Quadratic Solid–Shell Finite Elements for Geometrically ... - MDPI

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Quadratic Solid–Shell Finite Elements for Geometrically Nonlinear Analysis of Functionally Graded Material Plates Hocine Chalal and Farid Abed-Meraim *

ID

Laboratory LEM3, Université de Lorraine, CNRS, Arts et Métiers ParisTech, F-57000 Metz, France; [email protected] * Correspondence: [email protected]; Tel.: +33-3-87375-479 Received: 30 May 2018; Accepted: 17 June 2018; Published: 20 June 2018







Abstract: In the current contribution, prismatic and hexahedral quadratic solid–shell (SHB) finite elements are proposed for the geometrically nonlinear analysis of thin structures made of functionally graded material (FGM). The proposed SHB finite elements are developed within a purely 3D framework, with displacements as the only degrees of freedom. Also, the in-plane reduced-integration technique is combined with the assumed-strain method to alleviate various locking phenomena. Furthermore, an arbitrary number of integration points are placed along a special direction, which represents the thickness. The developed elements are coupled with functionally graded behavior for the modeling of thin FGM plates. To this end, the Young modulus of the FGM plate is assumed to vary gradually in the thickness direction, according to a volume fraction distribution. The resulting formulations are implemented into the quasi-static ABAQUS/Standard finite element software in the framework of large displacements and rotations. Popular nonlinear benchmark problems are considered to assess the performance and accuracy of the proposed SHB elements. Comparisons with reference solutions from the literature demonstrate the good capabilities of the developed SHB elements for the 3D simulation of thin FGM plates. Keywords: quadratic solid–shell elements; finite elements; functionally graded materials; thin structures; geometrically nonlinear analysis

1. Introduction Over the last decades, the concept of functionally graded materials (FGMs) has emerged, and FGMs were introduced in the industrial environment due to their excellent performance compared to conventional materials. This new class of materials was first introduced in 1984 by a Japanese research group, who made a new class of composite materials (i.e., FGMs) for aerospace applications dealing with very high temperature gradients [1,2]. These heterogeneous materials are made from several isotropic material constituents, which are usually ceramic and metal. Among the many advantages of FGMs, their mechanical and thermal properties change gradually and continuously from one surface to the other, which allows for overcoming delamination between interfaces as compared to conventional composite materials. In addition, FGMs can resist severe environment conditions (e.g., very high temperatures), while maintaining structural integrity. Thin structures are widely used in the automotive industry, especially through sheet metal forming into automotive components. In this context, the finite element (FE) method is considered nowadays as a practical numerical tool for the simulation of thin structures. Traditionally, shell and solid elements are used in the simulation of linear and nonlinear problems. However, the simulation results require very fine meshes to obtain accurate solutions due to the various locking phenomena

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that are inherent to these elements, which lead to high computational costs. To overcome these issues, many researchers have devoted their works to the development of locking-free finite elements. More specifically, the technology of solid–shell elements has become an interesting alternative to traditional solid and shell elements for the efficient modeling of thin structures (see, e.g., [3–8]). Solid-shell elements are based on a fully 3D formulation with only nodal displacements as degrees of freedom. They can be easily combined with various fully 3D constitutive models (e.g., orthotropic elastic behavior, plastic behavior), without any further assumptions, such as plane-stress assumptions. Based on the reduced-integration technique (see, e.g., [9]), they are often combined with advanced strategies to alleviate locking phenomena, such as the assumed-strain method (ASM) (see, e.g., [4]), the enhanced assumed strain (EAS) formulation (see, e.g., [10]), and the assumed natural strain (ANS) approach (see, e.g., [11]). Several FE formulations for the analysis of thin FGM structures have been developed in the literature. They can be classified into three main formulations: The shell-based FGM FE formulation, the solid-based FGM FE formulation, and the solid–shell-based FE formulation. The first formulation is considered as the most widely adopted approach for the modeling of 2D thin FGM structures. However, this approach requires specific kinematic assumptions in the FE formulation, such as the classical Kirchhoff plate theory, first and high-order shear theories, plane-stress assumption (see, e.g., [12–15]). The second approach is based on a 3D formulation of solid elements, in which a fully 3D elastic behavior for FGMs is adopted. In such an approach, some specific kinematic assumptions for thin plates, such as the classical Kirchhoff plate theory and the von Karman theory, are also adopted in the FE formulation (see, e.g., [16–18]). The third approach is based on the concept of solid–shell elements, which are combined with FGM behavior. Few works in the literature have investigated the behavior of thin FGM plates with this approach. Among them, the work of Zhang et al. [19], who investigated the piezo-thermo-elastic behavior of FGM shells with EAS-ANS solid–shell elements. Recently, Hajlaoui et al. [20,21] studied the buckling and nonlinear dynamic analysis of FGM shells using an EAS solid–shell element based on the first-order shear deformation concept. In this work, quadratic prismatic and hexahedral shell-based (SHB) continuum elements, namely SHB15 and SHB20, respectively, are proposed for the modeling of thin FGM plates. SHB15 is a fifteen-node prismatic solid-shell element with a user-defined number of through-thickness integration points, while SHB20 is a twenty-node hexahedral solid-shell element with a user-defined number of through-thickness integration points. These solid-shell elements have been first developed in the framework of isotropic elastic materials and small strains (see [22]), and recently coupled with anisotropic elastic–plastic behavior models within the framework of large strains for the modeling of sheet metal forming processes [23]. In this paper, the formulations of the quadratic SHB15 and SHB20 elements are combined with functionally graded behavior for the modeling of thin FGM plates. To achieve this, the elastic properties of the proposed elements are assumed to vary gradually in the thickness direction according to a power-law volume fraction. The resulting formulations are implemented into the quasi-static ABAQUS/Standard software. The performance of the proposed elements is assessed through the simulation of various nonlinear benchmark problems taken from the literature. 2. SHB15 and SHB20 Solid-Shell Elements 2.1. Element Reference Geometries The proposed SHB elements are based on a 3D formulation, with displacements as the only degrees of freedom. Figure 1 shows the reference geometry of the quadratic prismatic SHB15 and quadratic hexahedral SHB20 elements and the position of the associated integration points. Within the reference frame of each element, direction ζ represents the thickness, along which several integration points can be arranged.

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Figure 1. 1. Reference Reference geometry geometry of of (a) (a) quadratic quadratic prismatic prismatic SHB15 SHB15 element element and and (b) (b) quadratic quadratic hexahedral hexahedral Figure SHB20 element, element, and and position position of of the the associated associated integration integrationpoints. points. SHB20

2.2. Quadratic Quadratic Approximation Approximation for for the the SHB SHB Elements Elements 2.2. Conventional quadratic quadratic interpolation interpolation functions functions for for traditional traditional continuum continuum prismatic prismatic and and Conventional hexahedral elements are used in the formulation of the SHB elements. According to this formulation, hexahedral elements are used in the formulation of the SHB elements. According to this formulation, the spatial spatial coordinates coordinates xxi and and the the displacement displacement field field uui are are approximated approximated using using the the following following the interpolation functions: functions: interpolation

i

i

KK

xi =x x=iI xNIN (ξ,(ξη,,ηζ ,)ζ=) =∑ xxiI N  iI NII ((ξ,ξ ,ηη,, ζζ )), , i iI I I =I 1=1

(1) (1)

K

ui = diI NI (ξ, η, ζ ) = ∑K diI NI (ξ, η, ζ ), , ui = d iI N I (ξ ,η , ζ ) =I = 1 d iI N I (ξ ,η , ζ ) I =1

(2) (2)

where diI are the nodal displacements, i = 1, 2, 3 correspond to the spatial coordinate directions, and I diI are nodal displacements, to which the spatial coordinate directions, wherefrom 2, 3 correspond varies 1 to the K, with K being the numberi =of1,nodes per element, is equal to 15 for the SHB15 and I varies to K, with element. K being the number of nodes per element, which is equal to 15 for the element and from 20 for1the SHB20 SHB15 element and 20 for the SHB20 element. 2.3. Strain Field and Gradient Operator 2.3. Strain Field and Gradient Operator Using the above approximation for the displacement within the element, the linearized strain above approximation for the displacement within the element, the linearized strain tensorUsing ε canthe be derived as: tensor ε can be derived as:

1 1 (3) ε ij = u + u j, i = d N + d jI NI, i . 2 1i, j 21 iI I, j ε ij = ui , j + u j , i = diI N I , j + d jI N I , i . (3) By combining Equations (1) and2 (2) with the 2help of the interpolation functions, the nodal displacement vectorsEquations di write: (1) and (2) with the help of the interpolation functions, the nodal By combining displacement vectors di write: di = a0i s + a1i x1 + a2i x2 + a3i x3 + ∑ cαi hα , i = 1, 2, 3, (4) d i = a0 i s + a1i x1 + a2 i x 2 + a3i x 3 + αα cα i hα , i = 1, 2, 3 , (4)

(

)

(

)

T where , ·, ·x· ,) xiK nodal coordinate vectors. In Equation (4), index α goes ) are = ((xxi1i1, ,xix2 i2 , x, i 3x,i3 where xxi Ti = are the the nodal coordinate vectors. In Equation (4), index α goes from iK from 1 to 11 for the SHB15 element, and from 1 to 16 for the SHB20 element. In addition, vector 1 to 11 for the SHB15 element, and from 1 to 16 for the SHB20 element. In addition, vector sTT = (1, 1, · · · , 1) has fifteen constant components in the case of the SHB15 element, and twenty s = (1, 1, , 1) has fifteen constant components in the case of the SHB15 element, and twenty constant components vector for the SHB20 element. Vectors hα are composed of hα functions, which hα expressions are composed constant components vector for the SHB20 Vectors α functions, are evaluated at the element nodes, and theelement. full details of their canofbehfound in [23].which are evaluated the of element and theorthogonality full details of their expressions canthe beHallquist found in [23]. With the at help some nodes, well-known conditions and of [24] vectors ∂N the help of some well-known orthogonality conditions and of the Hallquist [24] vectors bi = With , where vector N contains the expressions of the interpolation functions NI , ∂xi |ξ =η =ζ =0 ∂ N the a ji and cNαi in Equation can be derived as: interpolation functions NI , the , where vector contains the(4)expressions of the bi =unknown constants ∂xi |ξ =η =ζ = 0

unknown constants a ji and cαi in Equation (4) can be derived as:

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a ji = bTj · di ,

cαi = γαT · di ,

(5)

where the complete details on the expressions of vectors γα can be found in [22]. By introducing the discrete gradient operator B, the strain field in Equation (3) writes:      ∇ s (u) =    

u x,x uy,y uz,z u x,y + uy,x uy,z + uz,y u x,z + uz,x

     dx      = B · d = B ·  dy ,   dz 

(6)

where the expression of the discrete gradient operator B is:      B=   

bTx + hα,x γαT 0 0 byT + hα,y γαT 0 bzT + hα,z γαT

0 byT + hα,y γαT 0 bTx + hα,x γαT bzT + hα,z γαT 0

0 0 bzT + hα,z γαT 0 T by + hα,y γαT bTx + hα,x γαT

     .   

(7)

2.4. Hu–Washizu Variational Principle The SHB solid-shell elements are based on the assumed-strain method, which is derived from. the simplified form of the Hu–Washizu variational principle [25]. In terms of assumed-strain rate ε, . interpolated stress σ, nodal velocities d, and external nodal forces fext , this principle writes . Z .T .T π ε = δε · σ dΩ − δd · fext = 0.

(8)

Ωe

The assumed-strain rate is expressed in terms of the discrete gradient operator B as: .

.

ε( x, t) = B · d.

(9)

Substituting the expression of the assumed-strain rate given by Equation (9) into the variational principle (Equation (8)), the expressions of the stiffness matrix Ke and the internal forces fint for the SHB elements are Z Ke = BT · Ce (ζ ) · B dΩ + KGEOM , (10) Ωe

fint =

Z

BT · σ dΩ,

(11)

Ωe

where KGEOM is the geometric stiffness matrix. As to the fourth-order tensor Ce (ζ ), it describes the functionally graded elastic behavior of the FGM material. Its expression is given hereafter. 2.5. Description of Functionally Graded Elastic Behavior In the framework of large displacements and rotations, the formulation of the SHB elements requires the definition of a local frame with respect to the global coordinate system, as illustrated in Figure 2. The local frame, which is designated as the “element frame” in Figure 2, is defined for each element using the associated nodal coordinates. In such an element frame, where the ζ-coordinate represents the thickness direction, the fourth-order elasticity tensor Ce (ζ ) for the FGM is specified.

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2. Element frame and global frame for the proposed SHB elements. Figure Figure 2. Element frame and global frame y for the proposed SHB elements.

x “global frame” In this work, a two-phase FGM is considered, which consists of two constituent mixtures of ceramic anda metal. The ceramic of the FGM can sustain very high gradients, In this work, two-phase FGMphase is considered, which consists of temperature two constituent mixtures of Figure 2. Element frameprevents and global the proposed SHB while the ductility of the metal phase theframe onset for of fracture due to theelements. cyclic thermal loading. ceramic and metal. The ceramic phase of the FGM can sustain very high temperature gradients, In such FGMs, the material at the bottom surface of the plate is fully metal and at the top surface of while the the ductility offully theaceramic, metal phase prevents the onset of consists fracture thetopcyclic thermal Inplate this iswork, two-phase FGM is considered, ofdue twoto constituent mixtures as illustrated in Figure 3. which Between these bottom and surfaces, the ofloading. metal. vary Theatceramic phasethrough of the FGM can plate sustain high gradients, In such ceramic FGMs, the material the bottom surface of thickness the is very fully metal and atproperties, the top surface of elastic and properties continuously the from metal to temperature ceramic the ductility ofas theillustrated metal phasein prevents of fracture to the cyclic thermal loading. according to a power-law volumethe fraction. The corresponding volume fractions for the the elastic the platewhile isrespectively, fully ceramic, Figure 3. onset Between thesedue bottom and top surfaces, In ceramic such FGMs, at the phase bottomf msurface of the plate is fully metal and at the top surface of phasethef c material and the metal are expressed as (see, e.g., [26,27]): properties vary continuously through the thickness from metal to ceramic properties, respectively, the plate is fully ceramic, as illustrated in Figure 3. Between these bottom and top surfaces, the n according to a properties power-law volume fraction.through The fractions for the ceramic phase 1  the thickness volume  z corresponding elastic vary continuously metal to ceramic properties, f c =  +  and f m = 1 − f c from , (12) t 2   f c and the metal phase f m are (see, e.g., [26,27]): respectively, according to expressed a power-lawas volume fraction. The corresponding volume fractions for the f c and the exponent, ceramic metal phase (see,toe.g., [26,27]): wherephase n is the power-law which greater than oras equal zero, and z ∈ [ − t 2, t 2] , with t nexpressed  f misare

z

1

0 , the material the thickness of the plate. For nf c== isn fully ceramic, while when n → ∞ the material is +  z 1  and f m = 1 − f c , f ct=  2+  and f m = 1 − f c , (12) fully metal (see Figure 4).

t

(12)

2

where nwhere is the npower-law exponent, which is greater than or equal to zero, and z ∈ [−t/2, t/2], with t the is the power-law exponent, which is greater than or equal to zero, and z ∈ [ − t 2, t 2] , with t thickness of the plate. For n = 0, the material is fully ceramic, while when ζ n → ∞ the material is fully the thickness of the plate. For n = 0 , the material is fully ceramic, while when n →η ∞ the material is metal (see Figure 4). fully metal (see Figure 4). Ceramic surface

z

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Figure 3. Schematic representation Ceramic surface of the functionally graded thin plate. SHB element

y

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Metal surface

x

Figure 3. Schematic representation of of the graded thin thin plate.plate. Figure 3. Schematic representation thefunctionally functionally graded

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n = 0.01 n = 0.1

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Dimensionless thickness 4. Volume fraction distribution of the ceramic phase as function of the power-law exponent n. Figure 4.Figure Volume fraction distribution of the ceramic phase as function of the power-law exponent n.

For an isotropic elastic behavior, the constitutive equations are governed by the Hooke elasticity law, which is expressed by the following relationship:

σ = λ tr ( ε ) 1 + 2μ ε , where 1 denotes the second-order unit tensor, λ and μ are the Lamé constants given by:

(13)

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For an isotropic elastic behavior, the constitutive equations are governed by the Hooke elasticity law, which is expressed by the following relationship: σ = λtr (ε)1 + 2µε,

(13)

where 1 denotes the second-order unit tensor, λ and µ are the Lamé constants given by: λ=

νE E and µ = , 2(1 + ν ) (1 − 2ν)(1 + ν)

(14)

with E and ν the Young modulus and the Poisson ratio, respectively. For FGMs that are made of ceramic and metal constituents, it is commonly assumed that only the Young modulus E varies in the thickness direction, while the Poisson ratio ν is kept constant. Therefore, the constant Young modulus in Equation (14) is replaced by E(z), whose value evolves according to the following power-law distribution: E(z) = ( Ec − Em ) f c + Em ,

(15)

where Ec and Em are the Young modulus of the ceramic and metal, respectively. 3. Nonlinear Benchmark Problems In this section, the performance of the proposed elements is assessed through the simulation of several popular nonlinear benchmark problems. The static ABAQUS/Standard solver has been used to solve the following static benchmark problems. More specifically, the classical Newton method is considered for most benchmark problems, aside from limit-point buckling problems for which the Riks arc-length method is used. To accurately describe the variation of the Young modulus through the thickness of the FGM plates, only five integration points within a single element layer is used in the simulations. For each benchmark problem, the simulation results given by the proposed elements are compared to the reference solutions taken from the literature. In the subsequent simulations, it is worth noting that the elastic properties of the metal and ceramic constituents of the FGM plates do not reflect a real metallic or ceramic material. Indeed, the terms metal and ceramic are commonly used in the literature to emphasize the difference between the properties of the FGM constituents (see, e.g., [15,26,27]). Regarding the meshes used in the simulations, the following mesh strategy is adopted: (N1 × N2 ) × N3 for the hexahedral SHB20 element, where N1 is the number of elements along the length, N2 is the number of elements along the width, and N3 is the number of elements along the thickness direction. As to the prismatic SHB15 element, the mesh strategy consists of (N1 × N2 × 2) × N3 , due to the in-plane subdivision of a rectangular element into two triangles. 3.1. Cantilever Beam Sujected to End Shear Force Figure 5a shows a simple cantilever FGM beam with a bending load at its free end. This is a classical popular benchmark problem, which has been widely considered in many works for the analysis of cantilever beams with isotropic material (see, e.g., [28,29]). The Poisson ratio of the FGM beam is assumed to be ν = 0.3, while the Young modulus of the metal and ceramic constituents are Em = 2.1 × 105 Mpa and Ec = 3.8 × 105 Mpa, respectively. Figure 5b illustrates the final deformed shape of the cantilever beam with respect to its undeformed shape, as discretized with SHB20 elements, in the case of fully metallic material. Figure 6 shows the load–deflection curves obtained with the quadratic SHB elements, along with the reference solutions taken from [15], for various values of the power-law exponent n. One recalls that fully metallic material is obtained when n → ∞ , and fully ceramic material for n = 0. Overall, the SHB elements show excellent agreement with the reference solutions corresponding to the various values of exponent n. More specifically, it can be observed

Em = 2.1× 105 MPa and Ec = 3.8 × 105 MPa , respectively. Figure 5b illustrates the final deformed

shape of the cantilever beam with respect to its undeformed shape, as discretized with SHB20 elements, in the case of fully metallic material. Figure 6 shows the load–deflection curves obtained with the quadratic SHB elements, along with the reference solutions taken from [15], for various values of11,the Materials 2018, 1046power-law exponent n. One recalls that fully metallic material is obtained when 7 of 15 n → ∞ , and fully ceramic material for n = 0 . Overall, the SHB elements show excellent agreement with the reference solutions corresponding to the various values of exponent n. More specifically, it thatcan thebebending behavior the FGM beamof lies thatlies of the fully that ceramic fully metal observed that the of bending behavior thebetween FGM beam between of theand fully ceramic beam, which is consistent with the power-law distribution of the Young modulus in the thickness and fully metal beam, which is consistent with the power-law distribution of the Young modulus in direction. Another advantage of the proposedofSHB elements SHB is that, using the same in-plane mesh the thickness direction. Another advantage the proposed elements is that, using the same discretization as in reference [15], five integration points throughpoints the thickness arethickness sufficient in-plane mesh discretization as inonly reference [15], only five integration through the for are the sufficient SHB elements, ten integration points have been considered [15] to simulate for the while SHB elements, while ten integration points have beeninconsidered in [15] this to simulate problem. this benchmark problem. benchmark

q

t = 0.1 mm

(a)

(b)

Figure 5. Cantilever beam: (a) geometry and (b) undeformed and deformed configurations.

Figure Cantilever beam: (a) geometry and (b) undeformed and deformed configurations. 8 of 17 Materials 2018, 11, 5. x FOR PEER REVIEW

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Metal from [15] Metal with SHB20 n = 5 from [15] n = 5 with SHB20 n = 2 from [15] n = 2 with SHB20 n = 1 from [15] n = 1 with SHB20 n = 0.5 from [15] n = 0.5 with SHB20 n = 0.2 from [15] n = 0.2 with SHB20 Ceramic from [15] Ceramic with SHB20

1.0

Metal from [15] Metal with SHB15 n = 5 from [15] n = 5 with SHB15 n = 2 from [15] n = 2 with SHB15 n = 1 from [15] n = 1 with SHB15 n = 0.5 from [15] n = 0.5 with SHB15 n = 0.2 from [15] n = 0.2 with SHB15 Ceramic from [15] Ceramic with SHB15

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Figure 6. Load–deflection curves for the cantilever beam. (a) Prismatic SHB15 element; (b) Figure 6. Load–deflection curves for the cantilever beam. (a) Prismatic SHB15 element; (b) hexahedral hexahedral SHB20 element. SHB20 element.

3.2. Slit Annular Plate 3.2. Slit Annular Plate In this section, the well-known slit annular plate problem is considered (see, e.g., [29–31]). The In this section, the well-known slitloaded annular is P, considered (see, [29–31]). annular plate is clamped at one end and by plate a line problem shear force as illustrated in e.g., Figure 7a. The annular plate is clamped at one end and loaded by a line shear force P, as illustrated in Figure R = 6 m R = 10 m The inner and outer radius of the annular plate are i and o , respectively, while 7a. The inner and outer radius of the annular plate are R = 6 m and R = 10 m, while the o the thickness is t = 0.03 m . The Poisson ratio of thei annular plate is ν = 0.3respectively, , while the Young thickness is t = 0.03 m. The Poisson ratio of the annular plate is ν = 0.3, while the Young modulus of modulus of the metal and ceramic constituents are Em = 21 GPa and Ec = 38 GPa , respectively. the metal and ceramic constituents are Em = 21 Gpa and Ec = 38 Gpa, respectively. Figure 7b illustrates Figure 7b illustrates the undeformed and deformed shapes of the annular plate, as discretized with the undeformed and deformed shapes of the annular plate, as discretized with SHB20 elements, in the SHB20 elements, in the case of fully metallic material. Figure 8 reports the load–out-of-plane vertical case of fully metallic material. Figure 8 reports the load–out-of-plane vertical deflection curves at deflection curves at the outer point A of the annular plate as obtained with the SHB elements, along the outer point A of the annular plate as obtained with the SHB elements, along with the reference with the reference solutions taken from [15]. One can observe that the SHB elements perform very well with respect to the reference solutions for all considered values of exponent n. Similar to the previous benchmark problem, the same in-plane mesh discretization as in [15] with only five integration points through the thickness has been adopted by the proposed SHB elements for this nonlinear test, while ten integration points have been considered in [15].

The inner and outer radius of the annular plate are R i = 6 m and R o = 10 m , respectively, while the thickness is t = 0.03 m . The Poisson ratio of the annular plate is ν = 0.3 , while the Young modulus of the metal and ceramic constituents are Em = 21 GPa and Ec = 38 GPa , respectively. Figure 7b illustrates the undeformed and deformed shapes of the annular plate, as discretized with 8 of 15 SHB20 elements, in the case of fully metallic material. Figure 8 reports the load–out-of-plane vertical deflection curves at the outer point A of the annular plate as obtained with the SHB elements, along with the reference solutions taken fromthat [15]. One can observeperform that thevery SHBwell elements performtovery solutions taken from [15]. One can observe the SHB elements with respect well with respect to the reference solutions for all considered values of exponent n. Similar to the the reference solutions for all considered values of exponent n. Similar to the previous benchmark previous benchmark the same as in-plane meshonly discretization as in [15] through with only problem, the same in-planeproblem, mesh discretization in [15] with five integration points the five integration points through the thickness has been adopted by the proposed SHB elements for this thickness has been adopted by the proposed SHB elements for this nonlinear test, while ten integration nonlinear test, while ten in integration points have been considered in [15]. points have been considered [15]. Materials 2018, 11, 1046

(a)

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Slit annular (a) geometry andundeformed (b) undeformed and deformed configurations. Slit7.annular plate:plate: (a) geometry and (b) and deformed configurations. Materials Figure 2018, Figure 11,7. x FOR PEER REVIEW 9 of 17 600

600 Metal from [15] Metal with SHB15 n = 5 from [15] n = 5 with SHB15 n = 2 from [15] n = 2 with SHB15 n = 1 from [15] n = 1 with SHB15 n = 0.5 from [15] n = 0.5 with SHB15 n = 0.2 from [15] n = 0.2 with SHB15 Ceramic from [15] Ceramic with SHB15

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(b)

Figure8.8.Load–deflection Load–deflection curves curves at plate. (a)(a) Prismatic SHB15 Figure at the the outer outerpoint pointAAfor forthe theslit slitannular annular plate. Prismatic SHB15 element; (b) hexahedral SHB20 element. element; (b) hexahedral SHB20 element.

3.3. Clamped Square Plate under Pressure 3.3. Clamped Square Plate under Pressure Figure 9a illustrates a fully clamped square plate, which is loaded by a uniformly distributed Figure 9a illustrates a fully clamped square plate, which is loaded by a uniformly distributed pressure. The length and thickness of the square plate are L = 1000 mm and t = 2 mm , respectively. pressure. The length and thickness of the square plate are L = 1000 mm and t = 2 mm, respectively. The Poisson ratio is ν = 0.3 , while the Young modulus of the metal and ceramic constituents are The Poisson5 ratio is ν = 0.3, while the Young modulus of the metal and ceramic constituents are E = 2 × 10 MPa and E = 3.8 × 105 MPa , respectively. Considering the problem symmetry, a quarter Em m= 2 × 105 Mpa and Ecc = 3.8 × 105 MPa, respectively. Considering the problem symmetry, a quarter of the plate is discretized. Figure 9b illustrates the undeformed and deformed shapes of the square of the plate is discretized. Figure 9b illustrates the undeformed and deformed shapes of the square plate, plate, as discretized with SHB20 elements, in the case of fully metallic material. The as discretized with SHB20 elements, in the case of fully metallic material. The pressure–displacement pressure–displacement curves for the SHB elements (where the displacement is computed at the curves for the SHB elements (where the displacement is computed at the center of the plate), along with center of the plate), along with the reference solutions taken from [15], are all depicted in Figure 10. the reference solutions taken from [15], are all depicted in Figure 10. The results obtained with the SHB The results obtained with the SHB elements, by adopting only five integration points in the thickness elements, by adopting only five integration points in the thickness direction and the same in-plane direction and the same in-plane mesh discretization as in [15], are in excellent agreement with the mesh discretization in required [15], are ten in excellent agreementintegration with the reference reference solutionsas that through-thickness points. solutions that required ten through-thickness integration points.

p

of the plate is discretized. Figure 9b illustrates the undeformed and deformed shapes of the square plate, as discretized with SHB20 elements, in the case of fully metallic material. The pressure–displacement curves for the SHB elements (where the displacement is computed at the center of the plate), along with the reference solutions taken from [15], are all depicted in Figure 10. The results obtained with the SHB elements, by adopting only five integration points in the thickness Materials 2018, direction 11, 1046 and the same in-plane mesh discretization as in [15], are in excellent agreement with the reference solutions that required ten through-thickness integration points.

9 of 15

p t

(b)

(a)

Figure 9. Clamped square plate: (a) geometry and (b) undeformed and deformed configurations. Materials 2018, x FOR PEER REVIEW 10 of 17 Figure 9. 11, Clamped square plate: (a) geometry and (b) undeformed and deformed configurations. 1.4

1.4

0.8

-3

P (×10 MPa)

1.0

0.6 0.4 0.2

Metal from [15] Metal with SHB20 n = 5 from [15] n = 5 with SHB20 n = 2 from [15] n = 2 with SHB20 n = 1 from [15] n = 1 with SHB20 n = 0.5 from [15] n = 0.5 with SHB20 n = 0.2 from [15] n = 0.2 with SHB20 Ceramic from [15] Ceramic with SHB20

1.2 1.0 0.8

-3

1.2

P (×10 MPa)

Metal from [15] Metal with SHB15 n = 5 from [15] n = 5 with SHB15 n = 2 from [15] n = 2 with SHB15 n = 1 from [15] n = 1 with SHB15 n = 0.5 from [15] n = 0.5 with SHB15 n = 0.2 from [15] n = 0.2 with SHB15 Ceramic from [15] Ceramic with SHB15

0.6 0.4

Mesh (8 × 8 × 2) × 1

0.0

0.2

Mesh 8 × 8 × 1

0.0 0

1

2

Displacement (mm)

(a)

3

0

1

2

3

Displacement (mm)

(b)

Figure 10. Load–deflection curves at the center point for the square plate. (a) Prismatic SHB15

Figure 10. Load–deflection curves at the center point for the square plate. (a) Prismatic SHB15 element; element; (b) hexahedral SHB20 element. (b) hexahedral SHB20 element.

3.4. Hinged Cylindrical Roof

3.4. Hinged Cylindrical Roof

Figure 11a shows a hinged cylindrical roof subjected to a concentrated force at its center. Two Figureof11a shows hinged cylindrical subjected to a concentrated at itst center. types roofs are aconsidered, thick androof thin, with thicknesses t = 12.7 force mm and = 6.35 Two mm, types of roofs are considered, thick and thin, with thicknesses t =geometric-type 12.7 mm andinstabilities t = 6.35 mm, respectively. respectively. Because this nonlinear benchmark test involves (limit-point buckling), the Riks path-following method is usedgeometric-type to follow the load–displacement curves beyond Because this nonlinear benchmark test involves instabilities (limit-point buckling), the limit points. The Poisson of the is ν = 0.3 , while curves the Young modulus of thepoints. the Riks path-following methodratio is used to cylindrical follow theroof load–displacement beyond the limit 3 103 MPa , of respectively. Owing metal and ceramic constituents c = 151 × The Poisson ratio of the cylindricalare roofEmis=ν70=×10 0.3,MPa whileand theEYoung modulus the metal and ceramic 3 Mpa, to the symmetry, one of the is modeled. FigureOwing 11b illustrates the constituents are Em =only 70 × 103quarter Mpa and Ec cylindrical = 151 × 10roof respectively. to the symmetry, undeformed and deformed shapes of the hinged cylindrical roof, as discretized with SHB20 only one quarter of the cylindrical roof is modeled. Figure 11b illustrates the undeformed and deformed elements, in the case of fully metallic material. The load–vertical displacement curves at the central shapes of the hinged cylindrical roof, as discretized with SHB20 elements, in the case of fully metallic point A of the thick and thin hinged cylindrical roofs are shown in Figures 12 and 13, and compared material. The load–vertical displacement curves at the central point A of the thick and thin hinged with the reference solutions taken from [30]. From these figures, it can be seen that the results cylindrical in Figures andelements 13, and are compared the reference obtainedroofs withare theshown proposed quadratic12SHB in goodwith agreement with thesolutions reference taken fromsolutions [30]. From these figures, it can be seen that the results obtained with the proposed quadratic for the different values of exponent n, corresponding to different volume fractions (from SHBfully elements are in good agreement with the reference solutions for the different values of exponent metal to fully ceramic). More specifically, the snap-through and snap-back phenomena, which are typically exhibited in such limit-point buckling reproduced the n, corresponding to different volume fractions (from problems, fully metalare to very fullywell ceramic). More by specifically, proposed SHBand elements. Note phenomena, that, for the thick roof t = 12.7exhibited mm), the in converged solutionsbuckling in the snap-through snap-back which are(i.e., typically such limit-point Figure 12 are obtained by using a mesh of (8 × 8 × 2) × 1 in the case of prismatic SHB15 elements, and problems, are very well reproduced by the proposed SHB elements. Note that, for the thick roof mesh of 8 × 8 × 1 with hexahedral SHB20 elements. As to the thin roof (i.e., t = 6.35 mm), finer (i.e., at = 12.7 mm), the converged solutions in Figure 12 are obtained by using a mesh of (8 × 8 × 2) × 1 meshes of (16 × 16 × 2) × 1 for the prismatic SHB15 elements, and 16 × 16 × 1 for the hexahedral SHB20 in the case of prismatic SHB15 elements, and a mesh of 8 × 8 × 1 with hexahedral SHB20 elements. elements are required to obtain converged results (see Figure 13). These mesh refinements are As to the thin roof (i.e., t = 6.35 mm), finer meshes of (16 × 16 × 2) × 1 for the prismatic SHB15 similar to those used by Sze et al. [29] for the thick and thin roof in the case of an isotropic material as elements, 16 × 16 × 1 for the hexahedral well asand for multilayered composite materials.SHB20 elements are required to obtain converged results (see Figure 13). These mesh refinements are similar to those used by Sze et al. [29] for the thick and thin roof in the case of an isotropic material as well as for multilayered composite materials.

Materials 2018, 11, x FOR PEER REVIEW Materials 2018, 11, x FOR PEER REVIEW

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=mm 12.7or mm or 6.35 12.7t mm 6.35 mmmm tt == 12.7 or 6.35 mm (a) (a)

(a)

(b)

(b) (b)

Figure 11. Hinged cylindrical roof: (a) geometry and (b) undeformed and deformed configurations.

Figure 11.11. Hinged and (b) (b)undeformed undeformedand anddeformed deformed configurations. Figure Hingedcylindrical cylindricalroof: roof: (a) (a) geometry geometry and configurations. Figure 11. Hinged cylindrical roof: (a) geometry and (b) undeformed and deformed configurations. 400

n = 0.2 from [30] Metal from [30] n = 0.2 with SHB15 Metal with SHB15 Ceramic n = 0.2 from [30] from [30] Metal350 from [30] n = 5 from [30] n = 0.2 from [30] Metal from [30] n = 5 with SHB15 n = 0.2 withCeramic SHB15with SHB15 Metal with SHB15 n = 0.2 with SHB15 Metal 300 with SHB15n = 2 from [30] Ceramic from [30] n = 5 from [30] n = 2 with SHB15 Ceramic from [30] n = 5 from [30] Ceramic with SHB15 n = 5 with 250 SHB15n = 1 from [30] Ceramic with SHB15 n = 5 with SHB15n = 1 with SHB15 n = 2 from [30] 200 [30] n = 0.5 from [30] n = 2 from n = 2 with SHB15 n = 2 with SHB15n = 0.5 with SHB15

400

300 300

Load ) ) Load(kN (kN

250 250 200 200 150 150

300

350 350 250

L oad ) L) oad (kN ) L oad(kN (kN

350 350

400

Load (kN )

400 400

n = 1 150 from [30] n = 1 from [30] n = 1 with SHB15 n = 1 with SHB15 n = 0.5100from [30] n = 0.5 from [30] n = 0.550 with SHB15 n = 0.5 with SHB15 0

300 300 200 250 150 250

n = 1 from [30] n = 1 from [30] n = 1 with SHB20 n = 1 with SHB20 n = 0.5 from [30] n = 0.5 from [30] n = 0.5 with SHB20 n = 0.5 with SHB20

100 200

200

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0

100 100

n = 0.2 from [30] Metal from [30] n = 0.2 with SHB20 Metal with SHB20 n =[30] 0.2 from [30] n = 5Metal from [30] from [30] Ceramic from n =SHB20 0.2 from [30] from [30] Ceramic with n = Metal 5 with SHB20 n = 0.2 with SHB20 Metal with SHB20 n = 0.2 with SHB20 n = Metal 2 from [30] with SHB20 Ceramic from [30] n = 5 from [30] n = 2 with SHB20 Ceramic from [30] n = 5 from [30] Ceramic with SHB20 = 5 with n = 1nfrom [30] SHB20 Ceramic with SHB20 n = 5 with SHB20 n = 1nwith = 2SHB20 from [30] 2 from n=n 0.5= from [30] [30] n = 2 with SHB20 n=n 0.5=with SHB20 2 with SHB20

400 350

Mesh 8 × 8 × 1

0

1000

40

10

20

30

40

Displacement (mm)

100

Displacement (mm)

(a)

(b)

50 50 Mesh 8 × 8 × 1 Figure 12. Deflection at the central point A under concentrated force for the thick hinged roof. (a) Mesh (8 × 8 × 2) × 1 Mesh 8 × 8 × 1 Mesh(b)(8hexahedral × 8 × 2) ×SHB20 1 prismatic SHB15 element; element. 0 0 0 10 20 30 40 10 20 30 40 0 10 20 30 40 10 20 30 40 Displacement (mm)

50 50 0 0 0 0

180

Displacement (mm) Displacement (mm)n = 0.2 from [30]

180

(a) (a)

Metal from [30] n = 0.2 with SHB15 Metal with SHB15 Ceramic from [30] n = 5 from [30] 140 Ceramic with SHB15 n = 5 with SHB15 120 n = 2 from [30] n = 2 with SHB15 100 n = 1 from [30] 80 n = 1 with SHB15 60 n = 0.5 from [30] n = 0.2 from [30] Metal40from [30]n = 0.5 with SHB15 n = 0.2 from [30]

160

160

Displacement (mm)

Metal from [30] Metal with SHB20 n = 5 from [30] n = 5 with SHB20 n = 2 from [30] n = 2 with SHB20 n = 1 from [30] n = 1 with SHB20 n = 0.5 from [30] n = 0.5 with from SHB20 [30] Metal

n = 0.2 from [30] (b) n = 0.2 with SHB20 (b) Ceramic from [30]

Load (kN )

80

60 180 180 40 n = 0.2 with SHB15 Metal from [30] 160 Metal with SHB15 n = 0.2 with SHB15 20 160 Ceramic from [30] Metal 20 with SHB15 n = 5 from [30] Ceramic from [30] 140 0 0 [30] Ceramic with SHB15 n = 5 from 140 n = 5 with SHB15 Ceramic with SHB15 -20 -20 SHB15 120 n = 5 with Mesh (16 × 16 × 2) × 1 n = 2 from [30] 120 n = 2 -40 from [30] -40 n = 2 with SHB15 100 0 10 20 30 40 n = 2 with SHB15 100 0 n = 1 from [30] Displacement (mm) 80 n = 1 from [30] n = 1 with SHB15 80 (a) n = 1 with SHB15 60 n = 0.5 from [30] 60 n = 0.5Figure from [30] 13. Deflection at the central point A under concentrated n = 0.5 with SHB15 40 n = 0.5prismatic with SHB15SHB15 element; (b) hexahedral SHB20 element.40 20 20 0 0 -20 Mesh (16 × 16 × 2) × 1 -20 Mesh (16 × 16 × 2) × 1 -40 -40 0 10 20 30 40 0 10 20 30 40 Displacement (mm)

Load ) ) Load(kN (kN

Load ) ) Load(kN (kN

180 180 160 160 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0 0 -20 -20 -40 -40 0 0

Load (kN )

140 Figure 12.Deflection Deflection at at the central point A under concentrated force for the thick hinged roof. (a) Ceramic with SHB20 Figure thecentral centralpoint point under concentrated for the thick roof. Figure12. 12. Deflection at the AA under concentrated forceforce for the thick hingedhinged roof. (a) 120 prismatic SHB15 element; (b) hexahedral SHB20 element. (a)prismatic prismaticSHB15 SHB15 element; hexahedral SHB20 element. element; (b)(b) hexahedral SHB20 element. 100

Displacement (mm)

(a) (a)

n = 0.2 from [30] n = 0.2 from [30] n = 0.2 with SHB20 Metal from [30] Metal with SHB20 n = 0.2 with SHB20 Ceramic from [30] Metal with SHB20 n = 5 from [30] Ceramic from [30] Ceramic with SHB20 n = 5 from [30] n = 5 with SHB20 Ceramic with SHB20 n = 5 with SHB20 Mesh 16 × 16 × 1 n = 2 from [30] n = 2 from [30] n = 2 with SHB20 20 30 40 n =102 with SHB20 n = 1 from [30] Displacement (mm) n = 1 from [30] n = 1 with SHB20 (b) n = 1 with SHB20 n = 0.5 from [30] n = 0.5for from [30]thin hinged roof. (a) force the n = 0.5 with SHB20 n = 0.5 with SHB20

Mesh 16 × 16 × 1 Mesh 16 × 16 × 1 10 10

20

30

20 30 Displacement (mm) Displacement (mm)

40 40

(b) (b)

Figure 13. Deflection at the central point A under concentrated force for the thin hinged roof. (a) Figure13. 13. Deflection Deflection at central AA under concentrated forceforce for the roof. (a) Figure at the the centralpoint point under concentrated forthin the hinged thin hinged roof. prismatic SHB15 element; (b) hexahedral SHB20 element. prismatic SHB15 element; (b) hexahedral SHB20 element. (a) prismatic SHB15 element; (b) hexahedral SHB20 element.

Materials 2018, 11, 1046 Materials 2018, 11, x FOR PEER REVIEW

11 of 15 12 of 17

3.5.3.5. Pull-Out of of ananOpen-Ended Pull-Out Open-EndedCylinder Cylinder Materials 2018, 11, x FOR PEER REVIEW 12 of 17 The well-known pull-out pull-out test cylinder is considered in this As The well-known test for for an anopen-ended open-ended cylinder is considered in section. this section. in Figure 14a, the cylinder is pulled by two opposite radial forces, which results in the Asillustrated illustrated in Figure 14a, the cylinder is pulled by two opposite radial forces, which results in the 3.5. Pull-Out of an Open-Ended Cylinder deformed shape shown in Figure 14b. The isotropic material case as well as the laminated composite deformed shape shown in Figure 14b. The isotropic material case as well as the laminated composite The well-known pull-out test for an open-ended cylinder is considered in this section. As material case have been considered by many authors inliterature the literature (see,[29,30,32]). e.g., [29,30,32]). The material case have considered by many authors the (see, e.g., The illustrated inbeen Figure 14a, the cylinder is pulled by in two opposite radial forces, which results in thePoisson ν = 0.3 Poisson ratio of the cylinder is , while the Young modulus of the metal and ceramic ratio of the cylinder is shown ν = 0.3, while14b. theThe Young modulus ofcase theasmetal ceramic constituents are deformed shape in Figure isotropic material well asand the laminated composite 9 9 E = 0.7 × 10 Pa and E = 1.51 × 10 Pa , respectively. Owing to the symmetry of constituents are 9 9 m c have Ebeen many authors in the literature (see, e.g., [29,30,32]). Em = 0.7material × 10 case Pa and 1.51 × 10byPa, respectively. Owing to the symmetry of theThe problem, c = considered = 0.3The theone problem, eighth the is modeled. The force–radial displacement curves at C Poisson ratio ofcylinder the cylinder is νcylinder , while the Young modulus of the metal ceramic only eighth only of theone isofmodeled. force–radial displacement curves at and points A, B and 9 9 E = 0.7 × 10 Pa and E = 1.51 × 10 Pa , respectively. Owing to the symmetry of constituents are A, Binand C (as14a), depicted in Figure obtained with the elements, are shown Figures m (aspoints depicted Figure obtained with14a), thec SHB elements, are SHB shown in Figures 15–17, in respectively, the problem, only one with eighththe of the cylindersolutions is modeled. Thefrom force–radial displacement curvesthat at the 15–17, respectively, along reference taken [13]. It can be observed along with the reference solutions taken from [13]. It can be observed that the developed SHB elements points SHB A, B and C (as depicted in Figurepass 14a), this obtained with the SHB elements, are shown in Figures developed elements successfully benchmark test as compared to the reference successfully pass this benchmark test as compared to the reference solutions. More specifically, the 15–17,More respectively, along the withtransition the reference solutions from [13]. It can be observed thatwhich the is solutions. specifically, zone in thetaken load–radial displacement curves, transition zone in SHB the load–radial displacement curves, which test is marked by thetosnap-through developed elements successfully pass this benchmark compared reference point, marked by the snap-through point, is well reproduced by both asprismatic and the hexahedral SHB is well reproduced byspecifically, both prismatic and hexahedral elements for the various values solutions. More the transition zone in the SHB load–radial displacement curves, which is of the elements for the various values of the power-law exponent n. Note that the converged solutions in marked by the n. snap-through is well reproduced by in both prismatic andare hexahedral SHB power-law exponent Note that point, the converged solutions Figures 15–17 obtained with the Figures 15–17 are obtained with the proposed elements by using only five integration points in in the elements for the various values of the power-law exponent n. Note that the converged solutions proposed elements by using only five integration points in the thickness direction, and meshes of thickness direction, and meshes of (24 × 36 × 2) × 1 and 12 × 18 × 1 in the case of the prismatic SHB15 Figures 15–17 are obtained with the proposed elements by using only five integration points in the (24element × 36 × and 2) ×hexahedral 1 and 12 ×SHB20 18 × 1element, in the case of the prismatic SHB15 element and hexahedral SHB20 for convergence thickness direction, and meshes of (24 × respectively. 36 × 2) × 1 and Hence, 12 × 18 ×the 1 inrequired the case ofmeshes the prismatic SHB15 element, respectively. Hence, the required meshes for convergence are coarser than those used by are coarser than used by Sze et al. [29]respectively. in the case Hence, of an isotropic material. element andthose hexahedral SHB20 element, the required meshes for convergence Sze et al.are [29] in the case of an isotropic coarser than those used by Sze etmaterial. al. [29] in the case of an isotropic material.

P/8 A

P/8 A

R = 4.953 m z R =z4.953 m y x

t = 0.094 mm t = 0.094

y

B B

x

C

C

(a)

(a)

(b)

(b)

Figure 14. Pull-out of an open-ended cylinder: (a) geometry and (b) undeformed and deformed Figure Pull-outof of open-ended cylinder: (a) geometry and (b) undeformed and Figure14. 14. Pull-out an an open-ended cylinder: (a) geometry and (b) undeformed and deformed configurations.

deformed configurations. configurations.

500

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Mesh (24 × 36 × 2) × 1

n = 0.2 with SHB15 0 Ceramic from [13] 0.0 0.5 1.0 Ceramic with SHB15

100

Metal from [13] Metal with SHB20 Metal from [13] n = 5 from [13] Metal with SHB20 n = 5 with SHB20 5 from n n= =2 from [13][13] 5 with SHB20 n n= =2 with SHB20 2 from n n= =1 from [13][13] n n= =1 with SHB20 2 with SHB20 n n= =0.51 from from[13] [13] n = 0.5 with SHB20 n = 1 with SHB20 n = 0.2 from [13] n = 0.5 from [13] n = 0.2 with SHB20 n = 0.5 with SHB20 Ceramic from [13] n = 0.2 with fromSHB20 [13] Ceramic

500 400

4

4

Pulling force (×10 )

400

Pulling force (×10 )

400

Metal from [13] Metal with SHB15 Metal from [13] n = 5 from [13] Metal with SHB15 n = 5 with SHB15 n = 5 from n = [13] 2 from [13] n = 5 with n =SHB15 2 with SHB15 n = 2 from n = [13] 1 from [13] n =SHB15 1 with SHB15 n = 2 with n = [13] 0.5 from [13] n = 1 from n = 0.5 with SHB15 n = 1 with SHB15 n = 0.2 from [13] n = 0.5 from [13] n = 0.2 with SHB15 n = 0.5 with SHB15 Ceramic from [13] n = 0.2 from [13]with SHB15 Ceramic

4 PullingPulling force (×10 force )(×10 )

500

400 300

300 200

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(a)

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n = 0.2 with SHB20 Ceramic from [13] 0.5 1.0 1.5 2.0 2.5 Ceramic withdeflection SHB20 Radial at point A (m)

(b)

0 2.0

Mesh 12 × 18 × 1

0.0

0.5

1.0

1.5

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Figure 15. Radial displacement at(m) point A under concentrated force forRadial the open-ended (a) deflection atcylinder. point A (m) Radial deflection at point A Prismatic SHB15(a) element; (b) hexahedral SHB20 element. (b)

Figure Radialdisplacement displacement at concentrated force for the cylinder. (a) Figure 15.15.Radial atpoint pointAAunder under concentrated force for open-ended the open-ended cylinder. Prismatic SHB15 element; (b) hexahedral SHB20 element. (a) Prismatic SHB15 element; (b) hexahedral SHB20 element.

Materials 2018, 11, 1046 Materials 2018, 11, x FOR PEER REVIEW Materials 2018, 11, x FOR PEER REVIEW

300 300 200 200

100 100

0 0 0 0

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4

Mesh (24 × 36 × 2) × 1 Metal from [13] Mesh (24 × 36 × 2) × 1 Metal from [13] Metal with SHB15 Metal with SHB15 n = 5 from [13] n = 5 from [13] n = 5 with SHB15 n = 5 with SHB15 n = 2 from [13] n = 2 from [13] n = 2 with SHB15 n = 2 with SHB15 n = 1 from [13] n = 1 from [13] n = 1 with SHB15 n = 1 with SHB15 n = 0.5 from [13] n = 0.5 from [13] n = 0.5 with SHB15 n = 0.5 with SHB15 n = 0.2 from [13] n = 0.2 from [13] n = 0.2 with SHB15 n = 0.2 with SHB15 Ceramic from [13] Ceramic from [13] Ceramic with SHB15 Ceramic with SHB15

4 Pulling Pullingforce force(×10 (×10) )

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4 Pulling Pullingforce force(×10 (×10) )

500 500

12 of 15 13 of 17 13 of 17

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0 0 0 0

1 2 3 4 1 Radial deflection 2 3 at point B (m) 4

Radial deflection at point B (m)

Metal from [13] Metal from [13] Metal with SHB20 Metal with SHB20 n = 5 from [13] n = 5 from [13] n = 5 with SHB20 n = 5 with SHB20 n = 2 from [13] n = 2 from [13] n = 2 with SHB20 n = 2 with SHB20 n = 1 from [13] n = 1 from [13] n = 1 with SHB20 n = 1 with SHB20 n = 0.5 from [13] n = 0.5 from [13] n = 0.5 with SHB20 n = 0.5 with SHB20 n = 0.2 from [13] n = 0.2 from [13] n = 0.2 with SHB20 n = 0.2 with SHB20 Ceramic from [13] Ceramic from [13] Ceramic with SHB20 Ceramic with SHB20

Mesh 12 × 18 × 1 Mesh 12 × 18 × 1

1 2 3 4 1 Radial deflection 2 3 at point B (m) 4

Radial deflection at point B (m)

(a) (a)

(b) (b)

Figure 16. Radial Radial displacement atatpoint B under concentrated force for the open-ended cylinder. (a) Figure displacementat point B under concentrated foropen-ended the open-ended cylinder. Figure16. 16. Radial displacement point B under concentrated forceforce for the cylinder. (a) prismatic SHB15 element; (b) hexahedral SHB20 element. (a)prismatic prismatic SHB15 element; hexahedral SHB20 element. SHB15 element; (b) (b) hexahedral SHB20 element.

300 300

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1 2 3 1 2 at point C (m) 3 Radial deflection

Radial deflection at point C (m)

(a) (a)

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4

Mesh (24 × 36 × 2) × 1 Metal from [13] Mesh (24 × 36 × 2) × 1 Metal from [13] Metal with SHB15 Metal with SHB15 n = 5 from [13] n = 5 from [13] n = 5 with SHB15 n = 5 with SHB15 n = 2 from [13] n = 2 from [13] n = 2 with SHB15 n = 2 with SHB15 n = 1 from [13] n = 1 from [13] n = 1 with SHB15 n = 1 with SHB15 n = 0.5 from [13] n = 0.5 from [13] n = 0.5 with SHB15 n = 0.5 with SHB15 n = 0.2 from [13] n = 0.2 from [13] n = 0.2 with SHB15 n = 0.2 with SHB15 Ceramic from [13] Ceramic from [13] Ceramic with SHB15 Ceramic with SHB15

4 Pulling Pullingforce force(×10 (×10) )

400 400

4

4 Pulling Pullingforce force(×10 (×10) )

500 500

300 300

200 200

100 100

0 0 0 0

Metal from [13] Metal from [13] Metal with SHB20 Metal with SHB20 n = 5 from [13] n = 5 from [13] n = 5 with SHB20 n = 5 with SHB20 n = 2 from [13] n = 2 from [13] n = 2 with SHB20 n = 2 with SHB20 n = 1 from [13] n = 1 from [13] n = 1 with SHB20 n = 1 with SHB20 n = 0.5 from [13] n = 0.5 from [13] n = 0.5 with SHB20 n = 0.5 with SHB20 n = 0.2 from [13] n = 0.2 from [13] n = 0.2 with SHB20 n = 0.2 with SHB20 Ceramic from [13] Ceramic from [13] Ceramic with SHB20 Ceramic with SHB20

1

Mesh 12 × 18 × 1 Mesh 12 × 18 × 1

2

3

1 2 at point C (m) 3 Radial deflection Radial deflection at point C (m)

(b) (b)

Figure 17. Radial displacement at point C under concentrated force for the open-ended cylinder. (a) Figure17. 17. Radial Radial displacement C under concentrated forceforce for the cylinder. (a) Figure displacementatatpoint point C under concentrated foropen-ended the open-ended cylinder. prismatic SHB15 element; (b) hexahedral SHB20 element. SHB15 element; (b) (b) hexahedral SHB20 element. (a)prismatic prismatic SHB15 element; hexahedral SHB20 element.

3.6. Pinched Hemispherical Shell 3.6. Pinched Hemispherical Shell 3.6. Pinched Hemispherical Shell It is worth noting that although the performance of the prismatic SHB15 element is similar to It is worth noting that although the performance of the prismatic SHB15 element is similar to isthe worth noting that although theasperformance ofin the SHB15 element is similar to that thatItof hexahedral SHB20 element, demonstrated theprismatic above nonlinear benchmark problems, that of the hexahedral SHB20 element, as demonstrated in the above nonlinear benchmark problems, ofthe themain hexahedral SHB20 element, asthe demonstrated in the above nonlinear benchmark motivation in developing prismatic solid–shell element is to use it for theproblems, mesh the main motivation in developing the prismatic solid–shell element is to use it for the mesh discretization of complex geometries. the Indeed, it is well-known complex cannot be the main motivation in developing prismatic solid–shellthat element is geometries to use it for the mesh discretization of complex geometries. Indeed, it is well-known that complex geometries cannot be discretized with only hexahedral elements, and require either an irregular mesh with prismatic discretization of complex geometries. Indeed, is well-known complex cannot discretized with only hexahedral elements, and itrequire either an that irregular meshgeometries with prismatic elements, or awith mixture based on a combination ofand prismatic and hexahedral elements. Inwith this section, beelements, discretized only hexahedral elements, require either an irregular mesh prismatic or a mixture based on a combination of prismatic and hexahedral elements. In this section, a hemispherical shell based is loaded alternating radial forces as shown in Figure 18a. Note that this elements, or a mixture on by abycombination of prismatic and hexahedral elements. In this a hemispherical shell is loaded alternating radial forces as shown in Figure 18a. Note that section, this benchmark problem has been considered in the literature for an isotropic material as well as athis a benchmark hemispherical shell is loaded by alternating radial forces as shown in Figure 18a. Note that problem has been considered in the literature for an isotropic material as well as a laminated composite material (see, e.g., [30]), while the for case of FGMs has not been considered yet. benchmark has been considered the while literature material as well as a laminated laminated problem composite material (see, e.g., in [30]), the caseanofisotropic FGMs has not been considered yet. Consequently, only the e.g., simulation results obtained with the proposed SHB elements corresponding composite material [30]), while case ofwith FGMs not been considered Consequently, Consequently, only(see, the simulation resultsthe obtained thehas proposed SHB elementsyet. corresponding to the fully metallic shell can be compared to the reference solution taken from [30]. only thefully simulation results withtothe SHB elements corresponding to the fully to the metallic shell canobtained be compared theproposed reference solution taken from [30]. metallic shell can be compared to the reference solution taken from [30].

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z

z y

x

y

A x

B

free (a)

A

B

(b)

free hemisphere: (a) geometry and (b) undeformed and deformed configurations. Figure 18. Pinched Figure 18. Pinched hemisphere: (a) geometry and (b) undeformed and deformed configurations. (a) (b)

The radius and thickness of the hemispherical shell are R = 10 m and t = 0.04 m , respectively. Figure 18. Pinched hemisphere: geometry (b), undeformed configurations. ν = 0.3 while theRYoung modulus metal The Poisson ratio of the hemispherical shell is and radius and thickness of the (a) hemispherical shell are =and 10deformed m and tof =the0.04 m,and respectively. 7 8 ceramic constituents are Em = 6.825 × 10 Pa and Ec = 1.46 × 10 Pa , respectively. Due to the

The The Poisson ratio thethickness hemispherical shell is ν = shell 0.3, while the Young of the metal and The radiusof the hemispherical are and ismodulus R = 10 m shell t = 0.04 m , respectively. symmetry, aand quarter of the of structure is discretized. hemispherical with a 7 Pa and The 8 Pa, discretized ceramic constituents are E = 6.825 × 10 E = 1.46 × 10 respectively. Due to the m hexahedral elements, mixture ratio of prismatic 90 Young SHB15 elements the and = 0.3consists , cwhile of the moduluslocated of theatmetal The Poisson of the and hemispherical shell is νwhich 7 points, see Figure 8 symmetry, quarter of the structure discretized. hemispherical shell is discretized with a topa of the hemisphere theis× load 18b) and 110 SHB20 elements for the ceramic constituents are(far Emfrom = 6.825 10 Pa and EThe c = 1.46 × 10 Pa , respectively. Due to the remaining area.and hexahedral elements, which consists of 90 SHB15 elements located at the mixture of prismatic symmetry, a quarter of the structure is discretized. The hemispherical shell is discretized with a The simulation results in terms of force–radial deflection at point A, for various values of the top of the hemisphere (far the load points, see consists Figure 18b) and 110 SHB20located elements for the mixture of prismatic andn,from hexahedral which 90the SHB15 power-law exponent are plotted inelements, Figure 19. This figure showsof that resultselements corresponding to aat the remaining area. top offully the metallic hemisphere (far from the load points, see Figure 18b) and 110 SHB20 elements for the shell (i.e., n → ∞ ), obtained by the combination of prismatic and hexahedral SHB The simulation results in terms of force–radial deflection at point A, for various values of the remaining area. elements, are in excellent agreement with those provided in [30] for an isotropic shell. Note that an equivalent in-plane mesh discretization has been used in [30], where a fully integrated shell element The simulation results in terms of force–radial deflection at point A, for various values of the power-law exponent n, are plotted in Figure 19. This figure shows that the results corresponding to with several points has considered. Moreover, Figure 19the reveals thatcorresponding for all values to a power-law exponent n, are plotted in been Figure 19. shows that results a fully metallic shellintegration (i.e., n→ ∞ ), obtained byThis the figure combination of prismatic and hexahedral SHB of the exponent n, the simulated load–radialby deflection curves lie between that of and the fully ceramic SHB fully metallic shell (i.e., ), obtained the combination of prismatic hexahedral n → ∞ elements,shell are in excellent agreement with those provided in [30] for an isotropic shell. Note that an and that of the fully metal shell, which is consistent with the numerical results found in the elements, are in excellent agreement with those provided in [30] for an isotropic shell. Note that an equivalentprevious in-plane mesh discretization has been used in [30], where a fully integrated shell element benchmark problems. equivalent in-plane mesh discretization has been used in [30], where a fully integrated shell element with several integration points has been considered. Moreover, Figure 19 reveals that for all values of with several integration points has been considered. Moreover, Figure 19 reveals that for all values the exponent n, the simulated load–radial curves lie between that of the fully ceramic shell Metaldeflection from [30] of the exponent n, the simulated 400 load–radial deflection curves lie between that of the fully ceramic Metal with SHB elements and that of the fully metal shell, which is consistent with numerical results found in the shell and that of the fully metal shell,n which iselements consistentthe with the numerical results found inprevious the = 5 with SHB n = 2 with SHB elements benchmark problems. previous benchmark problems. 300

P

n = 1 with SHB elements n = 0.5 with SHB elements n = 0.2 with SHB elements Metal from [30] Ceramic with SHB elements

400200

P

300100

200

0 0

Metal with SHB elements n = 5 with SHB elements n = 2 with SHB elements n = 1 with SHB elements n = 0.5 with SHB elements n = 0.2 with SHB elements Ceramic with SHB elements 1 2

3

4

Radial deflection at point A (m)

Figure 19. Load–displacement curves at point A for the pinched hemispherical shell, obtained with a 100 and hexahedral SHB20 elements. mixture of prismatic SHB15

0 0

1

2

3

4

Radial deflection at point A (m)

Figure 19. Load–displacement curves at point A for the pinched hemispherical shell, obtained with a

Figure 19. Load–displacement curves at point A for the pinched hemispherical shell, obtained with a mixture of prismatic SHB15 and hexahedral SHB20 elements. mixture of prismatic SHB15 and hexahedral SHB20 elements.

4. Conclusions In this work, quadratic prismatic and hexahedral solid–shell SHB elements have been proposed for the 3D modeling of thin FGM structures. The formulation of the SHB elements adopts the in-plane reduced-integration technique along with the assumed-strain method to alleviate various locking

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phenomena. A local (element) frame has been defined for each element, in which the thickness direction is specified. In this local frame, elastic properties of the thin structure are assumed to vary gradually through the thickness according to a power-law volume fraction distribution. The resulting formulations are implemented into the finite element software ABAQUS/Standard in the framework of large displacements and rotations. A series of selective and representative benchmark problems, involving FGM thin structures, has been performed to evaluate the performance of the SHB elements in geometrically nonlinear analysis. The results obtained with the SHB elements have been compared with reference solutions. Note that the state-of-the-art ABAQUS solid and shell elements have not been considered in the simulations, because these elements do not allow modeling of FGM behavior with only a single layer of elements. Overall, the numerical results obtained with the SHB elements showed excellent agreement with the available reference solutions. More specifically, the load–displacement curves for each benchmark test lie between that of the fully ceramic and fully metal behavior, which is consistent with the power-law distribution of the Young modulus in the thickness direction of the FGM plates. This good performance of the proposed SHB elements only requires a few integration points in the thickness direction (i.e., only five integration points), as compared to the number of integration points used in the literature to model thin FGM structures. Furthermore, it has been shown that the prismatic SHB15 element can be naturally combined with the hexahedral SHB20 element, within the same simulation, to help discretize complex geometries. Overall, the proposed SHB elements showed good capabilities in 3D modeling of thin FGM structures with only a single layer of elements and few integration points, which is not the case of traditional solid and shell elements. Author Contributions: H.C. conceived and performed the simulations. H.C. and F.A.-M. analyzed and discussed the results. Both authors contributed to the writing of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflicts of interest.

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