On consistent plate theories

Kienzler, Reinhold

doi:10.1007/s00419-002-0220-2

Cited by 77 publications

(68 citation statements)

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“…Such an approach was adopted by Kienzler [9,10] based on linear elasticity. By a procedure of 'pseudoreduction' of the resulting system to certain orders of h, the classical plate theory and a Reissner-type theory [11,12] were recovered.…”

Section: X(rmentioning

confidence: 99%

“…Finally, the plate equation (3.14) 1 furnishes an algebraic equation forr 0 upon using all the recursion relations 9) whereh = αh and the explicit expression of f 1 is omitted. Therefore, the quantityr 0 is a function of the parameterh for a fixed m. The algebraic equation (6.9) can be solved numerically without difficulty for any fixed m, and for some special values of m analytical solutions are available.…”

Section: An Example: Pure Finite-bending Problemmentioning

confidence: 99%

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On a consistent finite-strain plate theory based on three-dimensional energy principle

Dai

Song

2014

Proc. R. Soc. A.

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This paper derives a finite-strain plate theory consistent with the principle of stationary threedimensional potential energy under general loadings with a fourth-order error. Starting from the three-dimensional nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the three-dimensional field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a two-dimensional virtual work principle. An alternative approach based on a two-dimensional truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a twodimensional energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Compared with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of loadings, applicability to finite-strain problems and no involvement of non-physical quantities.

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Section: X(rmentioning

confidence: 99%

Section: An Example: Pure Finite-bending Problemmentioning

confidence: 99%

On a consistent finite-strain plate theory based on three-dimensional energy principle

Dai

Song

2014

Proc. R. Soc. A.

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“…These theories are based on the uniform approximation of the fundamental equations of elasticity [12][13][14]16]. In two previous papers [7,8], the uniform-approximation argument, which is proposed for the derivation of the plate equations, was extended to the requirement that it must be applied to the solution of the respective differential equations as well.…”

Section: Introductionmentioning

confidence: 99%

On Material Conservation Laws for a Consistent Plate Theory

Bose

Kienzler

2006

Arch Appl Mech

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In the paper, material conservation laws associated with a consistent second-order plate theory are derived, which takes shear deformations and strains in thickness direction of the plate into account. Three path-independent integrals are established. In the presence of inhomogeneities in the material (e.g., defects or cracks), energy-release rates due to the change of the configuration of such flaws can be calculated by these integrals. The resulting material forces may serve to assess the reliability of structures with cracks.

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“…On the other hand, the fact that the L 2 -norm of the basis polynomials decreases very fast for increasing n is in the later the key for the application of the truncation approach of the uniform-approximation technique (compare [1]). In the following, we assume a homogeneous material, i.e., E ijkl = const..…”

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confidence: 99%

A consistent second‐order plate theory for monotropic material

Schneider

Kienzler

2011

Proc Appl Math and Mech

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Mathematical homogenization (or averaging) of composite materials, such as fibre laminates, often leads to non-isotropic homogenized (averaged) materials. Especially the upcoming importance of these materials increases the need for accurate mechanical models of non-isotropic shell-like structures. We present a second-order (or: Reissner-type) theory for the elastic deformation of a plate with constant thickness for a homogeneous monotropic material. It is equivalent to Kirchhoff's plate theory as a first-order theory for the special case of isotropy and, furthermore, shear-deformable and equivalent to R. Kienzler's theory as a second-order theory for isotropy, which implies further equivalences to established shear-deformable theories, especially the Reissner-Mindlin theory and Zhilin's plate theory. Details of the derivation of the theory will be published in a forthcoming paper [3]. For an arbitrary three-dimensional body Ω ⊂ R 3 , which is assumed to be a bounded region, with the boundary regularity ∂Ω ∈ C 0,1 (this assumption allows to treat basically every body in engineering applications) and a boundary decomposition of the type ∂Ω = ∂Ω 0 ∪ ∂Ω N and ∂Ω 0 ∩ ∂Ω N = ∅, with ∂Ω 0 = ∅ relatively open and ∂Ω N relatively open, one can proof that the three-dimensional weak formulation of the mixed-boundary-value problem of linear elasticity (II) has a unique solution under very weak assumptions for the regularity of the given data, i.e., the prescribed displacement field u 0 on ∂Ω 0 , the prescribed traction-vector field g on ∂Ω N and the prescribed field of body force f . If we assume that the component functions of the fourth-order elasticity tensor E ijrs : Ω −→ R are continuous or piecewise constant and fulfill the symmetry relations E ijrs = E jirs , E ijrs = E ijsr and E ijrs = E rsij , so that in addition the associated 6 × 6-elasticity tensor is symmetric positive definite for all x ∈ Ω (which is a basic modeling assumption in continuum solid mechanics), we get:Theorem: Existence and uniqueness of the weak solution of the three-dimensional linear elasticity problemThen there exists an u 0 ∈ X that fulfills the weak displacement-boundary condition S u 0 (x) = u 0 (x) for almost all x ∈ ∂Ω 0 (with respect to the corresponding Lebesgue measure) and the problemsare equivalent and have a unique solution.Here S : W 1,2 (Ω) −→ W 1/2,2 (∂Ω) denotes the trace operator, X := W 1,2 (Ω) 3 and X 0 := {v ∈ X|∀i ∈ {1, 2, 3} :Sv i = 0 on ∂Ω 0 }. Problem (I) corresponds to the "principal of minimal potential energy" in a weak setting, which is usually the starting point of the derivation of consistent theories (e.g. in [1]). We use problem (II) as a starting point instead.We now assume our body to be a plate of constant thickness h. Therefore, let the mid-plane A ⊂ R 2 be a bounded region with ∂A ∈ C 0,1 and Ω :. For the boundary decomposition we assumeFurthermore, we introduce the plate parameter c :=, where a is a characteristic in-plane length of the plate, e.g., the diameter of A. The plate parameter is a dimensionles...

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On consistent plate theories

Cited by 77 publications

References 20 publications

On a consistent finite-strain plate theory based on three-dimensional energy principle

On a consistent finite-strain plate theory based on three-dimensional energy principle

On Material Conservation Laws for a Consistent Plate Theory

A consistent second‐order plate theory for monotropic material

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