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The imperative goal of this ebook is to narrate the random distributions of defects and fabric energy at the microscopic scale with the deformation and residual energy of fabrics at the macroscopic scale. to arrive this objective the authors thought of experimental, analytical and computational versions on atomic, microscopic and macroscopic scales.
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Extra info for Damage and Fracture of Disordered Materials (CISM International Centre for Mechanical Sciences)
47 Assume that U is nonnegative and it satisfies the domination principle. Then, the following are equivalent: (i) U is nonsingular; (ii) there is not a column of zeroes and no two rows of U are proportional; (iii) no two columns of U are proportional. Notice that if U satisfies the CMP, then the column j of U is zero if and only if Ujj D 0. So the condition of not having a column of zeroes is equivalent to have a positive diagonal. The same property holds for U that satisfies the domination principle (use that DU satisfies the CMP for some nonsingular diagonal matrix D).
For n D 2, we have that U has the form Â Ã ab U D ; cd where 0 < c Ä a; 0 < b Ä d , because U is a column diagonally dominant matrix. Hence, U is singular if and only if ad bc D 0, which implies that c D a; b D d , showing in this case the rows of U are equal and the columns are proportional. Now, we perform the inductive step. ii/ are equivalent for matrices with size at most n 1 for n 3 and we show the equivalence for any matrix U > 0 that satisfies CMP of size n. For the inductive step, we consider two different situations.
50 Let K D fi0 g and L ¤ ; be disjoint subsets of IN. Consider I D IN n L and assume that Q D PII satisfies that I Q is nonsingular. I Q/ 1 the potential associated with Q. 13) Proof By definition v takes the value 1 at K and 0 at L. K [L/. I Q/ D I and that P is reversible. Indeed, for any i different from i0 we have U i0 i D X Ui0 j Qji D j 2I D X Ui0 j Pji D j 2I i U i0 i0 i0 X X U i0 j j 2I i Pij D i U i0 i0 X i0 j 2I j Pij vj Pij vj ; j 2INWi j where in the last equality we have used the fact that v is zero on L.