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Extra resources for A Cauchy Problem for an Ultrahyperbolic Equation
4) The principal values of the stress tensor, referred to as the principal stresses, are equal to the roots iI , t2, t3 of its characteristic equation =0. 6) 46 1. Stress tensor where the symbol Y/-s implies no summation over s. 7) T = ti ee +tz ee +t3 ee, that is, the principal stresses t s on the surfaces with normal vectors ~ are normal whilst shear stresses are absent. 8) Here G sm = is' ';; since the principal axes play the role of the "old" axes. 5) simplifies as well tN! = hN1, tNZ = tzNz, tN3 = t 3N3, (Nk = N· ~) .
5 The necessary conditions for equilibrium of a continuum Let us consider an arbitrary volume V* bounded by a surface 0* . It is assumed that V* lies completely within volume V , and 0 * has no common points with surface 0 . The surface forces distributed over 0 * which are int ernal for V and external for V* are caused by th e stress st at e described by tensor T. 2) in which N denot es a unit vector of the external normal to 0* . There are two groups of necessary condit ions of equilibrium, namely equations of equilibrium in volume V and thos e on surface 0 .
10) disappear due to eq. 9). 7) ot s + pK = oX s (Otst oX s + pK t ) 1t. 6) contain six components of the symmetric stress tensor. ). The problem of equilibrium of a continuum is statically indeterminate. 42 1. 2) where tN is replaced by force F distributed over 0 N ·t=F. 15) where N, denote projections of the unit vector N on the coordinate axes. Let us agree to say that any particular solution of the equilibrium equations in the volume and on the surface determines a possible static state of the continuum.