Strain tensor

Similar to the (continuum) strain in soils, the strain signifies an averaged measure of the relative displacement between grains (particles) over a representative volume element (RVE). This abstraction captures the displacements arising from external forces and internal interactions, yielding a continuous and differentiable strain field that describes both the magnitude and direction of the averaged deformation.

Mathematical definition

In the context of three-dimensional continua, the strain tensor, \(\boldsymbol{\varepsilon}\), relates to the gradient of the displacement field, \(\boldsymbol{u}\). It is mathematically expressed as:

\[ \boldsymbol{\varepsilon} = \dfrac{1}{2} \left( \dfrac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}} + \left( \dfrac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}} \right)^T \right) \]

Where \(\dfrac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}}\) is the gradient of the displacement vector with respect to spatial coordinates and \(T\) denotes the transpose operation.

In index notation, the components of the strain tensor are expressed as:

\[ \varepsilon_{ij} = \dfrac{1}{2} \left( \dfrac{\partial u_i}{\partial x_j} + \dfrac{\partial u_j}{\partial x_i} \right) \]

For the normal strains:

\[ \begin{aligned} \varepsilon_{xx} &= \dfrac{\partial u_x}{\partial x} \\ \varepsilon_{yy} &= \dfrac{\partial u_y}{\partial y} \\ \varepsilon_{zz} &= \dfrac{\partial u_z}{\partial z} \end{aligned} \]

For the shear strains:

\[ \begin{aligned} \varepsilon_{xy} &= \dfrac{1}{2} \left( \dfrac{\partial u_x}{\partial y} + \dfrac{\partial u_y}{\partial x} \right) \\ \varepsilon_{xz} &= \dfrac{1}{2} \left( \dfrac{\partial u_x}{\partial z} + \dfrac{\partial u_z}{\partial x} \right) \\ \varepsilon_{yz} &= \dfrac{1}{2} \left( \dfrac{\partial u_y}{\partial z} + \dfrac{\partial u_z}{\partial y} \right) \end{aligned} \]

Geotechnical sign convention

Analogously to the sign convention followed for the stresses (compression = positive and tension = negative), the strains follow a similar convention where compressive strains (shortening) are positive, and tensile strains (elongation) are negative. This ensures a consistent representation when relating stresses to strains, especially in material constitutive models.

1D Example: Lengthening of a rod

Motivated by the discussion and questions about the calculation of strain in the lecture, I would like to explain the calculation of strain and the use of partial derivatives for this purpose with the help of a very simple example in the hope of clearing up any remaining ambiguities.

In a 1D scenario, a rod is considered to only deform along its length as depicted below.

The deformation is represented by the elongation or compression of the rod, and there's no shear strain in this simple scenario. Considering the displacement \(u(x)\) as a function of position \(x\) along the rod, strain at any given point along the length \(L\) of the rod is represented by

\[ \varepsilon_{xx} = \dfrac{\partial u}{\partial x} \]

But how does above equation relates to the common way to express strain in basic mechanics as the ratio of the change in length to the original length

\[ \varepsilon = \dfrac{\Delta L}{L} ~~~? \]

Let's imagine that we are looking at two points separated by a very small length \(\delta x\) along the rod. Due to the deformation, these points move and get displaced by \(u\) and \(u + \delta u\) respectively.

The change in length between these two points due to deformation is \(\delta u\), which can be approximated as the difference in displacements of the two points:

\[ \delta u = u(x + \delta x) - u(x) \]

Dividing both sides by \(\delta x\) we get:

\[ \frac{\delta u}{\delta x} = \frac{u(x + \delta x) - u(x)}{\delta x} \]

As \(\delta x\) approaches zero, this ratio gives us the local strain at point \(x\), and \(\frac{\delta u}{\delta x}\) converges to \(\frac{\partial u(x)}{\partial x}\). This is analogous to the concept of the derivative in calculus. So, the strain at a specific point \(x\) is:

\[ \varepsilon_{xx} = \frac{\partial u(x)}{\partial x} \]

The classical definition of strain, \(\frac{\Delta L}{L}\), gives an average measure of deformation over the entire length of the object. Meanwhile, \(\frac{\partial u(x)}{\partial x}\) provides a more precise, local measure of deformation at a specific point along the object. For a rod that's uniformly deforming, these two definitions converge, and the classical definition becomes a special case of the more general definition.

Partial derivative \(\partial\) vs. total derivative \(d\)

In the context provided, the use of \(\partial\) (partial derivative) instead of \(d\) (total derivative) might be a bit of over-specification, since we are considering a one-dimensional case. However, in mechanics, it's common to deal with multi-dimensional scenarios. When we look at 2D or 3D scenarios, displacements in one direction can depend on changes in multiple directions.

For instance, in a 2D case, the displacement \(u\) in the x-direction could be a function of both \(x\) and \(y\), i.e. \(u(x,y)\). In such cases, when we talk about how the displacement changes with \(x\) while keeping \(y\) constant, we'd use a partial derivative: \(\frac{\partial u}{\partial x}\).

Using the partial derivative notation helps preparing for these more complex situations, even if it's not strictly necessary in a 1D context. It's a way to emphasize that we're taking the derivative with respect to one specific variable, even though there may only be one variable in play at the moment.