having dimension
Four additional remarks help clarify the nature of the stress tensor.
1 With respect to any orthonormal basis , any linear transformation has a matrix representation with entries . For , this representation has the formFigure 2.6 A region in three‐dimensional space with unit outward normal vector field and the traction acting on the boundary .
2 In accordance with Exercise 2.4, with respect to any orthonormal basis, the diagonal entries represent forces per unit area acting in directions perpendicular to faces that are orthogonal to , , and , respectively. We refer to these entries as tensile stresses when they pull in the same direction as and as compressive stresses when they push in the opposite direction—namely inward—from . The off‐diagonal entries , where , are shear stresses.
3 A classic theorem in continuum mechanics reduces the angular momentum balance, which we do not discuss here, to the identity with respect to any orthonormal basis. In other words, the stress tensor is symmetric. See [4, Chapter 4] for details.
4 With respect to an orthonormal basis , the divergence of the tensor‐valued function has the following representation as a vector‐valued function:
Exercise 2.4 Consider the action of
Figure 2.7 A cube of material illustrating the interpretations of entries of the stress tensor matrix with respect to an orthonormal basis, from [4, page 109].
The differential momentum balance (2.9) generalizes Newton's second law of motion. The left side of Eq. (2.9) is proportional to mass
Based on this parallel, fluid mechanicians call
the inertial terms.
If we view the momentum balance as an equation for the velocity
2.3 Constitutive Relationships
The mass and momentum balance laws
furnish four scalar PDEs involving the 16 scalar functions required to specify
