Kinematics The Geometry of Deformation

Broadly speaking, our description of continuum mechanics will be divided along a few traditional lines. First, we will consider the kinematic description of deformation without searching for the attributes of the forces that lead to a particular state of deformation. Here it will be shown that the displacement fields themselves do not cast a fine enough net to sufficiently distinguish between rigid body motions, which are often of little interest, and the more important relative motions that result in internal stresses. These observations call for the introduction of other kinematic measures of deformation such as the various strain tensors. Once we have settled these kinematic preliminaries, we turn to the analysis of the forces in continua that lead to such deformation, and culminate in the Cauchy stress principle. 

Continuum theories have a number of generic features, one of which is the use of some set of field variables for characterizing the disposition of the system of interest. In the context of the thermodynamics of gases, pressure, volume and temperature may suffice while for electromagnetic media, the electric and magnetic 

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For both mathematical and physical reasons, there are many instances in which the spatial variations in the field variables are sufficiently gentle to allow for an approximate treatment of the geometry of deformation in terms of linear strain measures as opposed to the description including geometric nonlinearities introduced above. In these cases, it suffices to build a kinematic description around a linearized version of the deformation measures discussed above. Note that in component form, the Lagrangian strain may be written as... 

Yet another example of the geometry of deformation of interest to the present enterprise is that of structural transformation. As was evidenced in chap. 1 in our discussion of phase diagrams, material systems admit of a host of different structural competitors as various control parameters such as the temperature, the pressure and the composition are varied. Many of these transformations can be viewed from a kinematic perspective with the different structural states connected by a deformation pathway in the space of deformation gradients. In some instances, it is appropriate to consider the undeformed and transformed crystals as being linked by an affine transformation. A crystal is built up through the repetition... 

Several basic factors are involved in the process of modelling the mechanical behaviour of the LV by the FE method. These factors are the geometry of the ventricle including its kinematic boundary conditions, the extent of the deformation that the LV undergoes, the pressure distribution on the endocardium, the myocardial constitutive law as well as its anisotropy and the activation mechanism of the muscle, mainly manifested in the systolic phase. Furthermore, computer resources such as storage and computation time requirements should also be considered. 

The best designs of rheometers use geometries so that the forces/ deformation can be reduced by subsequent calculation to stresses and strains, and so produce material parameters. It is very important that the principle of material independence is observed when parameters are measured on the rheometers. The flow within the rheometers should be such that the kinematic variables and the constitutive equations describing the flow must be unaffected by any rigid rotation of both body and coordinate system - in other words, the response of the material must not be dependent upon the position of the observer. When designing rheometers, care is taken to see that the rate of deformation satisfies this principle for simple shear flow or viscometric flow. The flow analyzed can be considered as viscometric (simple shear) flow if sets of plane surfaces (known as shear planes) are seen to exist and each is moving past the other as a solid plane, i.e. the distance between every two material points in the plane remains constant. 

A short chronological review of the diverse available mechanical models of the LV precedes a critical reassessment of the various main factors involved in a finite element analysis of the ventricle. These factors constitute the three-dimensional geometry of the LV and its kinematical boundary conditions the extent of the deformation the ventricle undergoes the pressure distribution on the endocardium the myocardial constitutive law as well as its anisotropy, and the activation mechanism of the muscle. A rationale for developing an improved finite element model, gradually incorporating these factors, concludes the presentation. 

Dislocations occur in lattices other than those of atomic-scale crystals. The best known examples, no doubt, are those in the experiments of Bragg and Nye with rafts of soap bubbles [1]. Their work illustrated the geometry and kinematics of the edge dislocations in these two-dimensional hexagonal lattices, and vividly revealed how macroscopic plastic deformation of crystals is effected by the motion of large numbers of such dislocations. Ahead of the definitive identification of moving dislocations in atomic crystals by electron microscopy [2], Bragg and Nye s experiments had convinced many of the reality and the potential of what had initially been a purely theoretical concept [3-5]. 

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