A. W. Eberhardt Department of Orthopaedic Surgery, University of Minnesota, Minneapolis, UN 55455 and Department of Civil Engineering, Northwestern University, Evanston, IL 60201
L. M. Keer Department of Civil Engineering, Northwestern University, Evanston, IL 60201
J. L. Lewis Department of Orthopaedic Surgery, University of Minnesota, Minneapolis, MN 55455
V. Vithoontien Department of Civil Engineering, Northwestern University, Evanston, IL 60201
An Analytical Model of Joint Contact The stress distribution in the region of contact between a layered elastic sphere and a layered elastic cavity is determined using an analytical model to simulate contact of articulating joints. The purpose is to use the solution to analyze the effects of cartilage thickness and stiffness, bone stiffness and joint curvature on the resulting stress field, and investigate the possibility of cracking of the material due to tensile and shear stresses. Vertical cracking of cartilage as well as horizontal splitting at the cartilage-calcified cartilage interface has been observed in osteoarthritic joints. The current results indicate that for a given system (material properties y. and v constant), the stress distribution is a function of the ratio of contact radius to layer thickness (a/h), and while tensile stresses are seen to occur only when a/h is small, tensile strain is observed for all a/h values. Significant shear stresses are observed at the cartilage-bone interface. Softening of cartilage results in an increase in a/h, and a decrease in maximum normal stress. Cartilage thinning increases a/h and the maximum contact stress, while thickening has the opposite effect. A reduction in the indenting radius reduces a/h and increases the maximum normal stress. Bone softening is seen to have negligible effect on the resulting contact parameters and stress distribution.
Introduction Osteoarthritis is a degenerative disease which involves deterioration of the articular cartilage of synovial joints and alteration of the underlying calcified cartilage and subchondral bone [1]. In diseased joints, vertical splitting and fraying at the cartilage surface and within the cartilage layer has been observed [2], as well as horizontal cracking at the cartilagecalcified cartilage interface [3]. This degeneration process is believed to result from a combination of mechanical loading and biological weakening of the cartilage matrix [4]. To aid in understanding whether mechanical factors are primary or secondary in the evolution of the degeneration and cracking, knowledge of the stress distribution in the cartilage-subchondral bone system is needed. Such information can aid in the interpretation of experimental results, test hypothesis on cartilage breakdown, and indicate the effects of varying system material and geometrical parameters on degenerative processes. The determination of cartilage stresses in joints was first performed by Hirsch [5], who applied the Hertz solution for contact between two elastic spheres. Zarek and Edwards [6] generalized the solution for the case of contact between a rigid sphere and an elastic half-space to analyze the structure-func-. tion relationship of collagen. Sokoloff [7] modeled cartilage as an incompressible elastic medium to determine the "instantaneous Young's modulus." Hayes et al. [8] developed the solution for rigid spherical and flat-ended indentation of an Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division August 2, 1989; revised manuscript received June 15, 1990.
elastic layer on a rigid half-space, as a model of cartilage indentation. Hori and Mockros [9] used this analysis with indentation experiments and confined compression tests to determine the "short-time" shear modulus (y.) and Poisson's ratio (p) for healthy and degenerated cartilage. Using measurements taken at one second of loading, they obtained a range of values of 0.46 M P a < / i < 3 . 4 7 MPa, and 0.42
L. M. Keer Department of Civil Engineering, Northwestern University, Evanston, IL 60201
J. L. Lewis Department of Orthopaedic Surgery, University of Minnesota, Minneapolis, MN 55455
V. Vithoontien Department of Civil Engineering, Northwestern University, Evanston, IL 60201
An Analytical Model of Joint Contact The stress distribution in the region of contact between a layered elastic sphere and a layered elastic cavity is determined using an analytical model to simulate contact of articulating joints. The purpose is to use the solution to analyze the effects of cartilage thickness and stiffness, bone stiffness and joint curvature on the resulting stress field, and investigate the possibility of cracking of the material due to tensile and shear stresses. Vertical cracking of cartilage as well as horizontal splitting at the cartilage-calcified cartilage interface has been observed in osteoarthritic joints. The current results indicate that for a given system (material properties y. and v constant), the stress distribution is a function of the ratio of contact radius to layer thickness (a/h), and while tensile stresses are seen to occur only when a/h is small, tensile strain is observed for all a/h values. Significant shear stresses are observed at the cartilage-bone interface. Softening of cartilage results in an increase in a/h, and a decrease in maximum normal stress. Cartilage thinning increases a/h and the maximum contact stress, while thickening has the opposite effect. A reduction in the indenting radius reduces a/h and increases the maximum normal stress. Bone softening is seen to have negligible effect on the resulting contact parameters and stress distribution.
Introduction Osteoarthritis is a degenerative disease which involves deterioration of the articular cartilage of synovial joints and alteration of the underlying calcified cartilage and subchondral bone [1]. In diseased joints, vertical splitting and fraying at the cartilage surface and within the cartilage layer has been observed [2], as well as horizontal cracking at the cartilagecalcified cartilage interface [3]. This degeneration process is believed to result from a combination of mechanical loading and biological weakening of the cartilage matrix [4]. To aid in understanding whether mechanical factors are primary or secondary in the evolution of the degeneration and cracking, knowledge of the stress distribution in the cartilage-subchondral bone system is needed. Such information can aid in the interpretation of experimental results, test hypothesis on cartilage breakdown, and indicate the effects of varying system material and geometrical parameters on degenerative processes. The determination of cartilage stresses in joints was first performed by Hirsch [5], who applied the Hertz solution for contact between two elastic spheres. Zarek and Edwards [6] generalized the solution for the case of contact between a rigid sphere and an elastic half-space to analyze the structure-func-. tion relationship of collagen. Sokoloff [7] modeled cartilage as an incompressible elastic medium to determine the "instantaneous Young's modulus." Hayes et al. [8] developed the solution for rigid spherical and flat-ended indentation of an Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division August 2, 1989; revised manuscript received June 15, 1990.
elastic layer on a rigid half-space, as a model of cartilage indentation. Hori and Mockros [9] used this analysis with indentation experiments and confined compression tests to determine the "short-time" shear modulus (y.) and Poisson's ratio (p) for healthy and degenerated cartilage. Using measurements taken at one second of loading, they obtained a range of values of 0.46 M P a < / i < 3 . 4 7 MPa, and 0.42
