## NAl

Figure 4.1: Elongation of an Axial Rod

Figure 4.1: Elongation of an Axial Rod

3 We seek to quantify the deformation of the rod and even though we only have 2 variables (l0 and l), there are different possibilities to introduce the notion of strain. We first define the stretch of the rod as

This stretch is one in the undeformed case, and greater than one when the rod is elongated.

4 Using l0, l and A we next introduce four possible definitions of the strain in 1D:

Engineering Strain e

Natural Strain n

Lagrangian Strain E

Eulerian Strain E*

we note the strong analogy between the Lagrangian and the engineering strain on the one hand, and the Eulerian and the natural strain on the other.

5 The choice of which strain definition to use is related to the stress-strain relation (or constitutive law) that we will later adopt.

4.1.2 Small Strains in 2D

6 The elementary definition of strains in 2D is illustrated by Fig. 4.2 and are given by

Aux Auy

AY AX

)YxV

2 ,xy 2 V AY In the limit as both AX and AY approach zero, then

exx=

We note that in the expression of the shear strain, we used tan 0 « 0 which is applicable as long as 0 is small compared to one radian.

7 We have used capital letters to represent the coordinates in the initial state, and lower case letters for the final or current position coordinates (x — X + ux ). This corresponds to the Lagrangian strain representation.

0 0