Determination of cohesive laws in wood bonded joints under mode II loading using the ENF test

International Journal of Adhesion & Adhesives 51 (2014) 54–61

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Determination of cohesive laws in wood bonded joints under mode II loading using the ENF test F.G.A. Silva a, J.J.L. Morais b, N. Dourado b,n, J. Xavier b, F.A.M. Pereira b,c, M.F.S.F. de Moura c a

INEGI—Instituto de Engenharia Mecânica e Gestão Industrial, Rua Dr. Roberto Frias, 400, 4200-465 Porto, Portugal Universidade de Trás-os-Montes e Alto Douro, UTAD, CITAB, Departamento de Engenharias, Quinta de Prados, 5000-801 Vila Real, Portugal c Faculdade de Engenharia da Universidade do Porto, Departamento de Engenharia Mecânica, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal b

art ic l e i nf o

a b s t r a c t

Article history: Accepted 6 February 2014 Available online 28 February 2014

Cohesive laws of wood bonded joints under mode II loading were determined using two different methods. The direct method requires the establishment of the relation between the strain energy release rate under mode II loading (GII) and the corresponding crack tip shear displacement (CTSDII). An equivalent crack length procedure was used to assess the evolution of GII, with remarkable advantages relatively to classical data reduction schemes. The CTSDII was determined from displacements measured by digital image correlation. The cohesive laws were obtained by differentiation of a logistic function fitted to the GII ¼f(CTSDII) relation. The inverse method is based on an optimization procedure using a genetic algorithm with the objective to determine the cohesive law providing the best agreement between numerical and experimental load–displacement curves. Four parameters of the adjusted trapezoidal law were obtained in this optimization procedure. Both direct and inverse methods propitiated excellent agreement between numerical and experimental load–displacement curves, thus revealing their adequacy to determine cohesive laws of wood bonded joints under mode II loading. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Bonded joints Wood Cohesive laws End notched flexure (ENF) Digital image correlation

1. Introduction The use of bonded joints in wood structural applications is fundamental since it provides several advantages when compared to alternative connection methods. One example is the manufacturing of glued laminate timber (glulam) whose applications have been increasing since they lead to a more efficient structural application of wood. As a consequence, the development of accurate and suitable design methods for wood bonded joints acquires special relevancy. However, the application of the strength of materials or fracture mechanics based criteria in structural design presents some drawbacks. The stress based methods present mesh dependency during numerical analysis due to stress singularities that are normally present in the case of bonding. Fracture mechanics based approaches require the definition of an initial flaw or crack length. However, in many structural applications the damage initiation locus is not evident. On the other hand, the stress-based methods behave well at predicting damage initiation, and fracture mechanics has already demonstrated its accuracy in the crack propagation modelling. Cohesive zone models (CZM) overcome the referred drawbacks and exploit the advantages of each criterion. In fact CZM uses

n

Corresponding author. Tel./fax: þ 351 259 350356. E-mail address: [email protected] (N. Dourado).

http://dx.doi.org/10.1016/j.ijadhadh.2014.02.007 0143-7496 & 2014 Elsevier Ltd. All rights reserved.

a stress based analysis to model damage initiation and fracture mechanics to deal with damage propagation. Thus, it is not necessary to take into consideration that an initial defect and mesh dependency problems are overcome when appropriate refinement assuring several integration points under softening is assured. In addition, the use of cohesive models allows simulating damage onset and non-self-similar crack growth. Therefore, it is not necessary to assume an initial crack and damage propagation takes place without user intervention. They are usually based on a softening relationship between stresses and relative displacements between crack faces, thus simulating a gradual degradation of material properties. Cohesive elements must be positioned at the planes where damage is prone to occur, which is easy to identify in bonded joints. The application of cohesive zone models to wood and wood bonded joints is relatively recent [1-5]. In glulam, shear loads between lamellae are usually developed, especially in applications involving bending. As a result, fracture characterization of wood bonded joints under mode II loading is a crucial research topic. Consequently, the knowledge of the cohesive law under mode II loading is an important task since it allows the application of the cohesive models under these circumstances. Several authors have developed methodologies to determine the cohesive laws under mode I loading [6–9] and, in more recent times, some works dedicated to determination of cohesive laws under mode II loading have been published [10–12]. Leffler et al. [10] applied the so-called direct method, which consists of the

F.G.A. Silva et al. / International Journal of Adhesion & Adhesives 51 (2014) 54–61

experimental determination of the traction–deformation relationship for thin adhesive layers under mode II loading. Ouyang and Li [11] proposed an analytical solution for the interface shear fracture of end notched flexure (ENF) specimens, based on a cohesive zone model (CZM). Fernandes et al. [12] have used the direct method applied to the ENF test using an equivalent crack length procedure and digital image correlation to get the cohesive law of composite bonded joints. In this work, direct and inverse methods for evaluating the cohesive laws of wood bonded joints under mode II loading are presented. The direct method requires the monitoring of the crack shear displacement in mode II at its tip (CTSDII) and the continuous achievement of the corresponding strain energy release rate (GII) up to crack starting advance. The inverse method consists of a previous assumption of the cohesive law shape and evaluation of its cohesive parameters by a numerical procedure. An optimization strategy is used based on a developed genetic algorithm for minimizing the difference between the numerical and experimental load–displacement curves issuing from the ENF test. The cohesive law that leads to the best agreement between those curves is the one chosen to reproduce the fracture behavior under mode II loading. The two methods proved to be efficient on the determination of cohesive laws of wood bonded joints under mode II loading, although both present advantages and drawbacks that are discussed in detail in the present work.

2. Cohesive laws 2.1. Direct method The direct method is based on the relationship among the strain energy release rate in mode II (GII), the traction in mode II (τ) and the corresponding crack tip shear displacement (CTSDII). This relation can be given by [10] Z u τðuÞ du ; 0 r u r uu ð1Þ GII ðuÞ ¼ 0

where u represents the CTSDII. The differentiation of the previous equation provides the τ¼ f(u) relation τðuÞ ¼

dGII du

ð2Þ

which represents the cohesive law under mode II loading. This equation requires the determination of the evolution of the strain energy release rate in the course of the ENF test. In this work, GII was determined from a previously developed data reduction method based on the crack equivalent concept [13]. Considering the Timoshenko beam theory, the ENF specimen compliance (C¼δ/P) can be written as C¼

3a3 þ 2L3 8EL Bh

3

þ

3L 10GLR Bh

ð3Þ

where specimen dimensions (L, B, h, a), applied load (P) and displacement (δ) are identified in Fig. 1, and EL and GLR are the

55

longitudinal and shear moduli, respectively (L refers to the longitudinal and R refers to the radial anatomical directions of wood). Attending to the inherent variability of wood which has an impact on the elastic properties, the initial values of compliance C0 and crack length a0 are used to estimate an equivalent elastic modulus (Ef) for the specimen being tested using Eq. (3):  1 3a3 þ 2L3 3L C0  ð4Þ Ef ¼ 0 3 10GLR Bh 8Bh This procedure presents some advantages. In fact, besides the wood variability, there are several aspects intrinsic to the fracture test that are not accounted for the beam theory, as is the case of the presence of the adhesive, stress concentrations in the vicinity of the crack tip and contact between the specimen arms. These phenomena influence the specimen compliance and are indirectly taken into consideration by means of this Ef parameter (Eq. 4). The value of GLR is of minor relevancy and a typical value can be considered without affecting markedly the obtained results [13]. In the ENF fracture test the crack grows with its faces in contact with each other, which renders the clear identification of its tip difficult. In addition, modern ductile adhesives usually reveal a pronounced fracture process zone that should be accounted for toughness evaluation, since it is responsible for a non-negligible dissipation of energy. Owing to these aspects, the crack length monitoring in the course of the ENF test is not recommended. To overcome these difficulties an equivalent crack length (ae) approach can be pursued. The ae can be estimated from Eq. (3) using the current compliance C and the calculated Ef values (Eq. (4)):    1=3 Cc 3 2 Cc ae ¼ a0 þ 1 L3 ð5Þ 3 C 0c C 0c where Cc ¼ C 

3L 10BhGLR

and

C 0c ¼ C 0 

3L 10BhGLR

Finally, the evolution of GII ¼ f(ae) can easily be obtained by combining the Irwin–Kies equation: GII ¼

P 2 dC 2B da

ð7Þ

with Eq. (3), which leads to GII ¼

9P 2 a2e

ð8Þ

3

16B2 h Ef

This procedure provides the so called Resistance-curve (R-curve) which portrays the evolution of GII during the test, only requiring data issuing from the load–displacement curve. In order to get the cohesive law, the rigorous monitoring of the CTSDII is also of crucial importance. In the present case, this local relative displacement was continuously measured during the test by means of the digital image correlation technique (DIC) and synchronized with the evolution of GII [12]. DIC is a non-contact optical technique which provides the displacement field of a speckled pattern by correlating pair of images corresponding to R

R P,δ

a0

L d L

ð6Þ

2h

t

T B

L Fig. 1. Schematic representation of the ENF test.

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different mechanical states [14–17]. The CTSDII is defined as the relative displacement in crack direction measured with regard to the initial crack tip location. The CTSDII in mode II (u) was evaluated from the displacement field components as

function:

u ¼ jju þ –u– jj

where N represents the used number of points composing the P–δ curve obtained in the cohesive zone modelling, zti and zni the load values of a given point extracted from the numerical P–δ curve, and the corresponding experimental value, respectively. The design variables are coded in a binary string according to a predefined sequence. The algorithm first generates a set of random possible solutions (CZM parameters), also known as the initial population. Subsequently, after decoding, successive generations (identified as t) of CZM parameters (populations) are obtained based on how efficiently the error between the numerical (FEA) and the experimental P–δ curves is minimized. According to this strategy successively improved populations are obtained. The procedure has to take into account the set of constraints of the proposed problem in order to assure the generation of non-spurious solutions from the domain of the designed variables. This is guaranteed considering the following inequalities:

ð9Þ

where u þ and u  represent upper and lower displacement components parallel to crack extension, respectively.

2.2. Inverse method The inverse method is based on an optimization technique involving a genetic algorithm (GA). The main goal is to minimize an objective function which depicts the error between numerical (obtained by finite element analysis: FEA) and experimental load–displacement curves. The shape of the law is assumed a priori and the corresponding cohesive parameters are determined on the basis of the optimization procedure. In the present case, a trapezoidal law [18] with bilinear softening was considered (Fig. 2). Since some scatter was observed in the initial stiffness of the measured cohesive laws (direct method), the value used in the simulation of each specimen was the one which resulted from the experimental cohesive laws. The fracture energy in mode II (GIIc) must also be known in advance and should be considered as being the plateau value of the R-curve given by Eq. (8). In fact, the existence of propagation under constant energy reflects self-similar crack growth conditions that provide truthful characterization of material fracture. The area of the trapezoid (Fig. 2) is equated to this fracture energy (GIIc), thus allowing the definition of the ultimate relative displacement (uu) uu ¼

2GIIc  τu ðu2 þ u1  u0 Þ þ u1 τ2

ð10Þ

The parameters corresponding to local shear strength (τu), relative displacement of the second inflection point (u1) and the coordinates of the third inflection point (u2 and τ2) are obtained by the developed GA. This algorithm uses the evolutionary operators Selection, Crossover, Mutation and Elimination [19] to minimize the difference between the numerical and the experimental load– displacement curves, quantified by means of an objective function. The agreement is attained by identifying the combination of design variables (i.e., τu, u1, u2 and τ2, designated as b) used in the damage law, which permits to minimize the following objective

N

ΠðbÞ ¼ ∑ ‖zti  zni ‖2

ð11Þ

i¼1

τu 4 τ 2 ;

u1 4 u0 ;

u2 4u1 ;

τ u  τ 2 u2  u1 Z τ2 uu  u2

The last inequality (Eq. 12) has been established to assure that the slope of the last branch in the cohesive law is lower than the first one. The optimization procedure involves four main genetic operators known as Selection, Crossover, Mutation and Elimination. The first one (Selection) is used to gather in different groups potential solutions obtained through a ranking based on the value of ΠðbÞ. This procedure also permits to organize the solutions in pairs. The second genetic operator (Crossover) shares parts of binary string values of each pair to obtain new solutions. Mutation operator is used to perform modifications on the combination of the binary string belonging to the offspring. The reposition of the initial population size is accomplished by eliminating the worst solutions once a new ranking of all solutions has been performed (Elimination operator). The stopping criterion is based on the convergence of the best solution obtained in subsequent generations. Fig. 3 resumes the sequence of steps of the implemented genetic algorithm (GA). Fig. 2 also presents the numerical model used in the inverse method. A two-dimensional analysis was performed using a refined mesh in the region of crack growth with 8182 8-node plane stress elements and 240 6-node cohesive elements connecting the plane elements at the mid-thickness of the ENF specimen.

τ

GIIc

τu τ2

u u0

u1

u2

y

uu

ux = 0, uy

δ

Interface finite elements

x ux = 0, u y = 0

ð12Þ

u x = 0, u y = 0

Fig. 2. Trapezoidal cohesive law and finite element mesh details.

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A cohesive model was implemented in a user subroutine [18] of the commercial finite element software ABAQUSs. At the pre-crack region, cohesive elements were considered “opened” which means that they were only able to transmit normal compressive stresses since friction effects were neglected. Damage initiation and growth during the test were simulated considering the cohesive law issuing

t=1 Experimental P-δ curve

Number of bits Initial Population (Randomly produced)

Decoding t=t+1 FEA Genetic Operators: Selection, Crossover, Evaluation of Objective function Mutation and Elimination No

Yes End Fig. 3. Sequence of steps of the developed GA.

Table 1 Elastic properties of wood. Nominal elastic properties of Pinus pinaster Ait [20,21]

10.13

1.91

1.01

3. Experiments

νRT

GLR (GPa) GTL (GPa) GRT (GPa)

0.47 0.51 0.59 1.12

Four specimens for the ENF test were prepared considering maritime pine (Pinus pinaster Ait) oven dried to about 12% moisture. The specimens were cut from mature wood without defects. The nominal elastic properties of wood are presented in Table 1 [22,23]. The surfaces of the beams to be bonded were carefully polished with sandpaper and cleaned before bonding with the adhesive Prefere 3545 (E ¼4200 MPa and G ¼1520 MPa) from Dynea (a melanine–urea resin adhesive) used in timber constructions. The initial pre-crack was introduced by means of a thin Teflon film (thickness of 25 μm). Global specimen dimensions (Fig. 1) were 2L¼460 mm, a0 ¼ 162 mm, h¼15 mm, B¼ 20 mm, and t¼ 0.2 mm. The ENF tests were carried out on a conventional screw-driven Instron 1125 testing machine equipped with a load cell of 5 kN. A crosshead displacement rate of 5 mm/min was applied and the load–displacement (P–δ) curve was registered during the test with a frequency of 5 Hz (Spider 8 HBM acquisition system). The displacement fields at the crack tip (area 2h  d in Fig. 1) were also recorded by the DIC technique and correlated with load– displacement data to obtain the GII ¼f(u) curve. 3.2. Digital image correlation

Converged ?

νTL

from the optimization procedure. After a large number of iterations the best solution (the one that minimizes the objective function) is used to obtain the cohesive law by the inverse method.

3.1. Specimens preparation and fracture tests

State variables Input data: GA & FEA

EL (GPa) ER (GPa) ET (GPa) νLR

57

1.04

0.17

The photo-mechanical set-up is shown in Fig. 4(a). The ARAMIS DIC-2D system was used in this work [24]. A charge coupled device (CCD) camera coupled with a telecentric lens was used for image grabbing (Table 2). The optical system was connected to the cross-head movement of the testing machine, for ensuring continuous visualization of the region of interest during the ENF test. The deformation of an ENF specimen during crack propagation is shown in Fig. 4(b), in which mode II sliding can be macroscopically observed. The speckle pattern required for DIC measurements is painted ahead of the initial crack tip by means of an airbrush as shown in Fig. 4(c). As it can be seen a suitable size, uniformity, contrast and isotropy of the granular pattern were achieved at the scale of observation. In the reference image, subsets of 15  15 pixels2 were defined in a compromise between correlation and

Fig. 4. (a) Photomechanical set-up of the ENF test, (b) deformation of a ENF specimen in mode II crack propagation, and (c) speckle pattern typically used for DIC measurements (28.4  18.9 mm2).

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Table 2 Optical system components and measurement parameters. CCD camera Model Acquisition frequency (Hz) Lens Model Magnification (%) Field of view (1/1.80 0 ) (mm2) Working distance (mm) Working F-number Field depth (mm) Conversion factor (mm/pixel) Lighting Measurements Spatial resolution Resolution

Baumer Optronic FWX20 (8 bits, 1624  1236 pixels, 4.4 μm/pixel) 0.5 Opto Engineering Telecentric lens TC 23 36 0.24373 29.3  22.1 103.5 7 3 8 11.0 0.018 Raylux 25 white-light LED 15  15 pixel2 (0.27  0.27 mm2) 1–2  10  2 pixel (0.18–0.36 μm)

interpolation errors (Table 2). This corresponded to a displacement spatial resolution of 0.270 mm. The resolution associated with the measured displacements was estimated from standard deviation over reference fields obtained from rigid-body translation tests [24]. A displacement resolution in the range of 1–2  10  2 pixel (0.18– 0.36 μm) was determined (Table 2). The displacement fields provided by DIC were processed in order to determine the crack tip opening displacement (u) during test (Eq. (9)). To start with, in an image before crack propagation, the crack tip location was identified. The CTSDII was then evaluated from the displacement components parallel to the crack propagation measured in a pair of subsets located in upper and lower displacement components of the initial crack extremity. In this selection, the first pair was typically removed because erroneous measurements are obtained on the boundary discontinuity during crack surface separation.

Fig. 5. (a) R-curve (GII ¼f(ae)) of an experimental test and (b) GII ¼f(u) relationship with the fitted logistic function.

ones. Excellent agreement was obtained, thus proving the good performance of the proposed methodology on the determination of the cohesive laws.

4. Results and discussion 4.2. Cohesive laws—inverse method 4.1. Cohesive laws—direct method The first step of the method consists of the correlation between GII and CTSDII till self-similar crack propagation takes place. The evolution of GII is easily captured from the R-curve (Eq. (8)) and the monitoring of CTSDII is performed by a DIC method (Eq. (9)). A typical R-curve (GII ¼f(ae)) is presented in Fig. 5(a). The initial rising trend of GII ¼f(ae) corresponding to the formation of FPZ is followed by a clear plateau where self-similar crack growth takes place. In fact, the plateau region corresponds to crack growth with constant FPZ ahead of its tip, thus mimicking the crucial requirements for a truthful determination of fracture energy GIIc. Fig. 5(b) presents the corresponding GII ¼f(u) curve which was fitted by means of a logistic function GII ¼ A2 þ

A1  A2  α 1 þ uu0

ð13Þ

in order to obtain the experimental cohesive law after differentiation (Eq. (2)). The parameters A1 and A2 represent the initial and final values of GII (GIIc), respectively and α is a shape parameter. Fig. 6 shows the resulting experimental cohesive laws which were adjusted by trapezoidal laws with bilinear softening. These trapezoidal laws were subsequently used in numerical simulations of the four tested specimens and the resulting numerical load–displacement curves were compared with the experimental

The objective of the inverse method is to determine four cohesive parameters (τu, u1, u2 and τ2) using an optimization strategy based on the genetic algorithm presented previously. The main goal is to define the best cohesive law representing bonded joint fracture behavior under mode II loading. This is achieved by minimizing an objective function that quantifies the difference between numerical and experimental load–displacement curves. Fig. 7 plots the comparison between the cohesive laws resulting from the optimization strategy and the experimental ones issuing from the application of the direct method. In addition, the corresponding load–displacement curves reveal excellent agreement which underlines the good performance of the method on the determination of the cohesive laws of wood bonded joints.

5. Conclusions Two different methods were used to evaluate the cohesive law of wood bonded joints under mode II loading. The direct method is based on the correlation between the development of strain energy release rate (GII) as a function of the crack tip shear displacement under mode II loading (CTSDII), which was monitored using digital image correlation. An equivalent crack length procedure using the beam theory was used to obtain the Resistance-curve, with several advantages relative to classical methods.

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Fig. 6. Experimental cohesive laws obtained by the direct method and fitted trapezoidal ones and the corresponding P–δ curves for the four tested specimens.

In fact, the proposed methodology overcomes the difficulty associated with crack length measurement during the ENF test and allows accounting for the effect of a non-negligible fracture process zone characteristic of ductile adhesives. A logistic function was fitted to the GII ¼ f(CTSDII) curve and the experimental cohesive law was defined as its analytical derivative. Numerical analyses were performed, considering trapezoidal laws with

bilinear softening fitted to the experimental ones. It was verified that the resulting load–displacement curves present excellent agreement with the experimental ones which validates the proposed procedure. The inverse method is based on an optimization procedure using a genetic algorithm in order to obtain the cohesive law that minimizes the difference between numerical and experimental load–displacement curves. These laws were

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Fig. 7. Comparison between the inverse method and experimental cohesive laws and the corresponding P–δ curves for the four tested specimens.

compared to the experimental ones issuing from the direct method as well as the resulting load–displacement curves. An excellent agreement was obtained in both cases, which reveals the soundness of the inverse method.

The knowledge of cohesive laws in pure loading modes provides a fundamental tool to determine adequate failure criteria under mixed-mode loading, which is the common loading in real structures. This allows performing appropriate analysis of structural

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bonding details identified as being critical in a given structure using finite element analysis including cohesive zone modelling. Acknowledgments This work is supported by European Union Funds (FEDER/ COMPETE—Operational Competitiveness Programme) under Project FCOMP-01-0124-287 FEDER-022692 and by National Funds (FCT—Portuguese Foundation for Science and Technology) under Project PTDC/EME-PME/114443/2009 and Ciência 2008 program. References [1] Serrano E, Gustafsson PJ. Fracture mechanics in timber engineering—strength analyses of components and joints. Mater Struct 2006;40:87–96. [2] Danielsson H, Gustafsson PJ. A three dimensional plasticity model for perpendicular to grain cohesive fracture in wood. Eng Fract Mech 2013;98: 137–52. [3] de Moura MFSF, Silva MAL, Morais JJL, de Morais AB, Lousada JJL. Data reduction scheme for measuring GIIc of wood in end-notched flexure (ENF) tests. Holzforschung 2009;63:99–106. [4] Xavier J, Morais J, Dourado N, de Moura MFSF. Measurement of mode I and mode II fracture properties of wood-bonded joints. J Adhes Sci Technol 2011;25:2881–95. [5] Serrano E, Gustafsson PJ, Larsen HJ. Modelling of finger-joint failure in gluedlaminated timber joints. J Struct Eng 2001;127:914–21. [6] Sørensen BF. Cohesive law and notch sensitivity of adhesive joints. Acta Mater 2002;50:1053–61. [7] Andersson T, Biel A. On the effective constitutive properties of a thin adhesive layer loaded in peel. Int J Fract 2006;141:227–46. [8] de Moura MFSF, Gonçalves JPM, Magalhães AG. A straightforward method to obtain the cohesive laws of bonded joints under mode I loading. Int J Adhes Adhes 2012;39:54–9. [9] Silva FGA, Xavier J, Pereira FAM, Morais J, Dourado N, de Moura MFSF. Determination of cohesive laws in wood bonded joints under mode I loading using the DCB test. Holzforschung 2013;67:913–22.

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