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Strain compatibility equation in non-linear solid mechanics!!!

Submitted by ramdas chennamsetti on

We have six strain compatibility equations, which are obtained from strain-displacement relations by making an assumptions 'small strains'. Strain compatibility equations ensure a single valued and continuous displacemnet filed. These equations are used in stress based approach.

Now my queries are as following.

[1] Do we have strain compatibility equations for non-linear strain-displacement relations?

[2] Do we follow stress based approach in non-linear solid mechanics.

Potential for Strain energy

Submitted by ramdas chennamsetti on

Hi all,

When a conservative force does work, it is independent of the path, we define the potential and work done is given by  - (change in potential).

We define potentials for gravitational force, electrical force etc...

Assuming the body is linear elastic, internal forces, cause stresses in a body, are also conservative forces, whose work (strain energy) is independent of the path. Can we define potential for such internal forces? If so, we can calculate strain energy = -(change in potential).

You may kindly explain this.

Odd order governing equation - FE formulation

Submitted by ramdas chennamsetti on

Hi!!

We generally encounter governing equations of even order. In FE formulation we get a symmetric coefficient matrix 'A' (AX = B). I have a few doubts as follwing.

[a] Any odd order governig equations ? If so, you may please write.

[b] Say, it has a functional also, then, what's the order of that differentiation?

Body loads in wave propagation..

Submitted by ramdas chennamsetti on

Hi all,

[1] In solids, the wave propagation equation is obtained from stress equilibrium equations. We make use of constitutive and strain-displacement relations to convert these equations in terms of displacements

[2] In the above equations we assume that there are no body loads.

[3] The form of solution we assume for displacements is harmonic

[4] Plug these three displacements, u1, u2 and u3 in the equilibrium equations stated in [1].

[5] We end up with an Eigenvalue problem. This is nice.

Thin plate theory...

Submitted by ramdas chennamsetti on

Hi all!

I have a small doubt in the assumptions made in thin plate theory.

We make some of the following assumptions in thin plate theory (Kirchoff's classical plate theory) (KCPT).

[1] The normal stress (out of plane=> sigma(z)) is zero. and

[2] The vertical deflection 'w' is not a function of 'z' => dw/dz = 0

Now there are three stress components sigma(x), sigma(y) and sigma(xy). The other three stress components sigma(z), sigma(xz) and sigma(yz). This is like a plane stress.

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