# Research on the Fatigue Life Prediction Method of Thrust Rod

Authors: Guoyu Feng; Wenku Shi; Henghai Zhang; Qinghua Zu; Teng Teng; Zhiyong Wu
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Purpose of this paper is to investigate the fatigue life prediction method of the thrust rod based on the continuum damage mechanics. The equivalent stress used as damage parameters established rubber fatigue life prediction model. Through the finite element simulation and material test, the model parameters and the fatigue damage dangerous positions were obtained. By equivalent stress life model, uniaxial fatigue life of the V-type thrust rod is analyzed to predict the ratio of life and the life of the test was 1.73, within an acceptable range, and the fatigue damage occurring position and finite element analysis are basically the same. Fatigue life analysis shows that the method is of correct, theoretical, and practical value.


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## Body

## 1. Introduction

Thrust rod ball joint is a typical multidirectional rubber bearing element, and it is mainly used in quasistatic and fatigue strain environments. In complex work environments, rubber-metal composite ball joint suspension system may suffer from axial, radial, torsion, deflection, and other conditions to the role and impact loading [1–9]. Fatigue failure of the rubber material is one of the main forms of failure cause thrust rod; the thrust rod service life depends primarily on the fatigue life of the rubber ball joints [10–14]. Therefore, the study rubber ball joint fatigue life is the key thrust rod service life analysis.Studies have shown that the process of rubber dynamic fatigue can be divided into crack initiation and propagation and failure in three stages [15–17]. The first stage is a dramatic change in stress; the material appears to be rubber under stress softening phenomenon; second stage is the slowly varying stress or the internal surface of the rubber material to produce microcracks, often referred to as nuclear damage; third stage microcracks develop into cracks and continuously expand open until the rubber material is completely destroying a fracture phenomenon. The fatigue life of the rubber member is composed mainly of crack initiation life and crack propagation life in two parts [18–21].MARS research of natural rubber under cyclic loading cracks nucleation, expansion, and ultimate failure studies the relationship between crack growth and crack growth rate of the number of cycles and the energy release rate and summed up the crack growth life under uniaxial loading conditions and strain energy density function formula [22]. Woo and colleagues studied the maximum strain energy density and maximum strain Green rubber amount that are lifetime prediction parameters [23]. Rubber fatigue is a hot area; many scholars have carried out relevant research, but because the fatigue damage mechanism of rubber material is complex, the currently accepted and effective method for the prediction of standard rubber products and the fatigue life is not formed.The main thrust rod is subjected to low frequency and large load, once the rubber material cracks quickly expand, resulting in fatigue failure; hinge rubber layer and the ball were almost wrapped-up metal parts; the initial crack generation position cannot be directly observed or detected. Thus, the fatigue life of the thrust rod crack initiation is life based.


## 2. Rubber Fatigue Prediction Theory

### 2.1. Rubber Fatigue Failure Mechanism

Rubber fatigue failure is generated during repeated deformation when subjected to the elongation of its local deformation of rubber or stress exceeds the limit stress, and fatigue process starts repeatedly under load, so that the final destruction is reached. The starting point of fatigue failure is due to the rubber surface or internal homogeneity and internal defects or microcracks resulting in uneven distribution of stress and local stress concentration, causing the molecular chains, causing cracks in the spread and expansion caused by fatigue failure. Especially in the cyclical state of deformation, rubber molecular chain has a chance to relax and has a stress-strain lag angle, resulting in the fact that deformation of the rubber internal residual stress is increasing in the molecular chain always maintaining a certain stress gradient and fracturing more easily. Because of the fatigue failure, first at a certain point it produces tiny cracks; its starting point is called “fatigue source” and a source of fatigue crack from the beginning and gradually extended to four weeks.Deformation of rubber is generally limited deformation, fatigue with the development process, and the first increase in tensile strength, after the maximum value tends to decrease. After the element cracks within a small area from the opposite direction near the beginning of the crack tip, the stress increases, increasing rigidity and maximum stress at the crack tip; this value is known as the cohesive strength, and then stress is reduced [24]. Crack tip stress distribution is shown in Figure 1 [25].Figure 1
Distribution of crack tip stress.From the energy point of view, to overcome the stress fracture that is acting, we will expand the crack extension. And after cracks tear strength, dynamic modulus of elasticity and the loss factor is reduced at first, after a minimum turn increases, and the rubber in an external energy can continue this form of stress relaxation to be absorbed. Both for uniaxial loading and for multiaxial loading, the crack positions the main subject of the maximum principal stress directions. When tension crack propagation direction is simple, the direction of the maximum principal stress is perpendicular. When the maximum principal stress is greater than the critical stress, the material damage will occur, and the carrying capacity decreases [25].


### 2.2. Fatigue Life Prediction Model

Strain equivalence hypothesizes that the deformation stress acting on the material damage caused by the deformation is equivalent to the role of virtual nonmaterial damage for a one-dimensional problem; this principle is expressed as(1)ε=σE=σ~E~.Formula:ε is strain; σ is nominal stress; σ~ is a cross section on the effective stress; E is lossless elastic modulus; E~ is the elastic modulus of the material damage.Based on the principle of equivalent stress, Lemaitre and Chaboche proposed damaged material constitutive equation, the nominal stress tensor and effective stress tensorσ. Damage σ~ has a relation [26],(2)D=1-σσ~.Formula:D is a damage factor; there is no material damage when the initial damage factor is 0 and when the value is 1 damage to the formation of cracks occurs.When using a second class Piola Kirchhoff stress, according to the theory of finite strain and Lagrange, formula (2) can be expressed as(3)D=1-σiσ~i.Formula:σi is the current principal stress direction i;  σ~i is effective principal stress.Research shows that the calculation method of the rubber material equivalent stress is as follows [27]:(4)σf=σ12+Aσ22+Bσ32.Formula:σf is equivalent force; σ1,  σ2,  σ3 are for the three directions of principal stress. (5)σ1>0,σ1≥σ2≥σ3,A=1,σ2>00,σ2≤0,B=1,σ3>00,σ3≤0.When the principal stress is greater than zero, it is assumed that not only tensile stress crack initiation and propagation of rubber contribute to crack closure, but also stress does, and thus they do not contribute to crack initiation and propagation.We use effective stress parameters to assess the fatigue life of rubber parts, complex stress state, its geometry, and loading conditions unrelated to better reflect the actual situation of the fatigue life. This paper uses the equivalent stress as the fatigue damage parameter.Based on continuum mechanics, assuming that the rubber material is isotropic strain energy function describes the characteristics of rubber elasticity by unit volume. For Ogden model, the strain energy per unit volume is a function of [28](6)U=∑i=1N2μiαi2λ-1αi+λ-2αi+λ-3αi-3+∑i=1N1DiJ-12i,where N is the model order, μi,  αi,  1<i<n is temperature dependent material parameters, the value is determined by the test data, Di material parameters are determined by the initial bulk modulus, λi is the main rubber elongation ratio, and J is elastic volume ratio, considering the rubber is incompressible, and J=λ1λ2λ3=1.Under uniaxial stress state,λ1=λU,  λ2=λ3=λU-1/2.In the state of uniaxial tensile stressσU can be expressed as [29](7)σU=∂U∂λU.Formula:λU is a front uniaxial tensile elongation ratio state; U is strain energies.Under uniaxial stress state, the second-order model is Ogden strain energy function which can be expressed as [30](8)U=∑i=122μiαiλUαi+2λU-αi/2-3.In formulas (8) to (7), the stress can be obtained as expression σU:(9)σU=∑i=122μiαiλUαi-1-λU-αi/2+1=2μ1α1λUα1-1-λU-α1/2+1+2μ2α2λUα2-1-λU-α2/2+1.Considering only one-dimensional state, the strain can be expressed as a function of elongation,(10)εU=λU-1.According to (8), (9), and (10) expression for the stress can be derived:(11)σU=∑i=122μiαiεU+1αi+1-εU+1-αi/2+1.By stretching rubber samples tested for stress-strain data fitting can be drawn from material constitutive parametersμi,  αi.Formulas (11) to (2) can be obtained for the damage stress(12)σ~=∑i=122μiαi1-DεU+1αi+1-εU+1-αi/2+1.Formula for damage factor is [31](13)D˙=-∂φ∗∂y.In the formula,φ∗ is damage dissipation potential;  y is damage strain energy release rate.Let dissipation potential function be of the form [32](14)φ∗=ba+1-yba+1,where a,  b are material constants which can be determined by fatigue tests.Damage strain energy is a function of stress injury and damage strain energy release rate can be defined as(15)-y=∂U∂D=∂Uσ~∂D=∑i=13∂U∂λi∂λi∂D.By formula (2) the following can be derived:(16)∂σ~∂D=σ1-D2=∂σ~∂λi∂λi∂D.It can be further derived that(17)∂λi∂D=σ~1-D∂σ~∂λi-1.Under uniaxial stress state, the second-order Ogden model consists of (16) and (17) can obtain damage strain energy release rate which can be expressed as(18)-y=∂Uσ~∂D=2μ1α11-D2λUα1-1-λU-α1/2+12α1-1λUα1-2+α1/2+1λU-α1/2+2+2μ2α21-D2λUα2-1-λU-α2/2+12α2-1λUα2-2+α2/2+1λU-α2/2+2.Under uniaxial loading, effective stress injury isσf; then by damage factor (2), (8), (13), (14), and (18) can be derived:(19)D˙=σfb1-Da.Under a single cyclic loading, fatigue damage equivalent stress mainly consists of the scope and magnitude of maximum stress, strain amplitude being related.From this, the next single cycle fatigue damage can be expressed as(20)∂D∂N=σfb1-Da,where N is the cycle. Formula (20) is obtained by integrating both sides(21)∫0D1-DadD=∫0Nfσfb-1adN.Using the initial condition (N=0,D=0) and destroying the conditions (N=Nf,D=1), integral to obtaining the fatigue life expression,(22)Nf=ba1+aσf-a.


## 2.1. Rubber Fatigue Failure Mechanism

Rubber fatigue failure is generated during repeated deformation when subjected to the elongation of its local deformation of rubber or stress exceeds the limit stress, and fatigue process starts repeatedly under load, so that the final destruction is reached. The starting point of fatigue failure is due to the rubber surface or internal homogeneity and internal defects or microcracks resulting in uneven distribution of stress and local stress concentration, causing the molecular chains, causing cracks in the spread and expansion caused by fatigue failure. Especially in the cyclical state of deformation, rubber molecular chain has a chance to relax and has a stress-strain lag angle, resulting in the fact that deformation of the rubber internal residual stress is increasing in the molecular chain always maintaining a certain stress gradient and fracturing more easily. Because of the fatigue failure, first at a certain point it produces tiny cracks; its starting point is called “fatigue source” and a source of fatigue crack from the beginning and gradually extended to four weeks.Deformation of rubber is generally limited deformation, fatigue with the development process, and the first increase in tensile strength, after the maximum value tends to decrease. After the element cracks within a small area from the opposite direction near the beginning of the crack tip, the stress increases, increasing rigidity and maximum stress at the crack tip; this value is known as the cohesive strength, and then stress is reduced [24]. Crack tip stress distribution is shown in Figure 1 [25].Figure 1
Distribution of crack tip stress.From the energy point of view, to overcome the stress fracture that is acting, we will expand the crack extension. And after cracks tear strength, dynamic modulus of elasticity and the loss factor is reduced at first, after a minimum turn increases, and the rubber in an external energy can continue this form of stress relaxation to be absorbed. Both for uniaxial loading and for multiaxial loading, the crack positions the main subject of the maximum principal stress directions. When tension crack propagation direction is simple, the direction of the maximum principal stress is perpendicular. When the maximum principal stress is greater than the critical stress, the material damage will occur, and the carrying capacity decreases [25].


## 2.2. Fatigue Life Prediction Model

Strain equivalence hypothesizes that the deformation stress acting on the material damage caused by the deformation is equivalent to the role of virtual nonmaterial damage for a one-dimensional problem; this principle is expressed as(1)ε=σE=σ~E~.Formula:ε is strain; σ is nominal stress; σ~ is a cross section on the effective stress; E is lossless elastic modulus; E~ is the elastic modulus of the material damage.Based on the principle of equivalent stress, Lemaitre and Chaboche proposed damaged material constitutive equation, the nominal stress tensor and effective stress tensorσ. Damage σ~ has a relation [26],(2)D=1-σσ~.Formula:D is a damage factor; there is no material damage when the initial damage factor is 0 and when the value is 1 damage to the formation of cracks occurs.When using a second class Piola Kirchhoff stress, according to the theory of finite strain and Lagrange, formula (2) can be expressed as(3)D=1-σiσ~i.Formula:σi is the current principal stress direction i;  σ~i is effective principal stress.Research shows that the calculation method of the rubber material equivalent stress is as follows [27]:(4)σf=σ12+Aσ22+Bσ32.Formula:σf is equivalent force; σ1,  σ2,  σ3 are for the three directions of principal stress. (5)σ1>0,σ1≥σ2≥σ3,A=1,σ2>00,σ2≤0,B=1,σ3>00,σ3≤0.When the principal stress is greater than zero, it is assumed that not only tensile stress crack initiation and propagation of rubber contribute to crack closure, but also stress does, and thus they do not contribute to crack initiation and propagation.We use effective stress parameters to assess the fatigue life of rubber parts, complex stress state, its geometry, and loading conditions unrelated to better reflect the actual situation of the fatigue life. This paper uses the equivalent stress as the fatigue damage parameter.Based on continuum mechanics, assuming that the rubber material is isotropic strain energy function describes the characteristics of rubber elasticity by unit volume. For Ogden model, the strain energy per unit volume is a function of [28](6)U=∑i=1N2μiαi2λ-1αi+λ-2αi+λ-3αi-3+∑i=1N1DiJ-12i,where N is the model order, μi,  αi,  1<i<n is temperature dependent material parameters, the value is determined by the test data, Di material parameters are determined by the initial bulk modulus, λi is the main rubber elongation ratio, and J is elastic volume ratio, considering the rubber is incompressible, and J=λ1λ2λ3=1.Under uniaxial stress state,λ1=λU,  λ2=λ3=λU-1/2.In the state of uniaxial tensile stressσU can be expressed as [29](7)σU=∂U∂λU.Formula:λU is a front uniaxial tensile elongation ratio state; U is strain energies.Under uniaxial stress state, the second-order model is Ogden strain energy function which can be expressed as [30](8)U=∑i=122μiαiλUαi+2λU-αi/2-3.In formulas (8) to (7), the stress can be obtained as expression σU:(9)σU=∑i=122μiαiλUαi-1-λU-αi/2+1=2μ1α1λUα1-1-λU-α1/2+1+2μ2α2λUα2-1-λU-α2/2+1.Considering only one-dimensional state, the strain can be expressed as a function of elongation,(10)εU=λU-1.According to (8), (9), and (10) expression for the stress can be derived:(11)σU=∑i=122μiαiεU+1αi+1-εU+1-αi/2+1.By stretching rubber samples tested for stress-strain data fitting can be drawn from material constitutive parametersμi,  αi.Formulas (11) to (2) can be obtained for the damage stress(12)σ~=∑i=122μiαi1-DεU+1αi+1-εU+1-αi/2+1.Formula for damage factor is [31](13)D˙=-∂φ∗∂y.In the formula,φ∗ is damage dissipation potential;  y is damage strain energy release rate.Let dissipation potential function be of the form [32](14)φ∗=ba+1-yba+1,where a,  b are material constants which can be determined by fatigue tests.Damage strain energy is a function of stress injury and damage strain energy release rate can be defined as(15)-y=∂U∂D=∂Uσ~∂D=∑i=13∂U∂λi∂λi∂D.By formula (2) the following can be derived:(16)∂σ~∂D=σ1-D2=∂σ~∂λi∂λi∂D.It can be further derived that(17)∂λi∂D=σ~1-D∂σ~∂λi-1.Under uniaxial stress state, the second-order Ogden model consists of (16) and (17) can obtain damage strain energy release rate which can be expressed as(18)-y=∂Uσ~∂D=2μ1α11-D2λUα1-1-λU-α1/2+12α1-1λUα1-2+α1/2+1λU-α1/2+2+2μ2α21-D2λUα2-1-λU-α2/2+12α2-1λUα2-2+α2/2+1λU-α2/2+2.Under uniaxial loading, effective stress injury isσf; then by damage factor (2), (8), (13), (14), and (18) can be derived:(19)D˙=σfb1-Da.Under a single cyclic loading, fatigue damage equivalent stress mainly consists of the scope and magnitude of maximum stress, strain amplitude being related.From this, the next single cycle fatigue damage can be expressed as(20)∂D∂N=σfb1-Da,where N is the cycle. Formula (20) is obtained by integrating both sides(21)∫0D1-DadD=∫0Nfσfb-1adN.Using the initial condition (N=0,D=0) and destroying the conditions (N=Nf,D=1), integral to obtaining the fatigue life expression,(22)Nf=ba1+aσf-a.


## 3. Thrust Rod Finite Element Analysis

In order to obtain the equivalent stress and fatigue crack location, the finite element method is used to simulate the thrust rod, combined with rubber material specimen testing to verify stress calculations and then the fatigue life prediction.


### 3.1. Rubber Material Model

In order to obtain the constitutive properties thrust rod rubber material, project partners in accordance with national standards GB/T528-1998 vulcanized rubber or thermoplastic rubber tensile stress-strain properties using dumbbell specimen uniaxial tensile test. Rubber sample during a tensile test at room temperature is used to obtain a stress-strain curve, as shown in Figure2.Figure 2
Rubber material tensile stress-strain.Using ABAQUS software least squares fit function, the experimental data for the stress-strain curve are compared with several models, as shown in Figure3. Being only under uniaxial test data, second- order Ogden model can better reflect the mechanical response of the loading condition of the rubber material and the experimental data better. So choose second-order Ogden model as rubber constitutive model, model parameters obtained as shown in Table 1.Table 1
Ogden second-order model fitting parameters.
i μ i α i D i 1 0.0009 13.1737 0.01759 2 2.2885 −3.1471 0Figure 3
Rubber material model.


### 3.2. Finite Element Modeling Thrust Rod

The main thrust rod is comprised of a ball, ball joints, and sleeve. Spherical hinge has two stub ends; on both sides of the cover, the rubber body, the plastic layer, and the mandrel five are partially vulcanized together, wherein the rubber layer vulcanized under the load, the deformation by vibration and impact energy absorption part have complex loads; deformation is easily damaged element. Therefore, thrust rod life depends mainly on the fatigue life of the rubber body. When the meshing of rubber and plastic part is C3D8H unit, plastic layer material density is 1.2 × 10−6 kg/mm3, Poisson’s ratio is 0.40, and a modulus of elasticity is 1384 N/mm2. Metal materials used are C3D8R unit, density of 7.8 × 10−6 kg/mm3, Poisson’s ratio of 0.30, and an elastic modulus of 2.10 × 105 N/mm2. Thrust rod big end ball joint screw mandrel has 150 kN applied load; two small ball joint spindle fixed constraints were applied to extract data center reference point. V-type thrust rod finite element model is shown in Figure 4.Figure 4
V-type thrust rod finite element model.Ball joints in the radial load and deformation of rubber are shown in Figure5. One side of the rubber will produce compressive deformation (A location), while the other side will produce the tensile deformation of the rubber (B location). Because of the incompressible rubber characteristics, the area of the free end of the compression side rubber will expand outwardly and cause nearby rubber tensile deformation. If the compression is too large and rubber vulcanization of the mandrel and the expansion of the free surface at the end of the tensile stress happen, the other side of the rubber tension will produce tensile stress; tensile stress is caused by rubber and metal bonded face of the plastic and rubber cracking and accelerates the aging of the main factors of rubber.Figure 5
The radial loads by deformation of rubber ball joints schematic.To improve the fatigue resistance of the rubber certain precompression is carried out to reduce or even eliminate the tensile deformation caused by the radial load. Between the rubber body and the outer part, 2 mm and 5 mm, respectively, precompression are in radial and axial directions. Extraction of stress after assembly rubber strain distribution was done to analyze the performance of the original rubber bushing. Big-end rubber calculated maximum tensile principal stress is 13.86 Mpa and plastic maximum tensile principal stress is 6.861 Mpa, which are within the allowable range of stress.


### 3.3. Finite Element Results

Applying longitudinal load of 150 kN ball joints on both sides of the big end of the mandrel thrust rod, the rubber layer results obtained are shown in Figures6 and 7.Figure 6
The rubber layer maximum stress.Figure 7
Rubber layer maximum logarithmic strain.Finite element analysis results show that the longitudinal loading and fatigue failure occur in first location; cloud colour shows the deep region; the fatigue test results also validate the fact that the region is a dangerous area; first fatigue cracks easily. The maximum number of principal strains calculated value of 0.809. Extracting three directions of hazardous area rubber units of the main stress by formula (4) and (5) the calculated equivalent stress is as shown in Table 2.Table 2
Equivalent stress hazardous areas.
Unit number σ 1 σ 2 σ 3 σ f 578978 14.071 4.911 4.205 15.485 578759 8.542 2.482 0.554 8.913 578724 12.265 7.247 8.013 16.345 578754 8.189 3.050 4.066 9.638 563230 11.832 10.219 9.156 18.118


## 3.1. Rubber Material Model

In order to obtain the constitutive properties thrust rod rubber material, project partners in accordance with national standards GB/T528-1998 vulcanized rubber or thermoplastic rubber tensile stress-strain properties using dumbbell specimen uniaxial tensile test. Rubber sample during a tensile test at room temperature is used to obtain a stress-strain curve, as shown in Figure2.Figure 2
Rubber material tensile stress-strain.Using ABAQUS software least squares fit function, the experimental data for the stress-strain curve are compared with several models, as shown in Figure3. Being only under uniaxial test data, second- order Ogden model can better reflect the mechanical response of the loading condition of the rubber material and the experimental data better. So choose second-order Ogden model as rubber constitutive model, model parameters obtained as shown in Table 1.Table 1
Ogden second-order model fitting parameters.
i μ i α i D i 1 0.0009 13.1737 0.01759 2 2.2885 −3.1471 0Figure 3
Rubber material model.


## 3.2. Finite Element Modeling Thrust Rod

The main thrust rod is comprised of a ball, ball joints, and sleeve. Spherical hinge has two stub ends; on both sides of the cover, the rubber body, the plastic layer, and the mandrel five are partially vulcanized together, wherein the rubber layer vulcanized under the load, the deformation by vibration and impact energy absorption part have complex loads; deformation is easily damaged element. Therefore, thrust rod life depends mainly on the fatigue life of the rubber body. When the meshing of rubber and plastic part is C3D8H unit, plastic layer material density is 1.2 × 10−6 kg/mm3, Poisson’s ratio is 0.40, and a modulus of elasticity is 1384 N/mm2. Metal materials used are C3D8R unit, density of 7.8 × 10−6 kg/mm3, Poisson’s ratio of 0.30, and an elastic modulus of 2.10 × 105 N/mm2. Thrust rod big end ball joint screw mandrel has 150 kN applied load; two small ball joint spindle fixed constraints were applied to extract data center reference point. V-type thrust rod finite element model is shown in Figure 4.Figure 4
V-type thrust rod finite element model.Ball joints in the radial load and deformation of rubber are shown in Figure5. One side of the rubber will produce compressive deformation (A location), while the other side will produce the tensile deformation of the rubber (B location). Because of the incompressible rubber characteristics, the area of the free end of the compression side rubber will expand outwardly and cause nearby rubber tensile deformation. If the compression is too large and rubber vulcanization of the mandrel and the expansion of the free surface at the end of the tensile stress happen, the other side of the rubber tension will produce tensile stress; tensile stress is caused by rubber and metal bonded face of the plastic and rubber cracking and accelerates the aging of the main factors of rubber.Figure 5
The radial loads by deformation of rubber ball joints schematic.To improve the fatigue resistance of the rubber certain precompression is carried out to reduce or even eliminate the tensile deformation caused by the radial load. Between the rubber body and the outer part, 2 mm and 5 mm, respectively, precompression are in radial and axial directions. Extraction of stress after assembly rubber strain distribution was done to analyze the performance of the original rubber bushing. Big-end rubber calculated maximum tensile principal stress is 13.86 Mpa and plastic maximum tensile principal stress is 6.861 Mpa, which are within the allowable range of stress.


## 3.3. Finite Element Results

Applying longitudinal load of 150 kN ball joints on both sides of the big end of the mandrel thrust rod, the rubber layer results obtained are shown in Figures6 and 7.Figure 6
The rubber layer maximum stress.Figure 7
Rubber layer maximum logarithmic strain.Finite element analysis results show that the longitudinal loading and fatigue failure occur in first location; cloud colour shows the deep region; the fatigue test results also validate the fact that the region is a dangerous area; first fatigue cracks easily. The maximum number of principal strains calculated value of 0.809. Extracting three directions of hazardous area rubber units of the main stress by formula (4) and (5) the calculated equivalent stress is as shown in Table 2.Table 2
Equivalent stress hazardous areas.
Unit number σ 1 σ 2 σ 3 σ f 578978 14.071 4.911 4.205 15.485 578759 8.542 2.482 0.554 8.913 578724 12.265 7.247 8.013 16.345 578754 8.189 3.050 4.066 9.638 563230 11.832 10.219 9.156 18.118


## 4. Life Prediction and Verification

### 4.1. Fatigue Life Prediction

Rubber thrust rod element is according to the fatigue life of the life of the rubber dumbbell specimens to predict dumbbell test piece of rubber as shown in Figure8. The displacement control is used on hydraulic servo dynamic testing machine for specimen uniaxial tensile fatigue test, sine wave displacement control, and test frequency of 1 Hz.Figure 8
Rubber dumbbell specimens.For uniform stretching state in the lengthL of the rubber rod, taking into account the fact that the nearly incompressible rubber material is compressed, the configuration before and after deformation can be described as [33](23)x1=λX1,x2=λ-1/2X2,x3=λ-1/2X3,whereinXi,  xi  (i=1,2,3) were predeformed position coordinates of a point on the rubber specimen. λ is an X1 direction elongation, defined as the ratio of the length of the specimen before and after the deformation direction X1 of λ=l/L, l length after stretching. When the lever tension happens, λ>1; when the lever compression happens,0<λ<1.Axial load being along the rubber specimen, the deformation gradient rubber sheet specimen amount is [33](24)F=∂x1∂X1∂x1∂X2∂x1∂X3∂x2∂X1∂x2∂X2∂x2∂X3∂x3∂X1∂x3∂X2∂x3∂X3=λ000λ-1/2000λ-1/2.Thus, the right Green deformation tensorC and left Green deformation tensor B are equal [33],(25)B=C=FTF.Green-Lagrange strain tensorE and Almansi-Euler strain tensor e, respectively, by the left and right Green deformation tensors B and C derived.It follows that(26)E=12C-I=12λ2-1000λ-1-1000λ-1-1,e=12I-B-1=121-λ-20001-λ0001-λ.Therefore, their main value is the three elements of the corresponding diagonal tensor.When the lever is in a stretched condition,λ>1. According to engineering strain relations with each component of Green-Lagrange strain between the availability axially stretched rubber specimens, the maximum principal engineering strain is(27)εE=λ-1.Formula:E11 is tensor E elements in the first row and first column, also called the strain tensor components correspondingly.Logarithmic strain can be seen, and the direction ofX1 is the biggest main logarithmic strain εL [34],(28)εL=∫Lldll=ln⁡lL=ln⁡λ.Under constant amplitude alternating load, the maximum principal strain and the cycle change over time.Specimen ball joints rubber engineering stress and engineering strain relationships and effective stress and fatigue life relations are shown in Figures9 and 10. Test specimens stress and strain data by least squares fitting correlation coefficient R2 equals 0.99, to give power function equation (29).(29)y=2.45x2+2.68x-0.118.Figure 9
Engineering stress and engineering strain relations.Figure 10
Equivalent stress and life curve of dumbbell specimen.By dumbbell tensile fatigue test pieces fitting the data, equivalent stress calculated according to formula (4) and (5) and life prediction equation (22) can obtain injury parameters a and b which are a=2.83, b=2059 MPa. The finite element calculation to obtain the maximum thrust rod rubber logarithmic principal strain 0.809 into (28) calculated primary elongation; then the main elongation calculated values are substituted into formula (27) which calculated the maximum principal engineering strain by rubber test member of the engineering stress-strain relation (29), corresponding to the calculated stress test. Then the data into the life prediction equation (22) calculated the fatigue life of the thrust rod. Where the stress test is 18.18, the maximum value and the finite element equivalent stress are 18.118, and the error has 0.3% more consistent results, suggesting that the finite element calculation results are accurate, so you can use the finite element calculation results which were thrust rod life prediction. Calculate the predicted fatigue life of the thrust rod being 171,394 times numbered 563,230 units.


### 4.2. Fatigue Test

Thrust rod specimen bench life on the electrohydraulic servo fatigue test system applied 150 kN vertical alternating load on the specimen for the sine wave loading, loading frequency of 1 Hz, to maintain the test temperature by cooling fan around 25°C. In cycle alternating loads, the thrust rod stiffness reduced stiffness by 30% that is considered failed.By observing the rod fatigue test we found that, under cyclic loading, thrust rod big end ball joint rubber body and the metal core bond area near the first wrinkles increase fatigue cycles between the mandrel and the end cap and extruded rubber body phenomenon began to appear, gradually to the accumulation to both sides along the circumference, after the completion of 296,500 times of fatigue cycles of products in the form of a broken ring shown in Figure11.Figure 11
Extruded fail of rubber ball joints.Appearing near the rubber body bonded with the mandrel significant rubber extrusion, the ball joints are cut; its internal structure is shown in Figure12. Fatigue should be a source of rubber and metal curing adhesive (shown in the red), and fatigue damage is consistent with the location of the finite element analysis.Figure 12
Rubber fatigue failure location.The life expectancy is less than the thrust rod specimen measured fatigue life, which may have greater fatigue damage associated with the selected parameters; test life and life expectancy ratio is 1.73, and the fatigue life prediction in the range of 2 times the measured fatigue life prediction effect is more ideal. Factors that affect the life prediction results are geometric material constant fatigue prediction models obtained through fitting test and parameter fitting method used (there is a certain effect on prediction accuracy); prediction model for the damage equivalent stress range parameters calculated from the engineering stress and engineering strain verification, equivalent stress-dependent finite element results, and thus the accuracy of the finite element analysis also affect the life of the predicted value.


## 4.1. Fatigue Life Prediction

Rubber thrust rod element is according to the fatigue life of the life of the rubber dumbbell specimens to predict dumbbell test piece of rubber as shown in Figure8. The displacement control is used on hydraulic servo dynamic testing machine for specimen uniaxial tensile fatigue test, sine wave displacement control, and test frequency of 1 Hz.Figure 8
Rubber dumbbell specimens.For uniform stretching state in the lengthL of the rubber rod, taking into account the fact that the nearly incompressible rubber material is compressed, the configuration before and after deformation can be described as [33](23)x1=λX1,x2=λ-1/2X2,x3=λ-1/2X3,whereinXi,  xi  (i=1,2,3) were predeformed position coordinates of a point on the rubber specimen. λ is an X1 direction elongation, defined as the ratio of the length of the specimen before and after the deformation direction X1 of λ=l/L, l length after stretching. When the lever tension happens, λ>1; when the lever compression happens,0<λ<1.Axial load being along the rubber specimen, the deformation gradient rubber sheet specimen amount is [33](24)F=∂x1∂X1∂x1∂X2∂x1∂X3∂x2∂X1∂x2∂X2∂x2∂X3∂x3∂X1∂x3∂X2∂x3∂X3=λ000λ-1/2000λ-1/2.Thus, the right Green deformation tensorC and left Green deformation tensor B are equal [33],(25)B=C=FTF.Green-Lagrange strain tensorE and Almansi-Euler strain tensor e, respectively, by the left and right Green deformation tensors B and C derived.It follows that(26)E=12C-I=12λ2-1000λ-1-1000λ-1-1,e=12I-B-1=121-λ-20001-λ0001-λ.Therefore, their main value is the three elements of the corresponding diagonal tensor.When the lever is in a stretched condition,λ>1. According to engineering strain relations with each component of Green-Lagrange strain between the availability axially stretched rubber specimens, the maximum principal engineering strain is(27)εE=λ-1.Formula:E11 is tensor E elements in the first row and first column, also called the strain tensor components correspondingly.Logarithmic strain can be seen, and the direction ofX1 is the biggest main logarithmic strain εL [34],(28)εL=∫Lldll=ln⁡lL=ln⁡λ.Under constant amplitude alternating load, the maximum principal strain and the cycle change over time.Specimen ball joints rubber engineering stress and engineering strain relationships and effective stress and fatigue life relations are shown in Figures9 and 10. Test specimens stress and strain data by least squares fitting correlation coefficient R2 equals 0.99, to give power function equation (29).(29)y=2.45x2+2.68x-0.118.Figure 9
Engineering stress and engineering strain relations.Figure 10
Equivalent stress and life curve of dumbbell specimen.By dumbbell tensile fatigue test pieces fitting the data, equivalent stress calculated according to formula (4) and (5) and life prediction equation (22) can obtain injury parameters a and b which are a=2.83, b=2059 MPa. The finite element calculation to obtain the maximum thrust rod rubber logarithmic principal strain 0.809 into (28) calculated primary elongation; then the main elongation calculated values are substituted into formula (27) which calculated the maximum principal engineering strain by rubber test member of the engineering stress-strain relation (29), corresponding to the calculated stress test. Then the data into the life prediction equation (22) calculated the fatigue life of the thrust rod. Where the stress test is 18.18, the maximum value and the finite element equivalent stress are 18.118, and the error has 0.3% more consistent results, suggesting that the finite element calculation results are accurate, so you can use the finite element calculation results which were thrust rod life prediction. Calculate the predicted fatigue life of the thrust rod being 171,394 times numbered 563,230 units.


## 4.2. Fatigue Test

Thrust rod specimen bench life on the electrohydraulic servo fatigue test system applied 150 kN vertical alternating load on the specimen for the sine wave loading, loading frequency of 1 Hz, to maintain the test temperature by cooling fan around 25°C. In cycle alternating loads, the thrust rod stiffness reduced stiffness by 30% that is considered failed.By observing the rod fatigue test we found that, under cyclic loading, thrust rod big end ball joint rubber body and the metal core bond area near the first wrinkles increase fatigue cycles between the mandrel and the end cap and extruded rubber body phenomenon began to appear, gradually to the accumulation to both sides along the circumference, after the completion of 296,500 times of fatigue cycles of products in the form of a broken ring shown in Figure11.Figure 11
Extruded fail of rubber ball joints.Appearing near the rubber body bonded with the mandrel significant rubber extrusion, the ball joints are cut; its internal structure is shown in Figure12. Fatigue should be a source of rubber and metal curing adhesive (shown in the red), and fatigue damage is consistent with the location of the finite element analysis.Figure 12
Rubber fatigue failure location.The life expectancy is less than the thrust rod specimen measured fatigue life, which may have greater fatigue damage associated with the selected parameters; test life and life expectancy ratio is 1.73, and the fatigue life prediction in the range of 2 times the measured fatigue life prediction effect is more ideal. Factors that affect the life prediction results are geometric material constant fatigue prediction models obtained through fitting test and parameter fitting method used (there is a certain effect on prediction accuracy); prediction model for the damage equivalent stress range parameters calculated from the engineering stress and engineering strain verification, equivalent stress-dependent finite element results, and thus the accuracy of the finite element analysis also affect the life of the predicted value.


## 5. Conclusion

The theoretical equation for prediction fatigue life of rubber is established based on the equivalent stress. Continuum damage mechanics method can be used to predict fatigue life. The predicted life and experimental data have strong correlation. This supports the idea that the equivalent stress model based on continuum damage mechanics can be applied to study fatigue life for the thrust rod.



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*Source: 1254037-2016-10-27.xml*