Relation between Cartesian (or rectangular) components of stress

Consider the motion of a small rectangular parallelepiped in viscous fluid with centre at P(x,y,z) and it’s edges of length dx, dy, dz parallel to the fixed rectangular axes.

The mass (density´volume) i.e rdxdydz of the element remains constant and the element is assumed to move along with fluid. In relation to the surface at right angle to the axis of x, the force components per unit area (or stress component) at P(x,y,z) are P_xx, P_xy, P_xz.

The corresponding force components at the centre

                                                         of the face of area dydz far from the origin are 

The corresponding force components at the centre

                                                           of the face of area dydz near from the origin are 

acting in the fluid in the opposite sense of former.

1)The force component on this pair of parallel faces of area dydz through P₁ & P₂ at right angle to the axis of x may be compounded into a single force having components

                                 

acting at P(x,y,z) in the direction parallel to ox, oy & oz respectively together with couple of moments -p_xydxdydz  and +p_xydxdydz about oy and oz respectively.

2) The force component on this pair of parallel faces of area dxdz through P₁ & P₂ at right angle to the axis of y may be compounded into a single force having components

                           

   

acting at P(x,y,z) in the direction parallel to ox, oy & oz respectively together with couple of moments -p_yzdxdydz  and +p_yzdxdydz about oz and ox respectively.

3) The force component on this pair of parallel faces of area dxdy through P₁ & P₂ at right angle to the axis of z may be compounded into a single force having components

                             

acting at P(x,y,z) in the direction parallel to ox, oy & oz respectively together with couple of moments -p_zxdxdydz  and +p_zxdxdydz about oy and ox respectively.

* red indicate just changing the term

Thus taking into account, the surface forces on all six faces at parallelepiped, we see that they reduced into single force at P(x,y,z) having components

                                   

    together with a vector couple having components

                                          (p_yz - p_zy) dxdydz about ox

                                          (p_zx - p_xz) dxdydz about oy

                                          (p_xy - p_yx) dxdydz about oz

Note that the fact has been used here is that the moment of forces about an arbitrary axis must be zero when contracting the element to a point (i.e making edges shrink to zero), for instance

                                          (p_yz - p_zy) dxdydz =0

                                          (p_zx - p_xz) dxdydz =0

                                          (p_xy - p_yx) dxdydz =0

       Þ                               (p_yz - p_zy) =0

                                          (p_zx - p_xz) =0

                                          (p_xy - p_yx) =0

       Þ                               p_yz = p_zy

                                          p_zx = p_xz

                                          p_xy = p_yx

Hence nine rectangular components of stress at a point reduced to six.