Strain

Strain is a geometric description of how each point in a body has deformed relative to the undeformed state^2.6.

We track a point $x$ in the undeformed body to its corresponding point $\tilde{x}$ in the deformed body via a deformation $u(x)$ : $\tilde{x} = x + u(x)$ . We denote the distance between two nearby points in the undeformed state as $dx = (dx_1, dx_2, dx_3)$ and their images in the deformed state as $\tilde{dx} = (\tilde{dx}_1, \tilde{dx}_2, \tilde{dx}_3)$ . The quantity we wish to examine is the difference in these distances squared:

$$\vert\tilde{dx}\vert^2 - \vert dx\vert^2 = \tilde{dx}_i \tilde{dx}_i - dx_i dx_i.$$ (2.54)

If we take the mapping between undeformed and deformed to be $\tilde{x}(x) = \tilde{x}(x_1, x_2, x_3)$ , then we can express the differential quantities in the deformed state in terms of the undeformed variables. That is,

$$\tilde{dx}_i = \ensuremath{\frac{\partial \tilde{x}_i}{\partial x_1}} dx_1 + \... ...artial x_2}} dx_2 + \ensuremath{\frac{\partial \tilde{x}_i}{\partial x_3}} dx_3$$ (2.55)

$$= \tilde{x}_{i,j} dx_j.$$ (2.56)

The difference in squared distances becomes

$$\vert\tilde{dx}\vert^2 - \vert dx\vert^2 = \tilde{dx}_i \tilde{dx}_i - dx_i dx_i$$ (2.57)

$$= \tilde{x}_{i,j} dx_j \tilde{x}_{i,k} dx_k - dx_i dx_i$$ (2.58)

$$= (\tilde{x}_{i,j} \tilde{x}_{i,k} - \delta_{ij} \delta_{ik}) dx_j dx_k$$ (2.59)

$$= ((x_i + u_i)_{,j} (x_i + u_i)_{,k} - \delta_{jk}) dx_j dx_k$$ (2.60)

$$= ((\delta_{ij} + u_{i,j}) (\delta_{ik} + u_{i,k}) - \delta_{jk}) dx_j dx_k$$ (2.61)

$$= (u_{j,k} + u_{k,j} + u_{i,j} u_{i,k}) dx_j dx_k.$$ (2.62)

Interchanging the indices $i$ and $k$ , we finally get to the definition of the components of strain:

$$\vert\tilde{dx}\vert^2 - \vert dx\vert^2 = (u_{i,j} + u_{j,i} + u_{k,i} u_{k,j} ) dx_j dx_k$$ (2.63)

$$= 2 \varepsilon_{ij} dx_j dx_k$$ (2.64)

where the components of strain are defined as

$$\varepsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i} + u_{k,i} u_{k,j}).$$ (2.65)

The components of the gradient of the strain are often assumed to be small, allowing us to drop the last term of the strain:

$$\varepsilon_{ij} \approx \frac{1}{2}(u_{i,j} + u_{j,i}).$$ (2.66)

Note that it is the derivatives of the strain that are assumed to be small, not the strains themselves. Thus large deformations can be described with this definition of strain if the strain varies smoothly throughout a body. Strain is a symmetric tensor of second order.

Footnotes

... state^2.6

The undeformed state is sometimes called the rest state, but this is misleading, as the undeformed state does not have to be at rest - it could be moving uniformly as a rigid body.