Generalized continuum modeling of 2-D periodic cellular solids

Kumar, Raj; McDowell, David L.

doi:10.1016/j.ijsolstr.2004.06.038

Cited by 180 publications

(116 citation statements)

References 16 publications

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“…which are identical to those obtained for a triangular lattice in Kumar and McDowell (2004) and derived without considering second or higher-order terms in the Taylor-series expansion of displacements of Eqs. (19)- (21).…”

Section: Case 1: Rigid Nodessupporting

confidence: 81%

Elasto-static micropolar behavior of a chiral auxetic lattice

Spadoni

Ruzzene

2012

Journal of the Mechanics and Physics of Solids

295

158

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a b s t r a c tAuxetic materials expand when stretched, and shrink when compressed. This is the result of a negative Poisson's ratio n. Isotropic configurations with n % À1 have been designed and are expected to provide increased shear stiffness G. This assumes that Young's modulus and n can be engineered independently. In this article, a micropolarcontinuum model is employed to describe the behavior of a representative auxetic structural network, the chiral lattice, in an attempt to remove the indeterminacy in its constitutive law resulting from n ¼ À1. While this indeterminacy is successfully removed, it is found that the shear modulus is an independent parameter and, for certain configurations, it is equal to that of the triangular lattice. This is remarkable as the chiral lattice is subject to bending deformation of its internal members, and thus is more compliant than the triangular lattice which is stretch dominated. The derived micropolar model also indicates that this unique lattice has the highest characteristic length scale l c of all known lattice topologies, as well as a negative first Lamé constant without violating bounds required for thermodynamic stability. We also find that hexagonal arrangements of deformable rings have a coupling number N ¼1. This is the first lattice reported in the literature for which couple-stress or Mindlin theory is necessary rather than being adopted a priori.

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Section: Case 1: Rigid Nodessupporting

confidence: 81%

Elasto-static micropolar behavior of a chiral auxetic lattice

Spadoni

Ruzzene

2012

Journal of the Mechanics and Physics of Solids

295

158

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“…By an application of the Divergence theorem (see Kumar and McDowell (2004)) the seventh term becomes ( …”

Section: Variational Derivation Of the Micropolar Modelmentioning

confidence: 99%

Dispersive wave propagation in two-dimensional rigid periodic blocky materials with elastic interfaces

Bacigalupo

Gambarotta

2017

Journal of the Mechanics and Physics of Solids

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Dispersive waves in two-dimensional blocky materials with periodic microstructure made up of equal rigid units having polygonal centro-symmetric shape with mass and gyroscopic inertia, connected each other through homogeneous linear interfaces, have been analysed. The acoustic behavior of the resulting discrete Lagrangian model has been obtained through a Floquet-Bloch approach. From the resulting eigenproblem derived by the Euler-Lagrange equations for harmonic wave propagation, two acoustic branches and an optical branch are obtained in the frequency spectrum. A micropolar continuum model to approximate the Lagrangian model has been derived based on a second-order Taylor expansion of the generalized macro-displacement field. The constitutive equations of the equivalent micropolar continuum have been obtained, with the peculiarity that the positive definiteness of the second-order symmetric tensor associated to the curvature vector is not guaranteed and depends both on the ratio between the local tangent and normal stiffness and on the block shape. The same results has been obtained through an extended Hamiltonian derivation of the equations of motion for the equivalent continuum that is related to the Hill-Mandel macro homogeneity condition. Moreover, it is shown that the hermitian matrix governing the eigenproblem of harmonic wave propagation in the micropolar model is exact up to the second order in the norm of the wave vector with respect to the same matrix from the discrete model. To appreciate the acoustic behavior of some relevant blocky materials and to understand the reliability and the validity limits of the micropolar continuum model, some blocky patterns have been analysed: rhombic and hexagonal assemblages and running bond masonry. From the results obtained in the examples, it appears that the obtained micropolar model turns out to be particularly accurate to describe dispersive functions for wavelengths greater than 3-4 times the characteristic dimension of the block. Finally, in consideration that the positive definiteness of the second order elastic tensor of the micropolar model is not guaranteed, the hyperbolicity of the equation of motion has been investigated by considering the Legendre-Hadamard ellipticity conditions requiring real values for the wave velocity.

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“…This suggests that the chiral lattice indeed has a negative Poisson's ratio, but this depends on the particular configuration and approaches -1 but it is not exactly equal to this value. Following this minimum value, Poisson's ratio increases rapidly with a boundary-layer-like behavior to the limiting value of the triangular lattice for which ν = 1/3 [10].…”

Section: Resultsmentioning

confidence: 94%

“…The coefficients of such equations are compared to known equilibrium-equation forms such as Classical Elasticity Theory to obtain the equivalent Lamé constants. In a similar fashion, the method elucidated in [10], for elasto-static phenomena, relates Taylor-series-linearized displacements at the extremities of the unit cell to those of the central joint. The approximate kinematic variables are then employed to obtain an expression of the unit-cell strain energy in terms of geometric parameters of the lattice.…”

Section: Overview Of Methods To Determine the Me-chanics Of Cellular mentioning

confidence: 99%

An Isotropic Auxetic Structural Network With Limited Shear Stiffness

Spadoni

2011

Volume 8: Mechanics of Solids, Structures and Fluids; Vibration, Acoustics and Wave Propagation

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The chiral lattice is a unique structural network not symmetric to its mirror image, and with a negative Poisson's ratio. Previous investigations have considered this structural network for the design of superior structural components with sandwich construction, but these were limited by the in-plane Poisson's ratio predicted to be exactly −1. This paper presents estimates of the mechanical properties of the chiral lattice obtained from a multi-cell finite-element model. It is shown that the chiral lattice has a shear stiffness bound by that of the triangular lattice and it is very compliant to direct stresses. The minimum in-plane poisson's ratio is estimated to be ≈ −0.94. INTRODUCTIONCellular solids possess superior mechanical properties [1] that can be exploited for the development of novel structured materials. Their wide-spread employment in the aerospace, automotive, and naval industries, among others, is to be attributed primarily to unique physical properties: low relative density, low electrical conductivity, low Young's modulus and strength [1]. Low density facilitates the design components with high specific (mass-normalized) stiffness, and provides effective thermal insulation [1,2]. The latter is to be attributed to the low conductivity of the second phase, most often a gas. Low density is also ideal for naval applications where buoyancy and specific stiffness are required [1]. Low strength is very advantageous in applications where mechanical-energy absorption is paramount. This may be

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Generalized continuum modeling of 2-D periodic cellular solids

Cited by 180 publications

References 16 publications

Elasto-static micropolar behavior of a chiral auxetic lattice

Elasto-static micropolar behavior of a chiral auxetic lattice

Dispersive wave propagation in two-dimensional rigid periodic blocky materials with elastic interfaces

An Isotropic Auxetic Structural Network With Limited Shear Stiffness

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