Assume that F and M derive from two types of forces, namely, body forces f, such as gravitational attractions—defined such that force fdV acts on volume element dV (see Figure 1)—and surface forces, which represent the mechanical effect of matter immediately adjoining that along the surface S of the volume V being considered. Cauchy formalized in 1822 a basic assumption of continuum mechanics that such surface forces could be represented as a stress vector T, defined so that TdS is an element of force acting over the area dS of the surface (Figure 1). Hence, the principles of linear and angular momentum take the formswhich are now assumed to hold good for every conceivable choice of region V. In calculating the right-hand sides, which come from dP/dt and dH/dt, it has been noted that ρdV is an element of mass and is therefore time-invariant; also, a = a(x, t) = dv/dt is the acceleration, where the time derivative of v is taken following the motion of a material point so that a(x, t)dt corresponds to the difference between v(x + vdt, t + dt) and v(x, t). A more detailed analysis of this step shows that the understanding of what TdS denotes must now be adjusted to include averages, over temporal and spatial scales that are large compared to those of microscale fluctuations, of transfers of momentum across the surface S due to the microscopic fluctuations about the motion described by the macroscopic velocity v.

The nine quantities σ_ij(i, j = 1, 2, 3) are called stress components; these will vary with position and time—i.e., σ_ij = σ_ij(x, t)—and have the following interpretation. Consider an element of surface dS through a point x with dS oriented so that its outer normal (pointing away from the region V, bounded by S) points in the positive x_i direction, where i is any of 1, 2, or 3. Then σ_i1, σ_i2, and σ_i3 at x are defined as the Cartesian components of the stress vector T (called T⁽ⁱ⁾) acting on this dS. Figure 2 shows the components of such stress vectors for faces in each of the three coordinate directions. To use a vector notation with e₁, e₂, and e₃ denoting unit vectors along the coordinate axes (Figure 2), T⁽ⁱ⁾ = σ_i1e₁ + σ_i2e₂ + σ_i3e₃. Thus, the stress σ_ij at x is the stress in the j direction associated with an i-oriented face through point x; the physical dimension of the σ_ij is [force]/[length]². The components σ₁₁, σ₂₂, and σ₃₃ are stresses directed perpendicular, or normal, to the face on which they act and are normal stresses; the σ_ij with i ≠ j are directed parallel to the face on which they act and are shear stresses.

By hypothesis, the linear momentum principle applies for any volume V. Consider a small tetrahedron (Figure 3) at x with an inclined face having an outward unit normal vector n and its other three faces oriented perpendicular to the three coordinate axes. Letting the size of the tetrahedron shrink to zero, the linear momentum principle requires that the stress vector T on a surface element with outward normal n be expressed as a linear function of the σ_ij at x. The relation is such that the j component of the stress vector T is T_j = n₁σ_1j + n₂σ_2j + n₃σ_3j for (j = 1, 2, 3). This relation for T (or T_j) also demonstrates that the σ_ij have the mathematical property of being the components of a second-rank tensor.

Suppose that a different set of Cartesian reference axes 1′, 2′, and 3′ have been chosen. Let x₁′, x₂′, and x₃′ denote the components of the position vector of point x and let σ_kl′(k, l = 1, 2, 3) denote the nine stress components relative to that coordinate system. The σ_kl′ can be written as the 3 × 3 matrix [σ′], and the σ_ij as the matrix [σ], where the first index is the matrix row number and the second is the column number. Then the expression for T_j implies that [σ′] = [α][σ][α]^T, which is the defining equation of a second-rank tensor. Here [α] is the orthogonal transformation matrix, having components α_pq = e_p′ · e_q for p, q = 1, 2, 3 and satisfying [α]^T[α] = [α][α]^T = [I], where the superscript T denotes transpose (interchange rows and columns) and [I] denotes the unit matrix, a 3 × 3 matrix with unity for every diagonal element and zero elsewhere; also, the matrix multiplications are such that if [A] = [B][C], then A_ij = B_i1C_1j + B_i2C_2j + B_i3C_3j.

Equations of motion

Now the linear momentum principle may be applied to an arbitrary finite body. Using the expression for T_j above and the divergence theorem of multivariable calculus, which states that integrals over the area of a closed surface S, with integrand n_i f (x), may be rewritten as integrals over the volume V enclosed by S, with integrand ∂f (x)/∂x_i; when f (x) is a differentiable function, one may derive thatat least when the σ_ij are continuous and differentiable, which is the typical case. These are the equations of motion for a continuum. Once the above consequences of the linear momentum principle are accepted, the only further result that can be derived from the angular momentum principle is that σ_ij = σ_ji (i, j = 1, 2, 3). Thus, the stress tensor is symmetric.

Principal stresses

Symmetry of the stress tensor has the important consequence that, at each point x, there exist three mutually perpendicular directions along which there are no shear stresses. These directions are called the principal stress directions, and the corresponding normal stresses are called the principal stresses. If the principal stresses are ordered algebraically as σ_I, σ_II, and σ_III (Figure 4), then the normal stress on any face (given as σ_n = n · T) satisfies σ_I ≤ σ_n ≤ σ_III. The principal stresses are the eigenvalues (or characteristic values) s, and the principal directions the eigenvectors n, of the problem T = sn, or [σ]{n} = s{n} in matrix notation with the 3-column {n} representing n. It has solutions when det ([σ] − s[I ]) = −s³ + I₁s² + I₂s + I₃ = 0, with I₁ = tr[σ], I₂ = −(1/2)I 2/1 + (1/2)tr([σ][σ]), and I₃ = det [σ]. Here “det” denotes determinant and “tr” denotes trace, or sum of diagonal elements, of a matrix. Since the principal stresses are determined by I₁, I₂, and I₃ and can have no dependence on how one chooses the coordinate system with respect to which the components of stress are referred, I₁, I₂, and I₃ must be independent of that choice and are therefore called stress invariants. One may readily verify that they have the same values when evaluated in terms of σ_ij′ above as in terms of σ_ij by using the tensor transformation law and properties noted for the orthogonal transformation matrix.

Very often, in both nature and technology, there is interest in structural elements in forms that might be identified as strings, wires, rods, bars, beams, or columns, or as membranes, plates, or shells. These are usually idealized as, respectively, one- or two-dimensional continua. One possible approach is then to develop the consequences of the linear and angular momentum principles entirely within that idealization, working in terms of net axial and shear forces and bending and twisting torques at each point along a one-dimensional continuum, or in terms of forces and torques per unit length of surface in a two-dimensional continuum.