【作者】 易炜；

【导师】 王乘； 刘小虎；

【作者基本信息】 华中科技大学 ， 固体力学， 2017， 博士

【摘要】 材料的力学性能本质上取决于材料微观结构在不同尺度下的变形行为。多尺度数值方法可以高效而准确地描述材料微结构与材料性能之间的联系,建立材料的力学模型。MRCT(Multiresolution Continuum Theory)理论是一种应用范围较广并能考虑尺度效应的多尺度数值方法,但是目前在运用MRCT理论时存在三个问题:一是如何将现有的二维MRCT方法用于解决三维问题,二是如何考虑材料微结构的随机性,三是如何选定材料本构和获得材料参数。本文以骨材料为对象,建立了考虑材料微结构形貌随机性的三维MRCT多尺度计算方法,提出了代表体元RVE(Representive Volume Element)的三维建模方法及其特征尺寸的定量确定准则,用嵌套式稀疏网格配置点法考虑材料微结构形貌随机性,引入高阶的速度项和速度梯度项建立高阶MRCT(h-MRCT)控制方程,并以骨质材料为例进行了算例验证。本文取得的主要研究结果如下:(1)建立了基于MRCT的三维有限元计算框架及程序(MRCT-FEM)。MRCT控制方程较传统有限元法有较大不同,其最主要的特点在于引入了微应力项和微应力偶项。本文对MRCT控制方程在三维情况下的有限元列式进行了推导,给出了六面体等参单元的相关矩阵,编写了三维MRCT的有限元计算程序。(2)在MRCT-FEM计算中,代表体元RVE的建立至关重要。本文对骨试样进行了微CT扫描,获得了断层扫描图片数据,利用区域分离法(DSM)建立正交六面体网格离散的微结构有限元模型,研究了 DSM中不同的单元尺寸、朝向对有限元模型的影响;基于Mori-Tanaka平均场理论研究了 RVE尺寸的选取原则,得到RVE尺寸应选取为2.88倍内部特征尺寸。(3)材料微结构在一般情况下是未知的,往往只能知道孔隙率、密度等均匀化后的宏观信息,由于没有考虑微结构的随机性,这种以宏观平均量去度量微结构的方法存在较大误差。另一方面,为了考察微结构随机性,需要大量的微结构RVE,而通过大量扫描实验实际测量获得数据的方法缺乏可行性。本文提出了骨微结构重建的模拟方法,考虑了药物和应力刺激效应,并通过数值计算得到了大量的骨微结构样本数据。计算结果表明,提出的微观骨重建模拟方法能够很好地反映正常骨质结构在外界刺激的下的演化过程。通过对多个RVE采用相同步长并在界面上进行同步的方法,对人体宏观骨结构(股骨头)的微结构演化进行计算。(4)利用计算得到的大量骨微结构样本数据建立大量的RVE,通过加载周期性边界条件进行DNS数值模拟(Direct Numerical Simulation),得到各个尺度下材料广义等效应力-应变曲线。根据孔隙率对这些曲线进行分组,对不同孔隙率下的曲线数据进行拟合,获得了不同尺度下本构模型的参数。然后,通过MRCT-FEM方法模拟了骨试样的压缩试验,结果表明MRCT-FEM方法能够较好地模拟出骨试样在压缩过程中的软化行为。结合嵌套式稀疏网格随机配置法(NSSCM)和MRCT-FEM对考虑随机性后的压缩试验进行模拟,结果表明支反力极值大于0.18 kN且小于0.22 kN(压缩试验结果为0.2 kN)的概率只有0.223。(5)通过速度项和速度梯度项的二阶Taylor展开在MRCT的基础上引入高阶项,对虚动能项、虚动功率项、虚内能项、虚外功率项进行进一步修正,通过虚功率平衡得到高阶MRCT(h-MRCT)控制方程,随后给出了一维情况下MRCT-FEM和h-MRCT-FEM的有限元列式,并以由两种不同材料组成的一维杆的做为算例进行了计算分析。更多还原

【Abstract】 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 μCT 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 μCT 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.更多还原