Recasting theory of elasticity with micro-finite elements

In the classical theory of elasticity,a body is initially modeled as a homogeneous and dense assemblage of constituent ＂material particles＂.The kernel concept of elastic deformation is the displacement of the particle that associates the current configuration with the reference one.In this paper,we exploit an alternative constituent ＂micro-finite element＂,and use the stretch of the element as the essential quality to recast the theory of elasticity.It should be realized that such a treatment means that the elastic body can be modeled as a finite covering of elements and consequently characterized by a manifold.The recasting of the elasticity theory becomes more feasible for dealing with defects and topological evolution.

Evaluation of damage to trabecular bone of the osteoporotic human acetabulum at small strains using nonlinear micro-finite element analyses

Background With advance of age, alterations in bone quality, quantity and microarchitecture render osteoporotic trabecular bone become more sensitive to local failure. The aims of the present study were to clarify the extent to which the distribution of tissue-level stresses and strains was affected by structural changes and the extent to which osteoporotic acetabular trabecular bone was damaged at small strains. Methods Using a DAWING 4000A supercomputer, nonlinear micro-finite element （μFE） analyses were performed to calculate the tissue-level strains and stresses for each element in the trabecular bone of one osteoporotic acetabulum at small strains to quantify the tissue-level damage accumulation and mechanical properties. Results In contour plots of the tissue, maximum principal logarithmic strains, high tissue-level strains, both compressive and tensile, were observed in the osteoporotic trabecular bone at small apparent strains from 0.2% to 0.5% strain. The compressive apparent stress-strain curve showed typical nonlinear behavior and tangent modulus reduction with increasing strains. The microdamage curve suggested that microdamage began at 0.2% apparent strain in the osteoporotic trabecular bone and increased sharply, although very few microfractures occurred. The quartiles of the maximum principal logarithmic strains, minimum principal logarithmic strains and Von Mises stresses increased nonlinearly. For the inter-quartile range of the Von Mises stresses, a leap occurred at small strains ranging from 0.2% to 0.3% while microdamage commenced. Conclusions Extensive microdamage was primarily responsible for the large loss in apparent mechanical properties that occurred in the trabecular bone of the osteoporotic acetabulum at small strains. With increasing apparent strains, continuous nonlinear increments of tissue-level strains and stresses resulted in microdamage that propagated throughout the specimen with very few microfractures. Chin Med J 2009; 122（17）：2041-204 7

Age- and direction-related adaptations of lumbar vertebral trabecular bone with respect to apparent stiffness and tissue level stress distribution

The objective of this study was to study the age-related adaptation of lumbar vertebral trabecular bone at the apparent level, as well as the tissue level in three orthogonal directions. Ninety trabecular specimens were obtained from six normal L4 vertebral bodies of six male cadavers in two age groups, three aged 62 years and three aged 69 years, and were scanned using a high-resolution micro-computed tomography （micro-CT） system, then converted to micro- finite element models to do micro-finite element analyses. The relationship between apparent stiffness and bone volume fraction, and the tissue level yon Mises stress distribution for each trabecular specimen when compressed separately in the longitudinal direction, medial-lateral and anterior-posterior directions （transverse directions） were derived and compared between two age groups. The results showed that at the apparent level, trabecular bones from 69-year group had stiffer bone structure relative to their volume fractions in all three directions, and in both age groups, changes in bone volume fraction could explain more variations in apparent stiffness in the longitudinal direction than the transverse directions; at the tissue level, aging had little effect on the tissue von Mises stress distributions for the compressions in all the three directions. The novelty of the present study was that it provided quantitative assessments on the age and direction- related adaptation of Chinese male lumbar vertebral trabecular bone from two different levels： stiffness at the apparent level and stress distribution at the tissue level. It may help to understand the failure mechanisms and fracture risks of vertebral body associated with aging and direction for the prevention of fracture risks in elder individuals.

Tissue level microstructure and mechanical properties of the femoral head in the proximal femur of fracture patients

This study aims to investigate the regional variations of trabecular morphological parameters and mechanical parameters of the femoral head,as well as to determine the relationship between trabecular morphological and mechanical parameters.Seven femoral heads from patients with fractured proximal femur were scanned using a micro-CT system.Each femoral head was divided into 12 sub-regions according to the trabecular orientation.One 125 mm^3 trabecular cubic model was reconstructed from each sub-region.A total of 81 trabecular models were reconstructed,except three destroyed sub-regions from two femoral heads during the surgery.Trabecular morphological parameters,i.e.trabecular separation（Tb.Sp）,trabecular thickness（Tb.Th）,specific bone surface（BS/B V）,bone volume fraction（BV/TV）,structural model index（SMI）,and degree of anisotropy（DA） were measured.Micro-finite element analyses were performed for each cube to obtain the apparent Young＇s modulus and tissue level von Mises stress distribution under 1%compressive strain along three orthogonal directions,respectively.Results revealed significant regional variations in the morphological parameters（P〈0.05）.Young＇s moduli along the trabecular orientation were significantly higher than those along the other two directions.In general,trabecular mechanical properties in the medial region were lower than those in the lateral region.Trabecular mechanical parameters along the trabecular orientation were significantly correlated with BS/BV,BV/TV,Tb.Th,and DA.In this study,regional variations of microstructural features and mechanical properties in the femoral head of patients with proximal femur fracture were thoroughly investigated at the tissue level.The results of this study will help to elucidate the mechanism of femoral head fracture for reducing fracture risk and developing treatment strategies for the elderly.