Analogy between micropolar continuum and grid frameworks under initial stress

Bažant, Zdeněk P.; Christensen, Mike

doi:10.1016/0020-7683(72)90093-5

Cited by 164 publications

(90 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The behavior of this lattice is then described by couple-stress or Mindlin theory (Mindlin, 1963). So far couple-stress-elasticity behavior has been assumed a priori for certain configurations (Banks and Sokolowski, 1968) (incorrectly in this case, see remarks in Bažant and Christensen, 1972), or it was concluded based on experimental measurements Lakes, 1981, 1982). A hexagonal packing of rings therefore may provide a useful test bed to experimentally measure the implications of Mindlin theory.…”

Section: Comparison Of Chiral and Traditional Latticesmentioning

confidence: 99%

Elasto-static micropolar behavior of a chiral auxetic lattice

Spadoni

Ruzzene

2012

Journal of the Mechanics and Physics of Solids

293

171

View full text Add to dashboard Cite

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.

show abstract

Section: Comparison Of Chiral and Traditional Latticesmentioning

confidence: 99%

Elasto-static micropolar behavior of a chiral auxetic lattice

Spadoni

Ruzzene

2012

Journal of the Mechanics and Physics of Solids

293

171

View full text Add to dashboard Cite

show abstract

“…They are based on the Floquet-Bloch theorem, which enables to reduce the study of the whole structure to the study of the unit cell. Another approach consists in replacing a discrete structure by an equivalent continuum [13][14][15] . When the wavelength is much greater than the cell size, it is possible to use the rigorous asymptotic methods of homogenization which require no assumption about the nature of the continuum.…”

Section: Fig 1 Example Of Structuresmentioning

confidence: 99%

Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics

Chesnais

Boutin

Hans

2012

The Journal of the Acoustical Society of America

View full text Add to dashboard Cite

This work is devoted to the study of the wave propagation in infinite two-dimensional structures made up of the periodic repetition of frames. Such materials are highly anisotropic and, because of lack of bracing, can present a large contrast between the shear and compression deformabilities. Moreover, when the thickness to length ratio of the frame elements is small, these elements can resonate in bending at low frequencies, when compressional waves propagate in the structure. The frame size being small compared to the wavelength of the compressional waves, the homogenization method of periodic discrete media is extended to situations with local resonance and it is applied to identify the macroscopic behavior at the leading order. In particular, the local resonance in bending leads to an effective mass different from the real mass and to the generalization of the Newtonian mechanics at the macroscopic scale. Consequently, compressional waves become dispersive and frequency bandgaps occur. The physical origin of these phenomena at the microscopic scale is also presented. Finally, a method is proposed for the design of such materials.

show abstract

“…The approaches for solving these repetitive cell structures problems (Ba zant and Christensen, 1972) include (a) applying finite difference calculus and obtaining the exact solutions/series solutions or (b) making a continuous approximation. The continuous approximation could be conducted either on the motion of equation (or term as "continualization") via the Taylor series (Mindlin, 1972), or more accurately by the Pad e approximants (Andrianov and Awrejcewicz, 2001) or on the constitutive law for instance Eringen's nonlocal theory (Eringen and Edelen, 1972;Eringen, 1983), strain gradient theory (Mindlin 1964;Askes and Aifantis, 2011) and homogenization of a representative volume element (Odegard et al, 2002).…”

Section: Introductionmentioning

confidence: 99%

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

Duan

Challamel

Wang

et al. 2013

Journal of Applied Physics

View full text Add to dashboard Cite

The present study takes an analytical approach for solving the free vibration problem of a microstructured beam model, in which transverse displacement springs are added to allow for the transverse shear deformation effect in addition to the rotational springs. The exact vibration frequencies for the discrete microstructured beam model with simply supported ends are obtained via matrix decomposition. In addition, a general solution technique involving the use of Pad e approximants for the continualization procedure is proposed in order to obtain the continuous equivalent system for the discrete microstructured beam model. The analytical vibration solutions of the equivalent continuous system are obtained and their accuracy is assessed by using the exact solutions. It is found that the solutions of the equivalent continuous system have a first order accuracy when compared with the exact solutions of their discrete counterpart. The length scale coefficient in the nonlocal Timoshenko beam model is calibrated by using the analytical solutions. Two nonlocal Timoshenko beam models, i.e., the Wang model (without the length scale effect in the shear stress strain relation) and the Reddy model, are evaluated based on their ability to capture the nonlocal effect. V C 2013 AIP Publishing LLC. [http://dx

show abstract

Analogy between micropolar continuum and grid frameworks under initial stress

Cited by 164 publications

References 6 publications

Elasto-static micropolar behavior of a chiral auxetic lattice

Elasto-static micropolar behavior of a chiral auxetic lattice

Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

Contact Info

Product

Resources

About