An experimental evaluation of glass–polymer interfacial toughness

Engineering Fracture Mechanics 76 (2009) 2731–2747

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An experimental evaluation of glass–polymer interfacial toughness Francesco Caimmi *, Andrea Pavan Dipartimento di Chimica, Materiali e Ingegneria Chimica ‘‘Giulio Natta’’, Politecnico di Milano, P.za Leonardo da Vinci 32, I-20133 Milano, Italy

a r t i c l e

i n f o

Article history: Received 10 October 2008 Received in revised form 4 June 2009 Accepted 26 June 2009 Available online 4 July 2009 Keywords: Polymer matrix composites Interfacial fracture Glass–polymer interface Mixed-mode fracture

a b s t r a c t The evaluation of interfacial fracture toughness in glass fibre reinforced polymer composites is examined using macroscopic polymer–glass joints. Different bi-material systems providing various degrees of mixity were tested using various tests configurations. The test configurations were analysed in the frame of interfacial linear elastic fracture mechanics with the finite element method in order to investigate their properties and obtain relations between fracture mechanics parameters and joints’ geometry useful for the experimental data reduction. The analysis showed that the stress intensity factor phase angle characterizing mode mixity is almost constant with crack length for each test configuration examined: hence these tests lend themselves to be used to study the crack propagation behaviour when stable crack growth is observed. Joints made of combinations of E-glass, either treated or untreated with a surface finishing agent, and two polymers, poly (methylmethacrylate) and an epoxy resin, were employed. An attempt to model the fracture process of these systems with cohesive elements yielded not completely satisfactory results. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The paramount importance of the interaction between fibre and matrix at their interface in determining the mechanical properties of polymeric composites has been widely recognized. The effectiveness of the interaction depends strongly on the ‘‘strength” of the interfacial bond. Therefore a great deal of experimental and theoretical work has been undertaken to characterize interfacial behaviour and to quantify the interface’s mechanical resistance. Tests to study the interfacial adhesion proliferated during the years (see [1–6] for reviews and descriptions); among others we recall the single fibre pull-out, the single fibre push-out, the single fibre push-down, the fragmentation, the transverse tensile, the microbond, and the Outwater–Murphy compression test. These different tests try to reproduce to different extents the actual conditions experienced by fibres in fibrous composites, which in general are rather complex. Their intricacy is reflected in the range of analyses developed to interpret their results: for the single fibre pull-out test, for example, the analyses put forward span from the simple shear–lag theory (see e.g. [1]) to the highly complex fracture mechanics treatments given by Nairn [7] and by Hutchinson and Jensen [8]. An account of the various problems that arise in these tests and render their results uncertain is given in [3,9]: we mention residual stresses, friction, however complex and often inadequately known state of stress, experimental difficulties due to specimen small dimensions. As numerous as the tests employed to apprise the interface’s mechanical resistance are the parameters used to pin-point it: among them, failure stresses (both shear [1–3] and normal [6]), ‘‘adhesional pressure” [10], various definitions of fracture * Corresponding author. Tel.: +39 02 2399 3207; fax: +39 02 7063 8173. E-mail address: [email protected] (F. Caimmi). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.06.014

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Fig. 1. Notation used for the bi-material interface crack in 2D.

toughness [7,9,11–14]. While some of them are just heuristic parameters that can be used for ranking similar bi-material systems, what is needed are figures that can be used to predict failure in engineering design. Fracture mechanics provides figures of this kind, namely intrinsic properties of the bi-material combination. It also provides a framework in which complex stress states as those experienced in real composites can be treated with relative easiness through the concept of mixity. As a matter of fact it has been widely used in problems involving interfacial failures, since the pioneering works of Williams (e.g. [15]), and an extensive review of its applications is given in [12]. In this work we will adopt as a measure of the interface’s mechanical resistance the critical strain energy release rate as determined according to the theory of interfacial linear elastic fracture mechanics. We agree with Gupta et al. [9] when they state ‘‘The preferred test for the determination of interfacial properties should measure the state of adhesion directly using simple fundamentals procedures of mechanics relatively free of artifacts relying on complex models, based on unverifiable assumptions.” Therefore, following their example, we tried to measure the interfacial toughness using simpler macroscopic bi-material joints. In Ref. [9] special double cantilever beam tests were employed; asymmetric, bi-material DBC specimens have also been employed for this kind of measurement (e.g. [13]); special biaxial tests that could produce a wide range of mixities were used in [25]. For the purpose of this work some specially designed tests were used: their description is the object of Section 2. In Section 3 their properties are investigated through the use of the finite elements method (FEM). Section 4 is devoted to the discussion of the experimental results. Transferability of the results of macro-tests to the micro-scale typical of real composites is of course a question that should not be undervalued. Issues regarding, for example, the influence of surface roughness [12], have to be addressed. In this work we chose to tackle the question of transferability via the calibration of a cohesive zone model with data obtained by testing macroscopic joints. The cohesive zone model can be used to simulate via FEM the results from one, or more, of the conventional micro-scale tests previously cited. The use of the FEM would allow overcoming many of the above mentioned difficulties that usually make the data from these tests doubtful or difficult to interpret; for example, by means of the FEM the influence of residual stresses can be estimated, and the stress state along the interface can be precisely accounted for. Section 5 will present some initial results on this part of our work, which, however, is still in progress. 2. Tests overview 2.1. Background We deem it useful to recall some basic concepts in linearly elastic interfacial fracture mechanics that make a difference with respect to the case of homogeneous materials.1 With reference to Fig. 1 which depicts schematically a crack running along the interface between two different elastic half-spaces, the asymptotic solution for the general boundary value problem with stress vector prescribed at infinity predicts that the stress vector at the interface is given, in complex notation, by (see, e.g., [14,16])

K

ryy ðx; 0Þ þ irxy ðx; 0Þ ¼ pffiffiffiffiffiffiffiffiffi ðxÞie 2px

ð1Þ

where rij are the stresses, e is a number depending only on the values of the elastic constants of the two materials (see [14] for its expression) and K* is the complex stress intensity factor which represents the intensity of the stress field at the crack tip. As every complex number it can be expressed as the sum of a real and an imaginary part, i.e. 

K  ¼ K 1 þ iK 2

ð2Þ

1 We are assuming that interfacial cracks behave in accordance with the so called ‘‘open model” and not with the, perhaps more accurate, ‘‘closed model” developed after the works of Comninou [17]. A discussion of the approximations involved with this assumption is given in [14].

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Here, however, the real and the imaginary parts of the complex stress intensity factor can no longer be regarded as corresponding to symmetrical and skew-symmetrical stress distributions [16] as in the homogeneous case. For instance, consider the case K 1 ¼ 0: the inspection of Eq. (1) reveals that normal stresses are not in general vanishing at the interface, which is at variance with the homogeneous material case. K* units are Pa m0.5ie. The strain energy release rate G for an interfacial crack can be formally related to the stress intensity factor in a way similar to the one holding for the homogeneous material case, namely

 G ¼ vK  K

ð3Þ

where v is a function depending only on the elastic properties of the two materials (see [12] for its expression) and the overbar denotes complex conjugation. The phase angle of K*, namely

/ ¼ argðK  Þ

ð4Þ

is a crucial quantity in linear elastic interfacial fracture mechanics: the value of the critical strain energy release rate is, often highly, dependent on it [12]. To be meaningful, a value of interfacial toughness must be accompanied by the specification of the phase angle at which it was determined; otherwise it cannot be transferred to other cases.

2.2. Test configurations The basic idea behind the test configurations we used is straightforward. They simply consist of two rectangular cross section bars, one made of polymer and the other one made of glass, joined in such a way that a pre-crack is introduced during the preparation process. Since interfacial toughness depends on the phase angle, to asses such a dependence we made use of different tests configurations generating different phase angles. This can be achieved in various ways, the most obvious one being a change in the loading conditions. The actual phase angle obtained is to be checked ‘‘a posteriori”, e.g. by finite element analysis. A schematic representation of the four test configurations considered in this work is displayed in Fig. 2. Typical specimen dimensions are shown and the definition of some geometrical parameters we will use in the following is also indicated. With the materials combinations considered in this study the values of the phase angle / pertaining to each test configuration, as resulting from finite elements analysis, are indicated in Fig. 2 (the dependence of / on geometry will be extensively discussed in Section 3). Three of the loading conditions chosen (a–c in Fig. 2) resemble in some ways the peel-test (see e.g. [18]) although here the ‘‘peeled” arm is not very flexible. Configuration (a) gives / ffi 20° and represent the peeling of a glass bar from polymeric substrate which is rigidly held along its entire length so that vertical displacements are forbidden (in practice this is achieved by gluing the substrate to an aluminium plate; see the next section). As there is no symmetry across the plane of junction of the two materials, the simple exchange of the roles of the constituents (case c in Fig. 2) yields quite a different loading situation at the crack tip, with / ffi 40°. The difference between configuration (c) and configuration (b) lies just in the direction of the external load. The fourth configuration illustrated in Fig. 2 (case d), is similar to the conventional end loaded split (ELS) test and would provide a value for / near 70°. However specimens of this kind always failed cohesively inside the glass arm during the tests. We therefore stopped considering this test configuration and will not deal with it any further here.

a

b P

20 mm

P

P

hs

L

h

150 mm

P

a

c

d

Fig. 2. The four test configurations considered in this work. The holed gray squares represent aluminium blocks used to apply the load. The light grey lines represent initial crack length. Polymer: white. Glass: hatched.

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The choice of ‘‘peel-like” loading conditions (configurations a–c in Fig. 2) was justified by the fact that they are expected to provide the crack with an environment which changes only slightly during crack propagation thus maintaining a condition of constant phase angle. This issue will be further addressed in Section 3.2. In the following the nominal values of the phase angle, indicated in Fig. 2, will be used as labels to identify the different configurations. They are all negative: indeed there are some evidences that Gc(/) may not always be a symmetric function of / [12,13,24]). For the sake of precision we will therefore keep the minus sign in our notation, even if it might appear somewhat cumbersome. In comparison to the biaxial tests used in [24], our specimens allow for the determination of toughness over a more restricted range of /. However, they are amenable to testing with a uniaxial dynamometer.

2.3. Materials and samples preparation Four combinations of glass–polymer joints were prepared using two polymers and E-glass bars, either untreated or treated with a surface finishing agent. E-glass bars used were either untreated (we just degreased them with soap before joining) or treated with a methacrylate silane (3-methacryloxypropyl-trimethoxysilane). This is a sizing agent commonly employed in the sizing process of glass fibres for both epoxy and PMMA matrix composites [19]. The silane was dissolved in deionized water acidified with HCl at pH = 2.0. The 0.5 wt.% silane solution was stirred vigorously for 10 min and then left to rest for 10 min. Glass bars, previously washed with soap, were immersed into the solution for one hour. They were finally dried under vacuum at 100 °C for 20 min before joining. The width of the bars was 20 mm. The mean superficial roughness (Ra) of the untreated glass bar was 0.054 lm, measured with a contactless profilometer; it is of the same order of magnitude of the roughness of glass fibres [20]. The polymers we chose are: – a commercial bi-component epoxy system: Ampreg 26 by SP Systems, used with the so called (by the manufacturer) ‘‘slow hardener” which is mainly composed of polyamines, – a commercial poly (methylmethacrylate) (PMMA) grade (Repsol GlassTM by Repsol). The joints were made by adding the polymeric component onto the glass bar previously placed at the bottom of a silicon mould; before adding the polymeric component, however, part of the glass bar length was covered with a PTFE film (50 lm thick) to create an artificial pre-crack. The epoxy resin was poured into the mould in one shot, in such a quantity as to obtain a test specimen of the desired thickness (h or hs). The resin was then allowed to cure at room temperature for two weeks following manufacturer’s instructions. The PMMA component was instead formed of two parts: first a small amount of liquid methylmethacrylate (Struers Clarocit)2 was poured onto the glass bar placed in the mould, then a bar of solid Repsol Glass PMMA was placed on top of it and let it adhere to the glass bar, adhesion being promoted by the polymerization of the methylmethacrylate monomer (compare with [21]). The polymerization can be considered completed after 20 min. Such a technique of sample preparation should contain the possible introduction of internal stresses: the only stresses set in during sample preparation are those coming from the possible volume contraction occurring during polymerization. We were not able to measure them but no sign was revealed by examining the specimens under polarized light, so we deem them to be rather low. By contrast, hot welding a polymer bar onto the glass substrate produced joints which did not survive to the cooling phase, resulting in fracture of the joint at the polymer–glass interface or inside the glass arm. Such a preparation technique was therefore discarded.

2.4. Testing The load was applied to one arm of the test specimen (the ‘‘peel arm”) through a pin inserted into an aluminium block glued onto it while the other arm of the specimen is glued to an aluminium plate connected to the other grip of the testing machine via a linear bearing. The resulting peel fixture is shown in Fig. 3a. As the peel jig can be rotated to change peel angle, the same fixture can be used for all the test configurations used in this work. The tests were conducted on an Instron-1185 testing machine at a constant cross-head displacement rate of 1 mm/min at a temperature of 23 °C. Each test was video-recorded and, in case of stable crack propagation, the momentary crack length was measured from the images via an image analysis software. All specimens were pre-cracked from the PTFE film using the / ffi 40° configuration, which, as we shall see, is the most stable among the ones used in this work. 2 Actually, to reduce the volumetric shrinkage of the reacting compound during polymerization, this was not just pure methylmethacrylate but a suspension of methylmethacrylate, catalyst and PMMA.

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Fig. 3. The test setup. (a) Experimental setup, configuration for / ffi 40°, the polymer is epoxy. (b) The finite element model representing the same test configuration of (a), with boundary conditions shown. The arrow represents the load, while the arrowheads represent zero displacement in the direction of the arrowhead.

3. FEM analysis in LEFM context 3.1. FEM models details The specimens depicted in Fig. 2 were modelled using ABAQUSTM, version 6.7 [21]. The implicit solver available in the FE package was used. Materials constants used are given in Table 1. They were all measured in simple tension experiments with the exception of the value for Poisson’s coefficient m for the E-glass which was assumed. The specimens were modelled as 2D perfectly elastic objects in plane strain conditions. Geometrical non-linearities were neglected and no distinction was made between treated and untreated glass at this stage. Eight noded, reduced integration, quadrilateral elements were used in most of the FEM model. In a square region (side 3 mm) enclosing the crack tip fully integrated elements were used. This region was meshed with a structured mesh while a free meshing technique was adopted elsewhere. An example mesh for the configuration with / ffi 40° is shown in Fig. 3b. The aluminium load block was modelled as a rigid body with the load applied at the centre of the block; the aluminium plate was assumed as prescribing the boundary conditions (displacements) shown in Fig. 2. Perfect adhesion was assumed between parts of different materials, save removing this constraint from some nodes in order to simulate the crack. ABAQUSTM automatically calculates the values for the strain energy release rate G and for the real and imaginary parts of the stress intensity factor K* via J-integral evaluation; details can be found in [22] and in the references therein. Model compliance C was calculated as the ratio between the load-point displacement and the applied load.

Table 1 Material constants used in the finite element analysis. Material

E (GPa)

m

PMMA Epoxy E-glass

2.9 2.7 70

0.33 0.37 0.2

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3.2. Parametric analysis The main purposes of the FEM analysis were: – to provide guidelines for selecting a suitable geometry for testing, i.e. a geometry which provides a phase angle that remains constant as the crack advances. This is a particularly desirable feature, which does not often receive the deserved attention, with some noteworthy exceptions (e.g. [25–27]), – to determine G, / and C as functions of geometric parameters to be used for experimental data reduction and model validation. To pursue these objectives an extensive parametric study was carried out. Fig. 4a and b show some examples of how much compliance, C, and phase angle, /, are influenced by the substrate thickness hs. As this influence appears quite negligible, in the following we fixed hs value to 10 mm. The increase in compliance with substrate thickness can be related to the fact the prevailing stress state in the substrate is tension in the direction of the external load; its response is then similar to that of a uniaxially stretched bar. The configuration with nominal phase angle –40° corresponds practically to the case of a polymeric bar on a rigid substrate. Fig. 5a shows the strain energy release rate, G, normalized by the square of the load, P, as a function of a2/h3 for the PMMA case with / ffi 40°. The trend is perfectly linear as we would expect for the strain energy release rate of a beam of length a and thickness h (see e.g. [23]). The slope of the linear fit is 5.1 MN1; replacing PMMA Young’s modulus in the expression for the strain energy release rate of a cantilever one would obtain 5.2 MN1, very close to the previous figure. Similar results hold for the epoxy–glass case with / ffi 40° (not shown). Similar behaviour is displayed by the test with phase angle / ffi 20°, as shown in Fig. 5b for the epoxy–glass case. Here, however, small deviations from the beam-like behaviour can be seen at extreme crack lengths. The reason for this can be

a

0

φ [deg]

-10

-20

-30

-40 4

8

12

hs [mm] 30

b 3.0

26 2.4

φ ≅ −20° φ ≅ −30°

24

φ ≅ −40°

C [μm/N]

C [μm/N]

28

22

1.8 20 4

8

12

hs [mm] Fig. 4. Influence of hs on phase angle (a) and on compliance (b) for the epoxy–glass bi-material combination with a/L = 0.5.

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ascribed to edge effects. The configuration with / ffi 40° is instead not influenced by the edge presence for crack lengths up to 0.8L, this meaning that the lengths needed for the stress transfer from the upper to the lower arm are different in the two cases due to the different stiffness of the materials found above or below the crack plane in the different configurations. For a comprehensive discussion of this issue see [35]. For the configuration with / ffi 30°, a significant grouping of the variables a and h on which the strain energy release rate may be thought to depend on, as a2/h3 for the peel-like configurations, could not be found: as an example Fig. 5c shows the

a

0.08

PMMA-glass φ ≅ −40°

0.04

h [mm] 5 10 11 13 linear fit

2

G/P [1/(m N)]

0.06

0.02

0.00 0

2

4

6

8

2

10

12

14

3

a /h [1/mm]

b

epoxy-glass φ ≅ −20°

0.02

h [mm] 5 8 10 12

2

G/P [1/(m N)]

0.03

0.01

0.00 0

20

40

60 2

80

100

120

3

a /h [1/mm] 2

c

G/P [1/(m N)] 0.8 0.017

0.7

0.015 0.013

0.6

a/L

0.0094 0.0054

0.5

0.0034 0.0014

0.4

PMMA-glass φ ≅ −30°

0.3 0.2 6

8

10

12

14

h [mm] Fig. 5. Strain energy release rates results obtained from finite element analysis. (a) PMMA–glass, / ffi 40°. (b) epoxy–glass, / ffi 20°. (c) PMMA–glass, / ffi 30°.

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h [mm]

a

12

10

φ [deg]

8

-17 -18

4 0.2

-19

PMMA-glass φ ≅ −20°

6

0.3

0.4

0.5

0.6

0.7

-20

0.8

a/L

b

14

φ [deg]

h [mm]

12

10

-29.0 -29.7 8

PMMA-glass φ ≅ −30°

-30.3 -31.4

6 0.2

0.3

0.4

0.5

0.6

0.7

0.8

a/L

c PMMA-glass φ ≅ −40°

12

φ [deg] -35

10

h [mm]

-37 -39 -40

8

-41 6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

a/L Fig. 6. Contour plots of phase angle as a function of geometry for the system PMMA–glass. (a) Configuration with nominal phase angle / ffi 20°. (b) Configuration with nominal phase angle / ffi 30°. (c) Configuration with nominal phase angle / ffi 40°.

contour plot of G/P2 as a function of a/L and h for the case of PMMA–glass. It can be observed that in this case G decreases with increasing crack length, indicating the possibility of obtaining stable crack propagation. However (compare with Fig. 5a and b), G/P2 values are quite low; this is not a really efficient way of loading the joint. As a matter of fact, as we shall see, this will turn out to be the most unstable configuration among those tested. Phase angle dependency on geometry is shown in Fig. 6, where the PMMA case is shown. Its variations are generally rather small; the greatest changes, at any given thickness h, are found in the regions of high and low crack length, where there appeared to be edge effects. In the crack length region between a/L = 0.3 and a/L = 0.7, for a given thickness h, the phase

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angle is almost constant and the tests can be profitably used to measure constant phase angle toughness values. Similar results hold also if the epoxy is considered; the interested reader may find the complete set of graphs in [35]. The present numerical analysis obviously supplies only some discrete values of the wanted quantities. In order to render these results useful for experimental data reduction it would be convenient to have continuous analytical expressions. To this end, the results obtained from FEA were fitted with polynomials in the variables a/L, a/h and h. Good fits (residuals of the order of 1%) were obtained using polynomials of order up to four at most. The fits for both G/P2 and / are given in the Appendix A. Except for the cases regarding the strain energy release rate of the configuration with / ffi 40° mentioned above, there is no particular reason for choosing the order or the degree of completeness of the polynomials used, as there is no underlying model, such as the beam model: they were selected only on the basis of the quality of the fits provided.

3.3. Model validation To compare the output of finite element analysis with experimental data we will refer to the specimen compliance, C. To account for the influence of (minor) variations in specimen width B (out of plane), the product C B will be considered. In Fig. 7 some significant examples of this comparison are given. A good agreement is found for the configurations with phase angle / ffi 40° and / ffi 30° (Fig. 7c and b, respectively). Note that the tests at a nominal phase angle equal to 30° were conducted on samples of various height in order to put the FEA to test; anyway, as said before, for this configuration the dependence of G, and C, on a and h is not easily represented in a 2D graph. This confers to Fig. 7b its ‘‘scattered” appearance. It was observed previously that the configuration with / ffi 40° behaves as a cantilever as to the strain energy release rate; we can note here that it behaves as a cantilever also as to compliance (linearity in a3/h3). Discrepancies can be seen when / ffi 20° (Fig. 7a). FEM analysis predicts this last configuration to be very stiff, but the interfacial fracture energies determined experimentally are rather low and correspondingly low are the loads reached during the tests; moreover the peel fixture is not really stiff. Therefore machine plays and compliance can have a significant influence on the measured compliance. To check that we measured directly, by means of a videoextesometer, the displacements, v, of the points of the upper glass arm of the test specimen along the upper crack face. The results are shown in Fig. 8. As they appear quite in line with FEM predictions, we think we ought to trust the values obtained for / and G via FEM analysis in this case too, and use them together with the experimentally determined load for toughness evaluation.

4. Results Table 2 summarizes the results obtained experimentally on the four test configurations examined as to the stability in crack propagation. The epoxy-untreated glass system could not be assessed because of the impossibility of obtaining interfacial failure. Most of the other cases proved to be unstable, so only initiation toughness values could be determined. An example of a typical load–displacement trace obtained in a stable crack propagation case is reported in Fig. 9 (curve a); the initiation point, determined from the video-recorded images, is indicated with a arrow. Epoxy-treated glass joints tested with / ffi 40° showed some moderate stick slip, as it appears in Fig. 9 (curve b). Resolute stick slip behaviour was instead shown by the PMMA-treated glass system, see Fig. 9 (curve c); in such a case (unstable) fracture initiation is identified with the first sudden load drop. In Fig. 10 initiation toughness results are summarized. The two polymer-treated glass systems show (Fig. 10a and b) a significant dependence of toughness on /, while the PMMA-untreated glass system (Fig. 10c) seems insensitive in the range of phase angles we could explore. This last trend is analogous to the one found for PMMA/epoxy interfacial failures by Malyshev and Salganik [24]. A comparison with toughness data available in literature is not an easy task since often mixity values are not specified and there is a great variability both as to materials and as to surface finishing state. Nevertheless, we can loosely compare our data with those obtained in a series of biaxial experiments on an epoxy–glass system by Liechti and Chai [25]: the order of magnitude is the same as the one obtained in this work on the treated glass–epoxy system and also the steep trend increase with increasingly negative values of / is similar.3 It seems that is not possible to find toughness measurements on PMMA/glass interfaces, although composites made with these constituents are fairly common in biomedical applications. While the silane-treatment applied to glass seemed to have a detrimental effect on the glass–epoxy system, actually allowing us to get interfacial failure (whereas the interface was too strong to yield interfacial failure in the untreated glass counterpart), a noteworthy increment in interfacial toughness is obtained moving from the untreated to the treated-glass in the case of glass–PMMA joints (Fig. 10c and b, respectively). Depending on phase angle, toughness can become order of magnitude larger. 3 Please note that two different definitions of phase angle are used here and in [25]; they are simply related, with our figures being bigger of about 9° (see e.g. [14] for general conversion rules between different phase angle definitions).

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a

Experimental FEA

2

C B [mm /N]

0.6

epoxy-glass φ ≅ −20°

0.4

0.2

0.0 0.40

0.45

0.50

0.55

0.60

0.65

0.70

a/L

b

0.10

epoxy-glass φ ≅- 30°

2

C B [mm /N]

0.08

0.06

0.04

0.02

Experimental FEA 0.00 0.35

0.40

0.45

0.50

0.55

0.60

a/L

c

2.0

PMMA-glass φ ≅- 40°

2

C B [mm /N]

1.5

1.0

0.5

Experimental FEA 0.0 0

400

800 3

a /h

1200

3

Fig. 7. Comparison between experimental (solid symbols) and calculated values (hollow symbols) of compliance. Results are for various values of h, ranging from 5 to 12 mm. (a) Epoxy–glass, / ffi 20°. (b) Epoxy–glass / ffi 30°. (c) PMMA–glass, / ffi 40°.

The qualitative effect of the silane treatment observed in this work can be compared with some results obtained by standard micromechanical tests and reported in the literature. Anyway, it must be recalled that the interfacial shear strength obtained through micromechanical tests is a completely different quantity with respect to toughness. For the epoxy case the decrease in toughness is consistent with the shear strength results obtained by Tanoglu et al. [30], but at variance with the results obtained for instance by Debnath et al. [31], who observed an increase in interfacial shear strength after a treatment with a silane coupling agent. Once again the comparison is loose, because in the last two refer-

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Fig. 8. Comparison between calculated and measured displacements along the upper crack face for an epoxy-treated glass joint with / ffi 20°, a/L = 0.63, h = 9.8 mm. Continuous line: FEM prediction. Solid symbols: experimental data.

Table 2 Tests’ stability summary. Systems

Nominal phase angle / ffi 20°

/ ffi 30°

/ ffi 40°

PMMA-untreated glass PMMA-treated glass Epoxy-untreated glass Epoxy-treated glass

Unstable Unstable (No interfacial failure) Stable

Unstable Unstable (No interfacial failure) Unstable

Stable Unstable (stick slip) (No interfacial failure) Quasi-stable (moderate stick slip)

90

(c)

Load [N]

60

(b)

(a)

30

0 0

3

crosshead displacement [mm] Fig. 9. Examples of recorded load–displacement trace. Curve a (solid line, triangles): epoxy-treated glass, / ffi 20°, a0/L = 0.66, h = 9.85 mm. Curve b (solid line, circles): epoxy-treated glass, / ffi 40°, a0/L = 0.55, h = 9.4 mm. Curve c (solid line): PMMA-treated glass, / ffi 40°, a0/L = 0.43, h = 12.07 mm. Arrows: fracture initiation point.

enced works different epoxy systems and also different sizes (although based on the same silane) were used. The epoxy system used in reference [30] is probably closer to the one here employed than that used in [31]. On the other hand, improvements in interfacial shear strength were generally observed moving from untreated to 3methacryloxypropyl-trimethoxysilane treated glass fibres embedded in methylmethacrylate-based matrices [32], which are consistent with the present results. It is often affirmed that the transverse strength and the interlaminar shear strength (of long fibres composites) correlate with interfacial ‘‘strength”, when interfacial fracture is the failure mechanism triggering composite failure [1,3]. From the results obtained here, it is expected that a treatment with methylmethacrylate used should lead to improvements with respect to PMMA based composites with untreated fibres. This is actually the case for materials typically used in the dental field (see [33] and references therein).

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a

120

epoxy- treated glass

100

Unconstrained fit with eq.(5)

2

GC [J/m ]

80 60 40 20 0 -40

-30

-20

-10

0

-10

0

φ [deg]

b

250

PMMA-treated glass

2

Gc [J/m ]

200

150

100

50

Unconstrained fit with eq. (5) 0 -40

-30

-20

φ [deg]

c

6

PMMA-untreated glass 5

2

GC [J/m ]

4 3 2 1 0 -40

-30

-20

-10

0

φ [deg] Fig. 10. Experimental initiation toughness as a function of phase angle. Lines show fits to Eq. (5). (a) Epoxy-treated glass. (b) PMMA-treated glass. (c) PMMA-untreated glass.

However, care must always be exerted when making such comparisons since it is not uncommon for these rules of thumb to fail when other factors related to the complex structure of composites intervene; in this case, even an inverse correlations between the composites properties and the interface properties can be found (see e.g. [34] for the case of an epoxy system). Experimental Gc vs. / data were fitted to the function.

Gc ð/Þ ¼ a½1 þ ð1  kÞ tan2 ð/Þ

ð5Þ

F. Caimmi, A. Pavan / Engineering Fracture Mechanics 76 (2009) 2731–2747

2743

that has been often used to describe the dependency of toughness on phase angle [12]. In Eq. (5) a and k are purely fitting parameters, with k aimed at representing the ‘‘mode II” toughness contribution to the total toughness, for which its values should be constrained between 0 and 1. Indeed, Eq. (5) was proposed for materials with some peculiar combinations of elastic constants (namely, Dundurs’ b = 0 [12]). Here however, we do not deem k can be attached such a sense, and thus no constraints to its value was enforced. Good fits could be obtained with values of k not satisfying the aforementioned constraints. For the bi-material combination PMMA-untreated glass it would be possible to obtain a fit satisfying the condition 0 < k < 1, but since the trend is rather flat it wouldn’t be significant.

a 10

2

Gc [J/m ]

8 6 4

epoxy-treated glass φ ≅-20°

2 0 0

5

10

15

20

Δa [mm]

b 100

2

Gc [J/m ]

80

60

40

20

epoxy-treated glass φ ≅ -40°

0 0

10

20

30

40

50

Δ a [mm]

c

4

2

Gc [J/m ]

3

2

1

PMMA-untreated glass φ ≅ −40° 0 0

10

20

30

40

50

60

Δa [mm] Fig. 11. R-curves for the three systems yielding stable crack propagation. (a) Epoxy-treated glass, / ffi 20°. (b) Epoxy-treated glass, / ffi 40°. (c) PMMAtreated glass, / ffi 40°.

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For the three more stable crack propagation cases we could also determine propagation toughness values. Typical results are given in Fig. 11. While the PMMA–glass system shows no significant toughening effects (R-curvature) as the crack advances, some increment can be seen for the epoxy-treated glass couple, especially at / ffi 20°. The influence of the phase angle on crack propagation behaviour is noteworthy. 5. Cohesive zone modelling Cohesive zone modelling4 is the link we propose between the experimental results obtained at the macro-scale and the micro-scale typical of glass–fibre composites. To represent the interface we added a layer of four noded cohesive elements [21] to our 2D-FE models; an extrinsic formulation of the cohesive law with elastic behaviour until damage initiation and exponential softening thereafter, already implemented in ABAQUSTM [22], was assumed. Uncoupled normal and tangential behaviour was further assumed, as the intrinsic coupling of normal and tangential tractions typical of bi-materials problems should come just from compatibility requirements. The total cohesive energy C was assumed to be independent of the relative amount of normal and tangential tractions in the cohesive elements, as these, as long as we remain in a small scale yielding approximation, are uniquely determined by the phase angle of K*, which is almost constant in our macroscopic joints. Such a cohesive law is defined by five parameters: two elastic stiffnesses (normal and shear), two maximum cohesive stresses (normal and tangential) and a total cohesive energy. Damage is assumed to initiate when a quadratic criterion is satisfied, i.e. when

 2 hry i

rc

 þ

rxy sc

2 ¼1

ð6Þ

where rc and sc are the normal and tangential maximum stresses. Angular brackets are Macaulay brackets (i.e. if q P 0 then hqi ¼ q, else hqi ¼ 0). As to the evolution of the damage variable, among the various possibilities offered in ABAQUSTM, an energy driven damage evolution for the isotropic damage variable D was chosen, namely

D¼

Z

kuk

ku0 k

kreff ðu0 Þk dðku0 kÞ C  C0

ð7Þ

In the last equation there appear the effective tractions and displacement vectors, reff and u, the elastic energy at damage initiation C0 and the effective displacement vector at damage initiation u0. The effective tractions are defined as

reff ¼ fhry i; rxy g

ð8Þ

and the effective displacements are defined likewise. To identify the parameters we interfaced ABAQUSTM with MATLABTM: a MATLABTM optimization algorithm was used to supply ABAQUSTM with new parameters for the cohesive zone model. The optimization procedure seeks to minimize the difference between the experimentally measured load and the load output from the numerical simulations. In these simulations not the load but the load-point velocity was preferably imposed, because that was more convenient for generating the output to be used in the optimization routine. Results for the epoxy–glass system with / ffi 20° are shown in Fig. 12. Two sets of experimental data points are reported: circles are the ones used in the identification process. A very good agreement between the experimental results and FEM predictions was obtained in terms of global load–displacement traces (Fig. 12a) and a fair agreement was obtained in terms of crack’s length evolution (Fig. 12b). The identified parameters are given in Table 3; stiffness values are close to bulk polymer values and the cohesive energy is close to the measured toughness, i.e. Gc = 6.1 ± 1 J/m2. Less satisfactory results were obtained with other configurations. For example, in the case of the epoxy-treated glass system with / ffi 40° (Fig. 13), the numerical simulation was not able to catch the global response of our specimens. The response does not seem to be influenced by the value of the cohesive energy, but similar results were also obtained when varying the other parameters and using different shapes of the cohesive law (triangular). A model such as the one used here wouldn’t be able to simulate the small instabilities (‘‘stick–slip”) that can observed in Fig. 13. To properly address this matter, i.e. dynamic effects occurring locally at the tip of the crack, more refined modelling techniques are needed. This point is still under investigation and will be the subject of future work. 6. Summary and concluding remarks Tests that allow the determination of the interfacial toughness of polymer–glass joints were devised and analysed. The analysis assessed that they give an almost constant phase angle of the stress intensity factor with varying crack length, thus 4

Among the vast literature available on cohesive zone modelling we refer the reader to [28,29] and to the references therein.

F. Caimmi, A. Pavan / Engineering Fracture Mechanics 76 (2009) 2731–2747

a

epoxy-treated glass φ ≅ −20°

60

Load [N]

2745

40

20

, 0 0.0

0.5

FEM Experimental

1.0

1.5

crosshead displacement [mm]

b

epoxy-treated glass φ ≅ −20°

Δa [mm]

15

10

,

FEA Experimental

5

0 0

5

10

15

20

25

tprop [s] Fig. 12. Comparison between experiments and finite element simulation with cohesive elements. Epoxy-treated glass, / ffi 20°. (a) Two examples of load– displacement response. Circles: initial a/L = 0.52, h = 9.85. Squares: initial a/L = 0.66, h = 9.92. (b) Corresponding crack length vs. propagation time (test time–initiation time).

Table 3 Identified cohesive law parameters for the epoxy-treated glass bi-material system with nominal phase angle 20°. Normal stiffness, Ec Tangential stiffness, lc Normal maximum cohesive stress, rc Shear maximum cohesive stress, sc Cohesive energy, C

3 GPa 1 GPa 1 MPa 3 MPa 6.3 J/m2

allowing the study of crack propagation behaviour of interfacial cracks. Some suggestions on the crack length to be used during the experiments were also given. Although the configurations used allow exploring a relatively limited range of phase angles, the constancy of phase angle is a particularly desirable property, rarely studied in the context of bi-material testing. These tests were applied to study three bi-material systems and the dependence of toughness on stress intensity factor’s phase angle was determined over the range of phase angle experimentally accessible. As a by product, this study showed a clear influence of the glass surface finishing treatment on the measured interfacial toughness: it is detrimental in the case of epoxy–glass joints while it enhances toughness in the case of PMMA–glass joints. It also affects the variation of toughness with phase angle in the latter case. A cohesive zone approach was used to describe the interfacial fracture processes. The results obtained are only partly satisfactory indicating the need of further investigation.

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60

epoxy- treated glass φ ≅ −40°

2

Γ=120 J/m

Load [N]

40

20

2

Γ=30 J/m

cohesive energy

0 0

1

2

3

4

Crosshead displacement [mm] Fig. 13. Comparison between experiments (circles) and finite element simulations (lines) with cohesive elements. Epoxy-treated glass, / ffi 40°. Simulations with varying values of the total cohesive energy, as indicated.

Appendix A: fits for phase angle and strain energy release rate Here are reported the polynomial fits for the phase angle and the strain energy release rate for the various test configuration, classified on the base of the constituents and of nominal phase angle. They can be safely used for values of a/L 2 [0.2, 0.8] and h 2 [5, 13]. / values are in degrees, G/P2 in 1/ (Nm); a, L and h in (mm). Epoxy–glass, / ffi 20° 3

2

2

2

/ ¼ ð481a4 Þ=L4 þ ð5:68ha Þ=L3  ð896a3 Þ=L3  ð0:92h a2 Þ=L2 þ ð19:5ha Þ=L2 þ ð101  102 a2 Þ=ðh L2 Þ 2

2

þ ð416a2 Þ=L2  ð1:55h aÞ=L  ð7:71haÞ=Lð933aÞ=ðhaÞ þ ð87aÞ=L0:28h  9:54h þ 36:4932

ðA1Þ

G 3 2 ¼ ð7:95  102 a3 Þ=L3 þ ð37:3a2 Þ=ðh L2 Þ  ð0:4aÞ=ðhLÞ þ ð3:18aÞ=ðh LÞ  ð1:75  102 aÞ=L þ 7:96  103 p2

ðA2Þ

Epoxy–glass, / ffi 30° 3

2

3

3

/ ¼ ð87:5a4 Þ=L4  ð1:39ha Þ=L3 þ ð180a3 Þ=L3 þ ð3:26ha Þ=L2 þ ð47:9a2 Þ=ðh L2 Þ  ð133a2 Þ=L2 þ ð0:25h aÞ=L 2

3

2

 ð9:04h aÞ=L þ ð117aÞ=L þ ð1352aÞ=ðhLÞ  ð630aÞ=L þ 0:77h  10:2h þ 58:9h  154 G P2

ðA3Þ 3

¼ ð20:6  103 ahÞ=L  2:64  103 h  ð5:0  103 aÞ=L þ ð1:6  103 a2 Þ=L2  1:46=h þ ð0:61a2 Þ=ðL2 h Þ  0:146

ðA4Þ

Epoxy–glass, / ffi 40° 2

3

/ ¼ 220:965 þ 97:90h  18:154h þ 1:45857h þ 12:1706a4 =L4  33:1a3 =L3  ð4:37a3 hÞ=L3  145:6a2 =L2 3

2

2

þ ð5600a2 Þ=ðh L2 Þ þ ð43:8a2 hÞ=L2  ð1:82a2 h Þ=L2  ð1916aÞ=L þ ð3648aÞ=ðhLÞ þ ð355ahÞ=L  ð29:8ah Þ=L G P

2

¼ 1:93  103

a2 h

ðA5Þ ðA6Þ

3

PMMA–glass, / ffi 20° 3

2

2

3

/ ¼ ð16:7a4 Þ=L4 þ ð0:69ha Þ=L3  ð41:5a3 Þ=L3 þ ð0:47h a2 Þ=L2  ð9:5ha Þ=L2  ð1  103 a2 Þ=L þ ðh L2 Þ 3

2

3

2

þ ð82:7a2 Þ=L2 þ ð0:13h aÞ=L  ð4:70h aÞ=L þ ð57:6haÞ=L þ ð533aÞ=ðhLÞ  ð315:652aÞ=L þ 0:27h  3:3h þ 17:8h  50 G P2

3

ðA7Þ 2

¼ ð42:1a2 Þ=ðh L2 Þ  ð0:7aÞ=ðhLÞ þ ð3:2aÞ=ðh LÞ þ ð7:45  102 Þ=L  6:91  103

ðA8Þ

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F. Caimmi, A. Pavan / Engineering Fracture Mechanics 76 (2009) 2731–2747

PMMA–glass, / ffi 30° 3

3

3

2

/ ¼ ð6:88a4 Þ=L4  ð7:29ha Þ=L3  ð69:3a3 Þ=L3 þ ð7:13ha Þ=L2  ð2091a2 Þ=ðh L2 Þ  ð58:6a2 Þ=L2  ð0:19h aÞ=L þ ð4:18haÞ=L þ ð289aÞ=ðhLÞ  ð57:4aÞ=ðhLÞ  ð57:4aÞ=L þ 0:93h  3:69 G P2

ðA9Þ 3

¼ 0:11h  ð1:7  102 aÞ=L þ ð1:62  102 a2 Þ=L2  0:93 þ 3:5=h  ð3:8a2 Þ=ðL2 h Þ

ðA10Þ

PMMA–glass, / ffi 40° 2

3

/ ¼ ð142a4 Þ=L4  ð319a3 Þ=L3  ð3:07ha Þ=L2 þ ð120a2 Þ=ðh L2 Þ þ ð258a2 Þ=L2 þ ð2:94haÞ=L þ ð17:0aÞ=ðhLÞ 2

 ð99:6aÞ=L þ 1:07h  6:80h  11:2 G P

2

¼ 5:14  103

a2 h

3

ðA11Þ ðA12Þ

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