# On existence and uniqueness of weak solutions for linear pantographic beam lattices models

- 67 Downloads

## Abstract

In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. https://doi.org/10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

## Keywords

Strain-gradient elasticity Weak solutions Beam lattice Pantographic sheets Anisotropic Sobolev’s spaces## 1 Introduction

The strain-gradient theory of elasticity has its origin in the early works of some giants of continuum mechanics: see [1, 2, 3, 4, 5, 6] for historical developments in the mechanics of generalized continua, and it was developed further in the original works by Toupin [7] and Mindlin [8, 9]. The main conceptual tool for formulating these theories is given by the principle of virtual work and/or the principle of least action: indeed also continuum mechanics finds its more effective formulation when one bases its postulation on variational principles. This opinion was also shared by Hellinger, see [10, 11, 12] who, in his masterpiece “Fundamentals of the mechanics of continua”, showed, already with the knowledge available in 1913, that the unifying vision given by variational principles could allow for a effective presentation of all field theories.

These structures have been introduced in order to give an example of metamaterial which can undergo very large deformations still remaining in an elastic regime. First preliminary experimental and theoretical results are presented in [16, 22, 33, 34, 35]. These papers show that the concept underlying the design of pantographic metamaterials deserved to be developed and therefore its mathematical modelling is needed, for getting detailed predictions via suitably developed numerical codes. Being said metamaterials constituted by lattices of beams, their numerical analysis may be inspired by discrete or semi-discrete models, see [36, 37], or by continuum models see, e.g., [38, 39]. Also metamaterials with granular structures can be developed by considering heuristically homogenized continuum models as presented in [40, 41, 42, 43]. In order to perform effectively the numerical analysis of complex beam lattices, the most efficient methods are desired, (as those presented, e.g., in [44, 45, 46, 47, 48, 49, 50] and the references therein). Let us note that for polymer and metal materials when certain level of deformations is reached it could be important to take into account also inelastic phenomena [51, 52, 53, 54, 55]. Another source of inelastic behavior is the contact of the beams in the lattice and the related adhesion interactions [56, 57]. For current state of the pantographic metamaterials, we refer to [58, 59, 60, 61, 62, 63]. Beam lattices can be used as “meso-models” of cellular solids or regular open-cell foams which are widely used in engineering and tissue engineering: we therefore believe that the homogenized models introduced in the present paper can have a wider range of application. Remark also that (see e.g., [63]) in pantographic metamaterials some “phase segregations” or “phase transitions” have been observed: therefore it seems natural to assume that the mathematical techniques used in [64, 65, 66] are applicable also in the present context. A generalized form of Pott model has been used to simulate static and kinetic phenomena in foams and the biological morphogenesis [67, 68]. Pantographic sheets modelling is closely related to mechanics of networks, see [69, 70, 71] and the reference therein.

It has also to be investigated the whole damage mechanisms occurring in them, with methods which may be inspired to peridynamics, see, e.g., [72, 73, 74, 75, 76]. The relationship between peridynamics and higher gradient continuum theories, on the other hand, was already know to Piola himself [77, 78], see also [79], and we believe that it deserves to be fully investigated.

The main object of this paper is to prove a result of well-posedness of the deformation problem of linear elastic pantographic sheets deforming in space: i.e., bidimensional continua generalizing standard plate models, as their deformation energy not only depend on the second gradient of out-of-plane displacement but also on second gradients of in-plane displacements. We believe that this is an essential intermediate step in the study of large deformation of pantographic metamaterials or of composite reinforcements, in particular when wrinkling occurs. The experimental evidence about these phenomena, occurring in the forming of composite reinforcements, is discussed in [80, 81, 82, 83].

We believe that the insight gained by the results which we present here can supply a useful tool for developing the analysis of the more general nonlinear case. To this end, one may need more advanced techniques as used in the case of nonlinear theories of plates and shallow shells, see, e.g., [84, 85, 86, 87].

## 2 Derivation of a continuum model

*m*vertical fibers of length \(l_2\) and

*n*horizontal fibers of length \(l_1\) connected by short pivots of length

*h*, see Fig. 3. In what follows, we describe infinitesimal deformations of the lattice using the Euler–Bernoulli beam model. First, we introduce two kinematically independent fields of translations and rotations and then using the standard kinematic Euler–Bernoulli constraints we will replace rotations by derivatives of translations.

## 3 Strain energy density and equilibrium conditions

An important element of the analysis of existence and uniqueness of weak solutions is the study of the energy null-space, that is, set of admissible functions for which the given strain energy density is vanishing. For standard linear elasticity, such null-space reduces to the so-called infinitesimal rigid body motions, see [93, 94, 95]. For linear pantographic sheets considered within reduced strain-gradient elasticity, there are other possible energy-free solutions [28].

### 3.1 Energy-free deformations and rigid body motions: full model

Analyzing in-plane deformations of linear pantographic sheets, we demonstrated [28] the importance of shear energy of pivots to avoid additional energy-free shear deformations. To consider both in- and out-of-plane deformations, let us also underline the importance of the bending energies of pivots.

### 3.2 Energy-free deformations: pivot spring model

### 3.3 Energy-free deformations: fibers without torsional energy and pivots without bending energy

### 3.4 Energy-free deformations: perfectly connected fibers without torsion energy and pivots without bending energy

### 3.5 Equilibrium equations

## 4 Existence and uniqueness of weak solutions

For the proof of the existence and uniqueness of weak solutions, we use the same technique as in [28] which uses the anisotropic Sobolev’s spaces as the corresponding energy space for considered functionals. These functional spaces were introduced in [30, 103, 104, 105], see also [106]. They are generalizations of the Sobolev’s spaces [107, 108]. In what follows, we are restricted ourselves by the functionals which have a finite dimensional null-space and non-singular boundary conditions [28]. Our non-singular boundary conditions are nothing else as the Shapiro–Lopatinskii or complementary boundary conditions, see original works [109, 110] and [100, 108, 111, 112] for the mathematical definitions. For example, for strongly elliptic PDEs the Dirichlet- and von Neumann-type boundary conditions satisfy the Shapiro–Lopatinskii conditions.

Without loss of generality, we consider the problems in the dimensionless forms. We start from the simplest case.

### 4.1 Weak solutions for \( {\mathcal {W}}_{000}\)

### Definition 1

Using standard Riesz representation theorem and the Lax-Milgram theorem arguments [93, 94, 95, 113], we can prove the following

### Theorem 1

Let \(b_{1}\), \(b_{2}\), and \(b_{3}\) belong to the space \(L_{2}(\omega )\). There exists a weak solution \(\mathbf {u}^{*}\in E^{000}_0\) to the corresponding equilibrium problem (49)–(51), which for any \(\mathbf {w}\in E^{000}_0\) satisfies (56)

*c*. This solution belongs to the energy null-space, see (34).

These examples show that the consideration of out-of-plane deformations may lead to non-unique solutions. Indeed, for in-plane deformations of pantographic sheets given in Fig. 4, the solution of (49) and (50) is unique [28], whereas for out-of-plane deformations we have many solutions.

### 4.2 Weak solutions for \( {\mathcal {W}}_{00}\)

### 4.3 Remarks on other cases

Unlike two previous case, this technique cannot be applied straightforwardly to the pivot spring model introduced by (30). Indeed, for this model, the corresponding null-space is not finite dimensional. And vice versa, the full model with the strain energy density (12) can be analyzed similarly to these cases as its null-space is finite dimensional. So, energy-null solutions can be avoided choosing proper boundary conditions.

Let us also underline the crucial difference between the semi-discrete model (9) and its continual counterparts. The semi-discrete model corresponds to a system of linear ODEs. So its well-posedness is rather obvious. On the other hand, the continual “homogenized” models may loose this property. So, one should be aware of such situations.

## 5 Conclusions and future steps

- 1.
The placements of the two involved families of fibers are independent fields both defined in a bidimensional Lagrangian reference configuration and having images in the 3D Eulerian Euclidean affine space of positions;

- 2.
Both families of fibers are modelled as beams which can store deformation energy due to elongation, bending and twisting;

- 3.
The pivots between the fibers are assumed to be connected with two sections belonging to two different fibers (see Fig. 3), and these sections are assumed to behave as rigid bodies;

- 4.
The strain energy of these pivots depends on the relative displacement and rotations of the aforementioned fibers’ sections.

- 1.
The most suitable inertial terms to be added in the Lagrangian for considered systems;

- 2.
The properties of obtained dispersion formulas;

- 3.
The properties of eigenfrequencies and modal forms of finite pantographic specimens suitably constrained and excited.

Let us note that here we restricted ourselves to infinitesimal deformations. On the other hand, high flexibility of considered beam lattice structures results in necessity to analyze existence and uniqueness/non-uniqueness of the corresponding nonlinear boundary-value problems. Unlike classic plates and shallow shells [84, 85, 86, 87], where deflections are larger than in-plane displacements, in general, for a beam lattice in-plane displacements may have the same order of magnitude as out-of-plane ones. So, one can expect an essential nonlinearity of the corresponding boundary-value problems.

We expect that the presented mathematical analysis, which shows that the treatment presented in [28] can be extended to sheets deforming in 3D space, will guide us to treat the more complicated problems to be faced when considering pantographic sheets undergoing large deformations, see, e.g., [18, 22, 92]. In other words, we believe that for any linear or nonlinear physical model its linear mathematical counterpart that is a linear boundary-value problem with suitable boundary conditions, such as fixed boundary conditions, should have unique solution. Otherwise, this results not only in some pathological mathematical properties, but also in essential difficulties in numerical calculations.

This conclusion may be also useful for the mathematical analysis of other enhanced models of continua and structures, as for example polar, dipolar, non-local media and continue with additional kinematical descriptors [95, 116, 117, 118, 119, 120], where the enhanced kinematics may result in unusual floppy modes and corresponding constraints to external loading.

## Notes

## References

- 1.Maugin, G.A.: A historical perspective of generalized continuum mechanics. In: Altenbach, H., Erofeev, V.I., Maugin, G.A. (eds.) Mechanics of Generalized Continua. From the Micromechanical Basics to Engineering Applications, pp. 3–19. Springer, Berlin (2011)Google Scholar
- 2.Maugin, G.A.: Generalized continuum mechanics: various paths. In: Continuum Mechanics Through the Twentieth Century: A Concise Historical Perspective, Springer, Dordrecht, pp. 223–241 (2013)Google Scholar
- 3.Maugin, G.A.: Non-classical Continuum Mechanics: A Dictionary. Springer, Singapore (2017)zbMATHCrossRefGoogle Scholar
- 4.dell’Isola, F., Della Corte, A., Giorgio, I.: Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids
**22**(4), 852–872 (2017)MathSciNetzbMATHCrossRefGoogle Scholar - 5.Auffray, N., dell’Isola, F., Eremeyev, V.A., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids
**20**(4), 375–417 (2015)MathSciNetzbMATHCrossRefGoogle Scholar - 6.dell’Isola, F., Eremeyev, V.A.: Some introductory and historical remarks on mechanics of microstructured materials. In: dell’Isola, F., Eremeyev, V.A., Porubov, A. (eds.) Advances in Mechanics of Microstructured Media and Structures, pp. 1–20. Springer, Cham (2018)CrossRefzbMATHGoogle Scholar
- 7.Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal.
**11**(1), 385–414 (1962)MathSciNetzbMATHCrossRefGoogle Scholar - 8.Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal.
**16**(1), 51–78 (1964)MathSciNetzbMATHCrossRefGoogle Scholar - 9.Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct.
**4**(1), 109–124 (1968)zbMATHCrossRefGoogle Scholar - 10.Eugster, S.R., dell’Isola, F.: Exegesis of the introduction and sect. I from “Fundamentals of the Mechanics of Continua”** by E. Hellinger. ZAMM
**97**(4), 477–506 (2017)ADSMathSciNetCrossRefGoogle Scholar - 11.Eugster, S.R., dell’Isola, F.: Exegesis of Sect. II and III.A from “Fundamentals of the Mechanics of Continua” by E. Hellinger. ZAMM
**98**(1), 31–68 (2018)ADSMathSciNetCrossRefGoogle Scholar - 12.Eugster, S.R., dell’Isola, F.: Exegesis of Sect. III.B from “Fundamentals of the Mechanics of Continua” by E. Hellinger. ZAMM
**98**(1), 69–105 (2018)ADSMathSciNetCrossRefGoogle Scholar - 13.Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids
**24**(1), 212–234 (2019)MathSciNetCrossRefGoogle Scholar - 14.di Cosmo, F., Laudato, M., Spagnuolo, M.: Acoustic metamaterials based on local resonances: homogenization, optimization and applications. In: Generalized Models and Non-classical Approaches in Complex Materials 1, pp. 247–274. Springer, New York (2018)Google Scholar
- 15.Soubestre, J., Boutin, C.: Non-local dynamic behavior of linear fiber reinforced materials. Mech. Mater.
**55**, 16–32 (2012)CrossRefGoogle Scholar - 16.Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. ZAMP
**67**(4), 1–28 (2016)MathSciNetzbMATHGoogle Scholar - 17.Boisse, P., Colmars, J., Hamila, N., Naouar, N., Steer, Q.: Bending and wrinkling of composite fiber preforms and prepregs. A review and new developments in the draping simulations. Compos. Part B: Eng.
**141**, 234–249 (2018)CrossRefGoogle Scholar - 18.dell’Isola, F., Steigmann, D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast.
**118**(1), 113–125 (2015)MathSciNetzbMATHCrossRefGoogle Scholar - 19.dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev
**67**(6), 060804-1–21 (2016)ADSGoogle Scholar - 20.Sabik, A.: Direct shear stress vs strain relation for fiber reinforced composites. Compos. Part B: Eng.
**139**, 24–30 (2018)CrossRefGoogle Scholar - 21.Berrehili, Y., Marigo, J.-J.: The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers. Math. Mech. Complex Syst.
**2**(2), 181–207 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 22.dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenisation, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. Ser. A
**472**(2185), 20150790 (2016)ADSCrossRefGoogle Scholar - 23.Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro–macro models identification. Math. Mech. Complex Syst.
**5**(2), 127–162 (2017)MathSciNetzbMATHCrossRefGoogle Scholar - 24.Turco, E., Golaszewski, M., Giorgio, I., D’Annibale, F.: Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos. Part B: Eng.
**118**, 1–14 (2017)CrossRefGoogle Scholar - 25.Rahali, Y., Giorgio, I., Ganghoffer, J.F., dell’Isola, F.: Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci.
**97**, 148–172 (2015)MathSciNetzbMATHCrossRefGoogle Scholar - 26.Misra, A., Placidi, L., Scerrato, D.: A review of presentations and discussions of the workshop Computational mechanics of generalized continua and applications to materials with microstructure that was held in Catania 29–31 October 2015. Math. Mech. Solids
**22**(9), 1891–1904 (2017)zbMATHCrossRefGoogle Scholar - 27.Placidi, L., Giorgio, I., Della Corte, A., Scerrato, D.: Euromech 563 Cisterna di Latina 17–21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: a review of presentations and discussions. Math. Mech. Solids
**22**(2), 144–157 (2017)zbMATHCrossRefGoogle Scholar - 28.Eremeyev, V.A., dell’Isola, F., Boutin, C., Steigmann, D.: Linear pantographic sheets: existence and uniqueness of weak solutions. J. Elast.
**132**, 175–196 (2018). https://doi.org/10.1007/s10659-017-9660-3 MathSciNetzbMATHCrossRefGoogle Scholar - 29.Eremeyev, V.A., dell’Isola, F.: A note on reduced strain gradient elasticity. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds.) Generalized Models and Non-classical Approaches in Complex Materials 1, pp. 301–310. Springer, Cham (2018)CrossRefGoogle Scholar
- 30.Nikol’skii, S.M.: On imbedding, continuation and approximation theorems for differentiable functions of several variables. Russian Math. Surv.
**16**(5), 55 (1961)ADSMathSciNetCrossRefGoogle Scholar - 31.Kachala, V.V., Khemchyan, L.L., Kashin, A.S., Orlov, N.V., Grachev, A.A., Zalesskiy, S.S., Ananikov, V.P.: Target-oriented analysis of gaseous, liquid and solid chemical systems by mass spectrometry, nuclear magnetic resonance spectroscopy and electron microscopy. Russian Chem. Rev.
**82**(7), 648–85 (2013)ADSCrossRefGoogle Scholar - 32.Kashin, A.S., Ananikov, V.P.: A SEM study of nanosized metal films and metal nanoparticles obtained by magnetron sputtering. Russian Chem, Bull.
**60**(12), 2602–2607 (2011)CrossRefGoogle Scholar - 33.Alibert, J.-J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids
**8**(1), 51–73 (2003)MathSciNetzbMATHCrossRefGoogle Scholar - 34.Seppecher, P., Alibert, J.-J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys. Conf. Ser.
**319**(1), 012018 (2011)CrossRefGoogle Scholar - 35.Turco, E., Golaszewski, M., Cazzani, A., Rizzi, N.L.: Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete Lagrangian model. Mech. Res. Commun.
**76**, 51–56 (2016)CrossRefGoogle Scholar - 36.Eugster, S.R., Hesch, C., Betsch, P., Glocker, C.: Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int. J. Numer. Methods Eng.
**97**(2), 111–129 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 37.Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Contin. Mech. Thermodyn.
**30**, 1103–1123 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 38.Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math.
**103**(1), 1–21 (2017)MathSciNetzbMATHCrossRefGoogle Scholar - 39.Giorgio, I.: Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures. Zeitschrift für angewandte Mathematik und Physik
**67**(4), 95 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 40.Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn.
**28**(1–2), 215–234 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 41.Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst.
**3**(3), 285–308 (2015)MathSciNetzbMATHCrossRefGoogle Scholar - 42.Misra, A., Poorsolhjouy, P.: Grain-and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model. Mech. Res. Commun.
**81**, 1–6 (2017)CrossRefGoogle Scholar - 43.Misra, A., Poorsolhjouy, P.: Elastic behavior of 2D grain packing modeled as micromorphic media based on granular micromechanics. J. Eng. Mech.
**143**(1), C4016005 (2016)CrossRefGoogle Scholar - 44.Chróścielewski, J., Sabik, A., Sobczyk, B., Witkowski, W.: Nonlinear FEM 2D failure onset prediction of composite shells based on 6-parameter shell theory. Thin-Walled Struct.
**105**, 207–219 (2016)CrossRefGoogle Scholar - 45.Balobanov, V., Niiranen, J.: Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity. Comput. Methods Appl. Mech. Eng.
**339**, 137–159 (2018)ADSMathSciNetCrossRefGoogle Scholar - 46.Niiranen, J., Balobanov, V., Kiendl, J., Hosseini, S.B.: Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro-and nano-beam models. Math. Mech. Solids
**24**(1), 312–335 (2019)MathSciNetCrossRefGoogle Scholar - 47.Greco, L., Cuomo, M., Contrafatto, L., Gazzo, S.: An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Comput. Methods Appl. Mech. Eng.
**324**, 476–511 (2017)ADSMathSciNetCrossRefGoogle Scholar - 48.Chróścielewski, J., Schmidt, R., Eremeyev, V.A.: Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches. Contin. Mech. Thermodyn.
**31**(1), 147–188 (2019)ADSMathSciNetCrossRefGoogle Scholar - 49.Chróścielewski, J., Sabik, A., Sobczyk, B., Witkowski, W.: 2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis. Compos. Part B: Eng.
**165**, 335–353 (2019)CrossRefGoogle Scholar - 50.Maurin, F., Greco, F., Desmet, W.: Isogeometric analysis for nonlinear planar pantographic lattice: discrete and continuum models. Contin. Mech. Thermodyn.
**31**(4), 1051–1064 (2019)ADSMathSciNetCrossRefGoogle Scholar - 51.Alfano, G., De Angelis, F., Rosati, L.: General solution procedures in elasto/viscoplasticity. Comput. Methods Appl. Mech. Eng.
**190**(39), 5123–5147 (2001)ADSzbMATHCrossRefGoogle Scholar - 52.Palazzo, V., Rosati, L., Valoroso, N.: Solution procedures for \(j_3\) plasticity and viscoplasticity. Comput. Methods Appl. Mech. Eng.
**191**(8–10), 903–939 (2001)ADSzbMATHCrossRefGoogle Scholar - 53.Placidi, L., Barchiesi, E., Misra, A.: A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math. Mech. Complex Syst.
**6**(2), 77–100 (2018)MathSciNetzbMATHCrossRefGoogle Scholar - 54.Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A
**474**(2210), 20170878 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 55.Placidi, L., Misra, A., Barchiesi, E.: Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik
**69**(3), 56 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 56.Marmo, F., Toraldo, F., Rosati, A., Rosati, L.: Numerical solution of smooth and rough contact problems. Meccanica
**53**(6), 1415–1440 (2018)MathSciNetzbMATHCrossRefGoogle Scholar - 57.Nadler, B., Steigmann, D.J.: A model for frictional slip in woven fabrics. Comptes Rendus Mecanique
**331**(12), 797–804 (2003)ADSzbMATHCrossRefGoogle Scholar - 58.Golaszewski, M., Grygoruk, R., Giorgio, I., Laudato, M., Di Cosmo, F.: Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions. Contin. Mech. Thermodyn.
**31**, 1015–1034 (2019) ADSCrossRefGoogle Scholar - 59.Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.H.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Contin. Mech. Thermodyn.
**31**(1), 33–45 (2019)ADSMathSciNetCrossRefGoogle Scholar - 60.Barchiesi, E., Placidi, L.: A review on models for the 3D statics and 2D dynamics of pantographic fabrics. In: Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, pp. 239–258. Springer, Berlin (2017)Google Scholar
- 61.Turco, E., Misra, A., Sarikaya, R., Lekszycki, T.: Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling. Contin. Mech. Thermodyn.
**31**(1), 209–223 (2019) ADSMathSciNetCrossRefGoogle Scholar - 62.Misra, A., Lekszycki, T., Giorgio, I., Ganzosch, G., Müller, W.H., dell’Isola, F.: Pantographic metamaterials show atypical Poynting effect reversal. Mech. Res. Commun.
**89**, 6–10 (2018)CrossRefGoogle Scholar - 63.dell’Isola, F., Seppecher, P., Alibert, J.J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., Steigmann, D., Giorgio, I., Andreaus, U., Turco, E., Gołaszewski, M., Rizzi, N., Boutin, C., Eremeyev, V.A., Misra, A., Placidi, L., Barchiesi, E., Greco, L., Cuomo, M., Cazzani, A., Corte, A.D., Battista, A., Scerrato, D., Eremeeva, I.Z., Rahali, Y., Ganghoffer, J.-F., Müller, W., Ganzosch, G., Spagnuolo, M., Pfaff, A., Barcz, K., Hoschke, K., Neggers, J., Hild, F.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn.
**31**(4), 851–884 (2019)ADSMathSciNetCrossRefGoogle Scholar - 64.Carlen, E.A., Carvalho, M.C., Esposito, R., Lebowitz, J.L., Marra, R.: Droplet minimizers for the Gates–Lebowitz–Penrose free energy functional. Nonlinearity
**22**(12), 2919–2952 (2009)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 65.Eremeyev, V.A., Pietraszkiewicz, W.: The non-linear theory of elastic shells with phase transitions. J. Elast.
**74**(1), 67–86 (2004)zbMATHCrossRefGoogle Scholar - 66.Pietraszkiewicz, W., Eremeyev, V.A., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. J. Appl. Math. Mech.-ZAMM
**87**(2), 150–159 (2007)MathSciNetzbMATHCrossRefGoogle Scholar - 67.De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles. J. Stat. Phys.
**133**(2), 281–345 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 68.De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Coexistence of ordered and disordered phases in Potts models in the continuum. J. Stat. Phys.
**134**(2), 243–306 (2009)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 69.Atai, A.A., Steigmann, D.J.: On the nonlinear mechanics of discrete networks. Arch. Appl. Mech.
**67**(5), 303–319 (1997)ADSzbMATHCrossRefGoogle Scholar - 70.Luo, C., Steigmann, D.J.: Bending and twisting effects in the three-dimensional finite deformations of an inextensible network. In: Advances in the Mechanics of Plates and Shells, pp. 213–228. Springer, Berlin (2001)Google Scholar
- 71.Steigmann, D.J.: Continuum theory for elastic sheets formed by inextensible crossed elasticae. Int. J. Non-Linear Mech.
**106**, 324–329 (2018)ADSCrossRefGoogle Scholar - 72.Gao, Y., Oterkus, S.: Ordinary state-based peridynamic modelling for fully coupled thermoelastic problems. Contin. Mech. Thermodyn.
**31**, 907–937 (2019)ADSMathSciNetCrossRefGoogle Scholar - 73.Oterkus, E., Madenci, E.: Peridynamic analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct.
**7**(1), 45–84 (2012) CrossRefGoogle Scholar - 74.Oterkus, E., Madenci, E.: Peridynamic theory for damage initiation and growth in composite laminate. Key Eng. Mater.
**488**, 355–358 (2012)Google Scholar - 75.Diyaroglu, C., Oterkus, E., Oterkus, S., Madenci, E.: Peridynamics for bending of beams and plates with transverse shear deformation. Int. J. Solids Struct.
**69**, 152–168 (2015)CrossRefGoogle Scholar - 76.Diyaroglu, C., Oterkus, E., Oterkus, S.: An Euler–Bernoulli beam formulation in an ordinary state-based peridynamic framework. Math. Mech. Solids (2017). https://doi.org/10.1177/1081286517728424 CrossRefGoogle Scholar
- 77.dell’Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S. (Eds.): The complete works of Gabrio Piola: Volume I, vol. 38 of Advanced Structured Materials, Springer, Cham (2014)Google Scholar
- 78.dell’Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S. (Eds.), The complete works of Gabrio Piola: Volume II, vol. 97 of Advanced Structured Materials, Springer, Cham (2018)Google Scholar
- 79.dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids
**20**(8), 887–928 (2015)MathSciNetzbMATHCrossRefGoogle Scholar - 80.Giorgio, I., Harrison, P., dell’Isola, F., Alsayednoor, J., Turco, E.: Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches. Proc. R. Soc. A
**474**(2216), 20180063 (2018)ADSCrossRefGoogle Scholar - 81.Boisse, P., Hamila, N., Vidal-Sallé, E., Dumont, F.: Simulation of wrinkling during textile composite reinforcement forming. Influence of tensile, in-plane shear and bending stiffnesses. Compos. Sci. Technol.
**71**(5), 683–692 (2011)CrossRefGoogle Scholar - 82.Buet-Gautier, K., Boisse, P.: Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements. Exp. Mech.
**41**(3), 260–269 (2001)CrossRefGoogle Scholar - 83.Gelin, J.C., Cherouat, A., Boisse, P., Sabhi, H.: Manufacture of thin composite structures by the RTM process: numerical simulation of the shaping operation. Compos. Sci. Technol.
**56**(7), 711–718 (1996)CrossRefGoogle Scholar - 84.Ciarlet, P.: Mathematical Elasticity. Theory of Plates, vol. II. Elsevier, Amsterdam (1997)zbMATHGoogle Scholar
- 85.Ciarlet, P.: Mathematical Elasticity. Theory of Shells, vol. III. Elsevier, Amsterdam (2000)zbMATHGoogle Scholar
- 86.Vorovich, I.I.: Nonliner Theory of Shallow Shells. Applied Mathematical Sciences, vol. 133. Springer, New York (1999)Google Scholar
- 87.Lebedev, L.P., Vorovich, I.I.: Functional Analysis in Mechanics. Springer, New York (2003)zbMATHCrossRefGoogle Scholar
- 88.Svetlitsky, V.A.: Statics of Rods. Springer, Berlin (2000)zbMATHCrossRefGoogle Scholar
- 89.Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V.A., Pietras, D.: Deformation analysis of functionally graded beams by the direct approach. Compos. Part B: Eng.
**43**(3), 1315–1328 (2012)CrossRefGoogle Scholar - 90.Scerrato, D., Zhurba Eremeeva, I.A., Lekszycki, T., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM
**96**(11), 1268–1279 (2016)ADSMathSciNetCrossRefGoogle Scholar - 91.Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun.
**83**, 47–52 (2017)CrossRefGoogle Scholar - 92.Steigmann, D.J., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mech. Sin.
**31**(3), 373–382 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 93.Fichera, G.: Existence theorems in elasticity. In: Flügge, S. (ed.) Handbuch der Physik, vol. VIa/2, pp. 347–389. Springer, Berlin (1972)Google Scholar
- 94.Ciarlet, P.G.: Mathematical Elasticity. Three-Dimensional Elasticity, vol. I. North-Holland, Amsterdam (1988)zbMATHGoogle Scholar
- 95.Eremeyev, V.A., Lebedev, L.P.: Existence of weak solutions in elasticity. Math. Mech. Solids
**18**(2), 204–217 (2013)MathSciNetCrossRefGoogle Scholar - 96.Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010)zbMATHCrossRefGoogle Scholar
- 97.Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus. Première partie: théorie du second gradient. J. Mécanique
**12**, 236–274 (1973)zbMATHGoogle Scholar - 98.Fichera, G.: Linear Elliptic Differential Systems and Eigenvalue Problems. Lecture Notes in Mathematics, vol. 8. Springer, Berlin (1965)zbMATHCrossRefGoogle Scholar
- 99.Egorov, Y.V., Shubin, M.A.: Foundations of the Classical Theory of Partial Differential Equations. Encyclopaedia of Mathematical Sciences 30, vol. 30, 1st edn. Springer, Berlin (1998)CrossRefGoogle Scholar
- 100.Agranovich, M.: Elliptic boundary problems. In: Agranovich, M., Egorov, Y., Shubin, M. (eds.) Partial Differential Equations IX: Elliptic Boundary Problems. Encyclopaedia of Mathematical Sciences, vol. 79, pp. 1–144. Springer, Berlin (1997)CrossRefGoogle Scholar
- 101.Hörmander, L.: The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients. A Series of Comprehensive Studies in Mathematics, vol. 257. Springer, Berlin (1983)zbMATHGoogle Scholar
- 102.Palamodov, V.P.: Systems of linear differential equations. In: Gamkrelidze, R.V. (ed.) Mathematical Analysis. Progress in Mathematics, pp. 1–35. Springer, Boston (1971)Google Scholar
- 103.Besov, O.V., II’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. 1. Wiley, New York (1978)Google Scholar
- 104.Besov, O.V., II’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. 2. Wiley, New York (1979)Google Scholar
- 105.Besov, O.V., II’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Imbedding Theorems. Nauka, Moscow (1996). (in Russian)Google Scholar
- 106.Triebel, H.: Theory of Function Spaces III. Monographs in Mathematics, vol. 100. Birkhäuser, Basel (2006)zbMATHGoogle Scholar
- 107.Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. Academic Press, Amsterdam (2003)Google Scholar
- 108.Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972)zbMATHCrossRefGoogle Scholar
- 109.Lopatinskii, Y.B.: On a method of reducing boundary problems for a system of differential equations of elliptic type to a regular integral equation (in Russian. Ukrain. Math. Zhurnal.
**5**, 123–151 (1953)MathSciNetGoogle Scholar - 110.Shapiro, Z.Y.: On general boundary problems for equations of elliptic type (in Russian). Izv. Akad. Nauk SSSR. Ser. Math.
**17**, 539–562 (1953)Google Scholar - 111.Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math.
**12**(4), 623–727 (1959)MathSciNetzbMATHCrossRefGoogle Scholar - 112.Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math.
**17**(1), 35–92 (1964)MathSciNetzbMATHCrossRefGoogle Scholar - 113.Evans, L.C.: Partial Differential Equations. Graduate Series in Mathematics, vol. 19, 2nd edn. AMS Providence, Rhode Island (2010)Google Scholar
- 114.Polyanin, A.D., Nazaikinskii, V.E.: Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd edn. Chapman and Hall/CRC, Boca Raton (2016)zbMATHCrossRefGoogle Scholar
- 115.Laudato, M., Manzari, L., Barchiesi, E., Cosmo, F.D., Göransson, P.: First experimental observation of the dynamical behavior of a pantographic metamaterial. Mech. Res. Commun.
**94**, 125–127 (2018)CrossRefGoogle Scholar - 116.Eremeyev, V.A., Lebedev, L.P.: Existence theorems in the linear theory of micropolar shells. ZAMM
**91**(6), 468–476 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 117.Gharahi, A., Schiavone, P.: Uniqueness of solution for plane deformations of a micropolar elastic solid with surface effects. Contin. Mech. Thermodyn. (2019). https://doi.org/10.1007/s00161-019-00779-x
- 118.Marin, M., Öchsner, A.: An initial boundary value problem for modeling a piezoelectric dipolar body. Contin. Mech. Thermodyn.
**30**(2), 267–278 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar - 119.Marin, M., Öchsner, A., Taus, D.: On structural stability for an elastic body with voids having dipolar structure. Contin. Mech. Thermodyn. (2019). https://doi.org/10.1007/s00161-019-00793-z
- 120.Romano, G., Barretta, R., Diaco, M.: Iterative methods for nonlocal elasticity problems. Contin. Mech. Thermodyn.
**31**(3), 669–689 (2019)ADSMathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.