A Damage Model to Trabecular Bone and Similar Materials: Residual Resource, Effective Elasticity Modulus, and Effective Stress under Uniaxial Compression

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.

Damage Function of a Quasi-Brittle Material, Damage Rate, Acceleration and Jerk during Uniaxial Compression: Model and Application to Analysis of Trabecular Bone Tissue Destruction

“…Remark 1: Note that Equation ( 12) was justified without postulating the law of damage evolution at the development start. Nevertheless, using the logic of fracture mechanics, a model of damage accumulation of quasi-brittle materials under uniaxial compression was substantiated, in which the final Equation ( 12) corresponds to the Weibull distribution law, which is confirmed by the equivalence of Equations ( 5) [22] and ( 12) [8] under condition (13).…”

“…Using a different approach [8], it can be shown that for uniaxial compression, an analog of equation ( 5) can be obtained based on the assumption that the change in the effective area (d A) of the sample with the initial height H 0 and the initial cross-sectional area A 0 is proportional to the change in the elastic potential energy if the strain increases from ε to (ε + dε). In this case, the height of the sample varies from εH 0 to (ε + dε)H 0 .…”

“…The experimental data (Figure 4) can be transformed into the characteristics of the internal process, which include the damage of the sample material D (12) and the dependence of the apparent (σ) and effective ( σ) stresses on strain ε [8]: The function = ( ) (Figure 4) can be represented analytical…”

“…Within the framework of the abovementioned problem, the main issue in the presented research was to determine the speed, acceleration and jerk of the process of damage to a quasi-brittle material on the example of trabecular bone tissue under uniaxial compression. An analytical solution to this problem is proposed, which is obtained by standard methods of mathematical modeling of mechanical systems and is based on the use of the damage function justified in [8]. In addition to [8], a more detailed analysis of the damage function was carried out, which allowed us to obtain new analytical relations for determining the rate and other characteristics of the damage process of a quasi-brittle material.…”

“…An analytical solution to this problem is proposed, which is obtained by standard methods of mathematical modeling of mechanical systems and is based on the use of the damage function justified in [8]. In addition to [8], a more detailed analysis of the damage function was carried out, which allowed us to obtain new analytical relations for determining the rate and other characteristics of the damage process of a quasi-brittle material. A method of applying the obtained results to predict damage to the trabecular bone tissue during uniaxial compression is proposed.…”