Cartesian components infinitesimal volume element

Consider a fluid material body enclosing an infinitesimal volume element dV at the point p, as shown in Fig. 5.1. Choose any plane cutting through the volume element and let the cross section be denoted by dS. The direction perpendicular to the plane is regarded as the direction of the plane, indicated by the unit vector n. If there are forces applied to the body, a surface force f will be exerted on the plane of the volume element at the point p. In general, the directions of f and n are different. By dividing f by dS, we obtain the stress t (= fn/dS) exerted on the plane at the point p. The stress can be separated into the component perpendicular to the plane (the normal stress) and those parallel to the plane (the shear stresses). At the point p, we choose a Cartesian coordinate system with n as one direction and m and 1 in the plane as the other two. Then t may be expressed as... 

Transitioning from the stress state of a particle to the stress field of the continuum, the interaction of the Cauchy stress tensor components of neighboring points needs to be investigated. They have to satisfy the conditions of local equilibrium to be established with the aid of an arbitrary infinitesimal volume element. Such an element with faces in parallel to the planes of the Cartesian coordinate system is subjected to the volume force and on the faces to the components of the Cauchy stress tensor with additional increments in the form of the first element of Taylor expansions on one of the respective opposing faces. The balance of moments proves the symmetry of the stress tensor,... 

Any or all of these forces may result in local stresses within the fluid. Stress can be thought of as a (local) concentration of force, or the force per unit area that bounds an infinitesimal volume of the fluid. Now both force and area are vectors, the direction of the area being defined by the normal vector that points outward relative to the volume bounded by the surface. Thus, each stress component has a magnitude and two directions associated with it, which are the characteristics of a second-order tensor or dyad. If the direction in which the local force acts is designated by subscript j (e.g., j = x, y, or z in Cartesian coordinates) and the orientation (normal) of the local area element upon which it acts is designated by subscript i, then the corresponding stress component (ay) is given by... 

In tensor notation the three Cartesian directions x, y, and z are designated by suffixed variables i,j, k, l, etc. (Landau and Lifshitz 1970 Auld 1973). Thus the force acting per unit area on a surface may be described as a traction vector with components rj j = x, y, z. The stress in an infinitesimal cube volume element may then be described by the tractions on three of the faces, giving nine elements of stress cry (i, j = x, y, z), where the first suffix denotes the normal to the plane on which a given traction operates, and the second suffix denotes the direction of a traction component. 

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