Principal stretches

Question

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.

Answer 1

Just to be concrete, let's denote the stretched length $a$ so that $\Delta a=a-a_0$. The stretch actually has a more complicated definition in general, but in this case you can view it as just being the stretched length $a$ divided by the unstretched length $a_0$, i.e. $\lambda = a/a_0$. Manipulate these two expressions and you will find

\begin{equation} \lambda = 1+ \frac{\Delta a}{a_0}. \end{equation}

Some might call the term $\Delta a/a_0$ the normal strain in the direction of $\mathbf{P}$.