Isogeometric locking-free plate element: A simple first order shear deformation theory for functionally graded plates
Introduction
Functionally graded plates (FG plates) are a special type of composite structures with continuous variation of material properties between the top and bottom surfaces of the plate. Due to the advantageous mechanical behaviors of FG plates they are seeing increased use in a variety of engineering applications [1]. A significant number of studies have been performed to examine the mechanical behavior of FG plates see e.g., [2] for a review. It is widely accepted [2] that plate theories such as the first-order shear deformable theory (FSDT), sometimes also referred to as the Reissner–Mindlin theory, that take into account the shear-deformation effect are necessary to adequately capture the physical behavior of thick plates. Therefore the classical, or Kirchhoff plate theory, which does not capture the effect of shear deformations is not a suitable model for thick FG plates.
Historically the FSDT [3], sometimes also referred to as the Reissner–Mindlin theory, has been popular in computational mechanics for two main reasons: firstly, as mentioned above, it captures the extra physics of shear-deformation not present in the classical theory, and secondly, it relaxes the C1 continuity requirement of the classical theory to C0. This C0 continuity requirement is easier to satisfy using the low-order Lagrangian finite elements that form the basis of most finite element packages. However it is well-known that naïve numerical implementations of the standard FSDT using low-order Lagrangian shape functions typically suffer from shear-locking in the thin-plate or Kirchhoff limit resulting in totally incorrect solutions. Special techniques such as the MITC family of elements [4], assumed strain method [5], field consistent approach [6], smoothed finite element [7] with strain smoothing stabilization technique, are often applied to solve the shear-locking problem, but with additional expense and implementation complexity.
However, with the introduction of numerical methods relying on basis functions with natural C1 continuity such as NURBS in an isogeometric analysis framework (IGA) [8] and meshfree methods [9], [10] we believe that the physical accuracy and straightforward numerical implementation are no longer at odds. In this paper we develop a simple, efficient, robust and locking-free numerical method for thin through to thick shear-deformable plates by using a C1 continuity formulation that includes the effects of shear-deformation. We prove its efficacy by studying homogeneous and functionally graded plates.
The underlying differential equation in our formulation is based on the simple FSDT (S-FSDT) recently presented in [11], [12]. The key idea in the derivation of the S-FSDT is the decomposition of the transverse displacements in the FSDT into bending and shear parts before eliminating the rotation variables using the partial derivatives of the transverse bending displacement only. This weak formulation of the S-FSDT problem requires C1 continuity just like in the classical plate theory but also includes the shear deformable physics of the FSDT. Therefore as well as being viewed as a simple FSDT, this formulation could also be viewed as a classical theory augmented with the shear-deformable physics of the FSDT formulation. Furthermore because the rotation variables of the standard FSDT are eliminated in terms of the bending transverse displacements the resulting weak formulation contains only four variables rather than the usual five, resulting in reduced computational expense.
In the thin-plate limit the S-FSDT recovers the classical plate theory just like the standard FSDT. However, because the thin-plate limit is included naturally in the S-FSDT formulation there is no need to resort to special numerical formulations to eliminate shear-locking as in the standard FSDT; as long as the basis functions satisfy C1 continuity the formulation will be free from shear-locking. Other authors have also used modified plate formulations to ease the construction of numerical methods; recently, Brezzi et al. [13] introduced the twist-Kirchhoff theory that uses a partial Kirchhoff hypothesis to create a simple thin-plate finite element method. Cho and Atluri [14] use a change of variables from transverse displacement to shear stress to develop a meshfree method for the Timoshenko beam problem that is free from shear-locking. This type of approach has been extended by Tiago and Leitão [15] to the plate problem. Cen et al. [16] developed a simple hybrid displacement function element for analysis of thin and moderately thick plates. Based on the three-dimensional governing equation, Man et al. [17] presented a unified technique for solving both thick and thin plate problems by extending the scaled boundary finite element method.
Because of the requirement of C1 continuity we develop the S-FSDT within the framework of the isogeometric analysis (IGA) method proposed by Hughes et al. [8]. This method is becoming popular because of its many advantages, such as exact geometrical modeling, higher-order continuity, simple mesh refinement, and robustness and superior accuracy in comparison with the conventional finite element method. However, the primary reason for using the IGA method in this paper is to achieve the C1 continuity condition required by the weak form of the S-FSDT. As such, other numerical methods with natural C1 continuity such as meshfree methods [2], [9] are also be excellent candidates for the discretization of the S-FSDT.
The principle of IGA involves the adoption of CAD basis functions such as non-uniform rational B-spline (NURBS) functions as the shape functions of finite element analysis. The IGA has been successfully implemented in many engineering problems including structural vibrations [18], plates and shells [19], [20], [21], [22], [23], [24], fluid mechanics [25], fluid–structure interaction problems [26], damage and fracture mechanics [27], and structural shape optimization [28].
It is important to note that the usual IGA method also suffers from shear-locking when discretizing the standard FSDT problem, just like the standard Lagrangian finite element method. The most common remedy is to increase the polynomial order of consistency such that the basis functions are better able to represent the Kirchhoff limit. Echter and Bischoff [29] shows that this can result in sub-optimal convergence and reduce numerical efficiency in the IGA, and present a solution to the problem of shear-locking based on the Discrete Shear Gap (DSG) methodology for the 1D Timoshenko beam problem. Valizadeh et al. [19] use a modified shear correction factor dependent on the local discretization size to suppress shear locking. Beirão da Veiga et al. [30] present a method where the NURBS basis functions satisfy the Kirchhoff condition a priori. The resulting method is completely free of shear-locking, but requires a more complex basis function construction which involves a contravariant mapping for the basis functions interpolating the rotation variable. In contrast, the method we develop in this paper is considerably simpler and can be easily implemented using the existing functionality in open-source IGA frameworks such as igafem [31] and GeoPDEs [32].
In summary, the main objective of this study is to propose a new locking-free plate formulation for solving the static bending, buckling, and free vibration of both thin and thick FG plates. The new approach uses the high continuity of IGA to discretize the S-FSDT. The resulting S-FSDT-based IGA method has four degrees of freedom and is easy to implement within existing open-source IGA frameworks. We show the efficacy of the resulting method with extensive numerical examples focusing on functionally graded plates in static bending, free vibration, and buckling. We show the shear-locking free nature of the proposed method. The effects of boundary condition, gradient index, and geometric shape on the mechanical responses of FG plates are investigated numerically. The computed results are in typically within 1% of reference solutions in available in the literature.
The paper is structured as follows. Section 2 briefly presents the theoretical formulation. Section 3 describes NURBS-based isogeometric analysis in detail. Section 4 presents the validation of the locking-free characteristic of the proposed method. Section 5 shows the numerical results derived from the proposed IGA and in Section 6 we discuss the proposed method and suggest directions for future work.
Section snippets
Functionally graded plate
Consider a ceramic–metal FG plate with thickness h. The bottom and top faces of the plate are assumed to be fully metallic and ceramic, respectively. The xy-plane is the mid-plane of the plate, and the positive z-axis is upward from the mid-plane. In this study, Poisson’s ratio ν is constant and Young’s modulus E and density ρ vary through the thickness with a power law distribution:where n is the gradient index, z is the thickness coordinate variable
NURBS-based isogeometric analysis
In this section we give an overview of the NURBS basis function construction and derive the discrete weak form for the S-FSDT numerical formulation.
Validation of the fully locking-free property
A homogeneous square plate with length a and thickness h under a uniform load P = 1 N is considered to test the locking-free characteristic of the developed approach. The material parameters used for this particular study are Young’s modulus E = 1.092 × 106 N/mm2 and Poisson’s ratio ν = 0.3. The simply supported and clamped boundary conditions are considered. The rotations are obtained by using the derivatives of bending deflection ϕx = −∂ wb/∂x and ϕy = −∂wb/∂y; thus, the constraint on the rotations in the
Numerical applications
In this section, the static bending, free vibration, and buckling behavior of homogeneous and FG plates with different geometric shapes are examined by using the developed “S-FSDT-based IGA” model with cubic NURBS basis function. A 4 × 4 Gaussian quadrature scheme is used in each NURBS element to integrate the weak form. An original “FSDT-based IGA” computer code is implemented to the same problems for comparison purposes. In the following examples, a mesh of 16 × 16 control points is used for the
Conclusions
We presented a new locking-free plate formulation by using the characteristics of the NURBS-based IGA in combination with the S-FSDT theory for the study of homogeneous and nonhomogeneous functionally graded plates. Numerical examples for static bending, buckling and free vibration analysis were considered and their results were presented and discussed in detail. Aspects of the boundary conditions, gradient index, and geometric shape were also investigated. Our conclusions are as follows:
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The
Acknowledgments
This work was supported by Jiangsu Province Graduate Students Research and Innovation Plan (Grant No. CXZZ13_0235) and the National Natural Science Foundation of China (Grant No. 51179063). Jack S. Hale was supported by the Fonds National de la Recherche, Luxembourg under the AFR Marie Curie COFUND scheme and partially supported by the University of Luxembourg. Tinh Quoc Bui (ID No. P14055) was supported by the Grant-in-Aid for Scientific Research (No. 26-04055) – Japan Society for Promotion of
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