Research On Three-dimensional Multiresolution Continuum Theory And Its Applications In Bone Materials Considering The Microstructure Randomness

The mechanical properties of materials essentially depend on the deformation behavior of microstructure at different scales.Multiscale numerical method can efficiently and accurately fulfill the link between micro structure and material properties.MRCT(Multiresoluiton Continuum Theory)is a multiscale numerical method which can be applied in a wide range and can take into account scale effect.However,there are three major issues when implementing MRCT:the first one is how to use the existing two-dimensional MRCT method to solve three-dimensional problems;second one is how to considering the randomness of the material microstructure;and third one is how to select the material constitutive model and determine material parameters.In this dissertation,bone material was chosen as the research subject,a 3D finite element model constructing method of RVE(Representative Volume Element)named DSM(Domain separated Method)was proposed,a criterion which determines the size of the RVE was also proposed,a solving scheme which combines the sparse grid collocation method and 3D MRCT-FEM was proposed to take into account the randomness of microstructure,a higher order version of MRCT named h-MRCT was established by introducing the second order terms of Taylor expansion of velocity and velocity gradient.The main research contents and results are as follows:(1)A 3D MRCT finite element method solving framework(MRCT-FEM)was established and a corresponding program was developed.Compared to traditional continuum theory which contains only a single spatial scale,the main feature of MRCT control equation is that it contains microstress term and microstress couple term.Based on the MRCT control equation,the 3D finite element formulations were deduced.The correlation matrix forms of isoperimetric element were demonstrated.(2)In the MRCT-FEM solving framework,establishing RVE finite element model is essential.After performing ?CT scanning experiment on a bone specimen,tomography slice image data were obtained.By using proposed domain separation method(DSM),the RVE finite element model which was discrete by orthogonal hexahedral elements can be established.Furthermore,the impact of element size and element orientation on finite element model which established by using DSM was studied.Based on Mori-Tanaka mean field theory,a criterion which determines the size of the RVE was proposed,and the results shown that of RVE should be 2.88 times the internal characteristic length.(3)The micro structure of material is usually unknown.The macro information such as porosity and density,which are homogenized,can be known by performing CT scan experiment.Without considering the randomness of microstructure,using only homogenized variables to measure the microstructure is inaccurate.On the other hand,in order to investigate the randomness of microstructure,a large number of microstructure RVEs are required,but collecting RVE data through ?CT scan experiments is lack of feasibility.Thus,a microstructure remodeling simulation method was proposed,which takes into account the effects of strain and drugs stimulus.The results show that the proposed method can reflect the evolution of bone microstructure under external stimuli.Furthermore,by using the same step time and synchronization on the interface of RVEs,this method can also be implemented to simulate the evolution of human macro bone structure(femoral head).(4)Based on the microstructures which were obtained through bone microstructure remodeling simulation,a large number of RVEs were constructed.Direct numerical simulations(DNS)were carried out by applying a uniform periodic boundary condition.The generalized strain/stress data which existed at different scales were obtained from DNS results.According to the porosity,these data were divided into groups,and parameters of constitutive model were obtained by nonlinear fitting.Then,the uniaxial compression experiment of the bone specimen was simulated with MRCT-FEM.Results shown that the proposed MRCT-FEM revealed the softening behavior of bone specimen during compression.Finally,the nested sparse-grid stochastic collocation method(NSSCM)was implemented to MRCT-FEM solver for considering the randomness of microstructure.Results shown that the probability of the maximum value of reaction force within the range 0.18?0.22 kN(compression experiment result was 0.2 kN)was only 0.223.(5)By introducing second order terms of Taylor expression of velocity and velocity gradient,a higher order version of MRCT(h-MRCT)was derived.The virtual kinetic energy term,virtual kinetic power term,virtual internal power term and virtual external power term were modified correspondingly.According to virtual power equilibrium,the control equations of h-MRCT are obtained.Then 1D finite element formulations of MRCT-FEM and h-MRCT-FEM were deduced.At last,two sets of one dimensional bar which contain two different materials were simulated using FEM,MRCT-FEM and h-MRCT-FEM.