Assessment of the applicability of the Hertzian contact theory to edge-loaded prosthetic hip bearings

Abstract

The components of prosthetic hip bearings may experience in-vivo subluxation and edge loading on the acetabular socket as a result of joint laxity, causing abnormally high, damaging contact stresses. In this research, edge-loaded contact of prosthetic hips is examined analytically and experimentally in the most commonly used categories of material pairs. In edge-loaded ceramic-on-ceramic hips, the Hertzian contact theory yields accurate (conservatively, <10% error) predictions of the contact dimensions. Moreover, the Hertzian theory successfully captures slope and curvature trends in the dependence of contact patch geometry on the applied load. In an edge-loaded ceramic-on-metal pair, a similar degree of accuracy is observed in the contact patch length; however, the contact width is less accurately predicted due to the onset of subsurface plasticity, which is predicted for loads >400 N. The Hertzian contact theory is shown to be ill-suited to edge-loaded ceramic-on-polyethylene pairs due to polyethylene's nonlinear material behavior. This work elucidates the methods and the accuracy of applying classical contact theory to edge-loaded hip bearings. The results help to define the applicability of the Hertzian theory to the design of new components and materials to better resist severe edge loading contact stresses.

Introduction

The femoral head and acetabular liner of a prosthetic hip joint may not always function as the ideal ball-and-socket joint they are designed to be, which may be a root cause for clinically observed wear modes. Hip replacement surgery may alter the structural dimensions and the tissue constraints of a hip joint such that the ball and socket are held together more loosely than planned for in design. Thus, when examining the potential wear modes of prosthetic hips, it is important to consider that the joint has the potential to be damaged from adverse behaviors such as disassociation and eccentric contact.

Previous investigations have described various abnormal behaviors in prosthetic hip joints. Dislocation, with the ball (or head) fully exiting the socket, has been widely reported and is the extreme example (Lewinnek et al., 1978). Fluoroscopic studies have revealed smaller ball-socket separations (subluxation) during various hip motions (Lombardi et al., 2000, Dennis et al., 2001, Komistek et al., 2002). Such subluxation has been called microseparation (Nevelos et al., 2000) and micro-lateralization (Sariali et al., 2010). Implant malalignment and small femoral heads may contribute to femoral neck impingement on the liner's rim (Nadzadi et al., 2002, Crowninshield et al., 2004). Impingement may lever the head out of the socket, causing subluxation where the head bears upon the socket's edge (edge loading) (Scifert et al., 2001, Kluess et al., 2007). A relatively vertical cup orientation may cause edge loading (Mellon et al., 2010), and a recent clinical study found 34% of 1884 acetabular cups to be abducted above the ideal 45° maximum (Callanan et al., 2010). Thus, systematic laboratory investigations are needed to quantify the wear characteristics or other consequences of these ways that a prosthetic hip might deviate from ideal behavior.

Dislocation can severely scratch the femoral head (Bourne et al., 2005). Microseparation can cause ceramic component stripe wear (Walter et al., 2004, Yamamoto et al., 2005), and in laboratory tests, stripe wear has contributed to squeaking in ceramic hips (Taylor et al., 2007). Finite element analyses (FEA) of edge loading contact stresses from microseparation (Mak et al., 2002) and impingement (Kluess et al., 2007) have revealed markedly elevated contact stresses. Hip simulator wear tests imparting microseparation and edge loading have elicited clinically relevant stripe wear on ceramic prostheses (Nevelos et al., 2000, Manaka et al., 2004). Each of these effects is a potential failure mode in the sense that the bearings accrue potentially harmful damage.

In ideal concentric loading, the ball and socket are in conforming contact, meaning that their contacting surface radii are closely matched (e.g. <100 μm difference); thus, loads produce large contact areas and low contact stresses. On the contrary, adverse loading (e.g. edge loading) induces high contact stress because the ball and socket come into non-conforming contact, meaning that the contacting surfaces have radii that differ greatly. Under such conditions, loads generate smaller contact areas and higher contact stresses.

Although the Hertzian contact theory has been applied in the case of concentric ceramic-on-ceramic hip contact (Mak and Jin, 2002), its merits and shortcomings have not been assessed in abnormal states and multiple material couples. For analyses of adverse loading, the Hertzian theory may be useful and applicable provided that the following key assumptions of the theory are approximately satisfied: (1) the materials are homogeneous, linear elastic, and isotropic; (2) the surfaces are perfectly smooth and frictionless; (3) the surfaces are non-conforming; and (4) the contact dimensions are much smaller than the surface radii at the contact point (Hertz, 1882, Johnson, 1985). Considering these in the case of edge-loaded ceramic and metal hip prostheses, the following three preliminary observations are made:

• Items 1 and 2 are as closely approximated as they are in the case of concentric contact.

• For Item 3, the surfaces are less conforming than in the case of concentric contact.

• Thus, Item 4 is more likely to be valid than in the case of concentric contact.

Based on these observations, this study examines the hypothesis that the contact dimensions of edge-loaded ceramic and metal hip bearings can be accurately estimated using the Hertzian theory. For contrast, the study also examines the hypothesis that a plastic bearing material with nonlinear constitutive behavior will not be accurately modeled by the theory.