Abstract
Experimental research of bone strength remains costly and limited for ethical and technical reasons. Therefore, to predict the mechanical state of bone tissue, as well as similar materials, it is desirable to use computer technology and mathematical modeling. Yet, bone tissue as a bio-mechanical object with a hierarchical structure is difficult to analyze for strength and rigidity; therefore, empirical models are often used, the disadvantage of which is their limited application scope. The use of new analytical solutions overcomes the limitations of empirical models and significantly improves the way engineering problems are solved. Aim of the paper: the development of analytical solutions for computer models of the mechanical state of bone and similar materials. Object of research: a model of trabecular bone tissue as a quasi-brittle material under uniaxial compression (or tension). The new ideas of the fracture mechanics, as well as the methods of mathematical modeling and the biomechanics of bone tissues were used in the work. Compression and tension are considered as asymmetric mechanical states of the material. Results: a new nonlinear function that simulates both tension and compression is justified, analytical solutions for determining the effective and apparent elastic modulus are developed, the residual resource function and the damage function are justified, and the dependences of the initial and effective stresses on strain are obtained. Using the energy criterion, it is proven that the effective stress continuously increases both before and after the extremum point on the load-displacement plot. It is noted that the destruction of bone material is more likely at the inflection point of the load-displacement curve. The model adequacy is explained by the use of the energy criterion of material degradation. The results are consistent with the experimental data available in the literature.
Highlights
This article discusses the issues of biomechanical modeling of trabecular bone tissue using new ideas of fracture mechanics
It is taken into account that the structure of bone tissue in the process of its formation and renewal naturally and optimally adapts to biomechanical functions [1,2]
A sufficiently large mechanical impact on the bone is accompanied by the appearance of micro-cracks [10], the initial length of which usually corresponds to the characteristic size of the structural element of the bone [8]
Summary
This article discusses the issues of biomechanical modeling of trabecular bone tissue using new ideas of fracture mechanics. From the point of view of mechanics, at the macro level, bone tissue is an inhomogeneous anisotropic porous material. The characteristic sizes of bone structural units are in the range from 1 nanometer to 500 micrometers [8]. Stresses and deformations in the material of structural units increase, which can lead to bone fractures. A number of experimental methods for determining the ultimate stress are known: tests of bone tissue samples for compression, tension, three- or four-point bending, shear, and torsion [6]. From a methodological point of view, the phenomenological approach is used to construct a mathematical model of the behavior of bone tissue in the process of testing samples for compression and tension. To verify the results of the mathematical modeling, experimental data known from the literature were used
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