Variational Principles for Some Nonstandard Elastic Problems

1987 ◽  
Vol 54 (4) ◽  
pp. 768-771 ◽  
Author(s):  
R. T. Shield
Keyword(s):  
Boundary Condition ◽  
Elastic Foundation ◽  
Variational Principles ◽  
Rigid Punch ◽  
Elastic Bodies ◽  
Smooth Contact ◽  
Nonstandard Problems

Variational principles are derived for some nonstandard problems involving elastic bodies in smooth contact. For these problems, the portions of the surfaces where one boundary condition holds rather than another must be determined as part of the solution to the problem. Cases considered include a body containing a crack or delamination, indentation by a rigid punch, and contact with an elastic foundation.

scholarly journals Uniqueness for Elastic Crack and Punch Problems

1982 ◽  
Vol 49 (3) ◽  
pp. 516-518 ◽  
Author(s):  
R. T. Shield
Keyword(s):  
Elastic Foundation ◽  
Equilibrium States ◽  
Uniqueness Of Solution ◽  
Rigid Punch ◽  
Frictionless Contact ◽  
Elastic Bodies ◽  
Elastic Crack ◽  
Smooth Contact

Uniqueness of solution is shown for equilibrium states for elastic bodies in smooth contact. The cases considered include a body with a crack that may open only partially under load with parts of the faces in frictionless contact. Indentation of a body by a smooth rigid punch and contact with an elastic foundation are also treated.

The Boundary Condition Influence on a Stress-Strain State of a Corrugated Plate on an Elastic Foundation

2018 ◽  
Vol 931 ◽  
pp. 60-65 ◽  
Author(s):  
Aleksey N. Beskopylny ◽  
Elena E. Kadomtseva ◽  
Grigory P. Strelnikov
Keyword(s):  
Boundary Condition ◽  
Elastic Foundation ◽  
Strain State ◽  
Polymer Coatings ◽  
Stress Strain ◽  
Stress Strain State ◽  
Different Types ◽  
Deformed State ◽  
Decorative Material ◽  
Construction Practice

In this paper, we consider the influence of the conditions for fixing a wavy plate lying on an elastic foundation on its stressed-deformed state. The profiled plates are widely used in construction practice as fencing structures, for siding works, for roofing and others. The stress-strain state of the wavy plates varies depending on geometry, materials mechanical properties, foundation characteristics and boundary condition. Steel with polymer coatings, which make the sheets a decorative material, is increasingly used in individual and low-rise buildings. The elastic foundation is considered as Winkler base, so we suppose that the reaction of the base is directly proportional to the deflection of the plate at each point. The Bubnov-Galerkin method is used to determine the stress-strain state of the plate. To solve the problem, we use special orthogonal Legendre polynomials satisfying the boundary conditions: simply supported and clamped edges. The results of the calculations were compared for different types of fixation.

Mechanics Mechanism Analysis of Concrete Pavement Resonant Rubblization

2015 ◽  
Vol 667 ◽  
pp. 365-369
Author(s):  
Peng Chen ◽  
Xin Qiu ◽  
Qing Zhu ◽  
Chan Chan Ouyang
Keyword(s):  
Boundary Condition ◽  
Natural Frequency ◽  
Elastic Foundation ◽  
Thin Plate ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Concrete Pavement ◽  
Mechanism Analysis ◽  
Vibration Theory ◽  
Fundamental Natural Frequency

Based on the assumption of thin plate of elastic foundation and vibration theory, a method for calculating the fundamental natural frequency of cement slab is presented and the certain relationship between the fundamental natural frequency of cement slab and cement slab boundary condition is discussed. As well, according to the analysis results of fundamental natural frequencies of the typical cement pavements of China, the selected proposals of the excitation frequency of the resonant rubblization machine are presented .The research results provide a theory support to popularize resonant rubblization technology in overlaying and rebuilding engineering of the existed cement pavements in China.

The Incremental Variational Principles for Frictional Contact Problems of Linear Elasticity

1993 ◽  
Vol 60 (4) ◽  
pp. 982-985 ◽  
Author(s):  
G. Zboinski
Keyword(s):  
Variational Principles ◽  
Contact Problems ◽  
Complementary Energy ◽  
Energy Functionals ◽  
Variational Functional ◽  
Elastic Bodies ◽  
Static Contact ◽  
Special Cases ◽  
The Common ◽  
Type Potential

Four types of the most frequently used variational functional are employed in order to form the inequality principles of the kineto-static contact problem of two elastic bodies in the common relative motion. As the general case, the principle based on the Hu- Washizu functional is proposed. The principles formed with the Reissner type, potential energy, and complementary energy functionals are derived as the special cases.

Nonlocal and Nonlinear Friction Laws and Variational Principles for Contact Problems in Elasticity

1983 ◽  
Vol 50 (1) ◽  
pp. 67-76 ◽  
Author(s):  
J. T. Oden ◽  
E. B. Pires
Keyword(s):  
Variational Principles ◽  
Nonlinear Problems ◽  
Contact Problems ◽  
Nonlinear Friction ◽  
Metallic Surfaces ◽  
Uniqueness Of Solutions ◽  
Friction Laws ◽  
Elastic Bodies ◽  
Mathematical Difficulties ◽  
And Variational Principles

The use of the classical Coulomb law of friction in the formulation of contact problems in elasticity leads to both physical and mathematical difficulties; the former arises from the fact that this law provides a poor model of frictional stresses at points on metallic surfaces in contact, and the latter is due to the fact that the existence of solutions of the governing equations can be proved only for very special situations. In the present paper, nonclassical friction laws are proposed in an attempt to overcome both of these difficulties. We consider a class of contact problems involving the equilibrium of linearly elastic bodies in contact on surfaces on which nonlocal and nonlinear friction laws are assumed to hold. The physics of friction between metallic bodies in contact is discussed and arguments in support of the theory are presented. Variational principles for boundary-value problems in elasticity in which such nonlinear nonlocal laws hold are then developed. A brief discussion of the questions of existence and uniqueness of solutions to the nonlocal and nonlinear problems is given.

scholarly journals On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity

2020 ◽  
Vol 71 (6) ◽  
Author(s):  
Victor A. Eremeyev ◽  
Sergey A. Lurie ◽  
Yury O. Solyaev ◽  
Francesco dell’Isola
Keyword(s):  
Strain Gradient ◽  
Weak Solutions ◽  
Variational Principles ◽  
Gradient Elasticity ◽  
Deformation Energy ◽  
Well Posedness ◽  
Strain Gradient Elasticity ◽  
Elastic Bodies ◽  
Integration Schemes ◽  
Second Gradient

AbstractIn this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: only one extra second gradient elastic coefficient is introduced. The studied class of solids is therefore related to Korteweg or Cahn–Hilliard fluids. The postulated energy naturally induces the space in which the aforementioned well-posedness result can be formulated. In this energy space, the introduced norm does involve the linear combination of some specific higher-order derivatives only: it is, in fact, a particular example of anisotropic Sobolev space. It is also proven that aforementioned weak solutions belongs to the space $$H^1(div,V)$$ H 1 ( d i v , V ) , i.e. the space of $$H^1$$ H 1 functions whose divergence belongs to $$H^1$$ H 1 . The proposed mathematical frame is essential to conceptually base, on solid grounds, the numerical integration schemes required to investigate the properties of dilatational strain gradient elastic bodies. Their energy, as studied in the present paper, has manifold interests. Mathematically speaking, its singularity causes interesting mathematical difficulties whose overcoming leads to an increased understanding of the theory of second gradient continua. On the other hand, from the mechanical point of view, it gives an example of energy for a second gradient continuum which can sustain externally applied surface forces and double forces but cannot sustain externally applied surface couples. In this way, it is proven that couple stress continua, introduced by Toupin, represent only a particular case of the more general class of second gradient continua. Moreover, it is easily checked that for dilatational strain gradient continua, balance of force and balance of torques (or couples) are not enough to characterise equilibrium: to this aim, externally applied surface double forces must also be specified. As a consequence, the postulation scheme based on variational principles seems more suitable to study second gradient continua. It has to be remarked finally that dilatational strain gradient seems suitable to model the experimentally observed behaviour of some material used in 3D printing process.

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