Homogenization Applications In Bone Mechanics

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...