By Simo J.C.
A formula and algorithmic remedy of static and dynamic plasticity at finite lines in keeping with the multiplicative decomposition is gifted which inherits all of the positive aspects of the classical types of infinitesimal plasticity. the most important computational implication is that this: the closest-point-projection set of rules of any classical simple-surface or multi-surface version of infinitesimal plasticity incorporates over to the current finite deformation context with out amendment. particularly, the algorithmic elastoplastic tangent moduli of the infinitesimal thought stay unchanged. For the static challenge, the proposed classification of algorithms protect precisely plastic quantity alterations if the yield criterion is strain insensitive. For the dynamic challenge, a category of time-stepping algorithms is gifted which inherits precisely the conservation legislation of overall linear and angular momentum. the particular functionality of the method is illustrated in a couple of consultant huge scale static and dynamic simulations.
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Sin θ When θ = 0, since the right side is in the indeterminate form 0/0, we apply L H ospital s rule to determine the limit: sin2 nθ 2n sin nθ cos nθ = lim = 2n sin 0 = 0. θ→0 sin θ θ→0 cos θ lim Hence the two sides are equal at θ = 0 by the limit convention. 24) determine the closed-form sums of the following cosine and sine series: n cos θ = 1 + cos θ + · · · + cos nθ =? 25a) =0 n sin θ = sin θ + sin 2θ + · · · + sin nθ =? 25b) as the real and imaginary parts: n n =0 n ej z = =0 θ = n cos θ + j sin θ = =0 n cos θ + j =0 sin θ .
6 9 = 3 3 . It can be easily veri ed that y(t + To ) = y(t). , y(t + T ) = y(t). Since we have uniform spacing f = fk+1 − fk = 1/T , we may still plot Ak and Bk versus k with the understanding that k is the index of equispaced fk ; of course, one may plot Ak and Bk versus the values of fk if that is desired. 7. REVIEW OF RESULTS AND TECHNIQUES 13 3. A non-commensurate y(t) is not periodic, although all its components are periodic. For example, the function √ y(t) = sin(2πt) + 5 sin(2 3πt) √ is not periodic because f1 = 1 and f2 = 3 are not commensurate.
Note that f1 is the GCD (greatest common divisor) of the individual frequencies. In general, fk = k, and we need to distinguish periodic y(t) from non-periodic y(t) by examining its frequency contents. The conditions and results are given below. 1. The function y(t) is said to be a commensurate sum if the ratio of any two individual periods (or frequencies) is a rational fraction ratio of integers with common factors canceled out. 9 Hz, and the ratio fα /fβ = 2/3 is a rational fraction. 2. A commensurate y(t) is periodic with its fundamental frequency being the GCD of the individual frequencies and its common period being the LCM of the individual periods.