Presentation: Contact Mechanics of Impacting Slender Rods: Measurement and Analysis

Sanders, A., I. Tibbitts, D. Kakarla, S. Siskey, J. Ochoa, K. Ong, and R. Brannon. (2011). “Contact mechanics of impacting slender rods: measurement and analysis.” Society for Experimental Mechanics Annual Meeting. Uncasville, CT, June 13-16.


Images of a typical contact patch

To validate models of contact mechanics in low speed structural impact, slender rods with curved tips were impacted in a drop tower, and measurements of the contact and vibration were compared to analytical and finite element (FE) models. The contact area was recorded using a thin-film transfer technique, and the contact duration was measured using electrical continuity. Strain gages recorded the vibratory strain in one rod, and a laser Doppler vibrometer measured velocity. The experiment was modeled analytically using a quasi-static Hertzian contact law and a system of delay differential equations. The FE model used axisymmetric elements, a penalty contact  algorithm, and explicit time integration. A small submodel taken from the initial global model economically refined the analysis in the small contact region. Measured contact areas were within 6% of both models’ predictions, peak speeds within 2%, cyclic strains within 12 microstrain (RMS value), and contact durations within 2 µs. The accuracy of the predictions for this simple test, as well as the versatility of the diagnostic tools, validates the theoretical and computational models, corroborates instrument calibration, and establishes confidence thatthe same methods may be used in an experimental and computational study of the impact mechanics of artificial hip joint.

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Global model results comparison with analytical and experimental results for speed at the midpoint of one of the rods

Publication: Determining a Surrogate Contact Pair in a Hertzian Contact Problem

Sanders, A. P. and R. M. Brannon (2011). “Determining a Surrogate Contact Pair in a Hertzian Contact Problem.” Journal of Tribology 133(2): 024502-024506.

Hertzian substitution concept: An arbitrary contact pair (a) with given principal curvatures and orientation, is substituted with a simpler contact pair (b) consisting of a spheroid and a plane

Laboratory testing of contact phenomena can be prohibitively expensive if the interacting bodies are geometrically complicated. This work demonstrates means to mitigate such problems by exploiting the established observation that two geometrically dissimilar contact pairs may exhibit the same contact mechanics. Specific formulas are derived that allow a complicated Hertzian contact pair to be replaced with an inexpensively manufactured and more easily fixtured surrogate pair, consisting of a plane and a spheroid, which has the same (to second-order accuracy) contact area and pressure distribution as the original complicated geometry. This observation is elucidated by using direct tensor notation to review a key assertion in Hertzian theory; namely, geometrically complicated contacting surfaces can be described to second-order accuracy as contacting ellipsoids. The surrogate spheroid geometry is found via spectral decomposition of the original pair’s combined Hessian tensor. Some numerical examples using free-form surfaces illustrate the theory, and a laboratory test validates the theory under a common scenario of normally compressed convex surfaces. This theory for a Hertzian contact substitution may be useful in simplifying the contact, wear, or impact testing of complicated components or of their constituent materials.

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Publication: A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations

A. Sadeghirad, R. M. Brannon, and J. Burghardt

Three snapshots of the model with 248 particles in simulation of the radial expansion of a ring problem using: (a) CPDI method and (b) cpGIMP

A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram-shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4-node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems.

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Accelerated hip implant wear testing

3D model Experimental Setup

Rim cracking of polyethylene acetabular liners and squeaking in ceramic components are two important potential failure modes of hip implants, but the loads and stresses that cause such failures are not well understood. Contact stresses in hip implants are analyzed under worst case load conditions to develop new wear testing methods to improve the pre-clinical evaluation of next-generation hip implants and their materials. Complicated full-scale hip implant simulator tests are expensive and take months to complete. A primary goal of this work is to find inexpensive surrogate specimen shapes and loading modes that can, in inexpensive lab tests taking only a few hours, produce the same wear patterns as seen in full-scale prototype testing. Continue reading

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

Sanders, A. P. and R. M. Brannon (2011). “Assessment of the applicability of the Hertzian contact theory to edge-loaded prosthetic hip bearings.” Journal of Biomechanics 44(16): 2802-2808.


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 >400N. 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.

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Reciprocating edge-load wear test of artificial hip bearings

Movie: edge load wear test of hip prostheses

This short movie shows a novel wear test for artificial hip bearings. This test focuses on what may be the most severely damaging scenario that artificial hips experience in the human body. In some patients, the hip joint may partly separate under certain circumstances, and when that happens, the typically congruent contact between the ball and socket can quickly become a non-conforming contact between the ball and the socket’s edge. In that scenario, the contact between the ball and socket has the potential to cause increased wear, because the contact stress is much greater.

hip wear test set-up

The test apparatus applies a horizontal load to the femoral head (the ball), which forces the head into contact with the socket’s edge. Meanwhile, the acetabular liner is reciprocated up and down at a rate of 1 cycle per second. Several aspects of the test set-up may be configured as desired, including the magnitude of the horizontal force, the orientation of the acetabular liner, and the speed of reciprocation.

This test method is yielding valuable information about the performance of artificial hip bearings under worst case conditions. The findings of several focused studies will be published in the scientific literature relevant to the field of orthopedic implants and joint arthroplasty.

Animation showing set-up of femoral head reduction test

Animation: Biomechanical reduction test set-up

This short animation illustrates the set up of a femoral head reduction test. The items shown – a femur, a femoral stem with a femoral head, and an acetabular liner – will be tested in a manner that simulates the instantaneous reduction that occurs to a subluxed femoral head upon heel strike during human gait.

Continue reading

Conference Poster: Hertzian contact theory applied to edge-loaded ceramic-on-ceramic hip bearings: analysis and validation.

Sanders, A. P., and R. M. Brannon. (2010) “Hertzian contact theory applied to edge-loaded ceramic-on-ceramic hip bearings: analysis and validation.” Transaction of the 56th Annual Meeting of the Orthopaedic Research Society, March 6-9, New Orleans, LA, Poster 2258.


This work addresses the problem of wear of ceramic-on-ceramic bearing couples used in total hip arthroplasty. A wear pattern called stripe wear has been observed on retrieved ceramic femoral heads. It typically appears as a long, narrow, roughened area on affected implants. Evaluation of the wear stripe has revealed grain removal and material loss to depths of 30 mm. It has been theorized that such damage is caused by contact between the femoral head and the edge of the acetabular cup. An edge-loaded contact is much less congruent than normal contact, and it would cause increased contact stress, leading to the wear stripe. Edge-loading may be due to small separations of the head from the cup, as observed radiographically. Simulator wear tests involving such separation have yielded wear stripes similar to those on retrieved bearings.