A user-defined finite element for laminated glass

A user-defined finite element for laminated glass panels and photovoltaic modules based on a layer-wise theory J. Eisentr¨ agera,∗, K. Naumenkoa , H. Altenbacha , J. Meenenb a Otto-von-Guericke-Universit¨ at b Auf

Magdeburg, Universit¨ atsplatz 2, 39106 Magdeburg, Germany Aderich 3, 52156 Monschau, Germany

Abstract Laminated plates with glass skin layers and a core layer from polyvinyl butyral are widely used in the civil engineering and automotive industry. Crystalline or thin film photovoltaic modules are composed from front and back glass or polymer layers and a solar cell layer embedded in a polymeric encapsulant. For the structural analysis of such laminates, layer-wise theories (LWTs) for plates have been introduced in literature. In this paper, an extended LWT is proposed to assure the C0 continuity. Based on this theory, a finite element formulation and a user-defined quadrilateral serendipity element with quadratic shape functions and nine degrees of freedom (DOFs) is presented. The element is implemented using the Abaqus subroutine “User Element”. Benchmark problems are developed to examine the accuracy and efficiency of the proposed element for a wide range of material properties including the limiting cases of shear compliant and shear rigid laminates. An emphasis is placed on the influence of boundary conditions with respect to additional degrees of freedom as well as on accurate representation of boundary layer effects. Keywords: Photovoltaic module, layer-wise theory, FEM, User Element 2010 MSC: 74K20, 74S05

1. Introduction In the civil engineering and automotive industry, laminated plates with glass skin layers and a core layer from polyvinyl butyral (PVB) are widely used [1, 2, 3]. Crystalline or thin film photovoltaic modules currently available on the market are composed from front and back glass or polymer layers and a solar cell layer embedded in a polymeric encapsulant [4, 5, 6]. A lightweight design of photovoltaic modules includes front and back panels made from plastics. These skin layers are connected together by a transparent polyurethane (PUR), in which the solar cells are embedded [7]. Figure 1 illustrates different types of photovoltaic modules. For design of glass laminates and photovoltaic modules it is beneficial to analyze the suitability of materials like PVB, ethylene-vinyl acetate (EVA), or PUR for embedding solar cells. These encapsulates have to compensate different mechanical and thermal strains of bottom and top layers. Delamination between the layers must be avoided and solar cells have to be protected against air and water. Mechanical properties of soft encapsulate materials are usually affected by the manufacturing process. Furthermore, environmental effects can lead to changes in the mechanical behavior over time. Therefore a reliable assessment of the stiffness properties is only possible by the ∗ Corresponding

author Email address: [email protected] (J. Eisentr¨ ager) Preprint submitted to Composite Structures

testing of a prototype, e. g. by bending tests of a beam or a plate. To evaluate the test results, robust structural analysis methods are required to relate the global deformation state, for example the deflection, and the local stress and strain state, for example the transverse shear stresses and strains, to the applied external load. One feature of laminated glass plates or laminates used in photovoltaic industry is the layered composite with relatively stiff skin layers and relatively thin and compliant polymer encapsulant layer. Let GS be the shear modulus of the glass skin layer and GC the shear modulus of the polymeric core layer. The ratio of the shear moduli µ = GC /GS for materials used in photovoltaics is in the range between 10−5 and 10−2 , depending on the type of polymer and the temperature [4, 8, 7]. For classical sandwich applications, this ratio is in the range of 10−2 and 10−1 . In addition, in classical sandwich structures the face sheets are thin in comparison to the core, while in photovoltaic applications the face layers are thick and the core is thin. To analyse the behavior of laminated plates, various structural mechanics models are available. A widely used approach for sandwich and laminate structures is the firstorder shear deformation theory (FSDT) [9, 10]. The principal assumption of this theory is that any normal to the reference midplane of the plate behaves like a rigid line during the deformation. The rotation degrees of freedom are independent from the transverse displacements, in contrast to the Kirchhoff theory, where the lines are assumed July 24, 2015

lution. However, due to extreme differences in material properties of constituents and the relatively low thickness of the core layer, considerable numerical effort is required to obtain results with a desired accuracy [4, 22]. In particular, care should be taken for finite element meshing of the core layer in order to compute the transverse shear strains and the related stresses accurately. In addition to conventional shell and solid elements, continuum shell elements can also be applied to analyze laminates [23, 24]. The finite element (FE) code Abaqus, for example, offers continuum shell elements, which possess only displacement DOFs and use three-dimensional constitutive equations [15]. They include the linear elements SC6R (triangular, 18 DOFs per element) and SC8R (quadrilateral, 24 DOFs per element). However, at least three elements in the thickness direction are required to analyze a three-layer laminate. Therefore the total number of DOFs required for analysis of an entire laminate can increase significantly. Furthermore, continuum shell elements with quadratic shape functions are not available in Abaqus. In order to analyze laminated structures, zig-zag and layer-wise theories have been developed. A zig-zag theory approximates the displacements by piecewise functions with respect to the thickness coordinate such that the compatibility between the layers is fulfilled. Then the governing equations of the three-dimensional elasticity theory are reduced to the two-dimensional plate equations by means of variational methods or asymptotic techniques [25, 26, 27]. Within the layer-wise theory (LWT), equilibrium and constitutive equations are derived for individual layers. With constitutive assumptions for interaction forces and compatibility conditions, a model for the layered system is derived. For laminated beams with a core layer from soft polymers, LWTs are presented in [1, 2, 3, 4, 7], among others. To derive robust equations, the assumption is made that the glass skin layers deform according to the Bernoulli-Euler beam theory, i. e. the transverse shear deformations are negligible. The soft core layer carries out the transverse shear stress only, while the bending moment and the normal force are negligible. In [1, 4, 7], results of three-point-bending tests for beams with core layers from various polymers are presented. Closedform solutions for beams based on the LWT agree well with the experimental data. Furthermore, as shown in [3, 4, 7], the solutions according to the LWT agree well with the results of the three-dimensional finite element analysis. The LWT is more attractive if compared to the zig-zag approximations since the load transfer between the layers can be analyzed explicitly. In addition, with proper assumptions about stiffness and/or deformation of individual layers, the LWT can provide equations that are easier in comparison to the zig-zag theories, and can be solved in a closed analytical form. Recently, LWTs for laminated glass plates have been proposed in [20, 28]. The deformation of skin layers is described by the Kirchhoff plate theory, while the core is modeled as the shear layer. Despite the fact that

back sheet or glass encapsulant electrical conductor crystalline solar cells encapsulant front glass

(a)

back sheet or glass encapsulant electrical conductor thin film layer front glass (b)

reinforced plastic encapsulant electrical conductor crystalline solar cells encapsulant transparent plastic

(c) Figure 1: Types of photovoltaic modules. a) crystalline photovoltaic module [4], b) thin film photovoltaic module [4], c) lightweight photovoltaic module [7]

normal to the deformed midsurface. The advantage of this theory is the possibility to solve the governing differential equations in a closed analytical form for plates of various shapes. Closed-form or approximate analytical solutions for plates according to the FSDT are presented in [9, 10, 11, 12, 13, 14], among others. Furthermore, different types of shell elements are available for the analysis of layered structures [15]. A problem related to FSDT is to estimate effective characteristics of the layered system, in particular the properties related to the transverse shear deformation. Closed-form relationships are developed to find effective elastic stiffness of a laminate from the properties of layers, e. g. [16, 17]. However, numerical techniques are required to estimate the effective transverse shear deformation in the inelastic range, [18, 19]. It should be mentioned that for laminates with extreme differences in the stiffness of layers the FSDT fails to predict the deformation properties of the laminate correctly, as shown for example in [4, 7] for beams and in [20, 21] for plates. Laminated glass and photovoltaic modules can also be analyzed using the three-dimensional theory of elasticity and applying solid finite elements for the numerical so2

(a)

the LWT was found efficient, the majority of results published in the literature deal with beams, plate strips or plates with simply supported edges. The aim of this paper is to discuss a finite element formulation based on the LWT for the analysis of glass and photovoltaic laminates. To this end, a user-defined finite element is utilized inside the Abaqus code with the subroutine User Element (UEL) [15]. To develop and verify the user element, the following problems will be addressed: • In the literature on LWT related to laminated glass and photovoltaic structures, the skin layers are modeled by the Kirchhoff plate theory. With regard to the robust finite element formulation, theories having C 0 continuity should be preferred. In this paper, we extend previous approaches and develop a LWT by including transverse shear deformation for all layers.

(b)

• Material properties of the core layer like the Young’s modulus may vary essentially depending on the temperature, manufacturing conditions etc. The developed numerical approach must be efficient in analysis of stress and strain states for a wide range of material properties including the limiting cases of shear compliant and shear rigid laminates. To examine the results of the finite element analysis, benchmark solutions for plates based on LWT will be developed. • Glass and photovoltaic laminates are usually fixed by elastic frames such that the relative sliding of the skin layers is partly constrained. As shown in previous work, the LWT leads to a higher-order system of partial differential equations in comparison to the FSDT. Therefore boundary conditions must be formulated with regard to additional degrees of freedom. Two examples of a supported edge are presented in Fig. 2. For the case shown in Fig. 2a, the relative sliding of the top and bottom layers is allowed while for the case given in Fig. 2b the crosssection rotations are assumed to be equal and the relative sliding is constrained. To analyze the influence of these types of boundary conditions, closed-form and finite element solutions will be presented.

Figure 2: Deformation of a plate edge for two types of supports. a) support without rotational constraint, b) support with rotational constraint

vectors are represented by bold italic letters. Greek letters used for indices take the values 1 or 2. The Einstein summation convention is applied if indices appear twice in one term. The direct tensor calculus in the sense of Gibbs [29] and Lagally [30] is applied. A second-rank tensor is a finite sum of dyads of vectors, for example A = a ⊗bb +cc ⊗dd +. . .+ee ⊗ff . In analogy, a fourth-rank tensor A = a ⊗ b ⊗ c ⊗ d + e ⊗ f ⊗ g ⊗ h + . . . + i ⊗ j ⊗ k ⊗ l is a finite sum of tetrads of vectors. The basic operations for dyads and tetrads can be introduced as follows:

• Higher-order plate theories provide additional boundary layer effects, which are of importance for the strength analysis of laminates. Boundary layer solutions are well established within the framework of FSDT. For LWT, new benchmark problems will be developed to verify the finite element formulations with respect to the solutions rapidly varying in the neighborhood of boundaries or concentrated loads.

a, a ⊗ b · c = αa c · a ⊗ b = βbb,

2. Layer-wise theory

α=b·c β =c·a

(1) (2)

a ⊗ b × c = a ⊗ d, d = b × c a ⊗ b ·· c ⊗ d = αβ = γ, α = b · c , β = a · d a ⊗ b = γa a ⊗ b, a ⊗ b ⊗ c ⊗ d ·· e ⊗ f = αβa

(4)

α = d · e, β = c · f

(5)

(3)

Operations (1)-(5) are generalized for tensors and used in many textbooks on continuun mechanics and rheology, see for example [31, 32, 33].

The governing equations of the LWT are introduced using the tensor notation. Tensors of second-rank and 3

In the sequel, we analyze a rectangular plate that is composed of a top layer (index T), a core layer (index C), and a bottom layer (index B). A coordinate system with the orthonormal basis e 1 , e 2 , n and the corresponding coordinates x1 , x2 , z is applied. The origin of the z coordinate is located in the middle of the core layer, i. e. −hB − hC /2 ≤ z ≤ hC /2 + hT . hT , hC , and hB denote the thicknesses of the top, core, and bottom layer respectively. n as well as the tangential The transverse load q = qn load s = sαe α are applied on the top layer of the plate. Figure 3 illustrates free-body diagrams for individual layers, showing the stress resultants on the left-hand side and the interaction forces on the right-hand side. Here, the distributed forces ±ssK and ±qq K , defining the interactions between the corresponding layers, are shown. The index K indicates the layer and replaces the indices T, C, or B. The stress resultants are obtained through the integration of the corresponding stresses over the thickness:

formulated as follows:

NKαβ =

σαβ dz

σα3 dz

(6a)

(6b)

zσαβ dz

DK

In Eqs (6), N K denotes the membrane force tensor of the Kth layer, QK is the shear force vector, and M K represents the tensor of bending and twisting moments. Following [20], the equilibrium conditions for the layers can be formulated as follows:

∇ · Q T + qT + q = 0 ∇ · Q C + qB − qT = 0 ∇ · Q B − qB =0 hT (ssT + s ) = 0 2 h ∇ · LC − QC + C (ssB + sT ) = 0 2 hB ∇ · LB − QB + sB =0 2

∇ · LT − QT +

(9b)

(6c)

−hK /2

∇ · N T + sT − s = 0 ∇ · N C + sB − sT = 0 ∇ · N B − sB =0

χK

(9c)

6 12 DK νK P ⊗ P + 2 DK (1 − νK ) e α ⊗ P ⊗ e α 2 hK hK
+ eα ⊗ eβ ⊗ eα ⊗ eβ (10) 1 =DK νK P ⊗ P + DK (1 − νK ) eα ⊗ P ⊗ eα 2
+ eα ⊗ eβ ⊗ eα ⊗ eβ (11) =ΓK P (12)

CK =

hZK /2

MKαβ =

(9a)

ε K is the in-plane strain tensor, u K is the in-plane displacement vector, χ K denotes the curvature change tensor, ϕ K is the normal rotation vector, wK is the deflection, and γ K represents the transverse shear strain vector. The initial (0) (0) (0) stress resultants N K , L K , and Q K are taken into account in order to include thermal, swelling, and inelastic strains. CK and DK are the fourth-order membrane and bending stiffness tensors, respectively, while Γ K represents the second-order transverse shear stiffness tensor. If isotropic and homogeneous material behaviour is assumed, the stiffness tensors can be determined as follows [20]:

−hK /2

M K = MKαβ e α ⊗ n × e β ,

 1 ∇u K )T ∇u K + (∇ 2  1 ∇ϕ K )T = χKαβ e α ⊗ e β = ∇ϕ K + (∇ 2 e ∇ = γKα α = wK + ϕ K

(8c)

εK = εKαβ e α ⊗ e β =

γK

hZK /2

QKα =

(8b)

QK =

−hK /2

Q K = QKα e α ,

(8a)

(0) DK ·· χ K + L K (0) Γ K · γ K + QK

LK =

hZK /2

N K = NKαβ e α ⊗ e β ,

(0) N K = CK ·· εK + N K

ΓK

with: DK =

(7a) (7b)

EK h3K
, ΓK = κGK hK 2 12 1 − νK

(13)

DK is the bending stiffness and ΓK is the shear stiffness of the Kth layer. EK denotes the Young’s modulus, νK is the Poisson’s ratio, and GK is the shear modulus. κ denotes the shear correction factor [34], and P = e α ⊗ e α is the projector. Assuming that the layers are rigidly connected, the following kinematical constraints are valid [20]:

(7c) (7d) (7e) (7f)

hT hC ϕT = uC − ϕC 2 2 hB hC uB − ϕB = uC + ϕC 2 2 wB = wC = wT = w uT +

(7g) (7h) (7i)

(14a) (14b) (14c)

Now, the principle of virtual work (PVW) is applied to the LWT. In general, the PVW can be formulated as follows:

where ∇ = e α ∂(...) ∂xα is the Hamilton operator and L = M × n = Mαβ e α ⊗ e β . The constitutive equations can be

δWin = δWex 4

(15)

e1

e2

q n

T

s2 s1 MT21

QT1 NT12

NT21

MT11

sT1 sT2

qT

NT11 MT12

QT2 NT22

MT22

qT sT1 QC1 MC21

C

NC12

NC21 MC12 QC2 NC22

MC11

sT2 sB1

sB2

qB

NC11

MC22 qB sB1

sB2

QB1 B

MB21

NB12

NB21 MB12 QB2 NB22

MB11 NB11

MB22

Figure 3: Stress resultants and interaction forces on the laminated plate [20]

It states that the equilibrium conditions for a body are fulfilled if the virtual work of the internal forces δWin equals the virtual work of the external forces δWex after the body has been imposed to arbitrary virtual displacements that are compatible with the boundary conditions [35]. The PVW serves as basis for deriving the element stiffness relation for the user-defined element based on the LWT.

 Z  1 δWin = N T ·· δεεT + N B ·· δεεB + N C ·· (δεεT + δεεB ) 2 A  1 χT − hB δχ χB ) + (Q QT + Q C + Q B ) · δ∇ ∇w + (hT δχ 4  1 χT + L B ·· δχ χB + L C ·· + L T ·· δχ (δεεB − δεεT ) hC  1  hB hT χ χ ϕT + Q B · δϕ ϕB − δχ + δχ + Q T · δϕ B T hC 2 hC  1 uB − δu uT ) + QC · (δu hC  1  hB hT ϕ ϕ − δϕ + δϕ dA (16) B T hC 2 hC

5

Z δWex =

uT + n ∗B · δu uB + n ∗C · n ∗T · δu



1 uT + δu uB ) (δu 2

Using these new stress resultants and DOFs, one can simplify Eqs (16) and (17) as follows:    Z  1 1 χ ·· N + δεε∆ + Hδχ χ ·· N ∆ δWin = δεε + h∆ δχ 2 2

∂A

 1 ∗ ∗ ∗ ϕT − hB δϕ ϕB ) + (qT + (hT δϕ + qC + qB )δw 4  1 ϕT + m ∗B · δϕ ϕB + m ∗C · uB − δu uT ) + m ∗T · δϕ (δu hC   1  hB hT ϕ ϕ δϕ + δϕ ds − B T hC 2 hC Z
hT ϕ u + (17) 2 s · δϕ T − s · δu T + qδw dA

A

A

dA is the infinitesimal area element of the plate, and ds is the infinitesimal line element of the plate boundary. n ∗K = ∗ n∗Kαe α , m ∗K = m∗Kαe α , and qK are the external boundary forces and moments. In order to simplify the equations, new variables for the stress resultants are introduced:

δWex

∂Ap

1˜ ϕ∆ · n ∗C + q ∗ δw + δϕ ϕ · m ∗ + δϕ ϕ∆ · m ∗∆ + hδϕ 2  1 u∆ + Hδϕ ϕ + h∆ δϕ ϕ∆ ) · m ∗C dsp − (2δu hC Z  hT ϕ + δϕ ϕ∆ ) · s − (δu u + δu u∆ ) · s (δϕ + 2

N = N T + N B + N C, N ∆ = N T − N B (18a) Q = QT + QB + QC, Q∆ = QT − QB (18b) 1 1 N B − (hC + hT )N N T, L = L T + L B + L C + (hC + hB )N 2 2 L∆ = LT − LB (18c)

Ap

 + qδw dAp

N , Q , and L are the membrane force tensor, the shear force vector, and the moment tensor of the laminate respectively, while N ∆ , Q ∆ , and L ∆ represent the corresponding relative values. The stress resultants of the top and bottom layers are replaced by the new stress resultants of the laminate and the relative stress resultants, while the stress resultants of the core layer are not substituted. Furthermore, the following DOFs are introduced: 1 1 uT + u B ), u ∆ = (u uT − u B ) (u 2 2 1 1 ϕT + ϕ B ), ϕ ∆ = (ϕ ϕT − ϕ B ) ϕ = (ϕ 2 2 u =

1˜ χ∆ ·· N C + δγγ · Q + δγγ ∆ · Q∆ + hδχ 2 1 u∆ + Hδϕ ϕ + h∆ δϕ ϕ∆ ) · Q C − (2δu hC χ ·· L + δχ χ∆ ·· L ∆ + δχ  1 χ + h∆ δχ χ∆ ) ·· L C dA (21) − (2δεε∆ + Hδχ hC    Z  1 1 u + h∆ δϕ ϕ · n ∗ + δu ϕ · n ∗∆ u∆ + Hδϕ = δu 2 2

(22)

with: ˜ = 1 (hT + hB ), h∆ = 1 (hT − hB ), H = hC + h ˜ h 2 2

(23)

The index p is used to define the loaded parts of the plate surface and boundary. The mean and relative external ∗ , m ∗ , and boundary forces and moments n ∗ , n ∗∆ , q ∗ , q∆ ∗ m ∆ are computed with the external boundary forces and moments of the individual layers in analogy to Eqs (18). The proposed layer-wise plate theory includes nine DOFs: the mean in-plane displacements uα , the relative in-plane displacements u∆α , the mean cross-section rotations ϕα , the relative cross-section rotations ϕ∆α , and the deflection w. In contrast to the LWT, which applies the Kirchhoff theory to the skin layers [20], the theory presented here describes all three layers with the FSDT. For this reason, the virtual internal and external work depend only on the DOFs themselves and their first derivatives such that this model leads to a C 0 variational problem. This constitutes the principal advantage over the Kirchhoff plate elements which require C 1 continuity, i. e. also the first derivatives of the DOFs have to be continuous on the element boundaries. The last one imposes higher requirements concerning the shape functions and boundary conditions [36].

(19a) (19b)

u is the mean in-plane displacement vector, u ∆ is the relative in-plane displacement vector. In analogy, ϕ represents the mean cross-section rotation vector and ϕ ∆ is the relative cross-section rotation vector. The corresponding membrane strain tensors ε, ε∆ , the curvature change tensors χ, χ∆ , and the transverse shear strain vectors γ , γ ∆ are defined as follows:  1 1 ∇u )T = (εεT + εB ) ε= ∇u + (∇ (20a) 2 2   1 1 ∇u ∆ )T = (εεT − εB ) ε∆ = ∇u ∆ + (∇ (20b) 2 2   1 1 ∇ϕ )T = (χ χT + χ B ) χ= ∇ϕ + (∇ (20c) 2 2   1 1 ∇ϕ ∆ )T = (χ χT − χ B ) χ∆ = ∇ϕ ∆ + (∇ (20d) 2 2 1 1 ∇wT + ∇ wB ) + (ϕ ϕT + ϕ B ) = ∇ w + ϕ γ = (∇ (20e) 2 2 1 1 ∇wT − ∇ wB ) + (ϕ ϕT − ϕ B ) = ϕ ∆ γ ∆ = (∇ (20f) 2 2

3. Closed-form solution This section presents a closed-form solution which is based on the LWT presented in [20] for the laminate shown in Fig. 4. Three assumptions are made in [20] in addition 6

l1

with the auxiliary variable Ω: ET hT h2C 2 ) Γ ∇ · u ∆ + Hw (1 − νT C

Ω= x1

l2

Ψ and Φ are scalar potentials in the sense of potential theory [37] and defined as follows:

x2

q x1 z

ET hT h2C ∇ × u∆) · n (∇ 2 (1 + νT ) ΓC HET h3T hC Φ = Ω − Hw ˜− ∆w ˜ 2 2 (1 − νT ) κGC (h2T + 3H 2 )

Ψ =−

top core bottom

u∆ =

to the governing equations in Section 2 in order to derive a robust solution. First, the skin layers are assumed to be shear rigid. With the Kirchhoff kinematic hypothesis, it follows: (24)

2 1 − νT HqET h3T hC S− 2 ET hT 2 (1 − νT ) κGC D (h2T + 3H 2 )

=−

(26b) (26c)

(26d)

∆ (. . .) = (. . .),11 + (. . .),22 is the two-dimensional Laplace operator, Θ is the Airy function, cf. e. g. [9], and S is the potential of the tangential force vector s defined by: (28)

L

The generalized deflection w ˜ is introduced as follows [20]: H2 w ˜ 1+3 2 hT 



H = w + 3 2 Ω, hT

(33a)

2 (1 −

HqET h3T hC 2 νT ) κGC D (h2T

(33b)

(33c)

+ 3H 2 )

Equations (33a) and (33b) are well-known from the FSDT [9]. The scalar potential Ψ represents boundary layer effects, which have been examined for the FSDT e. g. in [13, 38]. The boundary layer effects are rapidly varying solutions in the vicinity of plate boundaries. Equation (33c) generalizes the FSDT to laminates with soft core layer. It requires to consider an additional boundary condition, for example a rotational constraint, as shown in Fig. 2. Since Equations (33b) and (33c) have a similar structure, one can suppose that the scalar potential Φ influences the solution particularly near the boundaries. This will be examined in Section 5. Once Eqs (33) are solved, the normal rotation vector of the core layer, the relative membrane force tensor, and the moment tensor can be computed as follows [20]:  1  ˜ ∇w u∆ − h∇ ϕC = − 2u (34) hC 24 N ∆ = 2 DT [(1 − νT ) ε∆ + νT tr ε∆ P ] (35) hT

(26a)

with the bending stiffness D for the laminate defined as follows:   H2 (27) D = 2DT 1 + 3 2 hT

∇S s = −∇

(32)

q D 4 (1 + νT ) κGC Ψ =0 ∆Ψ − ET hT hC
  2 κGC 2 1 − νT H2 ∆Φ − 1+3 2 Φ ET hT hC hT

For the sake of brevity, let us consider symmetric laminates, i. e. the top and bottom layer are from the same material. Then, Eqs (7)-(14) can be reduced to the following system of four differential equations [20]:

=

1 ∇Ω + ∇ × (Ψn n)] [∇ 2

∆∆w ˜=

(25)

∆∆Θ + (1 − νT ) ∆S = 0 q ∆∆w ˜= D 4 (1 + νT ) κGC Ψ =0 ∆Ψ − ET hT hC
  2 κGC 2 1 − νT H2 ∆Φ − 1+3 2 Φ ET hT hC hT

(31b)

Below, we set the tangential force to zero, s = 0 , and assume ν · N = 0 as a boundary condition, where ν is the outer normal vector to the plate boundary. In this case, Θ = 0 and only the bending state must be analyzed such that Eqs (26) can be simplified as follows:

Second, one supposes that the membrane stiffness of the laminate is primarily determined by the membrane stiffness of the skin layers. Nevertheless, the in-plane stresses and deformations of the core layer are not neglected. Third, the core layer is assumed moments-free: LC = 0

(31a)

Furthermore, the following relation holds true [20]:

Figure 4: Laminate subjected to uniform surface load

∇w ϕ T = ϕ B = −∇

(30)

= 2DT [(1 − νT ) Υ + νT tr Υ P ]

(36)

with: Υ =

(29) 7

i 1h ∇ψ )T , ∇ψ + (∇ 2

∇w − 6 ψ = −∇

H u∆ h2T

(37)

Let us introduce the following normalized coordinates x ¯1 , x ¯2 and the length-to-width-ratio ξ: x ¯1 =

x1 , l1

x ¯2 =

x2 , l2

ξ=

l2 l1

For the correction variables, the following series solutions are applied:

(38)

In order to derive the closed-form solution for a rectangular plate, a series approach is utilised, as proposed in [39] for Kirchhoff plates and extended in [11] to Mindlin plates. Let us assume that the edges x ¯1 = 0 and x ¯1 = 1 are simply supported such that the following boundary conditions are valid, Fig. 4: w = 0,

ϕC2 = 0,

M11 = 0,

N∆11 = 0

ΨC (¯ x1 , x ¯2 ) = ΦC (¯ x1 , x ¯2 ) =


ql14 x ¯1 (¯ x1 − 1) x ¯21 − x ¯1 − 1 24D ΨS (¯ x1 ) =0  Hl4 q cosh (β) − 1 sinh (β x ¯1 ) ΦS (¯ x1 ) = 4 1 β D sinh (β)  − cosh (β x ¯1 ) + 1

(40a)

(45b)

Rm (¯ x2 ) sin (λm x ¯1 )

(45c)

(46)

λ4m Ym (¯ x2 ) − 2ξ −2 λ2m Ym00 (¯ x2 ) + ξ −4 Ym0000 (¯ x2 ) = 0
00 2 2 2 Zm (¯ x2 ) − ξ λm + µ Zm (¯ x2 ) = 0
00 2 2 2 Rm (¯ x2 ) − ξ λm + β Rm (¯ x2 ) = 0

(40b)

(47a) (47b) (47c)

with: (40c)

(42a)

Ψ (¯ x1 , x ¯2 ) = ΨC (¯ x1 , x ¯2 )

(42b)

Φ (¯ x1 , x ¯2 ) = ΦS (¯ x1 ) + ΦC (¯ x1 , x ¯2 )

(42c)

d (. . .) d¯ x2

(48)

In what follows let us assume that the boundary conditions and the loads have the symmetry axis x ¯2 = 0. In this case, the general solutions to Eqs (47) are:

(41)

w ˜ (¯ x1 , x ¯2 ) = w ˜S (¯ x1 ) + w ˜C (¯ x1 , x ¯2 )

Ym (¯ x2 ) = Am cosh(λm ξ x ¯2 )+Bm λm ξ x ¯2 sinh(λm ξ x ¯2 ) (49a) p  Zm (¯ x2 ) = Cm sinh λ2m + µ2 ξ x ¯2 (49b) p  Rm (¯ x2 ) = Dm cosh λ2m + β 2 ξ x ¯2 (49c) where Am , Bm , Cm , and Dm are integration constants. The generalized deflection w ˜S and the scalar potential ΦS , cf. Eqs (40), are expanded in Fourier series as follows: ∞ X

w ˜S (¯ x1 ) =

The subscript C denotes the correction solutions. Inserting Eqs (42) into Eqs (33) yields the following system of homogeneous differential equations:

wm sin (λm x ¯1 ) ,

(50)

m=1,3,...

R1 wm =

w ˜S (¯ x1 ) sin (λm x ¯1 ) d¯ x1

0

R1

(43a)

(51) 2

sin (λm x ¯1 ) d¯ x1

0

(43b)

∞ X

ΦS (¯ x1 ) =

φm sin (λm x ¯1 ) ,

(52)

m=1,3,...

(43c) R1

with the parameter µ defined as follows: 4 (1 + νT ) κGC l12 µ2 = ET hT hC

Zm (¯ x2 ) cos (λm x ¯1 )

One may verify that with Eqs (45) the boundary conditions for simple supports (39) are satisfied. With Eqs (45), the partial differential equations (43) are reduced to the following ordinary differential equations with respect to the functions Ym , Zm , and Rm :

In order to mark the solutions referring to the plate strip, the subscript S is used. The above ansatz fulfils the differential equations (33) and the boundary conditions (39) for the simply supported edges. In order to fulfil the boundary conditions at the edges x ¯2 = ±0.5, the correction solution is applied as follows:

µ2 ∆ΨC − 2 ΨC = 0 l1 β2 ∆ΦC − 2 ΦC = 0 l1

(45a)

λm = πm

0

∆∆w ˜C = 0

m=1,3,... ∞ X

Ym (¯ x2 ) sin (λm x ¯1 )

with:

(39)

(. . .) = with the parameter β defined as follows:
  2 2 1 − νT κGC l12 H2 2 β = 1+3 2 ET hT hC hT

m=1,3,... ∞ X

m=1,3,...

For the edges x ¯2 = ±0.5, various boundary conditions can be applied. As a first approximation, the solutions of a simply supported plate strip under constant lateral load q can be applied [20]: w ˜S (¯ x1 ) =

∞ X

w ˜C (¯ x1 , x ¯2 ) =

φm = (44)

ΦS (¯ x1 ) sin (λm x ¯1 ) d¯ x1

0

R1 0

8

(53) sin2 (λm x ¯1 ) d¯ x1

With Eqs (45), (49), (50), and (52), the following functions can be formulated: w ˜ (x1 , x ¯2 ) =

∞ X

and   p ˜ m = Dm cosh 0.5ξ λ2m + β 2 . D The systems of equations can be solved numerically with computer programs like Mathcad or Matlab taking into account a finite number of terms for the series expansion. Let us note that the presented approach can be extended to other boundary conditions at x ¯2 = ±0.5.

[wm + Ym (¯ x2 )] sin (λm x ¯1 )

(54a)

Zm (¯ x2 ) cos (λm x ¯1 )

(54b)

[φm + Rm (¯ x2 )] sin (λm x ¯1 )

(54c)

4. Finite element formulation based on the layerwise theory and a user-defined element

The constants Am , Bm , Cm , and Dm are computed from the boundary conditions for the edge x ¯2 = 0.5. The boundary conditions for simply supported edges x ¯2 = ±0.5 are:

For the user-defined element, the quadrilateral twodimensional serendipity formulation with eight nodes and quadratic shape functions is chosen, cf. for example [40]. Then, Equations (21) and (22) are transformed into a matrix notation. In what follows, matrices are represented by bold roman upper-case letters, while bold roman lowercase letters are used for vectors. The usual procedure to derive the element stiffness relation from the PVW yields, e. g. [40, 41]:

Ψ (¯ x1 , x ¯2 ) = Φ (¯ x1 , x ¯2 ) =

m=1,3,... ∞ X m=1,3,... ∞ X m=1,3,...

w = 0,

ϕC1 = 0,

M22 = 0,

N∆22 = 0

(55)

With Eqs (54) and (29)-(37), a system of four algebraic equations can be obtained. By transforming these equations, one can show:

Ku = f ,

0 Zm (¯ x2 = 0.5) = 0

(56)

φm + Rm (¯ x2 = 0.5) = 0

(57)

M22 = 0,

N∆12 = 0,

N∆22 = 0

A

 AT +A 2 D CbA 2 B mb dA Z  h i T Dsym Ks = BT + A D A Bs Cs 3 s D s + 2D 3 ∆s

M22 = 0,

N∆12 = 0,

ϕC2 = ϕB2 = ϕT2

(61)

A

 sym  T T T D A N dA D A N + N A +2 B T A 4 Cs 4 s 3 Cs 4

(58)

(62)

B represents the strain matrices [40], and the matrices D are constitutive matrices. According to the introduced mean and relative variables, there are mean constitutive matrices D mb and D s , relative constitutive matrices D ∆mb and D ∆s , and constitutive matrices D Cm , D Cb , and D Cs for the core layer. N is the matrix of shape functions, and the superscript the symmetric part of a matrix:  sym represents  T sym 1 A = 2 A + A . A k , k = 1 . . . 4 are auxiliary matrices whose entries are constant and depend on the thicknesses of the layers. The vector u collects the nodal DOFs of the element:   u1   u =  ...  (63)

For the supported edges x ¯2 = ±0.5 with rotational constraint, c. f. Fig. 2b, the boundary conditions can be formulated as follows: w = 0,

(60)

K is the element stiffness matrix, while the indices m, b, and s denote the membrane, bending, and shear part, respectively. Since selective integration is applied, the stiffness matrix for the membrane and bending state and the stiffness matrix for the shear state are computed separately: Z  T Dsym K mb = B T mb D mb + 2D ∆mb + A 1 D CmA 1

Taking into account Eqs (49b), (54b), and (54c), it becomes obvious that the scalar potentials Ψ and Φ are zero for simply supported edges such that there is no boundary layer effect. This is well-known for the FSDT, e. g. [13, 38], and holds true for the LWT, too. Additionally, two different cases of free supports along the edges x ¯2 = ±0.5 are taken into account. For the supported edge shown in Fig. 2a in Section 1, the relative sliding of the top and bottom layer is allowed while for the case shown in Fig. 2b the cross-section rotations are assumed to be equal and the relative sliding is constrained. The boundary conditions for supported edges x ¯2 = ±0.5 without rotational constraint, c. f. Fig. 2a, are: w = 0,

K = K mb + K s

(59)

With Eqs (54) and (29)-(37), a system of four equations with respect to the constants Am , Bm , Cm , and Dm can be obtained from the boundary conditions (58) or (59). While solving these systems of equations numerically, errors may occur because the functions Zm , Rm , and their first derivatives evaluated for x ¯2 = 0.5 provide large numbers. In particular, this is the case if the core layer is very shear compliant or very shear-stiff. In order to overcome conditioning problems, one canintroduce the substitutions  p C˜m = Cm sinh 0.5ξ λ2m + µ2

u8

9

with: uT i =



square plate, the maximum deflection in the middle of the plate is determined as follows: u1i

u2i

u∆1i

...

ϕ1i

ϕ2i

u∆2i ϕ∆1i

wi ϕ∆2i

... 

(64)

wmax = 0.00406

The index i = 1 . . . 8 corresponds to the node number. The right-hand-side vector f includes surface loads q , line loads t , concentrated forces and moments p , and the influence of residual stresses f (0) : Z Z f = N Tq dAp + N TA 5t dsp + p + f (0) , (65) Ap

ql14 D

(66)

To compare the results of different plate theories with the solution (66), let us introduce the normalized deflection w: ¯

w=

Dwmax 0.00406ql14

(67)

The equations presented in Section 3 are evaluated numerically considering the first 10 terms in the series expansion, i. e. m = 1 . . . 10, and the material parameters, dimensions, and loads according to Table 1 are used. The material parameters for float glass and EVA are taken from [22]. The plate is meshed regularly with 400 elements, and selective integration has been applied in order to avoid shear locking effects. Finite element results are always extracted at the nodes. Two types of boundary conditions are applied to the edges x ¯2 = ±0.5, Fig. 2:

∂Ap

where A 5 is an additional auxiliary matrix. The integrals in Eqs (61), (62), and (65) are computed by a typical Gauss quadrature, cf. [40]. The Abaqus subroutine UEL is utilised to implement the user-defined element. The UEL defines the contribution of the element to the entire FE model. As soon as information about the user-defined element is required, Abaqus calls the subroutine. Thus, Abaqus provides the UEL with the current values of the DOFs and of the userdefined solution-dependent state variables (SDVs). After the computation, the current values of the DOFs and SDVs can be written to the output file. In a static finite element analysis (FEA), the UEL has to define the element stiffness matrix and the right-handside vector referred to the global coordinate system. In addition, the user can define SDVs, e. g. stresses or strains. Abaqus does not support visualisation of user-defined elements as the position and numbers of Gauss points are defined in the subroutine [15]. Nevertheless, in order to use Abaqus’s postprocessor, auxiliary elements are defined in the input file. The position and numbers of Gauss points of the auxiliary elements and the user-defined elements have to coincide such that the subroutine UMAT can transfer the SDVs to the auxiliary elements. Zero matrices are assigned as stiffness matrices to the auxiliary elements such that their material behaviour does not influence the results of the user-defined elements. All DOFs and the stress resultants are defined as output variables of the user-defined element. Furthermore, the user can choose between three different kinds of integration for the stiffness matrix and the right-hand-side vector: complete, selective, and reduced integration. Reduced and selective integration are also efficient methods to avoid shear locking effects [40, 42].

1. Free supports without rotational constraint. To derive the closed-form solution, the boundary conditions given by Eqs (58) are applied. For the FE analysis with the user-defined element, only the deflection is set to zero: w=0

(68)

2. Free supports with rotational constraint. Within the closed-form solution, the boundary conditions given by Eqs (59) are used. Compared to the boundary conditions for free supports without rotational constraint, the equality of cross-section rotations has to be taken into account additionally. Starting with the constraint ϕC2 = ϕB2 = ϕT2 and considering the kinematical constraints (14), one obtains: w = 0,

1 u∆2 + Hϕ2 = 0 2

(69)

Equation (69)2 is implemented into the FE model as a linear multi-point constraint. In the following, we refer to these boundary conditions as boundary conditions (BCs) 1 and 2. The edges x ¯1 = 0 and x ¯1 = 1 are simply supported such that the boundary conditions are given by Eqs (39). Because the deflection is set to zero and the closed-form solution defines the cross-section rotations of the bottom and top layer as the derivative of the deflection (ϕB2 = ϕT2 = −w,2 ), these cross-section rotations have to be set to zero in the FE model, too. In addition, the cross-section rotation of the core layer is equal to zero according to Eqs (39). Taking the kinematical constraints (14) into account, the boundary conditions for the FE model are:

5. Verification of solutions and analysis of boundary layer effects Verification based on the closed-form solution. The homogeneous shear rigid plate represents the first benchmark. In [39], a series solution is utilized for the simply supported and uniformly loaded rectangular Kirchhoff plate. For a

w = 0, 10

ϕ2 = 0,

ϕ∆2 = 0,

u∆2 = 0

(70)

Table 1: Material parameters, dimensions, and load of the laminate

EK [MPa] T C B Laminate

73 000 7.9 73 000

νK [−]

GK [MPa]

hK [mm]

0.3 0.411 0.3

ET 2(1+νT ) EC 2(1+νC ) EB 2(1+νB )

4.0 0.5 4.0

l1 [mm]

l2 [mm]

q [MPa]

1000

1000

10−4

w ¯ [-]

FSDT

LWT

4

Kirchhoff

5

3

2

1 10−2

Closed-form solution with BCs 1 UEL with BCs 1 Closed-form solution with BCs 2 UEL with BCs 2 10−1

100 β [-]

101

102

Figure 5: Normalized maximum deflection vs dimensionless parameter β

Figure 5 shows the normalized maximum deflection as a function of the dimensionless parameter β. One may observe that the curves coincide with the Kirchhoff asymptote w ¯ = 1 for β → ∞. Furthermore, the curves for the plates with BCs 1 and the curves for the plates with BCs 2 coincide for β > 20. Thus, in this case, the FSDT can be used to compute the deformation state, cf. [20]. Otherwise, the LWT should be applied. Higher values of β correspond to Kirchhoff (shear rigid) plates. In contrast, low values of β refer to shear compliant laminates such that the FSDT or the LWT have to be applied for the structural analysis. Therefore, the parameter β will be called shear rigidity parameter in the following. Figure 5 also indicates that the type of support, i. e. BCs 1 or 2, exerts great influence on the maximum deflection. As expected for laminates with soft core layer, the bearing with rotational constraint is stiffer than the bearing without rotational constraint. Furthermore, it becomes apparent that the results of the user-defined element and the closedform solution agree well. In [43], it is pointed out that the 8 node serendipity FSDT element shows good results for simply supported plates, but fails for clamped plates due to locking. For this

reason, a square clamped laminate under uniform transverse load has been analyzed with the user-defined element. The mesh comprises 40x40 elements, and the material parameters of the top layer in Table 1 are valid also for the other layers. All layers have the same thickness, while the thickness of the laminate is varied. The exact thin plate solution according to [39] serves as reference solution. Figure 6 reveals that the user-defined element underestimates the exact thin plate solution for high values of l1 h . Nevertheless, the user-defined element performs remarkably well compared to the results given in [43]. The chosen kind of integration does not influence the results highly even though the results of the fully integrated element deviate from the results of the partly integrated element for very thin laminates. Due to these findings, selective or reduced integration should be applied to avoid locking effects. Figure 7 shows selected results for the DOFs and stress resultants with the BCs 1. A good agreement between the results of the user-defined element and the closed-form solution can be observed. The results for the other DOFs and stress resultants are of the same accuracy. 11

0.0015 Reduced Integration Selective Integration Full Integration

Dwmax [−] ql14

0.0014 0.0014 0.0013

Exact thin plate solution 0.00126 0.0013 0.0012 101

102

103

104

l1 [−] h Figure 6: Performance of the user-defined element with varying span-to-thickness ratio for square clamped laminate under uniform transverse load

UEL

Closed-form solution ·10 5

0.1

ϕ2 [rad]

w [mm]

0.15

0.05

0

−5

0 −0.5

−4

−0.25

0

0.25

−0.5

0.5

−0.25

x2 [−]

0

0.25

0.5

0.25

0.5

x2 [−]

(a)

(b)

·10−2 4 Q2 [N mm−1 ]

M22 [N]

4

2

0 −0.5

−0.25

0

0.25

2 0 −2 −4 −0.5

0.5

−0.25

0

x2 [−]

x2 [−]

(c)

(d)

Figure 7: DOFs and stress resultants along the line x ¯1 = 0.5, BCs 1. a) deflection w, b) cross-section rotation ϕ2 , c) bending moment M22 , d) shear force Q2

12

F

f=

b

x

y z

ux [mm] 0.0152 0.0088 0.0023 -0.0041 -0.0105 -0.0170 -0.0234

ine

p

su

l rt po

Figure 8: Finite element mesh with 3D continuum elements and distribution of axial displacement ux for EC = 3.5 MPa [4]

uT1 x1 , x2 , z =

h 2




[mm]

0.0152 x1

0.0088

x2 z

0.0023 -0.0041 -0.0105 -0.0170 -0.0234

 Figure 9: Finite element mesh with user-defined elements and distribution of axial displacement uT1 x1 , x2 , z =

h 2



for EC = 3.5 MPa

w [mm] 1.1850 0.9151 0.6454 0.3756 0.1059 -0.1639 -0.4336

Figure 10: Finite element mesh with user-defined elements and distribution of deflection w for EC = 3.5 MPa

13

Table 2: Material parameters, dimensions, and load of the laminated beam [4]

EK [MPa] T C B Beam

70 000 0.01 . . . 100 70 000

νK [−]

GK [MPa]

hK [mm]

0.23 0.45 0.23

ET 2(1+νT ) EC 2(1+νC ) EB 2(1+νB )

2.92 0.4 2.92

a [mm]

b [mm]

F [N]

250

192

50

35

·10−2

3

3.5

2.5

3

|QC1 | [N mm−1 ]

2 wmax [mm]

l [mm]

1.5 1

2.5 β la1 = 1 β la1 = 10 β la1 = 100

2 1.5 1

0.5 0 10−2

UEL FEM 3D [4] Closed-form solution [4] 10−1

100 EC [MPa]

0.5

101

0

102

0

0.25

0.5 x1/a

Figure 11: Maximum deflection vs. Young’s modulus of the core layer

0.75 [−]

1

1.25

Figure 12: Shear force of the core layer vs. normalized axial coordinate

the range of 10−2 . . . 102 MPa. Figure 11 shows the maximum deflection of the beam as a function of the Young’s modulus of the core layer. The results of the user-defined element are in good agreement with the results of the 3D FE analysis and the closed-form solution for laminated beams. In addition, the user-defined element decreases the computational effort significantly since the number of DOFs is reduced. The 3D FE model comprises 70 661 nodes and 211 983 DOFs, while the model with user-defined elements has 2421 nodes leading to 21 789 DOFs. Thus, the number of DOFs is reduced by ≈ 90 %, and the computational effort decreases considerably. Accurate representation of the shear force in the core layer is critical since shear failure, creep, and delamination are related to Q C . For this reason, Figure 12 shows the shear force of the core layer depending on the normalized axial coordinate for different values of β la1 . As before, β la1 indicates the shear rigidity of the laminated beam. For the shear rigid beam with β la1 = 100, the Bernoulli-Euler or Timoshenko theory can be used. According to these theories, the shear force has a jump in the middle of the beam and in the support line. The results shown in Fig. 12 indicate that the LWT yields smooth curves, which are

Verification based on solution with solid elements. In [4], three-point-bending tests are discussed, and a closed-form solution is presented according to the LWT. Furthermore, results of an FE analysis with solid elements are presented. They show a good agreement with experimental data. Here, let us apply the model given in [4] to test the user-defined finite element. Figure 8 shows the in-plane displacement of the beam obtained by the use of 15 744 quadratic 20 node brick elements. The symmetry of the beam has been taken into account such that only one half is shown. Table 2 summarizes the dimensions, loads, and material parameters of the laminated beam. l is the half length of the beam, a is the distance between the support line and the centre of the beam, b is the width of the beam, and F is the force divided by the length such that it is applicable as line load. In comparison to Fig. 8, Figure 9 shows the in
plane displacement uT1 x1 , x2 , z = hT + 12 hC at the top of the plate obtained with 750 user-defined elements, while the support lines are represented by setting the deflection equal to zero. In addition, Figure 10 shows the deflection of the beam obtained with 750 user-defined elements. The Young’s modulus of the core layer is varied within 14

more easily manageable by the FEM. For low values of β la1 , the shear force has non-zero values outside the beam span. This is in agreement with results published in [4].

DOFs is reduced by 25 %. Let us note that continuum shell elements with quadratic shape functions are not available in Abaqus. The results of this paper are presented for symmetric laminates with three layers from isotropic materials taking elastic behaviour into account. The user-defined element is not restricted to symmetric laminates, and can be used for various photovoltaic modules designs, cf. Fig. 1. Inelastic behaviour with finite shear strains of the core layer should be taken into account in the future since encapsulant materials like EVA exhibit essential creep [44]. One feature of the LWT is the possibility to relax kinematical constraints and to apply advanced traction-separation constitutive laws for the interaction between the layers. The corresponding extensions will be presented in a forthcoming paper.

Boundary layer effects. To analyze the boundary layer effects, three laminated plates with three different values of the Young’s modulus of the core layer are considered. All other material parameters, the dimensions, and loads are given in Table 1. In the first example, EC = 0.79 MPa is assumed, in the second example EC = 7.9 MPa, and EC = 79 MPa is given in the third example. Figure 13 shows the relative membrane force N∆12 of the laminate and the shear force QC2 of the core layer along the path x ¯1 = 0.95 near the boundary of the plate, while BCs 2 are applied to the edges x ¯2 = ±0.5. The solution with the user-defined elements fulfils the boundary condition N∆12 = 0 at the supported edges with high accuracy. In the vicinity of the supported edges, both stress resultants increase/decrease rapidly towards the maximum/minimum values. The distance over which the boundary layer effect is observable depends on the parameters β and µ. With an increase of β, the distance over which the boundary layer effect is observable decreases and the deviations between the closed-form solution and the FEM increase in the vicinity of the minima and maxima. These results are in line with [21], where the boundary layer effects have been examined for Mindlin plates.

References [1] M. Z. A¸sik, S. Tezcan, Laminated glass beams: Strength factor and temperature effect, Computers & Structures 84 (2006) 364– 373. doi:10.1016/j.compstruc.2005.09.025. [2] Y. Koutsawa, E. M. Daya, Static and free vibration analysis of laminated glass beam on viscoelastic supports, International Journal of Solids and Structures 44 (2007) 8735–8750. doi: 10.1016/j.ijsolstr.2007.07.009. [3] I. V. Ivanov, Analysis, modelling, and optimization of laminated glasses as plane beam, International Journal of Solids and Structures 43 (2006) 6887–6907. doi:10.1016/j.ijsolstr.2006.02. 014. [4] S.-H. Schulze, M. Pander, K. Naumenko, H. Altenbach, Analysis of laminated glass beams for photovoltaic applications, International Journal of Solids and Structures 49 (2012) 2027–2036. doi:10.1016/j.ijsolstr.2012.03.028. [5] M. Paggi, S. Kajari-Schr¨ oder, U. Eitner, Thermomechanical deformations in photovoltaic laminates, The Journal of Strain Analysis for Engineering Design 46 (8) (2011) 772–782. doi: 10.1177/0309324711421722. [6] M. Corrado, M. Paggi, A multi-physics and multi-scale numerical approach to microcracking and power-loss in photovoltaic modules, Composite Structures 95 (2013) 630–638. doi: 10.1016/j.compstruct.2012.08.014. [7] M. Weps, K. Naumenko, H. Altenbach, Unsymmetric threelayer laminate with soft core for photovoltaic modules, Composite Structures 105 (2013) 332–339. doi:10.1016/j.compstruct. 2013.05.029. [8] U. Eitner, M. K¨ ontges, R. Brendel, Use of digital image correlation technique to determine thermomechanical deformations in photovoltaic laminates: Measurements and accuracy, Solar Energy Materials and Solar Cells 94 (8) (2010) 1346–1351. doi:10.1016/j.solmat.2010.03.028. [9] H. Altenbach, J. Altenbach, K. Naumenko, Ebene Fl¨ achentragwerke: Grundlagen der Modellierung und Berechnung von Scheiben und Platten, Springer, 1998. [10] R. Szilard, Theories and Applications of Plate Analysis, John Wiley & Sons, Hoboken, New Jersey, 2004. [11] K. Naumenko, J. Altenbach, H. Altenbach, V. K. Naumenko, Closed and approximate analytical solutions for rectangular Mindlin plates, Acta Mechanica 147 (2001) 153–172. doi: 10.1007/BF01182359. [12] J. N. Reddy, C. M. Wang, An overview of the relationships between solutions of classical and shear deformation plate theories, Composites Science and Technology 60 (2000) 2327–2335. doi:10.1016/S0266-3538(00)00028-2. [13] B. Brank, On boundary layer in the Mindlin plate model: Levy plates, Thin-Walled Structures 46 (2008) 451–465. doi: 10.1016/j.tws.2007.11.003.

6. Conclusions The aim of this paper is to present a finite element formulation and a user-defined element based on a layerwise theory, which can be utilized to analyze laminated glasses and photovoltaic modules. Starting from the governing equations of the LWT, a quadrilateral serendipity element with quadratic shape functions is developed and implemented inside Abaqus code. The element possesses nine DOFs: two mean in-plane displacements, two relative in-plane displacements, the deflection, two mean crosssection rotations, and two relative cross-section rotations. In order to verify the user-defined element, closed-form solutions for plates with various boundary conditions are derived. Furthermore, the results of analysis with solid elements are used to discuss the efficiency of the proposed approach. Verifications made for different types of boundary conditions and a wide range of material properties are successful and reveal the good performance of the userdefined element. The developed element is well applicable for limiting cases of shear compliant and shear rigid laminates. FE models with the developed element lead to a lower number of DOFs in comparison to solid or continuum shell elements available in Abaqus code. For example, if only one quadrilateral continuum shell element SC8R is applied for each layer of the laminate, 48 DOFs are required. With a four-node linear formulation of the developed element, 36 DOFs are used such that the number of 15

UEL

Closed-form solution

0.4 EC = 0.79 MPa β = 4.09 0.2

QC2 [N mm−1 ]

N∆12 [N mm−1 ]

·10−5

0 −0.2

EC = 0.79 MPa β = 4.09

5

0 −5

−0.4 −0.5

−0.25

0

0.25

−0.5

0.5

−0.25

0

0.25

0.5

x2 [−]

x2 [−]

EC = 7.9 MPa 0.5 β = 12.94

EC = 7.9 MPa β = 12.94

2 QC2 [N mm−1 ]

N∆12 [N mm−1 ]

·10−4

0 −0.5

0

−2

−0.5

−0.25

0

0.25

−0.5

0.5

−0.25

x2 [−]

0

0.25

0.5

x2 [−] ·10−4 5

EC = 79 MPa β = 40.91

QC2 [N mm−1 ]

N∆12 [N mm−1 ]

1

0

0

−5

−1 −0.5

EC = 79 MPa β = 40.91

−0.25

0

0.25

−0.5

0.5

−0.25

0

x2 [−]

x2 [−]

(a)

(b)

0.25

0.5

Figure 13: Stress resultants along the line x ¯1 = 0.95, BCs 2. a) relative membrane force of the laminate, b) shear force of the core layer

16

[14] H. Altenbach, V. A. Eremeyev, Direct approach based analysis of plates composed of functionally graded materials, Archive of Applied Mechanics 78 (10) (2008) 775–794. doi:10.1007/ s00419-007-0192-3. [15] Dassault Syst` emes, Abaqus 6.14 Online Documentation (2014). [16] H. Altenbach, V. Eremeyev, K. Naumenko, On the use of the first order shear deformation plate theory for the analysis of three-layer plates with thin soft core layer, ZAMM Journal of Applied Mathematics and Mechanicsdoi:10.1002/zamm. 201500069. [17] A. Sabik, I. Kreja, Thermo-elastic non-linear analysis of multilayered plates and shells, Composite Structures 130 (2015) 37–43. doi:http://dx.doi.org/10.1016/j.compstruct.2015. 04.024. [18] H. Altenbach, K. Naumenko, Shear correction factors in creepdamage analysis of beams, plates and shells, JSME International Journal Series A 45 (2002) 77–83. doi:10.1299/jsmea. 45.77. [19] C. Helfen, S. Diebels, A numerical homogenisation method for sandwich plates based on a plate theory with thickness change, ZAMM Journal of Applied Mathematics and Mechanics 93 (2– 3) (2013) 113–125. doi:10.1002/zamm.201100173. [20] K. Naumenko, V. A. Eremeyev, A layer-wise theory for laminated glass and photovoltaic panels, Composite Structures 112 (2014) 283–291. doi:10.1016/j.compstruct.2014.02.009. [21] J. Eisentr¨ ager, K. Naumenko, H. Altenbach, H. K¨ oppe, Application of the first-order shear deformation theory to the analysis of laminated glasses and photovoltaic panels, International Journal of Mechanical Sciences 96–97 (2015) 163–171. doi:10.1016/j.ijmecsci.2015.03.012. [22] U. Eitner, Thermomechanics of photovoltaic modules, Dissertation, Martin-Luther-University Halle-Wittenberg (2011). URL urn:nbn:de:gbv:3:4-5812 [23] S. Hosseini, J. J. C. Remmers, C. V. Verhoosel, R. de Borst, An isogeometric continuum shell element for non-linear analysis, Computer Methods in Applied Mechanics and Engineering 271 (2014) 1–22. doi:http://dx.doi.org/10.1016/j.cma. 2013.11.023. [24] S. Klinkel, F. Gruttmann, W. Wagner, A continuum based three-dimensional shell element for laminated structures, Computers & Structures 71 (1) (1999) 43–62. doi:http://dx.doi. org/10.1016/S0045-7949(98)00222-3. [25] E. Carrera, Historical review of Zig-Zag theories for multilayered plates and shells, Appl Mech Rev 56 (2) (2003) 287–308. doi: 10.1115/1.1557614. [26] F. Tornabene, N. Fantuzzi, M. Bacciocchi, E. Viola, Accurate inter-laminar recovery for plates and doubly-curved shells with variable radii of curvature using layer-wise theories, Composite Structures 124 (2015) 368–393. doi:http://dx.doi.org/10. 1016/j.compstruct.2014.12.062. [27] E. Carrera, A. Ciuffreda, A unified formulation to assess theories of multilayered plates for various bending problems, Composite Structures 69 (3) (2005) 271–293. doi:10.1016/j. compstruct.2004.07.003. [28] P. Foraboschi, Analytical model for laminated-glass plate, Composites Part B: Engineering 43 (5) (2012) 2094–2106. doi: 10.1016/j.compositesb.2012.03.010. [29] E. B. Wilson, Vector analysis, founded upon the lectures of J. W. Gibbs, Yale University Press, 1901. [30] M. Lagally, Vorlesungen u ¨ber Vektorrechnung, Geest & Portig, 1962. [31] K. Naumenko, H. Altenbach, Modeling of Creep for Structural Analysis, Springer, 2007. [32] T. Belytschko, W. K. Liu, B. Moran, K. Elkhodary, Nonlinear Finite Elements for Continua and Structures, Wiley, 2014. [33] H. Giesekus, Ph¨ anomenologische Rheologie: Eine Einf¨ uhrung, Springer Berlin Heidelberg, 1994. [34] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics 12 (1945) 69–77. [35] E. O˜ nate, Structural Analysis with the Finite Element Method.

[36] [37] [38]

[39] [40]

[41] [42] [43] [44]

17

Linear Statics: Volume 1: Basis and Solids, Lecture Notes on Numerical Methods in Engineering and Sciences, Springer, 2010. E. Hinton, D. R. J. Owen, G. Krause, Finite Elemente Programme f¨ ur Platten und Schalen, Springer, 1990. R. Courant, D. Hilbert, Methods of Mathematical Physics, Differential Equations, Wiley, 1989. A. Nosier, A. Yavari, S. Sarkani, A study of the edge-zone equation of Mindlin-Reissner plate theory in bending of laminated rectangular plates, Acta Mechanica 146 (2001) 227–238. doi:10.1007/BF01246734. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959. E. O˜ nate, Structural Analysis with the Finite Element Method. Linear Statics. Volume 2: Beams, Plates and Shells, Lecture Notes on Numerical Methods in Engineering and Sciences, Springer, 2009. K.-J. Bathe, P. Zimmermann, Finite-Elemente-Methoden, Springer, 2002. J. N. Reddy, Mechanics of Laminated Composite Plates. Theory and Analysis, CRC Press, 1997. O. C. Zienkiewicz, R. L. Taylor, The Finite Element method. Volume 2: Solid Mechanics, Butterworth-Heinemann, 2000. A. W. Czanderna, F. J. Pern, Encapsulation of PV modules using ethylene vinyl acetate copolymer as a pottant: A critical review, Solar Energy Materials and Solar Cells 43 (2) (1996) 101–181. doi:10.1016/0927-0248(95)00150-6.