0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 $\mathbf{P}$.

$\endgroup$
  • $\begingroup$ What would it be if I didn’t want to linearize the stretch? $\endgroup$ – sztr Dec 10 '17 at 0:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.