Accumulated crystal plasticity dissipation energy driven continuum damage two-scale model for fretting fatigue initiation life

Abstract

Fretting fatigue often occurs in the interfaces between components, subjected to complex multi-axial load states and high stress gradients at the contact edge region. For the prediction of fretting fatigue crack initiation and in-depth understanding of the crack initiation mechanism, it is essential to investigate the damage mechanisms across various scales and explore the underlying scale coupling mechanisms. By introducing a power-law based scale coupling relationship, a two-scale model of fretting fatigue crack initiation life is proposed by combining macroscopic continuum damage mechanics (CDM) with microscopic crystal plastic finite element method (CPFEM). The simulation results indicate that the predicted fretting fatigue initiation life shows better accuracy than the result predicted by single-scale CDM model. In case of low stress level the rate of accumulated dissipation energy can be clearly divided into two stages with turning points, whereas it exhibits a relatively uniform damage process under high stress level. Moreover, the proposed two-scale model partly provides physical explanation for fretting fatigue crack initiation based on the information from the microscale.

Keywords

Fretting fatigue crack initiation scale coupling crystal plasticity continuum damage mechanics

Introduction

Fretting fatigue refers to the combined effect of fretting and fatigue failure when two contact surfaces are simultaneously subjected to normal stress and axial cyclic loads. In engineering practice, there are many instances of failure due to fretting fatigue (Cai et al., 2020; Chowdhury et al., 2018; Croccolo et al., 2022). Compared to plain fatigue, the greatest difference lies in that fretting fatigue occurs in the area where two components are in contact, which will be affected by the complex multi-axial load conditions of the contact area and the stress gradient at the edge of the contact area. Relevant researches have demonstrated that under fretting conditions, the fatigue limit of materials can decrease down to a quarter of that under plain fatigue conditions (Lemaitre and Desmorat, 2005).

The fretting fatigue damage process generally consists of two stages: crack initiation and crack propagation. Many researchers have conducted separate studies on crack initiation (Bhatti et al., 2018; Wang C et al., 2022) and crack propagation (Martínez et al., 2017; Pereira and Abdel Wahab, 2017). Related studies (Araújo et al., 2008; Giner et al., 2014) suggest that the crack initiation and propagation in fretting fatigue are governed by different factors. In addition, under certain conditions, the crack propagation phase is very short, making crack initiation the dominant stage in the entire fatigue process (Hills and Nowell, 1994), for some engineering components, once the crack initiates, the component needs to be replaced. Therefore, research on the initiation stage of fretting fatigue cracks is necessary, and the methods for fretting fatigue crack initiation mainly include single-scale methods based on the continuum mechanics model and multi-scale methods based on crystal plasticity.

Single-scale methods based on the macroscopic continuum medium models mainly include two types (Bhatti and Abdel Wahab, 2018), one is based on empirical laws or physical observations, mainly including the critical plane approach (Vantadori et al., 2018; Wang Y et al., 2022), fretting specific parameter approach (Ding et al., 2011), and stress invariant approach (Burns and Parry, 1964). The other is the continuum damage mechanics (CDM) method (Attarha and Sattari-Far, 2023; Hojjati-Talemi and Wahab, 2013), derived from thermodynamic principles. Fretting fatigue belong to high-cycle fatigue where no macroscopic plasticity exists, CDM based fretting fatigue models usually assume that material is in a fully elastic state of stress (Hojjati-Talemi et al., 2014; Sun et al., 2017). Nevertheless, the primary cause of fretting fatigue crack initiation is the continuous advancement of micro-plastic zones under high stress gradients, the impact of plastic effects thus cannot be overlooked. Moreover, macroscopic continuum medium theory is unable to account for the detailed physical mechanisms of microstructure deformation, thus limiting its effectiveness in describing material damage evolution.

To account for the influence of microstructural mechanism, such as the morphology and size of grains, texture and the non-uniformity of grain plastic deformation, many researchers in recent years have employed the multi-scale methods based on crystal plasticity to capture local deformations at the crystal level (Abdolvand, 2022; K-S Li et al., 2021; Roters et al., 2010; Zghal et al., 2016). Compared to macroscopic continuum medium models, crystal plasticity methods allow for the prediction of microstructural features in the model. In the realm of fatigue, Manonukul and Dunne (2004) proposed the concept of the fatigue indicator parameter (FIP) within the framework of crystal plasticity. Subsequently, methods for studying FIP have rapidly advanced, forming such as accumulated plastic slip (Sweeney et al., 2014), accumulated dissipation energy (Cao et al., 2022; Cruzado et al., 2018a; He et al., 2024) and various types of FIPs improved by multi-axial fatigue life prediction criteria (Briffod et al., 2017; Hallberg et al., 2018; Przybyla et al., 2013). In addition, multi-scale methods based on crystal plasticity are also widely used in the field of fatigue crack growth, leading to the development of many short crack growth rate models (Aslan et al., 2011; Mao et al., 2022; Sun et al., 2016), which currently struggle to account for the long crack growth stage.

However, for multi-scale models studying the initiation of fatigue cracks, due to the high computational costs, these models are typically developed using representative volume elements (RVEs). They are usually computed under conditions that differ from the actual deformations experienced by large components. This leads to challenges in accurately depicting the multi-scale aspects of deformation, microstructural evolution, and mechanical response. With respect to fretting fatigue in metal materials, some researchers have combined crystal plasticity to study the impact of microstructure on fretting fatigue. Wang et al. (2021) proposed an energy-based criterion, studying the impact of microstructure on the initiation of fretting fatigue cracks in aluminum alloys. Minaii et al. (2019) used accumulated plastic slip as a FIP, studying the impact of grain size and parameters on the crack initiation life in 316 L stainless steel. Ashton et al. (2018) proposed a strain gradient crystal plasticity method for predicting the microstructure-sensitive length effects of crack initiation in ferrite-pearlite steel. However, most of these studies are traditional hierarchical multi-scale methods (see Figure 1(a)) based on submodel techniques, which simply replace the macroscopically interested region with a crystal plastic model, without considering the scale coupling mechanism between different scales, and cannot obtain deformation responses of different scales at the same time.

In fact, neither the single macroscopic continuum medium model nor the microscopic scale model can predict both the macroscopic damage evolution and microscopic mechanism of fretting fatigue. In recent years, an embedded multi-scale method (see Figure 1(b)) has been developed in various disciplines to address complex problems that are difficult to handle at a single scale. Many researchers have made contributions to the theory and applications of embedded multiscale methods (Hu et al., 2024; Jiang et al., 2021; Krairi et al., 2016; Sun et al., 2021). These methods aim to accurately describe the interaction of material responses at different scales, thereby predicting the performance and behavior of materials at different scales. Compared to hierarchical multi-scale methods, embedded multi-scale methods require dynamically inquiring into the microscale responses of material points during each macroscopic calculation, resulting in higher precision (Fang et al., 2022; Segurado et al., 2018). However, further research is needed on how to quantitatively characterize the coupling between multiple scales and how to overcome the issue of discrepancies in predictions at different scales.

In this work, by introducing a power-law based scale coupling relationship, a two-scale model of fretting fatigue crack initiation life is proposed by combining macroscopic CDM with CPFEM. Specifically, the macroscopic damage process of fretting fatigue crack initiation in cylindrical contact is modelled using the CDM method. Then, element deformation information of interest within the macroscopic model is extracted and assigned to the RVE of microscale as the boundary conditions to compute the evolution of microstructure. Through the proposed scale coupling relationship, the reverse influence of microstructural evolution on macroscopic damage was calculated, thereby considering the multiscale interactions and predicting the initiation life of fretting fatigue cracks. To validate the accuracy of this two-scale approach, comparisons were made with experimental results from the literature (Hojjati-Talemi et al., 2014). The results indicate a good agreement between the predicted initiation life of fretting fatigue cracks and the experimental results. Finally, the failure mechanism of the microscale RVE during the fretting fatigue crack initiation process is discussed. The results provide significant insights into further understanding of the process of fretting fatigue crack initiation.

Macroscale continuum damage model

Continuum damage mechanics model of fretting fatigue

Continuum damage mechanics employs the concept of effective stress to describe the damage evolution of materials, and predicts the crack initiation according to the damage variable. According to the principles of thermodynamics (Lemaitre and Desmorat, 2005), the density of elastic strain energy of damaged isotropic materials can be defined as:

W_{e} = \int d W_{e} = \int σ_{i j} d ε_{i j}^{e} = \frac{1}{2} a_{ijkl} ε_{i j} ε_{k l} (1 - D)

(1)

where

σ_{ij}

ε_{ij}^{e}

, are stress and elastic strain tensors respectively, and D is the damage variable. Assume that the material is isotropic and elastic, the elastic strain energy can be expressed by the concept of effective stress as follows (Lemaitre and Desmorat, 2005):

W_{e} = ρ ψ = \frac{1}{2 E (1 - D)} (\frac{2}{3} (1 + ν) σ_{e q}^{2} + 3 (1 - 2 ν) σ_{H}^{2})

(2)

where

ρ

, E and

ν

are the density, Young’s modulus and Poisson’s ratio, respectively

. σ_{eq}

is the von Mises equivalent stress and

σ_{H}

is the hydrostatic pressure.

ψ

is the thermodynamic potential. According to the thermodynamic damage law, the damage strain energy release rate Y is derived from the thermodynamic potential as (Lemaitre, 1996):

Y = ρ \frac{\partial ψ}{\partial D} = \frac{σ^{* 2}}{2 E {(1 - D)}^{2}}

(3)

$σ^{*}$ is defined as the multiaxial equivalent damage stress:

σ^{*} = σ_{e q} \sqrt{R_{ν}}

(4)

where,

R_{ν} = \frac{2}{3} (1 + ν) + 3 (1 - 2 ν) {(\frac{σ_{H}}{σ_{e q}})}^{2}

(5)

Based on the analysis of experimental results for several representative materials, the damage dissipation potential function F_D should be a nonlinear function related to the strain energy release rate Y, and its expression is as follows (Lemaitre, 1987):

F_{D} = \frac{S}{s + 1} {(\frac{Y}{S})}^{s + 1} (\dot{p} + \dot{π})

(6)

where S and s are material parameters,

\dot{p}

and

\dot{π}

represent the accumulated macro-plasticity strain and micro-plastic strain respectively. For high-cycle fatigue, the material often remains in the elastic stage without any macro-plasticity, meaning that the accumulated macro-plasticity strain

\dot{p} = 0

. However, due to the irreversibility of the accumulated micro-plastic strain, damage indeed occurs in the microscale. According to the thermodynamic laws of damage evolution, the damage can be derived from the dissipation potential function:

\dot{D} = \frac{\partial F_{D}}{\partial Y} = {(\frac{Y}{S})}^{s} \dot{π}

(7)

According to (Lemaitre, 1987), $\dot{π}$ can be expressed as a function of stress:

\dot{π} = {(\frac{σ_{e q}}{K (1 - D)})}^{m} \frac{1}{(1 - D)} | {\dot{σ}}_{e q} |

(8)

Substitute equations (3) and (8) into equation (7), the macroscale damage evolution law can be written as:

\dot{D} = {(\frac{σ_{e q}^{2} R_{v}}{2 S E {(1 - D)}^{2}})}^{s} {(\frac{σ_{e q}}{K (1 - D)})}^{m} \frac{1}{(1 - D)} | {\dot{σ}}_{e q} |

(9)

Integrate equation (9) with respect to time within one cycle, we have:

\frac{\partial D}{\partial N} = \int_{t_{1}}^{t_{2}} \dot{D} d t = \int_{t_{1}}^{t_{2}} | {\dot{σ}}_{e q} | {(\frac{σ_{e q}^{2} R_{v}}{2 S E {(1 - D)}^{2}})}^{s} {(\frac{σ_{e q}}{K (1 - D)})}^{m} \frac{1}{(1 - D)} d t

(10)

Assuming that D within one cycle is constant, and replace the R_v with ${\bar{R}}_{v} = (R_{v,max} + R_{v,min}) /2$ , then equation (10) becomes:

\frac{\partial D}{\partial N} = A \frac{(σ_{e q, \max} β - σ_{e q, \min} β) {\bar{R}}_{v}^{s}}{{(1 - D)}^{β}}

(11)

where

A = \frac{1}{β K^{m} {(2 SE)}^{s}}

and

β = 2 s + m + 1

are model parameters,

σ_{eq,max}

and

σ_{eq,min}

are the maximum von Mises stress and the minimum von Mises stress, respectively.

Finite element model for fretting fatigue

The experimental data used in this work are derived from Hojjati-Talemi et al. (Hojjati-Talemi et al., 2014) to validate the effectiveness of the proposed two-scale model. The schematic diagram of the experiment setup is shown in Figure 2. Both the specimen and the fretting pad are made of AA2024-T3 alloy. The tangential force Q between the specimen and the fretting pad is generated by the spring. The normal force F and the axial stress $σ_{axial}$ are exerted by the servo hydraulic press. All experimental loading conditions of fretting fatigue experiment and the number of failure cycles are shown in Table 1, where N_i refers to the fatigue crack initiation life, and N_f refers to the total fatigue life.

Table 1.

Fretting fatigue test data (Hojjati-Talemi et al., 2014).

Test No.	F(N)	Q(N)	$σ_{axial} (MPa)$	$σ_{reaction} (MPa)$	$N_{i} (cycles)$	$N_{f} (cycles)$
FF1	543	155.17	100	92.24	1041669	1407257
FF2	543	186.25	115	105.69	955469	1105245
FF3	543	223.7	135	123.82	277352	358082
FF4	543	195.55	135	125.22	322505	419919
FF5	543	193.7	160	150.32	187725	245690
FF6	543	330.15	190	173.49	120988	141890
FF7	543	322.1	205	188.9	96603	114645
FF8	543	267.15	220	206.64	82403	99607
FF9	543	317.85	220	204.11	71349	86647

The macroscopic finite element model is established by ABAQUS. Due to the symmetry of the experimental device and load, only half of the model needs to be built, as shown in Figure 3. The length of the specimen is 40 mm, the width is 5 mm, and the movement of the specimen in the direction perpendicular to its bottom is constrained. The size of the fretting pad is 10 mm × 10mm, the radius of the bottom arc is 50 mm, and the movement of the fretting pad is limited in the direction perpendicular to its side. The thickness of the specimen and the fretting pad is 4 mm. Apply a multi-point costraint (MPC) to the top of the fretting pad to avoid rotation of the fretting pad. The element type adopts linear quadrilateral reduced integration element (CPE4R), with a minimum mesh size of 0.01 mm in the contact area, which gradually increases away from the contact area. Master-slave contact algorithm is used to define the contact relationship between the fretting pad and the specimen. The circular surface of the fretting pad is defined as the main surface, and the top surface of the specimen is defined as the slave surface.

The loading process is shown in Figure 3. In the first step, contact load is applied and gradually increases to the maximum value, which remains unchanged in the following steps. In the remaining steps, axial stress and reaction force are applied to both sides of the specimen at the same time to match the cyclic loading conditions of the test. The reaction force can be calculated by the formula $σ_{reaction} = σ_{axial} - Q A_{s}$ , where A_s is the cross-sectional area of the specimen.

Numerical method

Continuum damage mechanics provides macroscopic field variables that reflect the average damage of structural materials. In step $n_{i +1}$ , the material behavior is updated by the damage variable D_i₊₁:

E_{i + 1} = (1 - D_{i + 1}) E_{0}

(12)

To reduce computational costs, it is typically assumed that damage remains constant over a certain number of cycles ( $Δ N$ jump steps). Based on the damage and stress from the step $n_{i}$ , the accumulation of damage in the step $n_{i +1}$ is calculated using the following formula:

D_{i + 1} = D_{i} + \frac{\partial D}{\partial N} Δ N

(13)

The subroutine USDFLD is used to calculate damage accumulation and stiffness reduction. To overcome the computational singularities that may occur due to the presence of (1−D) in the denominator of the equation, the damage limit is set as 0.95 instead of 1. Due to the large stress gradient in the fretting fatigue contact area, the critical distance theory method (Taylor, 2008) is used to determine the contact damage influence area to overcome the mesh dependency problem. The accuracy of this method has already been validated in our previous work (Lin and Xu, 2022). The macroscopic damage evolution law is only applied in the region indicated by L_c as shown in Figure 4. The formula for calculating the critical distance is as follows:

L_{c} = \frac{1}{π} {(\frac{Δ K_{t h}}{Δ σ_{0}})}^{2}

(14)

where L_c,

{Δσ}_{0}

and

{ΔK}_{th}

are the critical distance, the limit range of plain fatigue and the range of threshold stress intensity factor respectively.

The material parameters for the macroscopic continuum damage model are listed in Table 2. A comparison was made between the effects of various jump steps on the prediction accuracy of fretting fatigue crack initiation life in reference (Lin and Xu, 2022). It was discovered that when $Δ N = 10000$ , accuracy was met the requirement while reducing the computational cost. Therefore, the jump step number was determined to be 10000. The model parameters A and β were identified through four experiments (FF1, FF3, FF6, FF9), with the root mean square error (RMSE) used to quantify the error between experimental and simulated results:

RMSE = \sqrt{\frac{1}{n} {\sum_{j = 1}^{n} (N_{i, experiment} - N_{i, simulation})}^{2}}

(15)

where n is the number of scattered data. By minimizing the RMSE value, the model parameters A = 1.15e–21 and β = 5.9 can be obtained. The identified results are shown in Table 3.

Table 2.

The parameters required for CDM model (Hojjati-Talemi et al., 2014; Lemaitre and Desmorat, 2005).

Yield strength	$σ_{y}$	383 MPa
Tensile strength	$σ_{u}$	506MPa
Young’s modulus	E	72.1 GPa
Poisson coefficient	$ν$	0.33
Coefficient of friction	$μ$	0.65
Critical distance	$L_{c}$	0.05 mm
Damage law exponent	s	1.0
Damage law coefficient	S	1.7

Table 3.

Model parameters identification results.

Test no.	$σ_{axial} (MPa)$	$N_{i,experiment} (cycles)$	$N_{i,simulation} (cycles)$	Error
FF1	100	1041669	1190000	14.24%
FF3	135	277352	280000	0.95%
FF6	190	120988	90000	−25.61%
FF9	220	71349	70000	−1.89%

Crystal plasticity constitutive model and parameter identification

Crystal plasticity constitutive model

To reflect microscale mechanism unable to be explicitly considered in the macroscale, the crystal plasticity model was employed to describe the mechanism of polycrystalline mechanical response, heterogeneous deformation and microstructure evolution. The phenomenological constitutive law of DAMASK (Roters et al., 2019) user material subroutine of ABAQUS was used in this work. In the crystal plasticity model, the multiplicative decomposition of the deformation gradient F reads as:

F = F_{e} F_{p}

(16)

where F_e represents the rotation and elastic stretch of the lattice and F_p represents the plastic shear deformation along the crystalline slip plane. The plastic velocity gradient is defined as:

L_{p} = {\dot{F}}_{p} F_{p}^{−1} = \sum_{α=1}^{N} {\dot{γ}}^{α} m^{α} \otimes n^{α}

(17)

where

{\dot{γ}}^{α}

m^{α}

and

n^{α}

are the plastic shear rate of slip system α, the unit vector of the slip direction and the normal direction of the slip plane, respectively. N is the number of active slip system.

The plastic slip rate ${\dot{γ}}^{α}$ is calculated following the power-law flow equation proposed by Hutchinson (Hutchinson, 1976):

{\dot{γ}}^{α} = {\dot{γ}}_{0} {| \frac{τ^{α} - χ^{α}}{τ_{c}^{α}} |}^{n} sgn (τ^{α} - χ^{α})

(18)

with

{\dot{γ}}_{0}

the reference strain rate, n the strain rate exponent,

χ^{α}

the kinematic backstress,

τ^{α}

and

τ_{c}^{α}

respectively the resolved shear stress (RSS) and critical resolved shear stress (CRSS) on the slip system α. The introduction of a kinematic backstress is necessary to simulate cyclic deformation behavior, and the backstress evolution is defined by the Armstrong-Frederick (Frederick and Armstrong, 2007) model as follows:

{\dot{χ}}^{α} =C {\dot{γ}}^{α} -B |{\dot{γ}}^{α}| χ^{α}

(19)

where C and B are model constants, each associated with a dynamic hardening and a dynamic recovery. The influence of any slip system β on the hardening behavior of slip system α is given by (Bronkhorst et al., 1992[1662]):

{\dot{τ}}_{c}^{α} = {\sum_{β = 1}^{N} q^{α β} h_{0} (1 - \frac{τ_{c}^{β}}{τ_{c s}^{β}})}^{a} | {\dot{γ}}^{α} |

(20)

where h₀, a and

τ_{cs}^{β}

are the initial hardening coefficient, the hardening exponent and the saturated CRSS, respectively.

q^{αβ}

is the interaction coefficient of the slip systems, when α and β are coplanar systems,

q^{αβ} =1

while it is set to 1.4 for non-coplanar systems.

Calibration of CPFEM parameters of AA2024-T3 alloy

Crystal plasticity parameters are fitted through inverse analysis so that the simulated hysteresis curves match the experimental data under different cycle numbers. The low-cycle fatigue test and hysteretic curve of AA2024-T3 alloy were cited from (Li X et al., 2021). NEPER with a generalised Voronoi tessellation (GVT) method (Quey et al., 2011) was used to build a 3 D RVE with 400 grains. The model consists of 4913 linear hexahedral elements (C3D8), as shown in the Figure 5(a). Periodic boundary conditions (PBCs) were used to ensure that RVE deformation conformed to macroscopic behavior, as shown in the Figure 5(b). The PBCs are applied on the nodes of the RVE surfaces and satisfy the following equation constraints:

u_{x} (a, y, z) - u_{x} (0, y, z) + u_{ref, x} = 0

(21)

u_{y} (x, b, z) - u_{y} (x, 0, z) = 0

(22)

u_{z} (x, y, c) - u_{z} (x, y, 0) = 0

(23)

where u_x, u_y, u_z represent the components of displacement, while a, b, c denote the dimensions of RVE in the x, y, z directions, respectively. u_ref,x represents the predefined displacement value in the loading direction.

Figure 6 shows the stress-strain hysteresis curves from both experiments and simulations for the 1st, 7th, and 14th fatigue cycles. It can be seen that the simulation values are in good agreement with the experimental results. Accordingly, the crystal plasticity parameters are shown in Table 4. The elastic coefficients are taken from literature (Singh et al., 2022).

Table 4.

Crystal plasticity parameters for AA2024-T3 alloy. The elastic coefficients are taken from literature (Singh et al., 2022).

	Parameter	Unit	Value
Elastic constants	C ₁₁	GPa	106.75
	C ₁₂	GPa	60.41
	C ₄₄	GPa	28.34
Flow parameters	${\dot{γ}}_{0}$	s⁻¹	0.001
	n	–	50
Hardening parameters	$τ_{0}$	MPa	148
	$τ_{s}$	MPa	265
	a	–	2.25
	h ₀	MPa	1250
	A	MPa	3600
	B	–	480

Continuum damage mechanics-crystal plasticity FEM based two-scale model

Scale coupling law between CDM and CPFEM

Instead of a single scale method, this work integrates macroscopic CDM and microscopic CPFEM in a bidirectional manner to achieve the transfer of multiscale information in the model, reflecting the scale coupling mechanism of material deformation, and attempting to quantitatively characterize this mechanism. Such a two-scale model can predict both the macroscopic deformation behavior of the component and the microscopic structural evolution, thus achieving coupled predictions of multiscale responses in the fatigue crack initiation process. Figure 7 illustrates the characteristics of the two-scale model, which aims to reflect the microscopic mechanisms in macroscopic calculations of components. In order to impose the deformation response calculation results from the element with the maximum macroscopic damage to the microscale RVE, the microscale RVE is treated as an element with microstructure within the macroscale model. The boundary conditions of RVE are defined based on the theory of continuous displacement and linear interpolation in finite element method.

Existing findings (Cruzado et al., 2018b; Korsunsky et al., 2007) indicate that dissipation energy serves as the driving force for crack initiation, showing good agreement with the experimental data of crack initiation. To avoid the change in dissipation energy being negative, the back stress should also be considered. Therefore, the accumulated dissipation energy W is selected as the microscopic indicator parameter:

W = \sum_{α}^{N} \int_{0}^{t} (τ^{α} - χ^{α}) {\dot{γ}}^{α} d t

(24)

In order to extract the deformation history of the macroscopic element with the maximum damage value and apply it to the microscopic RVE to calculate the change in accumulated dissipated energy, a tessellated RVE with periodic microstructure (polycrystalline grains) is adopted. There are 200 grains in the RVE as shown in the Figure 8(a). The size of RVE, $L_{x} \times L_{y} = 1 mm × 1 mm$ . The RVE is discretized into $71 × 71 = 5041$ linear quadrilateral elements (CPE4), for the influence of RVE size and the mesh convergence analysis, please see Appendix 1. It is worth noting that due to the unavailability of the crystal orientation distribution of the test material, this study employs randomly generated grains instead of real grains to investigate the relationship between macroscopic damage and microscopic accumulated dissipation energy. This approach can contribute to the understanding of the universal relationship between macroscopic damage and microscopic dissipation energy, independent of specific grain size and orientation. However, it may fail to capture the influence of particular grain characteristics to certain extent. Additionally, the grain size and pole figure of the RVE used are shown in Figure 8(b) and (c). The grain sizes generated randomly in the model follow a normal distribution to represent the actual conditions of metallic materials. Figure 9 illustrates the inheritance of boundary conditions from the macroscopic element to the microscopic RVE. The boundary conditions for the RVE boundary nodes are implemented using linear interpolation of the displacement field.

Figure 10 shows the relationship between the damage value of the macroscopic damage maximum element and the maximum accumulated dissipation energy of RVE in FF1. It is worth noting that when the macroscopic damage reaches the critical value of 0.95, it will no longer increase, while the microscopic dissipation energy continues to accumulate. This will lead to certain errors in the numerical fitting process. Therefore, we choose to eliminate the data points where the damage reaches the critical value. As a result, the accumulated crystal plasticity dissipation energy was selected as the indictor parameter in the microscale, the scale coupling law with macro damage was proposed as follows:

D = d \cdot {| W_{RVE} - W_{0} |}^{p}

(25)

where D is the damage value of the integration point of the macroscopic element. W_RVE is the maximum accumulated dissipation energy of microscopic RVE, d, W₀ and p are three model parameters. It should be noted that when macroscopic cracks occur, i.e., the damage reaches the critical value, the maximum accumulated dissipation energy will reach its critical value W_crit. This scale coupling law establishes a bridge linking microscopic dissipation energy of RVE to macroscopic damage at material points. So that macro damage evolution process can be driven by the scale coupling law, instead of macro damage evolution laws.

Numerical procedures of the CDM-CPFEM based two-scale model

The overall flowchart of the proposed two-scale model is given in Figure 11. The numerical procedure can be briefly introduced as follows:

Start the two-scale prediction.

Calculate $n_{k}$ (k = 1, 2, 3…) cycles using macroscopic damage evolution model.

Extract the deformation history of the region of interest and apply it to the RVE to compute the microscopic accumulated dissipation energy W_k.

Calculates the damage D_k for $n_{k}$ cycles based on equation (25). Compare the predictions in macroscale and microscale. If any two-scale error is not allowed, return to step (b) to change the cycles $n_{k}$ . Otherwise, go to step (e).

Determine the calculated damage value D_k, if D_k < 0.95, bring it back to the macroscopic model and continue to execute step (b) for the next $n_{k}$ (k = k+1) cycles; if D_k ≥ 0.95, the initiation life is obtained by formula $N_{i} = ΔN \sum n_{k}$ .

Numerical results and discussion

Parameters of the two-scale model

Since d, W₀ and p of the two-scale model are unknown model parameters, experiments (FF1, FF3, FF6, FF9) were selected to identify these parameters. All model parameters d, p and W₀ are plotted as functions of shear stress ${Δσ = Q/A}_{s}$ on the contact surface and reaction force $σ_{reaction}$ , respectively, as shown in Figure 12, and Table 5 lists the model parameters for all experiments.

Table 5.

Summary of model parameters from nonlinear regression analysis.

Test no.	$N_{i} (cycles)$	$σ_{axial} (MPa)$	d	p	W ₀
Calibrated results
FF1	1041669	100	0.601	0.192	2.41E-10
FF3	277352	135	0.502	0.320	1.13E-03
FF6	120988	190	0.415	0.552	0.250
FF9	71349	220	0.423	0.524	0.470
Estimated results
FF2	955469	115	0.550	0.248	1.70E-06
FF4	322505	135	0.537	0.265	1.58E-03
FF5	187725	160	0.539	0.262	0.066
FF7	96603	205	0.420	0.533	0.376
FF8	82403	220	0.460	0.411	0.482

Fretting fatigue crack initiation of AA2024-T3 alloy

Figure 13 shows the comparison between numerical simulation and experimental results of initiation life in terms of the proposed two-scale model and macroscopic CDM model (Hojjati-Talemi et al., 2014). It can be seen that all the data points predicted by the proposed two-scale method fall within the scattering band of ±50% (2 N), indicating that the two-scale method has better accuracy in the prediction of fretting fatigue crack initiation life.

To validate the independence of the critical value of maximum accumulated dissipation energy W_crit from loading conditions, this study presents the maximum accumulated dissipation energy of the microscale RVE corresponding to all loading conditions and establish a constant value for W_crit, as shown in the Figure 14. It is observed that the average critical value fluctuates around 7.32 MJ/m³ under different conditions, with an error range within ±50%. Consequently, W_crit can be considered as a parameter independent of the loading conditions and reflecting the material properties.

As an illustration, Figure 15 presents the distribution of macroscopic damage and accumulated dissipation energy within the microscale RVE under different cycle numbers for the FF3 loading condition. The damage reaches its maximum at the contact boundary elements. By extracting the deformation history of these elements and applying it to the corresponding RVE, the distribution of dissipation energy can be obtained. The damage is then calculated using equation (25) and incorporated into the macroscopic model. As the number of cycles increases, damage at the contact edge gradually intensifies, and the corresponding RVE accumulated dissipation energy also increases, often occurring at or near grain boundaries. When the damage reaches 0.95, it indicates the completion of the fretting fatigue crack initiation process. The corresponding cycle number of 21 × 10⁴ is the predicted initiation life for FF3.

Moreover, Figure 16(a) and (b) show the evolution of the normal stress and shear stress on the contact surface under FF3 loading condition with the number of cycles. It can be seen that the distribution of normal stress and shear stress on the contact surface is consistent with Hertz contact theory (Hills, 1994). As the number of cycles increases, the normal stress jumps at approximately $x = 0.32 mm$ , suggesting this location experiences significant stress redistribution induced by damage accumulation. Figure 16(c) shows the evolution of the damage on the contact surface with respect to the number of cycles for FF3, revealing that the accumulation of damage first reaches a critical value at $x = 0.32 mm$ , suggesting that the initiation of fretting fatigue cracks predominantly occurs at this location.

In addition, Figure 17(a) shows the accumulated dissipation energy distribution map of RVE at the crack initiation of the FF3, and Figure 17(b) presents the accumulated plastic shear strain distribution map of RVE at the crack initiation of the FF3. The definition of accumulated plastic shear strain is as follows (Liu et al., 2021):

γ = \sum_{α}^{N} \int_{0}^{t} | {\dot{γ}}^{α} | d t

(26)

It can be observed that the distribution of accumulated dissipation energy and accumulated plastic shear strain is similar, with high levels appearing near the intersections and adjacent areas of grain boundaries in multiple grains. Moreover, as illustrated in Figure 17(c), the main concentration of maximum von Mises stress occurs at the intersection points of grain boundaries in multiple grains, suggesting that fatigue micro-cracks may occur in these stress concentrated areas.

Figure 18(a) illustrates the maximum accumulated dissipation energy in the RVE and the macroscopic fatigue damage for FF3, with respect to the number of loading cycles. It can be observed that both damage and accumulated dissipation energy increase gradually during the initial loading cycles and then accelerate significantly in the final stage. Notably, Figure 18(b) displays the changes in damage and accumulated dissipation energy process for specimens FF1, FF3, FF6 and FF9. Under the conditions of lower stress levels (FF1, FF3), the rate of dissipation energy can be clearly divided into two stages; the rate of accumulated dissipation energy is very slow during the macroscopic damage D < 0.5, but increases rapidly when D > 0.5. On the other hand, under conditions of higher stress levels (FF6, FF9), the rate of accumulated dissipation energy is more uniform during the damage process, with no apparent demarcation. It means that in the case of smaller load conditions, the microscopic plastic deformation is relatively small, and the dissipation energy of internal crystal defects accumulates slowly when damage is minor. But when the damage develops to a certain extent, the accumulation of material defects manifests greater plasticity, leading to a rapid increase in dissipation energy. However, under conditions of larger loads, even when the macroscopic damage is minor, the microscopic plastic deformation and stress of the internal defects in the material are larger, resulting in a relatively uniform dissipation energy speed throughout the damage process.

In fact, fatigue crack initiation and propagation exhibit a preference for specific planes, as illustrated in Figure 19. This figure specifically marks the first activated slip system in each grain and displays the relationship between equivalent stress and accumulated dissipation energy in grains G1 to G4 within the RVE for FF3. A noticeable jump in equivalent stress and accumulated dissipation energy occurs along the grain boundary at path A-A' and B-B'. This also shows that the heterogeneity of grain orientations is a primary contributor to the accumulation of material damage and the increase in premature failure.

Higher stress gradients in the contact area will lead to significant gradients in macroscopic damage as well, where damage exhibits a non-uniform distribution within the macroscopic elements. Figure 20 displays the accumulated dissipation energy distribution of RVE at the crack nucleation site for FF3. It is evident that the accumulated dissipation energy is primarily concentrated in the RVE corresponding to element B and shows a distinct band-like distribution. The formation of slip bands is a microscopic indicator of plastic deformation, which occurs when dislocations within the crystal lattice move under stress. The presence of such bands is a precursor to the crack nucleation and provides clues on the initial direction that cracks may form and propagate through the material.

Conclusion

Coupling the macroscopic continuum damage mechanics model and the microscopic crystal plasticity model, this study proposes a two-scale fatigue model and studies the crack initiation process of fretting fatigue in cylindrical contact. The main conclusions are as follows:

The proposed two-scale model considers the scale coupling mechanism and can predict the response of fretting fatigue crack initiation at different scales in a reasonable manner. The data points for predicting crack initiation life fall within the scatter band of ±50% (2N).

The macroscopic damage and the microscopic accumulated energy dissipation exhibit a power-law relationship, and with the increase of cyclic loading, the damage and the accumulated dissipation energy of the material show an obvious nonlinear growth trend. In addition, the inhomogeneity of grain orientation leads to the accumulation of dissipated energy mainly concentrated near the grain boundaries.

Under lower stress level conditions, the rate of accumulated dissipation energy can be clearly divided into two stages; the dissipation energy rate is slower during macroscopic damage D < 0.5, but increases rapidly when D > 0.5. Under higher stress level conditions, the rate of accumulated dissipation energy is more uniform during the damage process, without any noticeable demarcation point.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by National Natural Science Foundation of China (Program No. 52478209), Scientific Research Plan Projects of Shaanxi Education Department (Program No. 20JY032), and ‘Research on Flexible Photovoltaic Supporting Structure System’ from Xi’an Thermal Power Research Institute Co. Ltd., Huaneng Group.

ORCID iDs

Yuan Fang

Xu Yazhou

Appendix 1

By simulating the deformation history of the element with the maximum damage in FF9 using RVEs with four different numbers of elements, a convergence study was conducted to ensure that the mesh density was high enough to produce a reliable mechanical response from the model. Figure 21 shows the change in accumulated dissipation energy of the RVE with increasing loading cycles under the four different meshing conditions. The changes in accumulated energy dissipation under the four meshing conditions are similar. When using $71 × 71 = 5041$ and $90 × 90 = 8100$ elements, the peak difference in these quantities is within 2%, whereas the computation time significantly increases when using 5041 elements. Therefore, we have reason to believe that the CPFE model has converged when using 5041 elements, while also saving computational resources.

Simulations were conducted on four different sized RVEs under the strain history of the maximum damaged unit in FF9 to investigate the influence of RVE size on the simulation results. As shown in Figure 22, RVEs with four sizes of 1.5 mm × 1.5 mm, 1.2 mm × 1.2 mm, 1.0 mm × 1.0 mm, and 0.8 mm × 0.8 mm were simulated. The average grain diameter of all RVEs is the same, the grain orientations are randomly distributed, and the average number of elements per grain is constant. It can be seen from the figure that the evolution laws of the maximum accumulated dissipation energy of RVEs with different sizes are similar, but there are certain differences in their peaks, mainly due to the randomness of the grain morphology and orientation in RVEs of different sizes. Therefore, for a determined grain morphology size and orientation distribution, as long as the number of grains contained is sufficient, the influence of the RVE size can be ignored.

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