Vibrations of Thin Elastic Shells

Dikmen, M.

doi:10.1007/978-3-7091-2442-0_3

Cited by 11 publications

(16 citation statements)

References 58 publications

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“…(2.2)As is remarked in Dikmen[1] if the coordinate system happens to be a system of normal coordinates in the elastic configuration the angular momentum equations take on the particularly…”

mentioning

confidence: 89%

“…In this work we follow the notation used by Dikmen [1]; we view a shell as a three dimensional body, which we try to reduce to two dimensional consideration by introducing a suitable reference surface. A set of curvilinear coordinates 0 i, (i = 1, 2, 3) are chosen so that a reference surface within the shell may be represented by 0 3 = 0.…”

Section: Notation and Formulationmentioning

confidence: 99%

“…It is well known [1], [6] that if one can ignore all of the above quantities for n >/1 one obtains the so-called membrane theory of shells and that these equations may be treated by a reduction to the generalized analytic function theory. In the next section we shall consider the situation where the M ~, m ~, p~, etc.…”

Section: Notation and Formulationmentioning

confidence: 99%

“…Following Dikmen [1] we try to introduce bending into our model by retaining only those quantities for n < 2. On dropping the index 1, in component form these become If we ignore the effect of the terms 2 m and M'2 in equations (2.4) which is similar to the approach used in membrane theory we have that M "# = M #" and N "a = m ".…”

Section: Derivation Of Equations For N =mentioning

confidence: 99%

“…In this section we shall show that the system (2.31), (2.32), (2.33) may be put into the generalized hyperanalytic form [1], [3], [5], [8]. To this end we consider just the principal part of (2.31), namely IUx +Au,y =g.…”

Section: Reduction To Normal Formmentioning

confidence: 99%

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A function theory for thin elastic shells

Gilbert

1989

J Elasticity

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It is well known in the theory of elastic shells that a first order approximation using the shell thickness as an expansion parameter leads to the membrane theory of shells. The membrane equations have as solutions the generalized analytic functions. These functions have been exhaustively studied by Ilya N. Vekua [6], [7] and his students. R.P. Gilbert and J. Hile [3] introduced an extension of these systems to include elliptic systems of 2n equations in the plane and named the solutions of these systems generalized hyperanalytic functions.It is shown in this paper that the next order approximation to the shell, which permits, moreover, the introduction of bending, may be described in terms of the generalized hyperanalytic functions. It is strongly suspected that the higher order approximations may also be described in terms of corresponding hypercomplex systems. Notation and formulationIn this work we follow the notation used by Dikmen [1]; we view a shell as a three dimensional body, which we try to reduce to two dimensional consideration by introducing a suitable reference surface. A set of curvilinear coordinates 0 i, (i = 1, 2, 3) are chosen so that a reference surface within the shell may be represented by 0 3 = 0. For purposes of defining our notation let a be a surface embedded in R 3 which we represent in the form r = r(0~). The vectors a~ .'=r~ = (ar)/(O0 ~) are the base vectors. The first fundamental form of the surface is given by a~p d0~d0 a where the a~a.-=a~ "ap are the covariant components of the metric tensor. If the .4 ~p are the cofactors of the a~p we define the contravariant metric tensor as a~a:=.4~a/a where a is the determinant of a~p. The second fundamental form of the surface is defined through b~ a dO ~ d0a:= a3 • a~,a dO ~ d0 a = -dr" da3, with a3 = (al × a2)/([al )< a2l). The Christoffel symbols are given as usual by . i F~a ~ .= ~(a~,a + ap~,~ -a~a,~ ), and covariant differentiation is defined by , r'~ Tx. (1.1) T~I := T~ -The position vector of a generic point on the shell at time t is given by x = x(O ~, ~, t) where (0 l, 0 2) ~ i'~ is a point on the reference surface and ~ [0, h], in particular x(O ~, O, t)= r(O~, t). The methods of Dikmen [1] and

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“…(2.2)As is remarked in Dikmen[1] if the coordinate system happens to be a system of normal coordinates in the elastic configuration the angular momentum equations take on the particularly…”

mentioning

confidence: 89%

Section: Notation and Formulationmentioning

confidence: 99%

Section: Notation and Formulationmentioning

confidence: 99%

Section: Derivation Of Equations For N =mentioning

confidence: 99%

Section: Reduction To Normal Formmentioning

confidence: 99%

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A function theory for thin elastic shells

Gilbert

1989

J Elasticity

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Justification of the two‐dimensional equations of a linearly elastic shallow shell

Ciarlet

Miara

1992

Comm Pure Appl Math

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Ecole Superieure d'lngknieurs en Electrotechnique et Electronique AlbstractWe consider a problem in three-dimensional linearized elasticity, posed over a shell with a specific geometry, subjected to general loadings, and clamped on a portion of its lateral surface. We show that, as the thickness of the shell goes to zero, the solution of the three-dimensional problem converges to the solution of two-dimensional shallow shell equations. This approach, which provides in particular a mathematical definition of "shallowness", clearly delineates conditions under which a three-dimensional problem may be deemed asymptotically equivalent to a two-dimensional shallow shell problem.

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Bifurcation problems with O(2)⊕Z2 symmetry and the buckling of a cylindrical shell

Buzano

Russo

1986

Annali di Matematica pura ed applicata

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Vibrations of Thin Elastic Shells

Cited by 11 publications

References 58 publications

A function theory for thin elastic shells

A function theory for thin elastic shells

Justification of the two‐dimensional equations of a linearly elastic shallow shell

Bifurcation problems with O(2)⊕Z2 symmetry and the buckling of a cylindrical shell

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