Weak Local Minimizers in Finite Elasticity

Piero, Gianpietro Del; Rizzoni, Raffaella

doi:10.1007/s10659-008-9175-z

Cited by 17 publications

(38 citation statements)

References 19 publications

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“…Equivalently, one can derive estimates on the energy that ensure that the second variation of E is strictly positive at the deformation u h λ and, consequently, that u h λ is both linearization stable and a strict weak relative minimizer. For a compressible cylindrical solid in tension this technique was used by Spector [27] and, more recently, by Del Piero and Rizzoni [7] and Fosdick, Foti, Fraddosio, and Piccioni [10] (see, also, Del Piero [6]) to obtain estimates upon the values of λ where bifurcations cannot occur. Additionally, in [7] and [10] similar estimates are derived for compression (λ ≤ 1); incompressible materials are also considered in [7].…”

mentioning

confidence: 99%

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

Sivaloganathan

Spector

2011

J Elast

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Consider a cylinder (not necessarily of circular cross-section) that is composed of a hyperelastic material and which is stretched parallel to its axis of symmetry. Suppose that the elastic material that constitutes the cylinder is homogeneous, transversely isotropic, and incompressible and that the deformed length of the cylinder is prescribed, the ends of the cylinder are free of shear, and the sides are left completely free. In this paper it is shown that mild additional constitutive hypotheses on the stored-energy function imply that the unique absolute minimizer of the elastic energy for this problem is a homogeneous, isoaxial deformation. This extends recent results that show the same result is valid in 2-dimensions. Prior work on this problem had been restricted to a local analysis: in particular, it was previously known that homogeneous deformations are strict (weak) relative minimizers of the elastic energy as long as the underlying linearized equations are strongly elliptic and provided that the load/displacement curve in this class of deformations does not possess a maximum.

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confidence: 99%

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

Sivaloganathan

Spector

2011

J Elast

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“…In Fig. 1 we compare our estimates to that recently proposed in [4] for incompressible bodies: we see that our estimate is better for thick cylinders, whereas for slender cylinders the two estimates practically coincide. However, the effectiveness of an estimate from below depends on its "distance" from the actual critical load.…”

Section: Bound From Below Of the Hadamard Functionalmentioning

confidence: 51%

“…Indeed, by comparing our estimate λ LB to the upper bound estimate reported in [4], it is possible to conclude that for slender bodies, where the instability is mainly due to geometrical effects in presence of relatively small deformations, λ LB accurately approximates the critical load. Conversely, for small slenderness, when the instability is strongly related to the non-linearity of the constitutive equation, our lower bound estimate, although more efficient of that in [4], does not represent a good approximation of the critical load. Thus, there remains an open problem to enhance lower bound estimates when stable large deformations are anticipated.…”

Section: Bound From Below Of the Hadamard Functionalmentioning

confidence: 56%

Optimal bounds from below of the critical load for elastic solids subject to uniaxial compression

Castellano

Foti

Fraddosio

et al. 2015

Proc Appl Math and Mech

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In the recent papers [1,2] we studied a new procedure based on the Korn inequality for determining sufficient conditions for the Hadamard stability, aimed at determining optimal lower bound estimates for the critical load in bifurcation problems. Here, we discuss the effectiveness of our approach for the classical representative problem of uniaxial compression of a Mooney-Rivlin circular cylinder. We find that our lower bound estimate is effective and advantageous for applications, since it is easily implementable in numerical codes.

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“…In (Del Piero, 1979;Del Piero and Rizzoni, 2008) it is proved that if w 3 is frame-indifferent, then b 3 0 can be decomposed into…”

Section: Three-dimensional Energy Of a Mismatch Strained Interphasementioning

confidence: 99%

Asymptotic analysis of an adhesive joint with mismatch strain

Rizzoni¹,

Lebon²

2012

European Journal of Mechanics - A/Solids

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This paper proposes the study of the equilibrium problem, where two elastic bodies are bonded to a thin elastic film under mismatch strain conditions resulting in a state of residual stress. The asymptotic behavior of the film/adherent system is modeled as the film thickness tends to zero, using a method based on asymptotic expansions and energy minimization procedures. This method yields a family of non-local imperfect interface laws, which define a jump in the displacement and the traction vector fields. The amplitudes of the jumps turn out to be correlated with the state of residual stress and the elastic properties of the materials. As an example, the interface law is calculated at order zero in the case of a pure homogeneous mismatch strain and a thin isotropic film consisting of Blatz-Ko material

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Weak Local Minimizers in Finite Elasticity

Cited by 17 publications

References 19 publications

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

Optimal bounds from below of the critical load for elastic solids subject to uniaxial compression

Asymptotic analysis of an adhesive joint with mismatch strain

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