【作者】 侯亚君；

【导师】 朱兴华；

【作者基本信息】 吉林大学 ， 固体力学， 2004， 博士

【摘要】 如何以细观的力学模型来模拟骨的结构，分析骨的力学性能，不仅对生物力学的理论发展有重要意义，而且有重要的临床应用价值。特别是通过骨的细观力学分析及相应的实验研究，可以掌握在各种承载条件下各部位骨的刚度特性、强度特性和破坏机理，对于防止骨的损伤与骨折、对于骨的移植和再生，都有重要意义。 自上世纪七十年代发展起来的均匀化理论是一套严格的数学理论，一直是应用数学领域的研究课题之一。将其应用于细观力学，建立材料的宏观本构和其微结构的关系，并预测组织的生长和改变，被认为是一条有效的研究途径。它从构成材料的微结构的“胞元”(base cell)入手，假定胞元具有空间可重复性，同时引入宏观尺度和微观尺度，利用渐进分析的方法，从而可以详尽的考虑材料微结构的影响。 Crolet(1990，1993)将均匀化理论用于密质骨的分析中。Hollister et al分析了松质骨的几种胞元模式。Ko et al(1992)和Kohn et al(1992，1993)分别将均匀化理论用于分析螺纹型移植片、骨界面和孔状涂层的移植片、骨界面的力学性质，从而向我们展示了均匀化理论在生物力学的广阔应用前景。 对骨进行均匀化的基本思想是将骨组织看成是由骨的微结构(胞元)在宏观尺度上周期性重复构造而成的，并且在宏观(结构尺度)和微观(胞元尺度)两种尺度上描述骨结构。细观均匀化问题是定义在含单一骨胞元的区域上，宏观均匀化问题是定义在宏观骨区域内，具有等效材料常数的作用有面力和体力的一般线弹性问题。等效材料常数由细观均匀化问题的解确定。 本论文旨在加深和拓广均匀化理论在骨力学中的应用，试图用均匀化理论解决更广泛的问题，比如，松质骨的损伤问题，骨质疏松问题等等，因此可以说此选题具有较为重要的理论意义和应用价值。一~~~~~~~~一墓粼墓咎壑塞 本论文在查阅了大量国内外相关材料，掌握了均匀化理论应用最新进展的基础上，扩展了其在骨力学上的应用，主要做了以下几方面的工作:1.研究了松质骨弹性模量与表观密度之间的关系。 松质骨的两种最基本理想胞元模型是杆一杆结构和带孔板结构。我们把这两种“胞元”模型结合起来，统一为一种模型，即由杆一杆结构逐渐生长为带孔板结构。利用均匀化理论计算松质骨的弹性模量并讨论它与表观密度之间的关系。采用最小二乘拟合，得出弹性模量与表观密度的关系为E二l，78p’‘88(Gva)。 按照松质骨的胞元模型，可以推导出E与p的关系，对于杆一杆结构模型有E一c:p，，对于带孔板结构模型则有E二cZ尸+c3p的形式，我们的结果是对这两种模型用均匀化理论得出的，与理论导出的形式接近，另外M.Kasra和M.n.Grynpas(1995)I，，j根据理想化模型，利用有限元方法得出的结论是E=1.737尸78(Gpa)，此结果与我们的结论也非常接近，综上是否可以说这里的结论更合理些。2.研究了骨小梁的力学性质和松质骨的微结构形式对松质骨力学性质的影响。 详细地讨论了骨小梁的泊松比和弹性模量对松质骨力学性质的影响。指出:(l)骨小梁的泊松比变化时，松质骨的弹性模量变化很小且骨小梁的泊松比对松质骨力学性质的影响与表观密度有关;(2)松质骨的弹性模量与骨小梁的弹性模量近似正比例关系。 假设了松质骨若干种形式的微结构，讨论了松质骨力学性质与其微结构的关系，指出:当表观密度确定的情况下，孔洞为单连通域，微结构的形式可有多种，松质骨的弹性模量E与孔洞的周长有关，孔洞周长越大，弹性模量越小。从讨论的结果中可以看出，松质骨的实际微结构形式往往弹性模量最大，进而验证了骨的功能适应性。3.研究了松质骨的微裂纹对其力学性质的影响。 首先研究了骨小梁中微裂纹长度、密度和方向对骨小梁弹性模量的影响。指出:(l)骨小梁刚度与微裂纹的长度密切相关，当微裂纹的长度变大时，与微裂纹垂直方向上的弹性模量下降很快;(2)骨小梁刚度与微裂纹的密度有关，当微裂纹的密度增大时，三个方向上的弹性模量以相同的比率缓慢下降，对与裂纹垂直方向的弹性模量影响最大:(3)微裂纹的方向对骨小梁刚度有较大的影响;(4)骨小梁刚度与微裂纹的扩展方向也有密切关系。 其次，研究了骨小梁上的微裂纹对松质骨弹性模量的影响，指出:(l)微裂纹对与其垂直方向的松质骨弹性模量有显著影响，而对另外两个与裂纹平行方向的弹性模量影响很小;(2)在裂纹长度较大时，长度的增加使与裂纹垂直方向的弹性模量的下降变大;(3)微裂纹线密度对弹性模量的影响要小于发生微裂纹损伤骨小梁的数量的影响.4.研究了骨质疏松情况下骨的力学性质。 用均匀化理论讨论了发生骨质疏松时，骨量丢失和微结构的改变对松质骨和密质骨刚度的影响。 对于松质骨得出结论:(l)骨量丢失初期弹性模量下降的速度较快，大约是骨量丢失速度的2倍。当骨量丢失30%，弹性模量下降大约50%时以后，弹性模量下降的速度开始减缓(2)骨量丢失的位置也影响松质骨的力学性质，骨量沿骨小梁的表面横向丢失比纵向丢失更易导致松质骨弹性模量的下降。 对于密质骨指出:(1)当骨量丢失时密质骨的弹性模量会下?更多还原

【Abstract】 How to model bone structure and analyse the properties of bone with micromechanical model not only plays a significant role on developing biomechanics theory, but also has important clinical applied value. Specially, with micromechanical analysis to bone and corresponding experimental study ,we can know stiffness and strength of every parts of bone and its destroy mechanism under a various of load. This will paly a significant role on avoiding to damage and fracture bone , and on implantting and reproducing of bone.Homogenization theory ,which has developed since the 1970’s is proper mathematics theory and is one of study task on applied mathematics area all the time.it was thought of as a effective study way to find the relationship between macro-properties of composite and microstructure and to predict growth and change of tissue Homogenization makes it possible to take into account the effect of microstructure to material fully,which begins at base cell of microstructure and uses assumption of periodicity of base cell and the asymptotic expansion,and at the same time ,uses micro-scale and macro-scaleCrolet(1990,1993) applied homogenization to compact bone.Hollister et al analysed several cell models of trabecular bone,Ko et al (1992) and Kohn et al(1992,1993) respectively analysed mechanical properties to screw- shape and porous-shaped implant patch and bone-interfacing with homogenization. All these show us wide future of application of homogenization on biomechanics.The basic idea to apply homogenization to bone is: bone was assumed to be composite with a periodic microstructure and described with micro-scale and macro-scale. Micro homogenization is defined to base cell, macro homogenization is general linear elasticity problem defined to macro bone area with equivalent materialconstant and volume force and face force. Equivalent material constants are gave by solutions of micro homogenization .This paper aimed to develop application of homogenization on bone mechanics and tried to solve more wide problems such as bone damage and osteoporosis etc. this would be important not only on theory but also on application.The context mainly concerns with some aspects as follows:1. Investigation for the dependency of the stiffness of cancellous bone on apparent density based on the combination model of rod-rod structure and perforated plate structureThe structure of cancellous bone is cellular structure. Two basic models of cancellous bone structure include the model of rod-rod structure and the model of framework of perforated plate. This paper presented several models of structure which developed from model of rod-rod structure to model of framework of perforated plate , We computed elastic modulus of cancellous bone based on these modes using homogenization. The relationship between the elastic modulus(E) of cancellous boneand the apparent density( p )was investigated ,it is shown E = 1.78p1.88.2. Investigation for the effect of the elastic modulus of trabeculae poisson’s ratio and microstructure on cancellous boneThe results show that when the form of the microstructure and the proportion of the solid substance in the cell model are fixed,there exits certain relationship between cancellous bone elastic modulus and trabeculae elastic modulus.when the proportion of the solid substance in the cell model is small,the poisson’s ratio of trabeculae hardly affects the cancellous bone elastic modulus, besides that, we discussed the relationship between the form of the microstructure and the cancellous bone elastic modulus,and predicted the effect of hole edge’s length of cell on the cancellous bone elastic modulus3. Investigation for the mechanical proerties of cancellous bone with micro-cracksDuring the process of being loaded by some periodic loads, The cancellous bone will be damaged owing to fatigue. Some micro-cracks may appear on trabecular bone. The appearance of micro-cracks can decline the rigidity of the cancellous bone. In this paper,we used a rod-rod structure更多还原