Plastic Dynamical Model for Bulk Metallic Glasses

Based on previous experimental results of the plastic dynamic analysis of metallic glasses upon compressive loading, a dynamical model is proposed. is 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 inuences from the testing machine. is model analysis quantitatively predicts that the loading rate can inuence 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 Cu50Zr45Ti5 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.


Introduction
In [1], Cheng et al. investigate a plastic deformation and give the following model where σ p is the loading stress of the shear bands, d is the sample diameter, x is the shear sliding displacement, M is the equal e ective mass, and it is also the e ective 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, k (E/L)(1 + S), where L is the sample height, E is Young's Modulus, and S is the sti ness ratio of the sample to the testing machine and S K s /K M πd 2 E/4Lk M in [2].σ f 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.e 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 re ects the force resisting motion between the elements, which de nitely cannot be avoided [3].In the current study, the internal friction coe cient 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 friction where c is the internal friction coe cient and σ f is a complex function of the loading rate and temperature in the shear bands [1].Here, we assume σ f σ f 0 /(1 + Ax ′ ) according to [4], with σ f 0 taken as the yielding strength of the sample and A being a constant.So, (2) is translated into If σ p kpt, where p is the loading rate in [4][5][6][7], then (3) can be rewritten into a nonautonomous equation: Furthermore, we obtain where B � πd 2 /4M.In Section 2, we conduct a dynamic analysis of model (5). is 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 Cu 50 Zr 45 Ti 5 metallic glass.
On the other hand, if σ p � σ(0), where σ(0) is the initial internal stress, which is equal to the yield stress in [8][9][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.

Dynamical Analysis for Model (5)
Let x � z + pt.We have from (5) that and then Obviously, (8) has one equilibrium point at E ≔ (− (cp/k) − (σ f 0 /k + Akp), 0).Let A(z, y) be the coefficient matrix of the linearized system of (8) at an equilibrium point (z, y).en, we have at E e characteristic equation of which yields By analyzing, we obtain the following results.
, there exists an unstable periodic orbit of system (11) from the equilibrium point E. is is subcritical bifurcation.
A, there exists an unstable periodic orbit of system (11) from the equilibrium point E. is is subcritical bifurcation.
In the following, we consider Lyapunov exponent of system (5).By substituting z � t, 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 B(x, y, z) be the coefficient matrix of the linearized system of (17) at a trajectory (x, y, z).en, we have at the trajectory of ( 18) e characteristic equation of B(pt and then Define C � 4K M E/(4LK M + πd 2 E)Ak.By [12] (P.727), we obtain the following results.5) is the stable closed orbit for trajectory (18) 5) is hyperchaotic for trajectory (18).

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, e ∈ L 2 ([0, ω]) and e(t) ≡ g(0) − Kc, for all c ∈ R, g: (b, +∞) ⟶ R is a continuous function, and g(0 g may have a singularity of derivative at u � b, which means that where b is a constant and b < 0. Next, we embed (22) into the following equation family with a parameter λ ∈ (0, 1]: has no solution on zΩ ∩ R. en, ( 22) has at least one periodic solution on Ω.

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 Complexity solutions by applying mathematical methods.Based on eorems 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.
e 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.eorem 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].ese results in eorem 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 eorem 2.5, we can obtain a critical loading rate, p � (1/k)( , and the strain rate can be estimated about 10 − 3 s − 1 .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 4 × 10 − 3 s − 1 to 4 × 10 − 2 s − 1 in [15].e 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].e 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. ) e following lemma is Manásevich-Mawhin continuation theorem ([13], eorem 3.1).Lemma 3.1.([13], eorem 3.1) Let Ω be an open bounded set in the space ∀t ∈ R}.Suppose the following conditions are satisfied: (i) (24) has no solution on zΩ.(ii) e following equation Complexity 3