# Plastic Dynamical Model for Bulk Metallic Glasses

Authors: Shaowen Yao; Zhibo Cheng
Journal: Complexity
 (2019)

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## 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 aCu50Zr45Ti5 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.


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## Body

## 1. Introduction

In [1], Cheng et al. investigate a plastic deformation and give the following model(1)σp−kx−σfx′πd24=Mx″,where σp 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, k=E/L1+S, 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 S=Ks/KM=πd2E/4LkM in [2]. σf 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 friction(2)σp−kx−σfx′−γx′πd24=Mx″,where γ is the internal friction coefficient and σf is a complex function of the loading rate and temperature in the shear bands [1]. Here, we assume σf=σf0/1+Ax′ according to [4], with σf0 taken as the yielding strength of the sample and A being a constant. So, (2) is translated into(3)σp−kx−σf01+Ax′−γx′πd24=Mx″.Ifσp=kpt, where p is the loading rate in [4–7], then (3) can be rewritten into a nonautonomous equation:(4)kpt−kx−σf01+Ax′−γx′πd24=Mx″.Furthermore, we obtain(5)x″+γBx′+Bσf01+Ax′+Bkx=Bkpt,where B=πd2/4M. 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 Cu50Zr45Ti5 metallic glass.On the other hand, ifσp=σ0, where σ0 is the initial internal stress, which is equal to the yield stress in [8–10], from (3) and (5), we obtain(6)x″+γBx′+Bσf01+Ax′+Bkx=Bσ0.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)

Letx=z+pt. We have from (5) that(7)z″+γBz′+Bσf01+Ap+Az′+Bkz+γBp=0,and then(8)z′=y,y′=−γBy−Bσf01+Ap+Ay−Bkz−γBp.Obviously, (8) has one equilibrium point at E≔−γp/k−σf0/k+Akp,0. Let Az,y be the coefficient matrix of the linearized system of (8) at an equilibrium point z,y. Then, we have at E(9)A−γpk−σf0k+Akp,0=01−BkABσf01+p2−γB.The characteristic equation ofA−γp/k−σf0/k+Akp,0 is(10)fλ=λ2−ABσf01+p2−γBλ+Bk,which yields(11)λ1,2=ABσf0/1+p2−γB±ABσf0/1+p2−γB2−4Bk2.By analyzing, we obtain the following results.Theorem 2.1.
Ifγ≥Aσf0/1+p2+4kMπ/dπ, then the equilibrium point E is stable node; if γ≤Aσf0/1+p2−4kMπ/dπ, then the equilibrium point E is unstable node.Theorem 2.2.
IfAσf0/1+p2<γ<Aσf0/1+p2+4kMπ/dπ, then the equilibrium point E is stable focus; if Aσf0/1+p2−4kMπ/dπ<γ<Aσf0/1+p2, then the equilibrium point E is unstable focus. Moreover, system (8) has Hopf bifurcation with γ=Aσf0/1+p2.Next, from (8), we denote that(12)fz,y=y,gz,y=−γBy−Bσf01+Ap+Ay−Bkz−γBp.Letαγ=ABσf0/1+p2−γB/2. We know that the bifurcation value is αγ0=0 if γ=Aσf0/1+p2. And,(13)dαγdγγ=Aσf0/1+p2=−B2<0,i.e., αγ decreases with γ.By Andronov–Hopf bifurcation theorem ([11], P. 167), we have fz=0,gy=0,andfygz=−Bk<0. From(14)16Kr0=8A3B3/2k1/2σf01+p5>0,we have the following.Theorem 2.3.
Ifγ=Aσf0/1+p2, then the equilibrium point E is weak repell; if γ>Aσf0/1+p2, there exists an unstable periodic orbit of system (11) from the equilibrium point E. This is subcritical bifurcation.On the other hand, letασf0=ABσf0/1+p2−γB/2. We know that the bifurcation value is ασf0=0 if σf0=γ1+p2/A. And,(15)dασf0dσf0σf0=γ1+p2/A=AB21+p2>0,i.e., ασf0 is increasing about σf0.By Andronov–Hopf bifurcation theorem (see [11], P. 167), we have fz=0,gy=0,andfygz=−Bk<0. From(16)16Kr0=8A2γB3/2k1/21+p3>0,we have the following.Theorem 2.4.
Ifσf0=γ1+p2/A, then the equilibrium point E is weak repell; if σf0<γ1+p2/A, 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 z=t, nonautonomous system (5) can be rewritten into a three-dimensional autonomous system:(17)x′=y,y′=−Bγy−Bσf01+Ay−Bkx+Bkpz.z′=1.From (17), we know that there is a uniform solution (trajectory) in which the shear bands slide at the loading rate:(18)x=pt−σf0k+Akp−γpk,y=p,z=t.LetBx,y,z be the coefficient matrix of the linearized system of (17) at a trajectory x,y,z. Then, we have at the trajectory of (18)(19)Bpt−σf0k+Akp−γpk,p,t=010−Bk−Bγ+ABσf01+Ap2Bkp000.The characteristic equation ofBpt−σf0/k+Akp−γp/k,p,t is(20)gλ=λλ2−ABσf01+Ap2−Bγλ+Bk,and then(21)λ1=0,λ2,3=ABσf0/1+Ap2−Bγ±ABσf0/1+Ap2−Bγ2−4Bk2.DefineC=4KME/4LKM+πd2EAk. By [12] (P. 727), we obtain the following results.Theorem 2.5.
Ifp>CAσf0/γ−1, then (5) is the stable closed orbit for trajectory (18). If p<CAσf0/γ−1, then (5) is hyperchaotic for trajectory (18).


## 3. Periodic Solution for Model (6)

In this section, we prove the existence of a nonconstantω-periodic solution for model (6) by applying Manásevich–Mawhin continuation theorem. First, we consider the following differential equation with a singularity of derivative:(22)x″+Cx′+gx′+Kx=et,where K and C are positive constants, e∈L20,ω and et≡g0−Kc, for all c∈ℝ, g:b,+∞⟶ℝ is a continuous function, and g0≔σ<e≔maxt∈0,ωet, g may have a singularity of derivative at u=b, which means that(23)limu⟶b+∫1ugsds=+∞,where b is a constant and b<0.Next, we embed (22) into the following equation family with a parameter λ∈0,1:(24)x″+λCx′+gx′+Kx=λet.The following lemma is Manásevich–Mawhin continuation theorem ([13], Theorem 3.1).Lemma 3.1.
([13], Theorem 3.1) Let Ω be an open bounded set in the space X≔ϕ∈C1ℝ,ℝ:ϕt+ω=ϕt,ϕ′t+ω=ϕ′t,∀t∈ℝ. Suppose the following conditions are satisfied:(i)
(24) has no solution on ∂Ω.(ii)
The following equation(25)Fa≔1ω∫0ωg0+Kxt−etdt=0,has no solution on∂Ω∩ℝ.(iii)
The Brouwer degree of F(26)degF,Ω∩ℝ,0≠0.
Then, (22) has at least one periodic solution on Ω¯.Theorem 3.1.
Assume that condition (23) holds. If Kω/2π2<1, then equation (22) has at least one nonconstant periodic solution x with(27)x∈−e−σK−ω/2πe21−Kω/2π2,e−σK+ω/2πe21−Kω/2π2.Proof.
Suppose thatx is a solution of (24) for some λ∈0,1. Let t∗,t∗ be, respectively, the global maximum point and global minimum point of xt on 0,ω, then we obtain x′t∗=0, x′t∗=0, x″t∗≤0,andx″t∗≥0. Furthermore, we arrive at(28)x″t∗+λgx′t∗+Kxt∗=λet∗.
Sincex″t∗≤0 and g0=σ, then we obtain(29)xt∗≥et∗−σK≥e∗−σK≔D1,where e∗≔mint∈0,ωpt.
Similarly, we get(30)xt∗≤et∗−σK≤e−σK≔D2>0,since e−σ>0. From equations (29) and (30), x is a continuous function in ℝ, there exists a point τ∈0,ω such that(31)D1≤xτ≤D2.
Multiplying both sides of (24) by x″t and integrating over the interval 0,ω, we have(32)∫0ωx″t2dt+λ∫0ωCx′tx″tdt+λ∫0ωgx′tx″tdt+λK∫0ωxtx″tdt=λ∫0ωetx″tdt.
Substituting∫0ωCx′tx″tdt=0, ∫0ωgx′tx″tdt=0, and ∫0ωxtx″tdt=−∫0ωx′t2dt into (32) and applying the Hölder inequality, we see that(33)∫0ωx″t2dt=λK∫0ωx′t2dt+λ∫0ωetx″tdt≤K∫0ωx′t2dt+λ∫0ωetx″tdt≤K∫0ωx′t2dt+λ∫0ωet2dt1/2∫0ωx″t2dt1/2.
Using the Wirtinger inequality ([14], Lemma 2.4), it is clear that(34)∫0ωx″t2dt≤Kω2π2∫0ωx″t2dt+λ∫0ωet2dt1/2∫0ωx″t2dt1/2.
SinceKω/2π2<1 and ∫0ωx″t2dt≠0, it is easy to see that(35)∫0ωx″t2dt1/2≤λe21−Kω/2π2≔λM1′,where e2≔∫0ωet2dt1/2. From (31) and the Wirtinger inequality, we have(36)x≤D2+∫0ωx′tdt≤D2+ω∫0ωx′t2dt1/2≤D2+ωω2π∫0ωx″tdt1/2≤D2+ωω2πλM1′≤e−σK+ω/2πe21−Kω/2π2≔M1.
Sincex0=xω, there exists a point ξ∈0,ω such that x′ξ=0. Therefore, we have(37)x′t=x′ξ+∫ξtx″sds≤∫0ωx″tdt≤ω∫0ωx″t2dt1/2≤ωλM1′≤ωe2/1−Kω/2π2≔M2.
On the other hand, multiplying both sides of (24) by x″t, we get(38)x″t2+λgx′tx″t+λKxtx″t=λetx″t.
Letξ∈0,ω be defined in (37). For any ξ≤t≤ω, integrating (38) on ξ,t, we obtain(39)λ∫x′ξx′tgudu=λ∫ξtgx′sx″sds=∫ξtx″s2ds−λ∫ξtCx′sx″sds−λ∫ξtKxsx″sds+λ∫ξtesx″sds.
Furthermore, we see that(40)λ∫x′ξx′tgudu=λ∫ξtgx′sx″sds≤∫0ωx″s2ds+λC∫0ωx′sx″sds+λK∫0ωxsx″sds+λ∫0ωesx″sds.
From (35) and (36), applying the Hölder inequality, we have(41)λ∫x′ξx′tgudu≤∫0ωx″s2ds+λCx′ω∫0ωx″s2ds1/2+λKxω∫0ωx″s2ds1/2+λ∫0ωet2dt1/2∫0ωx″s2ds1/2≤λM1′2+λ2CM2ωM1′+λ2KM1ωM1′+λ2e2M1′.
The above inequality implies(42)∫x′ξx′tgudu≤M1′2+CM2ωM1′+KM1ωM1′+e2M1′≔M3′.
From (23), we know that there exists a constant M3>b such that(43)x′t≥M3,∀t∈ξ,ω.
The caset∈0,ξ can be treated similarly.
From (36), (37), and (43), we obtain(44)Ω=x∈X:x<M1andM3<x′t<M2∀t∈ℝ.
We know that (24) has no solution on ∂Ω as λ∈0,1, and when xt∈∂Ω∩ℝandxt=M1, from (31), we know that M1>D2; So, from (29), we see that(45)1ω∫0ωg0+Kx−etdt>0,since C∫0ωx′tdt=0. So, condition (ii) is also satisfied. Set(46)Hx,μ=μx+1−μ1ω∫0ωgx′+Kx−etdt,where x∈∂Ω∩ℝandμ∈0,1, we have(47)xHx,μ=μx2+1−μxω∫0ωgx′+Kx−etdt>0,and thus Hx,μ is a homotopic transformation and(48)degF,Ω∩ℝ,0=deg1ω∫0ωgx′+Kx−etdt,Ω∩ℝ,0=degx,Ω∩ℝ,0≠0.
So, condition (iii) is satisfied. By Lemma 3.1, there exists aω-periodic solution x with(49)x∈−e−σK−ω/2πe21−Kω/2π2,e−σK+ω/2πe21−Kω/2π2.
Finally, observe thatx is not a constant. Otherwise, suppose x≡c (constant), then we get x=et−σ/K, which contradicts the assumption et≡σ−Kc; 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 C=Bγ, gx′=Bσf0/1+Ax′, et=Bσ0, and K=Bk. It is easy to see that limx′⟶−1/A+∫1uBσf0/1+Ax′=+∞.Corollary 3.1.
Assume thatπd2k/4Mω/2π2<1, σ0>σf0, and σ0≠σf0+kc for all c∈ℝ hold. Then, (6) has at least one nonconstant ω-periodic solution x with(50)x∈−σ0−σf0k−ωπd2σ0ω/2π4M−πd2kω/2π2,σ0−σf0k+ωπd2σ0ω/2π4M−πd2kω/2π2.


## 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, p=1/kAσf0/γ−1, and the strain rate can be estimated about 10−3s−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 4×10−3s−1 to 4×10−2s−1 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.



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