I have a 2D rectangle with length $a_0$ and width $b_0$ and it is stretched uniaxially in the direction parallel to its length. With my measurement device, I can measure the displacement $\Delta a$ and the applied force $P$. The new length of the rectangle is $a$. I wish to compare the measurements with computations.
I have a formula for the engineering stress, which depends on material parameters and on the right Cauchy-Green strain. I understand, that in case of uniaxial stretch the principal stretches are $\lambda_1^2 := \lambda^2$, $\lambda_2^2 = 1/ \lambda$. From that I can express the engineering stress that I can compare to the measurements. What I am still missing is the connection between $\Delta a$ and $\lambda$. What is the connection between the displacement and the stretch ratio without linearization?
I was suggested to use $\frac{1}{2}(\lambda^2-1)=\frac{\Delta a}{a_0}$ to define the relation between these parameters. Is it a reasonable choice? I found no references for this formula.