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3.12: Stress Tensors

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    35264
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    Most of the force densities of concern in this text can be written as the divergence of a stress tensor. The representation of forces in terms of stresses will be used over and over again in the chapters which follow. This section is intended to give a brief summary of the differential and integral properties of the stress tensor.

    Suppose that the ith component of a force density can be written in the form

    \[ F_i = \frac{\partial{T_{ij}}}{\partial{x_j}}; \quad (\overrightarrow{F} = \nabla \cdot \overrightarrow{\overrightarrow{T}}) \label{1} \]

    Here, the Einstein summation convection is applicable, so that because the\( j\)'s appear twice in the same term, they are to be summed from one to three. An alternative notation, in parentheses, re-presents the same operation in vector notation. Much of the convenience of recognizing the stress tensor representation of a force density comes from then being able to convert an integration of the force density over a volume to an integration of the stress tensor over a surface enclosing the volume.This generalization of Gauss' theorem is easily shown by fixing attention on the ith component (think of \(i\) as given) and defining a vector such that

    \[ \overrightarrow{G}_i = T_{i1} \overrightarrow{i}_1 + T_{i2} \overrightarrow{i_2} + T_{i3} \overrightarrow{i}_3 \label{2} \]

    Then the right-hand side of Equation \ref{1} is simply the divergence of \(\overrightarrow{G}_i\) Gauss' theorem then shows that

    \[ \int_V F_i dV = \int_V \nabla \cdot \overrightarrow{G}_i dV = \oint_S \overrightarrow{G}_i \cdot \overrightarrow{n} da \label{3} \]

    or, in index notation and using the definition of \(\overrightarrow{G}_i\) from Equation \ref{2},

    \[ \int_V F_i dV = \oint_S T_{ij} n_j da \label{4} \]

    This tensor form of Gauss' theorem is the integral counterpart of Equation \ref{1}. Physically, Equation \ref{4} states that an alternative to integrating the force density in some Cartesian direction over the volume \(V\) is an integration of the integrand on the right over a surface completely enclosing that volume \(V\). The integrand of the surface integral can therefore be interpreted as a force/unit area acting on the enclosing surface In the ith direction. To distinguish it from a surface force density, it will be referred to as the "traction." It does not act on a physical surface and has physical significance only when integrated over a closed surface. It is simply the force/unit area that must be integrated over the entire surface to find the net force due to the volume force density

    \[ \Upsilon_i = T_{ij} n_j; \quad \overrightarrow{\Upsilon} = \overrightarrow{\overrightarrow{T}} \cdot \overrightarrow{n} \label{5} \]

    In vector notation and in terms of the traction \(\overrightarrow{\Upsilon}\), Equation \ref{4} is written as

    \[ \int_V \overrightarrow{F} dV = \oint_S \overrightarrow{\Upsilon} \cdot \overrightarrow{n} da \label{6} \]

    Figure 3.9.1 shows the general relationship of the traction and normal vector. The traction can act in an arbitrary direction relative to the surface.

    clipboard_e3dc9d692fe4c4ad7b2a6c3abf8f3fbd3.png
    Fig. 3.9.1. Schematic view of volume \(V\) enclosed by surface \(S\), showing traction acting on elements of surface

    To develop a physical interpretation of the stress tensor components, it is helpful to consider a particular volume \(V\) and surface \(S\) with surfaces having normals in the Cartesian coordinate directions. The cube shown in Fig. 3.9.2 is such a volume. Suppose that interest is in determining the net force on the cube in the x direction, from Equation \ref{4}. The required surface integration can then be broken into separate integrations over each of the cube's surfaces. For the integration on the right face, the normal vector has only an \(x\) component,so the only contribution to that surface integration is from \(T_{xx}\). Similarly, on the left surface, the normal vector is in the -x direction, and the integral over that surface is of \(-T_{xx}\). The minus sign is represented by directing the stress arrow in the minus \(x\) direction in Fig. 3.9.2. On the top and bottom surfaces, the normal vector is in the \(y\) direction, and the integration is of plus and minus \(T_{xy}\). Similarly, on the front and back surfaces, the only terms contributing to the traction are \(T_{xz}\). The stress tensor components represent normal stresses if the indices are equal, and shear stresses if they are unequal. In either case, the stress component acting in the ith direction on a surface having its normal in the jth direction is \(T_{ij}\).

    clipboard_ebb0ae4fe9c08a1057eed88a39620980a.png
    Fig. 3.9.2. Stress components acting on cube in the x direction.

    Orthogonal components are a familiar way of representing a vector \(\overrightarrow{F}\). In the coordinate system \((x_1,x_2,x_3)\) the components are denoted by \(F_j\). What is meant by a vector is implicit to how these components decompose into the components of the vector expressed in a second orthogonal coordinate system \((x_1^{‘},x_2^{‘} \cdot x_3^{‘})\) pictured in Fig. 3.9.3. The two coordinate systems are related by the transformation

    \[ x_k^{‘} = a_{kl} x_{l}; \quad \frac{\partial{x_k^{‘}}}{\partial{x_l}} = a_{kl} \label{7} \]

    Where \(a_{kl}\) is the cosine of the angle between the \(x_k^{‘}\) axis and the \(x_l\) axis.

    A component of the vector in the primed frame in the ith direction is then given by

    \[ F_i^{‘} = a_{ij} F_j \label{8} \]

    For example, suppose that \(i = 1\). Then, Equation \ref{8} gives the \(x_1^{'}\) component of \(\overrightarrow{F}^{'}\) as the projections of the components in the \(x_1, x_2, x_3\) directions onto the \(x_1^{'}\) direction. Equation \ref{8} summarizes how a vector transforms from one coordinate system onto another, and could be used to define what is meant by a "vector."

    Similarly, the components of a tensor transform from the unprimed to the primed coordinate system in a way that can be used to define what is meant by a "tensor." To deduce the transformation, begin with Equation \ref{8} using the divergence of a stress tensor to represent each of the force densities (Equation \ref{1}):

    \[ \frac{\partial{T_{ik}^{‘}}}{\partial{x_k^{‘}}} = a_{ij} \frac{\partial{T_{jl}^{‘}}}{\partial{x_l}} \label{9} \]

    Now, if use is made of the chain rule for differentiation, and Equation \ref{7}, it follows that

    \[ \frac{\partial{T_{ik}^{‘}}}{\partial{x_k^{‘}}} = a_{ij} \frac{\partial{T_{jl}^{‘}}}{\partial{x_k^{‘}}} \frac{\partial{x_{k}^{‘}}}{\partial{x_l}} = a_{ij} a_{kl} \frac{\partial{T_{jl}}}{\partial{x_k^{‘}}} \label{10} \]

    Thus, the tensor transformation follows as

    \[ T_{ik}^{‘} = a_{ij} a_{kl} T_{jl} \label{11} \]

    Useful conditions on the direction cosines \(a_{ij}\) are obtained by recognizing that the transformation from the primed frame to the unprimed frame, given generally by

    \[ F_j = b_{ji} F_i^{‘} \label{12} \]

    Involves the same direction cosines, because \(b_{ji}\), defined as the cosine of the angle between the \(x_j\) axis and the \(x_j^{‘}\) axis, is equal to \(a_{ij}\). Thus, Eqs. \ref{12} and \ref{8} together show that

    \[ F_i^{‘} = a_{ik} F_k = a_{ik} a_{lk} F_l^{‘} \label{13} \]

    clipboard_e2e9dd014cd030ee40a4cdba30b434e9b.png
    Fig. 3.9.3 Unprimed and primed coordinate systems. The geometric significance of the direction cosine \(a_{ij}\) is shown.

    And it follows that the direction cosines satisfy the condition that

    \[ a_{ik} a_{lk} = \delta_{il} \label{14} \]

    Where the Kronecker delta function \(\delta_{ik}\) by definition takes the values

    \[ \delta_{ik} = \begin{cases} 1 & \text{i = k} \\ 0 & \text{i \(\neq\) k} \end{cases} \label{15} \]

    Finally, suppose that a total torque rather than a total force is to be computed. By way of analogy to Equation \ref{6}, is there a way in which the integration of the torque density can be converted to an integration over the enclosing surface? With respect to the origin, the total torque on material within the volume \(V\) is

    \[ \overrightarrow{\tau} = \int_V \overrightarrow{r} \times \overrightarrow{F} dV \label{16} \]

    where \(\overrightarrow{r}\) is the vector distance from the origin. With \(\overrightarrow{F}\) given as the divergence of a stress tensor, Equation \ref{1}, and provided that\(\overrightarrow{\overrightarrow{T}}\) is symmetric \((T_{ij} = T_{ji})\), the tensor form of Gauss' theorem can be used to show that

    \[ \overrightarrow{\tau} = \oint_S \overrightarrow{r} \times (\overrightarrow{\overrightarrow{T}} \cdot \overrightarrow{n} ) da \label{17} \]

    The net torque is the integral over the enclosing surface of a surface torque density \(\overrightarrow{\Upsilon} \times \overrightarrow{T}\) (see Problem 3.9.1).


    This page titled 3.12: Stress Tensors is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James R. Melcher (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.