By B. F. Shorr (auth.)
This monograph offers intimately the radical "wave" method of finite point modeling of brief approaches in solids. robust discontinuities of tension, deformation, and pace wave fronts in addition to a finite value of wave propagation velocity over parts are thought of. those phenomena, comparable to explosions, shocks, and seismic waves, contain issues of a time scale close to the wave propagation time. software program programs for 1D and 2nd difficulties yield a lot better effects than classical FEA, so a few FORTRAN courses with the mandatory reviews are given within the appendix. The booklet is written for researchers, teachers, and complex scholars attracted to difficulties of numerical modeling of non-stationary dynamic tactics in deformable our bodies and continua, and likewise for engineers and researchers concerned designing machines and constructions, within which surprise, vibro-impact, and different unsteady dynamics and waves procedures play an important function.
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Additional resources for The Wave Finite Element Method
0= 0 I 0'=-1 c.. I 0'=-1 0=0 =1,2" .. ,n; (f,- = 0, uj,o = 0, vn+ = 0. a) ~ ~ The numerical procedure v=1 I~ v=11" f f f = 1,2, ... 8. 8a,b. After stopping the end x = 1 at the initial moment t = the compressive stress wave front travels to the free end x = 0. The whole rod stops at t = I, being entirely in compression. The rod rebounds at t = 2 when it restores the initial velocity, but in inverse direction, and is entirely unloaded. 28) and is independent of the rod length. 29) 28 1 Foundation of the Wave Finite Element Method and is independent of the impact velocity.
J. j -v'j. 1;. 44) The work introduced into the element in a time M; is l1Wij = A( a;V; -ajvj )M;. 45) B. o - aj - a; )(v; - vj) = O. ] = a·J, o +E(v+] -v-:-] )M. / L1x. t i -vi / =0. 48b) independently of velocities v~ values. This is in full agreement with the wave theory. 48b) are obtained. In other words, determination of the interval M; based on the law of the energy balance for the whole system provides the energy balance of each element. 2. 4. A. In the above, we have dealt with a homogeneous rod of a constant element length &.
2 I' 4 ~ .... "'---2 2 ,.. II. •. 8 6 II y·' lr ~I III 1\ v b) 10' - + (j n 2 o 0 1'''·2 \J , -2 A. "4 Ha. ,9 J c) III 1 ~ y,,; IIr. II! v~' I I\. 11110 l\r Fig. 6. 2 methods DMM and FEM for the elementary 1-D problems, especially in view of the fact that the exact solutions are known. 6a, b, c respectively. 1 Direct Mathematical Modeling of Wave Propagation in an Elastic Rod 25 curves 1 that coincide with the analytical solution are obtained by the DMM, the curves 2 - by traditional FEM. 025 .