PSI - Issue 50

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16th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures (MRDMS 2022) 16th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures (MRDMS 2022)

Acoustic characteristics of deformable metals Svetlana Barannikova*, Mikhail Nadezhkin, Polina Iskhakova Institute of Strength Physics and Materials Science SB RAS, 2/4, pr. Akademicheskii, Tomsk, 634055, Russia Acoustic characteristics of deformable metals Svetlana Barannikova*, Mikhail Nadezhkin, Polina Iskhakova Institute of Strength Physics and Materials Science SB RAS, 2/4, pr. Akademicheskii, Tomsk, 634055, Russia

© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers Abstract The stages of plastic flow in Fe-18 wt.% Cr-10 wt.% Ni structural steel were studied by performing non-destructive testing. In particular, the acoustic parameters of the steel were measured via uniaxial tensile tests. According to the changes in the velocity of propagation and attenuation of ultrasound in a wide temperature range, the stages of plastic deformation and destruction were clearly distinguished. Based on the results of this study, a non-destructive testing parameter characterizing the transition to the pre-destruction stage of the material was proposed. © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers Keywords: mecanical testing; non-destructive testing; stainless steels; ultrasonic Rayleigh waves; plastic deformation 1. Introduction Increasing requirements for the quality of parts imply the expansion of diagnostic and non-destructive testing methods. In that regard, the mechanical properties of the material are decisive in assessing the quality of products as well as the probability of failure along with safety and expediency of using parts under certain conditions. In this respect, relevant research is devoted to the diagnosis and control of damage in products with the aim of forecasting their operation life and reliability, both in normal and extreme operating modes. It is known that the stages of plastic deformation of metals are mainly determined by the development of the dislocation structure, which does not take into account the damage accumulation (Kobayashi (2010)). As the plastic deformation increases, the processes associated with damage accumulation in the material (e.g., pore nucleation and Abstract The stages of plastic flow in Fe-18 wt.% Cr-10 wt.% Ni structural steel were studied by performing non-destructive testing. In particular, the acoustic parameters of the steel were measured via uniaxial tensile tests. According to the changes in the velocity of propagation and attenuation of ultrasound in a wide temperature range, the stages of plastic deformation and destruction were clearly distinguished. Based on the results of this study, a non-destructive testing parameter characterizing the transition to the pre-destruction stage of the material was proposed. © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers Keywords: mecanical testing; non-destructive testing; stainless steels; ultrasonic Rayleigh waves; plastic deformation 1. Introduction Increasing requirements for the quality of parts imply the expansion of diagnostic and non-destructive testing methods. In that regard, the mechanical properties of the material are decisive in assessing the quality of products as well as the probability of failure along with safety and expediency of using parts under certain conditions. In this respect, relevant research is devoted to the diagnosis and control of damage in products with the aim of forecasting their operation life and reliability, both in normal and extreme operating modes. It is known that the stages of plastic deformation of metals are mainly determined by the development of the dislocation structure, which does not take into account the damage accumulation (Kobayashi (2010)). As the plastic deformation increases, the processes associated with damage accumulation in the material (e.g., pore nucleation and

* Corresponding author. Tel.: +7 (3822) 28-68-02 E-mail address: bsa@ispms.ru * Corresponding author. Tel.: +7 (3822) 28-68-02 E-mail address: bsa@ispms.ru

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers 2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers 10.1016/j.prostr.2023.10.019

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microcracks) shift toward the accumulation of defects until the limit values (Pelleg (2013)). The damage evolution is also a multi-stage process, depending on both the initial structural state and its further changes. Under the action of the applied stress, plasticity and failure occur almost simultaneously and proceed in an interconnected manner. At the onset of deformation, plastic strain forms a critical structure, determining the location and mechanism of the origin of microcracks. At the final stage, the emergence of a macrocrack is the leading process of plastic deformation. In turn, the intermediate stages in plastic deformation are determined by the evolution of microcracks. Technical diagnostics and assessment of accumulated plastic deformation are based on the well-known acoustic methods that consist in measuring elastic constants or elastic moduli along with elastic wave propagation velocities by pulsed and resonant tools, as well as determining relative values of the velocities of one longitudinal and two transverse waves by electromagnetic acoustic techniques (EMA, acoustic strain gauge method, acoustic emission testing, etc.). All these approaches allow one to control the state of the material, product or the whole structure (Badidi Bouda et al. (2000), Kumar et al. (2003), Torello et al. (2015), Murav’ev et al. (2017), Gorkunov et al. (2021)). In particular, the ultrasound velocity measurement under the direct impact of external and internal loads on the material or structure is one of such promising methods. The effectiveness of this technique follows from the fact that the acoustic waves "reflect" the structure of the matter under consideration, and their parameters change when certain defects (e.g., dislocations or interfaces) occur in the material. In addition, stresses of the I-th (macro-) and II th (micro-) kinds arising in the structures alter the velocity of ultrasound propagation depending on the applied loads. One of the most popular topics in the field of ultrasound research is the investigation of changes in the structure of materials as a result of plastic deformation, fatigue loading, and heat treatment. Numerous works have been dedicated to the influence of heat treatment or alloying elements on the propagation speed of ultrasound as well as its attenuation and dispersion (Hakan et al (2005), Hsu et al. (2004)). However, the relationship between the ultrasound velocity measured during loading and the acting deformations and stresses is still not sufficiently justified by physical arguments. This is because the issues that would definitely show how the change in the metal structure is related to the variation in the acoustic parameters of materials are barely worked out in this direction. Most studies are limited to measuring physical quantities proportional to the load, i.e., only in the elasticity range, whereas the elastic stress state of the material remains beyond the consideration (Palanichamy et al. (1995), Murthy et al. (2009)). In view of the above, monitoring the patterns of changes in the velocity of ultrasonic waves during mechanical tests with the aim of developing methods pre-determining the mechanical properties of a material and its structural evolution in the course of operation or mechanical testing is of decisive importance (Vasudevan et al. ( 2002)). In this work, the stages of plastic deformation and failure of a structural steel are investigated via non-destructive control methods over a wide temperature range. Special attention is paid to establishing the influence of temperature and load on the acoustic characteristics of the steel.

Nomenclature V

speed of ultrasonic Rayleigh waves

the attenuation coefficient

α

D V

damage parameter

strain stress



σ

2. Materials and methods Austenitic stainless steels are widely used in the chemical and petroleum industry, food engineering, and medical equipment. In this study, the experiments were performed on a polycrystalline Fe-18 wt.% Cr -10 wt.% Ni alloy with a grain size of ~ 12.5  m. Samples with dimensions of 40  5  2 mm were tensile on an “ Instron-1185: testing machine with a speed of 3.3  10 -4 s -1 at temperatures of 318, 297, 270, 254, 227, 211, and 180 K. The test temperature in the working chamber with the sample was set by the purge rate of nitrogen vapor from the Dewar

Svetlana Barannikova et al. / Procedia Structural Integrity 50 (2023) 33–39 S. Barannikova, M. Nadezhkin, P. Iskhakova / Structural Integrity Procedia 00 (2023) 000 – 000

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vessel and was controlled by a chromel-alumel thermocouple brought in contact with the sample. The stress-strain diagrams of Fe-Ni-Cr alloy samples covered the ranges of elastic and plastic deformations and failure. The mechanical and acoustic properties of the steel are given in Table 1. According to the data, a decrease in temperature led to an increase in the speed of ultrasound. Simultaneously with the recording of loading curves, the speed of ultrasonic Rayleigh waves V and the attenuation coefficient α were measured by means of a separate and combined sensing unit that consisted of emitting and receiving piezoelectric converters based on CTS-19 piezoceramics with a resonant frequency of 5 MHz (Murav'ev et al. (1996), Barannikova et al (2016), Lunev et al. (2018)). Besides, lowering the temperature caused an increase in the strength and speed characteristics of ultrasound. For comparison, the damage parameter was analyzed using the ultrasonic wave propagation speed as follows: D V = 1 – V/V 0 , (1) where V and V 0 are the ultrasound speeds in deformed and initial states, respectively. Table 1. Mechanical and acoustic properties (tensile strength σ B , yield strength σ 0.2 , relative elongation to failure  , and speed of ultrasound in the nondeformed steel V 0 ) of Fe-18 wt.% Cr -10 wt.% Ni alloy T, K σ B , MPa σ 0.2 , MPa  V 0 , m/s 318 598±3 289±2 0.71±0. 02 2689±3 297 786±2 257±3 0.74±0. 02 2849±3 270 882±2 211±3 0.5±0. 01 3003±3 254 896±3 220±2 0. 48± 0.01 3033±3 227 954±2 289±3 0. 43± 0.02 3143±3 211 992±3 353±2 0. 38± 0.02 3277±3 180 1090±2 362±3 0. 35± 0.01 3379±3 As a result of plastic deformation, structural changes occur in the temperature range under consideration, which are accompanied by variations of the martensitic α '-phase formed via the γ - α ' -phase transformation (Talonen et al (2005)). Simultaneously with the measurements of the velocity of ultrasonic waves, a magneto-phase analysis of samples, consisting in determination of the volume fraction of ferrite, was carried out using a multifunctional MVP 2M eddy current device (ferritometer). 3. Results and discussions Synchronous recording of strain- stress diagrams and measurements of velocity V and attenuation α of Rayleigh acoustic waves make it possible to obtain dependences of the propagation velocity of ultrasound on the magnitude of the total deformation  and the effective stress σ . Since the test temperature exerts a significant effect on the plasticity and strength, the data are further presented in dimensionless coordinates  /  f (  f refers to deformation to failure) and σ/σ B ( σ B is the ultimate strength). The measured velocity V of ultrasonic Rayleigh waves, attenuation coefficient α and volume fraction of martensite f α′ in accordance with the stain-stress diagram σ(  ) of Fe-18% Cr -10 % Ni alloy at 211 K are shown in Fig. 1, where the relative deformation  /  f is indicated along the horizontal axis. In these dependences, four stages (referred to as I, II, III, and IV) of changes in acoustic parameters can be distinguished throughout the investigated temperature range. The duration of the stages varies with a decrease in the test temperature. The first stage I reflecting the change in acoustic parameters for stainless steel is probably related to the elastoplastic transition of the sample material. This is evidenced by an increase in the propagation velocity V and a decrease in the attenuation coefficient α of ultrasonic waves (Fig. 1). At the yield point (stage II), the velocity V reaches its maximum value and then remains almost constant against the decrease in α . At the hardening stage III, there is the maximum growth rate of the martensitic α' -phase, which is due to the deformation-induced γ - α' -phase transformation, and this process is accompanied by a drastic decrease in the velocity of ultrasound propagation V

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with increasing attenuation coefficient α. At the last (pre-failure) stage IV, the growth rate of the martensitic α' phase reaches saturation, the velocity V decreases to a minimum value at the ultimate strength, and the attenuation coefficient α rises owing to the accumulation of defects and the development of cracks. The decline in the deforming force at the end of the stretching process, associated with the formation of a macroscopic neck, corresponds to a slight increase in V . An increase in the propagation velocity within this deformation region indicates the pre-critical state of the deformable material. Figure 2 displays the damage parameter D v , determined from the measured velocities of ultrasonic waves V in the deformed and initial state, as a function of relative deformation  /  f of Fe-18% Cr - 10% Ni alloy at different temperatures. In turn, Figure 3 depicts the D v (  /  f ) and D v (σ/σ B ) curves of the alloy at a test temperature of 180 K, revealing the differences in the relationships between the ultrasound velocity and strain/stress in the relevant plastic deformation regions.

Fig. 1. Strain-stress diagram σ(  ) and corresponding changes of martensite volume fraction f α′ , velocity V , and attenuation α of Rayleigh acoustic waves vs. relative deformation  /  f at T = 211 K

With an increase in the relative deformation  /  f , the values of D v in the entire temperature range increase according to the sigmoidal law. Herein, fluctuations in the room-temperature D v (  /  f ) dependence in the interval of 0.4-0.9 (Fig. 2, Curve 1) are associated with the Portevin – Le Chatelier effect, accompanied by the movement of single localized deformation fronts (Barannikova et al. (2004). Meanwhile, with a further decrease in temperature, this effect was not observed under the specified loading rate conditions.

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Fig. 2. Damage parameter D v as a function of relative deformation  /  f of Fe-18 % Cr -10 % Ni alloy at different temperatures: 1 – 297 K, 2 – 270 K, 3 – 254 K; 4 - 227 K; 5 – 211 K, 6 – 180 K

The detected change in the ultrasound velocity with an increase in the overall deformation correlates with the results earlier obtained on various steels as well as titanium and aluminum alloys exposed to loaded at room temperature (Barannikova et al (2016), Lunev et al. (2018)). The attenuation of ultrasound is influenced by various microstructural components during phase transformations. Variations in the attenuation of ultrasound were observed during heating and cooling of steels (Papadakis (1970)), which were associated with changing grain size of ferrite and austenite in the temperature range of phase transformation (between Ac 1 and Ac 3 ) (Palanichamy et al . (1995) ). In particular, it has been shown that the martensitic structure is the most attenuated microstructural phase, whereas the ferrite-pearlite structure is the least attenuated (Hsu et al. (2004)). A similar trend in the kinetics of phase transformation was found in the attenuation coefficient of ultrasound, which increased in order of martensite, bainite, perlite, and related phases (ferrite and cementite) (Vasudevan et al. (2002)). Meanwhile, these studies were performed on the nondeformed samples.

Fig. 3. (1) Stress as a function of relative deformation  /  f ; (2) damage parameter D v vs. relative stress ( σ / σ B ); (3) damage parameter D v vs. relative deformation  /  f of Fe-18 % Cr -10 % Ni alloy at T = 180 K

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According to the results of this work, the general regularity of the stepwise change in the velocity V of ultrasonic Rayleigh waves with a decrease in temperature consists in the increase at the first stage due to the elastoplastic transition, followed by the decrease at stages III and IV. In particular, the most significant peculiarity is a drop in the velocity of propagation V at stage III, which can serve as a diagnostic sign of the material failure. A noticeable increase in the attenuation coefficient α in stainless steel during the transition from stage III to stage IV occurs owing to the release of the martensitic phase (Talonen et al (2005)). In that regard, the transition from one stage to another can be detected by means of the damage parameter D v , calculated from the measured ultrasonic wave velocity. The course of the dependences D v (  /  f ) and D v (σ/σ B ) is probably related to the formation of structural fragments of different sizes and various levels of internal stresses in the sample volume due to martensitic transformation. The same changes in the deformation mechanism are reflected in the flow curve and the deformation hardening coefficient of the material under consideration (Pelleg (2013)). 4. Summary Ultrasonic measurements have a great potential for studying the kinetics of phase transformations that occur in real time in steels under loading at low temperatures. Both the speed and attenuation of ultrasound are strongly influenced by the test temperature, the degree of deformation, and the microstructure of the phase components of the steel. While the ultrasound velocity is mainly determined by the changes in elastic moduli associated with the degree of distortion of the crystal lattice and the texture of austenitic grains, the attenuation of ultrasound is associated with the size of structural elements and the volume fraction of phases in metastable austenitic stainless steel. Based on the behavior of acoustic characteristics, the stages of plastic flow and damage accumulation before the failure were distinguished in Fe-18% Cr - 10% Ni alloy. A non-destructive testing parameter was proposed for characterization of the transition to a pre- failure stage. Meanwhile, elucidating the specific mechanisms that affect the speed and attenuation of sound in deformable samples at low temperatures requires further research. Acknowledgements This work was supported by the Russian Science Foundation (grant no. 22-29-01608) . References Badidi Bouda, A., Benchaala, A., Alem, K., 2000. Ultrasonic characterization of materials hardness. Ultrasonics 38, 224 – 227. Barannikova S.A., Bochkareva A.V., Lunev A.G., Shlyakhova G.V., Zuev L.B., 2016. Changes in ultrasound velocity in the plastic deformation of high-chromium steel. Steel in Translation 46, 552-557. Barannikova, S.A., Danilov, V.I., Zuev, L.B., 2004. Plastic strain localization in Fe-3%Si single crystals and polycrystals under tension. Technical Physics 49, 1296 – 1300. Gorkunov, E.S., Povolotskaya, A.M., Zadvorkin, S.M., Putilova, E.A., Mushnikov A.N. The Effect of Cyclic Preloading on the Magnetic Behavior of the Hot-rolled 08G2B Steel under Elastic Uniaxial Tension. 2021. Research in Nondestructive Evaluation 32, 276 – 294. Hakan Gür C., Orkun Tuncer, B. 2005. Characterization of microstructural phases of steels by sound velocity measurement. Materials Characterization 55, 160 – 166. Hsu C.-H., Teng, Chen, Y.-J. 2004. Relationship between ultrasonic characteristics and mechanical properties of tempered martensitic stainless steel. Journal of Materials Engineering and Performance 13, 593 – 598. Kobayashi M., 2010. Analysis of deformation localization based on the proposed theory of ultrasonic wave velocity propagation in plastically deformed solids. International Journal of Plasticity 26, 107-125. Kumar, A., Jayakumar, T., Raj, B., Ray, K. K. 2003. Characterization of solutionizing behavior in VT14 titanium alloy using ultrasonic velocity and attenuation measurements. Materials Science and Engineering A 360, 58 – 64. Lunev A.G., Nadezhkin M.V., Barannikova S.A., Zuev L.B., 2018. Acoustic Parameters as Criteria of Localized Deformation in Aluminum Alloys. Acta Physica Polonica A 134, 342-345. Murav'ev V.V., Zuev L.B., Komarov K.L., 1996. Sound Velocity and Structure of Steels and Alloys. Nauka, Novosibirsk, pp. 181. Murav’ev, V.V., Baiteryakov, A.V., Len’kov, S.V., Zakharov, V.A. Correlation of rail structure with the Rayleigh -wave velocity and the coercive force. 2017. Steel in Translation 47, 561 – 563. Murthy, G.V.S., G. Sridhar, G., Kumar, A., Jayakum, T. 2009. Characterization of intermetallic precipitates in a Nimonic alloy by ultrasonic velocity measurements. Materials Characterization 60, 234 – 239. Palanichamy, P., Joseph, A., Jayakumar, T. Raj, B. 1995. Ultrasonic velocity measurements for estimation of grain size in austenitic stainless steel. NDT & E International 28, 179 – 185.

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Papadakis, E.P. 1970. Ultrasonic attenuation and velocity in SAE 52100 steel quenched from various temperatures. 1970. Metallurgical and Materials Transactions 1, 1053 – 1057. Pelleg J., 2013. Mechanical Properties of Materials. Springer, Dordrecht, pp. 634. Talonen J., Nenonen P., Pape G., Hanninen H., 2005. Effect of strain rate on the strain- induced γ → α′ -martensite transformation and mechanical properties of austenitic stainless steels. Metallurgical and Materials Transactions A 36, 421 – 32. Torello D., Thiele S., Matlack K. H., Kim J.-Y., Qu J., Jacobs L.J., 2015. Diffraction, attenuation, and source corrections for nonlinear Rayleigh wave ultrasonic measurements. Ultrasonics 56, 417-426. Vasudevan, M., Palanichamy, P. 2002. Characterization of microstructural changes during annealing of cold worked austenitic stainless steel using ultrasonic velocity measurements and correlation with mechanical properties. Journal of Materials Engineering and Performance 11, 169 – 176.

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16th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures (MRDMS 2022) An approach to determining parameters for active vibration control

of piezoelectric-based smart-systems Oshmarin D.A. a *, Iurlova N.A. a , Sevodina N.V. a a Institute of Continuous Media Mechanics UB RAS, 1, Akademika Koroleva Str.,Perm, 614013, Russia a

© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers Abstract. In this paper, based on the numerical solution to the problem of forced steady-state vibrations of piecewise homogeneous electro viscoelastic bodies, an analysis of the mechanical response of a system to the action of a number of parameters characterizing the applied mechanical or/and electrical action was carried out. The object of research was a cantilever rectangular plate with a piezoelectric element located on its surface. Functional relations that allow determining optimal parameters of electrical impact providing minimal value of a mechanical response were obtained for cases of action of two loading factors. A condition was established that allows reaching such a result: the resulting responses in case of action of a single load factor should be equal and loads must be applied in opposite phase. © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers Keywords: forced vibrations; numerical modelling; vibration control; complex dunamic moduli; electroelasticity; piezoelectric element. 1. Introduction Dynamic external influences on mechanical structures cause their vibrations, which may be undesirable or even dangerous, leading to damage, malfunctions, and the appearance of noise. This problem appears in a wide range of structures, such as buildings, bridges, aerospace systems, industrial machines, precision tools, robotic manipulators and others. Therefore, it is natural for researchers to control the dynamic behavior of structures, minimizing undesirable effects on structures. Currently, one of the promising areas of development in high-tech fields of

* Corresponding author. Tel.: +7(342)-237-83-08; fax: +7(342)-237-84-87. E-mail address: oshmarin@icmm.ru

2452-3216 © 2023 Oshmarin D.A., Iurlova N.A., Sevodina N.V. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers 10.1016/j.prostr.2023.10.044

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technology is the creation and application of smart-structures. Their main feature is the ability to react to external influences by changing the characteristics of the original object. There are two main strategies for controlling the mechanical behavior of structures: passive and active. Smart systems that have only sensors in their composition are called passive. The embedding of sensors allows monitoring the condition of the structure. For the manufacture of actively-controlled or adaptive smart structures, actuators are needed that can cause deformation of the main structure. Currently, a large number of functional materials are used as actuators, among which piezoelectric materials occupy one of the central places (Tani J Takagi T., Qiu J., 1998). The widespread use of piezoelectric materials, especially for controlling the mechanical behavior of structures, is explained by the presence of direct and inverse piezoelectric effects, which allows them being used as sensors as well as actuators. The second reason is in a possibility of creating an electrically conductive surface, technologically realized for piezoelectric materials, allows connecting various options of electrical circuits to the smart-structure (Ayres J.W., et al., 1996). There exist fundamentally different approaches to the active control of structural vibrations, which are discussed in detail, for example, in monographs (Fuller C.R., et al., 1997; Preumont A., 2011). Basically, the attention of researchers is focused on the development of the control system which receives a signal generated by the sensor, implementing its change according to a given law and supplying a modified and amplified signal to the actuator. From the other hand, researchers practically do not pay attention to the factors that influence the required response of a system and affect the efficiency of the control law. In some cases, knowing the mechanisms that condition relations between applied impact and resulting response can ease the way of constructing control design and decrease its hardware requirements. This work is devoted to determining the factors that govern mechanical response of a structure with a single piezoelectric element to external mechanical and electrical impacts. 2. Mathematical statement of the problem 1 2 3 V V V V    , in which parts 1 V and 2 V consist of elastic and viscoelastic elements, and part 3 V is the element with piezoelectric properties. All elements are perfectly bonded to each other. A part of the surface 3 el S of 3 V volume is electrode, i.e., covered with negligibly-thin conductive coating. The variational equation of motion of a such body is formulated based on relations of the linear elasticity theory, the linear viscoelasticity theory and quasi-static Maxwell's equations (Iurlova et al., 2019):       1 2 3 3 1 2 3 i i el ij ij i ij ij i V V ij ij i i i i e i i V S S u u dV u u dV DE uudV q dS pudS                               (1) We consider a piecewise-homogeneous body of volume ij  i j  are the components of the symmetric Cauchy stress tensor and the linear strain tensor; i u are the components of the displacement vector; 1 2 3 , ,    are the specific densities for materials of elastic, viscoelastic and piezoelectric parts of V ; S  is the part of the whole body surface, where components of surface load vector i p are applied;  is the electric potential, e q is the surface density of free charges at the electroded part 3 el S of the surface of piezoelectric elements. The elastic elements behavior is governed by Hooke’s law : The following notations are introduced: , i i D E are the vector components of the electric flux density and electric field intensity;

(2)

ij ijkl kl C   

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where 1, 2,3 i j k l  ). The physical relations for viscoelastic part 2 V of the body is written based on the relations of linear hereditary viscoelasticity (Kligman, E.P., Matveenko, V.P., 1997). In the problems of the forced steady-state vibrations this relations are used in the form of complex dynamic moduli: ijkl C is the tensor of elastic constants of material ( , , ,

 B G G iG G i B B iB B i                Re Im Re Im Re 2 , Ge , 1 1 ij ij g b s Re



(3)

,



.

Here , G B are the complex dynamic shear and bulk moduli (in the common case these quantities are the functions of frequency  ); , g b   are the correspondent tangents of the mechanical losses;  is the volumetric strain; , ij ij s e are the components of the deviators of stress and strain tensors. Relations for determining real and imaginary parts of complex moduli are given in (Kligman, E.P., Matveenko, V.P., 1997). For the elements of the piecewise-homogeneous body made of piezoelectric materials governing relations take the following form:

ijkl kl         ijk k ijk ij ki i э E C E  





ij

(4)

D

k

In the coupled problems of the electro-viscoelasticity mechanical and electrical boundary conditions are set in the corresponding form:

:

,

: S u U 

(5)

S

n p 



ij j

i

u

i

i









nel

el

:

0,

:

,

(6)

S

n D dS 

S







3

3

nel

S

3

Here S  and u S are the parts of the surface of the V volume where the stress tensor components

ij  and the

i U are set, j n are the components of the unit normal vector to the S  surface; 3 nel S

displacement vector components is the non-electrode surface, 3 zero-valued electric potential

el S is the electrode surface of the

3 V volume. On a part of electrode surface 3

el S the

0   is set, which means the grounding condition. Thus, the value of electric

potential   set at the rest part of 3 el S surface, determines potential difference. A solution to the forced steady-state vibration problem is sought in the form (8)     0 , e i t u x t u x  

(7)





where ( ) (,,), (,,),(,,),(,,) ux uxxx uxxx uxxx xxx   is the generalized state vector, containing the components of the displacement vector 1 2 3 , , u u u as well as the electric potential  ;  is the external excitation frequency. The numerical realization of the formulated problem is done in the commercial software package ANSYS. Complete mathematical statement of the problem and algorithm of its numerical implementation are described in detail in (Iurlova et al., 2019) 0 1123 2123 3123 1 2 3

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3. An object under study Here we consider a cantilever rectangular plate with dimensions: length l 1 = 500 mm, width b 1 = 60 mm, thickness h 1 = 2 mm (fig. 1). A piezoelectric element having length l p = 50 mm, width b p = 20 mm and thickness h p = 0.3 mm is attached to its surface. The upper and lower surfaces of the piezoelectric element are completely electroded. The position of the piezoelectric element is determined by its center of mass which is located on the plate along the symmetry axis at the distance of 55 mm from the clamped edge. The plate is made of the viscoelastic material which dynamic behavior is described by the complex dynamic shear and the bulk moduli. The real parts of the complex moduli are correspondingly equal to 8 Re 6.71 10 G Pa   and 10 Re 3.33 10 B Pa   . The imaginary parts are chosen as follows: 7 Im 6.71 10 G Pa   and 9 Im 3.33 10 B Pa   . These material properties correspond to ones that have reinforced polyurethane brand Ellastolan. Within the current research we assume that the components of complex moduli do not depend on frequency of vibrations in a frequency range under study. Specific density of the viscoelastic material is 1190 el   kg/m 3 .

Fig. 1. Computational sketch for a cantilever plate with attached piezoelectric element.

The piezoelectric element made of piezoelectric ceramics CTS-19 is polarized along z axis and has the following material properties: 10 11 22 10.9 10 C C    Pa, 10 13 23 5.4 10 C C    Pa, 10 12 6.1 10 C   Pa, 10 33 9.3 10 C   Pa, 10 44 55 66 2.4 10 C C C     Pa, C/m 2 , 33 14.9   C/m 2 , 51 42 10.6     m 2 , 9 11 22 8.2 10 э э     F/m, 9 33 8.4 10 э    F/m,  p = 7500 kg/m 3 . The electrodes of piezoelectric element supplied with an electric signal with prescribed characteristics of potential difference and electric current. 4. Mechanical response of the system to applied loading Let's consider the first three bending modes of vibrations of such a plate. These modes are realized at frequencies of 1.962 Hz, 12.006 Hz and 32.956 Hz correspondingly. We apply the following options of loading conditions: kinematic (components of displacement vector varying according to the harmonic law applied to the clamped end of the plate); force (a concentrated force varying according to the harmonic law is applied to the free end of the plate); potential difference or current applied to the electrode surfaces of the piezoelectric element. In the numerical implementation, both variants of mechanical loading are set using boundary conditions of the form (5), and variants of electrical loading are set using boundary conditions of the form (6). Figure 2 shows in logarithmic coordinates the dependences of the magnitude of the mechanical response of the plate on the magnitude of kinematic loading (Fig.3.a), force loading (Fig.3b), potential difference applied to the piezoelectric element (Fig.3c) and electric current (Fig.3d) for the three vibration modes under consideration. Here the blue line corresponds to the first mode, the red line corresponds to the second one and the green line corresponds to the third mode. A ratio z stat u u is accepted as a quantity that characterizes the mechanical response of the system. Here z u is maximal value of amplitude of component z u of displacement vector at the free end of the plate under forced

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vibrations; stat u is a value of component z u of displacement vector at the free end of the plate under static loading   0,0,1 F  N, applied to its free end.

a

b

c

d

Fig. 2. Relations between values of mechanical response and loading factors for (a) clamp movement, (b) force, (c) potential difference, (d) current

The results shown in Fig.2 correspond to the linear formulation of the problem and are well-known. However, the impact of two loading factors on the system simultaneously opens up the possibility of controlling the mechanical response of the system. In particular, by changing the angle of phase shift α between the operating loads. Since harmonic vibrations of the structure are considered, the greatest effect of the second loading factor is achieved in two cases: 0   (loads act in phase) and 180   (loads act in opposite phase). In the first case, it is assumed to observe an increase in the level of mechanical response, and in the second – a decrease. Due to the linearity of the mathematical formulation used, we expect that the resulting value of the mechanical response will be the sum of the values of the mechanical response from the action of each of the loading factors. Based on this assumption, we believe that the minimum oscillation amplitude will be observed in the case when the value of the control electrical signal cont U or cont I will provide the value of the amplitude     cont cont targ u U u I u   , which is realized under the action of a mechanical load. Thus, to determine the optimal characteristics of the control electrical signal, it is sufficient to know how the amplitude of the forced vibrations u depends on the magnitude of the electrical loading. Having dependencies   cont u U and   cont u I regardless its form (analytical, graphical, tabular, etc.), it is possible to obtain inverse dependencies, i.e. the dependences of the optimal values of the control signal on the magnitude of the mechanical response   cont U u and   cont I u . Using the inverse dependencies   cont U u and   cont I u , it is possible to determine the value of the optimal parameters of the control electrical signal to obtain targ u . Based on the results obtained earlier (Fig.2) the relations between the magnitude of the potential difference and the current and the magnitude of the mechanical response of the plate were determined. The plots of these relations for the three vibration modes under consideration are presented in logarithmic coordinates in Fig.3. The blue color

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in Fig.3 refers to the first mode of vibrations, red – to the second one, green – to the third mode, correspondingly. Due to the linearity of the chosen mathematical model, the resulting dependencies are also linear.

a

b

Fig. 3. Dependencies of optimal values of control electric impact on a magnitude of mechanical response foe the three vibration modes under study: (a) potential difference, (b) electric current

  cont U u and

  cont I u using the equation

Based on the plots from fig.3 we obtained explicit functional relations

  cont U u and

  cont I u take

of a straight line passing through two points. After some ease transforms the relations

the following form:

 



  u u I   1







2 u u U U    cont 1

1

2

1

I



cont

cont

cont

1 U I cont ;

1

(8)

U

I









cont

cont

cont

u u 

u u 

2 1

2 1

The data for obtaining functional relations (8) were chosen in such a way that we could have a possibility of using one and the same values of 1 u and 2 u for deriving the relations   cont U u and   cont I u . Thus, we took values     cont 1 cont 2 , U u U u and     cont 1 cont 2 , I u I u directly from the plots from fig.3. As a result, we obtain functions describing the relationship between the required mechanical response and the optimal parameters of the electrical signal for the three vibration modes under consideration, represented in Table 1. Thus, having the obtained dependences, we can calculate the value of the potential difference or current amperage required to achieve a given level of amplitude of forced vibrations of an electro-viscoelastic system.

  cont U u and

  cont I u for the three vibration modes under study.

Table 1. Dependencies

  cont I u

  cont U u

№ of mode









4

3 3 10 37.040   3 10 22.387   3 10 6.466     4 5

3

3 3 10 1.968 10    3 10 7.267 10    4

5

1 2 4

cont 1.232 10  

6.543 10 2.431 10





U U U

u            u u

I I

u       u 

   

cont



4

1

5

7.444 10 2.084 10





   

cont

cont



 5

5

5 cont 1.931 3 10 5.667 10 I u        

cont

The possibilities of controlling the amplitude of forced vibrations will be demonstrated by the example of the considered cantilever plate with a piezoelectric element located on its surface. Let's consider two variants of the force loading of this system, presented in Fig. 4:   ,0,0, x F F  , where 3 5 10 x F    N (fig.4a) and   0, ,0, y F F  , where 3 1 10 y F    N (fig.4b). Based on the solution to the problem of forced steady-state vibrations for the loading schemes represented at fig.4 we obtained frequency response plots (FRP) of the mechanical response / z stat u u u  for the first vibration mode.

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a

b

Fig. 4. Variants of mechanical loading used for demonstrating possibilities of the proposed approach

Table 2. Options of combination of loading factors.   ,0,0, x F F  ; var1 U   ,0,0, x F F  ; var1 I

  0, ,0, y F F  ; var2 U

  0, ,0, y F F  ; var2 U

3 1 10 

3 1 10 

Н ;

Н ;

y F

y F

 

 

Н ;

Н ;

3 5 10 

3 5 10 

x F

x F

 

 

I

var1 0 I 

var1 0 U 

0

0

U

I





var2

var2

3 1 10 

3 1 10 

Н ;

Н ;

y F

y F

 

 

Н ;

Н ;

3 5 10 

3 5 10 

x F

x F

 

 

II

1

7

opt U

var1 1.007 10 opt I   

var1 1.895 10  



В

А

2

8

opt U

var2 1.618 10 opt I   

3.046 10



В

А

 

var2

3 1 10 

3 1 10 

Н ;

Н ;

y F

y F

 

 

Н ;

Н ;

3 5 10 

3 5 10 

x F

x F

 

 

III

1 var1 1 10  

1

1 var1 0.5 10 no I   

7

no U



В

А

1

2

1 var2 1 10 no I   

8

no U

2 10



В

А

 

var2

3 1 10 

3 1 10 

Н ;

Н ;

y F

y F

 

 

Н ;

Н ;

3 5 10 

3 5 10 

x F

x F

 

 

IV

2 var1 3 10  

1

2 var1 1.5 10 no I   

7

no U



В

А

2

2

2

8

no U

no I

4 10

2 10





В

А

 

 

var2

var2

a

b

c

d

Fig. 5. FPRs of mechanical response of the system for the following loading cases: (a) force 

 ,0,0, x F F  and voltage var1 U ; (b) force   0, ,0, y F F  and amperage var2 I .

  0, ,0, y F F  and voltage var2 U ; (d) force

  ,0,0, x F F  and amperage var1 I ; (c) force

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5 var1 1.531 10 u   

According to the obtained results, maximal values of mechanical response are equal to

and

6 for the schemes represented at fig.4a and fig 4b correspondingly. Based on the relations represented in table 2 optimal values of potential difference and current amperage were calculated as 1 var1 1.895 10 opt U    V; 7 var1 1.007 10 opt I    A; 2 var 2 3.046 10 opt U    V; 8 var 2 1.618 10 opt I    A for the corresponding values of the mechanical response. Next for these values of electric loading we obtained FRPs for cases when two loading factors (force and electric load) act simultaneous. In order to demonstrate optimality of the calculated values opt U and opt I two additional series of calculations were performed. For these calculations the following (non optimal) values of voltage and amperage were chosen 1 1 var1 1 10 no U    V; 1 7 var1 0.5 10 no I    A; 1 2 var 2 2 10 no U    V; 1 8 var 2 1 10 no I    A; 2 1 var1 3 10 no U    V; 2 7 var1 1.5 10 no I    A; 2 2 var 2 4 10 no U    V; 2 8 var 2 2 10 no I    A. All options of combination of loading factors represented in table 2. Figure 5 shows the FRPS for all variants of combinations of loading factors presented in Table 3. The black dashed line in Figure 5 indicates the frequency response for the I variant of combinations of load values, the blue solid line for the II variant, the red dashed line for the III variant, the green dashed line for the IV variant. The results shown in Fig.5 confirm the fact that the values of the optimal parameters obtained on the basis of the ratios given in Table 2 are indeed optimal, since they provide the minimum amplitude of forced steady-state vibrations among all the considered combinations of loading factors. Thus, the proposed method really makes it possible to determine the optimal values of the parameters of the control electrical impact, at which the minimum amplitude of the forced steady-state vibrations of the system is achieved. 5. Conclusions In current research, on the basis of solving the problem of forced steady-state vibrations of piecewise homogeneous electro-viscoelastic bodies, the influence of the magnitude of the applied mechanical or electrical impact, as well as their combination on the mechanical response of a structure, was investigated. Numerical results obtained for the example of a cantilever plate with a piezoelectric element attached to its surface, shown that the magnitude of its mechanical response can be controlled quite effectively only with the combined action of mechanical and electrical loads. It is established that the resulting mechanical response of the structure in case of simultaneous action of two loading factors is an algebraic sum of mechanical responses to the action of each of the factors separately. An approach has been proposed for determining the parameters of the control electrical impact, which allows achieving a minimum level of mechanical response of a plate under forced steady-state vibrations in case of excitation by a mechanical load. The results obtained in the framework of this study can be the basis for the development of algorithms for solving problems of controlling the dynamic behavior of electro-viscoelastic structures. Acknowledgements The reported study was funded by RFBR and Perm region accordin g to the project № 19 -41-590007-r_a. References Iurlova N.A., Oshmarin D.A., Sevodina N.V., Iurlov M.A., 2019. Algorithm for solving problems related to the natural vibrations of electro viscoelastic structures with shunt circuits using ANSYS data. International Journal of Smart and Nano Materials, 10 (2), 156-176 Kligman, E.P., Matveenko, V.P., 1997. Natural Vibration Problem of Viscoelastic Solids as Applied to Optimization of Dissipative properties of Constructions. International Journal of Vibration and Control, 3(1), 87-102. Tani J., Takagi T., Qiu J., 1998. Intelligent Material Systems: Application of Functional Materials. Applied Mechanics Reviews, 51(8), 505-521 Ayres J.W., Lalande F., Rogers C.A., Chaudhry Z., 1996. Qualitative health monitoring of a steel bridge joint via piezoelectric actuator/sensor patches. SPIE Nondestructive Evaluation Techniques for Aging Infrastructure & Manufacturing, AZ 8р. Fuller C.R., Elliot S.J., Nelson P.A., 1997. Active Control of Vibration, Academic Press, London Preumont A., 2011. Vibration Control of Active Structures: An Introduction (3rd ed.), Springer-Verlag, Berlin. var 2 2.466 10 u   