Dynamic behaviors of phase transforming cellular structures

Accepted Manuscript Dynamic behaviors of phase transforming cellular structures Jingran Liu, Huasong Qin, Yilun Liu PII: DOI: Reference:

S0263-8223(17)31385-5 https://doi.org/10.1016/j.compstruct.2017.10.002 COST 8977

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Composite Structures

Received Date: Revised Date: Accepted Date:

30 April 2017 2 September 2017 2 October 2017

Please cite this article as: Liu, J., Qin, H., Liu, Y., Dynamic behaviors of phase transforming cellular structures, Composite Structures (2017), doi: https://doi.org/10.1016/j.compstruct.2017.10.002

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Dynamic behaviors of phase transforming cellular structures Jingran Liu, Huasong Qin and Yilun Liu* State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an, China, 710049 Shaanxi Engineering Research Center of Nondestructive Testing and Structural Integrity Evaluation, Xi’an Jiaotong University, Xi’an 710049, China *Corresponding author, [email protected]

Abstract: In this work, the dynamic behaviors of phase transforming cellular structures (PTCS) consisted of sinusoidal beam with bi-stable configurations are studied. A nonlinear spring-bead model is developed, where the mechanical properties of the sinusoidal beam are described by a nonlinear spring and the mass is concentrated at the bead. Therefore, the PTCS are modeled as a chain of nonlinear spring-bead. Based on the nonlinear spring-bead model, the quasi-static and dynamic behaviors of PTCS are theoretically and numerically studied, which shows abundant dynamic behaviors depending on the loading rate and damping coefficient. For a 2-layered PTCS, there are four deformation patterns, that is the snap through deformation of the sinusoidal beams for relative small loading rate and damping coefficient, the uniform deformation for relative large damping coefficient, and the pulse deformation of the springs near bi-stable configurations for relative large loading rate and small damping coefficient. While, for the multilayer PTCS, their dynamic behaviors are much more complicated due to the combined deformation of the multiple sinusoidal beams. However, the similar snap through and pulse deformation of the sinusoidal beams are also observed, respectively. To our knowledge, this work is the first time to systematically study the dynamic behaviors of PTCS. Keywords: Phase transforming cellular structures; Sinusoidal beam; Bi-stable configurations; Nonlinear spring-bead model; Dynamic behaviors

1. Introduction Energy absorbing materials and structures (EAMS) have attracted great interests in recent years thanks to their wide applications in personnel protection, protective packaging of delicate components, impacting mitigation in automobiles and aircraft [1-3], etc. Several mechanisms have been proposed to absorb the external impacting energy, such as plastic deformation [4-6], fragmentation of ceramics [4, 7], rate-dependent viscous dissipation [8], frictional dissipation [9-11] and phase transformation [12-16]. In all these strategies, cellular structures may be the most popular platform of EAMS due to their light-weight, high energy absorption density and large deformability. Traditional cellular structures dissipate external impacting energy via either plastic deformation or viscoelastic deformation of the parent materials. While, phase transforming cellular structures (PTCS) have multi-stable configurations and they can rapidly transform among these stable configurations under certain critical loads [6, 15, 17-21], so that a significant amount of energy can be dissipated during phase transformation of the cellular structures [14, 22]. Indeed, phase transforming behaviors have been widely observed in materials processing [23], unfolding of protein [14, 24], shape memory or magnetostrictive alloys [25, 26], etc. Materials showing phase transforming behaviors have a common characteristic that more than one stable configurations exist in their microscopic structures. Based on the same idea, PTCS are usually consisted of bi-stable elements, e.g. sinusoidal beams [15, 22], and the phase transformation of PTCS is realized through buckling deformation of the bi-stable elements [15, 19, 20, 22]. For example, Restrepo et al. [15] studied the quasi-static behaviors of two dimensional PTCS via approximate trilinear force-displacement relation. Findeisen et al. [22] established a spring-bar model to study the deformation and dissipation mechanisms of a three dimensional metamaterial consisted of buckling elements. As the elastic buckling deformation is fully recoverable, energy absorption of PTCS is reversible which make them good candidates for the impacting protective and vibration isolation structures [27]. However, the previous works mainly focused on the fabrication, design and

quasi-static mechanical behaviors of PTCS [22, 28-30], and systematical studies of their dynamic behaviors are still in lack. We think PTCS should have abundant dynamic behaviors as the buckling behaviors are very sensitive to the loading rate. Besides, for the practical applications of PTCS, such as impacting protection and vibration, they essentially work at dynamic loading. Therefore, it is of great importance to systematically explore the dynamic behaviors of PTCS. The PTCS studied herein is consisted of bi-stable sinusoidal beams and relative stiff frames adapted from Restrepo et al. [15], as shown in Fig. 1(a). The sinusoidal beam is described by y=(A/2)[1-sin(2π(x-λ/4)/λ)]. A representative volume element (RVE) is given in Fig. 1(b), where the thickness of the sinusoidal beam is t, wave amplitude is A and wavelength is λ. The thickness of the frame is 1.5t and the depth is b. Indeed, for the geometrical parameters studied in this work, the deformation of PTCS is mostly contributed by the sinusoidal beam and the frame can be treated as a rigid body. For the sinusoidal beam, its mechanical behaviors are predominately determined by the ratio of wave amplitude to beam thickness, i.e. Q=A/t, which was firstly defined by Qiu [31] and by constraining the rotation of the midpoint of the curved beam, the bi-stable configuration is observed for Q>2.31. Furthermore, for Q≥6, the force-displacement (F-D) curve can be approximately described by a trilinear relation, as schematically illustrated in Fig. 1(c). In general, there are three deformation stages during compression: at Stage 1 the compressive force increases with the displacement to a peak value (FI, DI); then at Stage 2 the compressive force decreases to a negative value (FII, DII); at Stage 3 the compressive force increases with the displacement again. Therefore, there are two stable equilibrium points of the RVE, that is the zero displacement point at Stage 1 and the zero compressive force point at Stage 3, as shown in Fig. 1(c). In this work, we firstly obtain the F-D relation of the RVE for Q=7 through finite element method (FEM) simulation via commercial software ABAQUS. Then, a nonlinear spring-bead model is proposed by precisely accounting for the F-D relation of RVE with the nonlinear spring. Based on the nonlinear spring-bead model, the quasi-static and dynamic behaviors of PTCS are analytically studied in Section 2. Due

to the complexity of the analytical solution under dynamic loading, the dynamic behaviors of PTCS with different loading rate and damping coefficient are further studied by numerically solving the nonlinear spring-bead model and a deformation phase diagram is given in Section 3. Finally, conclusions are given in Section 4.

Figure 1. (a) PTCS consisted of sinusoidal beams (Adapted from Restrepo et al. [15]). A representative volume element (RVE) is marked by a square. (b) The geometrical parameters of the RVE in (a). (c) Schematic illustration of the F-D relation of RVE (Q≥6). (d) The nonlinear spring-bead model.

2. Theoretical analysis In this section, a nonlinear spring-bead model is first proposed by precisely accounting for the F-D relation of the RVE with the nonlinear spring. Then, PTCS is modeled as a chain of nonlinear spring-bead. The quasi-static F-D relation of PTCS is deduced by minimizing the potential energy of the nonlinear spring-bead chain under a given compressive displacement, while the dynamic F-D relation of PTCS is obtained by solving the governing equations of the nonlinear spring-chain model. 2.1 Quasi-static behaviors The compression behaviors of the RVE is firstly studied via the Explicit procedure of the commercial FEM software ABAQUS. As shown in Fig. 2(a), the displacement of the bottom surface of RVE is fixed and the rotation of the two lateral

surfaces are also constrained due to the symmetrical deformation of RVE during compression. Indeed, the compression deformation of RVE is mainly contributed by the sinusoidal beam. The geometrical parameters of RVE are set as t=0.72 mm, A=5.04 mm (Q=7), λ=60 mm and depth b=20 mm adapted from Restrepo et al. [15]. The height of the vertical frame h is larger than A to avoid contact and densification during the dynamic loading in this work. In FEM simulation, the RVE is discretized by beam element and its constitutive material is described by linear-elasticity with Young’s modulus E=1 GPa and Poisson’s ratio ν=0.3. The compressive displacement is applied on the top surface of RVE and the corresponding compressive force is recorded to obtain the F-D relation of RVE. In addition, plastic deformation and fracture are ignored in our simulation.

Figure 2. (a) Schematic illustration of the FEM model of RVE. (b) Non-dimensional force-displacement relations of RVE obtained from FEM simulation (solid black line), nonlinear force-displacement relation described by Eq. (2) (dashed line) and trilinear force-displacement relation describe by Eq. (3) (dash-dotted line).

In previous works [31, 32], Qiu and his collaborators analytically derived the F-D relations of the curved beam under compression loading which are cubic relations at deformation Stage 1 and Stage 3 and linear relation at deformation Stage 2. In this work, based on the FEM results we have found the quadratic relations (for Stage 1 and Stage 3) as well as linear function (for Stage 2) are good enough to describe the F-D relation for Q=7. Especially, the tension deformation at Stage 1 and large compression deformation at Stage 3 can be well depicted, as shown in Fig. 2(b). Therefore, for simplicity of the following analysis, the quadratic F-D relation is

assumed as  F 2 FΙ + Ι2 ( D − DΙ ) , D < DΙ  DΙ   F − FΙ F ( D ) =  FΙ + ΙΙ ( D − DΙ ), DΙ ≤ D < DΙΙ D − D ΙΙ Ι   F 2 FΙΙ + Ι2 ( D − DΙΙ ) , D ≥ DΙΙ  DΙ 

(1)

where the end point of Stage 1 is FI=9.9N, DI=0.93mm, and the end point of Stage 2 is FII=-4.9N, DII=9.3mm. For the convenience of following discussion, the non-dimensional force and displacement are defined as f=F/FI and d=D/DI, so that Eq. (1) is reformulated as 1 − ( d − 1) 2 ,d < 1  F F −1  f ( d ) = 1 + ΙΙ Ι ( d − 1) ,1 ≤ d < DΙΙ DΙ D D − 1 ΙΙ Ι   F F + (d − D D )2 , d ≥ D D  ΙΙ Ι ΙΙ Ι ΙΙ Ι

(2).

Besides, F-D relation be further simplified as trilinear relation   ,d < 1 d  F F −1 f l ( d ) = 1 + ΙΙ Ι ,1 ≤ d < DΙΙ DΙ ( d − 1)  DΙΙ DΙ − 1  1 − FΙΙ FΙ ( d − DΙΙ DΙ ) ,d ≥ DΙΙ DΙ  FΙΙ FΙ + DΙIΙ DΙ − DΙΙ DΙ 

(3),

where DIII is the point that the compressive force equals FI. The comparison of Eq. 2, Eq. (3) and FEM results is presented in Figure 2(b). Note that the slopes of the trilinear relation at Stage 1 and Stage 3 are selected as the slopes between the point (0, 0) and (FI, DIII) and between the point (FII, DII) and (FI, DIII) a little different from the trilinear relation used in previous work [15], which we think is better to describe the overall F-D relation. Then, a nonlinear spring-bead model is proposed to study the quasi-static and dynamic behaviors of PTCS, where F-D relation of RVE is described by a nonlinear spring and its mass is concentrated at the bead. Thus, PTCS are modelled as a chain of nonlinear spring-bead, as shown in Fig. 1(d). The linear displacement loading is applied on the first bead and the end of the last spring is fixed. Restrepo et al. [15] studied the quasi-static behaviors of PTCS under displacement

loading via trilinear F-D relation and it was shown that the snap through behaviors only occurred for large number of layer of PTCS. Besides, the snap through deformation corresponds to the jumping of the sinusoidal beam between the deformation Stage 1 and Stage 3. In this work, both of the nonlinear F-D relation and trilinear F-D relation were employed to study the quasi-static behaviors of multilayer PTCS under displacement loading and the comparison between them was also made. The deformed configuration of multilayer PTCS is deduced by minimizing the potential energy of the nonlinear spring-bead chain. The non-dimensional displacement of ith bead is defined as ui= Ui/DI (Ui is the displacement of ith bead), the deformation of ith spring is di= ui - ui+1, and the non-dimensional external load at the first bead is p=P/FΙ (P is the external load applied on the first bead). Thus, non-dimensional potential energy of the nonlinear spring-bead chain is given by N

e p = ∑ ∫ f ( x ) dx i =1

di

(4).

0

The compressive displacement at the first bead is N

u1 = ∑ d i

(5).

i =1

Then, under compressive displacement u 1, the equilibrium configurations are given as N

∂[e p + λ (∑ d i − u1 )] = 0, ( j = 1, 2,..., N )

i =1

∂d j

(6),

N

∂[e p + λ (∑ d i − u1 )] i =1

=0

∂λ

where λ is the Lagrange multiplier. Furthermore, as the nonlinear spring-bead chain may have multiple equilibrium configurations, a stable equilibrium configuration should satisfy the following constraint N

∂ 2 [e p + λ ( ∑ d i − u1 )] zi

i =1

∂d i ∂d j

z j > 0 , i, j=1, 2, …, N

(7),

where zj is an arbitrary real vector. This means the second partial derivative matrix of

the potential energy must be positive definite. For N-layered PTCS, if all of the sinusoidal beams are at the deformation stage 1 or stage 3, i.e. u1NDIII/DI, there is only one equilibrium configuration that is the uniform deformation of every sinusoidal beam, i.e. d 1=d 2=…=d N=u1/N. Whereas for N≤u1≤NDIII/DI, there might be more than one equilibrium configurations at some given values of u1. For example, at the end of deformation stage 1 (u1=N), there are two equilibrium configurations that is the uniform deformation of d1=d2=…=dN=1 with compressive load p=1 and non-uniform deformation of d1=…=dk-1=dk+1=…=d N<1, dk>1 with compressive load p<1, where k is an arbitrary number from 1 to N. Here, we assume that the phase transforming of the sinusoidal beams from Stage 1 to Stage 2 or from Stage 1 to Stage 3 is sequential for quasi-static compression as evidenced by Restrepo et al. [15]. Therefore, there is at most one spring at Stage 2 and the changes of the number of nonlinear springs at Stage 1 and Stage 3 are sequential. Note that for dynamic compression, the assumption of the sequential phase transforming may be invalid which will be discussed in Section 3. Based on the above assumptions, if the kth spring jumps to Stage 2, the potential energy in Eq. (6) can be expressed as 2

2  d  k ( d − 1) e p = ∑ d  − i + 1 + 2 k + ( d k − 1) + , i ≠ k 2 3  3  i =1 N

2 i

(8),

where k2=(FII/FI-1)/(DII/DI-1), which is negative, represents the stiffness of the nonlinear spring at Stage 2. Therefore, the N+1 equations which derived from Eq. 5 can be expressed as  −di2 + 2d i + λ = 0, i ≠ k   k2 ( d k − 1) + 1 + λ = 0 ( N − 1) d + d − u = 0 i k 1 

(9).

Under u 1=N, the non-uniform deformation can be derived from Eq. (9) as  d k = 1 − k2 ( N − 1) 2   di = 1 + k2 ( N − 1) , i ≠ k

(10).

Note that Eq. (10) is valid only if dk≤DII/DI, namely the spring jumps to Stage 2, which requires N≤8. Otherwise, if N>8, which means the kth spring jumps to Stage 3,

the potential energy is expressed as 3

2

N D  k ( D / D − 1) DΙΙ 1  d  ( d − DΙΙ / DΙ ) FΙΙ  e p = ∑ di2  − i + 1 + k +  d k − ΙΙ  + 2 ΙΙ Ι + − ,i ≠ k 3 FΙ  DΙ  2 DΙ 3  3  i =1

(11).

Correspondingly, the governing equations are  −di2 + 2d i + λ = 0, i ≠ k  FΙΙ 2  +λ =0 ( d k − DΙΙ / DΙ ) + F Ι  ( N − 1) d i + d k − u1 = 0 

(12).

Then, the non-uniform deformation can be derived as 2 2 2  ( DII / DI )( N − 1) + 1 + ( N − 1) (1 − FII / FI ) ( N − 1) + 1 − ( DII / DI − 1)  2 d k = ( N − 1) + 1   2 2  ( N − DII / DI )( N − 1) + 1 − (1 − FII / FI ) ( N − 1) + 1 − ( DII / DI − 1) d = ,i ≠ k 2  i N − 1 + 1 ( ) 

(13).

Furthermore, based on Eq. (7) it is found the uniform deformation is unstable, while the non-uniform deformation is stable. Therefore, snap through deformation occurs at u1=N. The analysis of the quasi-static behaviors of PTCS using trilinear F-D relation is similar to that of nonlinear relation. Hence the quasi-static behaviors of multilayer PTCS with nonlinear relation, as well as trilinear relation for different number of layer are shown in Figure 3.

Figure 3. Comparisons of the quasi-static force-displacment relations of PTCS with nonlinear F-D relation and trilinear relation for (a ) N=4, (b) N=8, (c) N=9 and (d) N=20.

For multilayer PTCS, the number of layer has significant influence on its quasi-static behaviors under displacement loading. The whole system first deforms uniformly during compression, and snap through deformation occurs at FI that is one spring abruptly jumps to Stage 2 or Stage 3 and the other springs jump back to stage 1 accompanied by an abrupt drop of the applied force. If N≤8, one spring jumps to Stage 2 at FI for nonlinear relation, whereas for trilinear relation no snap through happens if N≤6, as illustrated in Fig. 3(a). If 815 for nonlinear relation or N>8 for trilinear relation, the spring jumps from Stage 1 to Stage 3 during the

whole compression, as shown in Figs. 3(c)-(d). For the spring jumping to Stage 2, the compressive force decreases as further compression of the system. The deformation of the spring at Stage 1 decreases, while the deformation of the spring at Stage 2 increases. Then, snap through deformation may occur again when the deformation of the spring at Stage 2 approaches to the end point of Stage 2 (FII, DII), so that it abruptly jumps to Stage 3, as shown in Figure 3(a)-(c) for the nonlinear F-D relation. Once the spring enters to the deformation Stage 3, the following mechanical behaviors of PTCS are similar to that of the firstly jumping to Stage 3 case (see Figure 3(d)) that is the deformation of the spring at Stage 1 and Stage 3 increases, following by the increasing of the compressive force. As the compressive force increases to FI, i.e. the springs at Stage 1 get to the point (FI, DI), the aforementioned snap through deformation occurs again. Repeating the similar deformation process, when all of the springs get to Stage 3, the deformation of PTCS becomes uniform again. Although the snap through behavior of multilayer PTCS is captured by using trilinear F-D relation, the nonlinear F-D relation is more suitable to describe the quasi-static behaviors of multilayer PTCS due to its capacity to capture the diversity of snap through deformation during compression loading (i.e. snap through from Stage 1 to Stage 2, Stage 2 to Stage 3 and Stage 1 to Stage 3).

2.2 Dynamic behaviors of PTCS

In order to study the dynamic behaviors of PTCS, a constant loading rate is applied to the first bead and the external force p acting on the first bead is also recorded for further analysis. As the deformation of PTCS is mainly contributed by the sinusoidal beams, the viscosity mainly comes from the deformation rate of the sinusoidal beams. Therefore, in this work we introduce a viscous damping parameter c to the spring and the viscous force of ith spring is c(u&i − u&i+1 ) . Taking

tc = mDΙ / FΙ as a characteristic time scale, the non-dimensional time is defined as t=T/tc, where T is the real time. Using all of the non-dimensional variations defined above, the non-dimensional governing equations of the nonlinear spring-bead model

are expressed as u&&i + ξ (u&i − u&i +1 ) + f ( d i ) = p  u&&i + ξ (2u&i − u&i −1 − u&i +1 ) + [ f ( d i ) − f ( d i −1 )] = 0 u&& + ξ (2u& − u& ) + [ f ( d ) − f ( d )] = 0 i i −1 i i −1  i

i =1 i = 2 ~ N −1

(14),

i=N

where ξ = c DΙ / mFΙ is the non-dimensional damping coefficient. For the constant

loading rate the displacement of the first bead is u1=v1t, where v1 is the non-dimensional constant loading rate. Due to the nonlinearity of F-D relation of sinusoidal beam and complex deformation patterns of multilayer PTCS, it is hard to obtain the analytical solution of Eq. (14) during the whole compression. Actually, there are 3N possible deformation stages for N-layered PTCS. So in this work we just show the analytical solutions of 2-layered structure. The analytical solution of dynamic behaviors for multilayer PTCS will be included in our future works. For 2-layered PTCS, using trilinear relation, i.e. Eq. (3), to study the dynamic behaviors of PTCS analytically. Eq. (3) can be reformulated as fl(d)=ad+b, where a and b are parameters determined by the deformation stage of the spring, i.e. a=1 and b=0 for Stage 1, a=(FII/FI-1)/(DII/DI-1) and b=1-(FII/FI-1)/(DII/DI-1) for Stage 2, and a=(1-FII/FI)/(DIII/DI-DII/DI) and b= FII/FI- [(1-FII/FI)/(DIII/DI-DII/DI)]DII/DI for Stage 3. Representing the trilinear relation of ith spring as fl(di)=aidi+bi, the governing equations of a 2-layered PTCS are degenerated into

ξ (u&1 − u&2 ) + a1 ( u1 − u2 ) + b1 = p

{

(15),

}

u&&2 + ξ (2u& 2 − u&1 ) + a2u2 + b2 −  a1 ( u1 − u2 ) + b1  = 0

(16).

By considering the constant loading rate at the first bead, the general solution of Eqs. (15) and (16) is u 2 = c1e λ1t + c2 e λ2t +

a1v1 b − b2 + ξ v1 2ξ a1v1 t+ 1 − a1 + a 2 a1 + a 2 ( a1 + a2 ) 2

(17),

where c1, c2 are two parameters determined by the initial displacement and speed of the second bead. Note that c1 and c2 have different values if the two springs are deformed to different stages. Therefore, we should continuously judge the

deformation stage of the springs which is very complicated if the number of spring is large. The exponential coefficients λ1 and λ2 in Eq. (17) are

λ1,2 = −ξ ± ξ 2 − ( a1 + a2 )

(18),

which are the eigenvalues of Eq. (16). According to Eq. (18), it is shown that the 2-layered PTCS is overdamped for ξ 2 − ( a1 + a2 ) > 0 , whereas it is underdamped for

ξ 2 − ( a1 + a2 ) < 0 . Therefore, a critical damping coefficient is ξ c = a1 + a2 . By assuming the first spring is at Stage 2 and the second spring is at Stage 1, i.e. a1=(FII/FI-1)/(DII/DI-1) and a2=1, the critical damping coefficient ξc is defined as

ξc = 1 +

FΙΙ / FΙ − 1 DΙΙ / DΙ − 1

(19).

The analytical results of the force-displacement relations for 2-layered PTCS are given in Fig. 4, which shows the loading rate and damping coefficient have significant effect on the dynamic behaviors of PTCS. In the following section, we will systematically study the dynamic behaviors of PTCS by numerically solving Eq. (14).

Figure 4. The dynamic force-displacement relations for 2-layered PTCS with different loading rates. (a) The damping coefficient is ξ/ξc =0.3 and (b) it is ξ/ξc =2, respectively. The solid lines are the numerical solutions of Eq. (14) and the dashed lines are the analytical solutions of Eq. (17).

3. Numerical analysis In this section, Eq. (14) are numerically solved by accounting for the practical F-D relation of sinusoidal beam via MATLAB software. The comparison between analytical and numerical solutions for a 2-layered PTCS is also given. In general, the analytical results agree well with the numerical results. However, as the analytical

solutions are deduced by assuming trilinear F-D relation of the sinusoidal beam at Stage 1 and Stage 3, discrepancy appears between the analytical and numerical results for the deformation of spring at these stages, as shown in Fig. 4. Especially, for the case of v1=2 and ξ/ξc =0.3 (blue lines in Figure 4 (a)), the spring has large fluctuation at Stage 3 corresponding to the large pulse force, and the analytical solutions cannot predict this type of phenomenon very well. In the following content, we focused on discussing the numerical results of the dynamic behaviors of PTCS with nonlinear F-D relation. 3.1 Dynamic behaviors of 2-layered PTCS

For 2-layered PTCS, the compression deformation of the first and second springs is shown in Fig. 5. In general, there are four deformation patterns depending on the loading rate and damping coefficient. The first deformation pattern occurs at relative small loading rate and damping coefficient, i.e. v1<0.8 and ξ/ξc<1.6. For the first deformation pattern, the first and second springs deform uniformly at the beginning of compression, and then snap through deformation occurs, that is the deformation of the first spring abruptly increases, whereas the deformation of the second spring abruptly decreases, and the uniform deformation is broken. However, when the deformation of the two springs is at Stage 3, it becomes uniform deformation again, as shown in regime I of Fig. 5. This phenomenon is similar to the quasi-static behaviors of 2-layered PTCS. As the damping coefficient increase, the snap through deformation occurs later. When the damping coefficient is ξ/ξc≥1.6, the snap through deformation disappears and the deformation of the two springs are almost identical during the whole compression, as shown in regime II of Fig. 5. This is because the non-uniform deformation is prevented by the large damping force. For v1≥0.8 and ξ/ξc<1.6, the deformation of the two springs is not equal during the whole compression, and the difference of the deformation between the two springs is larger for larger loading rate and smaller damping coefficient due to the combined effect of inertial and viscous effect, as shown in regimes III and IV of Fig. 5. Here, the spring deformation for the third (regime III) and fourth (regime IV) deformation patterns is analogous. However, for the third deformation pattern the first spring has pulse tension deformation at

Stage 1, while for the fourth deformation pattern the first or second spring has large pulse deformation at Stage 3, so that there is large negative pulse force for the third deformation pattern and positive pulse force for the fourth deformation pattern, as shown in Figs. 6(b) and 6(c).

Figure 5. The spring deformation of 2-layered PTCS with different loading rates and damping coefficients.

The dynamic force-displacement relations for 2-layered PTCS are shown in Fig. 6. For the first deformation pattern (regime I), due to the snap through deformation the compressive force abruptly change. However, different from the quasi-static behaviors of 2-layered PTCS, the snap through deformation doesn’t immediately occur at the ending point of Stage 1. Besides, the snap through deformation for dynamic loading is also larger than that of quasi-static loading, so that the change of compressive force is larger. As the damping coefficient increases, the 2-layered PTCS becomes uniform deformation (regime II), so that the force-displacement relation of relative small loading rate reproduces the quasi-static force-displacement relation of single sinusoidal beam, see the cases of ξ/ξc=1.6 and ξ/ξc=2 in Fig. 6(a). While, for

relative large loading rate with large damping coefficient, the viscous force contributes a large part of the compressive force, so that the force-displacement relation deviates from that of single sinusoidal beam and the value of compressive force is also larger, see the cases of ξ/ξc=1.6 and ξ/ξc=2 in Figs. 6(b) and 6(c). For v1≥0.8 and ξ/ξc<1.6, the compressive force has negative (regime III) or positive (regime IV) pulses due to the pulse deformation of the springs, as shown in Figs. 6(b) and 6(c). The pulse force is larger for smaller damping coefficient.

Figure 6. Dynamic force-displacement relations of 2-layered PTCS with different damping coefficients under different laoding rates (a) v1=0.1, (b) v1=1, (c) v1=2, respectively. The dashed

line is the quasi-static force-displacement relation of 2-layered PTCS.

Combining the deformation of the two springs and the corresponding force-displacement relations, the dynamic behaviors of 2-layered PTCS can be divided into four regimes, as shown in Fig. 7. For regime I, the 2-layered PTCS has snap through deformation and its force-displacement relation is similar to that of the quasi-static loading. For regime II, the 2-layered PTCS deforms uniformly and for relative small loading rate its force-displacement relation degenerates to the quasi-static force-displacement of single sinusoidal beam, while for relative large loading rate the viscous force contributes a large part of the compressive force. For regime III and IV, the spring has pulse deformation which leads to the pulse compressive force. Note that the damping coefficient is just an effective value of the PTCS which doesn’t correlate to a specific material. In our future works, experiments will be carried out to connect the material viscosity and structure damping coefficient.

Figure 7. Phase diagram of the dynamic behaviors of 2-laytered PTCS.

3.2 Dynamic behaviors of multilayer PTCS The dynamic behaviors of multilayer PTCS are very complicated due to the combined deformation of the multiple sinusoidal beams. Therefore, in this section, we only numerical method to study the dynamic behaviors of 9-layered PTCS with loading rate v1=0.1, 2 and damping coefficient ξ/ξc=0.1, 2 as examples to shed some useful hints on the dynamic behaviors of multilayer PTCS. The compressive force and

bead displacement under loading rate v1=0.1 are given in Fig. 8. For relative small loading rate and damping coefficient, i.e. v1=0.1 and ξ/ξc=0.1, the 9-layered PTCS has snap through deformation corresponding to the abrupt change of the compressive force and bead displacement, which is similar to the regime I of 2-layered PTCS. However, the 9-layered PTCS has significant vibration after each snap through deformation due to larger kinetic energy generated during the snap through deformation. For relative small loading rate and large damping coefficient, i.e. v1=0.1 and ξ/ξc=2, the 9-layered PTCS still has snap through deformation which is different from the dynamic behaviors of 2-layered PTCS with the same loading rate and damping coefficient. Besides, the snap through deformation may simultaneously occur at several springs corresponding to the larger displacement between adjacent peaks in the compressive force-displacement curves of Fig. 8(b).

Figure 8. Dynamic behaviors of 9-layered PTCS under loading rate v1=0.1. The force-displacement relations of the 9-layered PTCS with damping coefficient ξ/ξc=0.1 (a) and ξ/ξc=2 (b). The corresponding displacement of each bead with damping coefficient ξ/ξc=0.1 (c) and ξ/ξc=2 (d).

The compressive force and bead displacement under loading rate v1=2 are given in Fig. 9. Under relative large loading rate, the snap through deformation doesn’t occur for both of the small and large damping coefficient. For relative small damping

coefficient, i.e. ξ/ξc=0.1, the deformation of the 9-layered PTCS is sequential from the loading end to the fixed end, as shown in Fig. 9(c). This is because the loading speed is larger than the elastic wave propagation speed in PTCS and the compression deformation propagates from the loading end with the same loading speed. Besides, pulse deformation of the spring is also observed corresponding to the pulse compressive force in Fig. 9(a). For relative large damping coefficient, i.e. ξ/ξc=2, the deformation of the 9-layered PTCS is much more uniform, as shown in Fig. 9(d). However, the compressive force has also some pulses which means the springs have also pulse deformation. In general, as the degree of freedom increases for multilayer PTCS, the vibration is easier to occur after the snap through deformation, while the uniform deformation is harder to occur.

Figure 9. Dynamic behaviors of 9-layered PTCS under loading rate v1=2. The force-displacement relations of the 9-layered PTCS with damping coefficient ξ/ξc=0.1 (a) and ξ/ξc=2 (b). The corresponding displacement of each bead with damping coefficient ξ/ξc=0.1 (c) and ξ/ξc=2 (d).

4. Conclusions In summary, a nonlinear spring-bead model is developed to investigate the dynamic behaviors of PTCS consisted of bi-stable sinusoidal beams. Results have shown that the PTCS have abundant dynamic behaviors depending on the number of layer, loading rate v1 and damping coefficient ξ/ξc. For 2-layered PTCS, there are

generally four deformation patterns. Regime I represents the snap through deformation of the 2-layered PTCS occurring at relative small loading rate and damping coefficient. At this regime, its compressive force-displacement relation is similar to that of quasi-static loading. Regime II represents the uniform deformation occurring at relative large damping coefficient. For relative large loading rate and small damping coefficient, pulse deformation of the 2-layered PTCS occurs. For regime III, the pulse tension deformation occurs at Stage 1, so that the compressive force has negative pulse. While, for regime IV the large pulse deformation occurs at Stage 3 which leads to positive pulse force. Based on these results, a phase diagram of the dynamic behaviors of 2-layered PTCS is given. The dynamic behaviors of multilayer PTCS is very complicated due to the combined deformation of multiple sinusoidal beams. However, the snap through, uniform and pulse deformation are also observed in 9-layered PTCS. However, as the degree of freedom increases for multilayer PTCS, significant vibration is observed after the snap through deformation and the uniform deformation is also harder to occur. As the PTCS usually work at dynamic loading, the results presented in this work provide useful insights for the design and optimization of PTCS.

Acknowledgment: The authors acknowledge financial supports from the National

Natural Science Foundation of China (No. 11572239) and National Key Research and Development Program of China (No. 2016YFB0700300).

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