Identification and characterization of delamination in polymer coated metal sheet

ARTICLE IN PRESS Journal of the Mechanics and Physics of Solids 56 (2008) 3259–3276

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Identification and characterization of delamination in polymer coated metal sheet M.J. van den Bosch a,b, P.J.G. Schreurs a,, M.G.D. Geers a a b

Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands Netherlands Institute for Metals Research (NIMR), PO Box 5008, 2600 GA Delft, The Netherlands

a r t i c l e in fo

abstract

Article history: Received 14 May 2007 Received in revised form 15 April 2008 Accepted 9 July 2008

In this paper, a mixed numerical–experimental approach is adopted to quantitatively investigate and characterize delamination in polymer coated steel. The integral bimaterial system is analysed and special attention is given to the constitutive modelling of the polymer coating, the interface and the determination of all involved parameters. An extended cohesive zone model for large displacements is proposed, allowing for a mode-dependent behaviour in large deformations. Finally, peel tests are used to characterize the interface, whereby the interfacial properties are determined through an inverse parameter identification procedure. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Polymer coated steel Polymer metal interface Peel test Cohesive zone Parameter identification

1. Introduction Traditionally, products like aerosols, beverage and food cans, beer caps and luxury products are made from sheet metal. After the forming steps, the product is cleaned and lacquered to prevent corrosion, to give its surface a glossy appearance and to assure good printability. Currently, more and more products are made from polymer coated sheet metal, making subsequent lacquering superfluous. This naturally implies considerable cost savings and eliminates the emission of volatile organic compounds (Graziano, 2000; Chvedov and Jones, 2004). Because the coating is subjected to the same deformation processes as the metal substrate, delamination may occur, leading to the loss of protective and attractive properties of the product (Boelen et al., 2004) which is unacceptable. If delamination can be predicted, the processing routes, parameters and tooling can be adjusted to prevent it. The first step in this direction is to develop adequate constitutive models for coating, substrate and interface and accurately determine their (material) parameters through dedicated experiments. This paper focusses on the interface properties, properly identified using a combined numerical–experimental approach. The success of such an approach has been proven by many researchers. For example, Yang et al. (1999, 2001) characterized an interface with two different experimental set-ups. With these interfacial properties, the fracture of adhesive joints under mixedmode loading conditions was successfully simulated (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006). It has been emphasized that the delamination process between a metal substrate and a polymer coating generally takes place by the formation of fibrils, as shown in Fig. 1 (Van den Bosch et al., 2007a). Instead of the micromechanical modelling of an interface with individual fibrils, the delamination process can be described conveniently by a cohesive zone model, which projects all damage mechanisms in and around the delamination

 Corresponding author. Tel.: +31 40 2472778; fax: +31 40 2447355.

E-mail address: [email protected] (P.J.G. Schreurs). 0022-5096/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2008.07.006

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Fig. 1. SEM photograph of the delamination front showing the presence of fibrils.

front on the interface. This leads to a single constitutive relation, or cohesive zone law, between the traction and opening displacement of the interface describing the whole failure process. Cohesive zones are typically implemented in finite element codes as interface elements. A variety of cohesive zone models has been developed in recent years (for an overview see a.o. Chandra et al., 2002; De Borst, 2003; Van den Bosch et al., 2006). The majority of these cohesive zone models use a local coordinate system, i.e. a coordinate system that is oriented with respect to the (un)deformed geometry of the cohesive zone element. This local coordinate system is used to decompose the opening displacement in a normal and tangential contribution. It has been shown in previous work that this approach may result in ambiguous results when large openings occur at the interface (Van den Bosch et al., 2007a). The delamination of a polymer coating from a steel substrate typically involves very large displacements and deformations at the interface due to the formation of fibrils. Therefore, a large displacement formulation has been developed, which does not rely on a local coordinate system, thereby overcoming ambiguities that may result from a small displacement formulation. Accurate modelling of the mechanical behaviour of the polymer coating is of great importance to successfully combine delamination experiments and simulations used to indirectly characterize the interface. Therefore, this paper necessarily addresses the constitutive model of the polymer coating and the required experiments to determine its material parameters. Attention is given to the intrinsic stress–strain response of the coating, i.e. the macroscopic response in homogeneous deformation and experimental characterization techniques. Polymers typically show strain softening, which is a decrease in true stress after reaching the yield stress. In conjuncture with its strain rate-dependent behaviour this poses considerable modelling challenges. A vast amount of literature exists related to polymer coated metals. The majority of the research is focused on the corrosion protection or the corrosive delamination of the polymer. Little attention was given to the industrial forming processes of the polymer–metal laminate by means of wall-ironing (Van der Aa et al., 2000, 2001; Huang et al., 2001; Boelen et al., 2004) or deep-redrawing (Chang and Wang, 1998; Liu and Wang, 2004; Lee et al., 2007). To the best of our knowledge, no rigorous attempt was made to numerically predict the delamination of the polymer coating as relevant for the above mentioned forming processes. Therefore, a motivated and quantitative numerical–experimental method is here developed, enabling the accurate assessment of delamination of the coating. In some recent papers, the polymer–metal interface has been characterized with a numerical–experimental approach (Van Tijum and De Hosson, 2005; Van Tijum et al., 2007; Fedorov et al., 2007). In those papers, no attention was given to the properties of the coating and the interface was described by a small displacement cohesive zone model with five parameters, which cannot be related uniquely to the available experimental data. In this paper, a large displacement cohesive zone model is used with a reduced parameter set, allowing for a more rigorous parameter identification using a mixed numerical–experimental approach. A unified manner of characterizing interfaces by trying to make a parallel experimental analysis and a computational model (which is the simplified translation of our common understanding of the governing phenomena at a coarse scale) is almost never done in the way it is done here (including experimentally observed phenomena and length scales, within a 3D context, with large deformations, with mode mixity, etc.). Such a combined characterization method is the recommended scientific approach, since it does not help to carry out additional experiments if our interpretation thereof cannot be validated through a model. In this paper, first a suitable experimental method to characterize delamination is discussed. Then, the material system and delamination experiments are presented. Subsequently, the behaviour and constitutive model of the polymer are described and material parameters are identified. A cohesive zone model based on a large displacement formulation is proposed and its parameters are quantified in a mixed numerical–experimental identification procedure, linking up peel simulations to the conducted delamination tests.

2. Experimental set-up, materials and specimens A peel test is a frequently used experimental method for assessing the delamination of flexible laminates. The main requirement is that the delaminating coating must be able to withstand the peel force. The mechanics of the peel test has

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been reported extensively in literature (see Kinloch et al., 1994 and references therein). A peel test can be performed with standard mechanical testing equipment, which makes it a suitable experimental testing method to characterize the polymer coated sheet metal. The material system of interest is shown schematically in Fig. 2. The substrate is a batch annealed temper 52 soft deep-drawing steel. The engineering yield stress is 250 MPa, the ultimate (engineering) tensile stress is 310 MPa and the (engineering) linear strain at break is 25%. The steel is coated with chromium in order to improve the adhesion of the polymer layer by means of hydrogen bonds (Beentjes, 2004). The polymer coating consists of two layers: a 4 mm thick adhesion layer and a 26 mm thick poly-ethylene terephthalate (PET) layer. The adhesion layer is made from PETG (which is a modification of PET with glycol side groups) and several additives to make it stronger. The PET is extruded at its melt temperature (270 1C) and brought in contact with the substrate. Immediately after, the polymer coated steel is quenched in water. The fast cooling leads to a nearly amorphous polymer coating. The presence of residual stresses after the coating process has been investigated by accurately measuring the shrinkage of the coating after removing it from the steel by submerging it into a 18% hydrochloric acid solution. There was no measurable shrinkage found and thus it was concluded that residual stresses in the coating have been relaxed. Tensile tests have been performed on the coating material revealing that the coating is isotropic in-plane. It was also verified if the hydrochloric acid does not influence its mechanical properties by manufacturing free-standing PET films with different concentrations of hydrochloric acid and subsequently performing tensile tests. The results show no measurable difference in mechanical response and thus it was concluded that the acid does not influence PET. 2.1. Free-standing PET specimens The coating is next analysed as a homogeneous PET layer, because the difference in mechanical behaviour between PET and PETG is small (Dupaix and Boyce, 2005) and the PET layer is relatively thick compared to the PETG layer. Characterization of the mechanical behaviour of a PET coating is preferentially done by tensile tests rather than compression tests. Firstly, standard compression tests on thin polymer films are not trivially possible, and would require cylindrical bulk PET specimens, with possibly different mechanical properties. Secondly, tensile loading with inhomogeneous deformation dominates in the peel tests and thus characterization of the coating in similar loading conditions is expected to improve the accuracy. Nevertheless, compression tests will be required to capture the intrinsic softening behaviour of PET. In this context, nano-indentation has been assessed as a possible method to determine the mechanical properties of PET as well. However, the retrieval of material parameters remains not trivial. The deformation under the indentor tip is not uniform and it is essential to estimate the contact area between the indentor and the polymer in order to extract the correct properties. Even with a flat tip indentor, with a well defined contact area, the material parameters can only be accurately determined up to the yield point of the polymer (Den Toonder et al., 2006). To allow (tensile) testing of the coating, it needs to be detached from its steel substrate. Care has to be taken that the coating is not damaged during the removal and further specimen preparation. Dog bone shaped tensile specimens (see Fig. 3) are created by pushing a sharp punch through the coating into the steel before submerging the material in a hydrochloric solution. This results in well-dimensioned free-standing PET tensile specimens with almost no observable flaws along the edges. 2.2. Delamination specimens A sheet of adhesive tape was applied on the polymer coating to prevent damage of the layer during the preparation of the peel specimens. First, specimens of 40  8 mm were cut out of the polymer coated sheet. Second, the width of the specimen was reduced to 3.5 mm by grinding the edges with SiC-paper, which also removed possible burrs. Next a groove

Fig. 2. Structure and dimensions of the layered polymer-coated metal sheet.

Fig. 3. The geometry of the dog bone shaped, free-standing PET specimen, detached from its steel substrate. Dimensions are in mm.

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was milled in the back of the specimen with a table top precision cut-off machine, as shown in Fig. 4a. The specimens were clamped in a standard tensile tester and loaded uniaxially with a speed of 30 mm s1 until the steel substrate fractured at the location of the groove. The test set-up for the 0 peel tests and 90 peel tests is shown in Fig. 4b. The 90 peel specimens consist of two peel specimens of which half the lengths are adhesively bonded. The bonded part is placed vertically between two parallel steel plates, allowing vertical displacements only. The advantage of this set-up is that the peel angle remains consistently equal to 90 . Since the specimens present small variation in width and coating thickness (as investigated in Section 2.3), it is necessary to measure the width and coating thickness of every individual specimen before testing. The specimens were peeled in a Kammrath&Weiss microtensile stage with a clamp speed of 25 mm s1 and the reaction forces were measured with a 20 N (0:01 N) load cell. The temperature ð23  2  CÞ and the relative humidity ð27  9%Þ both influence the mechanical behaviour of PET (Dupaix and Boyce, 2005; Hu et al., 2005) and were monitored during the experiments. During testing, the tensile stage is installed in a closed plexiglass box to minimize environmental disturbances. The peel force F, measured during the peel test, is normalized to account for inevitable variations in the initial crosssection of the as-received coating A0 : F ¼

FA A0

(1)

where F  is the normalized peel force and A ¼ 3:5  0:030 ¼ 0:105 mm2 a reference cross-section, representative for the whole specimen series. Using F  restricts the scatter resulting from geometrical variations. However, the contribution of the interface only scales with the width of the specimen and not with the coating thickness. This second-order effect is verified to be small and therefore neglected.

2.3. Coating thickness The thickness of the coating was measured with a SensoFar Plm 2300 optical profiler in confocal mode. An example of a typical measurement is shown in Fig. 5. The top plane is the surface of the coating and the (rough) bottom plane is the surface of the substrate. The coating thickness of a polymer coated steel sheet was measured at 247 locations in a regular 19  13 grid with a spacing of 5 mm. At every location the average coating thickness was calculated from a thickness profile of 500  360 mm2 . In Fig. 6 the coating thickness profile is shown for a specimen of 60  85 mm2 . As can be seen from Fig. 6 the coating thickness varies considerably at the surface of the specimen. Another observation is the apparent anisotropy resulting from the oriented variations in coating thickness: perpendicular to the rolling direction of the steel, which is also the extrusion direction of the PET coating, the variation in the coating thickness is much larger than parallel to the rolling direction (parallel to the y-axis in Fig. 6).

Fig. 4. (a) The grooved specimen geometry (specimen is shown upside down). (b) Experimental set-up of a 0 peel tests and a 90 peel test.

Fig. 5. Surface profiles of the coating and the substrate ð100  100 mmÞ.

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Fig. 6. Thickness profile of a polymer coated sheet ð85  60 mmÞ. Extrusion direction of the PET is parallel to the y-axis.

Fig. 7. (a) Schematic representation of the intrinsic deformation behaviour of PET. (b) Schematic representation of the influence of the ageing time on the yield stress and the post-yield behaviour.

3. Modelling and characterization of the coating The intrinsic deformation behaviour is defined as the response of a material under homogeneous deformation. The general intrinsic behaviour of PET is visualized in Fig. 7a through its true stress–strain curve (or sðÞ curve), presenting several distinctive regimes. Initially, the material behaves in a (non-linear) visco-elastic manner with no permanent deformations. At increasing stress, the material starts to yield and deforms plastically. The plastic deformation leads to a structural evolution that reduces the material’s resistance against plastic deformation, i.e. strain softening. Finally, at large deformations the molecular chains become oriented, resulting in strain hardening. In general, polymers like PET show a time-dependent behaviour. An increase in strain rate _ results in an upward shift of the stress–strain curve, as shown in Fig. 7a. It leads to an increased yield stress but does not influence the slope of the curve in the hardening regime. When a polymer is cooled down from above its glass transition temperature T g ( 70  C for PET), the material is in a non-equilibrium thermodynamic state. Below T g, the molecular chains are still mobile and over time, the material gradually approaches equilibrium. This process is called physical ageing. The effect of physical ageing is an increase in yield stress but does not affect the behaviour at large strains, as shown in Fig. 7b. In Fig. 8, the behaviour of a free-standing PET film under tensile load is shown. During a tensile test the specimen is stretched (a.1)–(a.2). If at a location in the specimen the yield stress has been reached, a neck immediately initiates (a.3) causing an instant drop in the tensile force (see Fig. 8b). After that, the neck starts to propagate at a constant speed and at a constant tensile force (a.4)–(a.5). A schematic representation of the transition of a cross-section from the (undeformed) bulk material (P) into the neck (Q), is shown in Fig. 8c. The transition zones travel with a speed proportional to the speed of the clamps. The transition zone is a relatively small area where most of the deformation takes place and therefore, locally, high strain rates are achieved. The tensile sðÞ curve can be calculated from tensile test data by assuming incompressibility:

L þ W þ T ¼ 0

(2)

where L , W and T are the true strain in length (L), width (W) and thickness (T) direction, respectively. If isotropy in the cross-sectional plane is assumed (W ¼ T ), the current cross-section A follows from the initial thickness T 0 , initial width W 0 and the current width W, which is measured during a test: A ¼ WT ¼ WT 0

W W0

(3)

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Fig. 8. (a) Schematic representation of a uniaxial tensile test on free-standing PET films. (b) Force–displacement curve of a uniaxial tensile test on freestanding PET film. (c) Schematic representation of the cross-section from the ‘bulk’ film material (P), through the transition zone into the neck (Q).

Fig. 9. (a) Lumped parameter representation of the compressible Leonov material model. (b) Schematic stress–strain curve of the PET layer, revealing the intermolecular and network hardening contributions.

which leads to the true stress and true strain:   F W0 and  ¼ 2 ln s¼ A W

(4)

where F is the measured tensile force. 3.1. Constitutive modelling A 3D constitutive model is used for the description of the isotropic, large strain and time-dependent mechanical behaviour of the polymer coating. The model is implemented as a user-subroutine in the commercial finite element package MSC.MARC. It is a compressible generalization of the Leonov (1976) model, originally proposed by Baaijens (1991). In the model, a distinction is made between the contribution of secondary interactions between polymer chains, which determine the (visco-) elastic properties at small deformations and plastic flow, and the entangled polymer network, which governs the strain hardening. Accordingly, based on the original work of Haward and Thackray (1968), the total Cauchy stress r is decomposed into a driving stress rs and a hardening stress rr (Fig. 9):

r ¼ rs þ rr

(5)

The hardening is modelled with a neo-Hookean relation (Tervoort and Govaert, 2000):

rr ¼ Gr B~ d

(6) ~d

where Gr is the strain hardening modulus and B is the deviatoric part of the isochoric left Cauchy–Green deformation tensor. The driving stress is decomposed into a deviatoric stress rds and a hydrostatic stress rhs :

rds ¼ GB~ de and rhs ¼ kðJ  1ÞI

(7)

~ d the deviatoric part of the isochoric elastic left Cauchy–Green strain tensor, k the bulk where G is the shear modulus, B e ~ d results from the following modulus, J the volume change factor, and I the unity tensor. The evolution of J and B e

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kinematical equations: _J ¼ J trðDÞ

(8)



~ e ¼ ðDd  Dp Þ  B ~e þB ~ e  ðDd  Dp Þ B

(9)



~ e is the objective Jaumann rate of B ~ e and D represents the deviatoric part of the rate of deformation tensor. The Here, B plastic deformation rate tensor Dp is related to the deviatoric part of the driving stress rds by d

Dp ¼

rds 2Z

(10)

The viscosity Z was originally described by an Eyring (1936) relationship, which was extended later to incorporate pressure dependency and intrinsic strain softening (Govaert et al., 2000):   mp t¯ =t0 (11) ZðT; p; t¯ ; SÞ ¼ Z0;r ðTÞ exp expðSÞ t0 sinhðt¯ =t0 Þ where Z0;r denotes the zero-viscosity for the completely rejuvenated state, T is the temperature, m is the material parameter describing the pressure dependence and S is a state parameter. The hydrostatic pressure p, the characteristic stress t0 and the effective stress t¯ are defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kT 1 trðrds  rds Þ p ¼  trðrÞ; t0 ¼  ; t¯ ¼ (12) 3 V 2 with k Boltzmann’s constant and V  the activation volume. The state parameter S is decomposed into a factor Sa, which captures the thermo-mechanical history of the material, and Rg ðg¯ p Þ, which describes the softening kinetics (Klompen et al., 2005): Sðg¯ p Þ ¼ Sa Rg ðg¯ p Þ

(13)

where g¯ p is the effective plastic strain. The initial value of S equals Sa since the value of Rg is initially normalized to 1 and decreases towards 0 with increasing plastic strain. 3.2. Inverse parameter identification The material parameters can be determined by making use of an inverse mixed numerical–experimental approach. It has been shown before by several authors that the parameters of the constitutive model can be determined uniquely, see e.g. Meijer and Govaert (2003) and Klompen et al. (2005). In this section, first an overview is given of the material parameters and the required experimental methods to determine them. Next the numerical model is discussed, which is used to reproduce the test results from the materials description and then the material parameters are determined. The characteristic stress t0 describes the strain rate dependency of the yield stress, where t0 is a function of the strain rate (Foot et al., 1987). Therefore, to determine t0 , the strain rate range during peel tests needs to be known and tensile tests with those strain rates need to be performed. Within certain assumptions, t0 can be determined during the actual peel tests. The latter procedure is described in Section 4.2. The parameters Gr and Z0;r are determined from the tensile sðÞ curve of a free-standing PET film, where the sðÞ curve is constructed in a local point, see also Eq. (4). The thermo-mechanical history parameter Sa is determined from the experimentally measured shape of the transition zone, which is related to the softening instability. The softening parameters r 0 , r 1 and r 2 are determined from the strain softening regime in a sðÞ curve. Strain softening cannot be observed globally in tensile tests because, at the onset of yielding, the neck instantly grows and the tensile force immediately drops (see Fig. 8b). Hence, the measured force drives the propagating neck. In compression tests the softening regime can be measured because the specimen deforms homogeneously. To this purpose, cylindrical bulk PET specimens have been used to determine the softening parameters (Van der Aa et al., 2000). The pressure parameter m follows from the dependency of the yield stress on the hydrostatic pressure. Its value as well as Young’s modulus E and Poisson’s ratio n have been taken from the work of Van der Aa et al. (2000). 3.2.1. Computational model of the PET tensile test The material parameters Gr , Z0;r and Sa are identified from a tensile test on a free-standing PET film. Due to symmetry conditions only one-eighth of the free-standing PET specimen is modelled, as shown in Fig. 10. Consistent with the experimental conditions, a constant clamp speed (u_ x ¼ 164 mm s1 ) is applied. The model is discretized with (3D) eightnode hexahedral elements. First, the material parameter Sa is determined, which is related to the height of the yield drop. The transition zone reflects the strain localization and its geometry is directly related to the height of the yield drop and thus Sa . Therefore, the value of Sa is determined from this geometrical characteristic.

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Fig. 10. 3D FEM model of a free-standing PET film in tension, including boundary conditions. Only one-eighth is modelled (symmetry).

Z (μm)

10 6 2 -2 -6 -10

80 60

Y(

μm )

20

20 0

m )

X

(μ

60

40 40

0

Fig. 11. Measured surface profile in the transition zone of a PET tensile specimen, revealing the detailed geometry at the neck front.

The geometry in the transition zone was measured with a SensoFar optical imaging profiler and is shown in Fig. 11. The maximum slope of the transition zone equals 0:68 m=m (0:68 mm drop over a distance of 1 mm). The element discretization in the finite element model needs to be sufficiently detailed to capture the geometry of the transition zone correctly. Simulations with different element sizes have been carried out as shown in Fig. 12. Clearly, the calculated slope of the transition zone converges with decreasing element size and the converged value depends on the value of Sa . Interpolation of the data leads to Sa ¼ 12:4. The required local element size has to be smaller than 1 mm. The element size does also influence the strain rates at which the material locally deforms near the neck, see Fig. 13. With Sa ¼ 12:4 it follows that the maximum strain rate during the tensile test equals _ ¼ 7:46 s1 . Models with a coarse mesh will underestimate the strain rate and thus tend to underestimate the tensile force (due to the strain rate dependency of the material). A coarse discretization is therefore not suited to determine material parameters. The parameter Z0;r is determined by minimizing the difference between the simulated and experimental sðÞ response. Typically, the simulated sðÞ curves are retrieved from a specific location (e.g. point R) in the model of a free-standing PET film (shown in Fig. 10). Since computational analyses with element sizes below 1 mm are rather expensive, simulations on the basis of a single material point with the 3D constitutive model were carried out. Although the deformations during a tensile test are highly inhomogeneous due to the formation of a neck, the material point model directly reflects the desired sðÞ curve. The material is mostly deformed in the transition zone. The propagation speed of the transition zone is related to the clamp speed, which is constant. Obviously, all material points undergo the same deformation history, yet shifted in time, as dictated by the position of the transition zone. Hence, the sðÞ curve of a point in the material can be easily constructed by imposing the correct experimental loading path. The real evolution of the strain ðtÞ is shown in Fig. 14, which constitutes the imposed loading path on a single material point in tensile direction. The directions perpendicular to the tensile direction are stress free. The single material point model, with the correct loading conditions, is used in conjuncture with an automated parameter identification procedure to reveal the material parameter Z0;r. 3.2.2. Extracting material parameters An inverse parameter identification procedure is a frequently used tool to determine material parameters in complex models, where interacting parameters cannot be determined directly or trivially (Meuwissen et al., 1998; Geers et al., 1999; Bocciarelli et al., 2005; Cooreman et al., 2007). A least-squares objective function, relating experimental to numerical data, is thereby minimized using a gradient method, yielding the unknown parameters. In this section the results are given of the parameter identification by using the single material point model and experimental results. Before Z0;r can be determined, it is required to determine the value of t0 . To obtain its value, peel tests

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Slope [m/m]

100

10−1 Sa = 11 Sa = 12 Sa = 13 Simulation Experiment 10−2 10−7

10−6 Element size [m]

10−5

Fig. 12. Slope of the transition zone as a function of the element size.

Strain rate [s−1]

101

100

Sa = 11 Sa = 12 Sa = 13 Simulation

10−1 10−7

10−6 Element size [m]

10−5

Fig. 13. Strain rate in the transition zone as a function of the element size.

True strain [−]

1 0.8 0.6 0.4 0.2 0 0

0.2

0.4 Time [s]

0.6

0.8

Fig. 14. Strain–time history of a material point in the tensile specimen (t ¼ 0 upon entering the transition zone).

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and simulations are used, resulting in t0 ¼ 2:13 MPa. For reasons of clarity, the detailed description of this procedure is given in Section 4.2. The strain hardening parameter Gr is determined according to Klompen et al. (2005). The true stress is decomposed into three separate contributions as shown in Fig. 15:

sð_ ; S; Þ ¼ srej ð_ Þ þ Dsy ðSÞ þ sr ðÞ

(14)

where srej is the flow stress of the fully rejuvenated state (see Fig. 7b), sr is the strain hardening stress and Dsy is the yield drop that depends on the thermo-mechanical history of the material. The strain hardening stress follows a typical neoHookean response, i.e.:

sr ¼ Gr ðl2  l1 Þ

(15)

The hardening parameter Gr equals the slope of the hardening part of the curve shown in Fig. 16. This curve is derived from the sðÞ data and leads to Gr ¼ 11:45 MPa. The zero-viscosity of the rejuvenated material, Z0;r , is determined by fitting the model response of the rejuvenated material (Sa ¼ 0) on the strain hardening regime of the experimental curves (see Fig. 17), giving Z0;r ¼ 5:35  103 Pa s. Compression tests are required to determine the strain softening parameters. The driving stress ss can be isolated from the experimental (compression test) data by using Eqs. (14) and (15): ss ¼ srej ð_ Þ þ Dsy ðSÞ and Sðg¯ p Þ ¼ Sa Rg ðg¯ p Þ. Since Rg is taken to be equal to 1 at the onset of plastic deformation, Rg can be expressed as (Klompen et al., 2005) Rðg¯ p Þ ¼

ss  srej jDsy j

(16)

where g¯ p is the effective plastic strain for uniaxial compression: pffiffiffi g¯ p ¼ 3j_ jðt  ty Þ

(17)

The experimental data are shown in Fig. 18. Klompen et al. (2005) made use of the Carreau–Yassuda function to describe the softening characteristic: ð1 þ ðr 0 expðg¯ p ÞÞr1 Þðr2 1Þ=r1

(18)

ð1 þ r r01 Þðr2 1Þ=r1

true stress

Rg ðg¯ p Þ ¼

σr

Δσy

σrej λ2−λ-1 Fig. 15. Schematic representation of the decomposition of the true stress in the yield drop Dsy , the strain hardening stress sr and the flow stress of the fully rejuvenated state srej .

150

True stress [MPa]

Experiment Simulation 100

Gr

50

0 0

1

2

3

4

5

6

7

λ2−λ-1 [−] Fig. 16. The hardening parameter Gr equals the slope of the stress–strain curve for a large l.

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150

True stress [MPa]

Experiment Simulation 100

50

0 0

0.25

0.5 0.75 True strain [−]

1

Fig. 17. Simulation results with Sa ¼ 0 are matched with experiments to determine Z0;r .

1

Rγ [−]

0.8 0.6 0.4 0.2

0

0.1

0.2

0.3

0.4

0.5

Fig. 18. Rg as a function of g¯ of uniaxial compression tests at three strain rates and the best prediction of the Carreau–Yassuda function.

Table 1 Overview of the PET material parameters E (GPa)

n (–)

Sa (–)

Z0;r (Pa s)

t0 (MPa)

m (–)

Gr (MPa)

r 0 (–)

r 1 (–)

r 2 (–)

2.03

0.4

12.4

5:35  103

2.13

0.047

11.45

0.99

237

11.6

where r 0 , r 1 and r 2 are material parameters. The parameter identification of Eq. (18) with the experimental compression test data leads to: r 0 ¼ 0:99, r 1 ¼ 237 and r 2 ¼ 11:6. The material parameter that describes the pressure dependency is obtained from the work of Van der Aa et al. (2000) and equals: m ¼ 0:047. Due to the quasi-linear elastic behaviour of the model prior to yield, Young’s modulus E is determined such that the yield strain in the model approximately equals the experimental observed strain-at-yield. This leads to Young’s modulus of 2.03 GPa. Poisson’s ratio is taken from Van der Aa et al. (2000) and equals 0.4. In Table 1 an overview of the PET material parameters is given. 3.3. Interface model Because of the formation of fibrils at the interface, the delamination of a polymer coated steel involves large displacements and deformations. Instead of modelling individual fibrils, the delamination process can be described by a cohesive zone model, which projects all damage mechanisms in and around the crack tip on the interface. If opening is associated with fibrillation, as was observed in the peel tests, it is not physical to discriminate between normal and tangential directions as is commonly done in most cohesive zone models, because fibrils can only transfer a load along their axes. Moreover, the large displacements at the interface might lead to ambiguous results, if a normal–tangential decomposition is preserved (Van den Bosch et al., 2007a). Therefore, a large displacement formulation for a cohesive zone

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~: element has been developed that relies on a single relation between the traction ~ T and the opening displacement D ~Þ ~ T ¼~ f ðD

(19)

which is particularly relevant for the fibrillation studied here. The opening displacement is calculated between two points, on both sides of the interface, which coincided initially as sketched in Fig. 19. Xu and Needleman (1993) proposed an exponential cohesive zone law that has been improved to describe mixed-mode delamination better (Van den Bosch et al., 2006). The cohesive zone law used here is based on the normal traction relation of that exponential cohesive zone law:     f D D ~ ¼ D~ D e; T ¼ exp  (20) e; ~ T ¼ TðDÞ~

d

d

d

with ~ e a unit vector between two associated material points at the interface (see Fig. 19). The work-of-separation f is the dissipated energy after complete opening: Z 1 f¼ TðDÞ dD (21) D¼0

and d is the characteristic opening length for which T reaches the maximum value T max : T max ¼

f d expð1Þ

(22)

Eq. (20) is shown graphically in Fig. 20. The adopted cohesive zone model relies on the formation of fibrils, which have been observed when the interface is loaded in mode-I during a 180 peel test (see Fig. 1). It is experimentally cumbersome to visualize the presence of potential fibrils in other loading geometries. Therefore, the surface of the delaminated coating has been mapped with the optical profiler in order to find traces of snapped fibrils. In the case of the 0 peel test, no noticeable traces of fibrils were found, which might be due to other local deformation mechanisms, possibly dissipating a different amount of energy. The large displacement cohesive zone model is therefore extended to describe this mode-dependent work-of-separation. All geometrical considerations hereafter, only involve the opening vector and the individual geometries of each face. No assumptions are made on ‘intermediate’ planes as often the case for small displacements. The opening mode of the cohesive zone is quantified by a mode-mixity parameter d: d ¼ k~ d1  ~ d2 k

(23)

In a 2D cohesive zone, as shown in Fig. 21a, ~ d1 and ~ d2 are the components perpendicular to D of the normals ~ n1 and ~ n2 ~ on the two cohesive zone edges. In a 3D element, ~ d1 and ~ d2 are projected on Sn , which is a plane with normal direction D (see Fig. 21b).

Fig. 19. Vectorial interfacial opening between two initially bonded material points, consistent with the experimentally observed defibrillation process.

40

T [MPa]

30

20

10

0 0

1

2

3 Δ [μm]

4

5

6

Fig. 20. A traction–separation curve (Eq. (20)) with f ¼ 100 J m2 and d ¼ 1 mm.

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Fig. 21. The opening mode of a cohesive zone is determined from components (~ d1 and ~ d2 ) of the edge/face normals (~ n1 and ~ n2 ) perpendicular to the ~ ). opening displacement vector (D

α = 0.2 α = 0.0 α = −0.2

120

W [Jm−2]

110 100 90 80 0

30

60

90

β [°] Fig. 22. Dissipated energy as a function of the opening angle b. Mode-I corresponds to b ¼ 90 and mode-II to b ¼ 0 . f ¼ 100 J m2 .

50

α = 0.2 α = 0.0 α = −0.2

T [MPa]

40 30 20 10 0 0

2

4

6

8

10

Δ [μm] Fig. 23. Traction–opening displacement curves for cohesive zones, opened in mode-II, with different values for a. f ¼ 100 J m2 , d ¼ 1 mm.

Parameter d has a value between 0 (mode-I) and 2 (mode-II). Intermediate values of d represent a mixed-mode opening. Eq. (20) is extended with this mode-mixity d, where a parameter a controls the influence of mode-mixity behaviour: the interface is stronger (a40) or weaker (ao0) in mode-II than in mode-I. The more general, yet simple, cohesive zone law reads T¼

 









f D D d exp  exp a 2 d d d

(24)

To investigate the behaviour of a cohesive zone with Eq. (24), a single cohesive zone is opened under an angle b and the total dissipated energy is quantified. The two faces are kept parallel to each other and b is defined as the angle between one ~, as shown in Fig. 21a. The results are shown in Fig. 22. Obviously, the energy dissipated in of the faces and the vector D mode-I (b ¼ 90 ) is independent of a and equals f. In the case of mixed-mode or mode-II (b ¼ 0 ) opening of the cohesive zone, the total dissipated energy depends on a. In Fig. 23, Eq. (24) is plotted for three values of a in mode-II, where the total dissipated energy (area under the curve) depends on the parameter a.

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The cohesive zone model has been implemented as a user-subroutine in the finite element package MSC.MARC. The formulation and numerical details can be found in Van den Bosch et al. (2007a, b).

4. Peel tests In Section 2, it was argued that the peel test is an adequate experiment to characterize delamination in polymer coated steel. In this section, the interface is characterized using a mixed numerical–experimental parameter identification approach. The experimental test results are presented and the details of the finite element peel model are discussed. First, the material parameter t0 is determined from peel tests, after which the actual interface parameters are quantified.

4.1. Peel test set-up and modelling The experimental set-up of the peel test is described in Section 2.2. During the experiments the force and displacement are monitored. The normalized peel force as a function of the clamp displacement is shown in Fig. 24 for both the 0 and 90 peel tests. The average normalized peel force for the 0 peel test is F^ 0 ¼ 4:85 N with a standard deviation of 0:07 N. For the 90 peel test the average normalized peel force is F^ 90 ¼ 4:16 N, with a standard deviation of 0:06 N. The scatter in the experimental results is caused by small variations in experimental conditions and possible variations in specimen width, coating thickness, ambient temperature and relative humidity. Considering all these sources of scatter, the results seem remarkable consistent. Two models (3D and plane-strain) are created in a finite element solution context. Due to symmetry conditions, only half of the experimental set-up needs to be modelled, see Fig. 25 (Van den Bosch et al., 2007b). The 3D model is discretized with eight-node hexahedral elements and the plane-strain model with four-node quadrilateral elements. The steel substrate is modelled as a stiff linear elastic solid (for which a single element is sufficient). The cohesive zones are located between the coating and the substrate and have an initial zero thickness. The bottom nodes of all the cohesive zones are tied to the upper nodes of the steel element. The coating is loaded at the right side with a constant velocity of u_ x ¼ 12:5 mm s1 , which is half the experimental clamp velocity. A mesh size dependency study has been performed and it is concluded that the maximum element size (in peel direction) is 1 mm and the maximum displacement step size is 100 nm per time increment (Van den Bosch et al., 2007b).

Normalized peel force [N]

6 0° peel

5 4

90° peel

3 2 Experiments Simulations

1 0 0

5

10 15 Displacement [mm]

20

Fig. 24. Normalized peel force–displacement curves of the 0 and 90 peel experiments. The discrepancy during start-up between simulations and experiments is caused by the lower peel force required to delaminate the coating from the locally plastically deformed substrate.

Fig. 25. Geometry of the 3D peel test model with its associated boundary conditions.

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1.005 Plane strain 3D

F∗3D / F∗2D [−]

1

0.995

0.99

0.985 0.25

0.5

0.75 1 1.25 Model half−width [mm]

1.5

1.75

Fig. 26. Ratio between normalized peel forces of 2D and 3D simulations as a function of the 3D model half-width.

The determination of the interface or material parameters through the adopted parameter identification procedure requires a sequence of simulations. Using 3D peel simulations is computational expensive. Therefore, it is investigated whether a peel test model with a plane-strain geometry can be used instead. Several 3D simulations, with different widths, have been carried out and the ratio F 3D =F 2D of the normalized peel forces in 3D versus the ones in 2D (plane-strain) is shown in Fig. 26. Obviously, the wider the 3D model the more accurate the assumption of plane-strain is. Experiments were conducted with specimens that have a half-width of 1.75 mm. Fig. 26 indicates that this leads to an error in the normalized peel force of approximately 0.2%. This error is acceptable, since it falls well within the experimental measurement errors. The parameter identification simulations are therefore carried out with a plane-strain geometry to significantly reduce the calculation time, typically with a factor 15–50. 4.2. Extracting the characteristic stress and the interfacial parameters The characteristic stress t0 is the material parameter that describes the strain rate dependency of the yield stress. The strain rate is linearly related to the propagation speed of the transition zone, which is linearly related to the clamp speed. Assuming that the interface does not show a significant rate-dependent behaviour, the increase in peel force will be solely due to an increase in clamp speed. As sketched in Fig. 7, an increase in strain rate shifts up the whole stress–strain curve, including the yield stress. Therefore, it is assumed that, in the experimental velocity regime, the dependency of the peel force on the clamp speed is proportional to the dependency of the yield stress on the strain rate. In Fig. 27, the experimental normalized peel force is plotted as a function of the tensile stage clamp speed. The slope of the curves is solely determined by t0, whereas the absolute values of the peel force are a function of the coating and interface properties. Simulations are carried out with different values for t0 and the results are plotted in Fig. 27. The value of t0 , providing the best numerical–experimental approximations to the slope of the curve, is identified to be t0 ¼ 2:13 MPa, which is a realistic value (Foot et al., 1987). The interface parameters f and a are determined according to the inverse parameter identification procedure. In the 0 peel test set-up, the delamination front was tracked and yielded qualitative data. In the 90 set-up it was not possible to track the delamination front due to small scale set-up. However, it is not necessary to track the delamination front for the determination of the cohesive zone parameters. The input for the parameter identification procedure is the quasi-static normalized peel force level of both the 0 and 90 peel tests and because there are only two unknowns, the parameters can be determined uniquely. The characteristic cohesive zone length, d, follows from experimental observations of the delaminating interface and has a typical value of 1 mm (Van den Bosch et al., 2007a). The interface parameters are determined as f ¼ 194 J m2 and a ¼ 0:01. The force–displacement curves of the simulations are plotted in Fig. 24. The small value of a indicates that there is a small difference only in dissipated interfacial energy between mode-I and mode-II. If f would be determined with a ¼ 0, it follows that f ¼ 194:5 J m2 , still providing model predictions within the experimental range. The 0 peel force and especially the 90 peel force are very sensitive for variations in f or a. This sensitivity provides confidence about the accuracy of the quantified cohesive zone parameters. An initially large difference in steady-state peel force predicted by the simulations and observed in the experiments can be seen in Fig. 24 for the 90 peel tests. This discrepancy is caused by the lower peel force required to delaminate the coating from the locally plastically deformed substrate (due to breaking). The simulations assume that the substrate is

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Normalized peel force [N]

6.5 6

Experiments τ0 = 2.5 MPa

5.5

τ0 = 1.5 MPa

τ0 = 2.0 MPa τ0 = 1.0 MPa

5 4.5 4 3.5 10−1

100

101 Tensile speed

102

103

[μms−1]

Fig. 27. Normalized peel force versus peel speed of experiments and simulations with different values for t0.

elastic and thus no loss of adhesion is modelled. The cohesive zone model has been extended with a plastic straindependent work-of-adhesion and quantified according to the approach outlined in this article (Van den Bosch et al., 2007c). The quantification of the cohesive zone parameters has been verified by employing them in the simulation of coating delamination during an industrial deep-drawing process. The simulations are capable of predicting the delamination during deep-drawing at the correct location. The computational details and results are reported by Van den Bosch et al. (2008). Quantification of the cohesive zone parameters independently from the surrounding materials is hard to achieve. The main reason for this is that during interfacial separation a considerable amount of energy is dissipated in deforming the adjacent materials. As a result, different experimental set-ups do not provide consistent measurements of the work of adhesion, since the measured fracture energy typically consists of a contribution from (1) actual debonding at the surface, (2) dissipative deformation in the bonded material adjacent to the interface and (3) the interaction between both previous mechanisms. As a result, different values are reported in the literature for exactly the same material system. Even though the physicochemical bonding is largely identical in each of the tests used, the microscale dissipation enforced in each may be substantially different.

5. Conclusion In this paper, a combined numerical–experimental approach is successfully employed to characterize delamination in polymer coated steel. With respect to the delamination analysis, the following conclusions are drawn:

The peel test is a suitable technique to study the delamination of a PET coating from a steel substrate. Experimental results on the as-received material have shown that: (1) There are no measurable residual stresses present

in the coating. (2) The coating thickness varies considerably across the polymer coated sheet. (3) The coating is isotropic in-plane. (4) Submersion of PET in diluted hydrochloric acid does not influence the mechanical properties. Due to the large inelastic deformations of the coating during a peel test, it is important to establish an accurate constitutive model of the PET. Therefore, a non-linear visco-elastic constitutive model is adopted, for which the material parameters have been determined from a variety of tests in conjuncture with simulations. The identification of the viscosity of the rejuvenated state is achieved with a single material point model, which saves computational time. Strain rates in a uniaxial tensile test are inhomogeneous and particularly high in the transition zone (_ ¼ 7:46 s1 ). The latter necessitates the use of a fine discretization of the finite element models, with elements of less than a micron, to capture these high strain rates. The interface is described by a cohesive zone model that is based on a large displacement formulation. The description is extended with a mode-mixity parameter to enable discrimination between different opening modes. An extra parameter controls the ratio between the work-of-separation in mode-I and mode-II. It has been shown that a plane-strain model suffices to simulate the peel tests, since the difference with a 3D model in the predicted normalized peel force is only 0.2%. 0 and 90 peel tests are performed and the normalized peel force is used to determine the interface parameters with an inverse parameter identification procedure.

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Within this context, a number of somewhat more general original contributions have been realized, among which:

New experimental insights in the physical and mechanical behaviour of coating, interface and laminate. New insights in the computational assessment of the coating–interface–substrate system under complex, yet realistic, loading conditions.

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