Second-order linear plate theories: Partial differential equations, stress resultants and displacements

Kienzler, Reinhold; Schneider, Patrick

doi:10.1016/j.ijsolstr.2017.01.004

Cited by 14 publications

(37 citation statements)

References 15 publications

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“…Examining the growth behavior further and studying already derived consistent theories and their connection to other theories [3,4], we are able to show that even refined theories of Timoshenko/Reissner/Mindlin type and furthermore third-order shear deformable theories still reside in the range where 2 of 2 Section 4: Structural mechanics the geometric scaling factor dominates -even for not at all thin structures. Examining the growth behavior further and studying already derived consistent theories and their connection to other theories [3,4], we are able to show that even refined theories of Timoshenko/Reissner/Mindlin type and furthermore third-order shear deformable theories still reside in the range where 2 of 2 Section 4: Structural mechanics the geometric scaling factor dominates -even for not at all thin structures.…”

Section: Comparison Of Two Approachesmentioning

confidence: 72%

“…In contrary to the first approach, the elastic energy is now truncated after a certain power of the geometric scaling factor, which is usually very small ( 1).Although the power-law of the geometric scaling factor is not asymptotically dominant, it is dominant over the declination of the Taylor-coefficients prefactor for low orders of approximation. Examining the growth behavior further and studying already derived consistent theories and their connection to other theories [3,4], we are able to show that even refined theories of Timoshenko/Reissner/Mindlin type and furthermore third-order shear deformable theories still reside in the range where…”

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confidence: 80%

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Dual‐energy extension of the uniform‐approximation approach

Schneider

Kienzler

2019

Proc Appl Math and Mech

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In the talk an extension of the uniform-approximation approach, that relies on the truncation of the elastic energy after a certain power of a geometric scaling factor, towards an analogous truncation of the dual energy is proposed. On the one hand, this allows for a derivation of fully defined boundary value problems including compatible displacement boundary conditions. On the other hand, the extension enables us to provide an a priori error estimate for the systematic error of the arising approximative models with respect to the exact three-dimensional solution.The approach is compared to the approach of a fixed kinematic assumption for the displacement field which is widely used in engineering. We show that the later approach leads in general to a more complex model for a comparable approximation accuracy so that the consistent approximation approach is to prefer. The general idea of energetic approximationAnalytical models for thin structural members, like plate and beam theories are defined on two-or one-dimensional domains, like the mid-plane or the neutral axis. Therefore, in general they possess a systematic error compared to the three-dimensional continuum theory of elasticity -even if the model can be solved exactly. So called structured approaches aim to derive hierarchies of models with increasing approximation accuracy from the three-dimensional model of elasticity by dimension reduction techniques. One principal branch of structured approaches relies on the truncation of the elastic energy. The uniform approximation technique is one of the dimension reduction techniques belonging to this principal branch.The basic idea of energetic approximation can be motivated from a deeper lying theorem in elasticity that follows from Korn's inequality. It basically states that the quadratic error in the weak solution norm (component-wise H 1 ) of any admissible displacement field v to the exact solution u can be estimated via the difference in the elastic energy

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Section: Comparison Of Two Approachesmentioning

confidence: 72%

mentioning

confidence: 80%

Dual‐energy extension of the uniform‐approximation approach

Schneider

Kienzler

2019

Proc Appl Math and Mech

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“…For isotropy the system decouples and the original Reissner theory is recovered. Connections to other established plate theories are investigated in [12,13].…”

Section: The Orthogonal Decompositionmentioning

confidence: 99%

Modeling of anisotropic boundary layer effects in plate theory

Schneider

Kienzler

2018

Proc Appl Math and Mech

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The modeling of a Reissner-type plate theory for elastic, monoclinic material is presented. The basic theory is derived using the a-priori-assumption free uniform-approximation approach, which is based upon series expansions of the displacement field and a structured truncation of the elastic potential that gives rise to a hierarchy of approximating theories. An a-priori estimate for the approximation error shows that higher-order theories have indeed a higher rate of convergence with respect to the relative slenderness of the plate. Using a pseudo-reduction approach, the number of PDEs to be solved is reduced significantly. The resulting first-order theory is the classical monoclinic plate theory, whereas, the second order theory is not determined uniquely by the approach. Uniqueness is achieved by introduction of an orthogonal decomposition of higherorder gradients of the in-plane displacements. The final second-order theory coincides with the Reissner-Mindlin theory for the special case of isotropic material. The uniform approximation techniqueA monoclinic plate theory covering anisotropic edge effects is presented. For the special case of a linear elastic, isotropic material it equals the well established plate theory of Reissner [1,2] and could therefore be regarded as an anisotropic extension of this theory. All details of the modeling of the presented theory can be found in [3].The modeling is based on the so-called uniform-approximation technique, a modeling approach for consistent plate and shell theories that came up in the middle of the last century [4]. More recently, it was applied in [5,6] for the derivation of higher-order isotropic plate theories and the approach was extended towards monoclinic material in [7].The key idea of the uniform-approximation technique is to derive plate theories from the three-dimensional theory of linear elasticity, by the use of dimensionless coordinates and series expansions in thickness direction of the plate. Here, we use abstract Fourier-series expansions for the unknown quantities. To this end, a L 2 -basis of orthogonal polynomials is used that are scaled Legendre polynomials, firstly introduced in [7]. An advantage of the approach is that, on the one hand, the associated, generalized stress-resultants of the lowest order coincide with the classical shear forces and moments, while on the other hand, the stress-resultants in general turn out to be the Fourier coefficients of the stress field. This gives higher-order stress resultants a direct physical meaning and, moreover, it allows for a three-dimensional stress field reconstruction from the solution of the plate theory, which is a major advantage in comparison to other series expansions.It can be shown [7] that varying the first variation of the elastic potential with respect to the virtual displacement coefficients leads to an exact two-dimensional form of the three-dimensional equilibrium equations formulated in the generalized stress resultants. However, the arising infinite system of equations is intractabl...

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“…The well-known Koiter-and Naghdi-type shell theories are the first-and second-order members of the displacement-based, approximate shell models [30,36], which are originally elaborated on plates by Kirchhoff and Love, as well as Reissner and Mindlin, respectively, in [33,52]. The consistency of these shell models and many other ones is investigated in [24,25,53,54]. Additional modified, displacement-based shell theories are developed in [1,10,19,34,35,49].…”

Section: Introductionmentioning

confidence: 99%

Dual-mixed hp-version axisymmetric shell finite element using NURBS mid-surface interpolation

Tóth

Burmeister

2020

Acta Mech

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A new, general hp-version axisymmetric finite element is derived for the boundary value problems of thin linearly elastic shells of revolution, applying a complementary strain energy-based three-field dual-mixed variational principle. For the interpolation of the mid-surface geometry, non-uniform rational B-splines-NURBS-is used. The independent field variables of the weak formulation are the a priori non-symmetric stress tensor, the displacement vector, and the infinitesimal skew-symmetric rotation tensor. The theoretical model of the shell formulation is based on a consistent dimensional reduction process and a systematic variablenumber reduction procedure. The inverse of the unvaried three-dimensional constitutive equation is employed since neither the classical kinematical assumptions nor the stress hypotheses are built in the mathematical model; namely, both the through-the-thickness variation and the normal stress to the shell mid-surface are not excluded. The new hp axisymmetric shell finite element is tested by a representative model problem for extremely thin and moderately thick, singly and doubly curved shells of negative and positive Gaussian curvature. Following from the numerical experiments, the constructed hp-shell finite element gives locking-free results not only for the displacement but also for the stresses.

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Second-order linear plate theories: Partial differential equations, stress resultants and displacements

Cited by 14 publications

References 15 publications

Dual‐energy extension of the uniform‐approximation approach

Dual‐energy extension of the uniform‐approximation approach

Modeling of anisotropic boundary layer effects in plate theory

Dual-mixed hp-version axisymmetric shell finite element using NURBS mid-surface interpolation

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