Stress smoothing holes in planar elastic domains

Vigdergauz, Shmuel

doi:10.2140/jomms.2010.5.987

Cited by 7 publications

(10 citation statements)

References 11 publications

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“…as was already done in [5] for a finite number of strongly interacting holes in a plane. The reason is that, actually, the ESSs are the limiting case of the MSV with globally minimal zero variation as a result of the hoop stresses constancy.…”

Section: Introductionmentioning

confidence: 56%

Stress-smoothing holes in an elastic plate: From the square lattice to the checkerboard

Vigdergauz

2011

Mathematics and Mechanics of Solids

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Minimization of the hoop stresses variation was recently advanced to form a novel and efficient criterion of the stress state optimization in a statically loaded elastic domain with a finite number of free-form holes. As a radical generalization of the well-known equi-stress principle, it offers substantial analytical and numerical advantages over direct minimization of the local stress concentration factor. Here we extend the proposed criterion beyond its primary application to the shape optimization in a regular perforated structure with the checkerboard arrangement of identical traction-free holes. The averaging nature of the stresses variation allows its effective numerical implementation in a wide range of the governing parameters at a modest computational cost. The results obtained describe in depth the elastic response of the optimal structure to applied loads.

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Section: Introductionmentioning

confidence: 56%

Stress-smoothing holes in an elastic plate: From the square lattice to the checkerboard

Vigdergauz

2011

Mathematics and Mechanics of Solids

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“…The presented method of creating periodic structures allows for far richer formation of cell topology than the structures shown here made of only one fiber. For example, it is not difficult to modify the algorithm by introducing a few additional independent fibers or defining voids with different types of shape modeling Vigdergauz [74][75][76] cells.…”

Section: Recovery Of the Optimal Underlying Microstructuresmentioning

confidence: 99%

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

Czarnecki

Łukasiak

2020

Physica Status Solidi (b)

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The article is focused on recovering of two-phase microstructures of effective properties for 2D elasticity (see e.g., ref. [1]) predicted by two, stress-based topology optimization methods: isotropic material design (IMD) and cubic material design (CMD)-see e.g., refs. [2-5]. In these articles, the reconstruction of microstructures in subdomains where the auxetic properties [6-13] of the optimal, nonhomogeneous isotropic or cubic elastic material appear has been undertaken. In the present article, being an extension of the previous work, [14] the homogenization method based on parametric description of the shape of the domain filled by the given isotropic material inside the representative volume elements (RVEs), instead of the combinatorial search among the admissible microstructures, is implemented. Within the 2D setting discussed, the local cubic symmetry of the material reduces to the local symmetry of a square. The main goal is to recover isotropic microstructures or microstructures of symmetry of a square at arbitrary points of the body in a relatively simple way, considering the heterogeneity of the distribution of optimal elastic moduli. In the case of an isotropic material, the results obtained in ref. [14] are used as a starting point to the further search of the auxetic microstructures, realizing the Cherkaev-Gibiansky bounds. [15] In the case of 2D models of materials showing auxetic properties, a series of works devoted to their research at the level of regular atomic structures (e.g., hard cyclic hexamers [HCHs] or hard cyclic tetramers and many others) were particularly important in the development of the theory of such materials-see refs. [16-24]. Especially important in this respect is the ref. [16], in which a constant thermodynamic tension Monte Carlo (MC) simulations revealed the first planar, isotropic (and chiral) thermodynamically stable structure of negative Poisson's ratio in tilted phase formed by HCHs. In ref. [17], the exact solution for negative Poisson's ratio at high densities and in the static limit (i.e., at zero temperature) was obtained for planar, isotropic (and chiral) structure of cyclic hexamers interacting through nearest-neighbouring site-site inversepower potential. In ref. [18], MC simulations performed on hard, cyclic tetramers revealed an interesting feature of such a very simple model of molecules composed of four identical hard disks: depending on its shape, all possible material behavioursregarding negative Poisson's ratio could arise and thermodynamically stable auxetic (or partially auxetic) chiral structures of square symmetry could be spontaneously formed. Analyzing in ref. [19], other 2D cyclic trimers model of triatomic molecules in which "atoms" are distributed on vertices of equilateral triangles, the authors have found the exact solution at zero temperature for the planar auxetic structure which is isotropic but not chiral (soft cyclic trimers). The both chiral and nonchiral molecular models were studied in ref. [20], where the influence of molecular geome...

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“…\ x n b. Specifically, the variation of the hoop stresses serves as an optimization measure to be minimized by appropriately shaping the domain boundary L (the V-criterion [15]) V ½s uu À À ÀÀ À! fLg min :…”

Section: Basic Propertiesmentioning

confidence: 99%

“…In contrast, for shear-dominating loads, the equistressness makes no sense since any stress distribution should have a number of sign-changing points in locations predefined by the shear rotational antisymmetry. Instead, the less restrictive piecewise constant distribution of the hoop stresses (M-equistressness) js uu (t)j = Const, t 2 L ð3:11Þ was adopted and numerically shown to be attained in some non-trivial instances [7,10,15,17]. As each Walsh function is also piecewise constant, the WT of (3.11) involve only one or a few items thus providing an efficient stress optimization tool with respect to the M-equistressness.…”

Section: Basic Propertiesmentioning

confidence: 99%

Plane elastostatic stress analysis in complex variables: A wavelet processing perspective

Vigdergauz

2016

Mathematics and Mechanics of Solids

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The well-known Kolosov-Muskhelishvili (KM) representation of the Airy function for 2D stress analysis in complex variable terms is enhanced by combining it with Walsh wavelets decomposition. It allows us to perform general analytical derivations up to the maximum extent possible which, in turn, provides a basis for developing a new stress computation algorithm readily incorporated into the routine single scale KM scheme. The mathematical treatment of the wavelet application is supported by a number of examples where non-trivial closed-form solutions are known and serve as a benchmark for numerical simulations. The comparison shows that the proposed framework has better performance than the conventional Fourier transform, especially when it comes to non-smooth stress distributions. Downloaded from À c(t) = u(z) + zu 0 (z) À F(t) + 2B 0 t + G 0 t; t 2 L: ð2:6Þ Vigdergauz

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Stress smoothing holes in planar elastic domains

Cited by 7 publications

References 11 publications

Stress-smoothing holes in an elastic plate: From the square lattice to the checkerboard

Stress-smoothing holes in an elastic plate: From the square lattice to the checkerboard

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

Plane elastostatic stress analysis in complex variables: A wavelet processing perspective

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