Modeling of consistent second‐order plate theories for anisotropic materials

Schneider, Patrick; Kienzler, Reinhold; Böhm, Martin

doi:10.1002/zamm.201100033

Cited by 38 publications

(41 citation statements)

References 13 publications

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“…The N th-order stress-boundary conditions are formulated in moments of increasing order [6]. The first-order boundary conditions are the classical ones of the Kirchhoff theory [4], which are formulated in resulting forces and classical bending moments, only.…”

Section: Steigmannmentioning

confidence: 99%

Comparison of various linear plate theories in the light of a consistent second-order approximation

Schneider

Kienzler

2014

Mathematics and Mechanics of Solids

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The uniform-approximation technique in combination with the pseudo-reduction technique is applied in order to derive consistent theories for isotropic and anisotropic plates. The approach has already been used to assess and validate theories established in the literature, e.g., the theories of Reissner and Zhilin. In this contribution, we also present a comparison with the theories of Vekua, Ambartsumyan, Steigmann and Reddy's third-order theory. The current paper is a corrected and extended version of the one originally published in Kienzler and Schneider (Comparison of various linear plate theories in the light of a consistent second-order approximation.

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Section: Steigmannmentioning

confidence: 99%

Comparison of various linear plate theories in the light of a consistent second-order approximation

Schneider

Kienzler

2014

Mathematics and Mechanics of Solids

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“…In (26) the usual notation M nn = n α n β M αβ , M nν = n α ν β M αβ has been employed. Also, in (27) the symbol A k means jump of A across the k-th corner point. The standard shear forces Q α are not explicitly involved within (24)- (26); however, as it may be proved by the local equilibrium equations, the following equations hold true, that is,…”

Section: The Principle Of Virtual Powermentioning

confidence: 99%

“…Following a procedure similar to the one pursued in Sect. 4 for the principle of virtual power, the pertinent Euler-Lagrange equations can be found identifying themselves with the field equilibrium equations (25) and natural boundary conditions encompassed within (26) and (27). The mathematical details of this procedure are straightforward, hence they are omitted for brevity.…”

Section: Minimum Principlementioning

confidence: 99%

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A class of shear deformable isotropic elastic plates with parametrically variable warping shapes

Polizzotto

2017

Z Angew Math Mech

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A homogeneous shear deformable isotropic elastic plate model is addressed in which the normal transverse fibers are allowed to rotate and to warp in a physically consistent manner specified by a fixed value of a real non-negative warping parameter ω. On letting ω vary continuously (at fixed load and boundary conditions), a continuous family of shear deformable plates P ω is generated, which spans from the Kirchhoff plate at the lower limit ω = 0, to the Mindlin plate at the upper limit ω = ∞; for ω = 2, P ω identifies with the third-order Reddy plate. The boundary-value problem for the generic plate P ω is addressed in the case of quasi-static loads, for which a principle of minimum total potential energy is given. Taking the Kirchhoff deflection w K and the shear angle potentials G, H as basic unknown fields, the problem turns out to be governed by three uncoupled fourth-order PDEs, of which one coincides with the classical PDE for the Kirchhoff plate, the other two are PDEs of Helmholtz type. A closed-form representation of the general solution for the plate family is given in terms of (w K , G, H ) to within an arbitrary biharmonic function and an arbitrary pair of conjugate harmonic functions. These arbitrary functions are available to enforce the inherent boundary conditions. The connections between the present theory and other existing theories are pointed out. An application to a family of simply supported circular plates under uniform load is analytically worked out.

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“…(The two-dimensional equilibrium equations, i.e. for plates and discs, for a weak solution setting are given in [5].) For isotropic material the problem decouples into four independent subproblems (a rod-, a shaft-and two beam-problems with orthogonal loading directions) for the arbitrary three-dimensional load case.…”

Section: Consistent Refined Beam Theoriesmentioning

confidence: 99%

A‐priori error estimates for consistent, refined theories for thin structures

Schneider

Kienzler

2016

Proc Appl Math and Mech

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We propose an extension of the uniform approximation approach towards the simultaneous truncation of the associated dual energy. In the talk, we derive a hierarchy of approximating theories and corresponding a-priori error estimates for their solutions, for the special case of an isotropic, linear elastic, one-dimensional structural member with rectangular cross-section. The uniform approximation approachClassical theories for thin structures like the Euler-Bernoulli beam and the Kirchhoff plate theory were developed by the use of disputable a-priori assumptions already in the 18 th and 19 th century. Rigorous mathematical justifications for these theories are available which proof them to appear as a limit of the three-dimensional theory of elasticity, if the thickness of the structure tends to zero. Recent results mostly use the method of Γ-convergence, which does not deliver quantitative error estimates directly. The development of refined theories, i.e., theories for moderately thick structures, or higher precision, started already in the middle of the 20 th century. Nowadays refined theories are still an open topic which is under intensive development. Mathematical justifications for refined theories are missing -even for the most established theories like the Reissner-Mindlin plate theory.The uniform approximation approach for the a-priori assumption-free derivation of theories for thin structures from the three-dimensional theory of linear elasticity goes back to pioneer treaties of Naghdi [1] and Koiter [2]. Kienzler [3] interprets the approach as a truncation of the elastic energy after a maximum power of a characteristic parameter that describes the relative thickness of the structure.We extend his ideas by truncating the associated dual energy as well, in order to provide a mathematical justification for refined theories by the proof of quantitative error estimates. By extending a standard result from duality theory (based on Korn's inequality) towards general anisotropy, the square error of an arbitrary displacement field v from the (weak) solution of three-dimensional elasticity can be estimated via the difference E pot (v) − E dual (µ) of the elastic potential and the dual energy of an admissible stress field µ. By using a dimensionless formulation and (abstract) Fourier-series expansions in thickness direction for the displacement field, the potential and dual energy appear as infinite power series in parameters describing the relative thickness of the structure. When truncating both series after a certain power, a finite set of field equations and stressboundary conditions can be derived from the Euler-Lagrange equations of the potential energy, whereas the dual energy's Euler-Lagrange equations deliver matching displacement-boundary conditions. This generates a hierarchy of theories with regard to the truncation power and by virtue of the mentioned result from duality theory the corresponding quantitative apriori error estimates.If smooth solutions are available, one is enabled to reduce the n...

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Modeling of consistent second‐order plate theories for anisotropic materials

Cited by 38 publications

References 13 publications

Comparison of various linear plate theories in the light of a consistent second-order approximation

Comparison of various linear plate theories in the light of a consistent second-order approximation

A class of shear deformable isotropic elastic plates with parametrically variable warping shapes

A‐priori error estimates for consistent, refined theories for thin structures

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