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The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on simply connected bodies. More precisely, the problem may be stated in the following manner. [5] Figure 1. Motion of a continuum body. Consider the deformation of a body shown in Figure 1.
The material stiffness properties of these elements are then, through linear algebra, compiled into a single matrix equation which governs the behaviour of the entire idealized structure. The structure’s unknown displacements and forces can then be determined by solving this equation.
A single-displacement reaction, also known as single replacement reaction or exchange reaction, is an archaic concept in chemistry. It describes the stoichiometry of some chemical reactions in which one element or ligand is replaced by atom or group. [1] [2] [3] It can be represented generically as: + +
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. Equation of motion: , + = where the (), subscript is a shorthand for () / and indicates /, = is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.
The two reactions are named according tho their rate law, with S N 1 having a first-order rate law, and S N 2 having a second-order. [2] S N 1 reaction mechanism occurring through two steps. The S N 1 mechanism has two steps. In the first step, the leaving group departs, forming a carbocation (C +). In the second step, the nucleophilic reagent ...
Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), supported at both ends, is bent by a weight F placed at some intermediate point. The displacement x in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape.
The equations are written only for the small domain of individual elements of the structure rather than a single equation that describes the response of the system as a whole (a continuum). The latter would result in an intractable problem, hence the utility of the finite element method.
The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems.