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

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140494 ◽  
Author(s):  
Hui-Hui Dai ◽  
Zilong Song
Keyword(s):  
Finite Strain ◽  
Virtual Work ◽  
Plate Theory ◽  
Three Dimensional ◽  
Force Balance ◽  
Two Dimensional ◽  
Field Equations ◽  
Order Error ◽  
Energy Principle ◽  
Balance Structure

This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional 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 two-dimensional 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.

On a consistent dynamic finite-strain shell theory and its linearization

2018 ◽  
Vol 24 (8) ◽  
pp. 2335-2360 ◽  
Author(s):  
Zilong Song ◽  
Jiong Wang ◽  
Hui-Hui Dai
Keyword(s):  
Shell Theory ◽  
Finite Strain ◽  
Circular Cylindrical Shell ◽  
Shell Element ◽  
Momentum Balance ◽  
Orthotropic Materials ◽  
Bottom Surface ◽  
Position Vector ◽  
Field Equations ◽  
Order Error

In this paper, a dynamic finite-strain shell theory is derived, which is consistent with the three-dimensional (3-D) Hamilton’s principle with a fourth-order error under general loadings. A series expansion of the position vector about the bottom surface is adopted. By using the bottom traction condition and the 3-D field equations, the recursive relations for the expansion coefficients are successfully obtained. As a result, the top traction condition leads to a vector shell equation for the first coefficient vector, which represents the local momentum-balance of a shell element. Associated weak formulations, in connection with various boundary conditions, are also established. Furthermore, the derived equations are linearized to obtain a novel shell theory for orthotropic materials. The special case of isotropic materials is considered and comparison with the Donnell–Mushtari (D-M) shell theory is made. It can be shown that, to the leading order, the present shell theory agrees with the D-M theory for statics. Thus, the present shell theory actually provides a consistent derivation for the former one without any ad hoc assumptions. To test the validity of the present dynamic shell theory, the free vibration of a circular cylindrical shell is studied. The results for frequencies are compared with those of the 3-D theory and excellent agreements are found. In addition, it turns out that the present shell theory gives better results than the Flügge shell theory (which is known to provide the best frequency results among the first-approximation shell theories).

scholarly journals Three-dimensional stability of an elliptical vortex in a straining field

1984 ◽  
Vol 142 ◽  
pp. 451-466 ◽  
Author(s):  
A. C. Robinson ◽  
P. G. Saffman
Keyword(s):  
Steady State ◽  
Linear Stability ◽  
Cross Section ◽  
Dimensional Stability ◽  
Growth Rates ◽  
Finite Strain ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Elliptical Cross ◽  
Dimensional Instability

The three-dimensional linear stability of a rectilinear vortex of elliptical cross-section existing as a steady state in an irrotational straining field is studied numerically in the case of finite strain. It is shown that the instability predicted analytically for weak strain persists for finite strain and that the weak-strain results continue to be quantitatively valid for finite strain. The dependence of the growth rates of the unstable modes on the strain and the axial-disturbance wavelength is discussed. It is also shown that a three-dimensional instability is always more unstable than a two-dimensional instability in the range of parameters of most interest.

scholarly journals A refined dynamic finite-strain shell theory for incompressible hyperelastic materials: equations and two-dimensional shell virtual work principle

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200031 ◽  
Author(s):  
Xiang Yu ◽  
Yibin Fu ◽  
Hui-Hui Dai
Keyword(s):  
Shell Theory ◽  
Finite Strain ◽  
Virtual Work ◽  
Constitutive Relations ◽  
Strain Energy Function ◽  
Two Dimensional ◽  
Weak Formulation ◽  
Virtual Work Principle ◽  
Hyperelastic Materials ◽  
Shell Equations

Based on previous work for the static problem, in this paper, we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.

Effect of thermo-magneto-electro-mechanical fields on the bending behaviors of a three-layered nanoplate based on sinusoidal shear-deformation plate theory

2017 ◽  
Vol 21 (2) ◽  
pp. 639-669 ◽  
Author(s):  
Mohammed Arefi ◽  
Ashraf M Zenkour
Keyword(s):  
Temperature Distribution ◽  
Shear Deformation ◽  
Virtual Work ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Three Dimensional ◽  
Considerable Effect ◽  
Shear Deformation Plate Theory ◽  
Mechanical Bending ◽  
Magnetic Potentials

The nonlocal thermo-magneto-electro-mechanical bending behaviors of a three-layered nanoplate are presented in this study. The three-layered nanoplate includes a nano-sheet and two piezo-magnetic face-sheets at the top and the bottom. Temperature distribution is assumed linear along the thickness of the plate. The piezo-magnetic face-sheets are subjected to three-dimensional electric and magnetic potentials. The applied electric and magnetic potentials are applied at top of the face-sheets. The constitutive thermo-electro-magneto relations are derived based on the sinusoidal shear-deformation plate theory and nonlocal electro-magneto-elasticity. Using the principle of virtual work seven equations of the equilibrium are derived. The numerical results of this research indicate that some parameters have considerable effect on the bending behavior of three-layered nanoplate. Nonlocal parameter, applied electric and magnetic potentials, and temperature distribution are important parameters in this analysis.

Asymptotic Construction of Reissner-Like Models for Composite Plates With Accurate Strain Recovery

2001 ◽  
Author(s):  
Wenbin Yu ◽  
Dewey H. Hodges ◽  
Vitali V. Volovoi
Keyword(s):  
Plate Theory ◽  
Three Dimensional ◽  
Composite Plates ◽  
Present Theory ◽  
Laminated Plates ◽  
Constitutive Law ◽  
Two Dimensional ◽  
Normal Line ◽  
Plate Analysis ◽  
Line Analysis

Abstract The focus of this paper is to develop an asymptotically correct theory for composite laminated plates when each lamina exhibits monoclinic material symmetry. The development starts with formulation of the three-dimensional, anisotropic elasticity problem in which the deformation of the reference surface is expressed in terms of intrinsic two-dimensional variables. The Variational Asymptotic Method is then used to rigorously split this three-dimensional problem into a linear one-dimensional normal-line analysis and a nonlinear two-dimensional “plate” analysis accounting for transverse shear deformation. The normal-line analysis provides a constitutive law between the generalized, two-dimensional strains and stress resultants as well as recovering relations to approximately express the three-dimensional displacement, strain and stress fields in terms of plate variables calculated in the “plate” analysis. It is known that more than one theory that is correct to a given asymptotic order may exist. This nonuniqueness is used to cast a strain energy functional that is asymptotically correct through the second order into a simple “Reissner-like” plate theory. Although it is true that it is not possible to construct an asymptotically correct Reissner-like composite plate theory in general, an optimization procedure is used to drive the present theory as close to being asymptotically correct as possible while maintaining the beauty of Reissner-like formulation. Numerical results are presented to compare with the exact solution as well as a previous similar yet very different theory. The present theory has excellent agreement with the previous theory and exact results.

scholarly journals Graphic kinematics, visual virtual work and elastographics

2017 ◽  
Vol 4 (5) ◽  
pp. 170202 ◽  
Author(s):  
Allan McRobie ◽  
Marina Konstantatou ◽  
Georgios Athanasopoulos ◽  
Laura Hannigan
Keyword(s):  
Aspect Ratio ◽  
Recent Progress ◽  
Virtual Work ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Force Diagram ◽  
Graphic Statics ◽  
New Type ◽  
Structural Mechanisms ◽  
Displacement Diagram

In this paper, recent progress in graphic statics is combined with Williot displacement diagrams to create a graphical description of both statics and kinematics for two- and three-dimensional pin-jointed trusses. We begin with reciprocal form and force diagrams. The force diagram is dissected into its component cells which are then translated relative to each other. This defines a displacement diagram which is topologically equivalent to the form diagram (the structure). The various contributions to the overall Virtual Work appear as parallelograms (for two-dimensional trusses) or parallelopipeds (for three-dimensional trusses) that separate the force and the displacement pieces. Structural mechanisms can be identified by translating the force cells such that their shared faces slide across each other without separating. Elastic solutions can be obtained by choosing parallelograms or parallelopipeds of the appropriate aspect ratio. Finally, a new type of ‘elastographic’ diagram—termed a deformed Maxwell–Williot diagram (two-dimensional) or a deformed Rankine–Williot diagram (three-dimensional)—is presented which combines the deflected structure with the forces carried by its members.

Delamination Effects in Circular Sandwich Plates With Laminated Faces of General Layup and a “Soft” Core

2001 ◽  
Author(s):  
Oded Rabinovitch ◽  
Yeoshua Frostig
Keyword(s):  
Sandwich Plate ◽  
Virtual Work ◽  
Plate Theory ◽  
Constitutive Relations ◽  
Solution Procedure ◽  
Theory Of Elasticity ◽  
Soft Core ◽  
Sandwich Plates ◽  
Field Equations ◽  
Circular Sandwich Plates

Abstract The present study is concerned with the behavior of delaminated circular sandwich plates with a compressible “soft” core and composite laminated face sheets of general layup. The analysis follows the concepts of the High-Order Sandwich Plate Theory and employs the variational principle of virtual work for the derivation of the field equations of the fully bonded and delaminated regions. In the penny shaped disbonded region, the delaminated faces can slip horizontally with respect one to another, yet they may be in contact and resist vertical normal compressive stresses. The compressible core is considered using the 3D theory of elasticity and the lamination theory is employed for the modeling of the composite laminated face sheets. The formulation yields coordinate-dependent constitutive relations and governing equations and the solution procedure adopts the Galerkin method in the circumferential direction and the multiple-points shooting method in the radial direction. Numerical results regarding a typical delaminated sandwich plate are presented in terms of deformations and stresses. The results reveal the effect of the anisotropic laminated face sheets on the response of the structure and the influence of the delamination on the overall and, especially, the localized behavior of the plate.

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