E.F. Rybicki

On the mathematical analysis of stress in the human femur

Abstract A mathematical model for the behavior of the human femur is used to examine the effect of muscle forces on the resulting stresses and strain energy during a one-legged stance. The results show that tension in the tensor faciae latae can very effectively serve to counteract bending stresses in the femur while also reducing the strain energy. It was found that the choice of mathematical model to represent the behavior of the femur is important. Stresses in the femur are calculated using both beam theory and a continuum theory in the form of a finite element computer program. The analyses include joint and muscle loadings. The results of the study show that beam theory is appropriate for the calculation of stresses in the shaft of the femur, but gives an inaccurate picture of the stress distribution in the regions of the greater trochanter and the femoral head as well as the areas of muscle attachment. In these regions a continuum theory is required.

An Energy Release Rate Approach For Stable Crack Growth in the Free-Edge Delamination Problem

An experimental and analytical study was conducted to examine the free-edge delamination mode of failure in a (±30/±30/90/90) S Boron/ Epoxy laminate. The initiation and stable growth of the delamination for static tension loading conditions were identified using ultrasonic tech niques. A finite element representation for the test specimen was con structed using conventional elements. The interlaminar stress distribution at the laminate midplane was found to be in agreement with published results. The finite element representation was used to model the initiation and stable growth of the delamination. An applied force versus longitudinal strain curve was obtained from the model based on the data for the longi tudinal strain versus the extent of the delamination. The applied force versus longitudinal strain curve obtained from the model was between the two sets of data obtained from the specimens. The model also predicted the strain level at which the laminate showed a marked decrease in stiff ness. Energy release rates were evaluated by a simple computational scheme that does not require a special singular element or knowledge of the existence of a stress singularity in the solution. The results of these computations are combined with the experimental data and discussed in terms of the initiation and stable crack growth characteristics of the delamination.

Approximate Three-Dimensional Solutions for Symmetric Laminates Under Inplane Loading *

A three-dimensional equilibrium finite element analysis is used to obtain approximate stress solutions for some finite symmetric laminates under inplane loading. The analysis is based on a comple mentary energy formulation. Obtained stress solutions at the center section of the laminate are compared with solutions for uniform axial strain that are available in the literature. A comparison of the stress distributions obtained at the center and end sections of the laminate shows that a large τyz stress can occur at the end section and a large τxz can occur at the center section. The influence of laminate stacking sequence on the predicted σz is also presented.

In vivo and analytical studies of forces and moments in equine long bones

A study to evaluate the in vivo forces acting on equine long bones is described. Results for the maximum axial forces and bending moments are presented for the radius, metacarpus, tibia, and metatarsus. The forces and bending moments are obtained by combining in vivo strain gage readings and a mathematical model for the mechanics of the bones. Comparisons of equine activities including standing, walking, trotting, and getting up after the anesthetic indicate that one activity does not always produce the highest loads in all of the bones examined. However, for the cases considered here, recovery from the anesthetic produced the largest compressive forces in the metacarpus and the tibia. Also, large tensile forces were found during the foot-up positions during standing and trotting.

Effect of Stacking Sequence and Lay-Up Angle on Free Edge Stresses Around a Hole in a Laminated Plate Under Tension

One mode of failure in fiber reinforced laminated plates is delamination initiating at a free edge. Experimental results and theoretical stress investi gations found in the literature indicate that the interlaminar normal stress, σ z , is a factor in this mode of failure. In this paper, the stress behavior around a circular hole in a laminated plate is investigated using a three- dimensional finite element stress analysis. First, computed results for the tangential strain distribution around a circular hole for a (0/±45/0) s lamin ate under uniaxial loading are compared with laboratory data to provide a level of confidence in the analysis. Next, attention is focused on the inter laminar normal stress distribution around the free edge of the circular hole and the effects of changes in stacking sequence and lay-up angle. Initially, laminates with stacking sequences of (0/90) s are analyzed. Graphite/epoxy laminates of the type (02/±θ/∓θ) S , (±θ/∓θ/0 2) S , (902/±θ/∓θ) 2 , and (±θ/∓θ/902) S where θ is 30° , 45°, and 60° are also considered. Each lamin ate is subjected to a uniaxial stress loading distant to a hole.

Quantification of bone stresses during remodeling

Abstract An experiment to quantitate the effect of compressive stress upon bone remodeling rate was devised. Remodeling herein is described as the renewal process of bone separate from initial formation. A finite element mathematical model of the rabbit calvarium was devised. Actual rabbit calvaria were stressed utilizing a load cell monitored lever type loading device. They were periodically stained with fluorescing bone seeking dyes. Upon necropsy, histologic sections were prepared and the growth rate at known locations in the stressed area was measured by planimetry. Observed bone formation was subdivided into lamellar and fibrous. Lamellar bone formation (remodeling) rate remained above control levels until stresses of approximately 360 psi were reached; above this level remodeling rate fell to or below control. Fibrous bone exhibited higher formation rates which decreased with increasing stress but remained above control. At stresses approaching 1000 psi the technique could not measure formation rates due to the confused fibrous bone formation.

Mathematical and experimental studies on the mechanics of plated transverse fractures

Abstract A combined experimental and mathematical stress analysis approach to study the mechanics of a plated transverse fracture is described. Experimental results were obtained by placing strain gages on an equine third metacarpal and a bone plate. Two mathematical models for the mechanics of a bone and bone plate are presented. One is based on simple beam bending theory. The other is based on a more refined finite element representation for the deformations. Comparisons of experimental data and predicted results from the simple model are given and discussed. These comparisons provide a degree of confidence in the ability of a theoretical model to represent the actual deformations. The comparisons also show limitations of the simple beam model and the need for a more refined model such as a finite element representation. The resulting stress protection of the bone due to the bone plate and the percentage of load transferred through the plate are presented for a uniform loading condition. The advantages of a combined experimental and theoretical mechanics study are pointed out.

Mechanics of oblique fracture fixation using a finite-element model.

Abstract An investigation of the mechanics of a compression-plated oblique fracture using a finite-element stress analysis model is described. Three aspects of compression plating are considered: the effect of tension in the plate on stresses at the fracture site, the effects of placing bone screws across the fracture surface at different angles to the plane of the fracture, and the effect of a uniform end loading to represent weight bearing on the plated bone. In addition to stress distributions and deformations, the model predicts the contact area between the fractured surfaces. In one case, the predicted area shows good comparison with a range of values for a related case reported in the literature. Results of the model show that stress distributions at the fracture surfaces were not uniform and influenced by the tension in the plate, the bone screws, and the forces applied to the bone. The regions of contact between the fracture surfaces and the orientations of stress trajectories in terms of how these are influenced by plate tension, placement of bone screws, and applied loads are also shown. Clinically related aspects of the results are discussed.

Effects of cartilage stiffness and viscosity on a nonporous compliant bearing lubrication model for living joints

Abstract There are several possible lubrication mechanisms for living joints. One such mechanism, the squeeze film compliant bearing, is examined through the use of a computational model. The model is based on a finite element representation for the compliant surfaces. The lubricant is described by Reynolds equation. A minimization technique provides solutions to this inherently nonlinear problem, and alleviates difficulties in numerical convergence. In earlier work, the results of this model showed good correlation with both numerical results of another model and laboratory measurements for film thickness of a compliant sphere approaching a glass plate. In this paper, the model is used as a basis for conducting a parametric study of the influence of cartilage modulus and fluid viscosity on the lubrication characteristics. A load of 445 N was applied in the studies. Results of the compliant system are compared with those for a system with rigid surfaces representing a prosthesis. Based on the results obtained by the model and the ranges of parameters considered, it was found that changes in viscosity of the lubricant had a greater influence than changes in the cartilage modulus. The lubrication qualities of the compliant system were superior to those of the system with rigid surfaces if the same lubricant properties were used. However, lubrication viscosities could be found such that a compliant system with a low viscosity has poorer lubrication properties than a rigid system with a higher viscosity.

Measurements of healing at an osteotomy in a rabbit calvarium: The influence of applied compressive stress on collagen synthesis and calcification☆

Abstract Experimental data which establish a correlation between applied compressive stress in a rabbit calvarium and calcium formation, collagen synthesis, and histologic interpretation are presented. Estimates of collagen synthesis and calcium formation are obtained from measuring 45Ca and 3H proline uptakes. Stress distributions are obtained using a finite element analysis and photoelastic techniques. For the cases presented here, maximum 45Ca and 3H proline uptakes, as well as maximal histologic healing indices, are found at stress levels in the 10–30 psi range.