Forces Within Continua Stress Tensors

3 Forces and Balance Laws 2.3.1 Forces Within Continua Stress Tensors 

We now examine the question of how in continuum mechanics the forces due to material external to the region are communicated to it. Note that we will adopt the notation dQ to characterize the boundary of the region Q. In simplest terms, forces are transmitted to a continuum either by the presence of body forces or via surface tractions . Body forces are those such as that due to gravity which 

The Cauchy stress principle arises through consideration of the equilibrium of body forces and surface tractions in the special case of the infinitesimal tetrahedral volume shown in fig. 2.7. Three faces of the tetrahedron are perpendicular to the Cartesian axes while the fourth face is characterized by a normal n. The idea is to insist on the equilibrium of this elementary volume, which results in the observation that the traction vector on an arbitrary plane with normal n (such as is shown in the figure) can be determined once the traction vectors on the Cartesian planes are known. In particular, it is found that = crn, where a is known as the stress tensor, and carries the information about the traction vectors associated with the Cartesian planes. The simple outcome of this argument is the claim that 

Now that we have the notion of the stress tensor in hand, we seek one additional insight into the nature of forces within solids that will be of particular interest to our discussion of plastic flow in solids. As was mentioned in section 2.2.3, plastic deformation is the result of shearing deformations on special planes. Certain models of such deformation posit the existence of a critical stress on these planes such that once this stress is attained, shearing deformations will commence. To compute the resolved shear stress on a plane with normal n and in a direction s we begin by noting that the traction vector on this plane is given by 

To compute the component of this traction vector in the direction of interest (i.e. the resolved shear stress), it is necessary to project the force along the direction of interest via 

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Forces that are transferred from one particle to another, at the botmdary of and within the domain. These forces are expressed using the stress tensor. This is where the continuum concept intervenes. 

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