View Full Version : hyper/hypo-elasticity: question

Tom Impelluso
07-10-1999, 04:02 AM
Hi all,

This may not be the best group to ask this... if not, could you
suggest a more appropriate one?

>From what I understand, there are three classifications of elastic
material properites:

hyper-elastic (strain energy function, integral formulation)
elastic (stress/strain law, algebraic formulation)
hypoe-lastic (rate dependent law, differential formulation)

It seems that most constitutive laws for biological tissues are
written as hyperelastic.

Could someone explain why?

I will hazard a guess...

The field of biomechanics took off before the very widespread
development of large scale finite element codes. At such a time, the
hyper elastic formulation was easier to work with since it postulated
an integral formulation for ONE parameter -- strain energy -- from
which stress could be found by the simple act of differentiation..

It seems that a HYPOelastic formulation seems more natural yet
I continue to see papers pushing hyperelasticity.

I claim HYPO- seems more natural because it is a rate dependent formulation
which accounts for change of frame (Jaumann stress rate). This, it seems
to me, is essential when modelling human tissue which typically
undergoes large deformation...

and while (and *if* you're at it)... try this one one for size...
(because, this too, has been frustrating me...
but this next question is not as essential as the previous so
I am boxing it in between the dashes)
A Hyperelastic material is one which possesses a strain energy function.
An elastic material is one where there is a natural stress free state,
there is a one to one relationship between stress and strain.
A Hypoelastic material is one where the stress rates are linear
of the strain rates (thus becoming a differential equation).

Now, typically, when I see hyper, hypo, etc, I feel the need to
this in terms of comparatives... soft, softer, softest.... all embody
degrees of softness.

Yet, I try to search for comparatives in the three types of elasticity
and come out empty handed...

One person suggested that I view it this way:
Hyperelastic is one involving integration: the strain energy is the
INTEGRAL of stress, strain terms
Elastic is ALGEBRAIC
Hypoelastic is DIFFERENTIAL.

This only pushes my question back because then I cannot see how the
mathematical formulations introduce comparatives.

So... I try this...

If there is a strain energy function, then, internal stress work can
be related to the work of the applied loads. This definition causes
me to state that the material returns to the original configuration
if energy is conserved.


It seems that NOTHING in the definition of elastic or hypoelastic
states the RETURN to a stress free state.

OK... Now I must see how hypoelastic differs from elastic.
Well, in a hypoelastic material, we are relating rates of deformation
while in an elastic material we at least still have the one to one

I think...
But, basically, I must conclude I am still baffled and ask if someone is
willing to respond to me...
Surely, I will read the response and answer back with a question unless
perhaps you can, hopefully, put this issue to rest in one clean
answer... (which I am coming to doubt is possible)
[where am I going wrong?]

Another way to phrase this is to ask for a definition of ELASTIC
than has the potential to demonstrate gradations of elastiticity,
and exactly how the integral, algebraic, and derivative formulations
embody these gradation...


Could someone tackle these for me?


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