Vibration analysis of two-dimensional micromorphic structures using quadrilateral and triangular elements

PurposeThe paper aims to presents a numerical analysis of free vibration of micromorphic structures subjected to various boundary conditions.Design/methodology/approachTo accomplish this objective, first, a two-dimensional (2D) micromorphic formulation is presented and the matrix representation of this formulation is given. Then, two size-dependent quadrilateral and triangular elements are developed within the commercial finite element software ABAQUS. User element subroutine (UEL) is used to implement the micromorphic elements. These non-classical elements are capable of capturing the micro-structure effects by considering the micro-motion of materials. The effects of the side length-to-length scale parameter ratio and boundary conditions on the vibration behavior of 2D micro-structures are discussed in detail. The reliability of the present finite element method (FEM) is confirmed by the convergence studies and the obtained results are validated with the results available in the literature. Also, the results of micromorphic theory (MMT) are compared with those of micropolar and classical elasticity theories.FindingsThe study found that the size effect becomes very significant when the side length of micro-structures is close to the length scale parameter.Originality/value The study is to analyze the free vibrations of 2D micro-structures based on MMT; to develop a 2D formulation for micromorphic continua within ABAQUS; to propose quadrilateral and triangular micromorphic elements using UEL and to investigate size effects on the vibrational behavior of micro-structures with various geometries.

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

Size effects on the plastic shock formation in steady-state cavity expansion in porous ductile materials

In this paper, we have studied the hypervelocity expansion of a spherical cavity in an infinite medium modeled with theextension of the porous plasticity criterion of Gurson [23] developed by Chen and Yuan [7] to account for plastic strain gradient induced size effects. Following the self-similar, steady-state solution derived by Cohen and Durban [9] for size-independent porous materials, we have computed the critical cavity expansion velocity which leads to the emergence of plastic shock waves for a wide range of initial void volume fractions and different values of the length scale parameter that controls the effect of size. We have shown that size effects hinder the emergence of plastic shock waves, so that as the length scale parameter increases, the expansion velocity required for the plastic shock to be formed increases. In addition, while porosity favors the formation of plastic shocks, as shown by Cohen and Durban [9], our results indicate that the effect of initial void volume fraction on plastic shock wave formation decreases for size-dependent materials.

Phase-Field Fracture Modelling of Thin Monolithic and Laminated Glass Plates under Quasi-Static Bending

A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time-/temperature-dependent material properties we derived recently for two commonly used polymer foils made of polyvinyl butyral or ethylene-vinyl acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin–Francfort–Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. Therefore, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one/two cracks are localised using the quasi-static analysis, whereas multiple cracks evolve during the experiment. It was also observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.

Free Vibration Analysis of a Spinning Smart Piezoelectrically Actuated Heterogeneous Nanoscale Shell with Nonlocal Strain Gradient Theory

In this paper, a numerical procedure is proposed for analyzing the effects of length scale parameter, external electric field, angular speed and nonlocal parameter on the free vibration of a functionally graded piezoelectric cylindrical nanoshell. Nonlocal strain gradient theory (NSGT) is employed to study Eringen’s size-dependent effect and the length scale parameter. This new proposed method can be considered as a combination of Eringen’s nonlocal model and classical strain gradient theory. The obtained results show that this model can be used reliably for small-scale systems. The effects of boundary conditions, applied voltage, nonlocal parameter, rotational speed and length scale parameter on natural frequencies are presented. Compared to other elasticity theories, NSGT achieves the highest natural frequency and critical rotational speed and also a wider stability region. Doubling and tripling the length scale increases the natural frequency by approximately 1.8 and 2.6 times, respectively; while doubling and tripling the nonlocal parameter value reduces the natural frequency by approximately 1.2 and 1.4 times, respectively. Therefore, the natural frequency is more sensitive to the length scale parameter than the nonlocal parameter. Finally, it was shown that the critical angular speed goes up by increasing the length scale parameter, applied voltage, or nonlocal parameter.