True, natural or logarithmic strains

It may be noted that in the tensile test the following conditions apply:
*the principal strain increments all increase smoothly in a constant direction, i.e. dε1 always increases positively and does not reverse; this is termed a monotonic process;
*during the uniform deformation phase of the tensile test, from the onset of yield to the maximum load and the start of diffuse necking, the ratio of the principal strains remains constant, i.e. the process is proportional; and
*the principal directions are fixed in the material, i.e. the direction 1 is always along the axis of the test-piece and a material element does not rotate with respect to the principal directions.
If, and only if, these conditions apply, we may safely use the integrated or large strains defined in Principal strain increments. For uniaxial deformation of an isotropic material, these strains are
ε1 = ln(l/l0) ; ε2 = ln(w/w0 = (−1/2)ε1 ; ε3 = ln(t/to) = (−1/2)ε1