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# Complementary Strain Energy - Non-linearity

Hi all!

I read that the "cmplementary starin energy of a structure is not equal to sum of the complementary strain energies of it's components, if there is non-linearity like geometric"

That means for e.g. if I consider a truss stucture subjected to loading so that it undergoes geometric non-linearity, then the sum of the complementary strain energies from members is not equal to complementary strain energy of the structure.

I request somebody to explain why is it so??

Thanks and regards,

- Ramdas

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## Comments

## Hi Ramdas

Strain energy (or complimentary strain energy) are quantities

which are integrated over your system, thus the sum of the strain energy of

elements should be equal to the strain energy of the system.

On the other hand, even for a linearly elastic material the

strain energy is a quadratic function of stress/strain. So the sum of the

strain energy from force input F1 and F2 is different from the strain energy

from F1+F2.

## Yes, Strain energy of the

Yes, Strain energy of the system is not simplely the sum of it's components.

Besides, for some coupling system , such as gel(http://www.imechanica.org/node/1641 http://www.imechanica.org/node/1926 http://www.imechanica.org/node/3163 ). In general, equilibrium behavior of hydrogel can be derived from the free energy function W, which depends on both strain and ionic concentration, the free energy is assumed to be the sum of an elastic energy and an ionic concentration.( see papers of Prof. Suo zhigang's group). Is the free energy of gel the sum of strain energy and ionic energy? can this assumption be verified by experiments? I have doubted the validity of this assumption.

can anybody provide explanation?

thank you!

Ma lianhua

## Hi T. Markos, Thank you.

Hi T. Markos,

Thank you. I agree that strain energy of a component is obtained by integrating over the domain of the component. Similarly for a whole system, it has to be integrated over the whole domain.

Strain energy is a quadratic function, it doesn't hold good the superposition principle.

What I am asking is not regarding superposition principle. I am posing the question again..

"complementary starin energy of a structure is not equal to sum of the complementary strain energies of it's components, if there is non-linearity like geometric"

That means for e.g. if I consider a truss stucture subjected to loading so that it undergoes geometric non-linearity, then the sum of the complementary strain energies from members is not equal to complementary strain energy of the structure.

Here I am applyig a load 'F' on the structure. This load is being taken by components. Components undego deformation. For a linear system,

Total struture's strain energy = Sum of strain energies of all its components

I read that for a structure undergoing geometric non-liearuty, complimentary strain energy (U*) doesn't hold good the above equality.

With regards,

- Ramdas

## Hi

Hi

According to the energy conservation law, for a system only have elastic deformation, the sum of the complementary strain energies of it's components is equal to that of the whole structure, or else where the energy flows to?