Large elastic deformations of shells with the inclusion of transverse normal strain

Biricikoglu, V.; Kalnins, A.

doi:10.1016/0020-7683(71)90097-7

Cited by 21 publications

(4 citation statements)

References 9 publications

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“…The generalized kinematics (2) maintains the assumption that material fibers normal to the reference surface remain so under deformation, but allows for changes in thickness. One of the first appearances of this generalized theory in the nonlinear deformations of shells may be found in Biricikoglu and Kalnins [15], who considered the function to be linear in . Later, Chernykh [16] as well as Stumpf and Makowski [9] explored quadratic and more general dependence of on , arguing that the distribution of normal strains in the normal direction is essential for modeling large rotations and deformations of shells.…”

Section: Notation Kinematics Stretch Rotation and Curvaturementioning

confidence: 99%

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Energies for elastic plates and shells from quadratic-stretch elasticity

Vitral,

Hanna

2022

Preprint

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We derive stretching and bending energies for isotropic elastic plates and shells. Through the dimensional reduction of a bulk elastic energy quadratic in Biot strains, we obtain two-dimensional bending energies quadratic in bending measures featuring a bilinear coupling of stretches and geometric curvatures. For plates, the bending measure is invariant under spatial dilations and naturally extends primitive bending strains for straight rods. For shells or naturally-curved rods, the measure is not dilation invariant, and contrasts with previous ad hoc postulated forms. The corresponding field equations and boundary conditions feature moments linear in the bending measures, such that application of a pure moment results in isometric deformation of a unique neutral surface, a primitive behavior in agreement with classical linear response but not displayed by commonly used analytical models. We briefly comment on relations between our energies, those derived from a neo-Hookean bulk energy, and a commonly used discrete model for flat membranes. Although the derivation requires consideration of stretch and rotation fields, the resulting energy and field equations can be expressed entirely in terms of metric and curvature components of deformed and reference surfaces.

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Section: Notation Kinematics Stretch Rotation and Curvaturementioning

confidence: 99%

“…From (15) we may see that on the mid-surface ( = 0), the Biot strain is U + 1 NN − I. Expanding the threedimensional Biot strain in ,…”

Section: Quadratic-biot Elastic Theorymentioning

confidence: 99%

Energies for elastic plates and shells from quadratic-stretch elasticity

Vitral,

Hanna

2022

Preprint

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“…can, easily, be obtained to be considered in the next section 8 . The scalar product of g with e 3 through the right-hand side of Equation ( 59), recalling that e 3 ð¼ a 3 Þ, Equation (48), is normal to a , and therefore normal to g through Equation ( 53 ), yields…”

Section: Expressions Of Strainsmentioning

confidence: 99%

A Theory for Rubber-like Shells

Yükseler

2008

Journal of Elastomers & Plastics

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“…Berstein [13], Ebiciokog-lu [14], Biriciokoglu and Kalnins [15], and Pietraszkiewicz [16] considered further the deformation of thickness, asserting that the normal keeps straight but may be inclined and stretched, and the strain component ~J331 is invariant along the thickness. The stress moment of second-order was introduced in [14]and [16].…”

Section: Introductionmentioning

confidence: 99%

Studies of nonlinear theories for thin shells

Shen-rong

1983

Appl Math Mech

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In order to formulate the equations for the study here,the Fourier expansions upon the system of orthonormal polynomials are used.St may be considerably convenient to obtain the expressions of displacements as well as stresses directly from the solutions. Based on the principle of virtual work the equilibrium equations of various orders are formulated.In particular, the system of thirdorder is given in detail, thus providing the reference for accuracy analysis of lower-order equations.A theorem about the differentiation of Legendre series term by term is proved as the basis of mathematical analysis.Therefore the functions used are specified and the analysis rendered is no longer a formal one. The analysis will show that the Kirchhoff-Love's theory is merely of the first-order and the theory which includes the transverse deformation but keeps the normal straight is essentia~u oft he first order, too.

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Large elastic deformations of shells with the inclusion of transverse normal strain

Cited by 21 publications

References 9 publications

Energies for elastic plates and shells from quadratic-stretch elasticity

Energies for elastic plates and shells from quadratic-stretch elasticity

A Theory for Rubber-like Shells

Studies of nonlinear theories for thin shells

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