The modified version of strain gradient and couple stress theories in general curvilinear coordinates

KEYWORDSMicro-ring; Modi ed couple stress theory; Resonant frequency.Abstract. In this paper, based on the modi ed couple stress theory, the size-dependent dynamic behavior of circular rings on elastic foundation is investigated. The ring is modeled by Euler-Bernoulli and Timoshenko beam theories, and Hamilton's principle is utilized to derive the equations of motion and boundary conditions. The formulation derived is a general form of the equation of motion of circular rings and can be reduced to the classical form by eliminating the size-dependent terms. On this basis, the size-dependent natural frequencies of a circular ring are calculated based on the non-classical Euler-Bernoulli and Timoshenko beam theories. The ndings are compared with classical beam theories. Response of the micro-ring under application of static and dynamic loads is investigated and compared with the classical theories. Results show that when the thickness of the ring is in the order of the length scale of the ring material, the natural frequencies evaluated using the modi ed couple stress are considerably more than those predicted based on the classical beam theories, while the de ection and natural frequencies of the classical and non-classical beam theories approach one another for the rings with thickness much larger than the material length scale.

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“…Moreover, the components of the symmetric curvature tensor can be expressed in terms of the displacement components as [37]: …”

Section: Euler-bernoulli Ringsmentioning

confidence: 99%

Effect of size dependency on in-plane vibration of circular micro-rings

2017

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“…where A i are the vector contravariant components, B jk -the second rank tensor contravariant components, C k j -the components of second rank mixed tensor, D ijk -the contravariant components of third rank tensor and E k ji , F jk i are the components of the third rank mixed tensors. Γ jk i are the known Christoffel symbols of the second kind satisfying Γ Γ = jk i kj i and defined in orthogonal curvilinear coordinates as follows 28 :…”

Section: The Rmm Model In Orthogonal Curvilinear Coordinatesmentioning

confidence: 99%

“…Frobenius series is employed to get the analytical solution for the system of ordinary differential equations in (27) with the boundary conditions in Eq. (28). Since the point (0, 0, 0) is a regular point, one can express the functions u r and s rr in terms of the variable r as follows: Equation (31) is satisfied if we choose…”

Section: Half-space Involving Spherical Symmetrymentioning

confidence: 99%

“…(27) may be found in series form as in (30). Applying the boundary conditions (28), we find that the first boundary condition is automatically satisfied while the second one gives numerical Results Table 1 shows the values for the physical parameters 23 for an epoxy matrix and the inclusions materials. The following figures are obtained depending on the numerical values for materials parameters listed in Tables (1) and (2).…”

Section: Half-space Involving Spherical Symmetrymentioning

confidence: 99%

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Reduced micromorphic model in orthogonal curvilinear coordinates and its application to a metamaterial hemisphere

Dhaba

2020

Sci Rep

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The reduced micromorphic model (RMM) is used to study the effect of the applied force on a hemisphere made of phononic crystals that belongs to the metamaterials group. The strain tensor, the micro-strain tensor and the coupling between them are the kinematic relations used to measure the deformation and micro-deformation of the representative volume element of these materials. The free energy function, the constitutive relations, the field equations, and the boundary conditions are presented firstly in the Cartesian coordinate. Then, the orthogonal curvilinear coordinates are introduced as a general coordinate to describe the physical quantities included in the RMM. The spherical coordinates are deduced as a special case from the curvilinear coordinates to study the deformation and microdeformation for the hemisphere. The kinematic relations and the governing equations of the model are considered to changing with the radius of the hemisphere only. The analytical solutions of the field equations are also obtained by using the Frobenius series satisfying the given boundary conditions and consequently the value of the physical constants of the problem is determined. Numerical applications for the obtained results are introduced with discussion. The results showed that the displacement has a greater effect rather than the micro-strain, when it is measured relative to the classical physical quantities while the micro-strain has a greater effect rather than the displacement, when it is measured relative to the nanoscale physical quantities.Metamaterials are fabricated engineering materials with periodic internal structures. These materials have special properties that do not exist in other natural materials. They can be designed to prohibit the propagation of elastic waves in the bandgap frequency range effectively. This property has many potential applications in the vibration and noise reduction areas. Besides this property, there are many applications for metamaterials in the fields of technology, industrial engineering, telecommunication equipment, optical filters, medical devices and mobile communication systems and others.The main idea of constructing metamaterials is attributed to Sir J.C. Bose in 1898. He suggested the idea of the existence of artificial materials by carrying the microwave experiment on twisted structures 1,2 . Winston E. Kock 3,4 , used the optical radio waves properties to develop new type of antennas made of metamaterial. V. G. Veselago 5 , introduced a theoretical model that predicts the propagation of electromagnetic waves in left-handed materials. During the last few decades, more attention was given to metamaterials. This is because of their important applications in recent industries. Theoretical formulation and mechanical applications can be found in refs. [6][7][8][9] .The idea of building metamaterials was extended to include elastic and acoustic waves by Ding et al. 10 , where the authors proposed metamaterials with simultaneously negative bulk modulus and mass density. Wu et al. 1...

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“…Furthermore, R is the ring radius, and r is the distance of an arbitrary point from the neutral axis of the ring cross-section. Using this displacement eld, the non-zero components of the in nitesimal strain tensor, symmetric curvature tensor, and corresponding classical and higher order stresses [34] are calculated as follows:…”

Section: Modeling Of Circular Ringsmentioning

confidence: 99%

Vibrational characteristics of size dependent vibrating ring gyroscope

Karimzadeh

Ahmadian

2018

Scientia Iranica

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Abstract. In this paper, vibrational analysis of a size-dependent micro-ring gyroscope under electrostatic DC voltage is performed. Governing equations of size-dependent micro rings and corresponding nite-element formulation of circular micro ring along with eight half circular sti eners embedded inside the ring are derived based on the modi ed couple stress theory, Hamilton's principle, and in-extensionality approximation. Frequency analysis indicates that the obtained mode shape of the ring gyroscope is slightly di erent from the one previously reported in the literature. Size-dependent behavior of the gyroscope is studied, and related ndings con rm the gap between classic and non-classic natural frequencies and pull-in voltage when the ring thickness is in order of material length scale parameter. Two di erent orientations for the actuation electrodes of the micro-ring gyroscope are implemented, and the e ect of these orientations on the static de ection, pull-in instability, and device frequencies in the sense (45 direction) and drive (0 direction) directions is investigated. Results reveal that the pull-in phenomena take place under lower voltages for 0 and 45 orientation of electrodes in comparison with 0 orientation; frequency split occurs in higher voltages for 0 and 45 orientation. A comparison between nite-element numerical natural frequencies of single ring and previously obtained analytical ones shows excellent agreement.

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The modified version of strain gradient and couple stress theories in general curvilinear coordinates

Cited by 23 publications

References 24 publications

Effect of size dependency on in-plane vibration of circular micro-rings

Effect of size dependency on in-plane vibration of circular micro-rings

Reduced micromorphic model in orthogonal curvilinear coordinates and its application to a metamaterial hemisphere

Vibrational characteristics of size dependent vibrating ring gyroscope

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