Deformable Body Mechanics

All bodies undergo deformation when subjected to a force. In some cases, deformation may be ignored and these are problems of rigid body mechanics (Ch. 2). In other cases, deflections are of importance. Such problems are considered in this chapter, including  

Deformable body mechanics is sometimes referred to as mechanics of materials and strength of materials. 

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Deformable Body Mechanics 81 This is a quadratic equation whose maximum root is ... 

From this kind of continuum mechanics one can move further towards the domain of almost pure mathematics until one reaches the field of rational mechanics, which harks back to Joseph Lagrange s (1736-1813) mechanics of rigid bodies and to earlier mathematicians such as Leonhard Euler (1707-1783) and later ones such as Augustin Cauchy (1789-1857), who developed the mechanics of deformable bodies. The preeminent exponent of this kind of continuum mechanics was probably Clifford Truesdell in Baltimore. An example of his extensive writings is A First Course in... 

Classical lamination theory consists of a coiiection of mechanics-of-materials type of stress and deformation hypotheses that are described in this section. By use of this theory, we can consistentiy proceed directiy from the basic building block, the lamina, to the end result, a structural laminate. The whole process is one of finding effective and reasonably accurate simplifying assumptions that enable us to reduce our attention from a complicated three-dimensional elasticity problem to a SQlvable two-dimensinnal merbanics of deformable bodies problem. 

Mechanics is the physical science that deals with the effects of forces on the state of motion or rest of solid, liquid, or gaseous bodies. The field may he divided into the mechanics of rigid bodies, the mechanics of deformable bodies, and the mechanics of fluids. 

A rigid body is one that does not deform. True rigid bodies do not exist in nature however, the assumption of rigid body behavior is usually an acceptable accurate simplification for examining the state of motion or rest of structures and elements of structures. The rigid body assumption is not useful in the study of structural failure. Rigid body mechanics is further subdivided into the study of bodies at rest, stalks, and the study of bodies in motion, dynamics. 

Sommerfeld, A., Mechanics of Deformable Bodies, 1964 English translation Academic Press, New York, 1964. 

A. Sommerfield, Mechanics of Deformable Bodies (includes discussion of the perfect analogy between hydrodynamics and electrodynamics ). 

The mechanical equilibrium conditions applied to an arbitrary volume V, bounded by the closed surface A, and completely inside the deformable body give... 

As described above, instability of the interface between the electrolyte and molten metal is a significant problem that is one root cause of the energy inefficiency of Hall cells. Expressed simply, the interface is deformed by the electromagnetic body forces arising from the interaction between currents in the cell and the magnetic field. The currents are themselves affected by the interface position because it determines the distance between the top surface of the aluminum and the bottom of the anode. There is therefore the possibility that interface deformation leads to further interface deformation. Other mechanisms for generating waves at the interface may be significant, for example, the Kelvin-Helmholtz... 

As demonstrated by the foregoing two formulations, some problems taken from mechanics can be formulated by using only Newton s laws of motion these are called mechanically determined problems. The dynamics of rigid bodies in the absence of friction, statically determined problems of rigid bodies, and mechanics of ideal fluids provide examples of this class. Some other mechanics problems, however, require knowledge beyond Newton s laws of motion. These are called mechanically undetermined problems. The dynamics of rigid bodies with friction and the mechanics of deformable bodies provide examples of this class. 

Many biomedical devices are implanted in a mechanically dynamic environment in the body, which requires the implants to sustain and recover from various deformations without mechanical irritation of the surroimding tissue. 

The mechanics of a deformable body treated here is based on Newton s laws of motion and the laws of thermodynamics. In this Chapter we present the fundamental concepts of continuum mechanics, and, for conciseness, the material is presented in Cartesian tensor formulation with the implicit assumption of Einstein s summation convention. Where this convention is exempted we shall denote the index thus ( a). 

Schallamach, A. In Mechanics of the contact between deformable bodies. Delft University Press Enschede, Netnerland, 1974, pp. 359-376. 

Boucher, M. (1975) Signorini s Problem in Viscoelasticity , in The Mechanics of the Contact Between Deformable Bodies, ed. by A.D. de Pater, J.J. Kalker (Delft University Press) pp. 41-53... 

Deresiewicz, H. (1968) A note on Hertz theory of impact. Acta Mech. 6, 110-112 Doi, M., Edwards, S.F. (1986) The Theory of Polymer Dynamics (Clarendon Press, Oxford) Dugdale, D.S. (1960) Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100-104 Dundurs, J. (1975) Properties of Elastic Bodies in Contact , in The Mechanics of the Contact between Deformable Bodies, ed. by A. D. de Pater, J.J. Kalker (Delft University Press) pp. 54-66 Dundurs, J., Stippes, M. (1970) Role of elastic constants in certain contact problems. J. Appl. Mech. 37, 965-970... 

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