Dear all
I’m studying hypo-plastic constitutive law from beginning concepts
I traced concepts successfully until i faced **rate independent topic**
i read this topic in the prof. Wolfgangs paper (hypo-plasticity for beginners)
he said that “Rate-independence means that the gradient of the curve (stiffness) does not depend on the strain rate, Thus the stress rate must be precisely twice as large for a double strain rate” and in the fallowing he said that “the constitutive law may not have terms like (strain rate^2) (strain rate square )”
as you know the hypo-plastic constitutive law is non-linear in strain rate
my question is that how hypo-plastic constitutive law can fulfill the non-linearity (in strain-rate) with no term like (strain rate^2)
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Dear Sobhan, hypo equation is defined as stressrate=L:strainrate * N||strainrate||. Here, ||strainrate|| is Euclidean norm of strainrate, that is sqrt(strainrate:strainrate). You see that the equation is non-linear in strainrate (if strainrate changes direction, stressrate changes direction differently as the term with N only quantifies strainrate size, not direction). You also see that actually, units of ||strainrate|| are still 1/time, as there is sqrt(strainrate^2) involved, thus it is consistent with comment by prof. Wolffersdorff which you cited. I suggest to have a loom into my book “Modelling of Soil Behaviour with Hypoplasticity”, where fundamentals of hypoplasticity are explained in (hopefully) clear way. Regards David
thank you so much!