An isogeometric continuum shell element for modeling the nonlinear response of functionally graded material structures

Journal Pre-proofs An isogeometric continuum shell element for modeling the nonlinear response of functionally graded material structures Ning Liu, Xiang Ren, Jim Lua PII: DOI: Reference:

S0263-8223(19)33873-5 https://doi.org/10.1016/j.compstruct.2020.111893 COST 111893

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Composite Structures

Received Date: Revised Date: Accepted Date:

12 October 2019 3 January 2020 7 January 2020

Please cite this article as: Liu, N., Ren, X., Lua, J., An isogeometric continuum shell element for modeling the nonlinear response of functionally graded material structures, Composite Structures (2020), doi: https://doi.org/ 10.1016/j.compstruct.2020.111893

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An isogeometric continuum shell element for modeling the nonlinear response of functionally graded material structures Ning Liu, Xiang Ren, Jim Lua Global Engineering & Materials, Inc., Princeton, NJ 08540, USA

Abstract A geometrically nonlinear continuum shell element using a NURBS-based isogeometric analysis (IGA) approach is presented for the analysis of functionally graded material (FGM) structures. IGA offers a computationally efficient and geometrically exact representation of the original shell geometry and its underlying basis functions provide high-order continuity for the solution variables. The use of high-order smooth basis functions also alleviates shear and membrane locking phenomena that commonly occurred in shell structures. In addition, the developed continuum shell element features a precise description of the thickness-varying material properties in FGM in the sense that a set of desired high-order B-spline basis functions with sufficient number of quadrature points are employed for accurate through-thickness numerical integration. A simple power-law distribution function of the FGM is adopted in the current study. The performance of the proposed IGA solid shell element is demonstrated via a variety of nonlinear shell benchmark problems. The effect of the FGM power-law exponent on the geometrically nonlinear response of the shell structures is investigated as well. Keywords: isogeometric analysis; NURBS; functionally graded materials; thin shell; continuum shell element; geometrically nonlinear; large displacement.

1. Introduction Functionally graded materials (FGMs) have received a significant amount of attention since its advent in the 1980s. The main advantage is that they circumvent the stress concentration problem that frequently occurs at dissimilar material interfaces of composite structures and therefore eliminate the possibility of delamination failure. FGMs are usually manufactured with two or more phases of material constituents and their material properties grade smoothly from one end to another. Thanks to the multi-constituent compositions, FGMs can be made with the possession of the advantages of all its constituents. For instance, a common FGM is of the ceramic-metal type, of which the material exhibits strong thermal resistance due to ceramic and high material toughness due to metal. As a result, FGMs have been extensively used in various industrial applications such as aerospace and nuclear engineering. Composite plate and shell structures made of FGMs have been traditionally modelled using two strategies: one being the so-called equivalent single-layer (ESL) approach that equates a heterogeneous composite shell structure with a statically equivalent single layer model, the stiffness of which is calculated as a weighted average of each individual layer stiffness. Popular ESL theories include firstorder shear deformation theory (FSDT) and higher-order shear deformation theories (HSDT). In spite of its relative simplicity and the reduction of an actual three-dimensional problem into a two-dimensional one, ESL-based models fail to consider the discontinuity of the transverse strain components in the thickness of the shell body and therefore are not able to capture the through-thickness stress states that are essential in determining the failure patterns of composite structures. On the other hand, continuum modeling based layerwise theories [1] are built upon a separate displacement field expansion assumption for each layer and consider C0 displacement continuity at material interfaces. As a consequence,

continuum modeling based layerwise theories have the potential to capture the through-thickness variation of stress/strain states and provide a more thorough understanding of the stress and deformation responses of composite components. Thus, a continuum FGM shell model is beneficial in scenarios where an accurate stress distribution is required (e.g., failure analysis and design optimization). FGM shell modeling based on standard FEA approaches has been expansive in the past a few decades. We refer to these work [2,3] for excellent reviews in the advance of FGM shell modeling. In particular, Woo and Meguid [4] presented an analytical solution for the large-deflection analysis of thin rectangular shallow shell of FGM type subjected to transverse mechanical and thermal loadings. The geometrically nonlinear effect was considered in the von Karman sense. A similar approach based on HSDT was reported by Wali et al. [5,6] and Frikha and Dammak [7] with a discrete definition of double directors and also by Oktem et al. [8]. Pandey and Pradyumna [9] employed a layerwise theory for the modeling of FGM sandwich shell; however, the FGM layer was approximated with FSDT that fails to describe the volume fraction variation. Kim et al. [10] extended a four-node quasi-conforming shell element for geometrically nonlinear analysis of FGM plates and shells based on FSDT. The FSDT was also employed together with a matched asymptotic approach of the perturbation theory to analyze variable-thickness FGM cylindrical shells under internal pressure [11]. More recently, Tornabene and Viola [12] coupled FSDT with the generalized differential quadrature approach to study four-parameter FGM shells. This approach was also extended to HSDT with stress recovery [13] and for the analysis of variable-thickness FGM sandwich shells [14]. Later, the meshfree method was combined with a modified Sander’s nonlinear shell theory for the analysis of FGM shell panels under combined thermal-mechanical loading [15]. Zozulya and Zhang [16] expanded the field variables and FGM material parameters using Fourier series in the form of Legendre’s polynomials for axisymmetric cylindrical shell modeling. Cinefra et al. [17] adopted a nine-node quadratic shell element and Carrera’s Unified Theory for the analysis of FGM structures. The mixed interpolation of tensorial components was used to alleviate the membrane and shear locking phenomenon. Moreover, Reddy’s research group [18–20] utilized a set of high-order spectral/hp polynomial basis to formulate a seven-parameter solid FGM shell element. Along the line of solid FGM shell development, a modified first-order enhanced solid shell formulation [21] was recently reported, in which a parabolic shear strain distribution was imposed through the thickness in the compatible strain component. Reinoso and Blazquez [22] presented a seven-parameter first-order solid shell model that adopted a combination of the enhanced assumed strain and the assumed natural strain to mitigate locking. A similar work in the context of six-parameter thin shell and the neutral physical surface method was reported in [23]. Moreover, a generalized hybrid quasi-3D HSDT was proposed for FGM shell modeling [24]. It was further suggested that the non-polynomial HSDTs should be optimized for accuracy improvement. In another work, Srividhya et al. [25] developed a nonlocal stress-gradient FGM shell model in the HSDT framework. Beni et al. [26] employed a modified couple stress theory for the modeling of FGM cylindrical thin shells. In addition, a non-conforming triangular shell element was used in an HSDT setting for material distribution and sizing optimization analysis of FGM structures [27]. As opposed to computing the transverse normal/shear components through displacement derivatives, a mixed displacement/transverse stress approach was presented recently [28]. In this approach, the transverse stress equilibrium at the layer intersections was enforced via Lagrange multipliers, which inevitably increased the problem size. On the application side, FGM shell modeling has also been investigated in the context of free vibration [29–41] and dynamic response [42–44], instability [45–57], thermal-mechanical behavior [58–62], reinforcement with carbon nanotubes [63–69] and stiffeners [70–76], material elastoplasticity [77–79], variable thickness [80], active/passive control [81–83], nanoscale modeling [84], and extended FEM [85,86]. Despite the wide variety of work reported in literature, the FGM shell formulations were all developed under traditional finite element approaches. In this contribution, we extend the FGM shell modeling to the isogeometric analysis (IGA) setting. IGA was firstly put forward by Hughes et al. [87] in 2005 as a novel technology to seamlessly bridge computer-aided design (CAD) and finite element

analysis (FEA). Its essential idea is to adopt the same basis functions that are used in geometric design for discrete solution spaces. A number of the advantages of IGA include the elimination of the meshing process, tight interaction between CAD and FEA, geometrically exact numerical model description, improved computational accuracy and efficiency due to the use of high-order spline basis [88–94], among others. From a practical implementation point of view, the so-called Bézier extraction technique [95] provides a finite element data structure and therefore allows an efficient integration of the high-order spline model representation into the standard FEA code package. While a significant amount of work has been done in the IGA shell development ([96–108] to name a few), the analysis of FGM structures in IGA has been mostly limited to 2D plate analysis [1,109–125]. Aside from the studies in structural plates, Taheri and Hassani [126] have investigated the use of NURBS-based IGA for shape and material composition optimization of FGM structures, but the analysis was focused on 2D spaces as well. As far as IGA FGM shell analysis is concerned, investigations have been limited to a few studies. In particular, Nguyen et al. have attempted the use of NURBS and FSDT for the modeling of FGM shells reinforced by carbon nanotubes in static, dynamic and buckling analysis [127], as well as post-buckling analysis [128]. In the work of Tan et al. [129], the meshfree method was coupled with IGA for the analysis of FGM plates and shells, in which the IGA description was only applied at the domain boundaries and the interior was filled with meshfree nodes. In this context, the current work aims to develop an IGA continuum shell element suitable for modeling the nonlinear response of FGM shell structures. To the authors’ best knowledge, the current contribution represents the first effort that an IGA-based continuum shell element is developed for the analysis of FGM structures. The present IGA continuum shell model features an accurate FGM description via through-thickness quadrature point enrichment and a precise modeling of the geometrically nonlinear behavior of FGM shells. The rest of the paper is organized as follows: section 2 and 3 provide a brief introduction of functionally graded materials and NURBS basis construction, respectively. The IGA continuum shell formulation is presented in detail in section 4. To demonstrate the capability of the developed isogeometric FGM shell element, a variety of geometrically nonlinear FGM shell examples are shown in section 5 with various volume fraction exponents. In section 6, we draw some conclusions.

2. Functionally graded materials Functionally graded materials are characterized by the through-thickness gradual volume fraction variation from one material to another. A simple power-law distribution function is adopted here to describe the volume fraction variation: 𝑉𝑐 (𝑧) = (

2𝑧+𝑡 𝜅 2𝑡

𝑡

𝑡

2

2

) ,− ≤𝑧≤

(1)

with 𝜅 the volume fraction exponent and t the thickness of the shell. From eq. (1), the effective Young’s modulus at a thickness z can be calculated as, 𝐸(𝑧) = 𝐸𝑏𝑜𝑡 + (𝐸𝑡𝑜𝑝 − 𝐸𝑏𝑜𝑡 )𝑉𝑐 (𝑧)

(2)

where 𝐸𝑏𝑜𝑡 and 𝐸𝑡𝑜𝑝 are the Young’s modulus of the bottom and top materials, respectively. As an example, the volume fraction variation in the thickness direction is plotted in Fig. 1 with various volume fraction exponents, where we observe a drastic change in volume fraction when the power-law exponent 𝜅 becomes relatively large/small compared to one. This accounts for the reason why FGM modeling with traditional finite shell element is challenging.

Fig. 1. An example of the volume fraction variation through the thickness with different volume fraction exponents.

3. Fundamentals of Bézier, B-spline and NURBS In one dimension, the ith degree-n Bernstein polynomial can be defined in the following,

n n i Bin t    t i 1  t  i n

(3)

where t  0,1 ,    represents the binomial coefficient.  i  i!( n  i )!

n!

A Bézier curve S t  is then constructed as a linear sequence of n  1 Bernstein polynomials Bin t  and the associated set of control points bi , n 1

S t    Bin t bi

(4)

i 1

The B-spline basis functions are related to the Bernstein basis via the so-called Bézier extraction operator C [95] that is uniquely defined by a given knot vector in the parameter domain,

N t   CBt 

(5)

Through projection of the B-splines from ℝd to ℝd+1 using the weights of the associated control points, a degree-n NURBS curve can be constructed as,

Rin t  

N in t wi

(6)

ncp

 N t w j 1

n j

j

where ncp is the number of control points that are used to describe the curve. In two and higher dimensions, NURBS are merely a tensor product of their univariate counterparts. NURBS are standard basis functions in CAD and IGA to exactly describe curves, surfaces and solids as they are formed from rational polynomials.

4. Continuum shell formulation An isogeometric solid-like continuum shell element that employs separate through-thickness discretization from the in-plane curvilinear representation is formulated here. Specifically, the proposed shell element adopts NURBS for the in-plane representation, while a desired univariate B-spline is used for the thickness direction. Moreover, the thickness numerical integration is enriched with sufficient number of quadrature points for accurate description of the thickness-varying material properties of FGM. Geometrically nonlinear effect is also considered in a total Lagrangian framework. 4.1. Continuum shell kinematics

Fig. 2. Mapping between the parametric space, undeformed space and the deformed space. As illustrated in Fig. 2, the continuum shell kinematics can be described by a general mapping system between the parametric space Ą, the reference space Ω0, and the deformed space Ω. The position vector of an arbitrary material point in the undeformed space Ω 0 can be expressed in terms of the convective curvilinear coordinates as,

𝑋(𝜉1 , 𝜉 2 , 𝜉 3 ) = 𝑋 0 (𝜉1 , 𝜉 2 ) + 𝜉 3 𝑛(𝜉1 , 𝜉 2 ), 0 ≤ 𝜉 3 ≤ 1

(7)

where 𝑋 0 represents the projected point of 𝑋 onto the reference surface and 𝑛 is the surface normal director of the point. Without loss of generality, the bottom surface of the shell is herein treated as the reference surface. The C1-continuous shell geometry permits the determination of a set of basis vectors in the reference space in the following form, 𝜕𝑋

𝐴𝛼 = 𝜕𝜉𝛼0 , 𝛼 = 1,2 n0 =

A1 ×A2 ||A1 ×A2 ||

(8)

𝑡

where t is the shell thickness. It is convenient to introduce a set of co-variant tangent vectors to describe the motion of an arbitrary material point of the solid shell as, 𝐺𝛼 =

𝜕𝑋 𝜕𝜉 𝛼

= 𝐴𝛼 + 𝜉 3 𝑛0,𝛼 , 𝛼 = 1,2

(9)

𝐺3 = n0 where (∙),𝛼 denotes the partial derivative with respect to the curvilinear coordinates and the derivative of the surface normal is expressed as, 𝑛0,𝛼 =

𝐴̃3,𝛼⋅𝐴̅3−𝐴̃3⋅𝐴̅ 3,𝛼 𝐴2̅ 3

(10) ̃

̃

𝐴 ⋅𝐴 where 𝐴̃3 = 𝐴1 × 𝐴2, 𝐴̃3,𝛼 = 𝐴1,𝛼 × 𝐴2 + 𝐴1 × 𝐴2,𝛼 , 𝐴̅3 = ||𝐴1 × 𝐴2 || and 𝐴̅3,𝛼 = 3 ̅ 3,𝛼. 𝐴3

Now that the co-variant basis vectors are defined, their dual basis vectors (i.e., the contra-variant basis vectors) can be determined based on the following relation, 𝑗

𝐺𝑖 ⋅ 𝐺 𝑗 = 𝛿𝑖

(11)

with 𝛿𝑖𝑗 the Kronecker delta. The surface metric tensor of the first fundamental form is then written as, 𝐺 = 𝐺 𝑖𝑗 𝐺𝑖 ⊗ 𝐺𝑗 = 𝐺𝑖𝑗 𝐺 𝑖 ⊗ 𝐺 𝑗

(12) −1

where 𝐺 𝑖𝑗 = 𝐺 𝑖 ∙ 𝐺 𝑗 , 𝐺𝑖𝑗 = 𝐺𝑖 ∙ 𝐺𝑗 , and [𝐺 𝑖𝑗 ] = [𝐺𝑖𝑗 ] . Finally, the element-level volume under the reference configuration can be calculated in terms of the surface metric tensor in the parametric domain as, 𝑑𝑉0 = √det[𝐺𝑖𝑗 ]𝑑𝜉1 𝑑𝜉 2 𝑑𝜉 3

(13)

In order to define the stress and strain components, the deformation gradient between the reference and deformed spaces is needed,

𝐹 = 𝑔𝑖 ⊗ 𝐺 𝑖

(14)

where 𝑔𝑖 represents the co-variant tangent vector in the deformed configuration based on a total Lagrangian framework, 𝑔𝑖 = 𝑥,𝑖 = 𝐺𝑖 + 𝑢,𝑖 . Consequently, the surface Green-Lagrange strain tensor can be written as, 1

1

2

2

𝐸 = (𝐹 𝑇 ∙ 𝐹 − 𝐼) = (𝑔𝑖𝑗 − 𝐺𝑖𝑗 )𝐺 𝑖 ⊗ 𝐺 𝑗 = 𝐸𝑖𝑗 𝐺 𝑖 ⊗ 𝐺 𝑗

(15)

in which the strain components 𝐸𝑖𝑗 can be further expressed using the reference local tangent vectors and displacement derivatives as, 1

(16)

𝐸𝑖𝑗 = 2 (𝐺𝑖 ∙ 𝑢,𝑗 + 𝐺𝑗 ∙ 𝑢,𝑖 + 𝑢,𝑖 ∙ 𝑢,𝑗 )

Eq. (15) essentially describes the variation of surface metric tensors as a result of structural deformation. Eventually, the second Piola-Kirchhoff stress tensor and the Green-Lagrangian strain tensor are related through the Hooke’s law, 𝑆 = 𝐶 ∙𝐸

(17)

with 𝐶 the fourth-order FGM material elasticity tensor. Note that the stress and strain fields 𝑆 and 𝐸 are defined in terms of the contra-variant coordinates which are not necessarily orthonormal. Therefore, transformation from the curvilinear system to the element local coordinates (𝑒𝑖 , 𝑖 = 1,2,3) is needed, 𝑒 𝐸𝑖𝑗 = 𝐸𝑘𝑙 (𝐺 𝑘 ∙ 𝑒𝑖 )(𝐺 𝑙 ∙ 𝑒𝑗 )

(18)

The principle of virtual work is employed, in which the body force term is neglected for sake of brevity, 𝛿𝛱 = 𝛿𝑊𝑖𝑛𝑡 + 𝛿𝑊𝑒𝑥𝑡 = ∫𝛺 𝛿𝐸 ∙ 𝑆𝑑𝛺0 − ∫𝛤 𝑡0 ∙ 𝛿𝑢𝑑𝛤0 = 0 0

0

(19)

An incremental-iterative solution scheme (e.g., the Newton-Raphson method and arc-length method [130]) is usually employed to solve the above nonlinear system, where linearization of the internal virtual work is necessary for the computation of the tangent stiffness matrix, 𝛥𝛿𝑊𝑖𝑛𝑡 = ∫𝛺0 (𝛿𝐸 ∙ 𝛥𝑆 + 𝛥𝛿𝐸 ∙ 𝑆)𝑑𝛺0 = ∫𝛺0 (𝛿𝐸 ∙ 𝐶 ∙ 𝛥𝐸 + 𝛥𝛿𝐸 ∙ 𝑆) 𝑑𝛺0

(20)

where 1

1

𝛿𝐸𝑖𝑗 = 2 (𝐺𝑖 ∙ 𝛿𝑢,𝑗 + 𝐺𝑗 ∙ 𝛿𝑢,𝑖 + 𝛿𝑢,𝑖 ∙ 𝑢,𝑗 + 𝑢,𝑖 ∙ 𝛿𝑢,𝑗 ) = 2 (𝑔𝑖 ∙ 𝛿𝑢,𝑗 + 𝑔𝑗 ∙ 𝛿𝑢,𝑖 )

(21)

4.2. Discretized form Let the n in-plane NURBS basis functions be denoted by Ri and m through-thickness B-spline basis functions be denoted by Nj, the displacement variables can be described as follows, 1 2 3 𝑢(𝜉1 , 𝜉 2 , 𝜉 3 ) = ∑𝑛𝑖 ∑𝑚 𝑗 𝑅𝑖 (𝜉 , 𝜉 )𝑁𝑗 (𝜉 )𝑢𝑖𝑗

Adopting the Voigt notation, the variational strain tensor can be conveniently expressed as,

(22)

𝛿𝐸 𝑇 = [𝛿𝐸11 , 𝛿𝐸22 , 𝛿𝐸33 , 2𝛿𝐸12 , 2𝛿𝐸23 , 2𝛿𝐸13 ]

(23)

where each individual term is expanded and represented using the in-plane NURBS and throughthickness B-splines as, 𝑛 3 𝛿𝐸11 = 𝑔1 ∙ 𝛿𝑢,1 = ∑𝑚 𝑗 ∑𝑖 ∑𝑘 𝑔1 𝑒𝑘 𝑅𝑖,1 𝑁𝑗 𝛿𝑢𝑘 𝑛 3 𝛿𝐸22 = 𝑔2 ∙ 𝛿𝑢,2 = ∑𝑚 𝑗 ∑𝑖 ∑𝑘 𝑔2 𝑒𝑘 𝑅𝑖,2 𝑁𝑗 𝛿𝑢𝑘 𝑛 3 𝛿𝐸33 = 𝑔3 ∙ 𝛿𝑢,3 = ∑𝑚 𝑗 ∑𝑖 ∑𝑘 𝑔3 𝑒𝑘 𝑅𝑖 𝑁𝑗,3 𝛿𝑢𝑘

2𝛿𝐸12 = 𝑔1 ∙ 𝛿𝑢,2 + 𝑔2 ∙ 𝛿𝑢,1 =

𝑛 3 ∑𝑚 𝑗 ∑𝑖 ∑𝑘 𝑔1 𝑒𝑘 𝑅𝑖,2 𝑁𝑗 𝛿𝑢𝑘

(24) +

𝑛 3 ∑𝑚 𝑗 ∑𝑖 ∑𝑘 𝑔2 𝑒𝑘 𝑅𝑖,1 𝑁𝑗 𝛿𝑢𝑘

𝑛 3 𝑚 𝑛 3 2𝛿𝐸23 = 𝑔2 ∙ 𝛿𝑢,3 + 𝑔3 ∙ 𝛿𝑢,2 = ∑𝑚 𝑗 ∑𝑖 ∑𝑘 𝑔2 𝑒𝑘 𝑅𝑖 𝑁𝑗,3 𝛿𝑢𝑘 + ∑𝑗 ∑𝑖 ∑𝑘 𝑔3 𝑒𝑘 𝑅𝑖,2 𝑁𝑗 𝛿𝑢𝑘 𝑛 3 𝑚 𝑛 3 2𝛿𝐸13 = 𝑔3 ∙ 𝛿𝑢,1 + 𝑔1 ∙ 𝛿𝑢,3 = ∑𝑚 𝑗 ∑𝑖 ∑𝑘 𝑔3 𝑒𝑘 𝑅𝑖,1 𝑁𝑗 𝛿𝑢𝑘 + ∑𝑗 ∑𝑖 ∑𝑘 𝑔1 𝑒𝑘 𝑅𝑖 𝑁𝑗,3 𝛿𝑢𝑘

The above equations can further be conveniently described via the strain-displacement matrix 𝐵 and the displacement degrees of freedom (DOFs) 𝑢 as, 𝛿𝐸 𝑇 = 𝐵 ∙ 𝛿𝑢

(25)

where 𝐵 is a 6-by-[3(𝑝 + 1)2 (𝑞 + 1)] matrix, with p and q the in-plane NURBS and through-thickness B-spline order, respectively, 1 𝐵 = [𝐵𝑖𝑗𝑘

2 𝐵𝑖𝑗𝑘

3 𝐵𝑖𝑗𝑘

4 𝐵𝑖𝑗𝑘

5 𝐵𝑖𝑗𝑘

𝑇

6 𝐵𝑖𝑗𝑘 ]

1 𝐵𝑖𝑗𝑘 = 𝑔1 ∙ 𝑒𝑘 ∙ 𝑅𝑖,1 ∙ 𝑁𝑗 2 𝐵𝑖𝑗𝑘 = 𝑔2 ∙ 𝑒𝑘 ∙ 𝑅𝑖,2 ∙ 𝑁𝑗 3 𝐵𝑖𝑗𝑘 = 𝑔3 ∙ 𝑒𝑘 ∙ 𝑅𝑖 ∙ 𝑁𝑗,3 4 𝐵𝑖𝑗𝑘 = 𝑔1 ∙ 𝑒𝑘 ∙ 𝑅𝑖,2 ∙ 𝑁𝑗 + 𝑔2 ∙ 𝑒𝑘 ∙ 𝑅𝑖,1 ∙ 𝑁𝑗

(26)

5 𝐵𝑖𝑗𝑘 = 𝑔2 ∙ 𝑒𝑘 ∙ 𝑅𝑖 ∙ 𝑁𝑗,3 + 𝑔3 ∙ 𝑒𝑘 ∙ 𝑅𝑖,2 ∙ 𝑁𝑗 6 𝐵𝑖𝑗𝑘 = 𝑔3 ∙ 𝑒𝑘 ∙ 𝑅𝑖,1 ∙ 𝑁𝑗 + 𝑔1 ∙ 𝑒𝑘 ∙ 𝑅𝑖 ∙ 𝑁𝑗,3

As is discussed in Eq. (18), the above matrix 𝐵 needs to be transformed from the non-orthonormal curvilinear coordinate system to the element local coordinate system. Using the notation 𝑇𝑖𝑗𝑘𝑙 = 𝑡𝑘𝑖 𝑡𝑙𝑗 = (𝑔𝑘 ∙ 𝑒𝑖 ) (𝑔𝑙 ∙ 𝑒𝑗 ), this transformation is realized via the following, 𝐵𝑒 = 𝑇 ∙ 𝐵 where

(27)

𝑇1 𝑇=[ 3 𝑇

𝑇2 𝑇4

11 𝑇11 11 𝑇 = [𝑇22 11 𝑇33

22 𝑇11 22 𝑇22 22 𝑇33

1

1

𝑇2 =

2 1 2 1

] 33 𝑇11 33 ] 𝑇22 33 𝑇33

12 21 ) (𝑇11 + 𝑇11 12 21 ) (𝑇22 + 𝑇22

12 21 [2 (𝑇33 + 𝑇33 ) 11 11 𝑇12 + 𝑇21 11 11 𝑇 = [𝑇23 + 𝑇32 11 11 𝑇13 + 𝑇31 3

1 2 1 2 1 2

23 32 ) (𝑇11 + 𝑇11 23 32 ) (𝑇22 + 𝑇22 23 32 ) (𝑇33 + 𝑇33

22 22 𝑇12 + 𝑇21 22 22 𝑇23 + 𝑇32 22 22 𝑇13 + 𝑇31

1 2 1 2 1 2

31 13 ) (𝑇11 + 𝑇11 31 13 ) (𝑇22 + 𝑇22 31 13 ) (𝑇33 + 𝑇33 ]

(28)

33 33 𝑇12 + 𝑇21 33 33 ] 𝑇23 + 𝑇32 33 33 𝑇13 + 𝑇31

1 12 1 23 1 31 12 21 21 ) 23 32 32 ) 31 13 13 ) (𝑇12 + 𝑇21 (𝑇12 + 𝑇21 (𝑇 + 𝑇21 + 𝑇12 + 𝑇21 + 𝑇12 + 𝑇21 + 𝑇12 + 𝑇21 2 2 2 12 1 12 1 23 1 31 12 21 21 ) 23 32 32 ) 31 13 13 ) 𝑇 4 = (𝑇23 (𝑇23 + 𝑇32 (𝑇 + 𝑇32 + 𝑇32 + 𝑇23 + 𝑇32 + 𝑇23 + 𝑇32 + 𝑇23 + 𝑇32 2 2 2 23 1 12 1 23 1 31 12 21 21 ) 23 32 32 ) 31 13 13 ) ( (𝑇13 + 𝑇31 (𝑇 + 𝑇31 + 𝑇13 + 𝑇31 + 𝑇13 + 𝑇31 [2 𝑇13 + 𝑇31 + 𝑇13 + 𝑇31 ] 2 2 13 Now that all the terms have been expressed in the element local system, the geometrically linear part of the stiffness matrix can be written as, 𝑇

(29)

𝐾𝐿 = (𝐵𝑒 ) ∙ 𝐶 ∙ 𝐵𝑒

Next, we derive the geometrically nonlinear contribution to the stiffness formulation. From eq. (21), the derivatives of 𝛿𝐸𝑖𝑗 can be evaluated in the following, 1

𝐷𝛿𝐸𝑖𝑗 = 2 (𝐷𝑢,𝑖 ∙ 𝛿𝑢,𝑗 + 𝑔𝑖 ∙ 𝐷𝛿𝑢,𝑗 + 𝐷𝑢,𝑗 ∙ 𝛿𝑢,𝑖 + 𝑔𝑗 ∙ 𝐷𝛿𝑢,𝑖 )

(30)

By neglecting the higher-order terms, we obtain, 1

(31)

𝐷𝛿𝐸𝑖𝑗 = 2 (𝐷𝑢,𝑖 ∙ 𝛿𝑢,𝑗 + 𝐷𝑢,𝑗 ∙ 𝛿𝑢,𝑖 )

The derivation of each 𝛿𝐸𝑖𝑗 term involves directional derivatives and thus the final geometrically nonlinear part of the stiffness matrix becomes a block-diagonal matrix. 𝐾𝑁𝐿,𝑑𝑖𝑎𝑔

0

0

0

𝐾𝑁𝐿,𝑑𝑖𝑎𝑔

0

0

0

𝐾𝑁𝐿,𝑑𝑖𝑎𝑔

𝐾𝑁𝐿 = [

where

(32) ]

𝐾𝑁𝐿,𝑑𝑖𝑎𝑔 = 𝛹1 𝑇 ∙ 𝛹1 ∙ 𝑆11 + 𝛹2 𝑇 ∙ 𝛹2 ∙ 𝑆22 + 𝛹3 𝑇 ∙ 𝛹3 ∙ 𝑆33 + (𝛹2 𝑇 ∙ 𝛹1 + 𝛹1 𝑇 ∙ 𝛹2 ) ∙ 𝑆12

(33)

+ (𝛹3 𝑇 ∙ 𝛹2 + 𝛹2 𝑇 ∙ 𝛹3 ) ∙ 𝑆23 + (𝛹1 𝑇 ∙ 𝛹3 + 𝛹3 𝑇 ∙ 𝛹1 ) ∙ 𝑆31 In eq. (33), 𝛹𝑖 denotes the full 1-by-[(𝑝 + 1)2 (𝑞 + 1)] spline basis vector (i.e., the multiplication of each in-plane NURBS basis vector term by each through-thickness B-spline basis vector term). In specific, 𝛹1𝑘 = 𝑅𝑖,1 𝑁𝑗 , 𝛹2𝑘 = 𝑅𝑖,2 𝑁𝑗 , 𝛹3𝑘 = 𝑅𝑖 𝑁𝑗,3

(34)

where the superscript k represents the kth element of 𝛹, 𝑘 = 𝑖 + 𝑗(𝑝 + 1)2 . [𝑆𝑖𝑗 ] denotes the second 𝑒 ] pre-multiplied by the transpose of the transformation matrix 𝑇 in eq. Piola-Kirchhoff stress vector [𝑆𝑖𝑗 (27). Note that this is not intended to transform the second Piola-Kirchhoff stress tensor from the element local system to the curvilinear system, but is a result of the transformation of the strain tensor [𝐸𝑖𝑗 ] from curvilinear to element local coordinate system.

4.3. Numerical integration of FGMs Due to the drastic change in material properties of FGMs governed by the power-law distribution function, common approaches of Gauss integration (i.e., n-point Gauss integration for a univariate Bspline of order 2n-1 or less) may not be accurate. Therefore, the through-thickness integration should either be enriched with sufficient number of quadrature points or be represented by higher-order B-splines to match the power-law distribution function. The integration scheme in the current study is explained as follows: in each shell curvilinear plane, the same number of quadrature points is employed in each direction as the NURBS polynomial order, i.e., for a bi-cubic NURBS, a 3-by-3 integration rule is adopted. The thickness integration is treated differently from the in-plane integration, where a total of 5 integration points with just one thickness element are used to arrive at a converged solution in the cases show below. Nevertheless, it should be noted that the number of integration points needed for exact numerical integration is dependent on the power-law function order. Generally speaking, n-point Gauss quadrature is able to yield an exact integration for polynomials of degree 2n-1 or less. Alternatively, a higher-order B-spline basis (i.e., cubic or higher) could be used for the thickness description with equal number of quadrature points for numerical integration.

5. Numerical examples In this section, we apply the developed isogeometric continuum shell element to assess a number of benchmark FGM shell problems involving geometric nonlinearity. Specifically, a set of bi-cubic NURBS basis functions are employed for the in-plane curvilinear discretization and a quadratic B-spline is used to represent the thickness space. In regards to the solution scheme, the Newton-Raphson incrementaliterative solution procedure [130] is adopted. Additionally, all the numerical examples are modeled with isotropic materials first to verify the formulation and are then extended to FGM shell analysis. In all the FGM shell modeling, a ceramic-metal FGM material type is adopted with material properties 𝐸𝑐 = 380 𝐺𝑃𝑎 and 𝐸𝑚 = 70 𝐺𝑃𝑎 for ceramic and metal, respectively. In terms of the application of boundary conditions, they can be directly applied at the control points since these control points correspond to the end of the knot vectors and are therefore interpolatory. All the through-thickness control points are fixed when applying clamped boundary conditions, whereas only the control points on the mid-surface are fixed in simply supported boundary conditions. Symmetry boundary conditions are applied to all the corresponding through-thickness control points as well.

5.1. Annular FGM plate subject to tip distributed load The first numerical example is concerned with the classical annular plate that is clamped at one end and subjected to top distributed load at the other end. The internal and external radii of the annulus are 𝑟𝑖 = 6.0 𝑚 and 𝑟𝑒 = 10.0 𝑚, respectively. The plate has a constant thickness of 0.03 𝑚. The isotropic material properties are: Young’s modulus 𝐸 = 2.1 × 107 𝑃𝑎 and Poisson’s ratio 𝜈 = 0 . The tip distributed load is 𝑞 = 0.8 𝑁/𝑚. The detailed problem setup and deformed configuration can be found in Fig. 3, where we used a 32 × 8 in-plane mesh to obtain the final converged solution.

Fig. 3. The annular plate problem: (a) model setup and (b) deformed configuration at peak loads. For verification purposes, the force-displacement curves at point A, B and C (see Fig. 3) are plotted and compared to solutions provided by Sze et al. [131], as is illustrated in Fig. 4. We observe a very good agreement between our solutions and the reference solutions. Therefore, we conclude that the developed isogeometric continuum shell formulation is accurate.

Fig. 4. Load-displacement curves of the annular plate problem. Next, we investigate the performance of the proposed continuum shell element on modeling the nonlinear response of FGM shells with exactly the same geometry. As is aforementioned, the FGM of the ceramic-metal type is adopted here as an example. The deformation paths at point A, B and C are illustrated in Fig. 5-7 with various power-law exponents, respectively, where we observe a gradually stiffer response of the annulus plate with the decrease of the power-law exponent with the ceramic shell the stiffest. Note that the solutions of the FGM shell made of full ceramic and in the case of 𝜅 = 0.2 are compared to solutions available in [21] for verification purposes.

Fig. 5. Deformation path at point A of the annular plate with various power-law exponents.

Fig. 6. Deformation path at point B of the annular plate with various power-law exponents.

Fig. 7. Deformation path at point C of the annular plate with various power-law exponents. 5.2. Pinched FGM hemisphere In the second numerical example, we study the nonlinear response of the benchmark pinched hemisphere problem. A reference solution can be found in [102]. As is demonstrated in Fig. 8, the hemisphere is fixed at the vertex and subjected to two pairs of pinching and pull-out forces along the x and y axes. Due to symmetry, only a quarter of the full model is simulated. The geometry details and material properties of the model can be found in Fig. 8(a) and a relatively coarse NURBS control polygon is illustrated in Fig. 8(b).

Fig. 8. The pinched hemisphere test: (a) problem setup and (b) control polygon of a quarter of the model. A 16 × 16 in-plane mesh of the quarter hemisphere model is adopted to get the converged solution. For illustration purposes, the deformed configuration is plotted in Fig. 9, where Fig. 9(a) is colored in pull-out displacement and Fig. 9(b) colored in pinching displacement. The radial displacement vs. load relations at the points where external loads are applied (i.e., point A and B) are plotted in Fig. 10, where a convergence study is also performed using 14 × 14, 16 × 16, and 18 × 18 in-plane meshes. Again, our solutions match very well with reference solutions.

Fig. 9. The deformed configuration of the pinched hemisphere test colored in (a) pull-out displacement and (b) pinching displacement.

Fig. 10. Deformation paths of the pinched hemisphere problem.

Subsequently, the material is replaced to FGM with the same geometry to explore the nonlinear behavior of FGM shells with changing power-law exponents. Similar to the observation in the annular plate example, the hemisphere exhibits a much stiffer behavior as the power-law exponent decreases, while the response changes gradually from that of the pure metal model to the solution of the pure ceramic model.

Fig. 11. Deformation path at point A of the pinched hemisphere with varying power-law exponents.

Fig. 12. Deformation path at point B of the pinched hemisphere with varying power-law exponents.

5.3. Pull-out of a FGM cylindrical shell In the last example, the pull-out of a cylindrical shell is considered. As demonstrated in Fig. 13, the cylindrical shell is open-ended and is under two pull-out forces at the middle of the cylinder. A reference solution can be found in [131]. Due to symmetry, only one octant of the full cylinder is modeled. The problem details are drawn in Fig. 13(a) and the deformed configuration at peak loads is plotted in Fig. 13(b).

Fig. 13. The pull-out of a cylindrical shell problem: (a) model description and (b) deformed configuration at peak loads. In terms of the isotropic cylindrical shell analysis, a 12 × 12 in-plane mesh of the one-octant model is employed to arrive at a converged solution, and the deformation paths at point A, B and C are plotted in Fig. 14. We observe a perfect agreement between our solutions and the reference solutions.

Fig. 14. Deformation paths of the pull-out of the cylindrical shell problem. The nonlinear response of the FGM cylindrical shell is demonstrated here as well. In this case, the magnitude of the two concentrated pulling forces is 1.2 × 106 𝑁 . Fig. 15-17 illustrates the loaddisplacement relations at point A, B and C, respectively. We can see that, despite the quite challenging behavior of the FGM cylindrical shell, the proposed isogeometric continuum shell approach is capable of capturing the nonlinear response very well. Additionally, compared to the converged solution reported in [21] where a 16-by-16 mesh was used, our solution with the 12-by-12 mesh is computationally efficient.

Fig. 15. Deformation path at point A of the cylindrical FGM shell with varying power-law exponents.

Fig. 16. Deformation path at point B of the cylindrical FGM shell with varying power-law exponents.

Fig. 17. Deformation path at point C of the cylindrical FGM shell with varying power-law exponents.

6. Conclusion As a summary, we presented a geometrically nonlinear continuum shell formulation based on IGA that is suitable for the analysis of FGM structures. The developed continuum shell element characterizes a precise description of the thickness-varying material properties in FGM in the sense that a set of desired high-order B-spline basis functions with sufficient number of integration points are used for accurate through-thickness numerical integration. In addition, the geometrically nonlinear effect is considered in the formulation and therefore the proposed shell element is suitable for large-displacement analysis. The performance of the isogeometric continuum shell model is evaluated with a number of benchmark nonlinear shell problems and numerical results prove the accuracy of the proposed model. Moreover, the shell element is also applied to study challenging FGM problems and numerical solution demonstrates that the proposed formulation is a viable approach for modeling the nonlinear response of FGM shell structures.

Acknowledgement The authors are grateful for the support provided by the U.S. Naval Air Warfare Center (N68335-17C-0196) for which Dr. Gabriel Murray serves as the technical monitor.

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Ning Liu: Conceptualization; Data curation; Formal analysis; Methodology; Investigation; Software; Validation; Visualization; Writing Xiang Ren: Project administration; Reviewing and editing Jim Lua: funding acquisition; Reviewing and editing