Nominal Stress Tensor

Solid mechanics problems are easy to formulate in the referential description. However the balance laws (4.1.14) and (4.1.15) derived earlier are in the spatial description. We note that the stress-vector $\mathbf{f}$ in equations (4.1.5) and (4.1.6) is acting on a unit area in the present configuration of the body. If we can express this $\mathbf{f}$ in terms of the area in the reference configuration, we would then be able to write the balance laws in the referential description. Note that

$$f_i da = T_{ij} n_j da = T_{ij} da_j = T_{ij} J(F^{-1})_{Bj} dA_B$$ (4.3.1)

where we have substituted for $da_i$ in terms of $dA_B$ from eqn. (3.10.1). With the following definition

$$S_{iB} = JT_{ij}(F^{-1})_{Bj},$$ (4.3.2)

equation (4.3.1) becomes

$$f_ida = S_{iB}dA_B = S_{iB}N_BdA.$$ (4.3.3)

Thus $S_{iB}$ provides a measure of the present force acting on a unit area in the reference configuration. For this reason, $S_{iB}$ is called the nominal stress tensor, engineering stress tensor, or first Piola-Kirchhoff stress tensor. Note that $S_{iB}N_B$ equals the present

value of the force acting on the area in the present configuration at $P^\prime$ into which a unit area perpendicular to $\mathbf{N}$ at $P$ is deformed.

The stress tensors $S_{iB}$ and $T_{ij}$ are related by eqn. (4.3.2). However, to find one stress tensor from the other, one must know the deformation gradient. In the Mechanics of Materials course, the nominal stress tensor $S_{iB}$ and the engineering strain $e_{AB}$ are used to plot the stress strain curve. In general, the graph of the true stress $T_{11}$ versus the true strain $\epsilon_{11}$ will look quite different from that of $S_{11}$ versus $e_{11}$ in a simple tension test in which the load is applied in the $X_1$-direction, and $x_1$ and $X_1$ axes point in the same direction.

Substituting for $f_ida$ from (4.3.3) into (4.1.5) and writing $dm = \rho_0dV$, we obtain

$$\int \rho_0a_i dV = \int S_{iB}N_BdA + \int \rho_0g_i\ dV.$$ (4.3.4)

Now using the divergence theorem, we convert the surface integral on the right-hand side of (4.3.4) into the volume integral and thereby obtain

$$\int (\rho_0a_i - S_{iB,B} - \rho _0g_i)dV = 0.$$ (4.3.5)

This equation is to hold for all infinitesimal volume elements in the body. If the integrand is continuous which is assumed to be the case, then (4.3.5) can hold for every volume element in the body if and only if

$$\rho_0a_i = S_{iB,B} + \rho_0g_i,$$ (4.3.6)

or

$$\rho_0a_i = \frac{\partial S_{iB}}{\partial X_B} + \rho_0g_i.$$

This is Cauchy's first law of motion in the referential description. From (4.1.15) and (4.3.2) we obtain the following for the second law of motion.

$$[\mathbf{S}][\mathbf{F}]^T = [\mathbf{F}][\mathbf{S}]^T.$$ (4.3.7)

Thus the nominal stress tensor $S_{iB}$ need not be and, in general, is not symmetric.

The traction type boundary conditions in terms of $S_{iB}$ are written as

$$S_{iB}N_B = \hat f_i$$ (4.3.8)

in which $\hat f_i$ is the force acting at points on the boundary in the deformed configuration but is measured per unit area in the reference configuration.

The following stress tensor is introduced in the engineering literature:

$$\tilde S_{AB} = JF^{-1}_{Ai}T_{ij}(F^{-1})_{Bj}$$

or

$$[\tilde{\mathbf{S}}] = J[\mathbf{F}^{-1}][\mathbf{T}][\mathbf{F}^{-1}]^T = [\mathbf{F}^{-1}][\mathbf{S}].$$ (4.3.9)

The tensor $\tilde{\mathbf{S}}$ known as the second Piola-Kirchhoff stress tensor is symmetric; the symmetry of $\tilde{\mathbf{S}}$ is equivalent to the Cauchy's second law of motion. There is no easy physical interpretation of $\tilde{\mathbf{S}}$.