Mechanics elastic bodies

The new approach to crack theory used in the book is intriguing in that it fails to lead to physical contradictions. Given a classical approach to the description of cracks in elastic bodies, the boundary conditions on crack faces are known to be considered as equations. In a number of specific cases there is no difflculty in finding solutions of such problems leading to physical contradictions. It is precisely these crack faces for such solutions that penetrate each other. Boundary conditions analysed in the book are given in the form of inequalities, and they are properly nonpenetration conditions of crack faces. The above implies that similar problems may be considered from the contact mechanics standpoint. 

Contact mechanics deals with the deformation of solids in contact. Consider two elastic bodies, shown schematically in Fig. 3, of radii of curvature R[ and Rt, Young s moduli E and E2, and Poisson s ratios and V2. Define... 

Bernoulli and Euler dominated the mechanics of flexible and elastic bodies for many years. They also investigated the flow of fluids. In particular, they wanted to know about the relationship between the speed at which blood flows and its pressure. Bernoulli experimented by puncturing the wall of a pipe with a small, open-ended straw, and noted that as the fluid passed through the tube the height to which the fluid rose up the straw was related to fluid s pressure. Soon physicians all over Europe were measuring patients blood pressure by sticking pointed-ended glass tubes directly into their arteries. (It was not until 1896 that an Italian doctor discovered a less painful method that is still in widespread... 

The mechanical properties of a material play an important role in powder flow and compaction by influencing particle-particle interaction and cohesion, that is to say, by influencing the true area of contact between particles. For example, Hertz [26] demonstrated that both the size and shape of the zone of contact followed simply from the elastic properties of a material. Clearly then, the true area of contact is affected by elastic properties. From the laws of elasticity, one can predict the area of contact between two elastic bodies. More recent work has demonstrated, however, that additional factors must be taken... 

On the first line, SX is regarded as a function of X. On the second line, the first term is the work exerted from the outside on the surface, da and rij being the surface element and the outward unit normal vector, while the second term is the change within the elastic body due to mechanical disequilibrium. We divide the stress into two parts,... 

Many variables used and phenomena described by fracture mechanics concepts depend on the history of loading (its rate, form and/or duration) and on the (physical and chemical) environment. Especially time-sensitive are the level of stored and dissipated energy, also in the region away from the crack tip (far held), the stress distribution in a cracked visco-elastic body, the development of a sub-critical defect into a stress-concentrating crack and the assessment of the effective size of it, especially in the presence of microyield. The role of time in the execution and analysis of impact and fatigue experiments as well as in dynamic fracture is rather evident. To take care of the specihcities of time-dependent, non-linearly deforming materials and of the evident effects of sample plasticity different criteria for crack instability and/or toughness characterization have been developed and appropriate corrections introduced into Eq. 3, which will be discussed in most contributions of this special Double Volume (Vol. 187 and 188). 

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics variational formulations computational mechanics statics, kinematics and dynamics of rigid and elastic bodies vibrations of solids and structures dynamical systems and chaos the theories of elasticity, plasticity and viscoelasticity composite materials rods, beams, shells and membranes structural control and stability soils, rocks and geomechanics fracture tribology experimental mechanics biomechanics and machine design. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

Fracture mechanics approach. Fracture mechanics provides the basis for many modern fatigue crack-growth studies. AK is the stress intensity range (kjnilx-kjnm) where K is the magnitude of the mathematically ideal crack-tip stress field in a homogeneous linear-elastic body and is a function of applied load and crack geometry. 

A more common body is the plasto-viscoelastic, or Bingham body. Its mechanical model is shown in Figure 8-16C. When a stress is applied that is below the yield stress, the Bingham body reacts as an elastic body. At stress values beyond the yield stress, there are two components, one of which is constant and is represented by the friction ele-... 

Figure 8-16 Mechanical Models for a Plastic Body. (A) St. Venant body, (B) plasto-elastic body, and (C) plasto-viscoelastic or Bingham body.

Johnson, K. L. (1985). Contact Mechanics. Cambridge University Press, Cambridge. Kalker, J. J. (1990). Three dimensional elastic bodies in rolling contact. Kluwer Academic Publishers, Dordrecht. 

It is assumed that the limit equilibrium state is reached if cracks develop and increase on the surface of the body volume under action of external loadings. In linear fracture mechanics, Irwin s force criterion and an equivalent Griffith s energy criterion completely determine the equilibrium condition of a continuum elastic body with a crack [9],... 

According to linear fracture mechanics, fracture energy Gc of a linearly elastic body is given by... 

It should be emphasized that the concept of linear fracture mechanics can be applied only to materials with fine structure, such as a cement stone or fine-grained concrete, that is, to linear-elastic bodies. For such bodies, a condition of self-similarity for the prefracture zone exists... 

Contact mechanics is both an old and a modern field. Its classical domains of application are adhesion, friction, and fracture. Clearly, the relevance of the field for technical devices is enormous. Systematic strategies to control friction and adhesion between solid surfaces have been known since the stone age [1]. In modern times, the ground for systematic studies was laid in 1881 by Hertz in his seminal paper on the contact between soHd elastic bodies [2]. Hertz considers a sphere-plate contact. Solving the equations of continuum elasticity, he finds that the vertical force, F , is proportional to where S is the indentation. The sphere-plate contact forms a nonlinear spring with a differential spring constant k = dF/dS oc The nonhnearity occurs because there is a concentration of stress at the point of contact. Such stress concentrations - and the ensuing mechanical nonhnearities - are typical of contact mechanics. 

Such process is referred to as the elastic aftereffect and can be found in solid-like systems that reveal an elastic behavior. Elastic aftereffect is mechanically reversible the removal of applied stress results in gradual decrease of strain to zero due to the energy stored in the elastic element. The object thus restores its original shape. At the same time, in contrast to the case of a truly elastic body, the deformation of an object that follows Kelvin s model is thermodynamically irreversible due to the dissipation of energy in the... 

For concentrated emulsions and foams, Prin-cen [1983, 1985] proposed a stress-strain theory based on a two-dimensional cell model. Consider a steady state shearing of such a system. Initially, at small values of strain, the stress increases linearly as in elastic body. As the strain value increases, the stress reaches its yield value, then at stiU higher deformation, it catastrophically drops to the negative values. The reason for the latter behavior is the creation of unstable cell structure that provides the recoil mechanism. The predicted dependencies for modulus and the yield stress were expressed as ... 

Garagash, l.A. 1983. On brittle failure of elastic bodies with large number of cracks. In Mechanics of Tectonic Processes. Alma-Ata Nauka. 

Because of their chain-like structure, polymers are not perfectly elastic bodies, and deformation is accompanied by a complex series of long and short range coopraative molecular rearrangranraits. Consequently, the mechanical behavior is dominated by viscoelastic phenomraia, in contrast to materials such as metal and glass where atomic adjustmmts under stress are more localized and limited. 

For a cracked, non-linear elastic body one can define a parameter / that represents the mechanical energy release rate, i.e., /= -d U IAc. In this sense, / is analogous to the LEFM parameter G but for a non-linear rather than a linear elastic material. Rice (1968) has shown that / can be written as the line integral... 

The two extreme cases of mechanical behavior can be reproduced very well by mechanical models. A compressed Hookean spring can serve as a model for the energy-elastic body under load (Figure 11-11). On releasing the load, the compressed spring immediately returns to its original position. The relationship between the shear stress (021) = Oe, the shear modulus Ge, and the elastic deformation ye is given by Hooke s law [Equation (11-1)] ... 

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