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IIA. Ideal Material Having Uncontrollable Strains

The most general constitutive law relating the stress and strain, must be of the form

(IIA.01)

where e is the unit 3×3 matrix. We call this equation a natural constitutive law, on the grounds that it can be naturally derived from the very basic considerations about our representations of stresses and strains. Indeed, if our models of stress and strain are 3×3 matrices and, if the constitutive law is analytic, the equation (A.01) must be automatically in effect. For then the relation between the two matrices can be represented by a formal series reducible to a second order polynomial by the Hamilton-Cayley theorem. By the same token, that relation can as well be written with the places of stress and strain matrices interchanged. Thus, strain as a function of stress is also a quadratic function, only with other coefficients. This will be discussed in the next section.

The troubles with the constitutive law (A.01) come forth when characterizing the materials, for then stress and strain must not interchange freely without altering the physical interpretation of the constitutive law. More specifically, the equation (A.01) must be read as giving the possible states of strain under a given state of stress for a known material, or vice versa. However, there is an old experimental philosophy, borrowed from the practice on metals, saying that by knowing the states of stress and strain one can find the coefficients p₀, p₁, p₂. This is what we actually mean by material characterization. Often times these coefficients are considered pure material properties, but this restriction confuses the issues, sometimes with grave consequences mostly in engineering problems. Let us make this statement a little more explicit. No matter what these properties are, equation (A.01) shows that in each and every one of the loading experiments the principal directions of stress coincide with the principal directions of strain. On the other hand, if σ_1,2,3 are the principal values of stress matrix, and ε_1,2,3 those of the strain matrix, according to the constitutive law (A.01) we must have satisfied the system

(IIA.02)

Let us suppose that we can perform such experiments allowing us to measure all three principal values of strain and stress simultaneously. Their outcome will then allow us to calculate the material properties embodied in the coefficients p_0,1,2 from system (A.02). This system has a nontrivial unique solution if, and only if, the determinant

(IIA.03)

is non-null. Thus, the material parameters p₀, p₁, p₂ are uniquely determined, regardless of the character of imposed stress, by the solutions of the system (A.02) if, and only if, the resulting principal deformations are different from each other.

The difficulty is at once apparent if we observe that the parameters p₀, p₁, p₂, supposed to be pure material properties, depend actually on the values of principal stresses and on those of principal strains. According to the classical rigor, the material parameters must be objective or intrinsic, in the sense that they must not depend on experiment: all experiments should reveal the same values. At this moment, the concept of the state of a body comes to the rescue in both Physics and Engineering. Indeed, it is to be noticed that in a loading experiment, the stress and the strain actually characterize only a state of a body, among other physical parameters. Thus, p₀, p₁, p₂ as functions of the state, must also be functions of stress and strain, not only of the material. If we suppose that stresses and strains are tensors, then, for isotropic bodies, these functions depend only on some invariants constructed from those tensors. Without these two hypotheses, closely related to each other, the problem remains unsolved.

Let’s discard for the moment the assumption that the stresses and strains are tensors and the material is isotropic. However, the idea of state, as including stresses and strains in its definition cannot be discarded. One feels nonetheless necessary to define the constitutive law by intrinsic, purely material properties, with no dependence of the state of that material. These intrinsic properties are here naturally defined as special values of p₀, p₁, p₂ namely: intrinsic properties are those also found by some other kind of experiments than the loading ones, therefore even in the absence of any applied stress. According to the constitutive law (A.01) and its consequence (A.02), such properties will be exhibited by the solutions of the system

(IIA.04)

where the state of strain is the one observed in the case when no forces act upon material. This is usually termed as the natural state of the material. In other words, whenever the material is free of controlling forces and we notice deformations, we are in possession of some intrinsic properties that can be, at least partially, deduced by measuring these deformations. From (A.04) we have the following distinct cases:

(a). If the observed state of deformation is arbitrary, in the sense that has distinct eigenvalues, then the material is characterized by

(IIA.05)

This kind of material has no other response for any controlling stresses, but three uncontrollable deformations, all of which are measurable. Such a material is gas-like: for any imposed stress there is an uncontrollable internal strain.

(b). If the observed state of deformation is a material volumetric strain

(IIA.06)

then the material coefficients must satisfy the linear relation

(IIA.07)

Thus only two of them are independent when measuring the arbitrary volumetric strain. This material is nonlinearly elastic. The constitutive law can be written in the form

(IIA.08)

and contains three uncontrollable quantities, out of which one is measurable.

(c) The most general case of intrinsic properties possible under the natural constitutive law (A.01), is that where the observed state of strain under no controlling stresses is vector-like, i.e. has two equal eigenvalues, for instance

(IIA.09)

In this case the material properties are given by the system

(IIA.10)

so they are determined up to an arbitrary factor by

(IIA.11)

The constitutive law satisfied by such material will be

(IIA.12)

where K is an arbitrary constant. Such a material has three uncontrollable quantities, out of which two are measurable.

In closing this subsection, notice that, as long as we are interested in just the measurable quantities, a convenient way to express the matrix of a material exhibiting uncontrollable strains, is in the form of the tensor

(IIA.13)

where is a unit eigenvector, corresponding to the eigenvalue ε₁. Such a material has distinguished directional properties, with respect to the direction , and these properties are given by the eigenvalues ε₁ and ε₂. As a matter of fact, the equation (A.13) does contain all the previous two cases as particulars, if we agree to characterize the intrinsic material properties as deformations. Notice that this is an assumption independent of the constitutive description and must be secured by our capabilities of measurement. Thus we have this general conclusion: whenever a material deforms freely, i.e. under the action of no noticeable forces, its deformation matrix must be of the form given by equation (A.13), all the particular cases included. The deformations as well as the stresses are then manifestly tensors.

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