What Is Generalized Continuum Mechanics (GCM)?
We classify under the title “generalized continuum mechanics” all what is not covered in the restricted framework of the Cauchy model exposed in the prerequisite Chap. 1 under the title of “classical continuum mechanics”.
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What Is Generalized Continuum Mechanics (GCM)?
Introduction We classify under the title “generalized continuum mechanics” all what is not covered in the restricted framework of the Cauchy model exposed in the prerequisite Chap. 1 under the title of “classical continuum mechanics”. In a structured overview this generalization can be presented through the successive abandonment of the basic working hypotheses of standard continuum mechanics of Cauchy: that is, introduction of a density of bulk couple, of a rigidly rotating microstructure and couple stresses (Cosserat continua or micropolar bodies, nonsymmetric stresses), introduction of a truly deformable microstructure (micromorphic bodies), “weak” nonlocalization with gradient theories and the notion of hyperstresses, and the introduction of characteristic lengths, “strong” nonlocalization with space functional constitutive equations and the loss of the Cauchy notion of stress, and finally giving up the Euclidean and even Riemannian material background. We peruse these steps in this overview, referring the reader to specialized entries for technical details.
Asymmetric Stress This asymmetry may be due to the existence of body couples; the only known physical example of these couples relates to the case of electromagnetic deformable continua where the volume magnetization is not aligned with the local magnetic field M, or the dielectric polarization P is not aligned with the local electric field creating thus couples per unit volume in the form of vector products M H or P E in an obvious notation. Accounting for such terms in Eq. (1.7) will result in a deviation from the symmetry condition (1.2) with the existence of a nonzero skew part of the stress given by © Springer Nature Singapore Pte Ltd 2017 G.A. Maugin, Non-Classical Continuum Mechanics, Advanced Structured Materials 51, DOI 10.1007/978-981-10-2434-4_2
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What Is Generalized Continuum Mechanics (GCM)?
t½ ji ¼ M½i Hj
or
P½i Ej :
ð2:1Þ
In many materials this is strictly zero in reason of the proportionality of the field M in H or of P in E. Also, the situation described by Eq. (2.1) may be only transient as M may rapidly align with H or P with E. Of course interaction of electromagnetic fields with deformable matter may be much more complicated than that described by Eq. (2.1) involving both couple and force of electromagnetic origin, and an import of a specific energy. For a full development of this aspect in Galilean or relativistic dynamics we recommend the treatise of Eringen and Maugin (1990; reprint 2012).
Surface Couples This concept may be harder to imagine physically. But there is no opposition of principle to introduce in strict parallel with an applied surface traction (in the Cauchy model), an applied surface couple Cd per unit surface. This is an axial vector. A reasoning à la Cauchy will yield the introduction of the notion of couple stress m such that nj mji ¼ Cid :
ð2:2Þ
The object of induced component mji still is “axial” in its second index i. Accordingly, we can introdu
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