By W. Noll, R. A. Toupin, C. C. Wang, C. Truesdell
The time period "dislocation" is utilized in a number of diversified senses within the literature of mechanics. within the elassic paintings of VOLTERRA, WEINGARTEN, and SOMIGLIANA, it refers to specific strategies of the equations of linear elasticity, within which a con tinuous box of pressure doesn't correspond, globally, to a continuing box of dis placement. The configuration of the physique so acquired, even if that physique is freed from all load, is topic to inside pressure that doesn't vanish, and quite often no deformation of the physique as a wh oIe can convey it right into a relaxing configuration. however, if any sufficiently sm all a part of the physique is taken into account on its own, a configuration for it within which the tension is in all places 0 will be came upon instantly. during this paintings constitutiL"e assumptions give you the easy facts. those consist in prescribed relaxing configurations for every fabric aspect and in prescribed elastic moduli governing the reaction to deformation from the enjoyable configuration at each one fabric aspect. every little thing follows from those information, ineluding the dislocations current, if any. specifically, the typical boundary-value difficulties of linear elasticity should be set and solved for the dislocated body.
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Extra resources for Continuum Theory of Inhomogeneities in Simple Bodies: A Reprint of Six Memoirs
Let X be an isotropie solid partic1e. "J. ) is then called an undistorted state of X. The following theorem follows immediately from Theorems 4 and 5. Theorem 6. Let X be an isotropie solid particle. \l(et;~)R(t)T, (20-3) S (t) = Rio (t) SI. 4) S(t) = hl 2 (U/; V(t)). Tlze notation of Theorem 4 applies here. In addition, V = RU R T is the left strain tensor relative to M as a rejerenee and ~ 1'S the lejt strain tensor of the dejormatz'on from M to ""1. The junctionals St, hl 1 and St'2 satisfy the condihons (b), (e), and (d) of Theorem 5 for all orthogonal transformations Q.
The resulting equivalence dasses LI will be called deformations. J. With the law of composition thus defined, the deforrnations form a group denoted by p). , provided that GP = 0 is possible only for P = O. Two different linear transformations are never equivalent in the sense defined above. Henee a regular linear transformation defines a unique deformation and may thus be regarded as a special deformation. l'. The gradient at P = 0 of a loeal homeomorphism Cl is a loeal property of Cl and henee depends only on the equivalenee dass LI to whieh Cl belongs.
It turned out to be necessary, therefore, to develop eoncepts which deseribe the loeal behavior of a motion at a particular particle. This is done in Chapter I. Some of the eoncepts introdueed there are similar to those of the theory of "jets" introdueed by EHREsMANN [2J into modern differential geometry. The treatment of strain and related notions given in Chapter I is, I believe, more coneise and direct than previous treatments. It is based on the weIl known polar decomposition theorem for linear transformations.