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This paper investigates the static and dynamic responses of elastic cubic lattices, focusing on central and non-central interactions, specifically angular and volumetric interactions. The study aims to bridge the gap between discrete and continuous elasticity, addressing the capability of discrete elasticity approaches in capturing the macroscopic behavior of elastic continuous solids. Key topics include the calibration of equivalent Poisson's ratio, the positive definiteness of potential energy, and the mixed differential-difference equations derived from Hamilton's principle. The paper also explores the static responses of finite lattices under pure compression and pure shear modes, comparing them to their continuous asymptotic limits. Additionally, the discrete Lamé problem for the free vibration of a rectangular parallelepiped is solved for each lattice type, illustrating the convergence towards linear elastic continuous behavior. The findings highlight the importance of volumetric interactions in achieving a complete range of elasticity parameters, up to the incompressibility limit of 1/2. This comprehensive study provides valuable insights into the behavior of elastic materials, offering a deeper understanding of their static and dynamic responses.
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Abstract
This paper investigates statics and natural vibration of linear elastic cubic lattices, together with their continuum approximations. The lattice endowed with central and angular interactions, referred to as Gazis et al.’s, is considered first: since the stiffness of each lattice phase must be positive, the equivalent macroscopic Poisson’s ratio must be lower than its central limit 1/4. A volumetric interaction based on a volume-dependent internal pressure is introduced as an additional non-central interaction for a complete calibration of the equivalent Poisson’s ratio up to its incompressibility limit 1/2. This volumetric interaction can also be classified as Fuchs-type, providing a potential energy that depends on the volume variation of each cell. The mixed differential-difference equations of the associated lattice derive from Hamilton’s principle applied to the discrete energies. The algebraic properties of the stiffness matrix of the discrete cell provide information on the positive definiteness of the potential energy, for each lattice with central and non-central interactions. The convergence of this finite lattice towards a linear elastic continuous right parallelepiped is shown in several static loading schemes. The discrete Lamé problem for the free vibration of this parallelepiped is solved for all the considered lattices. It is concluded that discrete and continuum elasticity can be connected by this cubic lattice within a complete range of elasticity parameters.
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1 Introduction
This paper addresses the question on the capability of discrete elasticity approaches in capturing the macroscopic behaviour of elastic continuous solids. This question is related to the foundation of continuum elasticity and has been debated since the nineteenth century through the formulation of the so-called molecular theories of elasticity [5, 6, 10, 21]. For 3D continuous linear isotropic solids exhibiting pure central interactions, Navier [47] formulated a rare-constant elastodynamic theory yielding equal Lamé coefficients and Poisson’s ratio ν = 1/4. On the other hand, the energetic formulation of linear isotropic elastic continua leads to two independent elastic constants ([26]; [53]). This two-parameter formulation allows the identification of a free Poisson’s ratio that varies between − 1 and 1/2 to preserve the positive definiteness of the associated potential energy (see, e.g. Love [39]). Love [39] also mentioned that, even if these bounds for ν are theoretically admissible, negative values (characteristic of the materials called auxetic nowadays) would be, “for physical reasons, highly improbable”. The deviation of Poisson’s ratio from its rare-constant value 1/4 has been attributed to the contribution of non-central interactions at the molecular scale. At the end of the nineteenth century, Voigt and Poincaré presented molecular arguments considering non-central angular interactions and derived a two-parameter linear isotropic continuous elastic theory. Voigt [55, 56] considered the orientation of molecules, coupled to some additional moments, whereas Poincaré [50] introduced a three-body interaction potential in addition to pure central interactions, that is equivalent to an additional potential of angle variation.
A pivotal contribution in connecting discrete to continuous elasticity is the 3D cubic elastic lattice endowed with central and shear type non-central interactions by Born and von Kármán [4], for which they derived the mixed differential-difference equations. This non-central interaction can be also understood as frame-dependent and angular, connected to the axis of symmetry of the solid cube. This lattice model converges towards the above-described Navier’s rare-constant theory of elastodynamics [47] for pure central interactions, showing some consistency. However, the non-central interaction considered in Born and von Kármán [4] is inconsistent since it violates the principle of invariance under rigid rotations (see [36], [25]; [1, 32] for the requirement of a consistent lattice theory). Recently, Challamel et al. [13] demonstrated for 2D lattices that the mixed differential-difference equations postulated by Born and von Kármán [4] can be reformulated using a consistent objective angular interaction.
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An alternative model was developed in the 1940s for computing the response of elastic structures using a numerical truss method. Hrennikoff (1941, 1949) [28], [29] introduced a truss model consisting of bars, additional meta-cells, and auxiliary members that asymptotically converges towards linear isotropic elastic continua. When restricted to pure central interactions without additional meta-cells, Hrennikoff’s truss coincides with McHenry's [43]; the latter was developed for 2D analyses with Poisson’s ratio ν = 1/3 in plane stress [43], as foreseen by the 2D rare-constant constraint. Hrennikoff’s truss (1941) [28] holds for both 2D and 3D linear elasticity and has been effectively used to analyse elastic structures, accommodating a range of equivalent Poisson’s ratios. Remarkably, Hrennikoff’s has later been considered as the birth of the Finite Element Method [38, 57]. The 3D Hrennikoff’s lattice [28] can be regarded as a generalization of Born and von Kármán’s [4] cubic lattice when the latter is restricted to exhibit pure central interactions.
Another generalization of Born and von Kármán’s lattice, which includes consistent non-central interactions, is the lattice model developed by Gazis et al. [24]. This simple cubic lattice allows for central and objective angular interactions and converges towards the linear elastic cubic continuum in the asymptotic limit. Gazis et al.’s lattice has been successfully applied to the statics and dynamics of discrete elastic 2D problems ([9], [10], [11], [12] and 3D problems ([10], [48]). A limitation of this lattice with angular interactions is that the positivity requirement of each phase parameter (to ensure the positive definiteness of the associated potential energy) imposes a restriction on Poisson’s ratio in linear isotropic elasticity, i.e., ν < 1/4 for 3D, ν < 1/3 for plane stress.
In this paper, we propose a theory that can bridge the gap between discrete and continuous linear elasticity for the whole range of Poisson’s ratio (complete cubic lattice elasticity theory). It has been already shown for plane lattices that large Poisson’s ratios up to the 2D incompressibility limit ν = 1 can be achieved by considering surface interactions. Then, inspired by the 2D lattice model with surface interactions developed in Challamel et al. [11], we present herein a 3D lattice model that includes a volumetric interaction, and that can cover large Poisson’s ratios up to the incompressibility limit of 1/2 for 3D materials. This can be seen as the generalization of the 2D square lattice to the 3D cubic lattice, from surface to volume-dependent interactions. A volumetric interaction was introduced for 3D lattices by Fuchs [23], who attributed the failure of Cauchy’s conditions for central interactions in metals to the presence of electron gas [15, 16]. Fuchs’ model is based on a potential that is nonlinear in the volume change (a power function of order − 2/3, according to Fermi’s energy) [15, 16, 23, 54]. De Launay [15] assumed pressure to be proportional to the change in volume, which is implicitly associated with a quadratic potential. De Launay [15] obtained a simple equation, where the discrepancy with Cauchy’s relations for pure central interactions is proportional to the volumetric factor (in other words, the volumetric interaction is responsible for the discrepancies with the pure central configuration). Both Cousins [14] and Thomas [54] discussed the influence of the power exponent of the additional volumetric dependence that explains the discrepancy with Cauchy’s relations (see also [31]). Martin [42] discussed the validity of the volumetric contribution presented by Johnson [31] in terms of rotational invariance. Martin [42] also explicitly considered a quadratic potential for the volumetric dependence of the non-central contribution in the elastic energy (also called four-body interaction for the cubic lattice). More recently, Ekiz et al. [17] developed a 3D peridynamic model based on central (dubbed one-neighbour) and non-central interactions, including a volumetric one. This study on elastic lattices is closely related to molecular dynamics (MD) simulations, where bond stretching and angle bending interactions are coupled with long-range linear or nonlinear forces. Such simulations have been instrumental in modeling the mechanical behavior of 2D materials (see for instance [8]) (see for instance [8] for 2D materials). Recent works, such as those by Maalouf and Vel [40], have utilized numerical simulations with classical hyperelastic energy functions to derive mechanical properties across various crystal symmetries. Their findings on Poisson's ratio align closely with some results presented herein, although analyses pertaining to specific symmetries may not be directly comparable. In a subsequent study, Maalouf and Vel [41] introduced a computationally efficient molecular statics approach to simulate the nonlinear elastic response of materials. By incrementally deforming a unit cell, they computed variables of interest and provided linearized material constants. However, their results are specific to graphene and thus do not directly correspond to the current approach focused on simple cubic lattices.
The present paper on static and vibration responses of elastic cubic lattices has three parts. In the first part, we derive the mixed differential-difference equations of cubic lattices with central and angular interactions (Gazis et al.’s lattices) and those of cubic lattices with central and volumetric interactions. In both cases, the discrete field equations and the relevant natural and essential boundary conditions are derived by Hamilton’s principle applied to the discrete potential and kinetic energies that we build. The calibration of the lattice parameters for both lattices is discussed with respect to their asymptotic continuum limit. We shall show that, due to a loss of positive definiteness of the potential energy for high Poisson’s ratios, Gazis et al.’s lattice is mechanically consistent for limited values of ν. On the other hand, the positive definiteness of the potential energy associated to the volumetric interaction is ensured for all Poisson’s ratios up to the incompressibility limit ν = 1/2. We also checked the positive definiteness of the potential energy of both lattices for negative Poisson’s ratios, that is, even in the presence of phases with negative stiffness, thus paving the way to describe possible auxetic behaviours. There are some available studies on the stability of discrete or continuous solids incorporating negative stiffness phases, for instance those by Kochmann [33], Esin et al. [19] and Kochmann & Bertoldi [34].
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In the second part of this paper, some static and vibrating responses of both lattices having shape of right parallelepipeds are investigated. In statics, the uniaxial compression and pure shear tests provide homogeneous responses of both lattices for a large range of Poisson’s ratio; in addition, we check the response of each lattice specimen accounting for stiffness and loading corrections at the boundaries, as already introduced by Hrennikoff [28] for 2D and 3D finite elastic lattices. In free dynamics, we search the analytical and numerical vibrating response of a normally constrained right parallelepiped (discrete Lamé’s problem, [35]). Exact eigenfrequencies are found for the continuous isotropic limit in the case of shear-free boundary conditions (rigid-lubricated boundaries), as presented by Summerfeld [52], Nadeau [46] or Hill and Egle [27]. The vibration of the stress-free right parallelepiped, even in its continuum limit, is much more complex, as shown by Ekstein and Schiffman [18], Fromm and Leissa [22], Hutchinson and Zillmer [30] or Mindlin [45]. For the discrete Lamé’s problem, and for both families of cubic lattices, the natural modes are assumed in a trigonometric form, which is also the exact form of the asymptotic continuum. A related paper is the study of Bilek and Skala [3], who suggested the use of trigonometric normal modes for finite 3D lattices, and the link with infinite crystals.
In the final third part of this paper, we shall perform a parametric numerical analysis to assess the efficiency of the developed volumetric-dependent lattice model in capturing a large spectrum of behaviour in term of elasticity parameters.
2 Simple cubic lattice with central and angular interactions
The cubic lattice is first presented in presence of central and angular interactions (model of [24],see also [10, 48]). The mixed differential-difference equations of Gazis et al.’s [24] lattice can be deduced from the potential energy and kinetic energy of the parallelepiped lattice.
If n, m, p are integers, we consider a right parallelepiped lattice of length L1 = na along the ‘horizontal’ axis, width L2 = ma along the ‘vertical’ axis and depth L3 = pa along the third direction.
The kinetic energy can be equivalently written as the sum of internal and boundary contributions, as follows:
As per the boundary contribution, the masses of each particle equal M/2 inside the face, M/4 along the border of each face and M/8 at the corner of the parallelepiped. For the boundary face i = 0 the kinetic energy is then equal to:
The application of Hamilton’s principle to the Lagrangian L = K – U yields the mixed differential-difference equations of the 3D Gazis et al.’s lattice [10]:
For symmetry reasons, the same mixed differential equation is obtained for the two complementary equations.
The boundary conditions can be derived from the variational principle applied to the finite parallelepiped lattice (see for instance [48]). For instance, along the boundary face i = 0 of this lattice one obtains the variationally-based boundary conditions:
where the normal and the shear force in each direction on each face are derived naturally from the variational principle. The normal force \(N_{0,j,k}\) acting orthogonally to the face i = 0 is given by
The lattice parameters for central and non-central interactions are calibrated with respect to the equivalent linear isotropic continuum in the asymptotic limit. The mixed difference-differential equations are expanded using a continualization procedure and then compared to Navier’s partial differential equations in the low frequency regime. The relations \(u_{i,j,k} = u\left( {x = ai,y = aj,z = ak} \right)\),\(v_{i,j,k} = v\left( {x = ai,y = aj,z = ak} \right)\), \(w_{i,j,k} = w\left( {x = ai,y = aj,z = ak} \right)\) between the discrete and the equivalent continuous system hold true for sufficiently smooth displacement fields. Then, the following asymptotic expansion is assumed for the discrete displacement field around its considered reference node:
where ∂x, ∂y, ∂z are the partial derivatives with respect to x, y, z, respectively. Equation (11) is based on a Taylor-based asymptotic expansion of the discrete displacement field of a neighbour node around the pivotal one. The exponential operator in front of the continuous displacement field is often labelled as pseudo-differential. Such an expansion of difference operators was already known at the beginning of the nineteenth century (e.g., [7, 49]).
Equation (5) may be approximated by the continuous formulations at the long wave limit:
Furthermore, the stiffness of the cubic lattice is calibrated to asymptotically converge towards elastic linear isotropic solids at the continuum limit:
The same identification was obtained by Gazis et al. [24] with a calibration against continuum cubic elasticity, here restricted to isotropic elasticity as a particular case.
As expected, the rare-constant value of Poisson’s ratio is implied by pure central interactions:
which is valid for Poisson’s ratios − 1 < ν < 1/4. For Gazis et al.’s lattice, the potential energy remains positive definite for Poisson’s ratio range [0, 1/4], assuming that each constituent exhibits positive stiffness:
Consequently, the stiffness matrix is symmetric and positive definite when rigid body modes are constrained, ensuring Lyapunov stability of the unconstrained system. This stability is further supported by the symmetry and positive definiteness of the mass matrix, guaranteeing positive eigenfrequencies, according to Rayleigh’s quotient property (Rayleigh, 1894 [51]). In the Poisson’s ratio interval (− 1, 2/7), the local stiffness matrix of a unit cell remains positive definite under proper constraints, even if some phases exhibit negative stiffness, thereby maintaining system stability (see Appendix A). It is worth mentioning that 2/7≈0.286 and is slightly larger than 1/4 = 0.25, which means that the stiffness matrix does not lose its positive definiteness at the rare-constant value \(\nu\) = 1/4.
In the 3D lattice model introduced by Gazis et al. [24], which incorporates both central and angular interactions, the requirement for each lattice phase to possess positive stiffness imposes a constraint on the macroscopic Poisson's ratio. Specifically, to ensure positive stiffness in all lattice components, the effective Poisson's ratio must be less than 1/4. Within the range of 0.25 to approximately 0.286 (2/7), the overall stiffness matrix remains positive definite; however, the angular interactions exhibit negative stiffness. Theoretical studies have explored the feasibility of constructing lattices with such negative angular stiffness. For instance, Esin et al. [19] analyzed chains of oscillators incorporating negative stiffness elements, including normal, shear, and rotational springs, and established conditions under which the system remains stable. Furthermore, the concept of negative stiffness phases has been investigated in the context of composite materials. Kochmann [33] and Kochmann & Bertoldi [34] examined the potential of embedding negative stiffness inclusions within a positive stiffness matrix, demonstrating that such configurations can lead to materials with enhanced mechanical properties, provided the overall system is appropriately constrained to maintain stability.
In sum, the cubic lattice with central and angular interaction possesses macroscopic elastic parameters with Poisson’s ratio smaller than the rare-constant limit, i.e. \(\nu\) < 1/4, to ensure the positivity of each stiffness phase and a positive Poisson’s ratio. Auxetic behaviour induced by some lattice phase with negative stiffness can be obtained, and is compatible with the positive definiteness of the lattice cell stiffness matrix, as shown in Appendix A.
It is possible to expand the difference operators in the mixed differential-difference Eq. (5) at a higher level to derive a gradient elasticity constitutive law, which accounts (third person) for the lattice spacing contribution:
Equation (18) is a particular case of the one derived for cubic elasticity by Mindlin (1968) [44], who expanded the mixed differential-difference equations of Gazis et al.’s lattice with a calibration against the continuum cubic elasticity. Equation (18) can be also referred as a strain gradient cubic elasticity model, as studied by Lazar et al. [37] (see also [48] for the justification of such a classification). As pointed out by Mindlin (1968) [44] and Nguyen et al. [48], the associated energy of this lattice-based strain gradient elasticity model is not positive definite for the additional strain gradient terms.
3 Simple cubic lattice with central and volumetric interactions
The mixed differential-difference equations of the 3D elastic lattice with central and volumetric interactions can be deduced from the potential energy and kinetic energy of the parallelepiped lattice. The potential energy is given by
The last addend, due to the volumetric interaction, is a Fuchs-type volumetric potential, where the pressure is assumed to depend linearly on the volumetric variation. The variation of this potential is written for one cubic cell in the following form:
Figure 1 shows a 3D representation of the deformed cube, a cuboid, the variation of volume of which controls the volumetric elastic interaction. The potential is then expressed in a quadratic form of the volume variation, as considered by Martin [42]:
The same reasoning was followed by (Challamel et al., 2024-a [9]) for the introduction of a pressure-dependent surface interaction in 2D lattices.
The kinetic energy based on concentrated masses at each node is given by Eq. (2). The application of Hamilton’s principle to the Lagrangian L = K – U yields the mixed differential-difference equations of the 3D cubic lattice with central and volumetric interactions:
If the cubic lattice with central interaction also contains non-central interactions of angular and volumetric-type (Gazis et al.’s lattice with additional volumetric interaction), one obtains the mixed differential-difference equations:
In the case of pure central interaction augmented with volumetric interaction, but without angular interaction, the mixed differential-difference equations Eq. (23) can be rewritten as:
The boundary conditions can be derived from the variational principle applied to the finite cuboid lattice, as presented for the lattice of Gazis et al. For instance, along the boundary face i = 0 of the finite lattice one obtains the variationally-based boundary conditions Eq. (6), where the normal force \(N_{0,j,k}\) acting orthogonally to the face i = 0 is:
This partial differential equation is compared to Navier’s partial differential Eq. (13) for a linear elastic isotropic medium. As a result, the following lattice parameters are identified:
which is valid for Poisson’s ratios 1/4 < \(\nu\) < 1/2.
For the lattice with central and volumetric interactions, the potential energy remains positive definite within the Poisson’s ratio range [1/4, 1/2[, assuming that each constituent exhibits positive stiffness:
In this last case,\(\;\gamma_{v}\) plays the role of the Lagrange multiplier related to the volume constraint (see, e.g., [2] for elastic solids involving volume constraints).
Consequently, we can provide comments on the symmetry and definite positivity of the stiffness and mass matrices analogous to those reported previously. In the Poisson’s ratio interval (− 1, 1/2), the local stiffness matrix of a unit cell remains positive definite under proper constraints, even if some phases exhibit negative stiffness, thereby maintaining system stability (see Appendix C). For Poisson’s ratios in the interval (− 1, 1/2), the local stiffness matrix of one cell is symmetric and definite positive if the system is properly constrained to eliminate rigid modes, even if some phase may possess negative stiffness (see Appendix B). This ensures the stability of the unconstrained state for the same reasons. For such a lattice, there is a coincidence between the range of validity of the discrete and the continuum elastic models, associated with a Poisson’s ratio − 1 < \(\nu\) < 1/2 for the necessary stability criterion. In lattice models incorporating volumetric interactions, ensuring positive stiffness in each lattice phase imposes constraints on the macroscopic Poisson's ratio. Specifically, the effective Poisson's ratio must lie within the interval 1/4 < ν \(\nu\) 1/2. Outside this range, the volumetric interaction transitions from representing compressive forces to tensile ones, leading to a volumetric tension pressure within the unit cell. This phenomenon has been explored in the context of 2D pattern-forming pneumatic metamaterials. For instance, Faltus et al. [20] investigated pneumatically actuated 2D metamaterials, demonstrating that non-uniform internal pressures can induce structural instabilities, resulting in significant reductions in macroscopic stiffness and enabling active control over material properties.
In sum, the cubic lattice with central and angular interaction possesses macroscopic elastic parameters with Poisson’s ratio smaller than the rare-constant limit, i.e. for \(\nu\) ≤ 1/4, whereas the cubic lattice with central and volumetric interaction can capture Poisson’s ratio larger than the rare-constant limit, i.e. for \(\nu\) ≥ 1/4. This phenomenon is like what was observed for 2D lattices, where the central and angular interaction is associated to small Poisson’s ratios, whereas the central and surface interaction model captures larger values of Poisson’s ratio.
It is possible to expand the difference operator at a higher level to highlight a gradient elasticity constitutive law for the lattice with volumetric interaction:
4 Statics and dynamics of right parallelepiped lattices with volumetric interactions
In this section, statics and free vibration of right parallelepiped lattices are investigated under elementary loading and boundary conditions. We investigate both the statics of the cubic lattice of Gazis et al. [24] with central and angular interactions and those of the cubic lattice with central and volumetric interactions. Some exact solutions can be obtained in statics, for instance by considering the compression of an n × n × n cubic lattice under vertical loading, as shown in Fig. 2 (n = 2 considered in Fig. 2). The static response is obtained from a linear algebraic problem for the discrete displacement field, where the stiffness matrix is obtained from the nodal forces applied at each node of each cell (endowed with central and non-central interactions). The adaptation of equivalent forces and springs at the border was applied, as for McHenry’s or Hrennikoff’s trusses, for lattices with pure central interactions [43] and [28]). Whereas McHenry [43] only considered 2D lattices with pure central interactions, Hrennikoff [28] proposed both 2D and 3D lattices with central and additional central inner interactions to calibrate the macroscopic behaviour for general values of Poisson’s ratio. The cubic cell of Gazis et al.’s lattice with central and angular interactions is shown in Fig. 3, whereas the one with volumetric interaction is represented in Fig. 4. For the n × n × n cells problem, the strain response is homogeneous inside the specimen comprising repetitive cells, under the condition that the nodal loading and the springs at the border are corrected, as already stressed by McHenry [43] and Hrennikoff [28]. The truss undergoes pure compression with a prescribed displacement \(w_{i,j,n} = \overline{w}\) at the boundary upper face k = n:
The node at the centre of the lower face (i = n/2, j = n/2, k = 0) is assumed to be fixed (it is assumed that n is even for the considered lattice specimen).
As shown in Fig. 5, for both cubic lattices with central, angular or volumetric interactions, the discrete displacement field for such a repetitive specimen is:
Displacement field (normalized against the maximum vertical displacement) of lattice under compression: a Gazis et al. lattice, b lattice with volumetric interactions. c Deformed shape of lattice under compression.
Figure 5also highlights the deformed shape of the cubic lattice (with central, angular or volumetric interactions) under pure compression. The discrete homogeneous strain field in each horizontal or vertical link is given by:
As expected, the load–displacement relationship is size independent (for both cubic lattices with central and angular interactions and those with central and volumetric interactions), in the sense that the average stress–strain response is proportional to Young’s modulus for any number n of cells (Fig. 6):
It is essential to outline that the homogeneous response of the lattice is ensured by the correction of both the stiffnesses and of the nodal forces at the border, as shown in Fig. 2.
The pure shear behaviour of the cubic lattice (see Fig. 7) is also studied, following what was done for 2D lattices by Challamel et al. [11]. The 3D lattice exhibiting central and angular interactions (Gazis et al.’s) or the one exhibiting central and volumetric interaction is loaded by forces parallel to its boundaries, to perform a pure shear test. As already remarked for the pure compression test, the loading is corrected at the boundary nodes, following the suggestions of [28, 43] for the definition of equivalent nodal forces. The two lattices undergo pure shear when subjected to a prescribed displacement at the boundary nodes \(u_{i,j,n} = \overline{u}\):
Pure shear test of the cubic lattice (with angular or/and volumetric interactions); Horizontal shear forces on the upper surface and lateral forces on the two other surfaces (the last surfaces are free)
These results are confirmed by the numerical analysis of the pure shear of 2 × 2 × 2 cells shown in Fig. 8, where the loading along the edges follows the nodal loading representation. The deformed shape of the cubic lattice (with central, angular or volumetric interactions) under pure shear is shown in Fig. 8, and is consistent with the theoretical response predicted by Eq. (44).
Fig. 8
Displacement field of lattice in the shear test: (a) Gazis et al. lattice, (b) lattice with volumetric interactions. The maximum horizontal displacement is set to 1+\(\nu\). (c) Deformed shape of lattice under shear
Next, we study the vibration behaviour of a parallelepiped lattice with either central and angular interactions (Gazis et al.’s) or central and volumetric interactions (Fig. 9). This problem was solved by Nguyen et al. [48] for Gazis et al.’s lattice and it can be considered as a discrete Lamé’s problem (see [35], in which Lamé studied the vibration of an elastic isotropic parallelepiped continuum). The solution derived by Nguyen et al. [48] is valid only for sufficiently small values of Poisson’s ratio, whereas the ones presented herein using both angular and volumetric interactions cover a larger range of the equivalent macroscopic Poisson’s ratio.
Fig. 9
Lattice Lamé problem of a rectangular parallelepiped constrained along its sliding lattice surfaces (diagonal members are turned off for clearer illustration)
The discrete Dirichlet boundary conditions in Eq. (46) (simply supported edges) are completed by the discrete Neumann boundary conditions (null tangential forces along the edges).
Considering a harmonic vibration with circular frequency ω, the difference eigenvalue problem for Gazis et al.’s lattice derives from Eq. (5) (as in [48]):
where the parameters \(\left( {\alpha ,\beta ,\gamma_{a} } \right)\) are calibrated in Eq. (14). For the cubic lattice with volumetric interaction, the difference eigenvalue problem derives from Eq. (25) as:
where the calibrated parameters \(\left( {\alpha ,\beta ,\gamma_{v} } \right)\) are given in Eq. (31).
For a right parallelepiped with shear-free (rigid-lubricated) boundaries, it is possible to derive the exact eigenfrequencies of the 3D lattice, based on the following discrete field:
for \(i \in \left\{ {0,1,...,n} \right\}\); \(j \in \left\{ {0,1,...,m} \right\}\) and \(k \in \left\{ {0,1,...,p} \right\}\).
where k1, k2 and k3 are integers associated with the considered modes.
By introducing the discrete fields of Eq. (49) in the linear difference equation Eq. (47) for the 3D Gazis et al.’s lattice, one obtains the following linear system in the unknowns \(\left( {U,V,W} \right)\)
where the calibrated parameters \(\left( {\alpha ,\beta ,\gamma_{a} } \right)\) are given in Eq. (14). The other complementary difference equations are obtained analogously, so that we get the following linear system:
where the coefficients of the symmetric matrix aij = aji introduced in Eq. (51) can be expressed starting from the stiffness coefficients of the lattice (see also [48]):
The same method allows to calculate the eigenfrequencies of the 3D lattice with volumetric interaction. By introducing the discrete fields of Eq. (49) in the linear difference equation Eq. (48) for that lattice, one obtains the following linear system in the unknowns \(\left( {U,V,W} \right)\):
where the calibrated parameters \(\left( {\alpha ,\beta ,\gamma_{v} } \right)\) are given in Eq. (31). The coefficients aij = aji introduced in Eq. (51) can always be identified starting from the stiffness coefficients of the lattice with volumetric interaction:
It can be checked that the discriminant \(\Delta = 4p_{1}^{3} + 27q_{1}^{2} \le 0,\) for the parameters of interest, so that the three real solutions for the eigenfrequencies are:
The exact eigenfrequencies spectrum of the free vibration of the considered lattice can then be exactly determined from this algebraic method, and the ωi can be ordered as follows:
For the three families of eigenmodes, the eigenfrequencies are selected by the compatibility conditions (as detailed by Challamel et al.,[10] for the in-plane vibrations of a 2D rectangular elastic lattice; see also [48]):
Numerical results for the eigenfrequencies are shown in Tables 1, 2, 3 for the parallelepiped lattice with volumetric interaction. It is confirmed that these eigenfrequencies converge towards the continuous elastic ones for a sufficiently large number of cells. The convergence is from below for the 3D lattice with volumetric interaction, a result which is like the one observed for the 3D elastic lattice with angular interactions [48].
Table 1
Eigenfrequencies Ω - \(\nu\) = 1/3: converged numerical solutions for the cubic lattice with volumetric interaction
Mode
3D continuous solid
3D lattice with volumetric interactions
1
2.22
2.22
2
2.22
2.22
3
2.22
2.22
4
2.72
2.72
5
2.72
2.72
6
3.14
3.14
7
3.14
3.14
8
3.14
3.14
9
3.51
3.51
10
3.51
3.51
11
3.51
3.51
12
3.51
3.51
13
3.51
3.51
14
3.51
3.51
15
3.85
3.85
16
3.85
3.85
17
3.85
3.85
8
3.85
3.85
19
3.85
3.85
20
3.85
3.85
Table 2
FEM eigenfrequencies Ω of a continuous solid model - υ = 1/3
Mode
3D solid FEM solution using 8-node brick elements
Solution from Richardson extrapolation for 30×30×30 and 15×15×15 elements
10×10×10 elements
15×15×15 elements
20×20×20 elements
30×30×30 elements
1
2.24
2.23
2.23
2.22
2.22
2
2.24
2.23
2.23
2.22
2.22
3
2.24
2.23
2.23
2.22
2.22
4
2.74
2.73
2.73
2.72
2.72
5
2.74
2.73
2.73
2.72
2.72
6
3.16
3.15
3.15
3.14
3.14
7
3.16
3.15
3.15
3.14
3.14
8
3.16
3.15
3.15
3.14
3.14
9
3.60
3.55
3.53
3.52
3.51
10
3.60
3.55
3.53
3.52
3.51
11
3.60
3.55
3.53
3.52
3.51
12
3.60
3.55
3.53
3.52
3.51
13
3.60
3.55
3.53
3.52
3.51
14
3.60
3.55
3.53
3.52
3.51
15
3.90
3.87
3.86
3.85
3.85
16
3.90
3.87
3.86
3.85
3.85
17
3.90
3.87
3.86
3.85
3.85
18
3.94
3.89
3.87
3.86
3.85
19
3.94
3.89
3.87
3.86
3.85
20
3.94
3.89
3.87
3.86
3.85
Table 3
Eigenfrequencies Ω of 3D lattice with volumetric interactions - υ = 1/3
Mode
10×10×10 cells
15×15×15 cells
20×20×20 cells
30×30×30 cells
Solution from Richardson extrapolation for 30×30×30 and 15×15×15 cells
1
2.21
2.22
2.22
2.22
2.22
2
2.21
2.22
2.22
2.22
2.22
3
2.21
2.22
2.22
2.22
2.22
4
2.69
2.71
2.71
2.72
2.72
5
2.69
2.71
2.71
2.72
2.72
6
3.13
3.14
3.14
3.14
3.14
7
3.13
3.14
3.14
3.14
3.14
8
3.13
3.14
3.14
3.14
3.14
9
3.46
3.49
3.50
3.51
3.51
10
3.46
3.49
3.50
3.51
3.51
11
3.46
3.49
3.50
3.51
3.51
12
3.46
3.49
3.50
3.51
3.51
13
3.46
3.49
3.50
3.51
3.51
14
3.46
3.49
3.50
3.51
3.51
15
3.74
3.80
3.82
3.84
3.85
16
3.74
3.80
3.82
3.84
3.85
17
3.74
3.80
3.82
3.84
3.85
18
3.80
3.83
3.84
3.84
3.85
19
3.80
3.83
3.84
3.84
3.85
20
3.80
3.83
3.84
3.84
3.85
The eigenfrequencies solutions of the continuous right parallelepiped with shear-free boundary conditions (rigid-lubricated boundaries) are given in Summerfeld [52], Nadeau [46] or Hill and Egle [27]. For the continuous elastic linear isotropic parallelepiped, the cubic equation for the eigenfrequencies has double roots, which can be simply expressed by:
In the asymptotic limit (with an infinite number of particles in each direction), the cubic frequency equation exhibits a double root property. However, for a finite lattice with central, angular, and volumetric interactions, this property is no longer guaranteed, as the system generally has three distinct frequency roots. The continuous reference solution is presented in Table 1 for a parallelepiped and was also numerically obtained by a Finite Element Analysis (the analytical solution with double roots coincides with the lattice results in the asymptotic limit and with the FEM model for a sufficiently large number of 8-node 3D brick elements). For the two lattices (Gazis et al.’s and with volumetric interaction), the eigenfrequencies of the cubic lattice (parallelepiped with equal faces) are given in Tables 8 and 9. The analytical results obtained from the resolution of Cardano’s Eq. (59) exactly match the numerical ones obtained from the algebraic equation derived from the assembly of the stiffness and mass matrix of each model (see Tables 4, 5, 6, 7 for the cubic lattice with volumetric interaction). This confirms the consistency of the analytical solutions obtained for the discrete Lamé’s problem, considering both angular and volumetric interactions. Both lattice solutions approach the continuous one from below, as already remarked by (Challamel et al., [10]) for 2D lattices with angular interactions, or by Nguyen et al. [48] for 3D lattices with angular interactions. The same tendency is observed for the present lattice with both central and volumetric interactions. It is also confirmed that the lattice with volumetric interactions can cover a large spectrum of equivalent material parameters, with an equivalent Poisson’s ratio up to its incompressibility limit of 1/2. Table 8 also confirms the theoretical limit of the range of validity of Gazis et al.’s model for sufficiently small Poisson’s ratio (due to the requirement of positive definiteness of the stiffness matrix), for a Poisson’s ratio smaller than 2/7, as predicted in Appendix A. Furthermore, it is numerically observed that the condition \(- 1 < \upsilon < {2 \mathord{\left/ {\vphantom {2 7}} \right. \kern-0pt} 7} \approx 0.286\) is also a necessary condition of stability, as highlighted by the negative values of some eigenvalues of the stiffness matrix for \( \nu \) > \(0.286\). Consequently, the auxetic behaviour (negative Poisson’s ratio) of this lattice (central and angular interactions) prevails in the stable domain. This is also the case of the 3D lattice with volumetric interaction (Table 9), where stability is still preserved, as theoretically predicted, in the auxetic domain. As theoretically anticipated in Appendix C, a notable distinction arises in the stability behavior of lattices with volumetric interactions. Specifically, such lattices remain stable up to high values of the macroscopic Poisson's ratio, approaching the incompressibility limit of 1/2. This contrasts with lattices incorporating angular interactions, which exhibit stability constraints at lower Poisson's ratios due to the nature of their non-central force components. To investigate this further, eigenmodes for both central–angular and central–volumetric interaction lattices were analyzed. Tables 10 and 11 present the first five eigenmodes for each configuration. For a fair comparison, both 10 × 10 × 10 cubic lattices were assigned an identical macroscopic Poisson's ratio of 0.2. As detailed in Tables 8 and 9, the dimensionless fundamental eigenfrequency is 2.715 for the lattice with central and angular interactions, and 2.710 for the lattice with central and volumetric interactions. In both cases, this fundamental eigenfrequency corresponds to multiple modes (modes 1, 2, and 3), influenced by the specific non-central interactions present. Interestingly, while the eigenfrequencies differ slightly, the associated mode shapes remain identical across both lattice types. Furthermore, higher-order modes exhibit a convergence in behavior. The subsequent three modes in the higher frequency regime are independent of the Poisson's ratio, each exhibiting a common dimensionless eigenfrequency of 3.129 for both lattice configurations. This observation underscores the nuanced influence of non-central interactions on the dynamic properties of elastic lattices.
Table 4
Numerical and analytical Eigen frequencies Ω for a 30×30×30 lattice with volumetric interaction- υ = 1/3
Mode
Numerical solutions, Ω
Analytical solutions, Ω
1
2.22
2.22
2
2.22
2.22
3
2.22
2.22
4
2.72
2.72
5
2.72
2.72
6
3.14
3.14
7
3.14
3.14
8
3.14
3.14
9
3.51
3.51
10
3.51
3.51
11
3.51
3.51
12
3.51
3.51
13
3.51
3.51
14
3.51
3.51
15
3.84
3.84
16
3.84
3.84
17
3.84
3.84
18
3.84
3.84
19
3.84
3.84
20
3.84
3.84
Table 5
Analytical Eigenfrequencies Ω of a 30×30×30 lattice with volumetric interaction; selection of Eigenfrequencies - \(\nu\) = 1/3
k1
k2
k3
Ω1
Ω2
Ω3
Selected
1
0
0
1.57
1.57
3.14
Ω3
0
1
0
1.57
1.57
3.14
Ω3
0
0
1
1.57
1.57
3.14
Ω3
1
1
0
2.22
2.22
4.44
Ω3, Ω2
1
0
1
2.22
2.22
4.44
Ω3, Ω2
0
1
1
2.22
2.22
4.44
Ω3, Ω2
1
1
1
2.72
2.72
5.43
Ω3, Ω2, Ω1
2
0
0
3.14
3.14
6.27
Ω3
0
2
0
3.14
3.14
6.27
Ω3
0
0
2
3.14
3.14
6.27
Ω3
0
1
2
3.51
3.51
7.00
Ω3, Ω2
0
2
1
3.51
3.51
7.00
Ω3, Ω2
1
0
2
3.51
3.51
7.00
Ω3, Ω2
2
0
1
3.51
3.51
7.00
Ω3, Ω2
1
2
0
3.51
3.51
7.00
Ω3, Ω2
2
1
0
3.51
3.51
7.00
Ω3, Ω2
1
1
2
3.84
3.84
7.66
Ω3, Ω2, Ω1
1
2
1
3.84
3.84
7.66
Ω3, Ω2, Ω1
2
1
1
3.84
3.84
7.66
Ω3, Ω2, Ω1
Table 6
Numerical and analytical Eigenfrequencies Ω for a 10×10×10 lattice with volumetric interaction - \(\nu\) = 1/3
Mode
Numerical solutions, Ω
Analytical solutions, Ω
1
2.21
2.21
2
2.21
2.21
3
2.21
2.21
4
2.69
2.69
5
2.69
2.69
6
3.13
3.13
7
3.13
3.13
8
3.13
3.13
9
3.46
3.46
10
3.46
3.46
11
3.46
3.46
12
3.46
3.46
13
3.46
3.46
14
3.46
3.46
15
3.74
3.74
16
3.74
3.74
17
3.74
3.74
18
3.80
3.80
19
3.80
3.80
20
3.80
3.80
Table 7
Analytical Eigenfrequencies Ω of a 10×10×10 lattice with volumetric interaction; selection of Eigen frequencies - \(\nu\) = 1/3
k1
k2
k3
Ω1
Ω2
Ω3
Selected
1
0
0
1.56
1.56
3.13
Ω3
0
1
0
1.56
1.56
3.13
Ω3
0
0
1
1.56
1.56
3.13
Ω3
1
1
0
2.21
2.21
4.38
Ω3, Ω2
1
0
1
2.21
2.21
4.38
Ω3, Ω2
0
1
1
2.21
2.21
4.38
Ω3, Ω2
1
1
1
2.69
2.69
5.34
Ω3, Ω2, Ω1
2
0
0
3.09
3.09
6.18
Ω3
0
2
0
3.09
3.09
6.18
Ω3
0
0
2
3.09
3.09
6.18
Ω3
0
1
2
3.46
3.46
6.83
Ω3, Ω2
0
2
1
3.46
3.46
6.83
Ω3, Ω2
1
0
2
3.46
3.46
6.83
Ω3, Ω2
2
0
1
3.46
3.46
6.83
Ω3, Ω2
1
2
0
3.46
3.46
6.83
Ω3, Ω2
2
1
0
3.46
3.46
6.83
Ω3, Ω2
1
1
2
3.74
3.80
7.42
Ω3, Ω2, Ω1
1
2
1
3.74
3.80
7.42
Ω3, Ω2, Ω1
2
1
1
3.74
3.80
7.42
Ω3, Ω2, Ω1
Table 8
Numerical Eigenfrequencies Ω for a 10×10×10 Gazis et al. lattice
Mode
\(\nu\) =-0.75
\(\nu\) = -0.5
\(\nu\) = -0.25
\(\nu\) = 0
\(\nu\) = 0.2
\(\nu\) = 0.25
\(\nu\) = 0.275
\(\nu\) = 0.285
\(\nu\) = 0.286
\(\nu\) = 0.287
\(\nu\) = 0.3
\(\nu\) = 0.4
1
3.129
3.129
3.129
3.129
2.715
2.555
2.462
2.421
1.674
–
–
–
2
3.129
3.129
3.129
3.129
2.715
2.555
2.462
2.421
1.674
–
–
–
3
3.129
3.129
3.129
3.129
2.715
2.555
2.462
2.421
1.674
–
–
–
4
3.776
3.646
3.455
3.148
3.129
3.103
2.986
2.570
2.185
0.961
–
–
5
3.776
3.646
3.455
3.148
3.129
3.103
2.986
2.570
2.185
0.961
–
–
6
3.776
3.646
3.455
3.148
3.129
3.129
3.129
2.570
2.185
0.961
–
–
7
4.440
4.434
4.232
3.847
3.305
3.129
3.129
2.935
2.417
1.937
–
–
8
4.440
4.434
4.232
3.847
3.305
3.129
3.129
2.935
2.417
1.937
–
–
9
4.440
4.434
4.425
4.411
4.256
3.999
3.851
2.935
2.417
2.413
–
–
10
4.632
4.470
4.425
4.411
4.256
3.999
3.851
2.935
2.742
2.413
–
–
11
4.632
4.470
4.425
4.411
4.256
3.999
3.851
2.935
2.742
2.413
–
–
12
5.444
5.434
5.419
4.946
4.256
3.999
3.851
3.129
2.930
2.857
–
–
13
5.945
5.738
5.435
4.946
4.256
3.999
3.851
3.129
2.930
2.857
–
–
14
5.945
5.738
5.435
4.946
4.256
3.999
3.851
3.129
3.129
2.857
–
–
15
5.945
5.738
5.435
4.946
4.394
4.317
4.148
3.358
3.129
2.857
–
–
16
5.945
5.738
5.435
4.946
4.394
4.317
4.148
3.358
3.129
2.857
–
–
17
5.945
5.738
5.435
4.946
4.394
4.317
4.148
3.787
3.437
2.857
–
–
18
5.945
5.738
5.435
5.386
4.607
4.388
4.223
3.787
3.437
2.925
–
–
19
6.180
6.180
5.935
5.386
4.607
4.388
4.223
3.787
3.437
2.925
–
–
20
6.180
6.180
5.935
5.386
4.607
4.388
4.223
3.787
3.437
3.129
–
–
“–” indicates that the Eigenfrequency is an imaginary value
Table 9
Numerical Eigenfrequencies Ω for a 10×10×10 cubic lattice with volumetric interaction
Mode
\(\nu\) = − 0.75
\(\nu\) = − 0.5
\(\nu\) = − 0.25
\(\nu\) = 0
\(\nu\) = 0.2
\(\nu\) = 0.25
\(\nu\) = 0.275
\(\nu\) = 0.285
\(\nu\) = 0.286
\(\nu\) = 0.287
\(\nu\) = 0.3
\(\nu\) = 0.4
1
3.129
3.129
3.129
3.129
2.710
2.555
2.465
2.426
2.422
2.418
2.365
1.806
2
3.129
3.129
3.129
3.129
2.710
2.555
2.465
2.426
2.422
2.418
2.365
1.806
3
3.129
3.129
3.129
3.129
2.710
2.555
2.465
2.426
2.422
2.418
2.365
1.806
4
3.740
3.613
3.427
3.129
3.129
3.103
2.994
2.947
2.942
2.938
2.873
2.194
5
3.740
3.613
3.427
3.129
3.129
3.103
2.994
2.947
2.942
2.938
2.873
2.194
6
3.740
3.613
3.427
3.129
3.129
3.129
3.129
3.129
3.129
3.129
3.129
2.828
7
4.409
4.388
4.163
3.801
3.291
3.129
3.129
3.129
3.129
3.129
3.129
2.828
8
4.409
4.388
4.163
3.801
3.291
3.129
3.129
3.129
3.129
3.129
3.129
2.828
9
4.409
4.407
4.403
4.398
4.242
3.999
3.859
3.799
3.792
3.786
3.703
2.828
10
4.542
4.407
4.403
4.398
4.242
3.999
3.859
3.799
3.792
3.786
3.703
2.828
11
4.542
4.407
4.403
4.398
4.242
3.999
3.859
3.799
3.792
3.786
3.703
2.828
12
5.442
5.432
5.366
4.898
4.242
3.999
3.859
3.799
3.792
3.786
3.703
3.052
13
5.855
5.656
5.366
4.898
4.242
3.999
3.859
3.799
3.792
3.786
3.703
3.052
14
5.855
5.656
5.366
4.898
4.242
3.999
3.859
3.799
3.792
3.786
3.703
3.052
15
5.855
5.656
5.366
4.898
4.391
4.317
4.165
4.100
4.093
4.087
3.997
3.103
16
5.855
5.656
5.366
4.898
4.391
4.317
4.165
4.100
4.093
4.087
3.997
3.103
17
5.855
5.656
5.366
4.898
4.391
4.317
4.165
4.100
4.093
4.087
3.997
3.103
18
5.855
5.656
5.418
5.287
4.579
4.388
4.234
4.168
4.161
4.155
4.063
3.129
19
6.180
6.105
5.792
5.287
4.579
4.388
4.234
4.168
4.161
4.155
4.063
3.129
20
6.180
6.105
5.792
5.287
4.579
4.388
4.234
4.168
4.161
4.155
4.063
3.129
Table 10.
First 5 mode shapes of 10×10×10 lattice with angular interactions, \(\nu\) = 0.2. Diagonal members are turned off for clearer illustrations.
Table 11.
First 5 mode shapes of 10×10×10 lattice with volumetric interactions, \(\nu\) = 0.2. Diagonal members are turned off for clearer illustrations.
5 Concluding Remarks
In this paper, the static and dynamic responses of some linear elastic cubic lattices with central and non-central interactions were studied. Two non-central interactions, angular and volumetric, were considered. The calibration of the equivalent Poisson’s ratio of the linear isotropic continuum in the asymptotic limit is discussed for both cubic lattices that include non-central interactions, the lattice with central and angular interaction, and the one with central and volumetric interaction. For the 3D lattice with central and angular interactions (Gazis et al.’s lattice), the calibration is shown to be valid only for sufficiently small values of Poisson’s ratio to preserve the positive definiteness of the potential energy. A complete formulation of cubic lattices that asymptotically converge towards continuum isotropic elasticity with large values of Poisson’s ratio may be achieved introducing a volumetric interaction of Fuchs-type (volumetric dependent potential). It is shown that this volumetric interaction, physically equivalent to a volume-dependent internal pressure, covers a large variation of elasticity parameters, with a complete calibration of the equivalent Poisson’s ratio up to its incompressibility limit 1/2. The mixed differential-difference equations of the associated lattice with discrete volumetric interactions were derived applying Hamilton’s principle to the discrete energies. The positive definiteness of the lattice potential energy is also discussed, for each lattice with central and non-central interactions, from the algebraic property of the discrete cell stiffness matrix. The static responses of these finite lattices are studied for academic loadings (pure compression or pure shear modes) and are compared to their continuous asymptotic limits. The discrete Lamé problem for the free vibration of this rectangular parallelepiped is solved for each lattice with central, angular or volumetric interactions. The convergence of each finite lattice towards the linear elastic continuous rectangular parallelepiped is illustrated from this discrete eigenvalue problem. It is concluded that the connection between discrete and continuum elasticity can be achieved from this cubic lattice with a complete range of elasticity parameters.
Acknowledgements
Giuseppe Ruta gratefully acknowledges the financial support of the national grants PRIN 2022Y2RHHT-002 and PRIN PNRR P2022ATTAR of the Italian Ministry for University and Research.
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Appendix A: Positive definiteness of the stiffness matrix of Gazis et al lattice
For a cubic cell with 8 material nodes at the corners interacting with central and angular interactions (Gazis et al.’s cell), the local (one-cell) stiffness matrix associated with the nodal displacement vector can be determined by the stationarity of the associated potential with respect to each nodal component, thus leading to a 24 × 24 symmetric matrix KG.
The eigenvalues of this one-cell stiffness matrix, accounting for central and angular interactions, are listed below, and may be expressed as function of Lamé parameters λ and μ:
Six of the eigenvalues listed in Eq. (67) vanish, which is physically justified by the fact that, since the cell is unconstrained, the components of a spatial rigid body motion are undetermined. Consequently, the stiffness matrix is six times singular. The remaining nonzero eigenvalues all remain positive for:
Within the range \(- 1 < \upsilon < {2 \mathord{\left/ {\vphantom {2 7}} \right. \kern-0pt} 7}\), the multiple non-vanishing eigenvalues of the stiffness matrix remain positive, ensuring the positive definiteness of both the local stiffness matrix and the total elastic potential—a necessary condition for mechanical stability in the sense of Lyapunov. Thus, the condition \(- 1 < \upsilon < {2 \mathord{\left/ {\vphantom {2 7}} \right. \kern-0pt} 7}\) is a sufficient criterion for stability as it guarantees the symmetric and positive definite nature of the global stiffness matrix for an n × m × p cells system. Moreover, as shown in Table 4 for a 10 × 10 × 10 cell system, this condition \(- 1 < \upsilon < {2 \mathord{\left/ {\vphantom {2 7}} \right. \kern-0pt} 7} \approx 0.286\) is also a necessary condition of stability, as highlighted by the negative values of some eigenvalues of the stiffness matrix for \(\upsilon\) > \(0.286\).
Appendix B: Characterization of volumetric interaction
Consider a cell composed by four equal masses initially located at the vertexes of a cube with side of length \(a\), parallel to the axes \(x,y,z\) of a Cartesian frame, labelled as shown in Fig. 1.
Let us imagine that the plane facets of the cube are material surfaces, and that the four masses undergo displacements that let all straight segments traced in the facets and across the cube in this initial configuration turn into straight segments in a deformed configuration of the cube. Thus, for instance, in the facet on the \(xy\) plane one may write that
and analogously for the other displacement components on all facets of the cube and on all nodes. Then, the fields of the displacement components \(u,v,w\) of the material points on the facets of the cube parallel to the Cartesian coordinate axes are
Thus, the boundary of the deformed shape of the cube (as shown in Fig. 6) is defined through Eq. (71) and requires the specification of 24 quantities. These quantities may be either finite or infinitesimal, thus defining either a finite or infinitesimal deformation of the facets of the cube, hence of the cube itself.
Since we admitted that all straight segments remain straight in a deformation of the cube, irrespective of them being on the facets or across the volume, we may pose
and the analogous for \(v\left(x,y,z\right), w\left(x,y,z\right)\) simply by replacing \(v, w\) in the corresponding entries with the same subscripts; this symmetry is obviously a consequence of the initial symmetry of the cube and of the linearity assumed for the deformation of every straight segment.
We then have that the new placement of any material point inside the cube is given by
with \(\left\{{{\varvec{e}}}_{1},{{\varvec{e}}}_{2},{{\varvec{e}}}_{3}\right\}\) a basis of unit vectors consistent with the Cartesian frame adopted. Then, it is possible to evaluate, according to continuum mechanics theory,
Such an expression may be expanded if one admits that each displacement component at each node is linear in an evolution parameter on which the displacement is supposed to regularly depend, that is,
Let us now admit that there is a uniform pressure of intensity \(p\) acting on every facet of the cube in its initial configuration; if the facets undergo the displacement expressed by Eq. (71), and supposing that the displacement is infinitesimal, so that the pressure remains orthogonal to every facet of the cube, the inner pressure spends a total amount of work given by
Each couple of terms forming a difference in Eq. (79) expresses infinitesimal elongations of the links joining the nodes: for instance, \(- \delta w_{i,j,h} + \delta w_{i,j,1 + h} : = a \delta \varepsilon_{{i,j,\left( {h,h + 1} \right)}}\) is the infinitesimal elongation of the link initially oriented along the \(z\)-axis and joining the nodes \(\left(i,j,h\right)\) and \(\left(i,j,h+1\right)\) in terms of the unit infinitesimal strain \({\delta \varepsilon }_{i,j,(h,h+1)}\). Thus, one may re-write the result in Eq. (79) as follows
By grouping the unit strains along the Cartesian axes and averaging with respect to the denominator in the fraction facing the brackets, Eq. (80) becomes
with \({\overline{\varepsilon }}_{yy}, {\overline{\varepsilon }}_{xx}\) defined analogously. It is apparent that, by recalling Eqs. (75), (81) may be rewritten as
Appendix C: Positive definiteness of the stiffness matrix of the lattice with volumetric interaction
For a cubic cell with 8 material nodes at the corners interacting with central and volumetric interactions, the local (one-cell) stiffness matrix associated with the nodal displacement vector can be determined by the stationarity of the associated potential with respect to each nodal component, thus leading to a 24 × 24 symmetric matrix KV.
For the eigenvalues Eq. (86) we may provide comments like those in "Appendix A"; the stiffness matrix is six times singular. The remaining nonzero eigenvalues remain positive for:
Within the range \(- 1 < \upsilon < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2},\) i.e., the range of positive definiteness of the asymptotic isotropic elastic continuum, the (multiple) non-vanishing eigenvalues of the stiffness matrix remain positive, ensuring the positive definiteness of both the local stiffness matrix and the total elastic potential (necessary condition for mechanical stability in the sense of Lyapunov). Thus, the condition \(- 1 < \upsilon < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}\) is a sufficient criterion for stability as it guarantees the symmetric and positive definite nature of the global stiffness matrix for an n × m × p cells system. Moreover, as shown from Table 5 for a 10 × 10 × 10 cell system, this condition \(- 1 < \upsilon < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}\) is also a necessary condition of stability, as highlighted by the positive values of the eigenvalues of the stiffness matrix within this all domain of variation of the Poisson’s ratio (and the positivity of the eigenfrequencies).
1.
Alexander S (1998) Amorphous solids: their structure, lattice dynamics and elasticity. Phys Rep 296:65–236CrossRef
2.
Atluri SN, Reissner E (1989) On the formulation of variational theorems involving volume constraints. Comput Mech 5:337–344CrossRef
3.
Bilek O, Skala L (1978) From finite to infinite crystals: analytic solution of simple tight binding model of finite SC, FCC and BCC crystals of arbitrary size. Czech J Phys B 28:1003–1019CrossRef
4.
Born M, von Kármán T (1912) Über Schwingungen in Raumgittern. Physikalishe Zeitschrift 13:297–309 ((in german))
5.
Capecchi D, Ruta G, Trovalusci P (2010) From classical to Voigt’s molecular models in elasticity. Arch Hist Exact Sciences 64:525–559MathSciNetCrossRef
6.
Capecchi D, Ruta G, Trovalusci P (2011) Voigt and Poincaré’s mechanistic-energetic approaches to linear elasticity and suggestions for multiscale modelling. Arch Appl Mech 81:1573–1584CrossRef
7.
Cauchy A., Sur les différences finies et les intégrales aux différences des fonctions entières d’une ou de plusieurs variables, Exercices de mathématiques, page 155- 159, 1828.
8.
Chakraborty S. and Kumar H., Molecular dynamics simulations of two-dimensional materials, in: Yang E.H. Datta D., Ding J. and Hader G. (Ed.), Synthesis, Modeling, and Characterization of 2D Materials, and Their Heterostructures, Micro and Nano Technologies, Pages 125–148, 2020.
9.
Challamel N, Atanackovic T, Zhang YP, Wang CM (2022) A fractional nonlocal elastic model for lattice wave analysis. Mech Res Comm 126(103999):1–9
10.
Challamel N, Zhang YP, Wang CM, Ruta G, Dell’Isola F (2023) Discrete and continuous models of linear elasticity: history and connections. Continuum Mech Thermodyn 35(2):347–391MathSciNetCrossRef
11.
Challamel N, Ruta G, Nguyen HP, Wang CM, Zhang YP (2024) On angular and surface interactions in two-dimensional elastic lattices. Eur. J. Mech. Solids 104(105176):1–25MathSciNet
12.
Challamel N., Nguyen H.P., Wang C.M. and Ruta G., In-plane vibrations of elastic lattice plates and their continuous approximations, Mathematics, “Special Issue "Multiscale Mathematical Modeling", Andrianov I. and Kaplunov J. (Ed.), 12, 2312, 1–27, 2024b.
13.
Challamel N, Nguyen HP, Wang CM, Ruta G (2025) Derivation of Born/von Kármán difference equations through consistent lattice angular interactions. Chaos, Solitons Fractals 195(116251):1–29
14.
Cousins CSG (1971) New relations between elastic constants of different orders for structure-independent interactions with application to electrons in metals. J Physics F: Metal Physics 1(6):815–820CrossRef
15.
De Launay JD (1953) Lattice dynamics of body-centered and face-centered cubic metallic elements. J Chem Physics 21(11):1975–1986CrossRef
16.
De Launay J.D., Solid state physics, Vol. 2, Eds F. Seitz and D. Turnbull, Academic Press, pp. 219–303, 1956.
17.
Ekiz E, Steinmann P, Javili A (2022) From two- to three-dimensional continuum-kinematics inspired peridynamics: More than just another dimension. Mech Mater 173(104417):1–19
18.
Ekstein H, Schiffman T (1956) Free vibrations of isotropic cubes and nearly cubic parallelepipeds. J Appl Phys 27(4):405–412MathSciNetCrossRef
19.
Esin M, Pasternak E, Dyskin A (2016) Stability of chains of oscillators with negative stiffness normal, shear and rotational springs. Int J Eng Sci 108:16–33MathSciNetCrossRef
20.
Faltus O, Jirásek M, Horák M, Doškář M, Peerlings R, Zeman J, Rokoš O (2025) Towards active stiffness control in pattern-forming pneumatic metamaterials. European Journal of Mechanics / A Solids 112:105632CrossRef
21.
Foce F., The theory of elasticity between molecular and continuum approach in the XIXth century, in: P. Radelet de Grave, E. Benvenuto (Eds.), Between Mechanics and Architecture, Birkhaüser, 1995.
22.
Fromm JA, Leissa AW (1970) Free vibration of the rectangular parallelepiped. J Acous Soc Am 48:290–298CrossRef
23.
Fuchs K (1936) A quantum mechanical calculation of the elastic constants of monovalent metals. Proc. R. Soc. A: Math. Phys. Eng. Sci. 153(880):1893–1895
24.
Gazis DC, Herman R, Wallis RF (1960) Surface elastic waves in cubic crystals. Phys Rev 119:533–544MathSciNetCrossRef
25.
Gazis D.C. and Wallis R.F., Conditions for rotational invariance of a harmonic lattice, Phys. Rev., 151, 2, 578, 580, 1966 – see also Gazis and Wallis R.F., Erratum, Phys. Rev., 156, 1038, 1967.
26.
Green G., On the reflection and refraction of light at the common surface of two non-crystallized media, 1839; in: Mathematical papers, Ed. Norman Macleod Ferrrers, 245–269, London, MacMillan, 1971.
27.
Hill EVK, Egle DM (1982) The forced vibrational response of rectangular parallelepiped with rigid-lubricated boundaries. J Sound Vibration 80(1):61–69CrossRef
28.
Hrennikoff A (1941) Solutions of problems of elasticity by the framework method. J Appl Mech 8:A169–A175MathSciNetCrossRef
29.
Hrennikoff A (1949) Framework method and its technique for solving plane stress problems. IABSE Publications 9:217–248
30.
Hutchinson JR, Zillmer SD (1983) Vibration of a Free Rectangular Parallelepiped. J Appl Mech 50:123–130CrossRef
31.
Johnson R (1972) Relationship between two-body interatomic potentials in a lattice model and elastic constants. Phys Rev B 6(6):2094–2100CrossRef
32.
Keating PN (1966) Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure. Phys Rev 145(2):637–645CrossRef
33.
Kochmann D (2012) Stability criteria for continuous and discrete elastic composites and the influence of geometry on the stability of a negative-stiffness phase. Phys Status Solidi B 249(7):1399–1411CrossRef
34.
Kochmann D, Bertoldi K (2017) Exploiting microstructural instabilities in solids and structures: from metamaterials to structural transitions. Appl Mech Rev 69(050801):1–24
35.
Lamé G., Treizième leçon - Etats vibratoires du prisme rectangle, in : Leçons sur la théorie mathématique de l’élasticité des corps solides, Gauthier Villars, Paris, 1852, also 2nd Edn (1866).
36.
Lax M., E1. The relation between microscopic and macroscopic theories of elasticity, in: Lattice dynamics, Edited by R.F. Wallis, 583–596, Pergamon Press, New-York, 1965.
37.
Lazar M, Agiasofitou E, Böhlke T (2022) Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin-Mindlin anisotropic first strain gradient elasticity. Cont Mech Thermodyn 34:107–136MathSciNetCrossRef
38.
Liu WK, Li S, Park HS (2022) Eighty Years of the Finite Element Method: Birth, Evolution, and Future. Arch Comput Meth Engng 29:4431–4453CrossRef
39.
Love A.E.H., A Treatise on the Mathematical Theory of Elasticity – Vol. 1, Cambridge University Press Warehouse, London, 354 pages, 1892.
40.
Maalouf SR, Vel SS (2022) Nonlinear elastic analysis of 2D materials of arbitrary symmetries with application to black phosphorus. Mech Mat 165:104159CrossRef
41.
Maalouf SR, Vel SS (2023) Nonlinear elastic behavior of 2D materials using molecular statics and comparisons with first principles calculations. Physica E 148:1156332023CrossRef
42.
Martin JW (1975) Many-body forces in metals and the Brugger elastic constants. J Phys C: Solid State Phys 8(18):2837–2857CrossRef
43.
McHenry D (1943) A lattice analogy for the solution of stress problems, The Journal of Inst. Civil Engineers, London 2(5350):59–82CrossRef
44.
Mindlin R.D., Theories of elastic continua and crystal lattice theories, in: Kröner E. (Ed.), Mechanics of Generalized Continua, Proceedings of the IUTAM-Symposium on the Generalized Cosserat Continuum and the Continuum Theory of dislocations with applications, Freudenstadt and Stuttgart, Germany, p. 312–320, 1967, Published by Springer, 1968.
45.
Mindlin RD (1986) On vibrations of rectangular parallelepipeds. Int J Solids Structures 22(12):1423–1430CrossRef
46.
Nadeau G., Introduction to elasticity, New-York: Holt, Rinehart and Winston, Inc, pp. 271–274, 1964.
47.
Navier L., Sur les lois de l’équilibre et du mouvement des corps solides élastiques, Bulletin des sciences par la Société Philomatique de Paris pp 177–181, 1823 (in French).
48.
Nguyen HP, Challamel N, Wang CM (2025) Analytical solutions for free vibrations of rectangular cuboid elastic lattices and their continuous approximations. J Sound Vibration 600(118836):1–28
49.
Piola G (1836) Nuova analisi per tutte le questioni della meccanica molecolare. Memorie di matematica e fisica della Societá italiana delle scienze 21:155–163 ((in Italian))
50.
Poincaré H (1892) Leçons sur la théorie de l’élasticité. Carré, Paris
51.
Strutt J.W. (Lord Rayleigh), The Theory of Sound, Volume I, page 405, Dover, New-York, 1894.
52.
Summerfeld A., The mechanics of deformable bodies, Chapter VIII (8) "Supplement to the theory of elasticity", Academic Press, New-York, 1950.
53.
Timoshenko S., History of strength of materials with a brief account of the history of theory of elasticity and theory of structures, McGraw-Hill, 1953.
54.
Thomas JF (1971) Failure of the Cauchy relation in cubic metals. Scr Metall 5:787–790CrossRef
55.
Voigt V., L’état actuel de nos connaissances sur l’élasticité des cristaux. In : Rapport présenté au Congrès International de Physique, Rassemblés et publiés par Ch. Ed. Guillaume, L. Poincaré, 277–347, Paris, Gauthier-Villars, 1900.
56.
Voigt V., Theoretische Studien über die Elasticitätsverhältnisse der Kristalle, Abhandlungen der Gesellschaft der Wissenschaften Zu Göttingen—Mathematische Classe, 34, 1–100, 1887.
57.
Zienkiewicz O.C. and Taylor R.L., The Finite Element Method – Volume 1: The Basis, Fifth Edition, Butterworth-Heineman, Oxford, 2000.
