Compatibility of Strain Components

Previously, we proved that the stress tensor cannot vary arbitrarily within a region. In fact, its variation is constrained by Newton’s second laws. Now the question is: Can the components of strain vary arbitrarily? The answer is no. There are some restrictions on how strain may vary within a body. These restrictions, known as compatibility equations.

For small deformations, the strain components are related to the displacement field \(\mathbf{u}\) by \[ \epsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right). \] When the displacement field \(\mathbf{u}\) is known, the strain components can be easily calculated from the above equation. However, the inverse problem of determining the displacement field corresponding to a given strain field is far more complex.

There are six independent strain components \((\epsilon_{xx}, \epsilon_{yy}, \epsilon_{zz}\) , \(\epsilon_{yz}\), \(\epsilon_{xz}\), \(\epsilon_{xy}\)) and only three displacement components. This means that there are six equations for three unknown functions \(u_x, u_y, u_z\). Therefore, we do not expect that the system of equations has single-valued solutions if the functions \(\epsilon_{ij}\) are arbitrarily chosen.

Compatibility Equations

To ensure that a set of strain components corresponds to a physically possible displacement field, certain compatibility conditions must be satisfied. These conditions are derived by eliminating the displacement components \(u, v, w\) from the strain–displacement relations.

From the definitions of the strain components, we have: \[ \epsilon_{xx} = \frac{\partial u}{\partial x}, \quad \epsilon_{yy} = \frac{\partial v}{\partial y}, \quad 2\epsilon_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}. \] Taking appropriate second derivatives, we find: \[ \frac{\partial^2 \epsilon_{xx}}{\partial y^2} = \frac{\partial^3 u}{\partial x \partial y^2}, \quad \frac{\partial^2 \epsilon_{yy}}{\partial x^2} = \frac{\partial^3 v}{\partial y \partial x^2}, \quad 2\frac{\partial^2 \epsilon_{xy}}{\partial x \partial y} = \frac{\partial^3 u}{\partial x^2 \partial y} + \frac{\partial^3 v}{\partial y^2 \partial x}. \]

By combining these relations and recognizing that mixed partial derivatives are equal, we obtain: \[ \boxed{ \frac{\partial^2 \epsilon_{xx}}{\partial y^2} + \frac{\partial^2 \epsilon_{yy}}{\partial x^2} = 2\frac{\partial^2 \epsilon_{xy}}{\partial x \partial y}. } \]

This is one of the compatibility equations that the strain components must satisfy for a continuous and single-valued displacement field to exist.

Additional Relations

By cyclically permuting the coordinates \(x, y, z\), two additional relations of the same type can be derived.

Differentiating the three-dimensional strain–displacement relations gives expressions such as: \[ \frac{\partial^2 \epsilon_{xx}}{\partial y \partial z} = \frac{\partial^3 u}{\partial x \partial y \partial z}, \quad 2\frac{\partial \epsilon_{xy}}{\partial z} = \frac{\partial^2 u}{\partial y \partial z} + \frac{\partial^2 v}{\partial x \partial z}, \quad 2\frac{\partial \epsilon_{yz}}{\partial x} = \frac{\partial^2 v}{\partial x \partial z} + \frac{\partial^2 w}{\partial y \partial x}, \] and similar relations for the other components.

Combining these and eliminating \(u, v, w\), we obtain: \[ \boxed{ \frac{\partial^2 \epsilon_{xx}}{\partial y \partial z} = \frac{\partial}{\partial x} \left( - \frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{xz}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z} \right). } \]

By cyclic permutation of \(x, y, z\), we obtain three more such relations, giving a total of six compatibility equations in three dimensions.

\[ 2 \frac{\partial^2 \epsilon_{xy}}{\partial x \, \partial y} = \frac{\partial^2 \epsilon_{xx}}{\partial y^2} + \frac{\partial^2 \epsilon_{yy}}{\partial x^2}, \] \[ 2 \frac{\partial^2 \epsilon_{yz}}{\partial y \, \partial z} = \frac{\partial^2 \epsilon_{yy}}{\partial z^2} + \frac{\partial^2 \epsilon_{zz}}{\partial y^2}, \] \[ 2 \frac{\partial^2 \epsilon_{zx}}{\partial z \, \partial x} = \frac{\partial^2 \epsilon_{zz}}{\partial x^2} + \frac{\partial^2 \epsilon_{xx}}{\partial z^2}. \] \[ \frac{\partial^2 \epsilon_{xx}}{\partial y \, \partial z} = \frac{\partial}{\partial x} \left( - \frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z} \right), \] \[ \frac{\partial^2 \epsilon_{yy}}{\partial z \, \partial x} = \frac{\partial}{\partial y} \left( - \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z} + \frac{\partial \epsilon_{yz}}{\partial x} \right), \] \[ \frac{\partial^2 \epsilon_{zz}}{\partial x \, \partial y} = \frac{\partial}{\partial z} \left( - \frac{\partial \epsilon_{xy}}{\partial z} + \frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} \right), \]

For a simply connected region¹ (i.e., a material body without holes or discontinuities), the compatibility equations are both necessary and sufficient to ensure that the displacement field exists and is single-valued.

If the region is multiply connected (e.g., containing holes or voids), additional conditions must be applied to ensure compatibility throughout the body (see the following reference).

It is important to note that these equations are not needed when the displacement components \(u, v, w\) are treated as the primary unknowns—since they automatically satisfy the strain–displacement relations. However, when working directly with the strain components as the unknowns, the compatibility equations must be enforced to guarantee that the resulting strain field corresponds to a valid deformation.

Since the strain components describe the relative positions of points within a body, and rigid-body motion does not produce any strain, the displacement components can be determined only up to an arbitrary rigid-body motion. In other words, even if the strain components satisfy the compatibility equations, the displacement field is not uniquely determined.

Reference

Fung, Y. C. (1965). Foundations of continuum mechanics. Prentice-Hall.