d’Agostino, M.V., Barbagallo, G., Ghiba, I.-D., Eidel, B., Neff, P., Madeo, A.: Effective description of anisotropic wave dispersion in mechanical metamaterials via the relaxed micromorphic model. J. Elast. (2019). Accepted in this volume

17.

Diebels, S., Steeb, H.: Stress and couple stress in foams. Comput. Mater. Sci. 28(3–4), 714–722 (2003)

18.

Weinan, E., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci. 1(1), 87–132 (2003) MathSciNetMATH

19.

Ehlers, W., Bidier, S.: From particle mechanics to micromorphic media. Part I: Homogenisation of discrete interactions towards stress quantities. Int. J. Solids Struct. (2018). https://doi.org/10.1016/j.ijsolstr.2018.08.013 CrossRef

20.

Eidel, B., Fischer, A.: The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE² method. Comput. Methods Appl. Mech. Eng. 329, 332–368 (2018) ADS

21.

Cemal, A.: Eringen. Mechanics of micromorphic materials. In: Applied Mechanics, pp. 131–138. Springer, Berlin (1966)

22.

Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999) MATH

23.

Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple micro-elastic solids – I. Int. J. Eng. Sci. 2(2), 189–203 (1964) MathSciNetMATH

24.

Fischer, A., Eidel, B.: Convergence and error analysis of FE-HMM/FE² for energetically consistent micro-coupling conditions in linear elastic solids. Eur. J. Mech. A, Solids 77, 103735 (2019) ADSMathSciNetMATH

25.

Forest, S.: Mechanics of generalized continua: construction by homogenizaton. J. Phys. IV 08(4), 39–48 (1998)

26.

Forest, S.: Aufbau und Identifikation von Stoffgleichungen für höhere Kontinua mittels Homogenisierungsmethoden. Tech. Mech. 19(4), 297–306 (1999) MathSciNet

27.

Forest, S.: Homogenization methods and mechanics of generalized continua – Part 2. Theor. Appl. Mech. 28–29, 113–144 (2002) MATH

28.

Forest, S., Sab, K.: Cosserat overall modeling of heterogeneous materials. Mech. Res. Commun. 25(4), 449–454 (1998) MathSciNetMATH

29.

Forest, S., Trinh, D.K.: Generalized continua and non-homogeneous boundary conditions in homogenisation methods. Z. Angew. Math. Mech. 91(2), 90–109 (2011) MathSciNetMATH

30.

Ghiba, I.-D., Neff, P., Madeo, A., Münch, I.: A variant of the linear isotropic indeterminate couple-stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and orthogonal boundary conditions. Math. Mech. Solids 22, 1221–1266 (2016) MathSciNetMATH

31.

Ghiba, I.-D., Neff, P., Madeo, A., Placidi, L., Rosi, G.: The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics. Math. Mech. Solids 20(10), 1171–1197 (2014) MathSciNetMATH

32.

Gologanu, M., Leblond, J.-B., Perrin, G., Devaux, J.: Recent extensions of Gurson’s model for porous ductile metals Part II: a Gurson-like model including the effect of strong gradients of the macroscopic field. Contin. Micromech. 377, 97–130 (1997) MATH

33.

Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11(5), 357–372 (1963) ADSMATH

34.

Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 326(1565), 131–147 (1972) ADSMATH

35.

Hori, M., Nemat-Nasser, S.: Micromechanics: Overall Properties of Heterogeneous Materials, vol. 37. Elsevier, Amsterdam (2013) MATH

36.

Huet, C.: Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids 38(6), 813–841 (1990) ADSMathSciNet

37.

Huet, C.: An integrated micromechanics and statistical continuum thermodynamics approach for studying the fracture behaviour of microcracked heterogeneous materials with delayed response. Eng. Fract. Mech. 58(5–6), 459–556 (1997)

38.

Huet, C.: Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies. Mech. Mater. 31(12), 787–829 (1999)

39.

Hütter, G.: Homogenization of a Cauchy continuum towards a micromorphic continuum. J. Mech. Phys. Solids 99, 394–408 (2017) ADSMathSciNet

40.

Hütter, G.: On the micro-macro relation for the microdeformation in the homogenization towards micromorphic and micropolar continua. J. Mech. Phys. Solids 127, 62–79 (2019) ADSMathSciNet

41.

Jänicke, R., Diebels, S., Sehlhorst, H.-G., Düster, A.: Two-scale modelling of micromorphic continua. Contin. Mech. Thermodyn. 21(4), 297–315 (2009) ADSMathSciNetMATH

42.

Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40(13–14), 3647–3679 (2003) MATH

43.

Kouznetsova, V., Geers, M.G.D., Brekelmans, M.W.A.: Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Methods Eng. 54(8), 1235–1260 (2002) MATH

44.

Kouznetsova, V., Geers, M.G.D., Brekelmans, M.W.A.: Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput. Methods Appl. Mech. Eng. 193(48–51), 5525–5550 (2004) ADSMATH

45.

Lobos, M., Yuzbasioglu, T., Böhlke, T.: Homogenization and materials design of anisotropic multiphase linear elastic materials using central model functions. J. Elast. 128(1), 17–60 (2017) MathSciNetMATH

46.

Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944) MATH

47.

Madeo, A., Collet, M., Miniaci, M., Billon, K., Ouisse, M., Neff, P.: Modeling phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia. J. Elast. 130, 1–25 (2017) MathSciNetMATH

48.

Madeo, A., Ghiba, I.-D., Neff, P., Münch, I.: A new view on boundary conditions in the Grioli–Koiter–Mindlin–Toupin indeterminate couple stress model. Eur. J. Mech. A, Solids 59, 294–322 (2016) ADSMathSciNetMATH

49.

Madeo, A., Neff, P., d’Agostino, M.V., Barbagallo, G.: Complete band gaps including non-local effects occur only in the relaxed micromorphic model. C. R., Méc. 344(11), 784–796 (2016) ADS

50.

Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., Rosi, G.: Band gaps in the relaxed linear micromorphic continuum. Z. Angew. Math. Mech. 95(9), 880–887 (2014) MathSciNetMATH

51.

Madeo, A., Neff, P., Ghiba, I.-D., Rosi, G.: Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model. J. Mech. Phys. Solids 95, 441–479 (2016) ADSMathSciNet

52.

Mandel, J.: Plasticité classique et viscoplasticité. International Centre for Mechanical Sciences. Courses and Lectures (1971)

53.

Michel, J.-C., Moulinec, H., Suquet, P.M.: Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172(1–4), 109–143 (1999) ADSMathSciNetMATH

54.

Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964) MathSciNetMATH

55.

Münch, I., Neff, P., Madeo, A., Ghiba, I.-D.: The modified indeterminate couple stress model: why Yang et al.’s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless. Z. Angew. Math. Mech. 97(12), 1524–1554 (2017) MathSciNet

56.

Neff, P.: On material constants for micromorphic continua. In: Trends in Applications of Mathematics to Mechanics, STAMM Proceedings, Seeheim, pp. 337–348. Shaker–Verlag, Aachen (2004)

57.

Neff, P.: Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. R. Soc. Edinb., Sect. A, Math. 136(05), 997–1012 (2006) MathSciNetMATH

58.

Neff, P., Forest, S.: A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elast. 87(2–3), 239–276 (2007) MathSciNetMATH

59.

Neff, P., Ghiba, I.-D., Lazar, M., Madeo, A.: The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations. Q. J. Mech. Appl. Math. 68(1), 53–84 (2014) MathSciNetMATH

60.

Neff, P., Ghiba, I.-D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Contin. Mech. Thermodyn. 26(5), 639–681 (2014) ADSMathSciNetMATH

61.

Neff, P., Jeong, J., Fischle, A.: Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature. Acta Mech. 211(3–4), 237–249 (2010) MATH

62.

Neff, P., Jeong, J., Ramézani, H.: Subgrid interaction and micro-randomness – Novel invariance requirements in infinitesimal gradient elasticity. Int. J. Solids Struct. 46(25–26), 4261–4276 (2009) MATH

63.

Neff, P., Madeo, A., Barbagallo, G., d’Agostino, M.V., Abreu, R., Ghiba, I.-D.: Real wave propagation in the isotropic-relaxed micromorphic model. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473(2197) (2017)

64.

Neff, P., Münch, I.: Curl bounds Grad on SO(3). ESAIM Control Optim. Calc. Var. 14(1), 148–159 (2008) MathSciNetMATH

65.

Neff, P., Pauly, D., Witsch, K.-J.: Maxwell meets Korn: a new coercive inequality for tensor fields in $\mathbb{R}^{N\times \,N}$ with square-integrable exterior derivative. Math. Methods Appl. Sci. 35(1), 65–71 (2012) ADSMathSciNetMATH

66.

Neff, P., Pauly, D., Witsch, K.-J.: Poincaré meets Korn via Maxwell: extending Korn’s first inequality to incompatible tensor fields. J. Differ. Equ. 258(4), 1267–1302 (2015) ADSMATH

67.

Neumann, F.E.: Vorlesungen über die Theorie der Elasticität der festen Körper und des Lichtäthers. B.G. Teubner, Leipzig (1885) MATH

68.

Pecullan, S., Gibiansky, L., Torquato, S.: Scale effects on the elastic behavior of periodic and hierarchical two-dimensional composites. J. Mech. Phys. Solids 47(7), 1509–1542 (1999) ADSMathSciNetMATH

69.

Pham, K., Kouznetsova, V.G., Geers, M.G.D.: Transient computational homogenization for heterogeneous materials under dynamic excitation. J. Mech. Phys. Solids 61(11), 2125–2146 (2013) ADSMathSciNetMATH

70.

Reuß, A.: Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech. 9(1), 49–58 (1929) MATH

71.

Rokoš, O., Ameen, M.M., Peerlings, R.H.J., Geers, M.G.D.: Micromorphic computational homogenization for mechanical metamaterials with patterning fluctuation fields. J. Mech. Phys. Solids 123, 119–137 (2019) ADSMathSciNet

72.

Romano, G., Barretta, R., Diaco, M.: Micromorphic continua: non-redundant formulations. Contin. Mech. Thermodyn. 28(6), 1659–1670 (2016) ADSMathSciNetMATH

73.

Karam, S.: On the homogenization and the simulation of random materials. Eur. J. Mech. A, Solids 5, 585–607 (1992) MathSciNet

74.

Schröder, J.: A numerical two-scale homogenization scheme: the $\mathit{FE}^{2}$-method. Schröder, J., Hackl, K. (eds.) Plasticity and Beyond: Microstructures, Crystal-Plasticity and Phase Transitions, vol. 550, pp. 1–64. Springer, Berlin (2014)

75.

Smyshlyaev, V.P., Cherednichenko, K.D.: On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media. J. Mech. Phys. Solids 48(6–7), 1325–1357 (2000) ADSMathSciNetMATH

76.

Sridhar, A., Kouznetsova, V.G., Geers, M.G.D.: Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum. Comput. Mech. 57(3), 423–435 (2016) MathSciNetMATH

77.

Suquet, P.M.: Local and global aspects in the mathematical theory of plasticity. Plasticity today, 279–309 (1985)

78.

Suquet, P.M.: Effective properties of nonlinear composites. In: Continuum Micromechanics, pp. 197–264. Springer, Berlin (1997) MATH

79.

Trinh D.K., Janicke, R., Auffray, N., Diebels, S., Forest, S.: Evaluation of generalized continuum substitution models for heterogeneous materials. Int. J. Multiscale Comput. Eng. 10(6), 527–549 (2012)

80.

Voigt, W.: Lehrbuch der Krystallphysik (mit Ausschluss der Krystalloptik). B.G. Teubner, Leipzig (1910)

81.

Wang, C., Feng, L., Jasiuk, I.: Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid. J. Biomech. Eng. 131(12), 121008 (2009)

82.

Zohdi, T.I.: Homogenization Methods and Multiscale Modeling. Encyclopedia of Computational Mechanics (2004)

Titel: Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization
verfasst von: Patrizio Neff
Bernhard Eidel
Marco Valerio d’Agostino
Angela Madeo
Publikationsdatum: 16.10.2019
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 2/2020
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-019-09752-w

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 2/2020

Topological Defects and Metric Anomalies as Sources of Incompatibility for Piecewise Smooth Strain Fields

Boundary Value Problems for Euler-Bernoulli Planar Elastica. A Solution Construction Procedure

Discrete Cosserat Rod Kinematics Constructed on the Basis of the Difference Geometry of Framed Curves—Part I: Discrete Cosserat Curves on a Staggered Grid

On the Orientation Average Based on Central Orientation Density Functions for Polycrystalline Materials

Effective Description of Anisotropic Wave Dispersion in Mechanical Band-Gap Metamaterials via the Relaxed Micromorphic Model

Premium Partner

Marktübersichten