1 Introduction

A structural component serves as an essential part of a mechanical system that is typically subject to the cyclic loading during an industrial application. Ratcheting strain ε_γ is a plastic accumulation phenomenon occurring under asymmetrical stress cycling, and is important in the safety assessment and structure design of a component. An accurate prediction of the ratcheting behavior caused by cyclic loading requires the development of an appropriate constitutive model that includes the nonlinear relation between the stresses and strains.

In cyclic constitutive studies, a prediction of the ratcheting behavior is one of the most difficult aspects to achieve. As a basic model to simulate the ratcheting behavior, Frederick and Armstrong ^[1] proposed a well-known nonlinear kinematic hardening rule, called the A-F model. Because the A-F model over-predicts the ratcheting strain, a number of advanced nonlinear kinematic hardening rules have been developed based on this model. Chaboche et al. ^[2] assumed that the back stress is composed of several components that obey the A-F model, and developed a model that can describe a nonlinear plastic deformation curve much smoother than the A-F model ^[3-7]. With the Ohno-Wang model, Ohno and Wang ^[8-9] assumed that each component in kinematic hardening had a threshold for a dynamic recovery that was fully activated. The simulation results show that the Ohno-Wang model and its revisions ^[10-12] can predict much less accumulation of the ratcheting strains than the A-F model. To avoid the high-order nonlinearity of the kinematic hardening law with the Ohno-Wang Ⅱ model, Ohno and Abdel-Karim ^[13] developed the Ohno-Abdel-Karim model ^[13-14] which blended the A-F model ^[1] with the Ohno-Wang model ^[8] by introducing a ratcheting parameter μ. Moreover, an effective radial return algorithm for a nonlinear kinematic hardening law of the Ohno-Abdel-Karim model was proposed by Kobayashi and Ohno ^[15]. However, the Ohno-Abdel-Karim model can only simulate a constant ratcheting strain rate of far greater than zero after a finite cycle loading ^[16-17]. Therefore, the constant ratcheting parameter μ is replaced with an evolution equation, which is a function of the accumulated plastic strain ^[18-19]. When the value of the evolution equation is close to zero, a quasi-shakedown of the ratcheting can be properly simulated. Because ratcheting is the second cumulative phenomenon of strain under the asymmetric stress cycle loading, and a quantitative description of ratcheting is not easy to achieve, the development of more effective kinematic hardening models is still required.

Most metal materials present viscous characteristics and are very sensitive to their load history, environment, and properties. The establishment of a general form of a cyclic visco-plastic constitutive model based on the effects of certain viscous factors includes two parts, the choice of the viscosity function and the kinematic hardening rule ^[20-21]. Based on the existing unified visco-plastic constitutive model, for the development of a more complete visco-plastic model, an increased number of parameters are required, making the model more complex. The fractional calculus is an effective mathematical tool for describing the memory phenomena, and has been successfully applied to different disciplines ^[22-28]. Some early studies ^{[15, 29]} showed that the visco-elastic behaviors of materials were described much more accurately by a fractional model than by integer-order models (such as the Kevin, Voigt, and Maxwell models). However, a fractional model has rarely been used to describe the visco-plasticity, particularly the cyclic visco-plastic deformation. Sumelka ^[30] developed a fractional visco-plastic model based on the classical Perzyna's type visco-plasticity ^[31]. It was shown that the fractional visco-plastic strain evolution can simulate a change in volume in a flexible manner within the visco-plastic range without an additional potential assumption. The outstanding performance of a fractional memory index, which was discovered by Du et al. ^[25], is found not only in mechanics, but also in physics and biology. Sun et al. ^[32] incorporated fractional order into a traditional bounding surface plasticity theory describing the deformation behavior of a ballast under a significant amount of cyclic loading. Krasnobrizha et al. ^[33] developed a collaborative model composed of an elasto-plastic damage model and a fractional derivative model. It was shown that the memory phenomena of the fractional-order model can compensate for the deficiency of the elasto-plastic damage model in describing the hysteresis loops. Although the physical meaning of the fractional order can only be obtained from particular examples, it is still worth exploring in different disciplines.

In this study, using the backward Euler implicit stress integration algorithm, we find that the cyclic elasto-plastic constitutive model exhibits viscosity when the accumulated plastic strain rate and nonlinear kinematic hardening rule are described through a fractional order. Therefore, a novel fractional-order unified visco-plastic (FVP) constitutive model is proposed instead of the classical Perzyna's type visco-plasticity for the rate-dependent ratcheting behavior. Moreover, based on the Ohno-Abdel-Karim model, a new radial return method for the back stress developed to properly describe the unclosed hysteresis loops of the stress-strain is described. The capability of the FVP model to simulate rate-dependent ratcheting is discussed through a comparison with the experimental data ^[34]. It is found that the fractional order is a flexible mathematical tool for describing the viscous ratcheting behavior.

2 Fractional derivative

A fractional derivative is established based on the differential approximation recursive of the classical integer order derivative. Some of the most frequently used definitions for a fractional derivative are the Grünwald-Letnikov fractional derivative, the Riemann-Liouville fractional derivative, and the Caputo fractional derivative, which are equivalent under certain conditions. Among them, the forms of the initial conditions for the Caputo fractional-order differential equations are the same as those of the classical integer-order differential equations, which is an advantage in the modeling applied in engineering mechanics ^[35-36] and can be written as

(1)

Here, _a^C D_t^α u(t) is the Caputo fractional derivative of order α for the function u(t), n is a positive integer, a and t are the bounds of operation for _a^C D_t^α u(t), and Γ(x) is the classical Euler's gamma function, which plays an important role in the Caputo fractional derivative, and is defined as follows:

(2)

It should be mentioned that the gamma function satisfies a basic recurrence formula, namely,

(3)

In this study, the classical Caputo fractional derivative definition is incorporated into the cyclic elasto-plastic constitutive model. One of the most important advantages is that the Caputo fractional derivative can be approximated as a constant fractional derivative through the use of the implicit back Euler integration method in the FVP model, where the details of the derivation process are described in Subsection 4.1.

3 Main constitutive equations

In general, two different types of models, the cyclic elasto-plastic constitutive model (rate-independent) and the cyclic visco-plastic constitutive model (rate dependent), have been developed to describe the inelastic cyclic deformation behavior of a material. The state of stress in the cyclic elasto-plastic model is constrained within the elastic domain without the influence of the viscous force, and it is difficult to describe the viscous behavior of a metal material. Therefore, it is necessary to develop a general form of a cyclic visco-plastic constitutive model based on the effects of certain viscous factors, which essentially includes two fixed aspects. One is to choose the viscosity function, and the other is to develop kinematic hardening equations. For the traditional unified visco-plastic modeling, one can refer to previous studies ^[20-21]. In the fixed frame of a traditional unified visco-plastic model, a more complete visco-plastic model requires an increase in the number of parameters, which makes the model more complex. In this paper, a novel FVP constitutive model is proposed based on the elasto-plastic constitutive model.

3.1 Elasto-plastic constitutive model

In the framework of elasto-plasticity, the plastic deformation of the material is considered rate-independent. The total strain ε is composed of elastic strain ε^e that obeys Hooke's law, and the plastic strain ε^p controlled based on the associated plastic flow rule . Thus, the main equations of the elasto-plastic model are

(4)

(5)

(6)

(7)

where σ is the stress tensor, D^e is the elastic stiffness tensor, : is the double point inner product of the tensors, is the plastic multiplier, s is the stress deviation tensor of σ, n= is the direction tensor of the strain derived from Eqs. (6) and (7), and α is the deviatoric part of the back stress, which is the center of the yield surface, and can be decomposed into several parts ^[2],

(8)

where each component of α^j obeys the Ohno-Abdel-Karim model ^[13-14]. Thus, we have

(9)

(10)

where ξ^j and r^j are material constants, and μ^j=μ is the ratcheting parameter (0≤μ^j≤1), which is a function of the accumulated plastic strain ^[18],

(11)

where μ₀ and k are the parameters controlling the evolution rate of the ratcheting parameter μ and can be determined through trial and error ^[18]. Here, 〈 · 〉 is Macauley's bracket. In addition, for x≤0, 〈 x〉=0; for x>0, 〈 x 〉=x, and

(12)

Here, H is a Heaviside step function, and

(13)

Moreover, Y is the isotropic deformation resistance. The nonlinear isotropic hardening rule proposed by Chaboche and Nouailhas ^[5-6] is employed in this model,

(14)

where Y_sa is the saturated isotropic deformation resistance, γ is a material constant, p is the accumulated plastic strain, and

3.2 FVP constitutive model

The value of the ratcheting strain depends mainly on the viscous characteristics of the material. The greater the viscosity, the greater the ratcheting strain, and vice versa. Owing to the effect of the viscosity, the delayed memory of the creep will affect the increase in the accumulated plastic strain ^[37]. Therefore, the cumulative strain rate depends not only on the state of the current moment (classical integer-order derivatives definition), but also on the previous history (fractional derivatives definition). While a metal material undergoes the plastic deformation and then unloads, residual stresses occur at the grain boundaries between the grains, which leads to the Bauschinger effect ^[38-39] described using the kinematic hardening rule. For cyclic loading, the Bauschinger effect depends not only on the microscopic residual stress of the current cycle, but also on those of the previous cycles. In this work, considering the fractional-order memory index, we assume that the cumulative plastic strain p and the back stress α^j increase in the form of the fractional order used to describe the viscous behavior of the materials. Thus, we have

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

where tp is the cumulative time of the plastic deformation, ζ is the parameter controlling the strength of the memory index, and 0CDζtpβ is the Caputo fractional-order derivative operator with the order β (0 < β < 1). With an increase in the number of cyclic loading cycles, a crystal dislocation slip and micro cracks will occur in the microstructure of the material, and the viscous behavior and properties of the material will change accordingly [40]. The order β denotes the viscous characteristics of the material, which changes with the time of the loading and is assumed to be a function of the relative time between the plastic deformation and the total deformation,

(23)

Here, t is the cumulative time of the total deformation, and m is the parameter controlling the viscosity and can be determined by trial and error. For the meaning of the other parameters, refer to the elasto-plastic constitutive model.

4 Numerical implementation

In this section, the numerical implementation of the FVP constitutive model is elaborated upon, and a new implicit stress integration algorithm is derived based on the backward Euler integration and the radial return method ^{[5, 13-14, 41]}.

4.1 Discretization of FVP constitutive model

If adjacent steps are assumed from the state n to the state n+1, the FVP constitutive model in Eqs. (15)-(22) can be discretized through the backward Euler method as follows:

(24)

(25)

(26)

(27)

(28a)

where is assumed to be a constant for a given interval, and we can thus obtain

(28b)

(29)

(30)

and the kinematic hardening rule is discretized as follows:

(31)

(32)

where is assumed to be a constant for a given interval. Both sides of Eq. (20) are multiplied by Δ t_n+1, and with ξ^j=h^j/r^j, we can obtain

(33)

The isotropic hardening rule is discretized as follows:

(34)

(35)

4.2 Implicit stress integration algorithm

At the time t_n (the step n), we assume that the variables ε_n, ε_n^p, s_n, α_n, p_n, Δε_n+1, and Δ t_n+1 are known, and the variables ε_n^e and σ_n are obtained through the elastic stress-strain relationships ^{[16, 21]}. In this study, an elastic predictor and plastic corrector algorithm is used to solve the σ_n+1 (the step n+1) based on the discretized FVP constitutive equations.

4.2.1 Elastic predictor

The entire strain increment Δε_n+1 is assumed to be an elastic strain. Therefore, the tentative stress state can be solved through the elastic stress-strain relationships as follows:

(36)

The tentative stress state σ_n+1^* is discriminated based on the yield criterion in Eq. (30),

(37)

where

(38)

If F_n+1^* < 0, it is considered that the plastic strain occurs within the entire strain increment Δε_n+1. Therefore, the tentative stress σ_n+1^* is accepted as the actual stress state σ_n+1.

4.2.2 Plastic corrector

If F_n+1^*≥0, it is considered that the plastic strain occurs within the entire strain increment Δε_n+1, and the actual stress state σ_n+1 is solved through the plastic corrector. Therefore, substituting Eqs. (25) and (36) into Eq. (26), the following equation can be derived:

(39)

where D:Δε_n+1^p is the plastic corrector.

4.2.3 Nonlinear scalar equation

It can be seen from Eq. (39) that the actual stress state σ_n+1 can be readily obtained as long as Δε_n+1^p is solved. This problem can be reduced to an implicit nonlinear scalar equation, which is a function of Δ p_n+1, and can be solved using the successive substitutive algorithm ^{[15-16, 42-43]}. Based on the assumption of elastic isotropic and plastic incompressible plasticity, the plastic corrector D:Δε_n+1^p is equal to 2GΔε_n+1^p, and the expression of Eq. (39) in the deviatoric space can be written as follows:

(40)

where G is the modulus of rigidity. Using Eq. (31), we obtain

(41)

where α_n+1^j can be obtained from Eqs. (32) and (33),

(42)

and

(43)

(44)

It is worth noting that θ_n+1^j in Eq. (42), rather than Δp_n+1^j, is readily specified through the radial return method for integrating the back stress with the critical value ηr^j.

First, as illustrated in Fig. 1, the predicted magnitude of the back stress α^j from α_n^j to α_n+1^j is given by ignoring the critical surface f^j=0,

(45)

Then, the tentative state of α_n+1^#j is determined using the radial return method as follows:

(46)

where

(47)

If the tentative state α_n+1^#j is beyond the critical surface, i.e., f_n+1^#j=(α_n+1^#j)²-(r^j)²>0, α_n+1^#j is radially projected onto the critical surfaces to obtain α_n+1^j, where α_n+1^#j=:. Otherwise, α_n+1^#j is taken as α_n+1^j. Thus, we obtain

(48)

(49)

If α_n+1^j=θ_n+1^jα_n+1^#j, combining Eqs. (46), (48), and (49) with α_n+1^#j=c_n+1^jα_n+1^*j, θ_n+1^j can be written as follows:

(50)

Substituting Eq. (42) into Eq. (41) and using Eqs. (26) and (28b), we have

(51)

Substituting Eq. (51) into the yield function (30), and using F_n+1=0, we obtain

(52)

It should be noted that Y_n+1 and θ_n+1^j are functions of Δp_n+1 and Δp_n+1^j, respectively. Therefore, Eq. (51) is a nonlinear scalar equation for Δp_n+1, which can be solved using a successive substitution ^[15]. If Δ p_n+1 is found, the update states σ_n+1, ε_n+1^p, and α_n+1^j can be readily obtained using Eqs. (25)-(29) and (42). If T is the total time of the loading, the implicit stress integration algorithm can be generally illustrated using the flow diagram shown in Fig. 2.

5 Simulation and discussion

In this section, the predictive capability of the FVP constitutive model with regard to the rate-dependent behavior of the monotonic tension and uniaxial ratcheting at different rates is verified through a comparison with the corresponding experimental data ^[34].

5.1 Identification of FVP model parameters

The FVP constitutive model is developed from the Ohno-Abdel-Karim model based on the elasto-plastic framework, and therefore the material parameters ξ^j and r^j can be determined through the monotonic tensile experiments ^{[10-11, 17, 44-45]}, and the value of the material constant in Eq. (22) can be obtained with the experimental data ^{[5-6, 45]}. Three additional parameters (ζ, m, and η) are introduced in the FVP model for the rate-dependent ratcheting behavior. Figures 3-5 show the effect of different values of the parameters (ζ, m, and η) on the rate-dependent behavior. It can be seen in Figs. 3(a) and 3(b) that the value of the parameter ζ only affects the initial yielding point of the uniaxial tension and the initial value of the ratcheting strain, but cannot control the viscosity. The parameter m can control the rate-dependent behavior of the ratcheting strain without affecting the uniaxial tension, as shown in Figs. 4(a) and 4(b). By changing the value of the parameter η, the FVP constitutive model not only can control the rate-dependent behaviors of the uniaxial tension and the ratcheting strain, but also has the ability to adjust the initial value of the latter, as shown in Figs. 5(a) and 5(b). The values of the parameters (ζ and η) can be determined by trial and error based on the uniaxial tension at different strain rates. The parameter m is insensitive to the uniaxial tension, which can be determined through trial and error from the ratcheting strain. The ranges of empirical values of the three parameters are given as follows: ζ∈(0, 1), m∈(1.0, 5.0), and η∈(1.0, 2.0). The material parameters of the FVP constitutive model for M=8 are listed as follows:

5.2 Monotonic tensile

The rate-dependent behavior of monotonic tension is predicted using the FVP constitutive model. The predicted results are shown in Fig. 6. It can be seen that the FVP model is feasible for describing the rate-dependent behavior of the material, and the calculated results are in good agreement with the experimental data ^[34] in the previous studies.

5.3 Uniaxial rate-dependent ratcheting

The predicted results of uniaxial ratcheting are obtained using the FVP constitutive model at different stress rates (65 MPa, 13 MPa, and 2.6 MPa), as shown in Figs. 7(a)-7(c). The following conclusions can be drawn. (Ⅰ) The FVP constitutive model provides a good tool for simulation of the evolution of the ratcheting strain rate, i.e., the ratcheting strain rate decreases progressively as the axial ratcheting strain increases cycle by cycle. Finally, the ratcheting strain is accumulated at a very small constant ratcheting strain rate rather than under a shakedown of the ratcheting within the number of simulated cycles. (Ⅱ) The FVP constitutive model can reasonably predict the rate-dependent ratcheting behavior, which is not available with the elasto-plastic constitutive models. (Ⅲ) For the stress rates of 2.6 MPa/s and 13 MPa/s, the predicted results agree quite well with the experimental results, but not for 65 MPa/s. The reason for this phenomenon is that the three additional parameters (ζ, m, and η) introduced in the FVP model are determined through trial and error. The parameters determined using this method have a slight deviation from the optimal parameters. More effective optimization methods for determining the parameters will be developed in the future.

6 Conclusions

(i) Based on the framework of the elasto-plastic theory, a novel FVP constitutive model is developed for the rate-dependent ratcheting behavior, and a new implicit stress integration algorithm with fractional order is introduced using the backward Euler model and the radial return of the back stress. It is concluded that the cyclic elasto-plastic constitutive model exhibits viscosity when the accumulated plastic strain rate and nonlinear kinematic hardening rule are described based on the fractional order, which is different from the classical Perzyna's type visco-plasticity.

(ii) Three additional parameters, namely, the strength of the memory index ζ, the parameter controlling the material viscosity m, and the parameter controlling the critical value of dynamic recovery term η, are introduced in the FVP model for the rate-dependent ratcheting behavior, and the sensitivity of the FVP model to these three parameters is analyzed. This research shows that the three parameters can control the viscosity, the initial yielding point of the uniaxial tension, and the initial value of the ratcheting strain. Good simulation results can be obtained through an adjustment of these three parameters, and the ranges of empirical values of the parameters are given, i.e., the range of ζ is between 0.0 and 1.0, the range of m is between 1.0 and 5.0, and the range of η is between 1.0 and 2.0.

(iii) The developed FVP constitutive model provides a good simulation for the uniaxial ratcheting behaviors at different rates of stress, including the shape of the hysteresis loop and the value of the ratcheting strain, with the exception of that at a stress rate of 65 MPa/s. Therefore, we will focus on developing more effective optimization methods for determining the proper parameters in the future.