Numerical Methods And Simulation For Cavitation In Nonlinear Elasticity

Cavitation problem is directly related with the material failure research. It is a nonlinearbifurcation problem in essential. The cavitation condition, place, size and shape etc. areconnected with lots of geometrical and mechanical parameters, which presents greatchallenge to its mathematical modeling and numerical simulating. The research ofcavitation has aroused great interest for people studying in mathematical, mechanical,biological and medical ifelds.Basically, cavitation could be described by two kinds of models: one assumesthat cavitation appears in perfect materials, this model generally displays Laurentievphenomenon; another assumes that the cavities grow from small defects in the material.Defect models are usually preferred to be applied in numerical simulation, since itssolution has better regularity, and existing theory indicate that: when the prescribeddefects in the defect model coincide with the perfect moders cavitation places, perfectmodel solution could be approached by defect mod'el solution as the defects diametersapproach zero. However, the cavitation place is unknown in advance.If the cavitation places have been appointed, one of the main dififculties for sim?ulating cavitation is to get the higher order convergence results under the constraintof preserving orientation. The locally large expanding transformation around the cav?ity makes it hard for linear conforming ifnite element to preserve the orientation ofmesh, which leads to the fact that lots of freedoms are required when the initial holeis very small. In this thesis,we have devised radially curved triangle coupled withdual-parametric ifnite element method to solve the radially symmetric2dimensionalcavitation problem. The numerical experiments showed that we could use less freedomto simulate cavitation with small initial defect. Inspired by that, for general referenceconifgurations, we devised locally curved mesh together with quadratic isoparametricifnite element to compute the cavitation problem. We have numerically proved theif'nite elements robustness for preserving the orientation. Solution of higher order ac?curacy has been got with this method. Based on this algoirthm, we made the initialnumerical computation for conifgurational force in the cavitation problem. Our numer- ical experiments veriifed and extended the theory of Sivaloganathan and Spector on theconifgurational forces of cavities,as well as justiifed a crucial hypothesis of the theory.Another dififculty for cavitation simulation is to ifnd the cavitation places. Basedon the conifgurational force theory and its numerical extension, we have got the generalalgorithm for simulating cavitation. Using this method, the prescribed defect pointscould be adjusted according to its conifgurational forces, and ifnally we could get theperfect cavitation places. In this way, the defect model can be used to entirely simulatethe cavitation, and the Laurentiev dififculty is conquered at the same time.W,e also discussed the effect of geometrical parameter (the lfawssize, distributionetc.) and mechanical parameter (compressity of the material and unsymmetric load) forcavitation problem, in which we discovered the new bifurcation phenomenon: when theinitial defect is sufifciently small, critical values exist for all the above factors, and theifnal conifgurations are totally diverse for parameters greater than and less than thecritical values.Lots of numerical results and their comparison with theoretical and experimen?tal results showed that: the curved triangle coupled with isoparametric ifnite elementmethod can effectively simulate the cavitation phenomenon. The computation for con?ifgurational force and the correspondi'ng results coherence with the theory proved therobustness of our algorithm. Our method can simulate the cavitation phenomenon forextremely small lfaws (in the order of10₉,the smallest lfaw in other methods is inthe order for10₃to our knowledge). This method can be directly applied into the3dimensional problem. Since most theoretical research for cavitation are restricted tosingle hole problem, we believe that our numerical experiments for multiple holes canshed more light 'on peoples research.