Vibration of cracked functionally graded microplates by the strain gradient theory and extended isogeometric analysis

Highlights

• A new approach for cracked microplates with size effects based on strain gradient theory and extended isogeometric analysis.

• An efficient NURBS-based extended isogeometric analysis for vibration of cracked microplates with elements of $C^2$-continuity.

• New enrichment functions for near-tip asymptotic field in strain gradient regime.

• Benchmarking results for vibration of cracked microplates with size effects.

Abstract

In this study, the vibration behaviours of functionally graded microplates with cracks are investigated by means of a simple yet rigorous version of Mindlin’s generialised continuum and the extended isogeometric analysis (XIGA). The simplified strain gradient theory which includes one material length parameter and an additional micro-inertia term is employed to capture the size effects. Meanwhile, the displacement field of the plates is described using the refined plate theory with four unknowns and the XIGA in which enrichment functions are involved to effectively predict the responses of microplates with cracks. In addition, the IGA approach with highly smooth basis functions of non-uniform rational B-spline (NURBS) ensures a clean and efficient treatment of higher continuity requirements in the strain gradient theory. The benchmark numerical results show significant departure from those analysed by the classical continuum elasticity. Indeed, they reveal strong influences of microstructural characteristics on the vibration responses of microplates which are not shown in the platform of the classical theory and the influences are more pronounced as the size of the plates becomes comparable with the material length parameter.

Introduction

It is well known that the classical continuum theories are size independent and therefore they are unable to capture the small-scale effects. Those effects have been experimentally observed in settings where the sizes of the structure of interest are comparable to material’s microstructure lengths [1], [2]. This inability to capture the size effect is attributed to the absence of an internal length scale from the constitutive relation and the local characterisation of stress in the classical theories. The microstructural effects become significantly dominant in structures with very small dimensions which have been used in microelectronmechanical (MEMS) and nanoelectromechanical (NEMS) systems. Meanwhile, the generealised or higher-order continuum theories are equipped with the additional material lengths so that the small-scale effects can be captured efficiently. The theories that are able to account for the size effects include gradient theory by Mindlin [3], micropolar or Cosserat theory [4], couple stress theory by Toupin [5] and Koiter [6].

It is worth mentioning that since the gradient theory includes rotation and stretch gradients, it is more general than the couple stress one which is only based on the constrained rotation. Besides, the gradient theory and Cosserat theory are not similar due to the fact that the latter considers independent rotation components in addition to the displacement components which are used in the former [7]. There are many works have been done based on the couple stress theory [8], [9], [10] and Cosserat theory [11], [12], [13] to predict the behaviours of small-scale structures. Meanwhile, the generalised strain gradient theory by Mindlin which is also called, for the lowest-order theory, dipolar gradient theory or grade-two theory is considered as one of the most effective theories. The fundamental idea of this theory is to incorporate an internal displacement field to each particle of a continuum, i.e. the material particle is considered as a deformable medium. According to this theory, the gradients of strain are included in the strain energy density which implies the appearance of new material constants, in addition to two classical Lamé’s constants ($\lambda ,\mu$), and the material characteristic lengths. On the other hand, the expression of the kinetic energy density depends upon the micro-inertia term leading to the presence of the intrinsic material length that associated with the material microstructure. It is worth commenting that, at the early stage of the development, the gradient theories were highly complex with many independent parameters which discouraged researchers and engineers to consider them seriously for mechanics problems. Later, the theories were simplified and only one parameter in addition to the Lamé’s constants is involved. The comprehensive reviews on the development of the gradient theories can be found in the literature [7], [14], [15]. The gradient theories, both the general and the simplified ones, have been widely applied in many problem including stress concentration [16], [17], wave propagation [18], [19], plasticity [20], [21], fracture [22], [23] and static analysis [24].

In the context of plate theories, there is a great deal of works devoted to the development of reliable mathematical models that govern their behaviours. It is started with the classical plate theory or usually called the Kirchhoff-Love plate theory in which three degrees of freedom corresponding to displacements are involved. The fundamental idea of this theory is to neglect the shear deformations by assuming that the cross sections normal to the midplane remain normal during deformations. This theory appears to be simple and effective when dealing with thin plates of large length-to-thickness ratios where the transverse shear stresses and strains are negligibly small. However, classical plate theory shows its drawbacks in the analysis of thick plates in which the transverse shear components are significant. In order to overcome these shortcomings, the first-order shear deformation theory which is also known as Reissner-Mindlin plate theory is proposed acknowledging the existence of the shear deformations. Consequently, this theory with five unknowns (three displacements and two rotations) is capable of reliably predicting the structural behaviours of both thin and thick plates. However, the major issue of the theory is the necessity to calculate the shear correction factor to match the numerical results with the analytical ones. The procedure to determine this factor may not be able to be established due to the fact that it is problem dependent. Several higher-order plate theories have been proposed to eliminate this factor. Reddy [25] pioneered the use of the five-unknown third-order plate theory which incorporates shear deformations without using the shear correction factor. In an effort to simplify yet maintain all beneficial features of Reddy’s theory, Senthilnathan [26] deliberately split the transverse displacement into bending and shearing parts and employed their spatial derivatives to represent rotations. He derived the so-called refined plate theory (RPT) with four degrees of freedom. Recently, Karttunen et al. [27] worked on an approach that linked plate theories and elasticity solutions in which the exact 3D plate solution and the results for interior plate problem were presented. There is a great deal of studies that utilised the plate theories to estimate the structural responses of structures [28], [29], [30].

The attention to the combination of the plate theories and generalised continuum theories including their variations to investigate the behaviours of small-scale plates has been constantly increased over the last few years. Papargyri-Beskou and Beskos [24] conducted static, stability and dynamic analysis of Kirchhoff plates using the strain gradient theory (SGT) with one additional material constant. Zhang et al. [31] utilised the Fourier series to predict the mechanical behaviours of small-scale plate based on Kirchhoff theory. Farzam Dadgar-Rad [32] analysed the strain gradient Reissner-Mindlin plate using $C^0$ quadrilateral elements. Ji et al. [33] proposed a comparison of strain gradient theories which were used in the analysis of functionally graded (FG) circular plates. Khakalo and Niiranen [34], [35] employed the SGT for the analysis of micro/nano materials and structures as well as stress analysis around cylindrical holes with bi-axial tension. However, in most of the works, the micro-inertia term which appears in the kinetic energy density of the original SGT has been ignored. In addition, the solution procedures involve either analytical approach for limited study or finite element analysis with less requirement on the continuity of the inter-elements. It is worth noting that when a plate theory is combined with the SGT, the resulting equation are of sixth-order leading to third-order derivatives in the weak form, which requires at least $C^2$-continuity of the basis functions. Such higher order requirements can be efficiently handled by the isogeometric analysis (IGA) approach.

The IGA which is introduced by Hughes et al. [36] is capable of handling the high continuity requirements easily with appropriate basis functions. This newly developed method also aims at the integration of the design and analysis in industrial processes. This fundamental idea is fulfilled by employing the same basis functions to represent the geometries and to approximate the solution fields. Among the possibilities, the non-uniform rational B-splines (NURBS) is the most popular option of a basis function as it has a dominant establishment in the Computer Aided Design (CAD) industry. While NURBS can exactly represent complex geometries, the basis function is also able to approximate the unknowns with higher order of continuity effectively since it is highly smooth. Owing to this striking feature, NURBS can easily handle the sixth-order plate problems based on higher-order plate theory and generalised continuum theory where $C^2$ continuity is required. The self-contained reviews on the IGA can be found in a number of works in the literature [37], [38], [39], [40]. IGA has found its application in broad fields of solid mechanics including those with small-scale structures using beam/plate theories and generalised continuum theories and their variants [41], [42], [43], [44], [45], [46]. In attempts to deal with structural fractures, the ideas of extended finite element method (XFEM) [47] are incorporated in the platform of the IGA. While Benson et al. [48] presented the application of XIGA in dealing with the fracture mechanic problems, De Luycker et al. [49] showed the results of XIGA for linear fracture mechanics with good accuracy and better convergence rate. Ghorashi et al. [50] utilised the XIGA for the simulation of stationary and propagating cracks.

Regarding the investigation into plates with cracks, there is a well established body of work on the mechanical behaviours of such structures. Stahl and Keer [51] studied the vibration and buckling responses of cracked rectangular plates in which the dual series equations and the homogeneous Fredholm integral equations were considered. Bachene et al. [52] and Natarajan et al. [53] applied the XFEM to model the discontinuities and solve for the vibration responses of cracked plates. Tran et al. [54] analysed the vibrations of FG plates with cracks by means of the XIGA. However, most of the research dealt with normal-size plates using classical continuum theories without considering size effects. Recently, Liu et al. [55] studied the size-dependent effects of cracked plates based on Reissner-Mindlin plate theory and modified couple stress theory. However, as mentioned above, these theories have their own shortcomings in terms of choosing shear correction factor and accounting for the micro-inertia term.

In this study, in order to fill the existing gaps and enrich the research into small-scale plates with cracks, the vibration analysis of cracked FG microplates by means of the SGT and XIGA will be presented. While the simplified SGT with one internal length scale incorporating micro-inertia term is employed to efficiently capture the size-dependent effects, the displacement field of the plates is modelled by the RPT. In the combination of these particular continuum theory and plate theory, the sixth-order governing equations will be formed. Consequently, elements with $C^2$-continuities are required to be able to numerically solve the equations. This condition is effectively fulfilled in the platform of the NURBS-based XIGA in which Heaviside and novel enrichment functions are employed to model the discontinuities along the crack path and capture the near tip asymptotic field.

The outline of this study is as follows. Whereas Section 2 gives a brief review on the SGT and kinematics of plates, Section 3 provides the formulation of the XIGA and the discretisation procedure. Section 4 presents the convergence study and numerical results of cracked plates with various shapes, crack locations, and boundary conditions, as well as the effects of the scale parameters on the vibration responses. The study is closed with concluding remarks which are given in Section 5.