A common equation used to represent the onedimensional stres

A common equation used to represent the one-dimensional stress-strain behavior of collagenous tissues is: sigma = A(e^Bc - 1) where A and B are material coefficients to be determined; a is the stress applied and epsilon is resulting strain (unit less). Derive how the slope of the stress-strain curve is related to the applied stress. Examine and describe the effect of changing values of A and B on the stress-strain curve and its shape (i.e. values of 1 then 10, various combinations). Include the behavior of the slope vs stress curve with changes to A and B in your description. Even though A and B are phenomenological parameters, how might their magnitudes be reflective of structure within a given biological soft tissue?

Solution

given sigma = A( exp(Bepsilon)-1) = Aexp(Bepsilon)-A

(sigma+A) = Aexp (B epsilon)

ln((Sigma+A)/A) = (Bepsilon)

take derivatives: obtain dsigma/depsilon =B(( sigma+A)/A) = (B/A)(sigma+A)

Hence slope of the curve varies as B/A * applied stress.

rest is easy.

about significance of A and B , from the math expression A is an offset, B /A a modulus depending on Sigma.