A nonlinear analysis of surface acoustic waves in isotropic elastic solids

Hoxing Wu , Rongxing Wu , Tingfeng M , Zixio Lu , Honglng Li , Ji Wng ,

aPiezoelectric Device Laboratory, School of Mechanical Engineering & Mechanics, Ningbo University, Ningbo 315211, China

bNational Center for Nanoscience and Technology, Beijing 100190, China

Keywords:Surface Wave Nonlinear Galerkin Resonator

ABSTRACT With the fast evolution of wireless and networking communication technology, applications of surface acoustic wave (SAW), or Rayleigh wave, resonators are proliferating with fast shrinking sizes and increas- ing frequencies.It is inevitable that the smaller resonators will be under a strong electric field with induced large deformation, which has to be described in wave propagation equations with the considera- tion of nonlinearity.In this study, the formal nonlinear equations of motion are constructed by introduc- ing the nonlinear constitutive relation and strain components in a standard procedure, and the equations are simplified by the extended Galerkin method through the elimination of harmonics.The wave veloc- ity of the nonlinear SAW is obtained from approximated nonlinear equations and boundary conditions through a rigorous solution procedure.It is shown that if the amplitude is small enough, the nonlin- ear results are consistent with the linear results, demonstrating an alternative procedure for nonlinear analysis of SAW devices working in nonlinear state.

Surface acoustic wave (SAW) resonators are widely used in the electronic communication as a core component for filter and sen- sor applications [ 1 , 2 ], always demanding superior performance in functioning under various conditions.With the rapid development of wireless communication technology, the sizes of resonator de- vices continue to decrease under miniaturization schemes, which raise stricter requirements for their performance such as working frequency, accuracy, and stability.However, at the same time, there are also nonlinear effects mainly because small devices will in- evitably undergo large deformation through the interaction with a strong electric field [ 3 , 4 ], thus downgrading the device proper- ties as designed initially.Therefore, the nonlinear deformation and device performance effects need to be considered in the design phase through the calculation of properties of wave propagation with the consideration of structural and environmental factors in operations under extreme conditions and high mobility.This im- plies the SAW analysis in piezoelectric materials with structural complications such as electrodes and mountings at the presence of electric and thermal fields is essential in the precise design and it- erative optimization of devices.As a result, an accurate analysis of nonlinear SAW in anisotropic and piezoelectric materials of finite sizes will be critical in the development of mathematical proce- dure and solution technique.

The straightforward analysis of the nonlinear features and prop- erties of SAW are seldom studied due to difficulties in dealing with nonlinear differential equations of wave propagation.Conse- quently, nonlinear effects of SAW devices have been studied by au- thors with simplified models like coupling-of-modes (COM) [ 5 , 6 ], P-matrix [7-9] , the Mason model [ 10 , 11 ], or with the finite ele- ment method (FEM) [12-14] .Pang et al.derived nonlinearly cou- pled sets of piezoelectric field equations in the frequency domain, and implemented a solution procedure for solutions of these cou- pled equations with a 3D FEM model to study the nonlinear res- onances and generation of harmonics in SAW resonators [ 15 , 16 ].Maugin obtained the solitary-wave solutions in elastic solids for envelope signals in the nonlinear regime [4] .Qian studied and pre- sented formulations of nonlinear high-order SAW equation [17-19] .Clearly, there is a need of an approximate procedure for an accu- rate analysis of nonlinear effects in relation to device properties, and the results have to be verified with a more reliable and trusted method.The challenge has been known, and solutions from ap- proximate methods discussed above can provide answers to sat- isfy partial needs.As the continuation of effort s in finding accu- rate solutions of nonlinear waves in acoustic wave devices, a new procedure based on the popular approximate techniques such as Galerkin method or Rayleigh-Ritz method has been proposed and validated [20] .It is believed that the utilization of Galerkin method with an extension will offer another simple and reliable procedure to study the effects of nonlinear waves in finite solids due to wide applications in linear problems.With this objective in mind, this paper starts with Galerkin formulation of nonlinear SAW in elastic solids for simplification and the final results have been validated by the FEM.It is clear that a new procedure for the analysis of nonlinear waves has been established and validated for its utiliza- tion in the analysis of SAW devices which are in urgent needs for optimized design to improve functions in nonlinear states under many factors such as strong electric field and accelerations.

For the nonlinear SAW in elastic solids, it is natural to start with an infinite and isotropic solid as widely known.Although the quartz crystal material in piezoelectric devices is finite and anisotropic, it is necessary to understand the basic formulation and correct procedure for the analysis of propagation of nonlinear SAW.The device structures are finite with complications such as elec- trodes and mountings which have to be considered after the accu- rate analysis of SAW in elastic solids are made and validated.The formulation based on the fundamental theory of wave propagation of isotropic materials is provided in this paper as the overture and test of techniques.The extension of the formulation and solution of nonlinear SAW for anisotropic and piezoelectric solids with struc- tural complications will be made gradually in the same principle.

The semi-infinite isotropic elastic solid model for SAW is shown in Fig.1 , wherex1denotes the direction of wave propagation andx2is the depth.The material hasλandμas second-order elastic constants andρas density.Since the model is simplified to a two- dimensional plane, displacements of SAW are usually be expressed as

Fig. 1. A semi-infinite isotropic elastic solid.

whereui(i= 1,2,3)andtare displacements and time, respec- tively.This is the usual formulation of surface waves in a linear case.

Equations of motion of elastic solids with the coordinate system shown in Fig.1 are

Fig. 2. Linear and nonlinear wave velocities with different Poisson ratios atA= 10 ?4 m .

whereKijare components of Lagrangian stress tensor.The consti- tutive relation is

whereWis the strain energy density.For an isotropic solid,Wis usually written in terms of strain invariantsI1 ,I2 andI3 as [4]

wherel,mandnare the third-order elastic constants.The strain invariants are expressed by the strain tensors as [21]

With the consideration of large displacement gradients, the strain-displacement relations are

In this study, only the kinematic nonlinearity is considered in case of relatively large deformation, while the material nonlinear- ity is ignored because most isotropic materials are predominantly linear in general.

By substituting Eqs.(3) - (6) into Eq.(2) , it yields the nonlinear equations of motion as

As it can be seen, the expanded form of above nonlinear equa- tions is so complicated that it is impossible to obtain the analytical solution in closed form.Therefore, considerable efforts have been made in the simplification and approximation of the equations for approximate solutions, as it is discussed before with introductions of some popular techniques of approximation and adoptions of ap- proximate methods.

The core requirement of solutions from such nonlinear prob- lems is the relationship between vibration frequency and ampli- tudes, which is used to evaluate the effect of nonlinearity in de- vice function and applications.To solve for the velocity of non- linear SAW from above equations with amplitudes, the Galerkin method for approximate solutions of nonlinear differential equa- tions should be utilized.As it is known, Galerkin method is fre- quently employed to solve linear problems in elasticity including wave propagation because of the simple and reliable procedure, while nonlinear wave propagation problems in finite regions are also solved through some special techniques recently [ 20 , 22 , 23 ].Due to the periodicity of SAW, the elastic solid can be reduced to a unit cell of one wavelength in the wave propagating direction from the semi-infinite problem.It is assumed that displacements are in the standard form [24]

whereA,B,k,β,care amplitudes of displacements, wavenumber, decaying index, and phase velocity, respectively.By substituting Eq.(9) into Eq.(7) , the error from each equation is defined as

Since Eq.(7) is not satisfied automatically, approximate solu- tions have to be found.One approximate technique can be used for the simplification of nonlinear equations in Eq.(10) is the extended Galerkin method [22] , which is developed for the transformation and solution of nonlinear differential equation with periodicity.In addition to the known approximation of differential equations in physical domain with the popular Galerkin method, an extension is introduced for an approximation in the time domain with a weighted integration over one period of vibrations.The extended Galerkin method (EGM) has been proven and verified with a com- plete procedure and successful examples, and it will be applied to the nonlinear equations in Eq.(10) for approximate algebraic equa- tions to obtain solutions of frequency and amplitudes.Clearly, the choice of the EGM is critical to solutions of nonlinear SAW in elas- tic solids with the proposed simplification approach of a unit cell with one wavelength and weighted integration over time domain of one period also.Furthermore, the approximate technique is ir- relevant to materials and structures, thus making them applicable to problems of anisotropic and piezoelectric materials of acous- tic wave devices.The structural complications can be considered through the equivalent approach of EGM, or Rayleigh-Ritz method,

with the same integration scheme.At the end, an approximate pro- cedure for the analysis of nonlinear effects of actual SAW devices is established with extended Galerkin and Rayleigh-Ritz methods for the design and optimization to improve the nonlinear properties.

With the nonlinear equations of motion given in Eq.(7) and trial solutions in Eq.(9) , further simplifications and approxima- tions are needed to obtain unknown parameters.As discussed, the extended Galerkin method is applied to Eq.(10) with known dis- placements as weighting functions through

Without stressing the nonlinearity of equationsN1 andN2 and the addition of integration over time, it is exactly the Galerkin method for approximate solutions of differential equations.Also, it is not an overstatement as the Galerkin method is considered as the foundation of numerical methods such as the FEM which are very popular and powerful in these days.It is also important to point out the parameters are related through

whereTis the period andζis the wavelength.It is clear that the physical domain in this approximation is a unit cell of one wavelength length and infinite depth.A Galerkin approximation is made with a unit cell and one period of wave motion, which is not a usual procedure used in the analysis of wave motion in elastic solids and structures before, but it has been proven in SAW analy- sis that the unit cell model is reliable and effective in calculations.

After the integration and simplification with basic operations, nonlinear equations of Eq.(11) have been successfully converted to algebraic equations in the form of

with parameters

whereEis Young’s modulus,νis Poisson’s ratio,γis the am- plitude ratio ofBtoA, andcRis the relative SAW velocity.With known amplitudeA, Eq.(13) can be solved for different pairsβ1,γ1andβ2,γ2for differentcRby iteration method for thekth-order solution

and the displacements are now given as

This is exactly the displacements of linear SAW except the pa- rameters are obtained from nonlinear equations.Consequently, the nonlinear effect will be included in the solutions and further ma- nipulations of solutions will providing more information to evalu- ate the effect of nonlinearity.The correct amplitude ratio ofA2andA1is further defined as

The traction-free boundary conditions of elastic solids, which are also nonlinear equations of amplitudes and harmonics, can also be modified by the extended Galerkin method to

By substituting Eqs.(17) and (18) into Eq.(19) , the boundary conditions with parameters from solutions are simplified to

Now Eqs.(13) and (20) are the algebraic equations through the approximate procedure and they are going to be used for solutions of the parameters.

In a comparison with linear problems of SAW in elastic solids [25] , it is clear that the solution procedure is now quite differ- ent.In the linear procedure of wave propagation, the amplitudes of wave forms are indeterminant, and they are not needed to satisfy boundary conditions.In the case of nonlinear wave propagation, wave form solutions are dependent to amplitudes, and the num- ber of equations of coupled solution parameters is enlarged sig- nificantly.Furthermore, the nonlinear coupling of these equations makes the solution technique difficult to apply.A close examina- tion of the four equations for nonlinear parameters can reveal that there is no simple method to extract solutions from these algebraic equations.A search algorithm with brutal force has to be employed for solutions with prescribed amplitudes in a given frequency in- terval.If Rayleigh waves are the objective, the frequency has to be close to the known range of Rayleigh waves from linear solutions.It is a general knowledge that the frequency change even under a strong nonlinear effect should be limited, as can be proven from practical experiences.

In order to obtain the solution parameters from Eqs.(13) and (20) , a solution procedure, or algorithm, is outlined and imple- mented as shown below.Through changes of the value ofcRuntil both equations are satisfied, thencRis the normalized SAW veloc- ity with given amplitudes.

The procedure for the analysis of nonlinear SAW propagation is as follows:

1.Specify amplitudeA;

2.Specify the normalized wave velocitycRin Eq.(13) ;

3.Calculateβ1,γ1andβ2,γ2from Eq.(13) ;

4.Check if these solutionsA,cR,β1,γ1,β2,γ2satisfy boundary conditions in Eq.(20) ;

5.Solutions are obtained asA,cR,β1,γ1,β2,γ2,.

Now a complete procedure for the calculation of solution pa- rameters from equations of motion and boundary conditions of nonlinear SAW in a unit cell of isotropic elastic solid is established and tested.Basically, the extended Galerkin method has trans- formed the nonlinear differential equations to algebraic equations which can be solved more easily.The solution parameters, as listed above, are to be used to construct the wave forms of Rayleigh waves in the structure.Furthermore, the solutions will be used for formulation and calculation of other properties of the waves and structure.Of course, it should be emphasized that in this study, the nonlinear properties of material are not considered in the formu- lation and calculation and the problem is basically only the kine- matic nonlinearity.The constitutive or physical nonlinearity can be formulated accordingly with available material properties, if it is certain they cannot be neglected as in this study.Since the ma- terial properties are known along with some properties of waves, the nonlinear algebraic equations can be solved simultaneously as solutions of the nonlinear wave propagation.It is a typical non- linear system from nonlinear theory of wave propagation, and the formulation and approximation have to be validated with the unit cell model of one period length.

It is clear that Table 1 shows the relative SAW velocity with dif- ferent Poisson’s ratioνat a fixed amplitudeA= 10?4m , and Fig 2 .shows a more intuitive comparison of linear and nonlinear results with the velocity shift×106.If the am- plitudeAis small enough, there is no difference in propagation velocities between the linear and nonlinear SAW.

Table 1 Relative SAW velocity with different Poisson’s ratios atA= 10 ?4 m .

Table 2 and Fig.3 show the relative SAW velocity at different amplitudeAwith Poisson’s ratioν= 0.3 .The influence of nonlin-ear effect increases gradually with amplitude aboveA= 10?4m , which is a very large value.

Fig. 3. Variations of linear and nonlinear wave velocities under different amplitude withν= 0.3 .

Table 2 Relative SAW velocity at different amplitudes with Poisson’s ratioν= 0.3 .

Fig.4 shows the linear and nonlinear wave mode shapes in the thickness direction and a comparison with analytical and COMSOL results.It shows that the results from the nonlinear procedure pro- posed here are consistent with the FEM analysis and accurate.

Fig. 4. Displacement envelopes of nonlinear surface waves in comparison with linear case (a) Linear and nonlinear displacements in comparison with solutions from FEM with COMSOL withν= 0.3 andA= 10 ?6 m , (b) enlarged displacements for details.

It is clear that the nonlinear solutions can be obtained from the solutions of nonlinear algebraic equations, and the constructed wave forms are consistent with finite element analysis from non- linear equations of wave propagation.The solutions are obtained from isotropic materials with the kinematic nonlinearity, but the inclusion of material nonlinearity in the formulation is straightfor- ward and the solution procedure or algorithm remains the same.Furthermore, the same procedure including the formulation and transformation with the extended Galerkin method can be applied to anisotropic and even piezoelectric materials for more practical analysis related to SAW devices.In case of finite structures with structural complications, the Rayleigh-Ritz method can be used for the linear analysis, and its equivalence to the Galerkin method also implies that the extended Rayleigh-Ritz method can be uti- lized for nonlinear analysis of SAW resonators which are targeted in this study.The approach taken with the nonlinear formulation and eventual solutions is a novel procedure based the extended Galerkin method, representing a new technique for the solution of nonlinear differential equations with periodic nature.

The nonlinear properties of SAW resonators are urgently de- manded in device analysis and design as part of technology evolu- tion and improvement in connection with communication and sen- sor technology.The continuing miniaturization of SAW resonators signified the necessity of nonlinear formulation of wave propa- gation in close relationship with resonator design and optimiza- tion.With this objective, the analysis with a novel approximate technique, the extended Galerkin method, presented a reliable ap- proach to obtain nonlinear solutions of SAW in elastic solids, and eventually piezoelectric solids, as the essential part of a resonator structure.The analysis of nonlinear SAW is shown here with a val- idation by the finite element analysis, which is a standard pro- cedure in the structural analysis nowadays.The proven procedure based on the simplification of nonlinear equations can be further applied to resonator structures with the consideration of compli- cation factors such as discrete electrodes and electric voltage.This will be the needed analysis to improve resonator design with non- linear behavior under strong electrode field for certain applications with desired stable functions.The results also shown that the ex- tended Galerkin and Rayleigh-Ritz methods are capable of solving nonlinear differential equations from wave propagation and similar applications as a preferred technique for more challenging prob- lems.

Author Contribution

The first author (HXW) is mainly responsible for the formula- tion, derivation, calculation, verification, and revision of equations and data of the paper.The conception and evaluation of the proce- dure and algorithm is initiated by the corresponding author (JW).The third-author (RXW) contributed to the development of the for- mulation, derivation, and calculation through discussions with the first author.All other authors are in the discussion and review of the research project.

Declaration of Competing Interest

The Authors declares no competing financial or non-financial interests.

Data Availability

The data used and generated in this research will be publicly available from the publication’s website and author’s website.

Acknowledgments

This work is supported by the National Natural Science Foun- dation of China (Grant 11672142).Additional supports are from the Technology Innovation 2025 Program (Grant 2019B10122) of the Municipality of Ningbo, Research and Development Program of Key Disciplines of Guangdong Province (Grant 2020B0101040 0 02), and Research and Development Program in Key Disciplines of Hunan Province (Grant 2019GK2111).