COMSOL-based fatigue simulation of crankshaft

The crankshaft is the most valuable and precise single component of the automobile engine, and the force of the crankshaft is extremely complex. It works under the combined action of periodic changes of gas pressure, reciprocating inertia force, and torque, and bears huge bending and torsional alternating loads. It is of great significance to study the effects of bending, torsional, and compound loads on the strength and fatigue life of crankshaft. In this paper, a crankshaft simulation model is established according to the fatigue reliability theory, the fatigue reliability of the crankshaft is determined, the fatigue life is analyzed, the stress-strain simulation of the crankshaft is carried out by COMSOL, and the fatigue load spectrum and damage cloud map of the crankshaft are obtained. The results show that the structural parameter that has the greatest influence on the stress and deformation of the transition rounded radius. The position corresponds to the place of the crank pin where maximum stress occurs.


Introduction
The internal combustion engine is the leading power equipment of transportation, construction machinery, fishing vessels, and national defense equipment, and the internal combustion engine industry is an important basic industry of economic and national defense construction [1] .With the development of science and technology, higher requirements have been put forward for the reliability and power performance of internal combustion engines [2] .The crankshaft is an important part of the internal combustion engine [3] .Working at high temperatures for a long time, the crankshaft will produce alternating bending stress and torsional stress under the action of periodically changing dynamic load.Whether the crankshaft is reliable directly determines whether the internal combustion engine can work normally [4] .Therefore, no matter the research and development of new internal combustion engines or the optimization of existing internal combustion engines, the crankshaft strength should be strictly evaluated [5] .
Crankshaft stress analysis has always been a research hotspot for scholars at home and abroad [6] .Sun et al. used the Ansys finite element analysis software to model the crankshaft and study its static stress and strain [7] .Li et al. of Weichai Power also used the finite element analysis method to design two notched parts for fatigue analysis instead of a crankshaft [8] .Fonseca et al. established an explicit finite element analysis model with real boundary conditions for internal combustion engine crankshaft, simulated the convergent grid of crankshaft fillet radius, and obtained the distribution law of crankshaft stress along the fillet radius [9] .

Fatigue reliability analysis of the crankshaft
In this paper, the high cyclic stress region of a crankshaft is selected to determine the stress characteristics of its dangerous parts.Then the stress fatigue limit of the crankshaft is determined according to the properties of the steel material and the surface coefficient of the crankshaft.The fatigue reliability of the crankshaft is calculated according to the fatigue reliability theory, and the fatigue characteristics of the crankshaft are determined.

Theoretical basis of fatigue reliability
Fatigue reliability refers to the probability that no fatigue fracture occurs in the cyclic motion of an object [10] .The time of the cyclic movement of the object is set as t and the time of fatigue failure as T, then the fatigue reliability is expressed as P(t), that is, the fatigue reliability of the object within a certain time is: With the increase of the working time of the object, its fatigue reliability will gradually decrease, and the longer the set life of the object, the worse the fatigue reliability.At the same time, in the field of engineering design, the reliability of an object is determined by the generalized stress and generalized strength of the object, among which the generalized stress includes tensile stress, compression stress, shear stress, etc.The generalized strength is affected by materials, structural shape, processing technology, surface damage, and other factors.In this paper, the stress and strength are related by the stress intensity interference theory to calculate the reliability-related indicators of the crankshaft.The fatigue limit stress of the crankshaft under the set number of cycles and survival rate is determined by the steel material characteristics, and the reliability index is calculated by combining the actual stress of the crankshaft and the reliability theory, and then the reliability of the parts is determined.

Crankshaft fatigue limit
Crankshaft material, crankshaft structure, processing technology, and other factors will have a certain impact on the fatigue limit of the crankshaft, and the relationship between these factors and the fatigue limit is as follows: where , , and represent crankshaft shape, processing technology, and stress concentration coefficient, respectively.The crankshaft material is steel.In order to ensure sufficient fatigue reliability of the crankshaft, the life of the crankshaft is set to 10 9 .The fatigue reliability of the crankshaft is calculated according to the stress intensity interference theory.
The relevant parameters of steel fatigue limit are shown in Table 1: Fatigue-related parameters value Stress concentration factor ( , ) 1.9, 0.05 Surface machining factor ̅ , 0.9, 0.05 Fatigue limit ( ̅ , ) 430, 15 Size factor ( ̅ , )/ 0.9, 0.05 The reliability index of crankshaft ZR is expressed as: Fatigue reliability R is expressed as: =∅( ) (4) By substituting the fatigue limit parameters of steel, the reliability of the crankshaft is about 0.9 5 .

Fatigue cumulative damage theory
The fatigue cumulative damage theory is a summary of fatigue failure characteristics and rules.In order to analyze the fatigue life of the crankshaft, it is necessary to select an appropriate fatigue cumulative damage theory and a fatigue life analysis method to analyze and predict the fatigue life of the crankshaft [11] .At present, there are three kinds of fatigue cumulative damage theories widely used: (1) The linear fatigue cumulative damage theory.
In the theory of linear fatigue cumulative damage, fatigue and other factors such as structure, material, load, surface treatment method, surface medium, and temperature are regarded as independent factors, so they can be linearly accumulated.The cumulative damage of the object is regarded as a constant, and the load sequence has no effect on the fatigue damage However, when the cumulative damage is too large, the object will undergo fatigue failure.Provided that the crankshaft has p different loads in actual working conditions, the cycle number of each load is ni, the total cycle number of material corresponding to each load is Ni, and the fatigue damage corresponding to each load is Di, then: According to the fatigue cumulative damage theory, the direct addition of the fraction Di represents the accumulation of various loads.When the sum of the fatigue damage fraction of different loads is equal to 1, it means that the fatigue cumulative damage of the crankshaft reaches the threshold value, and the crankshaft fatigue failure can be considered at this time.However, according to practical experience, the sum of Di is not always equal to 1, so in order to fit the actual situation, the fatigue damage critical value in the simulation can be appropriately adjusted according to the test experience.The revised fatigue damage critical value used in this paper is: In the formula above, e is the critical value of fatigue damage, and D is the total fatigue damage.
(2) The bilinear fatigue cumulative damage theory.The fatigue process is divided into two stages: crack generation and crack expansion, which involve different linear cumulative damage models to calculate.In the two processes of crack generation and crack expansion, the fatigue life is calculated as follows: Among them, N1 and N2 are the ultimate fatigue life, N1 represents the lowest stress, N2 represents the highest stress, and Z and ε are the coefficients determined by the fatigue life ratio in the load spectrum.When the life of fatigue damage N1 under the lowest stress is exhausted, the life of N2 under the highest stress is calculated until the fatigue damage reaches 1, that is, fatigue failure occurs.
According to the nonlinear fatigue cumulative damage theory, in terms of the physical concept of damage, it can be considered that different load sequences in the fatigue damage process will have a serious impact on the cumulative damage.Load cycle fatigue damage is defined as: where a and b are constants, and m and n are the cycles of fatigue damage and given stress, respectively.
Before fatigue failure of the crankshaft structure occurs, p cyclic damage under different loads will occur.According to the nonlinear fatigue cumulative damage theory, the total fatigue damage is: where D is the total fatigue damage and p is the cyclic damage under the i-th load.

COMSOL simulation
COMSOL Multiphysics is based on the finite element method and uses partial differential equations to realize multi-physical field simulation [12] .With COMSOL Multiphysics to simulate fatigue damage of the crankshaft, the real force situation of the crankshaft in cyclic motion can be intuitively observed.COMSOL Multiphysics has an integrated fatigue module, which can generate fatigue characteristic curves of crankshaft materials, select different parts of the crankshaft, and simulate them by setting rigid or flexible bodies, so as to better observe fatigue damage and predict fatigue life.In addition, COMSOL Multiphysics can also evaluate the fatigue of objects through steady state, transient, frequency domain, and random vibration.

Fatigue characteristic curve of crankshaft
When the crankshaft is working, it is more in a high-cycle fatigue state of low stress, and the S-N curve can better represent the fatigue characteristics of the crankshaft.The S-N curve reflects the relationship between the fatigue life of the crankshaft and the stress amplitude.Different materials have different fatigue characteristics of the crankshaft.It can be intuitively seen from the S-N curve that the fatigue characteristics of the crankshaft are mainly divided into three stages.When the number of fatigue life cycles is less than 10 3 , it is a nonlinear relationship, which is the low-cycle fatigue force of the crankshaft.When the number of fatigue life cycles is between 10 3 and 10 6 , the relationship is linear, and then it is the high cycle fatigue force of the crankshaft.When the fatigue life cycle number is greater than 10 6 , it is the sub-fatigue force of the crankshaft.The crankshaft steel material will not fail easily at this time.The crankshaft fatigue characteristic curve is shown in Figure .1.

Fatigue life analysis of crankshaft
According to fatigue reliability basis, the fatigue characteristic curve of the crankshaft, and fatigue cumulative damage theory analysis, simulation was carried out.

Conclusion
By analyzing the fatigue reliability and life of the crankshaft, the S-N curve of the crankshaft material is constructed, and the fatigue load spectrum and fatigue damage cloud map of the crankshaft are obtained by simulation.According to the simulation results, the crankshaft prone to fatigue damage is the position where the crankshaft is stressed most during the cyclic operation, and the simulation results are completely consistent with the calculated results.

3. 2 .
Fatigue load spectrum of crankshaftThe load-time history of the crankshaft during the cyclic operation is called the load spectrum.The load numerical curves of the three crank pins with the largest load on the crankshaft were analyzed through COMSOL Multiphysics simulation.On this basis, the load data required for fatigue life analysis was established, namely, the fatigue load spectrum of each crank pin in the set working cycle of the crankshaft.The fatigue analysis load spectrum is formed in the fatigue analysis module and matched with the static simulation load.The simulation of crankshaft in different fatigue stress stages is shown in Figure.2, Figure.3, and Figure.4.

Figure. 5
shows the damage cloud image of the crankshaft without the fatigue cycle, and Figure.6 to Figure.8 show the damage cloud image of early, middle, and late fatigue cycles, respectively.It can be seen from the fatigue life and damage cloud map of the crankshaft that the distribution of fatigue damage and fatigue life is consistent with the stress distribution of the crankshaft.The fatigue damage of the stress concentration part of the crankshaft is greater than other parts, and the fatigue life is smaller than other parts.The closer the crankshaft is to the power unit, the smaller the fatigue life of the crank pin is.The part with the least life is at the transition corner between the crank pin and the crank arm, corresponding to the position where the crank pin stress is the greatest.