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III. A Category of Real Materials

A real material is, in fact, more complicated than those described above. For, to characterize a material, we have a limited power: if we just observe the material, we are actually not sure that there are not forces under which the unassisted deformation takes place. On the other hand, if we perform experiments on it, we need bodies of pure material of given shapes, and creating these shapes may induce uncontrollable strains. Thus both cases before are highly idealized, and we have to decide the existence of pure material properties some other way. However, the conclusions we arrived at are very important, in that they allow us to characterize a general material, and thus to discover its characteristics in their utmost generality. Let us reiterate these conclusions once more: in ideal situations as described above, both intrinsic stress and strain matrices have to be tensors. This fact is obvious from the forms of those matrices as given by equations (IIA.13) and (IIB.10). As we have seen, the intrinsic material properties are given by measurements on either deformations or stress, revealing some kind of vector. Two quantities are enough for each type of ideal material.

A little digression: this case is specific for matrices that we term here as equivalent to a vector field. We understand this equivalence in the following way: let be a vector field, and let’s construct the following matrix

(III.01)

Because v_k are the components of a vector, supposing α and β as scalars, gives v_ij as the components of a second order tensor. One of the principal values of this tensor, namely α, is double. The other principal value, different from α, is given by

(III.02)

Notice some interesting features of this kind of tensor. First of all, if either β or v_k are null, v_ij is a purely spherical tensor, of the form given in (IIA.06) or (IIB.04). Secondly, if we calculate the eigenvector of v, corresponding to the eigenvalue (III.02), we find out that this eigenvector is , up to a normalization factor. This property is independent of the parameter α, and this is what we mean by the above mentioned equivalence: given the vector we can directly construct the tensor v as a family of two-parameter matrices having it as an eigenvector.

The point of attack of the problem of characterizing a real material is given by the equations (IIA.13) and (IIB.10). They just give the intrinsic states of ideal materials in two extreme situations. Then we can reason in the following way: a real material will be characterized by intrinsic properties having, to a certain extent, both types of properties; for, no material accessible to our observation is without shape. This fact can be expressed in a few ways. Here we choose to express it in a matrix form, by writing the intrinsic matrix of a general material as a linear combination of (IIA.13) and (IIB.10):

(III.03)

where λ and μ are real parameters, describing the degree of ”ideality”, with deformations and stresses defined by

(III.04)

where and are some “intrinsic” vectors. This tensor contains four measurable quantities: λ, μ, and the lengths of the two intrinsic vectors. Written at length, the tensor (III.03) is

(III.05)

It is easy to see that it has three real eigenvalues. Indeed, its orthogonal invariants are

(III.06)

where we denoted

(III.07)

The eigenvalues of t_ij can then be calculated as the roots of the corresponding characteristic equation, and they are

(III.08)

It turns out that the pair from (III.07) gives one eigenvector of t and the corresponding eigenvalue. The other two eigenvectors of t are orthogonal, and located in the plane of the vectors and .

An interesting point of this representation is that the tensor (III.05) is invariant with respect to a two-parameter group of transformations in the plane of vectors and , as given by

(III.09)

where the coefficients are the entries of the matrix

(III.10)

with Φ an arbitrary angle.

We note that, if t is a stress tensor, then is the traction normal component on the surface element of outside normal , so that if the surface moves with the relative speed with respect to the material, the scalar quantity

(III.11)

is the rate of transfer of work due to tractions through that surface element. By this equation, the transfer vector defined as

(III.12)

is just a work rate exercised by stresses. Recalling that the stress is of the form (III.05) above, the transfer vector becomes

(III.13)

It is only for that we have and thus the possibility of a definition of mechanical momentum in the classical sense of transport theory. The eigenvalue of the stress is in this case the analog of a mass, for we can write

(III.14)

Otherwise, in the general case, the main trend of the description should be to prove the practicality of the general transfer vector as a momentum vector.

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