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متن کامل


اطلاعات دوره: 
  • سال: 

    1402
  • دوره: 

    3
  • شماره: 

    1
  • صفحات: 

    21-52
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    46
  • دانلود: 

    8
چکیده: 

در مقاله حاضر، کمانش نانو ورق های تابعی مدرج مستطیلی با در نظر گرفتن اثر سطح بررسی شده است. به منظور تعریف خواص مواد از مدل موری-تاناکا بهره برده شده است که بر اساس این مدل خواص به طور پیوسته در راستای ضخامت تغییر می کند. میدان جابجایی با استفاده از تئوری های تغییر شکل برشی اصلاح شده بدست آمده اند که در این تئوری ها بر خلاف تئوری کلاسیک ورق ها اثر اینرسی دورانی و تغییر شکل های برشی عرضی در نظر گرفته شده است. جهت در نظر گرفتن اثرات مذکور از توابع مختلف مانند توابع نمایی، مثلثاتی، هایپربولیکی و پارابولیکی در راستای ضخامت بهره گرفته شده است. برای در نظر گرفتن اثرات غیر محلی و اثرات سطح به ترتیب از تئوری های الاستیسیته ی غیر محلی و تئوری الاستیسیته ی سطح استفاده شده است. معادلات حاکم بر حرکت با استفاده از اصل همیلتون بدست آمده اند و از روش حل گلرکین جهت حل این معادلات استفاده شده است. برای بررسی صحت نتایج به دست آمده، نتایج حاصل از این تحقیق با نتایج منتشر شده در مقالات معتبر مقایسه شده است. درانتها اثر پارامتر های مرتبط با اثر سطوح مانند تنش باقیمانده سطح، ثوابت الاستیسیته ی سطح و همچنین پارامتر هایی مانند نسبت ضخامت به طول، نسبت عرض به طول، اندیس توانی، پارامتر غیر محلی بر روی بار بحرانی کمانش سیستم بررسی می گردد.

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اطلاعات دوره: 
  • سال: 

    1396
  • دوره: 

    4
  • شماره: 

    2
  • صفحات: 

    151-160
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    915
  • دانلود: 

    257
چکیده: 

هدف این مقاله مطالعه ارتعاش آزاد نانو ورق مستطیلی مرکب پیزوالکتریک تحت بار الکترومکانیکی شامل نیروی محوری و ولتاژ خارجی بر اساس تئوری های تغییر شکل برشی اصلاح شده نمایی و مثلثاتی به همراه تئوری الاستیسیته غیرمحلی و شرایط مرزی چهار طرف تکیه گاه ساده است. در تئوری غیرمحلی تنش در هر نقطه تابعی از کرنش در تمامی نقاط محیط می باشد. این تئوری اهمیت بسزایی در ساختارهای با ابعاد میکرو و نانو دارد. درتئوری تغییرشکل برشی اصلاح شده نمایی و تئوری تغییر شکل برشی اصلاح شده مثلثاتی، توابع نمایی و مثلثاتی در راستای ضخامت، جهت در نظر گرفتن تاثیرات تغییر شکل برشی عرضی و اینرسی دورانی بکار رفته است. از تئوری الاستیسیته غیرمحلی جهت در نظر گرفتن پارامتر مقیاس طول کوچک نانو ورق مستطیلی مرکب پیزوالکتریک استفاده شده است. نانو ورق مرکب پیزوالکتریک مورد مطالعه در این تحلیل ورق مستطیلی ساخته شده از ماده مرکب پیزوالکتریک PZT 4 متشکل از ترکیبات بلوری سرب، زیرکونیم و تیتانیم جهت دست یابی به خواص فلز- سرامیک و همچنین خاصیت پیزوالکتریک می باشد. با بکارگیری اصل همیلتون معادلات دیفرانسیل حاکم بر ارتعاش نانو ورق مستطیلی مرکب پیزوالکتریک به دست آمده و برای به دست آوردن پاسخ فرکانسی از حل دقیق ناویر استفاده شده است. در نهایت تاثیر پارامتر مقیاس نانو، نیروهای محوری، ولتاژ خارجی و نسبت های هندسی روی فرکانس های طبیعی بی بعد شش مد ارتعاشی اول نانو ورق مستطیلی مرکب پیزوالکتریک مورد بحث و بررسی قرار گرفته است.

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بازدید 915

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اطلاعات دوره: 
  • سال: 

    2022
  • دوره: 

    15
  • شماره: 

    1
  • صفحات: 

    21-28
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    82
  • دانلود: 

    0
چکیده: 

In this paper, a Mindlin Rectangular nanoplate model is developed for the bending and vibration analysis of a graphene nanoplate based on a modified couple stress theory. In order to consider the small scale effects, the modified couple stress theory, with one length scale parameter, is used. In modified couple stress theory, strain energy density is a function of strain tensor, curvature tensor, stress tensor and symmetric part of couple stress tensor. After obtaining the strain and kinetic energy, external work and substituting them in the Hamilton’ s principle, the main and auxiliary equations of the nanoplate are obtained. Then, by manipulating the boundary conditions the governing equations are solved using Navier approach for bending and vibration of the nanoplate. The bending rates and dimensionless bending values under uniform surface traction and sinusoidal load and different mode frequencies are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The effect of material length scale, length, width and thickness of the nanoplate on the bending and vibration ratios are investigated and the results are presented and discussed in details.

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بازدید 82

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مرکز اطلاعات علمی Scientific Information Database (SID) - Trusted Source for Research and Academic Resources
اطلاعات دوره: 
  • سال: 

    1396
  • دوره: 

    4
  • شماره: 

    1
  • صفحات: 

    109-119
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    1107
  • دانلود: 

    215
چکیده: 

لطفا برای مشاهده چکیده به متن کامل (PDF) مراجعه فرمایید.

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بازدید 1107

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اطلاعات دوره: 
  • سال: 

    2014
  • دوره: 

    6
  • شماره: 

    1
  • صفحات: 

    98-121
تعامل: 
  • استنادات: 

    1
  • بازدید: 

    583
  • دانلود: 

    0
چکیده: 

In the present work, the free vibration behavior of Rectangular graphene sheet under shear in-plane load is studied. Nonlocal elasticity theory has been implemented to study the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) subjected to shear in-plane load. The SLGSs is embedded on a viscoelastic medium which is simulated as a Visco-Pasternak foundation. Using the principle of virtual work, the governing equations are derived for the Rectangular nanoplates. Differential quadrature method (DQM) is employed and numerical solutions for the vibration frequency are obtained. The influence of surrounding elastic medium, material property, aspect ratio, nonlocal parameter, length of nanoplate and effect of boundary conditions on the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) is studied. Six boundary conditions are investigated. Numerical results show that the vibration frequencies of SLGSs are strongly dependent on the small scale coefficient and shear in-plane load. The present analysis results can be used for the design of the next generation of nanodevices that make use of the vibration properties of the graphene.

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بازدید 583

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اطلاعات دوره: 
  • سال: 

    2018
  • دوره: 

    10
  • شماره: 

    2
  • صفحات: 

    399-414
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    290
  • دانلود: 

    0
چکیده: 

In this paper, axisymmetric free vibration analysis of a circular Nano-Plate having variable thickness was studied. The variation in thickness of plate was considered as a linearly in radial direction. Nonlocal elasticity theory was utilized to take into account size-dependent effects. Ritz functions was utilized to obtain the frequency equations for simply supported and clamped boundary. To verify accuracy of Ritz method, differential transform method (DTM) also used to drive the size dependent natural frequencies of circular Nano-Plates. The validity of solutions was performed by comparing present results with those of the literature for both classical plate and nano plate. Effect of nonlocal parameter, mode number and taper parameter on the natural frequency are investigated. Results showed that taper parameter has significant effect on the non-dimensional frequency and its effects on the clamped boundary condition is more than simply support.

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بازدید 290

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مرکز اطلاعات علمی Scientific Information Database (SID) - Trusted Source for Research and Academic Resources
اطلاعات دوره: 
  • سال: 

    1403
  • دوره: 

    13
  • شماره: 

    25
  • صفحات: 

    3-32
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    15
  • دانلود: 

    0
چکیده: 

In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating Rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of generalized differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

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بازدید 15

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نویسندگان: 

GHADIRI M. | SHAFIEI N. | ALAVI S.HOSSEIN

اطلاعات دوره: 
  • سال: 

    2017
  • دوره: 

    9
  • شماره: 

    2
  • صفحات: 

    319-337
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    342
  • دانلود: 

    0
چکیده: 

The nanostructures under rotation have high promising future to be used in nano-machines, nano-motors and nano-turbines. They are also one of the topics of interests and it is new in designing of rotating nano-systems. In this paper, the scale-dependent vibration analysis of a nanoplate with consideration of the axial force due to the rotation has been investigated. The governing equation and boundary conditions are derived using the Hamilton’s principle based on nonlocal elasticity theory. The boundary conditions of the nanoplate are considered as free-free in y direction and two clamped-free (cantilever plate) and clamped-simply (propped cantilever) in x direction. The equations have been solved using differential quadrature method to determine natural frequencies of the rotating nanoplate. For validation, in special cases, it has been shown that the obtained results coincide with literatures. The effects of the nonlocal parameter, aspect ratio, hub radius, angular velocity and different boundary conditions on the first three frequencies have been investigated. Results show that vibration behavior of the rotating nanoplate with cantilever boundary condition is different from other boundary conditions.

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بازدید 342

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نویسندگان: 

FAROUGHI SHIRKO | GOUSHEGIR SEYED MOHAMMAD HOSSEIN

اطلاعات دوره: 
  • سال: 

    2016
  • دوره: 

    2
  • شماره: 

    1
  • صفحات: 

    1-20
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    170
  • دانلود: 

    0
چکیده: 

In this paper, the Ritz method has been employed to analyze the free inplane vibration of heterogeneous (non-uniform) Rectangular nanoplates corresponding to Eringen’ s nonlocal elasticity theory. The non-uniformity is taken into account using combinations of linear and quadratic forms in the thickness, material density and Young’ s modulus. Two-dimensional boundary characteristic orthogonal polynomials are applied in the Ritz method in order to examine the nonlocal effect, aspect ratio, length of nanoplate and non-uniformity parameters on the vibrational behaviors of the nanoplate. Results are verified with the available published data and good agreements are observed. The outcomes confirm apparent dependency of inplane frequency of nanoplate on the small scale effect, non-uniformity, aspect ratio and boundary conditions. For instance, frequency parameter decreases by increasing the nonlocal parameter in all vibration modes; the frequency parameters increase with length and aspect ratio of nanoplates. Furthermore, the effect of nonlocal parameters on the frequency parameter is more prominent at the higher aspect ratios. Finally, the effect of nonlocal parameter on the in-plane modes is also presented in this analysis.

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اطلاعات دوره: 
  • سال: 

    2021
  • دوره: 

    14
  • شماره: 

    2
  • صفحات: 

    85-92
تعامل: 
  • استنادات: 

    0
  • بازدید: 

    287
  • دانلود: 

    0
چکیده: 

In this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nano In this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff nanoIn this study, the characteristics of Rectangular Kirchhoff Nano-Plate plate plate vibrations are investigated using a modified couple stress theory. 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To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the vibrations are investigated using a modified couple stress theory. To consider the effects of smalleffects of smalleffects of smalleffects of smalleffects of smalleffects of smalleffects of small effects of smalleffects of smalleffects of smalleffects of smalleffects of smalleffects of smalleffects of small-scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) scale, the modified couple stress theory proposed by Young (2002) is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, is used as it has only one length scale parameter. In modified couple stress theory, the s the sthe sthe strain energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, train energy density is a function of the components strain tensor, curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of Nano-Plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof Nano-Plate with plate with plate with plate with plate with plate with plate with the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method used in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier method used in this study is the Navier method used in this study is the Navier method used in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier method used in this study is the Navier method used in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier method andandand the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated results are presented and discussed in details results are presented and discussed in details results are presented and discussed in details results are presented and discussed in details results are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in details.

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