applied sciences Article
Evaluation of Interlaminar Stresses in Composite Laminates with a Bolt-Filled Hole Using a Linear Elastic Traction-Separation Description Yong Cao *, Yunwen Feng, Xiaofeng Xue, Wenzhi Wang * and Liang Bai School of Aeronautics, Northwestern Polytechnical University, P.O. Box 120, Xi’an 710072, Shaanxi Province, China;
[email protected] (Y.F.);
[email protected] (X.X.);
[email protected] (L.B.) * Correspondence:
[email protected] (Y.C.);
[email protected] (W.W.); Tel.: +86-29-8846-0383 (Y.C.) Academic Editor: Peter Van Puyvelde Received: 6 December 2016; Accepted: 16 January 2017; Published: 18 January 2017
Abstract: Determination of the local interlaminar stress distribution in a laminate with a bolt-filled hole is helpful for optimal bolted joint design, due to the three-dimensional (3D) nature of the stress field near the bolt hole. A new interlaminar stress distribution phenomenon induced by the bolt-head and clamp-up load, which occurs in a filled-hole composite laminate, is investigated. In order to efficiently evaluate interlaminar stresses under the complex boundary condition, a calculation strategy that using zero-thickness cohesive interface element is presented and validated. The interface element is based on a linear elastic traction-separation description. It is found that the interlaminar stress concentrations occur at the hole edge, as well as the interior of the laminate near the periphery of the bolt head. In addition, the interlaminar stresses near the periphery of the bolt head increased with an increase in the clamp-up load, and the interlaminar normal and shear stresses are not at the same circular position. Therefore, the clamp-up load cannot improve the interlaminar stress distribution in the laminate near the periphery of the bolt head, although it can reduce the magnitude of the interlaminar shear stress at the hole edge. Thus, the interlaminar stress distribution phenomena may lead to delamination initiation in the laminate near the periphery of the bolt head, and should be considered in composite bolted joint design. Keywords: interlaminar stresses; filled-hole compression testing; traction-separation laws; bolted joints; cohesive element
1. Introduction Interlaminar stresses can lead to damage in the form of delamination and matrix cracking of laminate, especially at the free-edges of a laminate [1,2]. Accurate and proper design of bolted composite joints requires the determination of the local stress field within the laminate [3]. Furthermore, the stress field in the vicinity of a bolt hole is generally considered to be three-dimensional (3D) [4–6]. Thus, it is important to take interlaminar stresses into account in composite bolted joint design. Due to various kinds of bolted composite joints, a filled-hole laminate was used as a means to characterize the stress distribution of bolted joints in this paper. Filled-hole strength has been widely used to generate design data to establish allowables for composite bolted joint analysis [7]. Numerous investigators have studied the stress fields in laminates with a bolt. Camanho and Matthews [8] presented a review of the previous investigation into the stress and strength analysis of mechanically fastened joints in fiber-reinforced plastics. They concluded that significant issues such as the effects of friction, clearance or interference and the contact area on the stress distribution around a pin-loaded hole. They also suggested that a 3D model was required to accurately predict the strength
Appl. Sci. 2017, 7, 93; doi:10.3390/app7010093
www.mdpi.com/journal/applsci
Appl. Sci. 2017, 7, 93
2 of 15
of mechanically fastened joints. Following that study, Ireman [9] developed a 3D finite element model (FEM) of composite bolted joints to determine non-uniform stress distribution through the thickness of laminate in the vicinity of a bolt hole. McCarthy et al. [10] analyzed the effects of bolt hole clearance on the mechanical behavior of composite bolted joints using a 3D FEM, and studied radial stress through thickness variations at the hole. Caccese et al. [11] investigated the influence of stress relaxation on clamp-up load in hybrid composite-to-metal bolted connections by experiment, and the pressure distribution at the free face of the laminate under different clamp-up load was obtained. Egan et al. [12] provided an analysis of the stress distribution at a countersunk hole boundary using the nonlinear finite element code Abaqus. Feo et al. [13] examined the stress distribution among the different bolts by varying the number of rows of bolts as well as the number of bolts per row. The results of their study indicated that in the presence of washers, the stress distributions in the fiber direction and varying fiber inclinations decreased for each value of maximum bearing stresses. Zhao et al. [14] proposed a determination method of stress concentration relief factors for failure prediction of composite joints. Moreover, the in-plane stress distribution around the hole edge for open-, filled- and loaded-hole laminates was discussed in their study. These studies were mainly aimed at the investigation of stress distributions at bolt hole edge or in-plane stress distributions. However, the experiments outlined in References [15,16] have shown that damage might initiate near the edge of the bolt head or washer in composite laminates with bolts. Whitney et al. [3] investigated the singular stress fields near the contact surfaces of the bolt head and laminate in a composite bolted joint, and gave all six stress distributions in the position; however, they did not focus on interlaminar stresses. It is also noteworthy that interlaminar stresses are a stress component along the thickness direction of the laminate, therefore, the interlaminar stress distribution along the thickness direction of laminate with a bolt-filled hole should also be investigated. An understanding of geometry-based effects and interlaminar stress distributions are helpful in explaining the failure mode of laminates with a bolt. Several levels of modeling can be used for the evaluation of interlaminar stresses. Pagano [17] provided an exact solution for a simply supported plate subject to sinusoidal pressure load, which had been considered a benchmark problem. The approaches based on the equivalent single layer (ESL) theories, layer-wise theories and zigzag theories are introduced in these References [1,18,19]. The sets of ESL refer to the Classical Laminated Plate Theory (CLPT), the First Order Shear Deformation Plate Theory (FSDT) and the Higher Order Deformation Plate Theory (HSDT). The limitations of these theories, especially the ESL model, are introduced in References [20–22]. In addition, some other reviews and extensive assessments were investigated in References [23,24]. In a filled-hole laminate, the stress state near the hole is a complex 3D problem. Generally, numerical methods based on 3D elasticity theory are usually used to accurately evaluate interlaminar stresses under complex boundary. The 3D elasticity theory includes traditional 3D elasticity formulations and layerwise theories [1,25–27]. In this method, the laminate is modeled as 3D elements [20,28]. However, the element type and mesh density are significant factors that can affect the calculation accuracy of FEM. If the laminate is discretized with the 3D solid element, a further challenge is that the transverse shear stresses are still discontinuous at the layer interface without enough elements in thickness direction. To overcome these limitations, some multiple model methods, such as the interface model, are used. Dakshina Moorthy and Reddy [29] developed an interface model based on the penalty function method and obtained the interlaminar stresses using this approach. At present, the cohesive zone model (CZM) is widely used in the interface modeling [30–32]. Some of the CZMs are described by the so-called traction-separation law (TSL). The TSL is derived from trial-and-error finite element computations [33] and originally designed to be used to model complex fracture mechanisms at the crack tip. Against this background, there are two main contributions in this paper. Firstly, a new interlaminar stress distribution phenomenon induced by the bolt-head and clamp-up load was investigated, in accordance with experimental failure evidence of composite laminate with a bolt-filled hole [16]. This analysis can yield a better understanding of stress distributions in bolt joints and improve
Appl. Sci. 2017, 7, 93
3 of 15
Appl. Sci. 2017, 7, 93
3 of 16
reliability and part life. Secondly, the interface model based on TSL was employed to evaluate interlaminar stresses. This interlaminar modeling approach widely appliedhas in composites analyses; employed to evaluate stresses. has Thisbeen modeling approach been widelyfailure applied in however, less attention has beenhowever, dedicated to attention its role inhas stress calculation. composites failure analyses; less been dedicated With to its this role approach, in stress the interlaminar stress distribution canthe beinterlaminar evaluated efficiently. calculation. With this approach, stress distribution can be evaluated efficiently. 2. Linear Elastic Traction-Separation ofthe theInterface Interface Model 2. Linear Elastic Traction-SeparationConstitutive Constitutive Behavior Behavior of Model In this approach, a zero-thicknesscohesive cohesive interface interface element introduced intointo the finite In this approach, a zero-thickness elementwas was introduced the finite element computations for composite materials. The zero-thickness means that the geometric element computations for composite materials. The zero-thickness means that the geometric thickness thickness of cohesive elements is equal zero, not the constitutive thickness. Since the initial of cohesive elements is equal to zero, andtonot theand constitutive thickness. Since the initial geometric geometric thickness of the interface element is zero, the deformation state of the zero-thickness thickness of the interface element is zero, the deformation state of the zero-thickness element cannot be element cannot be described by the classical definition of strain. Instead, the constitutive response of described by the classical definition of strain. Instead, the constitutive response of the zero-thickness the zero-thickness interface element is established in terms of a linear elastic traction-separation interface elementInisgeneral, established in termstraction-separation of a linear elasticmodel traction-separation In general, description. the available consists of threedescription. parts: linear elastic the available traction-separation model consists of three parts: linear elastic traction-separation; traction-separation; damage initiation criteria; and damage evolution laws [34,35]. However, only damage initiation criteria; and damage evolution [34,35]. However, initially linear elastic initially linear elastic behavior is considered due tolaws stress distribution being only the only concern in this paper. behavior is considered due to stress distribution being the only concern in this paper. The constitutive response of interface this interface was obtained a procedure introduced by The constitutive response of this was obtained usingusing a procedure introduced by Camanho Camanho and Davila [36]. Consider a typical interface that uses a traction-separation description and Davila [36]. Consider a typical interface that uses a traction-separation description between two two as shown Figure 1. The two coincident nodes of the interface arenodes, a pair and plies between as shown in plies Figure 1. The in two coincident nodes of the interface element areelement a pair of of nodes, and each interface element has four pairs of nodes, and each node has three degrees of each interface element has four pairs of nodes, and each node has three degrees of freedom (DOFs); freedom (DOFs); thus, there are 24 DOFs in total for each interface element. The nodes global thus, there are 24 DOFs in total for each interface element. The nodes global displacement vector uN is displacement vector uN is defined as follows: defined as follows: n o
+ + + − − − uN u =N uix u,ixu,iyu,iyu, iz uiz, u, uixix, ,uuiyiy,,uuiziz
TT
(1)
(1)
+ + + − − − wherewhere uix , uiy ofofthe interfaceelement element uix, u, u , uixiz,,uuiyix,,uuiziy ,are uiz the are displacements the displacements thei nodes i nodes of of the the interface in in thethe x, x, y, iziy, u and z directions.
y, and z directions.
4+ Interface element
2+ 4-
z,3 y,2 x,1
Element of upper layer
3+
1+
1-
32-
Element of lower layer
1. Zero-thickness cohesive interfaceelement. element. FigureFigure 1. Zero-thickness cohesive interface
The relative displacement u for each pair of nodes can be described as
The relative displacement ∆u for each pair of nodes can be described as u uk uk ( I1212 | I1212 )uN ∆u = u+ − u− = (− I12×12 I12×12 )u N
(2)
k k where I1212 is the identity diagonal matrix.
(2)
Converting u to the relative displacement at the interface local coordinate, defining the
where I12×12 is the identity diagonal matrix. T δn , δs ,δt atinthe relative displacements local coordinates. n, sdefining and t represent Converting ∆u to the tensor relativeδkdisplacement interface local Subscript coordinate, the relative T the local 3-direction, respectively, in which beswritten as δ k cann, displacements tensor δkthe = 1, δt } in local coordinates. Subscript and t represent the local {δand n , δs2-direction, 3-direction, the 1- and 2-direction, respectively, in which δ can be written as δ Bu k (3) k
k
where B is the geometric matrix. δk = B∆uk Unlike continuum elements, the interface element stiffness matrix before softening onset penalty stiffness whererequires B is thethe geometric matrix. K of the interface material [36]. The penalty stiffness K relates to the
(3)
elementcontinuum traction stresses the element relative displacements tensor δ konset : k the n , interface , t to element Unlike elements, stiffness matrix before softening requires s the penalty stiffness K of the interface material [36]. The penalty stiffness K relates to the element traction stresses σk = {σn , σs , σt }T to the element relative displacements tensor δk : T
Appl. Sci. 2017, 7, 93
4 of 16
Appl. Sci. 2017, 7, 93
4 of 15
k K k
(4)
.
σk =traction-separation Kδk . (4) From Equation (4), it can be seen that constitutive behavior is not stress-strain constitutive behavior. The interface element can be visualized as a spring between the From Equation (4), it can be seen that traction-separation constitutive behavior is not stress-strain initially coincident nodes of the element. After softening onset, the initially coincident nodes will constitutive behavior. The interface element can be visualized as a spring between the initially open or slide relative to each other [37]. Due to the zero-thickness interface element describing coincident nodes of the element. After softening onset, the initially coincident nodes will open or slide interlaminar behavior, the three traction stresses can be considered as corresponding interlaminar relative to each other [37]. Due to the zero-thickness interface element describing interlaminar behavior, stresses. In addition, the penalty stiffness K is the interface element constitutive parameter, which the three traction stresses can be considered as corresponding interlaminar stresses. In addition, has been introduced by some researchers [38]. The value of K is usually expected to be higher than the penalty stiffness K is the interface element constitutive parameter, which has been introduced by the elastic modulus of the material, but it should also be noted that a larger value for K in FEM some researchers [38]. The value of K is usually expected to be higher than the elastic modulus of the might lead to a bad convergence [31]. material, but it should also be noted that a larger value for K in FEM might lead to a bad convergence [31].
3. Verification of the Interface Model with a Benchmark Problem 3. Verification of the Interface Model with a Benchmark Problem The interlaminar stress calculation strategy based on linear elastic traction-separation behavior The interlaminar stress calculation strategy based on linear elastic traction-separation behavior was validated by the consideration of a benchmark problem. This benchmark problem was was validated by the consideration of a benchmark problem. This benchmark problem was analyzed analyzed by Pagano [17], where the 3D exact elasticity solution of a square, simply supported, by Pagano [17], where the 3D exact elasticity solution of a square, simply supported, symmetric symmetric cross-ply [0/90/0] laminated plate under a bisinusoidally distributed transverse load q, cross-ply [0/90/0] laminated plate under a bisinusoidally distributed transverse load q, was provided. was provided. As shown in Figure 2, the span and thickness of the laminate is denoted by “a” and As shown in Figure 2, the span and thickness of the laminate is denoted by “a” and “h”, and q can be “h”, and q can be described as described as πx / )a)sin( πy / a)) q =q q0qsin (πx/a sin(πy/a (5) 0 sin( where q00 is a constant. A laminate S =S a/h = 4= are considered, andand can can be considered as a thick laminateof ofspan-to-thickness span-to-thicknessratio ratio = a/h 4 are considered, be considered as a laminate. The thickness of the ply is 0.25 when a =when 3 mm.a The following lamina material thick laminate. The thickness of the plymm is 0.25 mm = 3 mm. The following laminaproperties material 6 psi, E 6 = E = 106 psi,6 G are used: Eare 25 ×E10 =12G=23G= 0.2 psi, νν1212= = ν23 = 0.25. properties 1 = 25 × 102 psi, 3E2 = E3 = 10 psi, 23 = 0.2×× 1066 psi, ν13ν=13ν= 23 = 0.25. The 1 =used: 12 G The resulting stresses are normalized according to the following formulae: resulting stresses are normalized according to the following formulae:
11 τ,xzτ,yzτ yz = τ yz τxz ττ , τ, yz xzxz qq 0S 0S
y
q
(6)
h
z a
x a
Figure 2. Square laminated plate subjected sinusoidally distributed transverse load. Figure 2. Square laminated plate subjected sinusoidally distributed transverse load.
The interlaminar stresses analyses were performed using the commercial FE code, Abaqus The interlaminar stresses analyses were performed using the commercial FE code, Abaqus (Version 6.11, Dassault Systemes Simulia Corp., Providence, RI, USA, 2011). A 3D 8-node linear (Version 6.11, Dassault Systemes Simulia Corp., Providence, RI, USA, 2011). A 3D 8-node linear brick, brick, with an incompatible modes element, was used to model the ply. The interface element based with an incompatible modes element, was used to model the ply. The interface element based on the on the linear traction-separation description was introduced into two adjacent plies. This linear traction-separation description was introduced into two adjacent plies. This constitutive response constitutive response is readily available in Abaqus and therefore does not require user-defined is readily available in Abaqus and therefore does not require user-defined subroutines. This 3D FE subroutines. This 3D FE model is generated using uniform meshes, where the in-plane mesh model is generated using uniform meshes, where the in-plane mesh consisted of 30 × 30 elements. consisted of 30 × 30 elements. For comparison purposes, two kinds of mesh density in the thickness For comparison purposes, two kinds of mesh density in the thickness directions are considered, namely directions are considered, namely coarse mesh and fine mesh, respectively. For the case, a = 3 mm, coarse mesh and fine mesh, respectively. For the case, a = 3 mm, S = 4, the former of which is just S = 4, the former of which is just one element for each ply in the thickness direction, and the latter of one element for each ply in the thickness direction, and the latter of which indicates four elements for which indicates four elements for each ply. According to the recommended rage of the interface each ply. According to the recommended rage of the interface stiffness by Reference [38], K is taken stiffness by Reference [38], K is taken as 5 × 106 N/mm3. The distribution of various as 5 × 106 N/mm3 . The distribution of various non-dimensionalized stresses τxz , τyz respectively, non-dimensionalized stresses τ , τ yz respectively, through the thickness of the plate at the through the thickness of the platexzat the position where each stress attained a maximum is shown in
Appl. Sci. 2017, 7, 93 Appl. Sci. 2017, 7, 93
5 of 15 5 of 16
position where each stress attained a maximum is shown in Figure 3. In this procedure, the Figure 3. In this procedure, the interlaminar stresses were calculated by the interface element, which interlaminar stresses were calculated by the interface element, which used the traction-separation used the traction-separation description. The transverse stresses in layers were analyzed based on 3D description. The transverse stresses in layers were analyzed based on 3D elasticity and extrapolated elasticity and extrapolated from the element integration point. from the element integration point.
Thickness coordinate, z/h
(a)
z/h=0.667
z/h=0.333 Exact Coarse mesh Refine mesh
Transverse shear stress,`τxz
Thickness coordinate, z/h
(b)
z/h=0.667
z/h=0.333 Exact Coarse mesh Refine mesh
Transverse shear stress, `τyz
Figure 3.3.Nondimensionalized Nondimensionalizedtransverse transverse shear stress distributions through the thickness of a Figure shear stress distributions through the thickness of a square τ square laminates (a = 3 mm, S = 4): (a) ; (b) . τ xz yz laminates (a = 3 mm, S = 4): (a) τxz ; (b) τyz .
As shown in Figure 3, for the coarse mesh, the percentage error of τ at the point of interface, As shown in Figure 3, for the coarse mesh, the percentage error of τxzxz at the point of interface, compared with with the 2.9%. However, However, the the percentage percentage error error at at the the compared the exact exact solution solution is is 4.3%, 4.3%, and and ττyzyz isis 2.9%. maximum value exactissolution is 21%, 20.9%, τ xz τand points ofofmaximum value of τof with thewith exactthe solution 21%, 20.9%, respectively. τ yz compared xz and yz compared Refined mesh in thickness direction can improve the precision of the transverse stress calculation respectively. Refined mesh in thickness direction can improve the precision of the transverse stress precision within the ply, for the value of τxz and τyz of , theτ percentage error compared with calculation precision within themaximum ply, for the maximum value xz and τ yz , the percentage error the exact solution is 3.1% and 5.3%, respectively, and the stresses obtained at interface are very close to compared with the exact solution is 3.1% and 5.3%, respectively, and the stresses obtained at the exact solution. In summary, for the transverse stresses at interface (namely interlaminar stresses), interface are very close to the exact solution. In summary, for the transverse stresses at interface both of these meshing strategies showed a reasonable accuracy (within 5%). It was noted that the stress (namely interlaminar stresses), both of these meshing strategies showed a reasonable accuracy distributions within the ply predicted by the coarse mesh model showed considerable error for thick (within 5%). It was noted that the stress distributions within the ply predicted by the coarse mesh plate. As the transverse stresses at the interface are only considered in this paper, the modeling strategy model showed considerable error for thick plate. As the transverse stresses at the interface are only with coarse mesh in thickness direction is still used in this paper. Furthermore, the interlaminar shear considered in this paper, the modeling strategy with coarse mesh in thickness direction is still used stress τxz and τyz distribution at the 0/90 interface without additional normalization based on the in this paper. Furthermore, the interlaminar shear stress τ xz and τ yz distribution at the 0/90 coarse mesh strategy is shown in Figure 4. This stress distribution states are consistent with the results interface without in Reference [20]. additional normalization based on the coarse mesh strategy is shown in Figure 4. This stress distribution states are consistent with the results in Reference [20].
Appl. Sci. 2017, 7, 93 Appl. Sci. 2017, 7, 93
6 of 15 6 of 16
Appl. Sci. 2017, 7, 93 (a)
6 of 16
(b) (b)
(a)
4. Interlaminar Interlaminarshear shearstress stress distributions at the [90/0] interface a [0/90/0] laminate =3 Figure 4. distributions at the [90/0] interface of a of [0/90/0] laminate (a = 3(amm, Figure 4. Interlaminar shear stress distributions at the [90/0] interface of a [0/90/0] laminate (a = 3 mm, = 4), obtained from a linear elastic traction-separation description interface: (a) ; (b) τ xzτyz τ yz . S =mm, 4),S obtained from a linear elastic traction-separation description interface: (a) τ ; (b) . xz S = 4), obtained from a linear elastic traction-separation description interface: (a) τ xz ; (b) τ yz .
of of Composite 4. FEM 4. FEM CompositeLaminates Laminateswith withaaBolt-Filled Bolt-Filled Hole Hole AA filled-hole composite laminate analyze the the interlaminar interlaminarstress stressdistributions distributions A filled-hole filled-holecomposite compositelaminate laminatewas was selected selected to to analyze within the laminate near the bolt. to The geometric model is designed according the geometry within the laminate near the bolt. The geometric model is designed according to the geometry of of thethe filled-hole specimens References [16,39]. Henceforth in “filled-hole laminate” will in Henceforth in this paper, filled-hole specimens in References [16,39]. Henceforth in this paper, “filled-hole laminate” will bebe used to to refer toto the laminate the specimens specimenswith withuntightened untightened bolts will used refer the laminatewith withtightened tightened bolts, bolts, and the bolts will bebe special notes. special notes. FE modelwas developed inin Abaqus predict the stress distributions near 3D FE model in Abaqus to predict the interlaminar interlaminar stressstress distributions near AA 3D3D FE model wasdeveloped developed Abaqus to predict the interlaminar distributions hole edge of the filled-hole composite laminate. consists ofof the laminate, thethe hole edge of the filled-hole composite laminate. This 3D 3D FE FE model model consists thebolt, bolt, near the hole edge of the filled-hole composite laminate. This 3D FE model consists oflaminate, the bolt, and interface basedon on the the linear elastic traction-separation behavior. The bolt and nutnut and thetheinterface based elastic traction-separation behavior. The bolt and laminate, and the interface based onlinear the linear elastic traction-separation behavior. The bolt and simplified as a whole in the FE model is shown in Figure 5. The stacking sequence of the laminate is is simplified as a whole in the FE model is shown in Figure 5. The stacking sequence of the laminate [0 2 /45/90/−45/0/45/0/−45/0]s. One linear solid element per ply in the thickness direction was used to One linear solid element per ply the direction waswas used to [022/45/90/−45/0/45/0/−45/0]s. /45/90/−45/0/45/0/−45/0]s. One linear solid element perinply inthickness the thickness direction used model the laminate,and andthe theinterface interfaceelement element was was introduced introduced between between plies. The bolt was defined model thethe laminate, introduced betweenplies. plies.The Thebolt bolt was defined to model laminate, and the interface element was defined as isotropy material, andsolid linear solid element element was also As stress as as isotropy material, and linear solid was also employed. employed. Ashigh highinterlaminar interlaminar stress isotropy material, and linear element was also employed. As high interlaminar stress concentrations concentrations were expected near the hole, a high mesh refinement was used in this area, as seen concentrations werethe expected nearmesh the hole, a highwas mesh refinement was this area, were expected near hole, a high refinement used in this area, asused seen in Figure 5. as seen in Figure 5. in Figure 5. D D Interfaces in the Interfacesregion in the clamp-up clamp-up region d
L
L
W
d
Bolt
Bolt
W
F0 Laminate
F0
Laminate P
Uy=Uz=0
Uy=Uz=0 P
Uy=Uz=0
Uy=Uz=0
P
F0
P
F0
Figure 5. Finite element mesh and modeling strategy for a filled hole compression (FHC) specimen, Figure 5. Finite element mesh and modeling strategy for a filled hole compression (FHC) specimen, and showing boundary conditions. Figure 5. Finite elementconditions. mesh and modeling strategy for a filled hole compression (FHC) specimen, and showing boundary and showing boundary conditions.
As shown in Figure 5, the laminate dimensions are L = 32 mm by W = 32 mm, the hole of the As shown Figure 5, are Lof = 32 mm by is W = 32 mm, the hole of laminate has ain ofthe D =laminate 6.35 mm, dimensions and the thickness lamina mm.the Thehole bolt of hasthe As shown indiameter Figure 5, the laminate dimensions are L = the 32 mm by Wt == 0.25 32 mm, the laminate has a diameter of D = 6.35 mm, and the thickness of the lamina is t = 0.25 mm. The bolt has a shank diameter D bolt = 6.35 mm, and head diameter d = 11.3 mm. Elastic, isotropic material laminate has a diameter of D = 6.35 mm, and the thickness of the lamina is t = 0.25 mm. The bolt has a shank diameter Dbolt = 6.35 mm, and head with diameter d = 11.3 Elastic, material properties properties are used section, E = 109 GPamm. ν = 0.3.isotropic laminate is made of a shank diameter Dboltfor= the 6.35bolt mm, and head diameter d =and 11.3 mm. The Elastic, isotropic material arematerial used forplies the bolt with E 109 GPa and νelastic = 0.3. and The orthothropic. laminate is made materialmaterial plies that that section, are idealized as=homogeneous, The of following properties are used for the bolt section, with E = 109 GPa and ν = 0.3. The laminate is made of used: E11 = 181elastic GPa, and E22 =orthothropic. E33 = 10.3 GPa, 12 = G13 = material 7.17 GPa,properties G23 = 4.13are GPa, areproperties idealized are as homogeneous, TheGfollowing used: material plies that are idealized as homogeneous, elastic and orthothropic. The following material ν12 GPa, = ν13 =Eν 0.25. and denote the fiber, transverse, E11and = 181 =Subscripts 10.3 GPa, 1, G 2, = G 3= 7.17 GPa, G = 4.13 GPa,and andthickness ν12 = ν13directions, = ν23 = 0.25. 2223== E properties are used: E3311 = 181 GPa, E1222 = E1333 = 10.3 GPa, 23 G12 = G13 = 7.17 GPa, G23 = 4.13 GPa, respectively. Subscripts 1, 2, and 3 denote the fiber, transverse, and thickness directions, respectively. and ν12 = ν13 = ν23 = 0.25. Subscripts 1, 2, and 3 denote the fiber, transverse, and thickness directions, The Coulombfriction frictionmodel modelwas was adopted adopted for and thethe bolt. The Coulomb for the the contact contactbetween betweenthe thelaminate laminate and bolt. respectively. Based on previous investigations [40], the friction coefficients between bolt shank and laminate is is Based on previous investigations [40], the friction coefficients between bolt shank and laminate The Coulomb friction model was adopted for= the contact betweenand theloading laminate and theare bolt. μ = 0.1, and between bolt head and laminate is μ 0.43. The boundary conditions µ = 0.1, and between bolt head and laminate is µ = 0.43. The boundary and loading conditions are also Based on previous investigations [40], the friction coefficients between bolt shank and laminate is μ = 0.1, and between bolt head and laminate is μ = 0.43. The boundary and loading conditions are
Appl. Sci. 2017, 7, 93 Appl. Sci. 2017, 7, 93
7 of 16 7 of 15
also shown in Figure 5. The in-plane surface traction load P = 100 N/mm2 are applied to the both ends of the model. The bolt load is defined in terms of a concentrated2 force, and different pre-load shown in Figure 5. The in-plane surface traction load P = 100 N/mm are applied to the both ends F0 are applied on the bolt. of the model. The bolt load is defined in terms of a concentrated force, and different pre-load F0 are applied on the bolt. 5. Results 5. Results 5.1. Interlaminar Stress Distributions in the Filled-Hole Laminate near the Hole 5.1. Interlaminar Distributions in the Filled-Hole Laminate near at thethe Holeinterface in the filled-hole A numericalStress solution of interlaminar stress distributions laminate near the solution hole is shown in Figurestress 6. Analyses were performed in the of two levels A numerical of interlaminar distributions at the interface incondition the filled-hole laminate of clamp-up load, namely non-tightened bolt case and tightened bolt case. In the former case, the near the hole is shown in Figure 6. Analyses were performed in the condition of two levels of clamp-up clamping force is assumed as F 0 = 10 N, the latter as F 0 = 2000 N. The interlaminar shear stress load, namely non-tightened bolt case and tightened bolt case. In the former case, the clamping forceτ yzis assumed as F0in= this 10 N, the latter as Fof 2000 N. The interlaminar shearofstress not present in this is not present paper because similar distribution feature remarks τ xz . τThe yz isfollowing 0 =the paper of the similar distribution feature of τxz . The following remarks can be highlighted: can be because highlighted: 1. 1.
2. 2.
3. 3.
Forthe thenon-tightened non-tightenedbolt bolt case case (clamping (clamping force force FF00 == 10 10 N), N), interlaminar interlaminar stress stress concentrations concentrations For mainly occur around the hole edge (Figure 6a,b, Position I), and tapers out at the interior of of the the mainly occur around the hole edge (Figure 6a,b, Position I), and tapers out at the interior laminate. This prediction is consistent with the previous phenomenon that laminated composites laminate. This prediction is consistent with the previous phenomenon that laminated often exhibit transverse concentrations near material near and geometric discontinuities composites often exhibit stress transverse stress concentrations material and geometric (the so-called free-edge effect) [1]. In spite of the bolt, Position I can still be considered as discontinuities (the so-called free-edge effect) [1]. In spite of the bolt, Position I can still be geometric discontinuities, and the free-edge effects are observed here. considered as geometric discontinuities, and the free-edge effects are observed here. Forthe thetightened tightenedbolt boltcase case(clamping (clampingforce force FF00 ==2000 2000N), N),the theresults resultsshow show that that the the interlaminar interlaminar For stress concentrations in the filled-hole laminate occur at Position I, as well as the interior of the the stress concentrations in the filled-hole laminate occur at Position I, as well as the interior of laminatenear nearthe theperiphery peripheryof ofthe thebolt bolthead head (Figure (Figure 6c,d, 6c,d, Position Position II). II). Note Note that that Position Position IIII of of laminate the laminate is not a free edge. However, traditionally, the free edge effect is only a concern the laminate is not a free edge. However, traditionally, the free edge effect is only a concern near the free laminate, andand decay to zero theas distance from thefrom free edge increases. near free edge edgeofofthethe laminate, decay to as zero the distance the free edge σz is a negative in thevalue clamp-up in other words, σz iswords, interlaminar compressive increases. a negative in theregion, clamp-up region, in other σ z is value σ z is interlaminar stress. According Reference the compression interlaminar stress is able to delay compressive stress. to According to[41], Reference [41], the compression interlaminar stress is ablethe to delamination initiation. delay the delamination initiation. shouldalso alsobe benoted noted that that the the stress stress concentrations concentrations of of σσz and τxz near Position II are not in the ItItshould z and τ xz near Position II are not in same circumference locations. The former at at Position II II is is shown 6c,d. The latter is the same circumference locations. The former Position shownininFigure Figures 6c,d. The latter located near Position II, which is away from the hole in the in-plane direction, this position in the is located near Position II, which is away from the hole in the in-plane direction, this position laminate is defined as Position III, as III, presented in Figure 6d. 6d. in the laminate is defined as Position as presented in Figure (b)
(a)
σz
1
Ⅱ
Ⅱ
Ⅰ
τ xz
Ⅰ Z 3
2
Ⅰ Z
X
X
Y
Y
0/0 interface
0/0 interface
(c)
σz
(d) 1
Ⅱ
Ⅱ
Ⅰ
Ⅱ
Ⅰ
Ⅱ
τ xz
Ⅱ Ⅲ
Ⅲ
Ⅰ Ⅱ Ⅰ
Ⅰ Z 2
Z
X 3
X
Y
Y 0/0 interface
0/0 interface
Figure 6. Stress Stressdistributions distributions at interface the interface the filled-hole near(a)the hole: (a) Figure 6. at the in the in filled-hole laminate laminate near the hole: Interlaminar σ z , withbolt; Interlaminar non-tightened bolt; shear (b) Interlaminar shear stress τ xz , with normal stressnormal σz , withstress non-tightened (b) Interlaminar stress τxz , with non-tightened bolt;
σ z , (d) τ xz ,1—Bolt; (c) σz , with tightened τxztightened , with tightened 2—Interface; 3—Quarter clamp-up non-tightened bolt; (c) bolt; with bolt; (d)bolt. with tightened bolt. 1—Bolt; 2—Interface; region; I—The hole edge; II—The projection boundary of the periphery of the bolt head on the laminate.
Appl. Sci. 2017, 7, 93
8 of 16
Appl. Sci. 2017, 7, 93clamp-up region; I—The hole edge; II—The projection boundary of the periphery of the8 of 15 3—Quarter
bolt head on the laminate.
5.2. Stress Distributions at Each Interface around the Hole Edge and the Periphery of Bolt Head 5.2. Stress Distributions at Each Interface around the Hole Edge and the Periphery of Bolt Head According to our numerical calculations, interlaminar stress concentrations occurred at Position According to our numerical calculations, interlaminar stress concentrations occurred at I, as well as the interior of the laminate near Position II. The interlaminar shear stress distributions Position I, as well as the interior of the laminate near Position II. The interlaminar shear stress at each interface around Position I with the tightened bolt (F0 = 2000 N) are shown in Figure 7. distributions at each interface around Position I with the tightened bolt (F0 = 2000 N) are shown in The interlaminar normal stress is not presented in this subsection, due to the assumption of that Figure 7. The interlaminar normal stress is not presented in this subsection, due to the assumption interlaminar compressive stress is able to delay delamination. This assumption is in agreement with of that interlaminar compressive stress is able to delay delamination. This assumption is in the view of Reference [41]. In addition, shear stress distributions play a significant role in determining agreement with the view of Reference [41]. In addition, shear stress distributions play a significant the mechanical behavior of multi-direction laminates. Therefore, we only present the shear stress role in determining the mechanical behavior of multi-direction laminates. Therefore, we only distributions here. Furthermore, the data points of the stress distribution curve are the absolute values present the shear stress distributions here. Furthermore, the data points of the stress distribution of the magnitude of the interlaminar shear stresses. curve are the absolute values of the magnitude of the interlaminar shear stresses. (a) 35
, MPa
30 25
Stress xz
20 15 10 5 100 80 60
40
An 20 gl e 0-20 , d -40 eg -60 -80 -100
0/0
0 0/0 -45/0 0/-45 e 45/0 at 0/45 in m -45/0 la 90/-45 he 45/90 ft o 0/45 ce
In
rfa te
0/0 0/45 45/90 90/-45 -45/0 0/45 45/0 0/-45 -45/0 0/0
Middle interface
y Ⅰ θ x Ⅱ
Outer interface
(b) 70
50
20
Stress yz
40 30
10 100 80 60
40
20 An 0 gl e -20 ,d -40 eg -60 -80 -100
0/0
0 0/0 -45/0 0/-45 e 45/0 at 0/45 in -45/0 m la 90/-45 he 45/90 ft o 0/45 e
, MPa
60
c rfa te Outer interface In
0/0 0/45 45/90 90/-45 -45/0 0/45 45/0 0/-45 -45/0 0/0
Middle interface
y Ⅰ θ x Ⅱ
τ yz . Outer Figure Interlaminar shear stress distributions xz ; (b) Figure 7.7.Interlaminar shear stress distributions aroundaround PositionPosition I: (a) τxz ; I:(b)(a) τyz .τOuter interface—The interfaces located outsidelocated the filled-hole laminate, such as the such 0/0 and 0/45 Middle interface—The interfaces outside the filled-hole laminate, as the 0/0 interface. and 0/45 interface. interface—The interfaces near the symmetry plane of laminate, as thesuch −45/0 interface. Middle interface—The interfaces near the symmetry plane of such laminate, as and the 0/0 −45/0 and 0/0 interface.
As seen in Figure 7, the magnitude of interlaminar shear stresses at the 0/0 interface are lower As seen Figure 7, the in magnitude of area interlaminar shear stresses theinterfaces 0/0 interface than those of in other interfaces most of the around Position I. Theseat0/0 not are onlylower refer than those of other interfaces in most of the area around Position I. These 0/0 interfaces not only to the middle interface, but also the outer interface. The value of the interlaminar shear stresses of refer to the middle alsointhe outer Thethe value theIninterlaminar these 0/0 interface dointerface, not exceedbut 2 MPa most of theinterface. area around hole of edge. contrast, the shear outer
Appl.Appl. Sci. 2017, 7, 937, 93 Sci. 2017,
9 of 15 9 of 16
stresses of these 0/0 interface do not exceed 2 MPa in most of the area around the hole edge. In
0/45contrast, interfacethe or the middle −45/0 interface exhibits−45/0 a higher magnitude of interlaminar shear stresses. outer 0/45 interface or the middle interface exhibits a higher magnitude of In general, a higher shear stresses occurs at all non 0/0 interface at Position I. This phenomenon interlaminar shear stresses. In general, a higher shear stresses occurs at all non 0/0 interface at is similar problems of interlaminar that of exist at the free edge laminate. PositiontoI.the Thisclassical phenomenon is similar to the classicalstress problems interlaminar stress thatofexist at edgeproblems of laminate. The two classical problems ofininterlaminar stress arelaminates included exhibit in The the two free classical of interlaminar stress are included Reference [42]. [±θ] [42]. [±θ] laminatesand exhibit only shear-extension coupling, and [0/90] laminates exhibit onlyReference shear-extension coupling, [0/90] laminates exhibit only a Poisson mismatch between layers. only a Poisson mismatch between layers. Both of these two problems exist at the 0/45 interface Both of these two problems exist at the 0/45 interface where adjacent layers of the laminate are subjected where layers the interface laminate are subjected to an interlaminar axial load, so the 0/45stress interface exhibits to an axialadjacent load, so the of 0/45 exhibits obvious shear concentrations. obvious interlaminar shear stress concentrations. In addition, due to a match in the elastic In addition, due to a match in the elastic properties between the 0/0 interface adjacent layers, the properties between the 0/0 interface adjacent layers, the interlaminar shear stress at 0/0 interface is interlaminar shear stress at 0/0 interface is lower than at other interfaces. lower than at other interfaces. The interlaminar shear stress distributions curve at Position III is shown in Figure 8. As seen in The interlaminar shear stress distributions curve at Position III is shown in Figure 8. As seen in Figure 8a, the of τof interface of the is higher thanthan thatthat of the xz in τ xzthe Figure 8a, magnitude the magnitude in outer the outer interface of laminate the laminate is higher of middle the ◦ and −90◦ , and there is a decrease in thickness direction interfaces at Position III, except near θ = 90 middle interfaces at Position III, except near θ = 90° and −90°, and there is a decrease in thickness fromdirection outside from to inside. τyz to shows a similar tendency excepted nearexcepted θ = 0◦ . Anear larger of the outside inside. a similar tendency θ =magnitude 0°. A larger τ yz shows interlaminar shear stress τyz is also exhibited at the 0/45 interface. Therefore, the maximum value of magnitude of the interlaminar shear stress τ yz is also exhibited at the 0/45 interface. Therefore, the τxz or τyz at the outer 0/0 and 0/45 interfaces are much higher than those of the middle −45/0 and maximum value of τ xz or τ yz at the outer 0/0 and 0/45 interfaces are much higher than those of 0/0 interfaces. It is interesting to note that, at Position I (Figure 7), regardless of whether it is the outer the middle −45/0 and 0/0 interfaces. It is interesting to note that, at Position I (Figure 7), regardless interface or the middle, large interlaminar shear stresses at the 0/45 interface and small interlaminar of whether it is the outer interface or the middle, large interlaminar shear stresses at the 0/45 shear stresses at 0/0 interface are found. However, at Position III, the magnitude of interlaminar interface and small interlaminar shear stresses at 0/0 interface are found. However, at Position III, stresses appears to have little relation to the elastic properties of adjacent layers and exhibit a higher the magnitude of interlaminar stresses appears to have little relation to the elastic properties of value at the outer adjacent layers interfaces. and exhibit a higher value at the outer interfaces. (a) 0/0 0/45 45/90 90/-45 -45/0 0/45 45/0 0/-45 -45/0 0/0
20
8 4
100 80 60 40 An 20
gl e
0 -20 eg -40-60 -80 -100
d
Appl. Sci. 2017, 7, 93
0/0
Stress, xz
12
MPa
16
0 0/0 -45/0 0/-45 e 45/0 at in 0/45 -45/0 am l e 90/-45 th 45/90 of e 0/45 c
rfa te
In
Middle interface
y θ x
Ⅰ
Ⅲ Ⅱ
Outer interface
10 of 16
(b) 0/0 0/45 45/90 90/-45 -45/0 0/45 45/0 0/-45 -45/0 0/0
12
Stress yz
8
, MPa
10
6 4 2 0 100 80 60 40 A 20
ng le
,
0 -20 de -40 g -60 -80 -100
0/0
0/0 -45/0 0/-45 e 45/0 at in 0/45 m -45/0 a l 90/-45 he 45/90 ft o 0/45 ce
rfa te
In
Middle interface
Outer interface
y Ⅰ
θ x
Ⅲ Ⅱ
τ xz τ; xz Figure Interlaminarshear shear stress stress distributions Position III:III: (a) (a) (b) τ yzτ.yz . Figure 8. 8.Interlaminar distributionsaround around Position ; (b) 5.3. Stress Distributions at the 0/45 Interface near the Hole under Different Clamp-Up Load Generally, the plies elastic properties will have a momentous effect on the interlaminar stresses. However, for the laminate with tightened bolts, bolt clamp-up load seems to play a more important role for interlaminar stress distributions in the laminates near the bolt head, than in the
An 20 0 gl e -20 ,d eg -40-60 -80 -100
0/0
y
i 0/45 -45/0 am el 90/-45 h t 45/90 of 0/45 ce rfa te
In
θ x
Ⅰ
Ⅲ Ⅱ
Outer interface
Appl. Sci. 2017, 7, Figure 93 8. Interlaminar shear stress distributions around Position III: (a)
τ xz ; (b) τ yz .
10 of 15
5.3. Stress Distributions at the 0/45 Interface near the Hole under Different Clamp-Up Load 5.3. Stress Distributions at the 0/45 Interface near the Hole under Different Clamp-Up Load Generally, the plies elastic properties will have a momentous effect on the interlaminar Generally, the plies elastic properties will have a momentous effect on the interlaminar stresses. stresses. However, for the laminate with tightened bolts, bolt clamp-up load seems to play a more However, for the laminate with tightened bolts, bolt clamp-up load seems to play a more important important role for interlaminar stress distributions in the laminates near the bolt head, than in the role forproperties. interlaminar stress distributions in the laminates the bolt thandistribution in the ply properties. ply To further understand the clamp-up loadnear F0 effect on head, the stress around ToPosition further understand the clamp-up load F effect on the stress distribution around I and 0 I and Position III, numerical calculation was performed at different clampingPosition forces. The Position III, numerical calculation was performed at different clamping forces. The interlaminar stress interlaminar stress distributions in the 0/45 interface at Position I and Position III are only given in distributions in thedue 0/45 Position I and Position are only given this subsection, due this subsection, to interface the 0/45 at interface exhibiting a highIIImagnitude of theininterlaminar shear to stress the 0/45 interface exhibiting a high magnitude of the interlaminar shear stress in this position. in this position. The variation of the interlaminar stresses around Position I at the 0/45 The variation the interlaminar around Positionstresses I at thearound 0/45 interface presented interface areofpresented in Figurestresses 9, and the interlaminar Position are III at the 0/45 in Figure 9, and interlaminar around Position III at the 0/45 interface are shown in Figure 10. interface arethe shown in Figurestresses 10. s33-hole
(a)
5
Interlaminar normal stress z, MPa
0 -5 -10 -15 -20 -25 -30 -35 -40 -45 -50
y Ⅰ
-55 -60
θ x
10 N Clamp-up 2000 N Clamp-up 6000 N Clamp-up
-65 -70
Ⅱ
-75
Appl. Sci. 2017, 7, 93
-90
-80
-70
-60
-50
-40
-30
-10
0
10
50
y Ⅰ
10 N Clamp-up 2000 N Clamp-up 6000 N Clamp-up
45 40
θ x Ⅱ
35 30 25 20 15 10
(c)
100
Interlaminar shear stress yz, MPa
Interlaminar shear stress xz, MPa
(b)
-20
20
30
Angle , deg
(b) S13 hole
90
40
50
60
70
80
11 of 16
90
(c) s23 edge 10 N Clamp-up 2000 N Clamp-up 6000 N Clamp-up
80
y Ⅰ 70
θ x
60
Ⅱ
50 40 30 20 10
5
0
0 -90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
Angle , deg
20
30
40
50
60
70
80
90
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
60
70
80
90
Angle , deg
Figure Stress distributions 0/45interface interfaceatatPosition PositionI Iatatdifferent differentclamping clampingforces: forces:(a) (a)σzσ;z(b) ; (b) Figure 9. 9. Stress distributions inin 0/45 τxz ; (c) ττxzyz;. (c) τ yz .
seen Figure9,9, AsAs seen inin Figure
The interlaminar normal stress σ z rises with increased clamp-up load in all regions of The interlaminar normal stress σz rises with increased clamp-up load in all regions of Position I Position I at the 0/45 interface (Figure 9a). In addition, σ z also shows interlaminar at the 0/45 interface (Figure 9a). In addition, σz also shows interlaminar compressive stress at compressive stress I forN.the case of F0 = 6000 N. Position I for the caseatofPosition F0 = 6000 A higher clamping force results in lower interlaminar shear stresses for most of the region • A higher clamping force results in lower interlaminar shear stresses for most of the region around around Position I (Figure 9b,c). This area is in the region around −80° < θ
