The  paper  is  devoted  to  the  theoretical  study  of  the  mechanical  behavior  of materials in the friction pair of hip resurfacing endoprosthesis. The investigation was based on three-dimensional computer simulation by the movable cellular automaton (MCA) method, which is a representative of the methods of particles in the mechanics of materials. The results indicate the promising use of metallic alloys with biocompatible ceramic coatings in friction pairs to increase the service life of hip resurfacing

Eremina, Galina M.

Smolin, Alexey Yu.

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Numerical model of the mechanical behavior of coated materials in the friction pair of hip resurfacing endoprosthesis

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CONTRIBUTED SESSIONSS - Bio Medical EngineeringNumerical odel of the Mechanical Behavior of Coated Materials in the Friction Pair of Hip Resurfacing EndoprosthesisG.M. Eremina, A.Y. Smolin a d M.O. EreminVI International Conference on Particle-based Methods – Fundamentals and ApplicationsPARTICLES 2019E. Oñate, M. Bischoff, D.R.J. Owen P. Wriggers, , & T. Zohdi (Eds)NUMERICAL MODEL OF THE MECHANICAL BEHAVIOR OFCOATED MATERIALS IN THE FRICTION PAIR OF HIPRESURFACING ENDOPROSTHESIS GALINA M. EREMINA², ALEXEY YU. SMOLIN¹¹ National Research Tomsk State University (TSU)pr. Lenina 36, 634050, Tomsk, Russia E-mail: asmolin@ispms.tsc.ru , web page: http://www.tsu.ru² Institute of Strength Physics and Materials Science (ISPMS)Siberian Branch of Russian Academy of Sciencespr. Akademicheskii 2/4, 634055, Tomsk, RussiaE-mail: rector@tsu.ru , web page: http://www. ispms.ruKey words: hip resurfacing endoprosthesis, coatings, scratch test, friction, simulation, movable cellular automaton method, Movable Cellular Automata, Particle-Based s.Abstract. The  paper  is  devoted  to  the  theoretical  study  of  the  mechanical  behavior  ofmaterials in the friction pair of hip resurfacing endoprosthesis. The investigation was based onthree-dimensional computer simulation by the movable cellular automaton (MCA) method,which is a representative of the methods of particles in the mechanics of materials. The resultsindicate the promising use of metallic alloys with biocompatible ceramic coatings in frictionpairs to increase the service life of hip resurfacing.1 INTRODUCTIONFor the treatment of pathologies of the hip joint in the modern world, endoprosthesis (EP)are widely used. Two types of endoprosthesis are used for hip: total hip replacement and hipresurfacing arthroplasty. For the treatment of osteoporosis and other diseases of the hip joint,a hip resurfacing arthroplasty is preferable.  The advantage of using hip resurfacing is thepossibility of maintaining healthy bone and the possibility of revision operations [1].An important factor affecting the durability of the hip joint endoprosthesis is wear in pairof friction “acetabular cup-femur head”. In the case of resurfacing, just the surface of thefemur head is replaced by the cap, which is specially shaped like a mushroom [2-3]. Thedifference in the diameters of the hip resurfacing cup and the acetabular cup averages about 2mm [4]. Polyethylene-metal friction pairs were the first generation of a pair of materials usedin a friction pair of hip resurfacing endoprosthesis, however, a large degree of wear led tolook for other materials suitable for use in the friction pair [5]. Metal friction pairs based oncobalt-chromium alloys are suitable for the manufacture of endoprosthesis even for youngphysically active people, since due to their mechanical behavior they are able to withstandhigh dynamic loads. However, during wear, metal endoprosthesis excrete particles that induceinflammatory processes in the surrounding tissues [6–7]. The use of titanium alloys is a goodalternative to the use of cobalt-chromium alloys due to its biocompatible properties, however,these  alloys  are  prone  to  increased  wear.  Another  group  of  materials  with  good197Galina M. Eremina, Alexey Yu. Smolinbiocompatibility are ceramic products, however, this type of prosthesis is applicable mainly topeople with low and medium physical activity due to their tendency to brittle failure underhigh dynamic loads. Therefore, at present, the directions of hip resurfacing endoprosthesisbased  on  metallic  titanium  alloys  with  ceramic  coatings  [8],  which  combine  goodbiocompatibility and wear resistance, are being actively developed.Endoprosthesis testing of has two main stages: preclinical and clinical trials. Clinical trialsare  carried  out  by installing  the  endoprosthesis  in  a  living  human  body.  However,  whenconducting  clinical  trials  there  is  a  danger  that  poor-quality  or  incorrectly  chosenendoprosthesis may adversely affect the patient's health, therefore, great attention is paid topreclinical trials in the development of endoprosthesis. Preclinical studies of the mechanicalbehavior of the endoprosthesis can be divided into experimental and theoretical. Experimentalstudies  are  tests  using  a  technological  installation  that  simulates  the  dynamic  loadingexperienced by the  endoprosthesis.  Theoretical  studies  of  the  mechanical  behavior  of  theendoprosthesis  using  computer  simulation  make  it  possible  to  investigate  the  mechanicalbehavior of endoprosthesis taking into account the influence of various factors.Most of the work on modeling the mechanical behavior of materials in friction pairs isdevoted to total hip replacement. Works on modeling hip resurfacing are mainly devoted tostudying  the  mechanical  behavior  of  a  single-component  endoprosthesis  and  the  system“bone-endoprosthesis” under dynamic loads without rotation in a friction pair [9-11]. In [12],a force action scheme with rotation of the head in the acetabular cup with and without frictionis presented. However, there is no work to simulate the rotational motion of the resurfacingcap in the acetabulum; therefore, it is important to build a numerical model for the rotation ofthe  resurfacing  cap  in  the  acetabulum  consisting  of  one-component  and  two-componentmaterials.This paper proposes a numerical study of the mechanical behavior of contacting elementsof a friction pair consisting of a homogeneous material and a material with a coating duringrotational motion using the method of movable cellular automata.2 METHOD OF MOVABLE CELLULAR AUTOMATAMCA is a  new efficient  numerical  method in particle  mechanics that  is  different  frommethods in  the traditional  continuum mechanics  [13,14].  Within the frame of MCA, it  isassumed that any material is composed of a certain amount of elementary objects (automata)which interact among each other and can move from one place to another and rotate, therebysimulating a real deformation process. The automaton motion is governed by the Newton-Euler equations:¿¿mid2Ridt2=∑j=1NiF ijpair+FiΩ,Ĵidωidt2=∑j=1NiM ij¿{¿(1)where Ri, ωi, mi and Ĵi are the location vector, rotation velocity vector, mass and moment2198Galina M. Eremina, Alexey Yu. Smolinof inertia of ith automaton respectively, F ijpair is the interaction force of the pair of ith andjth automata, FiΩ is the volume-dependent force acting on ith automaton and depending onthe  interaction  of  its  neighbors  with  the  remaining  automata.  In  the  latter  equation,M ij=q ij(n ij× F ijpair )+K ij  , here  qij is the distance from the center of  ith automaton to thepoint of its interaction (“contact”) with  jth automaton,  nij=(R j− Ri )/rij  is the unit vectordirected  from the  center  of  ith  automaton  to  the  jth  one  and  rij is  the  distance  betweenautomata centers, K ij  is the torque caused by relative rotation of automata in the pair.The forces acting on automata are calculated using deformation parameters, i.e. relativeoverlap, tangential displacement and rotation, and conventional elastic constants, i.e. shearand bulk moduli. A distinguishing feature of the MCA method is calculating of forces actingon the automata within the framework of multi-particle interaction [15,16], which providesfor an isotropic behavior of the simulated medium regarded as a consolidated body rather thana granular medium. Moreover, stress tensor components can be calculated for the automatontaking  into  account  all  the  forces  acting  on  the  automaton  [15-17],  which  enables  therealization of various models of the plastic behavior of materials developed in the frame ofcontinuum mechanics.A pair of elements might be considered as a virtual bistable cellular automaton, whichpermits simulation of fracture and cracks healing and micro welding by the MCA. In thiswork, a fracture criterion based on the threshold value of von Mises stress was used [15,16]. Acriterion  based on the threshold value of  plastic  work was used for  making a new bondbetween contacting automata [16].  Switching of a  pair  of automata from bonded to non-bonded state and vice versa would result in a changeover in the forces acting on the elements;in particular, non-bonded automata would not resist moving away from one another. 3 DESCRIPTION OF THE MODEL3.1 Material characterizationFrom the literature [18] we chose the following values for the material properties of thetitanium alloy Ti6Al4V: density ρ = 4420 kg/m3, shear modulus  G = 41 GPa, bulk modulusK = 92 GPa, Young’s modulus E = 110 GPa, yield stress σy = 0.99 GPa, ultimate strength σb =1.07 GPa and ultimate strain εb = 0.10. The mechanical properties of the TiN coating [19]:density ρ = 5220 kg/m3, shear modulus  G = 104 GPa, bulk modulus  K = 129 GPa, Young’smodulus E = 258 GPa, yield stress σy = 3.00 GPa, ultimate strength σb = 3.5 GPa and ultimatestrain εb = 0.075.3.2 Geometry of the model and scheme of loadingIn the case of a one-component material, the geometric model consists of a hemispheresimulating the resurfacing cap for femur head, with an external diameter D_ext_cap=36 mmand an interior  diameter  D_int_cap=33 mm, a hemisphere simulating the acetabulum cupinsert,  with  an  outer  diameter  of  D_ext_insert=41  mm  and  an  iterior  diameter  ofD_int_insert=38 mm, and also a prismatic shell for the insert imitating the surrounding bonetissue  (fig.  1).  In  the  case  of  a  two-component  coated  material,  hollow hemispheres  areadditionally specified for the resurfacing cap with an outer diameter of 35.9 mm and an inner3199Galina M. Eremina, Alexey Yu. Smolindiameter of 33.1 mm (fig.2, a).Figure 1. Schematic representation of the model of friction pair of the resurfacingendoprosthesis in cross-sectionThe load is applied by specifying the translational and rotational velocity of the resurfacingcap automata, corresponding to its rotation as an absolutely rigid body around the axis ofsymmetry of the corresponding sphere and parallel to the axis X. a bFigure 2. General view of the model for friction pair of hip resurfacing endoprosthesis (a) and itscross-section with loading parameters (b), represented by automata packingIn this case, the value of the corresponding rotational velocity gradually increases from 0to 10 1/s. The bottom layer of the automata of the cylindrical shell of the bone tissue is rigidlyfixed (fig 2, b).4200Galina M. Eremina, Alexey Yu. Smolin4 SIMULATION RESULTSWhen simulating a single rotational cycle, the maximum reaction force was not greaterthan 3000 kN, which corresponds to the load of a walking man, and the angle of rotation ofthe resurfacing cap was 120°, which is typical for standard daily physical activities for anordinary healthy person.abFigure 3: Distributions of mean stress in the friction pair of the hip resurfacingendoprosthesis of titanium alloy (a) and titanium alloy with TiN coating (b)The simulation results showed that in the case of a friction pair of a homogeneous materialin the zone of contact interaction of the acetabulum insert and resurfacing cap and behind it atextreme positions (edge of the acetabulum) a large are of tensile stress with a maximum value5201Galina M. Eremina, Alexey Yu. Smolinnot  reaching  990 MPa appeared  in  the  cap.  Such  a  load  exceeds  the  yield  strength  and,therefore, can lead to rapid wear of the surface of the resurfacing cap of the femur head of thejoint (fig.3, a). These results are consistent with the data on the stress distribution in the metalhead obtained in [20]. At the same time, the stress in the acetabulum insert did not exceed100 MPa.In the case of a coated endoprosthesis, a zone of tensile stresses with a maximum value of1.1  GPa  was  observed  in  the  contact  zone,  but  this  area  was  significantly  smaller  andconcentrated  mainly  in  the  coating  (fig.3,  b).  In  addition,  when  using  two-componentmaterials in the friction pair, there was no noticeable increase in stress values in the cap whenit  was in  the  extreme positions.  Consequently,  the  use  of  titanium alloys  with  a  ceramiccoating allows avoiding premature wear at the extreme positions of the femur head in theacetabulum. In the insert consisting of titanium alloy and coating, the value of compressivestresses reached 300 MPa. It should be noted that the magnitude of such stresses is not criticalfor the coating.In general, the results of the numerical simulations suggest that the use of coated materialsin the friction pair of hip resurfacing endoprosthesis can help avoiding premature wear of theendoprosthesis.5 CONCLUSIONSA three-dimensional model of the mechanical behavior of materials in a friction pair of hipresurfacing  endoprosthesis  during  rotational  motion  is  presented  based  on  the  method  ofmovable cellular automata. The obtained simulation results allow us to draw the followingconclusions.The greatest value of tensile stresses in the cap made of titanium alloy during rotation isobserved at the extreme positions of the resurfacing cap, in the case of the cap made of TiNcoated titanium, this tendency is avoided.The use of a hardening coating would help to avoid premature wear of the contactingelements of the hip resurfacing endoprosthesis.6 ACKNOWLEDGMENTSThe reported study was funded by RFBR according to the research project No 18-38-00323(the  models  developed  and  the  simulations)  and  framework  of  the  Russian  FundamentalResearch  Program of  the  State  Academies  of  Sciences  for  2013-2020  (Priority  directionIII.23, the modifications of the MCA software for importing CAD models).REFERENCES[1] Vendittoli, P.A., Lavigne, M., Girard, J., Roy A.G. A randomised study comparing resection ofacetabular bone at resurfacing and total hip replacement. 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Numerical model of the mechanical behavior of coated materials in the friction pair of hip resurfacing endoprosthesis

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