Some Two-dimensional Non-classical Models of Anisotropic Plates

2020 ◽  
pp. 75-94
Author(s):  
Alexander K. Belyaev ◽  
Nikita F. Morozov ◽  
Peter E. Tovstik ◽  
Tatyana P. Tovstik
Keyword(s):  
Two Dimensional ◽  
Anisotropic Plates ◽  
Classical Models

Interface Cracks in Kirchhoff Anisotropic Thin Plates of Dissimilar Materials

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Xu Wang ◽  
Peter Schiavone
Keyword(s):  
Thin Plate ◽  
Interfacial Crack ◽  
Dissimilar Materials ◽  
Thin Plates ◽  
Two Dimensional ◽  
Structure And Properties ◽  
Intensity Factors ◽  
Crack Tip Field ◽  
Anisotropic Plates ◽  
Appl Math

This paper investigates the problem of stretching and bending deformations of a Kirchhoff anisotropic thin plate composed of two dissimilar materials bonded along a straight interface containing a crack. Our analysis makes use of the Stroh octet formalism developed recently by Cheng and Reddy (Cheng and Reddy, 2002, “Octet Formalism for Kirchhoff Anisotropic Plates,” Proc. R. Soc. Lond., A458, pp. 1499–1517; Cheng and Reddy, 2003, “Green’s Functions for Infinite and Semi-Infinite Anisotropic Thin Plates,” ASME J. Appl. Mech., 70, pp. 260–267; Cheng and Reddy, 2004, “Laminated Anisotropic Thin Plate With an Elliptic Inhomogeneity,” Mech. Mater., 36, pp. 647–657; Cheng and Reddy, 2005, “Structure and Properties of the Fundamental Elastic Plate Matrix,” J. Appl. Math. Mech., 85, pp. 721–739) for Kirchhoff anisotropic plates. It is found that the interfacial crack-tip field consists of a pair of two-dimensional oscillatory singularities, which are explicitly determined. Two complex intensity factors are proposed to evaluate the two oscillatory singularities.

Two dimensional displacement and stress fields for tri-material V-notches and sharp inclusions in anisotropic plates

2020 ◽  
Vol 80 ◽  
pp. 103927
Author(s):  
Shanlong Yao ◽  
Michele Zappalorto ◽  
Wei Pan ◽  
Changzheng Cheng ◽  
Zhongrong Niu
Keyword(s):  
Stress Fields ◽  
Two Dimensional ◽  
Anisotropic Plates

Linear and nonlinear stability of floating viscous sheets

2011 ◽  
Vol 683 ◽  
pp. 112-148 ◽  
Author(s):  
G. Pfingstag ◽  
B. Audoly ◽  
A. Boudaoud
Keyword(s):  
Surface Tension ◽  
Surface Forces ◽  
Two Dimensional ◽  
Buckling Mode ◽  
Geometrical Nonlinearities ◽  
Flat Sheet ◽  
Long Time ◽  
Classical Models ◽  
Linear And Nonlinear ◽  
The Stability

AbstractWe study the stability of a thin, Newtonian viscous sheet floating on a bath of denser fluid. We first derive a general set of equations governing the evolution of a nearly flat sheet, accounting for geometrical nonlinearities associated with moderate rotations. We extend two classical models by considering arbitrary external body and surface forces; these two models follow from different scaling assumptions, and are derived in a unified way. The equations capture two modes of deformation, namely viscous bending and stretching, and describe the evolution of thickness, mid-surface and in-plane velocity as functions of two-dimensional coordinates. These general equations are applied to a floating viscous sheet, considering gravity, buoyancy and surface tension. We investigate the stability of the flat configuration when subjected to arbitrary in-plane strain. Two unstable modes can be found in the presence of compression. The first one combines undulations of the centre-surface and modulations of the thickness, with a wavevector perpendicular to the direction of maximum applied compression. The second one is a buckling mode; it is purely undulatory and has a wavevector along the direction of maximum compression. A nonlinear analysis yields the long-time evolution of the undulatory mode.

scholarly journals A two-dimensional nonlinear theory of anisotropic plates

2000 ◽  
Vol 32 (7-8) ◽  
pp. 855-875 ◽  
Author(s):  
R.P. Gilbert ◽  
T.S. Vashakmadze
Keyword(s):  
Nonlinear Theory ◽  
Two Dimensional ◽  
Anisotropic Plates

Stroh-Like Complex Variable Formalism for the Bending Theory of Anisotropic Plates

2003 ◽  
Vol 70 (5) ◽  
pp. 696-707 ◽  
Author(s):  
C. Hwu
Keyword(s):  
Complex Variable ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Plate Bending ◽  
Two Dimensional ◽  
New Formalism ◽  
Anisotropic Plates ◽  
Present Formalism ◽  
Bending Problems ◽  
Almost All

Based upon the knowledge of the Stroh formalism and the Lekhnitskii formalism for two-dimensional anisotropic elasticity as well as the complex variable formalism developed by Lekhnitskii for plate bending problems, in this paper a Stroh-like formalism for the bending theory of anisotropic plates is established. The key feature that makes the Stroh formalism more attractive than the Lekhnitskii formalism is that the former possesses the eigenrelation that relates the eigenmodes of stress functions and displacements to the material properties. To retain this special feature, the associated eigenrelation and orthogonality relation have also been obtained for the present formalism. By intentional rearrangement, this new formalism and its associated relations look almost the same as those for the two-dimensional problems. Therefore, almost all the techniques developed for the two-dimensional problems can now be applied to the plate bending problems. Thus, many unsolved plate bending problems can now be solved if their corresponding two-dimensional problems have been solved successfully. To illustrate this benefit, two simple examples are shown in this paper. They are anisotropic plates containing elliptic holes or inclusions subjected to out-of-plane bending moments. The results are simple, exact and general. Note that the anisotropic plates treated in this paper consider only the homogeneous anisotropic plates. If a composite laminate is considered, it should be a symmetric laminate to avoid the coupling between stretching and bending behaviors.

A Unified Formalism of Two-Dimensional Anisotropic Elasticity, Piezoelectricity and Unsymmetric Laminated Plates

2005 ◽  
Vol 72 (3) ◽  
pp. 422-431
Author(s):  
Wan-Lee Yin
Keyword(s):  
General Solution ◽  
Constitutive Relations ◽  
Anisotropic Elasticity ◽  
Complex Variables ◽  
Laminated Plates ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Eigenvalues And Eigenvectors ◽  
Infinite Domain ◽  
Anisotropic Plates

A unified formalism is presented for theoretical analysis of plane anisotropic elasticity and piezoelectricity, unsymmetric anisotropic plates, and other two-dimensional problems of continua with linear constitutive relations. Complex variables are used to reduce the governing differential equations to algebraic equations. The constitutive relation then yields an eigenrelation, which is easily solved explicitly for the material eigenvalues and eigenvectors. The latter have polynomial expressions in terms of the eigenvalues. When the eigenvectors are combined after multiplication by arbitrary analytic functions containing the corresponding eigenvalues, one obtains the two-dimensional general solution. Important results, including the orthogonality of the eigenvectors, the expressions of the pseudometrics and the intrinsic tensors, are established here for nondegenerate materials, including the case of all distinct eigenvalues. Green’s functions of the infinite domain, and of the semi-infinite domain with interior or edge singularities, are determined explicitly for the most general types of point loads and discontinuities (dislocations).

Sign in / Sign up

Export Citation Format

Share Document