ProtoQuant.com

IIB. Ideal Materials Having Uncontrollable Stresses

So far we discussed about the natural constitutive law as given in the form of the equation (A.01), characterizing materials that can flow freely, i.e. exhibit deformations under no stress. From the very same point of view we can discuss a natural constitutive relation of a form revealing what is happening when we control the deformation. The control of deformation is not always at hand. However let us just say that it is a matter of scale: there is a certain scale where we can control the deformation. The difference with respect to the previous situation is that here we need to consider the bodies explicitly, i.e. in their shape; otherwise we cannot talk about controlled deformation. In this case the constitutive equation has to be written down as

(IIB.01)

Then, the natural state of such a material will be characterized by the system of equations

(IIB.02)

corresponding to no strain response. Thus we can have as before:

(a’) Materials exhibiting no strain under an arbitrary state stress

(IIB.03)

Such a material has no strain in response to any stress. It can be termed as infinitely rigid, in the following sense: to the arbitrary internal stress there always corresponds a null strain; this is also valid for any superimposed state of stress. The model corresponds to the old idea of ether, as medium supporting waves of a very high speed.

(b’) Materials exhibiting no strain under a hydrostatic stress (incompressible)

(IIB.04)

In this case knowing the pressure σ₀ we have the following relation for the coefficients q_k

(IIB.05)

showing that only two of the three material coefficients are independent. This material is a perfect stationary incompressible liquid. The constitutive law can be written in the form analogous to (A.08):

(IIB.06)

(c’) Materials exhibiting no strain under a vector-like stress state

(IIB.07)

in which case q_k are determined to a factor by

(IIB.08)

The constitutive law characteristic to such a material is analogous to (A.12), and it reads

(IIB.09)

where the constant K₁ has dimensions of a stress. This material is a perfect metal. Indeed, the relation (B.09) with σ₁ + σ₂ = 0 has been found by Bell (Bell, 1968, 1973) to be characteristic for metals in large deformations.

Again, as long as we are interested in just measurable quantities characterizing such a material, then its intrinsic stress tensor assumes the following convenient representation, similar to (A.13)

(IIB.10)

where is, again, a unit vector corresponding to the eigenvalue σ₁. One can say that the general characteristic of materials exhibiting no strain under stress is of the form (B.10), all the particular cases included.

Pages: 1 2 3 4 5 6 7 8 9