An Ordinary State-Based Peridynamic Model for Fatigue Cracking of Ferrite and Pearlite Wheel Material

applied sciences

Article An Ordinary State-Based Peridynamic Model for Fatigue Cracking of Ferrite and Pearlite Wheel Material

Junzhao Han and Wenhua Chen *

Department of Mechanical Engineering, Zhejiang University, Hangzhou 310027, China; [email protected] * Correspondence: [email protected]

Received: 19 May 2020; Accepted: 19 June 2020; Published: 24 June 2020

Abstract: To deal with a new-developed ferrite and pearlite wheel material named D1, an alternative ordinary state-based peridynamic model for fatigue cracking is introduced due to cyclic loading. The proposed damage model communicates across the microcrack initiation to the macrocrack growth and does not require additional criteria. Model parameters are verified from experimental data. Each bond in the deformed material configuration is built as a fatigue specimen subjected to variable amplitude loading. Fatigue crack initiation and crack growth developed naturally over many loading cycles, which is controlled by the parameter “node damage” within a region of finite radius. Critical damage factors are also imposed to improve efficiency and stability for the fatigue model. Based on the improved adaptive dynamic relaxation method, the static solution is obtained in every loading cycle. Convergence analysis is presented in smooth fatigue specimens at different loading levels. Experimental results show that the proposed peridynamic fatigue model captures the crack sensitive location well without extra criteria and the fatigue life obtained from the simulation has a good correlation with the experimental results.

Keywords: peridynamic model; ferrite and pearlite wheel material; fatigue cracking; crack initiation and propagation; fatigue life

1. Introduction Fatigue of metallic materials is a cumulative and irreversible process [1–3]. Under cyclic loading, the new dynamic balances are broken and the voids are generated, so that the damage will be initiated and accumulated [4–6]. The whole fatigue life is essentially a multiscale phenomenon [7–11], depending strongly on the material, geometry of the loaded body, the external loaded forces conditions, environmental factors, etc. The life prediction schemes of ferrite and pearlite wheel material under multiaxial loading have been through extensive research both in industry and scientific institution [12–14]. Modeling of fatigue crack growth in metallic materials encounters many problems [15–17]. Based on the framework of traditional continuum theory, the finite element method (FEM) models or various modified versions aim to find the kinetic relations among parameters near the fatigue crack tip that characterize the fatigue crack evolution [18–20]. By redefining the body, the crack is treated as a boundary so that extra criteria, such as fatigue crack growth speed and direction which guide the crack path, are necessary [21–23]. To solve these problems, Linear Elastic Fracture Mechanics (LEFM) use Cohesive Zone Elements (CZE) to deal with Mode-I crack mode and mixed-mode fracture [24–27]. The number of cohesive elements increases with decreasing mesh size, yet the size of the continuum region remains the same. Frequent redefining the body is difficult and costly, especially with multiple interacting cracks. The extended finite element method (XFEM), using the local enrich functions, permits the cracks to propagate on any element surface without remeshing in

Appl. Sci. 2020, 10, 4325; doi:10.3390/app10124325 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 4325 2 of 23 every incremental crack growth [28–32].Both the FEM model and the XFEM method need extra criteria such as the time of forming fatigue cracks, fatigue propagation speed, direction, branch, arrest, etc. The final fatigue fracture mechanisms are associated with grain boundaries, dislocations, microcracks, anisotropy, etc [33–35]. Each of them plays an important role at a specific length scale. It is difficult to obtain the location of fatigue initiation in advance, let alone the extra criteria from the experimental data [36–38]. In contrast, a peridynamic theory (PD) uses the spatial integral equations as opposed to derivatives of the displacement field to compute internal force acting on a material particle [39–41]. Material damage is part of the constitutive model. Without the need for some special crack growth criteria, peridynamics allows the cracks to propagate at multiple sites with natural paths not only along the element boundary in a consistent framework. Furthermore, the PD theory can link the micro to macro length scales [42–44]. The fatigue crack model with peridynamics was originally proposed by Oterkus, Guven, and Madenci [45] and substantially improved by and Askari [46]. Two phases of fatigue failure: crack initiation, fatigue propagation, are included in a consistent fatigue model and the parameters of the model are calibrated separately with experimental data. In ref [47]. the dynamic relaxation method is used to obtain the static solution. Nucleation and growth of a helical fatigue crack are demonstrated by using an aluminum alloy rod. Zhang [48] presented the conjugate gradient energy minimization method to obtain the static solution and applied this fatigue model to two-phase composite materials. The results show that the peridynamic fatigue crack model can deal with multiple crack without extra criteria to guide the crack path [49–51]. However, these peridynamic fatigue models are mainly focus on the simple linear elastic property. And the fatigue crack criteria just use the critical bond elongation technic so that it is difficulty to build effective links between the actual physical parameters and the current models. Moreover, few peridynamic fatigue models take into the cases of multiaxial fatigue failure. To bridge the micro fatigue crack initiation to the macro fatigue crack growth, an alternative ordinary state-based peridynamic model for fatigue cracking is proposed. Based on the thermal disturbance of atomic motion, the theoretical foundation for the peridynamic fatigue model is built under cyclic loading. Fatigue model parameters are verified from S-N data of ferrite and pearlite wheel material. Each bond in the deformed material configuration is treated as a fatigue specimen subjected to variable amplitude loads. Bond damage accumulated over time, according to the cyclic strain in the bond that its progressive failure is characterized by a history variable called “wear-out life”. A bond will be broken when the variable reaches the entire life. Fatigue crack initiation and crack growth formulated naturally over many loading cycles which is controlled by the parameter “node damage” within a region of finite radius. Critical damage factors are also imposed to improve efficiency and stability for the fatigue model. Based on the improved adaptive dynamic relaxation method, the static solution is obtained in every loading cycle. Convergence analysis is presented in the smooth fatigue specimens at different loading levels. Experimental results show that the peridynamic results capture the crack sensitive location well without extra criteria. Fatigue lifetimes obtained from the simulation have a good correlation with the experimental results.

2. Mechanism of Fatigue Damage Evolution

2.1. Fatigue Damage Evolution From the point of a material’s atomic structure, the atoms vibrate in high frequency (about 1012~1013 Hz) near the equilibrium position. Each atom in the material has certain energy whose value is random and diﬀerent from other atoms. When the active energy of an atom exceeds a certain value, the atom Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 22

where k is the Boltzmann constant, 푇 is thermodynamic temperature, 푄 is the critical active energy that is the smallest value of an atom escaping from the equilibrium position. As shown in Figure 1a, without external loading, although the critical energy 푄 is larger than other loading condition, there is still some atoms escaping from the original equilibrium position and the escaped atoms reach to a new dynamic vacant site equilibrium state. Hence, in such a loading condition, damage evolution will be negligible. Appl. Sci. 2020, 10, 4325 3 of 23 As shown in Figure 1b, with the static loading, the critical active energy Q is described as follows: 2 휏푏 + 훼휏푏휏 + 훽휏 푄 = (2) will escape from its original equilibrium position and2 a퐺 void will be generated. The probability of an atom’swhere active 휏푏 is energyideal material above a shear certain strength, value Q 휏is is described the external as follows: shear stress, 퐺 is the shear modulus, 훼 is the constant associated with stress concentration factor, 훽 is the constant associated with the Q amplitude of the static loading. CombiningP =Equationsexp (1) and (2), the probability of an atom’s active(1) −kT energy above a certain value Q under static loading is described as follows: where k is the Boltzmann constant, T is thermodynamic휏 + 훼 temperature,휏 휏 + 훽휏2 Q is the critical active energy that 푃 = exp (− 푏 푏 ) (3) is the smallest value of an atom escaping from the equilibrium2퐺푘푇 position. whereAs shownit can be in seen Figure that1a, the without probability external of an loading, atom escaping although from the critical the equilibrium energy Q positionis larger under than otherstatic loading loading condition, increased therecompared is still with some no atoms external escaping loading from condition. the original The displacement equilibrium position u0 of an atom and theaway escaped from atoms the original reach to equilibrium a new dynamic position vacant is invariable. site equilibrium Therefore, state. the Hence, voids in generated such a loading by the condition,thermal disturbance damage evolution can be annihilated. will be negligible.

(a) No external loading (b) Static loading (c) Cyclic loading

FigureFigure 1. 1.The The thermal thermal disturbance disturbance of of atomic atomic motion. motion. ( a(a)) No No external external loading; loading; ( b(b)) Static Static loading; loading; ( c()c) CyclicCyclic loading. loading.

AsAs shown shown in in Figure Figure1b, 1c with, with the staticthe cyclic loading, loading, the critical the displacement active energy ofQ anis described atom away as from follows: the original equilibrium position is alternating. The voids generated by the thermal disturbance cannot 2 τb + ατbτ + βτ be annihilated so that the damage will Qbe= initiated and accumulated. (2) 2G 2.2. Fatigue Damage Evolution Law where τb is ideal material shear strength, τ is the external shear stress, G is the shear modulus, α is the constantThe associated change rate with caused stress by concentration atomic escape factor, is describedβ is the constant as follows: associated with the amplitude of the static loading. Combining Equations (1) and (2), the probability푄 of an atom’s active energy above a 푣 = 퐶exp (− ) (4) certain value Q under static loading is described as follows:푘푇 where C is the material constant that contributes to the change rate caused by the probability of 2 ! atomic escape. τb + ατbτ + βτ P = exp (3) There is a direct correlation between the− current2GkT stress and the fatigue damage under cyclic loading for the critical energy The damage evolution rate of fatigue under such loading condition is where it can be seen that the probability of an atom escaping from the equilibrium position under static described as follows [52]: loading increased compared with no external loading condition. The displacement u0 of an atom away 푑퐷 푄(휎 , 휎 , 퐷 ) from the original equilibrium position is invariable.= 퐶exp (− Therefore,푎 푚 ) the voids generated by the thermal(5) 푑푁 푘푇 disturbance can be annihilated. where 휎 is cyclic stress amplitude, 휎 is mean stress, D is the current damage quantity. As shown푎 in Figure1c, with the cyclic푚 loading, the displacement of an atom away from the original equilibrium position is alternating. The voids generated by the thermal disturbance cannot be annihilated so that the damage will be initiated and accumulated.

2.2. Fatigue Damage Evolution Law The change rate caused by atomic escape is described as follows:

Q v = C exp (4) −kT where C is the material constant that contributes to the change rate caused by the probability of atomic escape. Appl. Sci. 2020, 10, 4325 4 of 23

Appl. Sci. 20There20, 10, isx FOR a direct PEER correlationREVIEW between the current stress and the fatigue damage under cyclic4 of 22 loading for the critical energy The damage evolution rate of fatigue under such loading condition is 3. Peridynamicdescribed as Theory follows [for52]: Fatigue Cracking ! dD Q(σa, σm, D) = C exp (5) 3.1. Basic Theory dN − kT

whereAccordingσa is cyclic to peridynamic stress amplitude, theory,σm isthe mean physics stress, ofD isa thematerial current body damage at a quantity. point interacts with all points within its horizon, as shown in Figure 2. The equation of motion of the material point 퐱 in 3. Peridynamic Theory for Fatigue Cracking (푘) the deformed configuration is described as follows [53]: 3.1. Basic Theory 휌 퐮̈ (퐱 , 푡 ) = ∫ (퐓( 퐱 , 푡 )〈 퐱 − 퐱 〉− 퐓( 퐱 , 푡 )〈 퐱 − 퐱 〉) 푑ℋ + 퐛(퐱 , 푡 ) (푘) (푘) (푘) (푗) (푘) (푗) (푘) (푗) (퐱(푘)) (푘) According toℋ peridynamic− theory, the physics− of a material body at a point interacts with all (6) (퐱(푘)) points within its horizon, as shown in Figure2. The equation of motion of the material point x(k) in the wheredeformed 퐛(퐱(푘),푡 conﬁguration) is the body is describedload vector, as follows 퐓(퐱(푘) [,53푡)]: and 퐓(퐱(푗),푡) are the force vector states, ℋ(퐱 ) − − (푘) Z is the horizon of.. point 퐱(푘), 퐮(퐱(푘),푡) is the displacement vector at time t. ρ(k)u x(k), t = T x(k), t x(j) x(k) T x(j), t x(k) x(j) d (x ) + b x(k), t (6) As shown in Figure 2, the bond¯ 휉 strain between¯ material points 퐱(k) and 퐱 is described ( ) 푘푗h − i− h − i H (푘) (푗) H x(k) as follows: where b x(k), t is the body load vector, T x((|k퐲), t − and퐲 T|− x| (퐱j), t − are퐱 the|) force vector states, (x ) is the ¯ (푗) (푘)¯ (푗) (푘) H (k) 푠(푘)(푗) = (7) horizon of point x(k), u x(k), t is the displacement| vector퐱(푗)−퐱 at(푘 time)| t.

FigureFigure 2. 2.TheThe ordinary ordinary state state-based-based peridynamicperidynamic model. model.

As shown in Figure2, the bond ξ strain between material points x and x is described As shown in Figure 3, all of the relativekj position vectors in the horizon(k )of point(j) 퐱(푘), (퐲(푗)−퐲(푘)) (푗= 1 as, 2 , follows:. . . , ∞) is described as follows: y(j) y(k) x(j) x(k) − 퐲 (−1) − 퐲−(푘) s = (7) (k)(j) 퐘(퐱(푘),푡) = x({j) x(⋮k) } − 퐲 − − 퐲 (8) (∞) (푘) As shown in Figure3, all of the relative position vectors in the horizon of point x(k), y(j) y(k) (퐲(푗)−퐲(푘)) = 퐘(퐱(푘),푡)〈퐱(푗)−퐱(푘)〉 − (j = 1, 2, ... , ) is described as follows:{ − ∞ where 퐘(퐱(푘),푡) is the deformation vector state. Equation (8) means that the response of a material −   y y    (1) (k)  point 퐱( ) depends collectively on the deformation  of all− bonds connected to the point. 푘   .   Y x(k), t =  .   ¯   (8)      y( ) y(k)   ∞ −   y( ) y( ) = Y x(k), t x(j) x(k) j − k ¯ h − i where Y x(k), t is the deformation vector state. Equation (8) means that the response of a material ¯ point x(k) depends collectively on the deformation of all bonds connected to the point.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 22

3. Peridynamic Theory for Fatigue Cracking

3.1. Basic Theory According to peridynamic theory, the physics of a material body at a point interacts with all

points within its horizon, as shown in Figure 2. The equation of motion of the material point 퐱(푘) in the deformed configuration is described as follows [53]:

휌 퐮̈ (퐱 , 푡 ) = ∫ (퐓( 퐱 , 푡 )〈 퐱 − 퐱 〉− 퐓( 퐱 , 푡 )〈 퐱 − 퐱 〉) 푑ℋ + 퐛(퐱 , 푡 ) (푘) (푘) (푘) (푗) (푘) (푗) (푘) (푗) (퐱(푘)) (푘) ℋ − − (6) (퐱(푘))

where 퐛(퐱(푘),푡) is the body load vector, 퐓(퐱(푘),푡) and 퐓(퐱(푗),푡) are the force vector states, ℋ(퐱 ) − − (푘)

is the horizon of point 퐱(푘), 퐮(퐱(푘),푡) is the displacement vector at time t. As shown in Figure 2, the bond 휉푘푗 strain between material points 퐱(푘) and 퐱(푗) is described as follows:

(|퐲(푗)−퐲(푘)|−|퐱(푗)−퐱(푘)|) 푠(푘)(푗) = (7) |퐱(푗)−퐱(푘)|

Figure 2. The ordinary state-based peridynamic model.

As shown in Figure 3, all of the relative position vectors in the horizon of point 퐱(푘), (퐲(푗)−퐲(푘)) (푗= 1 , 2 , . . . , ∞) is described as follows: 퐲(1) − 퐲(푘)

퐘(퐱(푘),푡) = { ⋮ } − 퐲 − 퐲 (8) (∞) (푘)

(퐲(푗)−퐲(푘)) = 퐘(퐱(푘),푡)〈퐱(푗)−퐱(푘)〉 { −

where 퐘(퐱(푘),푡) is the deformation vector state. Equation (8) means that the response of a material Appl. Sci.−2020, 10, 4325 5 of 23 point 퐱(푘) depends collectively on the deformation of all bonds connected to the point.

Figure 3. Deformation vector state Y x( ), t ¯ k As shown in Figure4, the force vector state T x(k), t including an inﬁnite-dimension array force ¯ density vectors, t(k)(j), is described as follows:

z : z α < r (9) { | − | }    t(k)(1)     .  T x(k), t =  .  (10) ¯      t(k)( )  ∞ where the force density vector, t(k)(j), that the material point at location x(j) exerts on the material point at location x(k) can be expressed as: t(k)(j) u(j) u(k), x(j) x(k), t = T x(k), t x(j) x(k) (11) − − ¯ h − i Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 22

FigureFigure 4. 4.ForceForce vector vector statestateT 퐓x(k퐱),(t푘.),푡). ¯ −

3.2. Energy-Based Failure Criterion

As shown in Figure 5, the force vector states 퐓(퐱(푘),푡) and 퐓(퐱(푗),푡) are opposite in direction − − and parallel to their deformed relative deformed position. However, their values are not equal.

Figure 5. Force vector state and the bond 휉.

As shown in Figure 6, the force density 푓 = (푡[퐱(푘),푡]−푡[퐱(푗),푡]) exhibits non-linearity under (푘푗) − − cycle loading, therefore the total energy density stored in the bond 휉 can be expressed as follows.

휀휉(∆푡) 휔휉 = ∫ (푓(푘푗 )) 푑휀 (16) 0 휉

Figure 6. Critical bond strain damage model.

As shown in Figure 7, all points 푃푑표푤푛 along the dashed line 푙, connected to all points 푃푢푝 across fracture plane 훾 of unit area and the volume above the fracture plane. The energy density required to open the plane 훾 is as follows: 훿 2휋 훿 cos−1(푧/ 휉 ) 2 푤훾 = ∫ ∫ ∫ ∫ 푤푐휉 sin휙푑휙푑휉푑휃푑푧 (17) 0 0 푧 0 where 푤푐 is the critical energy density, 푤훾 is the energy release rate defined in the continuum damage mechanics theory. Appl. Sci. 2020, 10, 4325 6 of 23

As for ordinary state-based peridynamic theory, the scalar state of the force density vector can be expressed as   d   ωx ωe   t = 3kθ −− + 15G −−   A A     −   A = ωx x   · f or3D   − − θx−   d   e = e −   3   − − −ωx e   ·   θ = 3 −−  A (12)  or    ωx ωed     t = 2k0θ −− + 8G −−   A A Appl. Sci. 2020, 10, x FOR PEER REVIEW   − 6 of 22     A = ωx x f or2D   ·   − − θx−   d   e = e 3− − − − where ω is the inﬂuence function, θ is the volume dilatation, ed is the deviatoric extension state and − − e is the extension scalar state, k is the bulk modulus, G is the shear modulus, k0 can be expressed respectively− as:  E  planestress k =  2(1 υ) (13) 0  E−  2(1+υ)(1 2υ) planestrain − where E is the elastic modulus, υ is the Poisson ratio. For elastic solid material, the strain energy density of material point x(k) in the peridynamic state model can be expressed as 1 15G W = Bθ2 + ωe e (14) 2 2A − − · where B is deﬁned as: Figure 4. Force( vector15G state 퐓(퐱(푘),푡). k f or3D − B = − 9 (15) k 15G f or2D − 4 3.2. Energy-Based Failure Criterion 3.2. Energy-Based Failure Criterion

As shown in Figure 5, the force vector states 퐓(퐱(푘),푡) and 퐓(퐱(푗),푡) are opposite in direction As shown in Figure5, the force vector states T x−(k), t and T x(j),−t are opposite in direction and ¯ ¯ and parallelparallel to their to their deformed deformed relativerelative deformed deformed position. position. However, However, their values their are values not equal. are not equal.

Figure 5. Force vector state and the bond ξ. Figure 5. Force vector state and the bond 휉. h i h i As shown in Figure6, the force density f = t x , t t x , t exhibits non-linearity under (kj) (k) − (j) − − As showncycle loading, in Figure therefore 6, the the force total energydensity density 푓 stored= (푡 in[퐱 the(푘),푡 bond]−푡[ξ퐱can(푗),푡 be])expressed exhibits as non follows.-linearity under (푘푗) − − Z ( ) cycle loading, therefore the total energy density εstoredξ ∆t in the bond 휉 can be expressed as follows. ω = f dε (16) ξ 휀 (∆푡 ) (kj) ξ 휉0 휔휉 = ∫ (푓(푘푗 )) 푑휀 (16) 0 휉

Figure 6. Critical bond strain damage model.

Figure 4. Force vector state 퐓(퐱(푘),푡). −

3.2. Energy-Based Failure Criterion

Figure 5. Force vector state and the bond 휉.

휀휉(∆푡) (16) Appl. Sci. 2020, 10, 4325 휔휉 = ∫ (푓(푘푗 )) 푑휀 7 of 23 0 휉

FigureFigure 6. Critical 6. Critical bond bond strain damage damage model. model.

As shown in Figure7, all points Pdown along the dashed line l, connected to all points Pup across As fracture shown plane in Figureγ of unit 7, area all and points the volume 푃푑표푤푛 above along the the fracture dashed plane. line The energy푙, connected density required to all points to 푃푢푝 across fractureopen the plane plane γ 훾is asof follows:unit area and the volume above the fracture plane. The energy density

required to open the plane 훾 is as follows: 1 Z δ Z 2π Z δ Z cos− (z/ξ) −1 2 wγ = 훿 2휋 훿 cos (푧/휉)wcξ sin φdφdξdθdz (17) 0 0 z 0 2 푤훾 = ∫ ∫ ∫ ∫ 푤푐휉 sin휙푑휙푑휉푑휃푑푧 (17) 0 0 푧 0 where wc is the critical energy density, wγ is the energy release rate deﬁned in the continuum damage Appl.where Sci. 20 푤20mechanics푐, 10is, x the FOR theory. critical PEER REVIEW energy density, 푤훾 is the energy release rate defined in the continuum7 of 22 damage mechanics theory.

FigureFigure 7. 7. FractureFracture surface. surface. 3.3. Fatigue Crack Tip Deformation Analysis 3.3. Fatigue CrackAs shown Tip Deformation in Figure8, there Analysis are three kinds of bonds near the model-I crack tip: broken bonds, core As shownbonds, in and Figure partially 8, damaged there are bonds. three For kinds the core of bonds, bonds the near bond strainthe modelscore∗ is the-I crack largest tip: compared broken bonds, with the partially damaged bonds [46]. Since the material is linear elastic-perfectly plasticity,∗ the core core bonds,bond and strain partially can be described damaged as: bonds. For the core bonds, the bond strain 푠푐표푟푒 is the largest K compared with the partially damaged bondss∗ (δ) =[46sˆcore]. Since the material is linear elastic(18) -perfectly core √ plasticity, the core bond strain can be described as: E δ where K is the stress intensity factor, E is elastic modulus, sˆcore is a dimensionless parameter. ∗ 퐾 푠푐표푟푒(훿) = 푠푐표푟푒̂ (18) 퐸√훿

Where 퐾 is the stress intensity factor, 퐸 is elastic modulus, 푠푐표푟푒̂ is a dimensionless parameter.

Figure 8. The bonds near the crack tip.

As shown in Figure 9, there is a plastic zone near the crack tip, the length of the plastic zone is ∗ 푟푝 . For the linear elastic fracture mechanics, the strain in the plastic zone can be described as:

퐾 ∗ 푠(푧)~ 0 ⩽ 푧 ⩽ 푟푝 (19) 퐸√2휋푧 Combining Equations (17) and (18), a function 푓̂ can be described as: 푧 1 푓̂ ( ) ~ (20) 훿 푠푐표푟푒̂ √2휋푧/훿 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 22

Figure 7. Fracture surface.

3.3. Fatigue Crack Tip Deformation Analysis As shown in Figure 8, there are three kinds of bonds near the model-I crack tip: broken bonds, ∗ core bonds, and partially damaged bonds. For the core bonds, the bond strain 푠푐표푟푒 is the largest compared with the partially damaged bonds [46]. Since the material is linear elastic-perfectly plasticity, the core bond strain can be described as:

∗ 퐾 푠푐표푟푒(훿) = 푠푐표푟푒̂ (18) Appl. Sci. 2020, 10, 4325 퐸√훿 8 of 23 Where 퐾 is the stress intensity factor, 퐸 is elastic modulus, 푠푐표푟푒̂ is a dimensionless parameter.

FigureFigure 8.8. The bonds near the crack tip.tip.

As shownshown inin FigureFigure9 ,9, there there is is a plastica plastic zone zone near near the the crack crack tip, tip, the the length length of theof the plastic plastic zone zone is risp∗ . ∗ For푟푝 . For the the linear linear elastic elastic fracture fracture mechanics, mechanics, the the strain strain in the in the plastic plastic zone zone can can be describedbe described as: as:

퐾 ∗ 푠(푧)~ K 0 ⩽ 푧 ⩽ 푟푝 (19) s(z) 퐸√2휋푧 0 6 z 6 rp∗ (19) ∼ E √2πz Appl. Sci. 20Combining20, 10, x FOR Equation PEER REVIEWs (17) and (18), a function 푓̂ can be described as: 8 of 22 푧 1 푓̂ ( ) ~ (20) 훿 푠푐표푟푒̂ √2휋푧/훿

FigureFigure 9. 9. PlasticPlastic zone zone at fatiguefatigue crack crack tip. tip. ˆ 4. Trans-CombiningScale Peridynamic Equations Fatigue (17) and Model (18), a function f can be described as: z 1 In ref. [46], Silling and Askari proposedfˆ a fatigue cracking model which can simulate two(20) phases δ ∼ in fatigue failure: crack initiation, crack propagation.sˆcore √ The2πz /evolutionδ law for the bond “remaining life” is calibrated4. Trans-Scale with S Peridynamic-N curve data Fatigue during Model the fatigue crack nucleation phase and with Paris’ law during the fatigue crack growth phase. Inspired by this work, in this section, a trans-scale peridynamic In ref. [46], Silling and Askari proposed a fatigue cracking model which can simulate two phases fatigue model is built based on the mechanism of fatigue. Macroscale is directly depicted by the tans- in fatigue failure: crack initiation, crack propagation. The evolution law for the bond “remaining scale peridyanmic model. Each bond in the body is defined as the ideal fatigue test specimen under life” is calibrated with S-N curve data during the fatigue crack nucleation phase and with Paris’ law variableduring loads. the fatigue crack growth phase. Inspired by this work, in this section, a trans-scale peridynamic fatigue model is built based on the mechanism of fatigue. Macroscale is directly depicted by the 4.1. Bondtans-scale Damage peridyanmic and Point model.Damage Each within bond the in Horizon the body is deﬁned as the ideal fatigue test specimen under variable loads. 4.1.1. Bond Fatigue Damage and Failure Bond damage in a peridynamic material body is defined by bond breakage. The simplest criterion is that when the bond strain, shown in Equation (21), exceeds the critical value, the micro- potential between two material points 퐱(푘) and 퐱(푗) will be removed away. As shown in Figure 5, the spring-like bond 휉 is subjected to variable force density in its two ends during each loading cycle exert on the body ℛ. Fatigue bond damage is tracked by a history-dependent variable 푑푘푗(휉,푡). 1 푖푓 휉 푖푠 푏푟표푘푒푛 푑 (휉, 푡 ) = { (21) 푘푗 휑(휉, 푠 , 푡 ) 표푡ℎ푒푟푤푖푠푒 where 휑(휉, 휀 , 푡 ) is the normalized damage function, depending on the current bond strain 푠 and time t.

4.1.2. Point Damage and Fatigue Cracking For a given material point, the point damage is defined as the weighted ratio of the number of eliminated bonds interactions to the total number of initial interactions within its family. The fatigue damage of a point 퐱(푘) during each loading cycle is described as: ( ) ′ ∫ℋ 휑 휉,푠,푡 푑푉 퐱(푘) 휙(퐱(푘)) = (22) ∫ℋ 푑푉′ 퐱(푘) where 푑푉′ is an incremental volume for the material point connecting the point 퐱(푘) within the ℋ 휉 horizon (퐱(푘)). As shown in Figure 10, the failure of one bond 푘푗 in the peridynamic body leads to the incremental point damage for 퐱(푘) and 퐱(푗) . Therefore, the force density acting among the material points will be redistributed leading to the autonomous damage of neighboring bonds. The progressive failure of bond damage leads to the crack surface 풫푐푟푎푐푘 in the peridynamic material body. Appl. Sci. 2020, 10, 4325 9 of 23

4.1. Bond Damage and Point Damage within the Horizon

4.1.1. Bond Fatigue Damage and Failure Bond damage in a peridynamic material body is deﬁned by bond breakage. The simplest criterion is that when the bond strain, shown in Equation (21), exceeds the critical value, the micro-potential between two material points x(k) and x(j) will be removed away. As shown in Figure5, the spring-like bond ξ is subjected to variable force density in its two ends during each loading cycle exert on the body . Fatigue bond damage is tracked by a history-dependent variable d (ξ, t). R kj ( 1 i f ξ is broken d (ξ, t) = (21) kj ϕ(ξ, s, t) otherwise

where ϕ(ξ, ε, t) is the normalized damage function, depending on the current bond strain s and time t.

4.1.2. Point Damage and Fatigue Cracking For a given material point, the point damage is deﬁned as the weighted ratio of the number of eliminated bonds interactions to the total number of initial interactions within its family. The fatigue damage of a point x(k) during each loading cycle is described as: R ϕ(ξ, s, t)dV0 x H (k) φ x(k) = R (22) dV x 0 H (k)

where dV0 is an incremental volume for the material point connecting the point x(k) within the horizon ( ). As shown in Figure 10, the failure of one bond ξ in the peridynamic body leads to the H x(k) kj incremental point damage for x(k) and x(j). Therefore, the force density acting among the material points will be redistributed leading to the autonomous damage of neighboring bonds. The progressive Appl. Sci. 20failure20, 10, of x bondFOR PEER damage REVIEW leads to the crack surface in the peridynamic material body. 9 of 22 Pcrack

FigureFigure 10. The 10. The progressive progressive failure failure inin the the peridynamic peridynamic body. body. 4.2. Trans-Scale Fatigue Model 4.2. Trans-ScaleAs shownFatigue in Model Figure 11a,b, The peridynamic solid undergoes cycle loading acting on the body As shownboundary. in ForFigure a given 11a bond,b, Theξkj peridynamicwithin the horizon solid of undergoes point x(k), the cycle force loading density actingacting on on the the body spring-like bond varies in a noncyclic way. The spectrum loading of the bond strain ε , shown in boundary. For a given bond 휉 within the horizon of point 퐱 , the force densitykj acting on the Figure 11d, varies irregularly푘푗 with every loading cycle acting on the(푘) body boundary. Deﬁnes the 휀 spring-likemaximum bond andvaries minimum in a noncyclic bond strains way. in the The current spectrum cycle smax loadingand smin of. the bond strain 푘푗, shown in Figure 11d, varies irregularly with every loading cycle acting on the body boundary. Defines the maximum and minimum bond strains in the current cycle 푠max and 푠min.

Figure 11. Fatigue model for a given bond. 휉푘푗 . (a) Tensile and compress cycle loading; (b) Peridynamic solid; (c) The bond 휉푘푗; (d) Spectrum loading

As shown in Equation (5), to express the damage accumulation rate of a given bond explicitly, the concise representation of the critical active energy 푄(휎푎,휎푚,퐷) is obtained by using inverse analysis through the damage fatigue failure process and damage fatigue law. As shown in Figure 12, the normalized fatigue damage quantity can be expressed by fictitious time 푡푓 as follows:

휆(퐱,휉,푡푓) = 푁퐶exp(푚휀) { 푁 (23) 푡푓 = 푁푓 where 푁푓 is the cycle number to failure, 푁 is the current cycle number, 휀 = |푠max−푠min| is the current cycle bond strain in the bond, 퐶 and 푚 are positive parameters corresponding to the material property. The normalized fatigue damage quantity 휆(퐱,휉,푡푓) is defined as the consumption life of the bond. It involves the loading cycle 푁 increases. 푑휆 (푁) = 퐶exp(푚휀 ) {푑푁 (24) 휆(0) = 0 The bond breaks at the loading cycle 푁 that Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 22

Figure 10. The progressive failure in the peridynamic body.

4.2. Trans-Scale Fatigue Model As shown in Figure 11a,b, The peridynamic solid undergoes cycle loading acting on the body

boundary. For a given bond 휉푘푗 within the horizon of point 퐱(푘), the force density acting on the spring-like bond varies in a noncyclic way. The spectrum loading of the bond strain 휀푘푗, shown in Figure 11d, varies irregularly with every loading cycle acting on the body boundary. Defines the Appl. Sci. 2020, 10, 4325 10 of 23 maximum and minimum bond strains in the current cycle 푠max and 푠min.

Figure 11. 11.Fatigue Fatigue model model for fora given a given bond. ξ bondkj.(a.)휉 Tensile푘푗 . (a) andTensile compress and cycle compress loading; cycle (b) Peridynamicloading; (b) solid;Peridynamic (c) The solid; bond ξ(ckj) ;(Thed) Spectrumbond 휉푘푗; loading.(d) Spectrum loading

As shown in Equation (5), to express the damage accumulation rate rate of of a a given given bond bond explicitly, explicitly, the concise representation of the critical active energy Q(σa, σm, D ) is obtained by using inverse analysis the concise representation of the critical active energy 푄(휎푎,휎푚,퐷) is obtained by using inverse throughanalysis thethrough damage the fatiguedamage failure fatigue process failure and process damage and fatiguedamage law. fatigue law. As shown in Figure 1212,, thethe normalizednormalized fatigue damage quantity can be expressed by ﬁctitiousfictitious time t f as follows: time 푡푓 as follows:   λ x, ξ , t f = NC exp(m ε)  휆(퐱,휉,푡푓) = 푁퐶exp(푚휀)  N (23)  t f =  { N푁f (23) 푡 = 푓 푁 where N is the cycle number to failure, N is the current푓 cycle number, ε = smax s is the current f | − min| cyclewhere bond 푁 strainis the in cycle the bond, numberC and to failure,m are positive 푁 is the parameters current correspondingcycle number, to휀 the= |푠 material− 푠 property.| is the 푓 max min Thecurrent normalized cycle bond fatigue strain damage in thequantity bond, 퐶λ andx, ξ , t푚f isare deﬁned positive as parameters the consumption corresponding life of the to bond. the

Itmaterial involves property. the loading The cyclenormalizedN increases. fatigue damage quantity 휆(퐱,휉,푡푓) is defined as the consumption life of the bond. It involves the loading( cycle 푁 increases. dλ (N) = C exp(mε) dN푑휆 (24) (푁)(=) 퐶exp(푚휀 ) {푑푁 λ 0 = 0 (24) Appl. Sci. 2020, 10, x FOR PEER REVIEW 휆(0) = 0 10 of 22 The bond breaks at the loading cycle N that The bond breaks at the loading cycle 푁 that 휆λ(푁N) >⩾11 (25) (25)

FigureFigure 12. 12. SchemeScheme of of fatigue fatigue damagedamage for for each each bond. bond.

For the fatigue crack initiation phase, parameter 퐶 and 푚 are calibrated by the S-N curve expressed in exponent form, as shown in Figure 13. Equation Error! Reference source not found. means that the cycle 푁 is the first bond breaks. Parameter 퐶 and 푚 are set to 퐶 = 퐶 { 1 (26) 푚 = 푚1 Combining Equations (23), Error! Reference source not found. and (26), the first bond in the entire domain ℛ breaks with 휀1 is the cyclic strain range.

푁1퐶1exp(푚1휀1) = 1 (27) where 푁1 is the smallest cycle at which this bond breaks and 휆(푁1) = 1. The first bond breaks in the fatigue crack initiation phase when its fatigue cycle number 푁 becomes larger than 푁1. 1 푁1 ⩾ (28) 퐶1exp(푚1휀1)

When the first bond breaks with the cyclic bond 휀1 in fatigue crack initiation, new static solutions for other bonds are calculated in the same cycle. If those bonds strain ranges are larger than

휀1, then the bonds break and new static solutions continue until no more bonds break at the same cycle. The largest bond strain is calculated for the next cycle number by using the static solution in the deformed peridynamic material configuration.

Figure 13. Calibration parameters for the initiation phase.

For the fatigue crack growth phase, parameter 퐶 and 푚 are calibrated by the well-known Paris law. Parameter 퐶 and 푚 are set to Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 22

휆(푁) ⩾ 1 (25)

Figure 12. Scheme of fatigue damage for each bond.

When the first bond breaks with the cyclic bond 휀1 in fatigue crack initiation, new static For the fatigue crack initiation phase, parameter C and m are calibrated by the S-N curve expressed solutionsin for exponent other form,bonds as are shown calculated in Figure 13in. the Equation same (25) cycle. means If those that the bonds cycle N strainis the ﬁrstranges bond are breaks. larger than 휀 , then the bonds break and new static solutions continue until no more bonds break at the same 1 Parameter C and m are set to ( C = C cycle. The largest bond strain is calculated for the next1 cycle number by using the static (26)solution in = the deformed peridynamic material configuration.m m1

FigureFigure 13. Calibration 13. Calibration parameters parameters for for the the initiation initiation phase. phase. Combining Equations (23), (25) and (26), the ﬁrst bond in the entire domain breaks with ε is R 1 For thethe cyclic fatigue strain crack range. growth phase, parameter 퐶 and 푚 are calibrated by the well-known Paris law. Parameter 퐶 and 푚 are set to N1C1 exp(m1ε1) = 1 (27)

where N1 is the smallest cycle at which this bond breaks and λ(N1) = 1. The ﬁrst bond breaks in the fatigue crack initiation phase when its fatigue cycle number N becomes larger than N1.

1 N1 > (28) C1 exp(m1ε1)

When the first bond breaks with the cyclic bond ε1 in fatigue crack initiation, new static solutions for other bonds are calculated in the same cycle. If those bonds strain ranges are larger than ε1, then the bonds break and new static solutions continue until no more bonds break at the same cycle. The largest bond strain is calculated for the next cycle number by using the static solution in the deformed peridynamic material configuration. For the fatigue crack growth phase, parameter C and m are calibrated by the well-known Paris law. Parameter C and m are set to ( C = C 2 (29) m = m2 As shown in Figure 14, a fixed bond ξ normal to the axis of the growing model-I fatigue crack. During the cycle loading, the deformation in the vicinity of the crack tip is constant and the crack growth rate is defined as da/dN in each loading cycle. The relative distance between the bond and the crack tip is expressed as follows. da z = x N (30) − dN Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 22

퐶 = 퐶 { 2 (29) 푚 = 푚2 As shown in Figure 14, a fixed bond 휉 normal to the axis of the growing model-I fatigue crack. During the cycle loading, the deformation in the vicinity of the crack tip is constant and the crack growth rate is defined as 푑푎/푑푁 in each loading cycle. The relative distance between the bond and the crack tip is expressed as follows. 푑푎 푧 = 푥 − 푁 (30) Appl. Sci. 2020, 10, 4325 푑푁 12 of 23 where 푧 = 0 means that the crack tip is on the bond 휉, 푥 is the spatial coordinate along the crack axis. Thewhere bondz consumption= 0 means that thelife crack at 푧 tip= is0 onby the integrating bond ξ, x is its the first spatial derivative coordinate to along distance: the crack axis. The bond consumption life at z = 0 by integrating its first derivative to distance: 훿 푑휆 푑푁 휆(훿) = 휆(0) +Z∫ 푑푧 (31) δ d푑푁λ dN푑푧 λ(δ) = λ(0) + 0 dz (31) dN dz where 휆(훿) = 0 and 휆(0) = 1, for the damage evolution0 law is effective within the horizon of the crack where tip. λ(δ Combing) = 0 and λ (0) =Equation1, for thes damage Error! evolution Reference law is effective within source the horizon not of the found. crack and Error! Referencetip. Combing source Equations not found. (30) and (31)leads leads to to Z 훿 푑푎da δ m푚22ε휀 == 퐶C22 ∫ 푒e dz푑푧 (32) (32) 푑푁dN 00 Recall theRecall well the-known well-known Paris Paris law law for for fatigue fatigue crack growth: growth: 푑푎 da 푀 = c푐((∆∆k푘)M) (33) (33) 푑푁dN where 푐 and 푀 are constant coefficients. Comparing Equations (32) and (33), two experimental data where c and M are constant coefficients. Comparing Equations (32) and (33), two experimental data 퐶 푚 points arepoints selected are selected to determine to determine the the constants constants C22 andandm 2. 2.

FigureFigure 14. 14.TheThe bond bond 휉ξ normal to to the the model-I model crack.-I crack. 4.3. Transition from Fatigue Crack Initiation to Growth 4.3. Transition from Fatigue Crack Initiation to Growth As shown in Equation (24), the fatigue crack initiation and growth phase are depicted in a single As shownmodel. The in Equation mechanism (24), of fatigue the initiationfatigue crack and growth initiation are diﬀ anderent. growth During thephase fatigue are crack depicted initiation, in a single model. Theeach bond mechanism strain is independent of fatigue of initiation the cycle number. and growth However, are a bond different. strain is changing During over the time fatigue as crack initiation,the each bond bond transit strain to the is fatigue independent crack growth of phase.the cycle As shownnumber. in Figure However, 15, for a the bond given str pointain xis(j )changing, within the horizon of point x(k), the bond ξkj (denoted as green) transit to the fatigue crack growth over time as the bond transit to the fatigue crack growth phase. As shown in Figure 15, for the given phase as the point damage φ x(j) satisﬁes the following conditions. point 퐱(푗), within the horizon of point 퐱(푘), the bond 휉푘푗 (denoted as green) transit to the fatigue crack growth phase as the point damage 휙(퐱φ(푗x))(j )satisfies> 0.5 the following conditions. (34)

휙(퐱(푗)) ⩾ 0.5 (34) Appl. Sci. 2020, 10, 4325 13 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 22

(a) Without damage between point 퐱(푗) and point 퐱(푘)

(b) The fatigue crack initiation within the horizon of points 퐱(푗) and 퐱(푘)

Figure 15. FigureTransition 15. Transition from crack from crack initiation initiation to topropagation propagation.. ( a()a Without) Without damage damage between between point x(j )pointand 퐱(푗) point x ;(b) The fatigue crack initiation within the horizon of points x and x . and point 퐱(푘); (b(k) The fatigue crack initiation within the horizon of points(j) 퐱((k푗)) and 퐱(푘) 5. Numerical Solution Method

5.1. Adaptive Dynamic Relaxation for Static Solution The peridynamic control equation for fatigue cracking is integral-diﬀerent. As for cyclic loading simulation, only the maximal loading condition is necessary. Whenever some bonds break in a loading cycle, the new static solution is calculated at the same cycle until no bonds break. To solve quasi-static or static problems, the Adaptive Dynamic Relaxation is used for the fatigue crack simulation problems. The peridynamic equation of motion for all material points is expressed as follows

.. . DU(X, t) + ζDU(X, t) = F(U, U0, X, X0) (35)

where D is the fictitious diagonal density matrix and ζ is the damping coefficient, X and U are initial position and displacement vector for all the material points in the configuration body.

 T n o  X = x( ), x( ), ... , x( )  n 1 2 N o (36)  T  U = u x(1), t , u x(2), t , ... , u x(N), t

where N is the number of all the material points in the conﬁguration body. Combing Equations (6) and (11), the vector F can be expressed by its kth component.

XM F = t t v V + b (37) (k) (k)(j) − (j)(k) cj (j) (k) j=1 Appl. Sci. 2020, 10, 4325 14 of 23

where M is the total number of material points within the horizon of a material point x(k), vcj is the volume correction factor for the material point x(j). The velocities and displacements for the next time step are expressed as follows.

 n n 1/2 1 n  n+1/2 ((2 ζ ∆t)V − +2∆tD− F )  V = − n  2+ζ ∆t (38)  Un+1 = Un + Vn+1/2∆t where n is the nth iteration. The diagonal elements of the density matrix, D, can be expressed as:

1 X γ > ∆t2 (39) kk 4 Jkj j where is the stiﬀness matrix of the current structure. As for the small displacement assumption. Jij  X XM ξ e  2 (k)(j) 4δ 1 ad δ o = · (v V +vcjVj + b (40) Jkj 2 ck k j j=1 ξ(k)(j) ξ(k)(j)  ξ(k)(j) where e is a unit vector.

5.2. Equivalent Stress Intensity Factor Each step of crack growth requires a driving force, which guides the fatigue crack path. Naturally, this driving force named stress intensity factor. Because of the complexity in mathematics and physics, solving the three-dimensional dynamic stress intensity factors is certainly limited in mechanics. In the state-based peridynamic theory, there is no concept of this driving force. To better characterize the propagating crack-tip ﬁelds, an equivalent stress intensity factor Jequ can be expressed as follows [54]: R ∂ui Jequ = Wdy Ti ds ΓP R − ∂x = 2E (W dy w ds) (41) (1 v2) i i i − i=I,II,II − where x = x1, y = x2 and z = x3 are 3D Cartesian coordinates with origin at the fatigue crack tip, J is a contour integral evaluated counterclockwise along a crack surface, Ti = σijnj is the traction vector along the crack surface, with the outward unit normal vector nj and σij, ui is a displacement vector and ds is an element of the crack surface. By taking the surface integration, the relationship between the displacement and fatigue crack criteria is built. Each step of crack growth can be obtained from the relationship.

5.3. Flowchart for the Fatigue Cracking Simulation Based on the comparison and analysis of the classical fatigue crack simulation algorithm and the peridynamic one, the simulation process for the fatigue crack initiation and growth is deduced, as shown in Figure 16. Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 22 vector along the crack surface, with the outward unit normal vector 푛푗 and 휎푖푗, 푢푖 is a displacement vector and 푑푠 is an element of the crack surface. By taking the surface integration, the relationship between the displacement and fatigue crack criteria is built. Each step of crack growth can be obtained from the relationship.

FigureFigure 16. Flowchart 16. Flowchart of of the the fatigue cracking cracking simulation. simulation. 6. Experiment 6. Experiment 6.1. Experimental Equipment

6.1. ExperimentalAs Equipment shown in Figure 17, the experiment was conducted on a fatigue test machine with a ﬂoor model biaxial servo-hydraulic dynamic test system, which provides axial and torsion load on a specimen in an integrated biaxial actuator. The system has an axial load capacity of 250 KN ( 56 kip) and a torque As shown in Figure 17, the experiment was conducted on ±a fatigue± test machine with a floor capacity of 2000 Nm ( 17700 in-lb), with an axial actuator stroke of 100 mm and a rotary stroke of 90 . ± ± ◦ model biaxial servoAs shown-hydraulic in Figure 18 dynamic, the loading test system, system, which comeswhich with provides console software axial and to provide torsion full load on a specimen insystem an integrated control from biaxial a personal actuator. computer (PC),The includingsystem waveformhas an axial generation load incapacity both axes, of calibration, ±250 KN (±56 kip) and a torquelimit capacity set up, andof ±2000 status monitoring. Nm (±17700 The loadingin-lb), waveformwith an axial is sine actuator and the loading stroke frequency of 100 is mm 2HZ. and a rotary stroke of 90°. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 22

Appl. Sci. 2020, 10, 4325 16 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 22

Figure 17. Fatigue test machine for the combined tension and torsion.

As shown in Figure 18, the loading system, which comes with console software to provide full system control from a personal computer (PC), including waveform generation in both axes, calibration, limitFigure setFigure up, 17. 17.Fatigue andFatigue status test test machine monitoring. machine for forthe theThe combined combined loading tension tension waveform and and torsion. torsion. is sine and the loading frequency is 2HZ. As shown in Figure 18, the loading system, which comes with console software to provide full system control from a personal computer (PC), including waveform generation in both axes, calibration, limit set up, andSE M status spe c monitoring.imen chambe r The loading waveform is sine and the loading frequency is 2HZ. loading force charge specimen oscilloscope system transducer amplifier SEM specimen chamber

loading force chaDrgAeC data specime nmicroprocessor oscilloscope system transducer ampmliofideurle recorder

fatigue test DAC data micropprorocceedsusroer module recorder Figure 18. HardwareHardware configuration conﬁguration of the test system system.. 6.2. Specimen Size Requirements andfatig Loadingue test Parameters pr ocedure As shown in Figure 19, a smooth specimen with a length of 210 mm is machined cutting from a high-speed wheel. TheFigure diameter 18. Hardware of the grasping configuration side is of 25 the mm. test system The minimum. diameter of the test specimen is 7 mm, which also means that it is the weak part of the specimen without considering other factors. The maximum percentages of the various speciﬁed elements are shown in Table1. The basic Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 22

6.2.Appl. Specimen Sci. 2020, 10 Size, x FOR Requirements PEER REVIEW and Loading Parameters 16 of 22

6.2. SpecimenAs shown Size in RequirementsFigure 19, a smooth and Loading specimen Parameters with a length of 210 mm is machined cutting from a high-speed wheel. The diameter of the grasping side is 25 mm. The minimum diameter of the test Appl. Sci.As 2020shown, 10, 4325in Figure 19, a smooth specimen with a length of 210 mm is machined cutting from17 of 23a specimen is 7 mm, which also means that it is the weak part of the specimen without considering high-speed wheel. The diameter of the grasping side is 25 mm. The minimum diameter of the test other factors. The maximum percentages of the various specified elements are shown in Table 1. The specimen is 7 mm, which also means that it is the weak part of the specimen without considering mechanicalbasic mechani propertiescal properties for the for wheel the wheel material material are shown are shown in Table in2 .Table The loading2. The loading parameters parameters are shown are other factors. The maximum percentages of the various specified elements are shown in Table 1. The inshown Table in3 andTable the 3 loadingand the loading path is shown path is in shown Figure in 20 Figure. 20. basic mechanical properties for the wheel material are shown in Table 2. The loading parameters are shown in Table 3 and the loading path is shown in Figure 20.

Figure 19. Fatigue specimen specimen..

Table 1. The maximum percentages of the various speciﬁed elements. Table 1. The maximumFigure percentages 19. Fatigue of thespecimen various. specified elements. C% Si% Mn% P% S% Cr% Cu% Mo% Ni% V% (Cr + Mo + Ni)% C% Si% Mn%Table 1.P% The maximumS% percentagesCr% Cu% of theMo% various Ni%specified V% elements (Cr. + Mo + Ni)% 0.530.53 0.99 0.99 0.96 0.96 0.010 0.010 <0.002<0.002 0. 0.0303 0.036 0.036 <0.005<0.005 0.16 0.16 <0.005<0.005 <0.50<0.50 C% Si% Mn% P% S% Cr% Cu% Mo% Ni% V% (Cr + Mo + Ni)%

0.53 0.99 0.96Table 0.010 2. Basic <0.002 properties 0. of 03 the wheel0.036 material<0.005 at room0.16 temperature.temperature<0.005 . <0.50

Name E/GPa Tableσ0.2/MPa 2. Basic σpropertiesb/MPa ofδ the% wheelFerrite material Hardness at room/HV temperatureFerrite Pearlite. HardnessPearlite/HV Name E/GPa σ0.2/MPa σb/MPa δ% D1 178.530 553.230 956.380 16. 21 210 Hardness/HV 250Hardness /HV NameD1 E/GPa 178.530 σ0.2/MPa 553.230 σb/MPa 956.380δ% Ferrite 16. Hardness 21/HV 210Pearlite Hardness 250/HV D1 178.530 553.230 956.380Table 3.16. Loading 21 parameters210. 250 Table 3. Loading parameters. σa/MPa τa/MPa σequ/MPaTable 3.R Loadinga Fmax parameters/KN T.max /N.m /° f/HZ

σa/MPa 340τa/MPa 196.3σequ /MPa480.83 R√a 3 F9.6max /KN T8.3max /N.m 90 ϕ/◦ 2 f/HZ σa/MPa τa/MPa σequ/MPa Ra Fmax/KN Tmax/N.m /° f/HZ √ 340340 196.3 196.3480.83 480.83 √3 3 9.69.6 8.3 8.390 902 2

Figure 20. Loading path.

6.3. Fatigue Crack Path and Morphology Figure 20. Loading path.path.

6.3. Fatigue The fracture Crack Path morphology and Morphology and the angle between the fatigue crack path and the central axis are shown in Figure 21. The fracture morphology and the angle between the fatigue crack path and the central axis are shown in Figure 21.. Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 22

Figure 21. Macro fracture characteristics of the smooth specimen.

6.4. Results Analysis As shown in Figure 22, using an electron microscope, it can be seen that the angle between the main long fatigue crack and cylinder axis line 푙푐 is about 훼 = 59.5°. As shown in Table 4, the average fatigue cycle is 15,992 for the fatigue specimens, and the

Appl.average Sci. 2020 angle, 10 ,between 4325 the main long fatigue crack and cylinder axis line 푙푐 is 61.54°. 18 of 23 Appl. Sci.As 20shown20, 10, xin FOR Figure PEER 23,REVIEW the angle between the main long fatigue crack and cylinder axis line17 of 22is about 58.5°. This simulation result has a good correlation with the average test result, as shown in Figure 22.

As shown in Figure 24, the value of the equivalent stress intensity factor 퐽푒푞푢 becomes larger as the number of iterations increases.

Table 4. Fatigue cycles and angles.

NO N/Cycles α/° D1-1 16239 57.8 D1-2 15860 61.5 D1-3 17963 70.1 FigureD1-4 21. Macro fracture characteristics16938 of the smooth specimen.50.2 6.4. Results Analysis D1-5 15608 53.6 6.4. Results Analysis D1-6 15656 67.5 As shown in Figure 22, using an electron microscope, it can be seen that the angle between the As shown in FigureD1 22,-7 using an electron microscope,15320 it can be seen 62.8that the angle between the main long fatigue crack and cylinder axis line lc is about α = 59.5◦. main long fatigue crackD1 and-8 cylinder axis line 푙14352푐 is about 훼 = 59.5°. 68.8 As shown in Table 4, the average fatigue cycle is 15,992 for the fatigue specimens, and the

average angle between the main long fatigue crack and cylinder axis line 푙푐 is 61.54°. As shown in Figure 23, the angle between the main long fatigue crack and cylinder axis line is about 58.5°. This simulation result has a good correlation with the average test result, as shown in Figure 22.

As shown in Figure 24, the value of the equivalent stress intensity factor 퐽푒푞푢 becomes larger as the number of iterations increases.

Table 4. Fatigue cycles and angles.

NO N/Cycles α/° D1-1 16239 57.8 D1-2 15860 61.5 D1-3 17963 70.1 D1-4 16938 50.2 D1-5 15608 53.6 D1-6 15656 67.5 D1-7 15320 62.8 Figure 22. The angle and morphology of fatigue crack. D1-8 14352 68.8 As shown in Table4, the average fatigue cycle is 15,992 for the fatigue specimens, and the average angle between the main long fatigue crack and cylinder axis line lc is 61.54◦.

Table 4. Fatigue cycles and angles.

NO N/Cycles α/◦ D1-1 16,239 57.8 D1-2 15,860 61.5 D1-3 17,963 70.1 D1-4 16,938 50.2 D1-5 15,608 53.6 D1-6 15,656 67.5 D1-7 15,320 62.8 D1-8 14,352 68.8

As shown in Figure 23, the angle between the main long fatigue crack and cylinder axis line is about 58.5◦. This simulation result has a good correlation with the average test result, as shown in Figure 22. Figure 22. The angle and morphology of fatigue crack. Appl. Sci. 2020, 10, 4325 19 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 22 Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 22

FigureFigure 23. SimulationSimulation results results based based on on the the proposed proposed peridynamic peridynamic fatigue fatigue model. model. (a) The (a) fatigue The fatigue crack Figure 23. Simulation results based on the proposed peridynamic fatigue model. (a) The fatigue crack crackpropagation propagation in nstep in nstep= 1; = (b1) ;(Theb) The fatigue fatigue crack crack propagation propagation in innstepnstep==5;5 ;((cc) )The The fatigue crack crack propagation in nstep = 1; (b) The fatigue crack propagation in nstep = 5; (c) The fatigue crack propagationpropagation inin nstepnstep == 20;(; (d)) TheThe fatiguefatigue crackcrack propagationpropagation inin nstepnstep == 30;(; (e)) TheThe fatiguefatigue crackcrack propagation in nstep = 20; (d) The fatigue crack propagation in nstep = 30; (e) The fatigue crack propagationpropagation inin nstepnstep==35;35; ( f()f) The The fatigue fatigue crack crack propagation propagation in in nstep nstep==45.45 propagation in nstep = 35; (f) The fatigue crack propagation in nstep = 45 As shown in Figure 24, the value of the equivalent stress intensity factor Jequ becomes larger as the number of iterations increases.

(a) Jequ at crack front in nstep = 1 (b) Jequ at crack front in nstep = 5 (a) Jequ at crack front in nstep = 1 (b) Jequ at crack front in nstep = 5

Figure 24. Cont. Appl. Sci. 2020, 10, 4325 20 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 22

(e) Jequ at crack front in nstep = 35 (f) Jequ at crack front in nstep = 45

FigureFigure 24. 24.Equivalent Equivalent stress stress intensity intensity factors factors inin eacheach specified speciﬁed iteration. iteration. (a ()a J)equJequ at crackat crack front front in in nstepnstep= =1;(1b; )(bJequ) Jequat at crack crack front front in innstep nstep= 5=;(5c;) (Jcequ) Jequat at crack crack front front in innstep nstep= 20=;(20d;) (Jdequ) Jatequ crackat crack front in nstepfront= in30; nstep (e) Jequ=at30 crack; (e) Jequ front at crack in nstep front= in35; nstep (f) Jequ=at35 crack; (f) Jequ front at crack in nstep front= in45. nstep = 45

7. Conclusions Conclusions (1) ToTo evaluate evaluate and and predict predict the the fatigue fatigue life life of ofspecimens specimens under under combined combined tension tension and andtorsion, torsion, the peridynamicthe peridynamic fatigue fatigue model model was proposed was proposed and the and fatigue the cracks fatigue initiate cracks and initiate propagate and propagate naturally withoutnaturally extra without criteria. extra criteria. (2) TheThe proposed proposed trans trans-scale-scale fatigue fatigue model model has has no no notion notion of of scale, scale, or or it it releases releases the the scale scale constraints. constraints. TheThe evolution evolution of of fatigue fatigue crack crack shows shows micro micro and macro and scale macro including scale crackincluding initiation, crack propagation, initiation, propagation,and fracture. and fracture. (3) TheThe short short diffusion diﬀusion-generated-generated and and propagating propagating crack crack converges converges and and grows grows into into the the main main crack, crack, whichwhich expands expands to to the the macroscopic macroscopic visible visible morphology. morphology. (4) TheThe macroscopic macroscopic morphology morphology of of fatigue fatigue was was highly highly consistent consistent with with the acactualtual experimental resultsresults and and extended extended along along the the direction direction of of 61.54° 61.54◦ untiluntil the the final ﬁnal fracture. fracture. (5) TheThe natural natural propagation propagation of of the the crack crack can can be be realized realized without without the the need need for for additional crack crack propagationpropagation criteria, criteria, and and the the distribution distribution and and quantitative quantitative analysis analysis of of the the fatigue fatigue life life can can be obtained.be obtained.

Author Contributions: Contributions: Conceptualization,Conceptualization, W.C.; Wenhua methodology, Chen; methodology, J.H. and W.C.; Junzhao formal Han analysis, and Wenhua J.H. and Chen; W.C.; formalinvestigation, analysis, W.C.; Junzhao resoures, Han J.H. and and Wenhua W.C.; writing-original Chen; investigation, draft preparation, Wenhua Chen; J.H. and resoures, W.C.; writing-review Junzhao Han and and Wenhuaediting, J.H.Chen; and writing W.C.; visualization,-original draft J.H.; preparation, supervision, Junzhao W.C.; Han funding and acquisition,Wenhua Chen; W.C.; writing All authors-review have and read editing, and Junzhaoagreed to Han the publishedand Wenhua version Chen; of visualization, the manuscript. Junzhao Han; supervision, Wenhua Chen; funding acquisition, Wenhua Chen; All authors have read and agreed to the published version of the manuscript. Appl. Sci. 2020, 10, 4325 21 of 23

Funding: This research was partially supported by the Joint Funds of the National Natural Science Foundation of China (Grant No. U1334204). Conﬂicts of Interest: The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article.

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