Exponential decay estimates for a class of nonlinear Dirichlet problems

Horgan, Cornelius O.; Olmstead, W. E.

doi:10.1007/bf00280597

Cited by 15 publications

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“…For this reason and since we wish to obtain an explicit bound for the constant Q(1/<J) appearing in (4.36), it is convenient to appeal to a direct proof of (4.36) for the rectangular domain R_ of concern here. Similar issues were considered in [41], In what follows for simplicity we present a derivation of the scalar version of (4.36). The same arguments for vector functions would yield (4.36) with the same value for the Sobolev constant.…”

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confidence: 93%

A Saint-Venant principle for a theory of nonlinear plane elasticity

Horgan¹,

Payne²

1992

Quart. Appl. Math.

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Abstract. A formulation of Saint-Venant's principle within the context of a restricted theory of nonlinear plane elasticity is described. The theory assumes small displacement gradients while the stress-strain relations are nonlinear. We consider plane deformations of such an elastic material occupying a rectangular region. The lateral sides are traction-free while the far end is subjected to a uniformly distributed tensile traction x > 0. The near end is subjected to a prescribed normal and shear traction. If this end loading were also that of uniform tension, then one possible corresponding stress state throughout the rectangle is that of uniform tension. When the near end loading is not uniform, the resulting stress field is expected to approach a uniform tensile state with increasing distance from the near end. This result is established here using differential inequality techniques for quadratic functionals. It is shown that, under certain constitutive assumptions, an energy-like quadratic functional, defined on the difference between the deformation field and the uniform tensile state, decays exponentially with distance from the near end. The estimated decay rate (which is a lower bound on the actual rate of decay) is characterized in terms of the load level x , the domain geometry, and material properties. The results predict a progressively slower decay of end effects with increasing load level x . The mathematical issues of concern involve spatial decay of solutions of a fourth-order nonlinear elliptic partial differential equation.

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A Saint-Venant principle for a theory of nonlinear plane elasticity

Horgan¹,

Payne²

1992

Quart. Appl. Math.

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“…3 Although they do not bear directly on Saint-Venant's principle, some results concerning spatial decay for nonlinear second-order elliptic partial differential equations may be found in[5]-[8] 4. 3 Although they do not bear directly on Saint-Venant's principle, some results concerning spatial decay for nonlinear second-order elliptic partial differential equations may be found in[5]-[8] 4.…”

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The effect of nonlinearity on a principle of Saint-Venant type

Horgan

Knowles

1981

J Elasticity

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This paper establishes a principle of Saint-Venant type associated with finite anti-plane shear of a cylinder whose cross-section is a semi-infinite strip. The long sides of the strip are traction-free, and the short side carries an arbitrarily distributed shear traction. At the infinity in the strip, the deformation is prescribed to be one of simple shear, and the associated shear stress is uniform. The analysis is based on the fully nonlinear theory of finite elastostatics and is carried out for a special class of homogeneous, isotropic incompressible materials. It is shown that, along the parallel sides of the strip, the nonvanishing component of shear stress differs from its average value (taken across the strip) by an exponentially decaying function of the distance from the end. A lower bound is given for the rate of decay.

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Opial Inequalities in Several Independent Variables

Agarwal

Pang

1995

Opial Inequalities With Applications in Differential and Difference Equations

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Exponential decay estimates for a class of nonlinear Dirichlet problems

Cited by 15 publications

References 18 publications

A Saint-Venant principle for a theory of nonlinear plane elasticity

A Saint-Venant principle for a theory of nonlinear plane elasticity

The effect of nonlinearity on a principle of Saint-Venant type

Opial Inequalities in Several Independent Variables

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