1.6. Shear Strain

The 2 dimensional shear strain is the average decrease of the angle between the sides AB and BC.

Shear strain refers to how the shape of a material changes due to stress. Specifically, it measures the change in angle between two sides of a material when subjected to shear forces. Here, 'sides AB and BC' indicate two edges of a shape that are originally at a certain angle. As these sides slide against each other, the angle decreases, which we characterize as shear strain.

Imagine a deck of cards. If you push the top half of the deck sideways while keeping the bottom half stationary, the angle between the edges of the cards changes. This change in angle is a practical example of shear strain.

The two components of the shear strain rate can be defined as: $\frac{d\theta}{dt} = \frac{v_x}{d_x} + \frac{v_y}{d_y}$, which is the rate of decrease of the angle of deformation.

This equation provides a mathematical representation of how quickly the angle between two sides changes. $v_x$ and $v_y$ are velocities in the respective directions, while $d_x$ and $d_y$ denote the distances. By examining how these variables change over time, we quantify the shear strain rate, which is crucial for understanding material behavior under stress.

Continuing with the deck of cards analogy, if you push the cards faster, they will deform more quickly, leading to a larger shear strain rate. The faster you push, the quicker the angle between card edges decreases.

The dilatation or extensional strain is defined as the ratio of the change in length to the original length.

Dilatational strain focuses on volume change and is defined as how much longer or shorter a material becomes when forces are applied. The formulation compares the change in length of a segment to its original length to provide a dimensionless measure of extension. This measure helps engineers and scientists understand how materials respond to different loads.

Consider a rubber band. As you stretch it, its length increases. The dilatational strain is the ratio of the increase in length of the rubber band to its original length, which shows how much the band deforms.

Along each axis, the extensional strain can be expressed as: $\epsilon_{xx} = \frac{\partial u}{\partial x}$, $\epsilon_{yy} = \frac{\partial v}{\partial y}$, and $\epsilon_{zz} = \frac{\partial w}{\partial z}$.

This section describes how to calculate extensional strain for three-dimensional objects. The terms $\epsilon_{xx}$, $\epsilon_{yy}$, and $\epsilon_{zz}$ correspond to the strains experienced in the x, y, and z directions, respectively. By taking the partial derivatives of the displacement with respect to each spatial coordinate, we find how the material expands or contracts in each direction under loading.

Think of a balloon being inflated. As air fills the balloon, it expands in all directions. The change in size in each direction (up/down, left/right, forward/backward) can be thought of as $\epsilon_{xx}$, $\epsilon_{yy}$, and $\epsilon_{zz}$ respectively, showing how the balloon’s shape and volume change.

--