Plastic Dynamical Model for Bulk Metallic Glasses

Abstract

Based on previous experimental results of the plastic dynamic analysis of metallic glasses upon compressive loading, a dynamical model is proposed. This model includes the sliding speed of shear bands in the plastically strained metallic glasses, the shear resistance of shear bands, the internal friction resulting from plastic deformation, and the influences from the testing machine. This model analysis quantitatively predicts that the loading rate can influence the transition of the plastic dynamics in metallic glasses from chaotic (low loading rate range) to stable behavior (high loading rate range), which is consistent with the previous experimental results on the compression tests of a metallic glass. Moreover, we investigate the existence of a nonconstant periodic solution for plastic dynamical model of bulk metallic glasses by using Manásevich–Mawhin continuation theorem.

1. Introduction

In [1], Cheng et al. investigate a plastic deformation and give the following modelwhere is the loading stress of the shear bands, d is the sample diameter, x is the shear sliding displacement, M is the equal effective mass, and it is also the effective inertia of the machine-sample system (MSS) when responding to the stress gradient and is an empirical parameter estimated to be of the order of 10–100 kg for a typical MSS. And, , where L is the sample height, E is Young’s Modulus, and S is the stiffness ratio of the sample to the testing machine and in [2]. is the shear resistance along the shear plane. As the driving forces exceed the static shear resistance for one block, shear sliding will occur corresponding to the formation of one shear band.

The above model in [1] is considered an ideal situation, i.e., without internal friction. However, when a solid material undergoes plastic deformation, the internal friction reflects the force resisting motion between the elements, which definitely cannot be avoided [3]. In the current study, the internal friction coefficient of the model materials, i.e., metallic glasses [1], was measured by an elastic modulus and internal friction meter (NIHON Techno-Plus Company, Japan), which is 0.0012 at room temperature.

Motivated by this problem, we improve model (1) with internal frictionwhere γ is the internal friction coefficient and is a complex function of the loading rate and temperature in the shear bands [1]. Here, we assume according to [4], with taken as the yielding strength of the sample and A being a constant. So, (2) is translated into

If , where p is the loading rate in [4–7], then (3) can be rewritten into a nonautonomous equation:

Furthermore, we obtainwhere . In Section 2, we conduct a dynamic analysis of model (5). This model analysis quantitatively predicts that the loading rate can influence the transition of the plastic dynamics in metallic glasses from chaotic (low loading rate range) to stable behavior (high loading rate range), which is consistent with the previous experimental results on the compression tests of a metallic glass.

On the other hand, if , where is the initial internal stress, which is equal to the yield stress in [8–10], from (3) and (5), we obtain

Using Manásevich–Mawhin continuation theorem, we obtain the existence of a periodic solution for model (6) in Section 3. Moreover, we give the existence of upper and lower bounds of the periodic solution of this equation.

2. Dynamical Analysis for Model (5)

Let . We have from (5) thatand then

Obviously, (8) has one equilibrium point at . Let be the coefficient matrix of the linearized system of (8) at an equilibrium point . Then, we have at E

The characteristic equation of iswhich yields

By analyzing, we obtain the following results.

Theorem 2.1. If , then the equilibrium point E is stable node; if , then the equilibrium point E is unstable node.

Theorem 2.2. If , then the equilibrium point E is stable focus; if then the equilibrium point E is unstable focus. Moreover, system (8) has Hopf bifurcation with .

Next, from (8), we denote that

Let . We know that the bifurcation value is if . And,i.e., decreases with γ.

By Andronov–Hopf bifurcation theorem ([11], P. 167), we have . Fromwe have the following.

Theorem 2.3. If , then the equilibrium point E is weak repell; if , there exists an unstable periodic orbit of system (11) from the equilibrium point E. This is subcritical bifurcation.

On the other hand, let . We know that the bifurcation value is if . And,i.e., is increasing about .

By Andronov–Hopf bifurcation theorem (see [11], P. 167), we have . Fromwe have the following.

Theorem 2.4. If , then the equilibrium point E is weak repell; if , there exists an unstable periodic orbit of system (11) from the equilibrium point E. This is subcritical bifurcation.

In the following, we consider Lyapunov exponent of system (5). By substituting , nonautonomous system (5) can be rewritten into a three-dimensional autonomous system:

From (17), we know that there is a uniform solution (trajectory) in which the shear bands slide at the loading rate:

Let be the coefficient matrix of the linearized system of (17) at a trajectory . Then, we have at the trajectory of (18)

The characteristic equation of isand then

Define By [12] (P. 727), we obtain the following results.

Theorem 2.5. If , then (5) is the stable closed orbit for trajectory (18). If , then (5) is hyperchaotic for trajectory (18).

3. Periodic Solution for Model (6)

In this section, we prove the existence of a nonconstant ω-periodic solution for model (6) by applying Manásevich–Mawhin continuation theorem. First, we consider the following differential equation with a singularity of derivative:where K and C are positive constants, and , for all , is a continuous function, and , may have a singularity of derivative at , which means thatwhere b is a constant and .

Next, we embed (22) into the following equation family with a parameter

The following lemma is Manásevich–Mawhin continuation theorem ([13], Theorem 3.1).

Lemma 3.1. ([13], Theorem 3.1) Let be an open bounded set in the space . Suppose the following conditions are satisfied:(i)(24) has no solution on .(ii)The following equation has no solution on .(iii)The Brouwer degree of FThen, (22) has at least one periodic solution on .

Theorem 3.1. Assume that condition (23) holds. If , then equation (22) has at least one nonconstant periodic solution x with

Proof. Suppose that x is a solution of (24) for some . Let be, respectively, the global maximum point and global minimum point of on , then we obtain , , Furthermore, we arrive atSince and , then we obtainwhere
Similarly, we getsince . From equations (29) and (30), x is a continuous function in , there exists a point such thatMultiplying both sides of (24) by and integrating over the interval , we haveSubstituting , , and into (32) and applying the Hölder inequality, we see thatUsing the Wirtinger inequality ([14], Lemma 2.4), it is clear thatSince and , it is easy to see thatwhere . From (31) and the Wirtinger inequality, we haveSince , there exists a point such that . Therefore, we haveOn the other hand, multiplying both sides of (24) by , we getLet be defined in (37). For any , integrating (38) on , we obtainFurthermore, we see thatFrom (35) and (36), applying the Hölder inequality, we haveThe above inequality impliesFrom (23), we know that there exists a constant such thatThe case can be treated similarly.
From (36), (37), and (43), we obtainWe know that (24) has no solution on as , and when , from (31), we know that ; So, from (29), we see thatsince . So, condition (ii) is also satisfied. Setwhere we haveand thus is a homotopic transformation andSo, condition (iii) is satisfied. By Lemma 3.1, there exists a ω-periodic solution x withFinally, observe that x is not a constant. Otherwise, suppose (constant), then we get , which contradicts the assumption ; so, the proof is complete.
Next, we apply Theorem 3.1 to the plastic dynamical model of bulk metallic glasses (6). Model (6) is of form (22) with , , , and . It is easy to see that .

Corollary 3.1. Assume that , , and for all hold. Then, (6) has at least one nonconstant ω-periodic solution x with

4. Conclusion and Discussion

In conclusion, we establish a model considering the internal friction during the plastic deformation and investigate how the parameters influence the stability of the system. Meanwhile, we prove the existence of chaotic and periodic solutions by applying mathematical methods. Based on Theorems 2.1 and 2.2, for larger internal friction coefficient, the plastic system manifests a stable state, while for smaller internal friction coefficient, the system becomes unstable. The increasing of the friction coefficient improves the resistance of the motion. As a result, it requires more energy for the plastic deformation, which means the state of the system will not be changed easily, reflecting a stable state.

Theorem 2.5 shows that the plastic dynamics transits from chaotic to stable state as the loading rate increases. For larger loading rate, the system evolves into a stable state. It is corresponding to that the self-organized critical behavior happens at the larger strain rate [15]. While for lower loading rate, the system is chaos, which is corresponding to the chaotic behavior happens at lower strain rate [15]. These results in Theorem 2.5 are consistent with the analysis based on the experimental data considering the loading rate is linearly dependent to the strain rate. Based on the result in Theorem 2.5, we can obtain a critical loading rate, , and the strain rate can be estimated about . It is quite accordant with the results that the plastic dynamic behavior changes from chaotic to self-organized critical behavior as the strain rate increases from to in [15].

The stick-slip system shows rich dynamic behaviors such as chaos and quasi periodic solution [16, 17]. In this paper, we prove that there is a periodic solution based on mathematical theory, and the periodic solution is accordant with the sinusoidal density variations in shear bands [18]. The chaotic behavior is a result of the shear band instabilities [19]. We illustrate the plastic dynamics transits from chaos to stable state applying nonlinear dynamic theory and demonstrate how the parameters influence the plastic dynamics, which helps us to clarify the internal mechanism of plastic deformation for bulk metallic glasses.

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

YSW and CZB declare that they have no competing interests.

Authors’ Contributions

YSW and CZB contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (no. 11501170), China Postdoctoral Science Foundation funded project (no. 2016M590886), Fundamental Research Funds for the Universities of Henan Province (NSFRF170302), Young backbone teachers of colleges and universities in Henan Province (2017GGJS057), and Education Department of Henan Province project (20B110006).