The Influence of the Shape and Rigidity of an Elastic Inclusion on the Transverse Flexure of Thin Plates

1943 ◽  
Vol 10 (2) ◽  
pp. A69-A75
Author(s):  
Martin Goland
Keyword(s):  
Stress Distribution ◽  
Future Study ◽  
Rigid Inclusion ◽  
Elastic Inclusion ◽  
Thin Plates ◽  
Elastic Plates ◽  
Perforated Plates ◽  
Circular Rigid Inclusion ◽  
Special Cases ◽  
Elastic Inclusions

Abstract The purpose of this paper is to investigate the influence of several types of inclusions on the stress distribution in elastic plates under transverse flexure. An “inclusion” is defined as a close-fitting plate of some second material cemented into a hole cut in the interior of the elastic plate. Depending upon the properties of the material of which it is composed, the inclusion is described as rigid or elastic. In particular, the solutions presented will deal with the effects of circular inclusions of differing degrees of elasticity and rigid inclusions of varying elliptical form. Since the rigid inclusion and the hole are limiting types of elastic inclusions, and the circular shape is a special form of the ellipse, plates with either a circular hole or a circular rigid inclusion are important special cases of this discussion. It is hoped that the present analysis of several types of inclusions will aid in a future study of perforated plates stiffened by means of reinforcing rings fitted into the holes.

Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies

2015 ◽  
Vol 22 (4) ◽  
pp. 737-750 ◽  
Author(s):  
AM Khludnev ◽  
L Faella ◽  
TS Popova
Keyword(s):  
Rigid Inclusion ◽  
Existence Of Solutions ◽  
Elastic Inclusion ◽  
Joint Point ◽  
Elastic Bodies ◽  
Rigidity Parameter ◽  
Elastic Inclusions ◽  
Type Boundary ◽  
Crack Faces ◽  
Inequality Type

This paper concerns an equilibrium problem for a two-dimensional elastic body with a thin Timoshenko elastic inclusion and a thin rigid inclusion. It is assumed that the inclusions have a joint point and we analyze a junction problem for these inclusions. The existence of solutions is proved and the different equivalent formulations of the problem are discussed. In particular, the junction conditions at the joint point are found. A delamination of the elastic inclusion is also assumed. In this case, the inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between the crack faces. We investigate the convergence to infinity and zero of a rigidity parameter of the elastic inclusion. It is proved that in the limit, we obtain a rigid inclusion and a zero rigidity inclusion (a crack).

scholarly journals Analytical Investigation of Viscoelastic Stagnation-Point Flows with Regard to Their Singularity

2021 ◽  
Vol 11 (15) ◽  
pp. 6931
Author(s):  
Jie Liu ◽  
Martin Oberlack ◽  
Yongqi Wang
Keyword(s):  
Stress Distribution ◽  
Stress Field ◽  
Stagnation Point ◽  
Analytical Solutions ◽  
Wide Spectrum ◽  
Momentum Conservation ◽  
Stagnation Point Flow ◽  
Model Parameters ◽  
Viscoelastic Models ◽  
Special Cases

Singularities in the stress field of the stagnation-point flow of a viscoelastic fluid have been studied for various viscoelastic constitutive models. Analyzing the analytical solutions of these models is the most effective way to study this problem. In this paper, exact analytical solutions of two-dimensional steady wall-free stagnation-point flows for the generic Oldroyd 8-constant model are obtained for the stress field using different material parameter relations. For all solutions, compatibility with the conservation of momentum is considered in our analysis. The resulting solutions usually contain arbitrary functions, whose choice has a crucial effect on the stress distribution. The corresponding singularities are discussed in detail according to the choices of the arbitrary functions. The results can be used to analyze the stress distribution and singularity behavior of a wide spectrum of viscoelastic models derived from the Oldroyd 8-constant model. Many previous results obtained for simple viscoelastic models are reproduced as special cases. Some previous conclusions are amended and new conclusions are drawn. In particular, we find that all models have singularities near the stagnation point and most of them can be avoided by appropriately choosing the model parameters and free functions. In addition, the analytical solution for the stress tensor of a near-wall stagnation-point flow for the Oldroyd-B model is also obtained. Its compatibility with the momentum conservation is discussed and the parameters are identified, which allow for a non-singular solution.

The Quasi-Static Analysis for Two-Dimensional Thin Plates

2011 ◽  
Vol 255-260 ◽  
pp. 166-169
Author(s):  
Li Chen ◽  
Yang Bai
Keyword(s):  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Solution Space ◽  
Expansion Method ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Elastic Plates ◽  
Various Boundary Conditions ◽  
Eigenfunction Expansion Method ◽  
Concentration Phenomena

The eigenfunction expansion method is introduced into the numerical calculations of elastic plates. Based on the variational method, all the fundamental solutions of the governing equations are obtained directly. Using eigenfunction expansion method, various boundary conditions can be conveniently described by the combination of the eigenfunctions due to the completeness of the solution space. The coefficients of the combination are determined by the boundary conditions. In the numerical example, the stress concentration phenomena produced by the restriction of displacement conditions is discussed in detail.

Research on Vibration Control of Thin Plate Based on Pre-Stressing

2018 ◽  
Author(s):  
Cheng Zhang ◽  
Jian-run Zhang ◽  
Xi Lu
Keyword(s):  
Differential Equation ◽  
Stress Distribution ◽  
Tensile Stress ◽  
Vibration Control ◽  
Thin Plate ◽  
Natural Frequencies ◽  
Control Method ◽  
Dynamic Stiffness ◽  
Thin Plates ◽  
Shape Functions

The weak dynamic stiffness of thin plate is one of the important factors that limit the use of thin plate. Improving the dynamic stiffness of thin plate is one of the effective methods for the vibration control of thin plate. In this paper, the influence of pre-stress on the vibration characteristics of thin plate is studied. A vibration control method of thin plate based on pre-stress is proposed. The vibration differential equation of quadrate thin plate under pre-stressing is established. Using the Galerkin principle, the natural frequencies corresponding to the shape functions of the quadrate thin plates under pre-stressing in different distribution forms are obtained. By comparison, it is found that pre-stressing on the thin plate can change the dynamic stiffness of thin plate. In particular, tensile stress can increase the dynamic stiffness of thin plate while compressive stress can reduce the dynamic stiffness of the thin plate. The greater the pre-stress, the more obvious the effect. In the end, the requirements of the pre-stress distribution which can improve the dynamic stiffness of thin plate effectively are derived.

An elastic harmonic inclusion near a rigid harmonic inclusion loaded by a couple

2019 ◽  
Vol 84 (3) ◽  
pp. 555-566
Author(s):  
Xu Wang ◽  
Liang Chen ◽  
Peter Schiavone
Keyword(s):  
Rigid Inclusion ◽  
Elastic Inclusion ◽  
Complex Loading ◽  
Solution Procedure ◽  
External Loading ◽  
Surrounding Matrix ◽  
The Matrix ◽  
Mapping Techniques ◽  
Loading Parameter ◽  
Harmonic Inclusion

AbstractWe use conformal mapping techniques to solve the inverse problem concerned with an elastic non-elliptical harmonic inclusion in the vicinity of a rigid non-elliptical harmonic inclusion loaded by a couple when the surrounding matrix is subjected to remote uniform stresses. Both a size-independent complex loading parameter and a size-dependent real loading parameter are introduced as part of the solution procedure. The stress field inside the elastic inclusion is uniform and hydrostatic; the interfacial normal and tangential stresses as well as the hoop stress on the matrix side are uniform along each one of the two inclusion–matrix interfaces. The tangential stress along the interface of the elastic inclusion (free of external loading) vanishes, whereas that along the interface of the rigid inclusion (loaded by the couple) does not. A novel method is proposed to determine the area of the rigid inclusion.

Electric-current-induced thermal stress around a non-circular rigid inclusion in a two-dimensional nonlinear thermoelectric material

2020 ◽  
Vol 231 (11) ◽  
pp. 4603-4619
Author(s):  
Hai-Bing Yang ◽  
Chuan-Bin Yu ◽  
Jie-Yao Tang ◽  
Jian Qiu ◽  
Xiao-Qing Zhang
Keyword(s):  
Thermal Stress ◽  
Electric Current ◽  
Thermoelectric Material ◽  
Rigid Inclusion ◽  
Two Dimensional ◽  
Circular Rigid Inclusion

Stress analysis of a debonding and a crack around a circular rigid inclusion

1986 ◽  
Vol 32 (3) ◽  
pp. 169-183 ◽  
Author(s):  
Norio Hasebe ◽  
Mikiya Okumura ◽  
Takuji Nakamura
Keyword(s):  
Stress Analysis ◽  
Rigid Inclusion ◽  
Circular Rigid Inclusion

Shape Control of Flexural Vibrations of Circular Plates by Shaped Piezoelectric Actuation

2003 ◽  
Vol 125 (1) ◽  
pp. 88-94 ◽  
Author(s):  
Manfred Nader ◽  
Hubert Gattringer ◽  
Michael Krommer ◽  
Hans Irschik
Keyword(s):  
Shape Control ◽  
Analytic Solution ◽  
Thin Plates ◽  
Elastic Plates ◽  
Circular Plates ◽  
3 Dimensional ◽  
Dimensional Finite Element ◽  
Flexural Vibrations ◽  
External Forces ◽  
Kirchhoff Theory

Vibrations of smart elastic plates with integrated piezoelectric actuators are considered. Piezoelastic layers are used to generate a distributed actuation of the plate. A spatial shape function of the piezoelastic actuators is sought such that flexural vibrations induced by external forces can be completely nullified. An analytic solution of this problem is worked out for the case of clamped circular plates with a spatially constant force loading. The Kirchhoff theory of thin plates is used to derive this analytic solution. Our result is successfully validated by means of coupled 3-dimensional finite-element computations.

Sign in / Sign up

Export Citation Format

Share Document