PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Numerical study on energy transmission for soft materials andmetamaterials structures by structural intensity method
Rong Huang1; Ziqian Li1; Yiheng Xue1; Zishun Liu1;
1

International Center for Applied Mechanics, State Key Laboratory for Strength and Vibration of MechanicalStructures, Xi’an Jiao Tong University, China

ABSTRACTIn this work, we studied the energy transmission and dynamic performance of multiple structures from theaspect of energy flow by using structure intensity (SI) method. Compared with the traditional vibration or
dynamic analysis methods, the SI method considers the internal force and displacement in the structures, andcan accurately reflect the effect of the excitation source, the damper or virtual damper locations and theenergy flow on the structures. In this study, numerical simulation system was built to represent different
structures under different loading and damping cases. We get the corresponding results of structural intensity
distribution and plot the SI streamline for each case. We find that the structure intensity distribution can besignificantly affected by adjusting the material parameters and setting damper in specific locations. Based on
the SI energy distribution of the engineering materials, soft materials and metamaterials structures, westudied the methods to how to reduce the vibration of the structure or identify the potential damage positions
by using its main energy propagation paths. In addition, the effect of structural damping on the energy flowtransmission performance of metamaterials structures is discussed for swelling process. We hope that ourworks could provide researchers new ideas of vibration control for similar new smart structures in potential
industry applications.
Keywords: Structure Intensity, metamaterials, energy flow

1. INTRODUCTION
The study on structural intensity of vibration and vibro-acoustic characteristic in different
structures has been investigated extensively. The concept of structural intensity was first introduced asa quantifier for the structural vibro-acoustic analysis in 1970s by Noiseux.(1) Analogous to theconcept of acoustic intensity in acoustic medium, the structural intensity can be expressed as the resultof stress and the velocity, a direction sum of the scalar product of the force vectors and theircorresponding velocity vectors. Pavic developed the formulation of structural intensity and applied it
to complex structure entity, which led to the growing interest to this field for the next few decades inapplied mechanics community.(2) Hambric developed the computational approach of 3D structuralintensity through the finite element method.(3) Gavric and Pavic evaluated the structural intensity
fields of a rectangular plate to identify the source and the sink of the energy flow.(4) Xu et al.(5) andLiu et al.(6) firstly successfully introduced the structural intensity streamline concept into platestructures. Furthermore, Liu et al.(6) theoretically derived the formulations for structural intensity
streamline according to the concept of fluid mechanics streamline. The structural intensity streamline
can be used to clearly display the structural intensity flow paths, which provide the visualization
expression of the energy sources and sinks in structures.
The structural intensity streamline concept was then introduced into different engineering
applications. Guo et al.(7) studied the SI distribution of beam structures under random vibration. Liu etal.(8) applied the SI methods into medical field to study the vibrational characteristics of human bodysnoring identification. In noise control aspect, Liu et al.(9) adopted the SI method to propose a basic
method of active and passive interior noise control for box structures, which would be used for vehicle
cabins. Ang et al(10) presented a potential for controlling cabin noise of automobiles and armouredvehicles. Besides, the SI streamline concept can also be helpful for the detection of cracks. For1

zishunliu@mail.xjtu.edu.cn8081

example, Tian et al. (11)applied the SI method to detect crack of offshore platform. Lee et al.(12)studied the diversion of energy flow near crack tips of a vibrating plate. Overall, the study in structuralintensity method still has a great potential and can be applied into multiple complex engineeringproblems.
This study is mainly an extension of our previous work for the structure intensity method. We
will investigate the potential application for structure intensity method in the study of fracture andfatigue problems of soft materials (such as hydrogel), metamaterials structures. The SI method may
useful to investigate their vibration characteristics and failure mode under multiple loading caseswith different structures. We will first propose the vibration mode method to calculate the structuralintensity of cylindrical shell structures theoretically and investigate the vibro-acoustic behaviors ofcylindrical shell structures. Then the SI distribution for notched hydrogel plate will be briefly
introduced. Currently, the fracture and fatigue problems of hydrogel soft materials become the newresearch trend, what is the SI distribution when the dynamic loading is applied? The SI distributionsof a notched hydrogel plate and a kind of metamaterials structures are provided in this shortsummary paper. For notched hydrogel plate, our results demonstrate the effect of Mullins-effect forhydrogel. For metamaterials structure, we found that the SI distribution is significantly changed
during deformation process.

2. RESULTS
2.1

Using SI approach to characterize vibro-acoustic behavior of the cylindrical shellstructures
2.1.1 Formulation of structural intensity by vibration mode shapes
From our study, we found the structural intensity is an infinite weighted sum of function ofvibration mode shapes. Simplify the detailed formulation to weight coefficients about load conditions
and distribution functions of vibration mode shapes, the SI of the cylindrical shell structure along axial
direction and circumferential direction can be further expressed as follow:






I x   0 2  C  f mn  f rs  X mnrs  x,  m 1 n 1 r 1 s 1I   0 2  C  f mn  f rs   mnrs  x,  (1)m 1 n 1 r 1 s 1

where C is model coefficient;

0 is exciting frequency; f mn  f rs are weight coefficients;

X mnrs , mnrs are distribution functions; m, n, r, s are wave numbers. The wave numbers indicates thenumber of waves on the vibration shapes and represents a definite modal. We use (m, n) to represent
the modality of a force condition, using (r, s) for a displacement of the modal. Finally, the (m, n, r, s)
can represent a modal state of the structural intensity component. In this equation, the weightcoefficient components show the load conditions and reflect the weight coefficient of time.
The structural intensity streamlines are a family of curves that are instantaneously tangent to thevector of the structural intensity. For steady vibration, the structural intensity streamlines are thevibrational power flow paths and can effectively assess vibrational sources and sinks. The structuralintensity streamline in the thin cylindrical shell surface can be defined as (6):dr  I  r, t   0(2)where r is particle position; I is structural intensity vector. The structural intensity vectors thatlocated on the streamlines are perpendicular to r and parallel to dr. For the cylindrical shell, thedifferential equation describing is:dx Rd

 0.
Ix
I

(3)

Every particular point that locate on the cylindrical shell corresponds with a structural intensity
vector, and the vector is tangential to a streamline at this point. Therefore the group of streamlines candescribe the power flow travel paths on the cylindrical shell.

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2.1.2 Vibro-acoustic behavior of the cylindrical shells
Once we get the above formulation, we can calculate the structural intensity for different loadingcase. If we make the loop-integration of structural intensity and applied it on the internal surface ofvibrating cylindrical shell, which is the coupling surface of vibration and sound, structural intensity
energy can be obtained. And at mean time the volume integral of internal acoustic intensity is
considered as sound energy in sound field. Once we get this two-energy value, we find that the internal
acoustic pressure field has the close connection with structural intensity in the interface ofstructural-acoustic coupling. From figure 1, we know that the structural intensity energy and theradiation sound energy almost follows the same trend when varying damping cases and exaction
frequencies. Therefore, it provides a method to control the sound radiation energy for a vibrating
structure by changing its structural intensity energy.

(a) Excitation frequency 20 Hz(b) Excitation frequency 56 Hz

Figure 1 Structural intensity and sound energy trends in various cases
Based on this result, we can further provide a method which can reduces the acoustic energy
through properly putting damper position and redistributing the structural intensity in the cylinder
shell is developed. For cylinder shell structures, there are some eddies pattern in the structuralintensity streamline, and these eddies have potential of collecting energy. When the damper is p ut inthe eddy position, the stored energy will be released and a new sink appears. According to source andsink positions, we can modify the structural intensity distribution so as to reduce the acoustic energy ofcylinder shell structure.(13)2.2 Vibration performance of slotted plate by using SI method
We have also investigated the SI distribution of a given simply supported slotted plate undersinusoidal force loading condition. The effect of slotted depth on SI distribution of slotted plate and its
vibration control method has been investigated. It is easily found that the slot depth can significantly
affect the SI distribution of the slotted plate. According to the SI distribution, setting the position of
the damper close to the excitation source can obviously affect the vibration performance of the slottedplate. As an alternative to traditional dampers, the ease and effectiveness of using a constrainedhydrogel layer have been considered to dissipate energy. The simulation shows that hydrogel materials
can effectively reduce the maximum stress and maximum displacement of the slotted plate. Thishighlights the potential applications of the hydrogel in vibration control systems , and it proves thathydrogel layers can be applied to dissipate energy. (14)2.3 Using SI to detect Hydrogel crack propagation for fatigue test caseWe established a hydrogel model to simulate pure shear fatigue test, and the size of crack tip is
0.4mm. For dynamic analysis, we taken time period as 1s. During this time step, the model will be
stretched twice as shown Figure 2.

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Symmetric boundary
condition

Figure 2 The schematic of hydrogel crack model. The model is a 1/2 up-down symmetric model.
The SI streamline for every time frame is shown in Figure 3a-c. The streamline distributions vary
dramatically at a different time frame. At t=0.15s the distribution of streamline is gentle. But at
highly stretched state, streamline distribution becomes complex, and a virtual source and a virtual
sink of streamline will appear. Besides, SI streamline can represent the energy flow within material.
In Figure 3a-b, the applied excitation does work to the hydrogel system, hence we can see that theenergy flows form upper line to the lower line and the crack tip. In Figure 3c, the model is restoring
its deformation and releasing stored elastic energy, thus the energy is flowing out of the hydrogel
model.

(a) t=0.15s(b) t=0.25s(c) t=0.35s
Figure 3 The distribution of SI streamline at different time frame. (a) t=0.15s, (b) t=0.25s, there are a virtual
source and a virtual sink (c)t=0.35s, there is also a virtual source.
2.4

Energy flow transformation for metamaterial structure under dynamic loading duringswelling of hydrogel
We also build a metamaterial structure as shown in Figure 4. In this case, we first use
Static-General step to simulate the process of hydrogel swelling. For different swelling state, wethen analyze their SI fields.
ClampedClampedActivate(Displancement)(a)(b)
Figure 4. (a) The schematic of the metamaterial structure and its representative volume element. (b) The
boundary condition in unit cell model.
As shown in Figure 4a, the green area can absorb water and then swell. We select two equilibriumstates (before swelling and swollen state) in the Static-General step. Then we export the models ofthese two states into a new case with Dynamic analysis. Here, the time period is 0.1s and we hold thetwo upper corners and apply sinusoid excitation at the lower corner as shown in Figure 4b. From
Figure 5, we plot the SI streamlines for different swelling state. We can easily see that the SI
streamline distributions are much more complicated than that in hydrogel swelling for normalstructure. It is as the result of the complex structure of the metamaterial deformation. Besides, in thiscase, there are much more vortexes with metamaterial. We also show the effect of the Mullins effecton SI field. We find that viscoelasticity indeed influences the SI distribution for soft materials, that is,
Mullins effect can weaken vortexes within the material and result in energy dissipation. This finding8084

may be useful for further metamaterial structure control.
Ideal elastic
Before swelling

SwollenPossess Mullins-effect
Before swelling
Swollent=25mst=50mst=80msFigure 5 The distribution of SI streamlines on the metamaterial structure.

3. CONCLUSIONSThis paper summarized some SI potential applications for vibrational structures, fracture of soft
materials and metamaterial structure. For regular structure such as cylinder shell and plate structures,
we found there are some eddies pattern, which have potential of collecting energ y in the structuralintensity. Therefore, the vibration energy can be adjusted by setting damper s to create new sinks in SI
streamline and release stored energy. The SI streamline can also clearly demonstrate the energy
distribution and dissipation pattern for soft material in fracture mode as well as metamaterial structure
under complex loading conditions. This could provide researchers new ideas when analyzing
deformation mechanisms and failure mode for similar smart structures (hydrogel and shape memorypolymers) in potential industry applications.(15, 16)

ACKNOWLEDGEMENTS
Authors are grateful for the support from the National Natural Science Foundation of China
through grant numbers 11572236 and 11820101001.

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