Inverse problems of the plane theory of elasticity

Черепанов, Г. П.

doi:10.1016/0021-8928(75)90085-4

Cited by 126 publications

(153 citation statements)

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“…This leads, through the use of Kolosov-Muskhelishvili potentials, to a problem in complex variables for the shape of the inclusions. The problem is very similar to that addressed by Cherepanov (1974). Using his ideas we derive an explicit representation for the shape of the inclusions in terms of elliptic functions.…”

Section: Introductionmentioning

confidence: 89%

“…Indeed, this is probably what Vigdergauz had in mind when he first sought a periodic array of "equally strong inclusions", c.f. (Banichuk, 1977;Cherepanov, 1974;Eldiwany and Wheeler, 1986;Vigdergauz, 1983;Wheeler and Kunin, 1982). We explore this issue in section 5.2.…”

Section: Introductionmentioning

confidence: 99%

“…Here as in Part I we apply the method of Cherepanov (1974). We recall that the remaining unknown potential ψ 2 satisfies (2.11) and (2.13):…”

Section: Introductionmentioning

confidence: 99%

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Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Grabovsky

Kohn

1995

Journal of the Mechanics and Physics of Solids

123

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For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. S. Vigdergauz has shown the existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from * This work was done while Y. G. was a student at the Courant Institute. † The work of R. V. K. was partially supported by ARO contract DAAL 03-92-G-0011 and NSF grants DMS-9404376 and DMS-9402763. 1 isotropic components in two space dimensions. This "Vigdergauz microstructure" consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

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Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Grabovsky

Kohn

1995

Journal of the Mechanics and Physics of Solids

123

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“…The absolutely nontrivial and purely analytical example here consists of the equistress shapes (ESS) [Cherepanov 1974;Vigdergauz 1976;Banichuk 1977] along which the hoop stresses are simultaneously uniform [Cherepanov 1974] and globally minimal [Vigdergauz 1976]. In other words, an ESS is optimal with respect to both criteria, which, for brevity, will be referred to as and V , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the equistressness (1-5) for a general geometry and the M-equistressness (1-7) for a single hole were first derived as a stationary point of the variation of the strain energy integral over the solid phase with moving boundaries [Cherepanov 1974]. In contrast, we formulate the V-criterion as an essential relaxation of equistressness rather than variationally.…”

Section: Introductionmentioning

confidence: 99%

Stress smoothing holes in planar elastic domains

Vigdergauz

2010

J. Mech. Mater. Struct.

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The actual elastostatic problem of optimizing the stress state in a two-dimensional perforated domain by proper shaping of holes is considered with respect to minimization of the global variations of the boundary hoop stresses. This new criterion radically extends the rather restrictive equistress principle introduced by Cherepanov and results in a favorable response of the structure to an external load, with neither local stress concentrations nor underloading of other parts of the boundary. Mathematically, the variations provide an integral-type assessment of the local stresses which requires less computational effort than direct minimization of the stress concentration factor. The proposed criterion can thus be easily incorporated in the numerical optimization scheme previously developed by the author in the closely related context of energy optimization. It includes an efficient complex-valued direct solver and a standard evolutionary optimization algorithm enhanced with an economical shape parametrization tool. The effectiveness of the proposed scheme is illustrated through numerical simulations.

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Optimal shape of a grain or a fibre cross‐section in a two‐phase composite

Shenfel'D

Givoli

Vigdergauz

2004

Commun. Numer. Meth. Engng.

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SUMMARYThe shape of grains or of cross-sections of ÿbres in a two-phase elastic material has an important in uence on the overall mechanical behaviour of the composite. In this paper a numerical scheme is devised for determining the optimal shape of a two-dimensional grain or of a ÿbre's cross-section. The optimization problem is ÿrst posed mathematically, using a global objective function, and then solved numerically by the ÿnite element method and a specially designed global optimization scheme. Excellent agreement is obtained with analytical results available for extreme cases. In addition, optimal shapes are obtained under more general conditions.

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Inverse problems of the plane theory of elasticity

Cited by 126 publications

References 0 publications

Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Stress smoothing holes in planar elastic domains

Optimal shape of a grain or a fibre cross‐section in a two‐phase composite

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