The Linear Theory of Elasticity

Gurtin, Morton E.

doi:10.1007/978-3-662-39776-3_1

Cited by 516 publications

(418 citation statements)

References 312 publications

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“…Thanks to (23), (24), and (25), the estimates (21), (22) are proved in the case T = B (or T = λB, λ > 0). If T = B, there holds…”

Section: «Ffmentioning

confidence: 99%

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A Notion of Capacity Related to Elasticity: Applications to Homogenization

Bellieud

2011

Arch Rational Mech Anal

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Abstract. We study a notion of capacity related to elasticity which proves convenient to analyze the concentrations of strain energy caused by the rigid displacements of some infinitesimal parts of an elastic body in two or three dimensions. By way of application, we investigate the behavior of solutions to initial boundary value problems describing vibrations of periodic elastic composites with rapidly varying elastic properties. More specifically, we analyze a two-phase medium whereby a set of heavy stiff tiny particles is embedded in a softer matrix. This task is set in the context of linearized elasticity.

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“…Thanks to (23), (24), and (25), the estimates (21), (22) are proved in the case T = B (or T = λB, λ > 0). If T = B, there holds…”

Section: «Ffmentioning

confidence: 99%

“…By passing to the limit as ε → 0 in the first and third terms of the last inequalities, taking (24) into account, we infer…”

Section: «Ffmentioning

confidence: 99%

A Notion of Capacity Related to Elasticity: Applications to Homogenization

Bellieud

2011

Arch Rational Mech Anal

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“…Moreover, by assumption, we have f goF~u~df~ ~ CR'~8 (9) 4 By virtue of (2) and (6), for any (symmetric) second order tensors ~ii, "Y~i and for any positive ~, we have >_ _<-q 1 5 Here we are interested in the case of small e, so that the integral makes sense. However, in the case of large ~, by slightly modifying the previous arguments [6] it can be proved that the theorem still holds.…”

Section: <~(½) Fagq~k~e~ek~d~+(ivf2r2/2) Fa ~Dglgr2-"-2~ <~(½) Fa Gc~ek~ Dfl + C/r~ ~mentioning

confidence: 99%

“…E. Gurtin and E. Sternberg in a paper of 1961 [1] (cf. also [4,5]) have shown, among other things, some uniqueness theorems for the linear elastic equilibrium problem on exterior domains, in the homogeneous isotropic case. More preciesly, they proved that for uniqueness to hold it suffices just to assume that either the displacement or the strain of the "difference" state tends to zero uniformly at infinity.…”

Section: Introductionmentioning

confidence: 99%

On the well-posedness of the equilibrium problem for linear elasticity in unbounded regions

Galdi

Rionero

1980

J Elasticity

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In this paper we establish some continuous dependence and uniqueness theorems for equilibrium solutions of the equations of general anisotropic linear elasticity in exterior domains. The method we employ is that of the weight function which we introduced in previous papers. However, this is the first example where the method is applied to a static problem. The above theorems are obtained by allowing the strain to be unbounded at large spatial distances. In some cases, no growth condition is assumed. Moreover, the displacement and the elasticities are also possibly allowed to grow.

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“…The content of this contribution is restricted to the linear theory of elasticity. For the basics, the reader is referred to Hetnarski and Ignaczak [1], Gurtin [2] Lurie [3].…”

Section: Introductionmentioning

confidence: 99%

Bending moment tracking and the reduction of the axial stress in vibrating beams by piezoelectric actuation

Schoeftner

2017

Acta Mech

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This contribution focuses on bending moment tracking in slender beam-type structures that are equipped with piezoelectric actuators. Bending moments are associated with the axial stress, which is the dominant stress component of laterally excited beam structures. If the maximum value exceeds a certain tensile stress limit, the structures will crack or be irreparably damaged. In the present contribution, a piezoelectric bimorph beam is considered and it is investigated in which manner the piezoelectric actuation devices have to be distributed along the beam length, such that a certain bending moment distribution is obtained. This is called bending moment tracking. First, the basic equations of a piezoelectric bimorph beam are recalled and the differential equations for the bending moment are derived. Then a positive semi-definite integral depending on the error of the bending moment is defined, which is the difference between the actual and the desired bending moment. The results of the derivations are conditions for the eigencurvature due to the piezoelectric actuation, such that a certain bending moment distribution is achieved. Approximate solutions for the eigencurvature are also presented for the lower-and for the high-frequency domain. The theory is verified by a support-excited piezoelectric bimorph. First, the frequency response curves for the deflection, the bending moment and the axial stresses are calculated. Then the responses due to a sinusoidal excitation are computed, showing that the suggested control algorithm enables the reduction of the bending moment and also of the axial stress in a satisfactory manner.

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The Linear Theory of Elasticity

Cited by 516 publications

References 312 publications

A Notion of Capacity Related to Elasticity: Applications to Homogenization

A Notion of Capacity Related to Elasticity: Applications to Homogenization

On the well-posedness of the equilibrium problem for linear elasticity in unbounded regions

Bending moment tracking and the reduction of the axial stress in vibrating beams by piezoelectric actuation

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