Numerical Approach to Predict the Lifespan of a Prosthesis

This study consists of modeling and simulating the behavior of a prosthetic leg adapted to running using the finite element method, therefore the work is divided into three phases. In the first stage, the study is focused on the modeling of prostheses and their contact with the ground, followed by a phase of simulation of its behavior. The final phase of this work is oriented to the study of the influencing factors on the overall behavior of the prosthesis: number of folds, mechanical properties and an estimate of the service life. The evaluation is determined on the uncertainty of the stresses due to modeling errors, and the dispersion of the data obtained from the fatigue tests [1]. The results indicate that the accuracy of the prosthesis fatigue assessment is strongly related to the accuracy of the stress estimation. Additionally, a relationship between the probability of fatigue failure and the number of stroke cycles has been suggested, in order to provide a benchmark for prosthetics to determine the time interval for prosthesis inspection. a Prosthesis.


Introduction
Vertical reaction force c) Horizontal reaction force. According to [4].
The tibia of a sports amputee, the phases of running (the support phase, the stride or the aerial phase "flight stage") are identical to those of a normal person ( Figure 1). The prosthesis replaces the foot and the tibia [2]. Thus, it must ensure the function of locomotion of these anatomical elements removed by amputation. In the support phase, the prosthesis behaves as follows: a) Damping phase: the prosthesis is contracted with a value determined by the elastic characteristics of the material, and depending on the body weight, b) Following the initial support phase, a process of displacement of the energy stored at the peaks of the elastic blade is triggered. c) Energy is initially stored at the tip of the blade. At this stage, this energy produced by the amputee runner is equivalent to that generated by the gluteal muscles at the hip of an able-bodied runner. The scene is finalized by the forward propulsion of the rider's body. The prosthesis is used as an identical passive adaptive dynamic system which can increase performance, if used correctly, and is capable of making simple harmonic movement, is a trampoline [3].
However, a trampoline has non-linear stiffness, which allows it to be used easily by athletes of varying mass. The inertia of the athlete falling from a height stores a large amount of deformation energy in the sheet and the springs under the effect of their great movement. This deformation energy is then recovered by the mass, which tries to return to the initial state. However, under the effect of damping and loss of energy, this mass will never regain its original height without assistance or without additional energy input. If such a mass was able to generate a cyclic rebound action, and if the frequency of this action matches the frequency of the perceived mass/spring system, then the athlete will continuously bounce without leaving the sheet at the natural frequency of the system [4].
In the case of a trampoline, if the input energy per cycle is greater than the system energy loss per cycle, the athlete will be able to gain height otherwise impossible to reach without assistance and simply by using their ankles [5]. This result in a higher takeoff speed, which means higher kinetic energy, resulting in higher potential energy stored in the system. In this work, we consider the hypothesis that walking or running at a frequency that corresponds to the natural frequency of the ESR system of the foot and the body mass which has a mechanical advantage if used in synchronization, as in a trampoline. . This allows for improved performance if the foot is used at its optimal dynamic characteristics.  Cheetah Flex foot type prosthesis with these accessories is shown in Figure 2B.

Boundary Condition
The boundary conditions applied in this modelization are defined in the following way: a) The plate, which models the ground, is fixed b) The reference point (RP) which controls the movement of the prosthesis has two degrees of freedom, a movement along the vertical Y-axis and a rotation around the Z-axis.

c)
A force concentrated in the Y direction.

Mechanical Properties
The Flex Foot Cheetah sports prosthesis is made from the composite material pre-impregnated with flat stress carbon fibers [6]. It is possible to simply define the mechanical characteristics as "lamina" for which six properties are required. These properties are obtained from the literature of composite materials such as HexaTOOLS® composites, and they are illustrated in the table below (Table 1). Since there is no precise information on the rolling sequence, we will first assume that the prosthesis is composed by the sequence of the following layers: two layers of tissue with on the size of each section. The latter is composed of several films of carbon fiber base prepreg. In our study, we adopted this fiber arrangement methodology.

Six-Section Model
In this model we increased the number of layers in the third and fourth sections in order to identify their influence on the stiffness or deformation of the prosthesis. The number of strata and the arrangement of the fibers of this model are well mentioned and configured successively in Table 2, and in Figure 3. The configurations below show the orientation of the fibers and the layer thickness of each section.

Results Analysis
The evolution of the stresses during the time of the contact is presented in Figure 4, this curve shows that the maximum stress value is about 324 MPa at 60% of the time of the contact. Note from the previous figure that the stresses are concentrated at the third and fourth section where the curvature of the prosthesis is located ( Figure 5).  Miner's law is written: Where n is the number of the cycle specified at the i-th stress acting on the structure, N is the life of the material subjected to a stress σ, and D is the coefficient of damage, which is equal to 1 at the time when the part is break, and it is often considered to be a random variable that follows the lognormal distribution, . r e K σ σ = (2) σr : the actual constraint.
K : the uncertainty on the constraint which is assumed to be a variable follows the log-normal law.
σe : the constraint obtained by simulation by finite. In order to calculate N1 and N2, Wöhler's S-N curve model, which is suitable for most materials, is adopted here:  Table 3. In order to identify the values of m and of ξ, the computation of the data regression of the fatigue test of the material is done. This calculation consists in finding: The graph of the S-N curve is shown in Figure 6. It can be seen from the curve that the Wöhler model roughly matches the test data, and that the endurance limit is approximately 150 MPa.

Fatigue Reliability Analysis
With the substitution of equations (2) and (4) Generally, reliability analysis considers the limit state (ultimate limit) to define fatigue. For an ultimate limit state, the resistance or fatigue strength is represented by a certain degree of structural resistance, which represents a maximum value of the structural resistance. Damage occurs when the predicted load is greater than the breaking limit. In our case, the limit state equation characterizing the rupture of the prosthesis is defined by: According to the theory of fatigue probabilities: Where Pf is the conversation of fatigue. Because ξ, D and K are random variables which apply the lognormal distribution, therefore G (X) is subjected to a normal distribution. Thus, Pf is written: Where µ: expectation of G, τ: standard deviation of G, , φ : the probability density of the normal distribution, β: the safety index in the reliability analysis, the higher it is, the more robust the structure will be.

Results Analysis
In evaluating the fatigue strength of the prosthesis, the standard deviation and expectation of G were calculated as follows: is a very complex phenomenon for composite materials, which can be formed by different failure modes [9]. Our study assumes that the rupture of the prosthesis occurs at the level of the third section.
In addition, experimental validation is strongly requested to justify the various hypotheses used during this study.