Abstract—- The purposes of this article were to obtain mechanical properties of the dry femur cortical bonesamples through a tensile load and stress concentration factor approach and to provide simulations topredict experimental behaviors based on manipulations of certain properties and parameters of thebiomaterial. Since bone samples have characteristics and geometries, the development of a mathematicalmodel was necessary to describe the combination of stresses interacting in the bone when a tension load isapplied. The samples have average diameters and lengths of 0.5 and 2 inches respectively and were testedusing a 10 kN Universal Tensile Machine to determine mechanical properties such as yield and ultimate stress,young module, fracture, among others. Several simulations were conducted to evaluate failure criteria like“Von Mises”, “Tresca” and “Tsai-Wu”. Finally, was concluded that 83% of the data obtained from the 22 samplesobserved in the “Stress-Strain” charts showed a directly proportional relationship. Keywords: mechanical properties, stress-strain curve, stress concentration factor, failure criteria simulationSimulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical BonesISSN-E: 2697-3650Velez. Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones Alex J. Velez-Cruzhttps://orcid.org/0000-0002-9289-5256alvelez@pupr.edu Polytechnic University of Puerto Rico San Juan, Puerto RicoResumen—Los propósitos de este artículo fueron obtener propiedades mecánicas de muestras secas enhueso fémur a través de fuerzas en tensión y del factor de concentración de esfuerzos y proveersimulaciones para predecir comportamientos experimentales basados en manipulaciones de ciertosparámetros y propiedades. Dado que las muestras tienen geometrías características, fue necesariodesarrollar un modelo matemático para describir las combinaciones de esfuerzos que interaccionaban en elhueso cuando se aplica una carga de tensión. Las muestras tienen diámetros promedios y longitudes de 0.5y 2 pulgadas respectivamente y fueron evaluadas utilizando una Maquina de Tensión Universal paradeterminar propiedades mecánicas como esfuerzos ultimo y de fluencia, módulo de elasticidad, entre otras.Varias simulaciones fueron ejecutadas para evaluar criterio de fallas tales como “Esfuerzo Von Mises”,“Tresca” y “Tsai-Wu”. Finalmente, se concluyó que 83% de los datos obtenidos de 22 muestras observadas engráficas “Esfuerzo-Desplazamiento” mostraron una relación directamente proporcional.Palabras clave: propiedades mecánicas, curva de esfuerzo y desplazamiento, factor de concentración deesfuerzo, simulación de criterio de fallaSimulación de los Efectos de Combinaciones de Esfuerzo y Factor de Concentración de Esfuerzo en Hueso Femoral Cortical8Recibido(16/04/2022), Aceptado(15/05/2022)Minerva JournalVol.3, Issue. 8, (pp. 8-19)https://doi.org/10.47460/minerva.v3i8.60
I.Introduction The characterization of biomaterials is necessary to determine the mechanical, chemical, and electrical, amongother interesting properties of the material [1]. The mechanical property is obtained from a mechanicaldestructive test, called the tensile test. Basically, it is when a pulling force or tension is applied to materialsuntil it fails or breaks, providing information about the Yield Strength , Ultimate Tensile Strength ,Ductility (D), Young's modulus (E), and Poisson's ratio (ν) of the material [2] [3] [4]. Bone is composed of three different types of bones, cortical, trabecular (cancellous), and marrow bones.Cortical bone is dense and solid and surrounds the marrow space, whereas trabecular bone is composed of ahoneycomb-like network of trabecular plates and rods interspersed in the bone marrow compartment [5].Those bones are separated into two main elements, the cellular component, and an extracellular matrix. Thematrix, which is responsible for the mechanical strength of the bone tissue, is formed by an organic and amineral phase, but a liquid component is also present [6]. By weight, bone contains approximately 60%mineral, 10% water, and about 30% collagenous matrix. The mineral component influences the stiffness of thebone, whereas the collagen network contributes significantly to its fracture properties. Typically, engineers consider three basic tasks when biomaterials are being evaluated. The first, understandingthe properties of the materials (strength, fatigue, among others); second, the analysis of the response of thestudy material when is subject to external loads (Free Body Diagrams) and third; the determination of theweakest areas of the material (stress concentration factors) [7]. The intended research is oriented to obtainthe stress-strain relation of dry canine cadaveric cortical bone samples using the stress concentration factor rrrrr analysis.The Kscf is the ratio of the highest stress (σmax) to reference stress (σref) of the gross cross-section. Thisexperimental factor shall be considered as part of the engineering analysis on the stress-strain curve since themechanical properties of the biomaterial can be affected directly. During this research will be seen acombination of the stresses interacting on bone samples when they are subject to axial loads. Based on thedestructive tensile test and the stress concentration factor approach, it is expected to see normal, bendingand shear stresses influencing the behavior of the stress-strain curve.Therefore, the target of this research will be focused to perform a simulation through a Computer Aid Drafting(CAD) tool (CREO Parametric) with the intent to use a failure criterion (Von Miss Stress, Tresca, Tsai-Wu, etc.) todetermine their critical values before a fracture occurs and to compare those values among them. Also, willserve to model the interaction of the principal stresses and the effect that those stresses have on thebehavior of the stress-strain curve. The new stress-strain curve obtained from the simulation will be comparedagainst the experimental curve obtained from the tensile tests. Finally, error calculations are documented toanalyze, describe, and predict the accuracy and precision of the proposed model as well as how well isbehaving.9Minerva JournalISSN-E: 2697-3650Velez. Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones Vol.3, Issue. 8, (pp. 8-19)
II. DEVELOPMENTA. Mechanical Properties of the BoneThe diverse forms and geometries of cortical and trabecular bones result in different mechanical properties.The mechanical properties of the bone vary according to species, size, age, among other characteristics andparameters. The mineral content in a bone shows little changes with increasing age, and this behavior isobserved in its stiffness. In contrast, the energy absorbed (toughness) during the fracture of bone decreasessignificantly with increasing age, which contributes to an inverse proportional relationship. The mineral phasemost likely imparts stiffness to the bone, whereas the collagen network contributes significantly to its fractureproperties [6]. Cortical bone is an anisotropic material, meaning that its mechanical properties vary according to the directionof load. The strength and tensile/compressive moduli of cortical bone along the longitudinal direction aregreater than those along the radial and circumferential directions. Nowadays, minor fluctuations in mechanicalproperties have been observed in the radial versus circumferential direction, recommending that cortical bonecan be considered as a transversely isotropic material. When samples receive tension along the longitudinaldirection, cortical bone shows a bilinear stress-strain response in which a distinct yield point separates alinearly elastic region and a region of linear hardening that ends abruptly at a fracture strain of less than 3.Cortical bone specimens loaded in the transverse direction fail in a more brittle manner compared with thoseloaded in the longitudinal direction [8]. B. Stress-Strain CurveThe following diagram has the intent to provide detailed background information regarding the behavior of thefemur bone material when is subject to tension loads.10Minerva JournalISSN-E: 2697-3650Fig. 1. Stress-Strain Diagram [9] In Fig. 1 above, point A represents the proportional limit, which the slope of this line is better known as theYoung’s Modulus. For segment AB, the material may still be elastic in the sense that the deformations arecompletely recovered when the load is removed, and this point B is called the elastic limit or yield point. PointA and segment AB are part of the elastic region, which is governed by Hook’s law. Beyond point B, Velez. Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones Vol.3, Issue. 8, (pp. 8-19)
C. Principal Stresses The engineering measures of stress (σ) and strain (ε) are determined from the measured load (P) anddeflection (δ) using the original specimen cross-sectional area and length as:11Minerva JournalISSN-E: 2697-3650When the stress (σ) is plotted against the strain (ε), an engineering stress-strain curve such as that shown inFig. 1 is obtained [10]. In the early phase of the stress-strain curve, various materials obey Hooke’s law to areasonable approximation, so that stress is proportional to strain with the constant of proportionality beingthe modulus of elasticity or Young’s modulus (E) [11]:Since bones vary in geometry, a representation of the mathematical model is needed to explain the physicalphenomena occurring during the experimental tests. Therefore, the following equation related to thecombination of principal stresses (σt) (normal, bending, and shear) will be briefly discussed and furtherimplemented in the proposed cortical bone model. The kscf, σn, σb and τ are provided as follows:where stress concentration factor (kscf) is the ratio of the highest stress (σmax) to reference stress (σref) ofthe gross cross-section: where F is the applied normal force and A is the cross-sectional area of the specimen:where F is the applied normal force, r is radio, and I is the inertia moment A is the cross-sectional area of thespecimen:and assuming a cylindrical hollow element, τ, can be approximated as described in (7), where V is the shearstress value and A is the cross-sectional area of the specimen. Thus, substituting (4), (5), (6) and (7) into (3), thefinal combined stresses equation is represented as follows: Velez. Simulation of Combined Stresses and Stress Concentration Factor Effects on a Femur Cortical Bones Vol.3, Issue. 8, (pp. 8-19)
D. Principal StrainsOn the other hand, principal strains (maximum and minimum normal strains) shall be considered as part ofthe cortical bone behavior, which is obtained from differentiating axial , and lateral with respect to θ.Then, the general equation for the total principal strains present in the experiment