# Adhesion along metal–polymer interfaces during plastic deformation

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## Abstract

In this paper a numerical study is presented that concentrates on the influence of the interface roughness that develops during plastic deformation of a metal, on the work of adhesion and on the change of interface energy upon contact with a glassy polymer. The polymer coating is described with a constitutive law that mimics the behavior of Poly-Ethylene Terephthalate. It includes an elastic part, a yield stress, softening and hardening with increasing strains. For the interface between the metal and the polymer a mixed-mode (mode I and II) stress-separation law is applied that defines the interface energy and an interaction length scale. At the onset of deformation the surface of the substrate has a self-affine roughness characterized by the so-called Hurst exponent, a correlation length and an rms roughness amplitude, that evolves as a function of increasing strain. The findings are the following: the interface energy decreases until the strain at yield of the polymer coating. Interestingly, after yielding as the polymer starts to soften macroscopically, the decreasing average stress levels result in partial recovery of the interface energy at the interface. At higher strains, when macroscopic hardening develops the recovery of the interface stops and the interface energy decreases. The effect of coating thickness is discussed as well as the physical relevance of various model parameters.

## Keywords

Shear Band Interface Energy Polymer Coating Cohesive Zone Hurst Exponent## Introduction

Polymer-coated metal sheets are rather recent products of steel manufacturers that are used in various applications in food and automotive industry. In the manufacturing process severe plastic deformation is used to obtain the final shapes of the end products. A drawback of plastic deformation is the intrinsic roughening of the surface of the metal caused by dislocation activity. This paper concentrates on the implications of the roughening process for the mechanical properties of the combined metal–polymer system. Clearly the subject is closely related to that of a large number of papers discussing the impact of roughness on the work of adhesion *W*, e.g [1, 2, 3, 4, 5, 6, 7]. However, this paper sets itself apart from earlier research because it emphasizes on: First, the evolution of roughness of the metal as a mechanical loading mechanism of a metal–polymer interface; second, the coupling between the metal substrate and the polymer coating using a stress-separation law; and third, the polymer behavior including yielding, softening and hardening. In the following these points are briefly discussed in the framework of the current understanding.

*C*(

*r*,ɛ) of the interface may be defined as [8]

*w*above a certain lateral correlation length ξ, and a Hurst exponent

*H*characterizing the self-affine geometry below ξ. For a uniaxially deforming metal these surface parameters are a function of the applied strain \({\varepsilon}\). From experiments parameterized fits for \({w(\varepsilon)}\), \({H(\varepsilon)}\) and \({\xi(\varepsilon)}\) may be determined (see also [9, 11]).

The topics addressed here are the work of adhesion *W* between such a self-affine roughening metal–polymer system, the dependence of *W* on the parameters *w*, *H* and ξ, and the evolution of *W* as a function of uniaxial strain \({\varepsilon}\).

*W*is the subject of this paper.

*m*) and a polymer (

*p*) coming into contact across an area

*A*

_{0}the work of adhesion

*W*per unit area is defined as:

_{m}and γ

_{p}represent the surface energies and γ

_{pm}the interface energy. During uniaxial deformation at a strain\({\varepsilon}\) the nominal (projected) contact area

*A*

_{nom}is given by:

*A*

_{real}at a strain \({\varepsilon}\) is larger than

*A*

_{nom}(\({\varepsilon}\)) and is given by

*d*is the layer thickness, \({E_{\rm pol}}\) is the Young’s modulus of the polymer. Due to the substantial difference of the elastic moduli the metal can be regarded as a rigid solid and is omitted from the energy balance.

Second, \({U_E^h (\varepsilon)}\) represents the elastic energy contribution of the roughening interface.

The term \({G(\varepsilon)=\int {G(\vec{r})}dA(\vec{r})/A_{\rm nom}}\) takes into account the two competing effects of roughening on the interface energy, on the one hand an increase in real contact area, and on the other hand local delamination caused by the stresses acting on the interface.

*W*enters in a natural way as

*G*

_{0}again contains all surface and interface terms. One can now also define

The separate contributions of interface and bulk to the work of adhesion and their dependence on the strain are a key issue. In the following our numerical approach is described that takes into account all the aforementioned features.

## Model description

The numerical model describes the interface between steel and PET and it will capture the following three aspects: Polymer deformation; roughening of the metal surface; interaction between metal and polymer across the surface. The representation of the PET layer is a finite element model built from quadrilaterals representing a size of 1 × 1 μm^{2} [12, 13]. The initial length of the system is 2049 μm and the initial thickness *h* of the PET film is 60 μm unless mentioned otherwise. In the calculations the ‘metal’ substrate is taken to be rigid. This is a reasonable approximation, since the elastic modulus of steel is typical of the order of 200 GPa and therefore the thin polymer coating is expected to have little effect on its roughening behavior.

### Polymer constitutive behavior

### Roughening of the metal surface

One of the simplifications in the approach proposed here is to parameterize the roughness evolution of a metal surface as a function of strain, and to assume that the roughness evolution at a polymer–metal interface is essentially identical to this because of the large difference in elastic moduli of typical metals (∼100 s of GPa) and glassy polymers (∼1 GPa).

*C*

_{1}= 6.1 and \({\xi_{0}=35}\) μm. The Hurst exponent

*H*was found to be insensitive to strain and is taken to be constant \(H(\varepsilon)= H_{0}= 0.6\). The correlation length

*ξ*increases with the strain in the tensile direction. A qualitatively similar behavior was also found for other materials (Fe, Al, [10]). From experimental results least square fits to Eq. 10 were performed [9]. A recursive refinement algorithm [11] was used to simulate surfaces with the characteristics described by Eq. 10. A detailed example of the roughness evolution in the numerical model is displayed in [9], a few stages of which are apparent from Fig. 3.

### Interaction between metal and polymer

^{2}, working distance \({\Delta_{0}=300}\) nm and σ

_{ n}

^{max}= 36.8 MPa. To illustrate the key features of the cohesive zone \({\sigma_n(\Delta_n,0)}\) is shown in Fig. 4.

*G*

_{0}

^{CZ}refers to

*G*

_{0}of the cohesive zone (see Eq. 8), \({\sigma_{i,n}^{\rm CZ}}\) is the stress normal and \({\tau _{i,t}^{\rm CZ}}\) the stress parallel to the interface element

*i*. In this work we are interested in \({w(\varepsilon)}\) and therefore in \({G(\varepsilon)}\) and \({U(\varepsilon)}\) . In the context of the numerical model we define \({G(\varepsilon)}\) as follows:

*i*running over the discrete elements in the model, \({A_i (\varepsilon)}\) and \({G_i^{\rm CZ}(\varepsilon)}\) are the surface area and the interface energy of the i-th element, respectively.

Since all energies in the following are derived from the numerical model the superscript CZ will be dropped.

### Simulation of deformation

*x*= 0 displacements along

*x*are imposed while displacements along

*y*are free. Similar boundary conditions are applied at

*x*=

*L*(

*ɛ*). Along the interface displacements in the substrate are constrained in all directions. Displacements in the polymer are not restricted and coupled to those in the substrate by the stress-separation laws incorporated in the cohesive zone. At each strain step the positions of the substrate nodes are updated in

*x*(to reflect the increase in strain) and

*y*(to reflect the increase in roughness).

## Results

A typical result of the calculation is shown in Fig. 3. The figure shows \({\sigma_{xx}}\) distribution in the PET at strains for three different stages depicted in Fig. 2: elastic I, softening II, and hardening III. Clearly visible are localized shear bands and roughening of the polymer surface during the softening and hardening regime.

A series of calculations was carried out in which the parameters describing the roughness and the parameters describing the cohesive zone were varied. For these calculations \({w(\varepsilon)}\) and \({G(\varepsilon)}\) were studied and are discussed below.

### General characteristics of *G*(\(\varepsilon\))

*G*

_{0}.

Figure 7a shows the reduction in interface energy as indicated in Eq. 14. This effectively removes the effect of increasing area and gives an indication of the decrease in
\(G(\varepsilon)\)caused by the loading of the interface. At the onset
\(\Delta G(\varepsilon)\) decreases rapidly (approximately linearly with *w*^{2}). At the yield point of the polymer the rate of decrease of
\(\Delta G(\varepsilon)\)reduces considerably, and in fact for low \({w_{\rm sat}}\)the interface partly recovers due to the softening of the polymer layer, and the reduced stresses acting on the interface as a cause of that.

Figure 7b shows \(A_{\rm real}(\varepsilon)/A_{\rm nom}\)which exhibits a monotonous increase due to roughening up to a maximum where the increase in nominal area starts dominating the effects of the increase of \({w(\varepsilon}\)).

*G*(*ɛ*) as function of* w*,* H* and \({\xi}\)

*H*. A low value of

*H*results in rapid fluctuation on a short length scale, and this leads to a larger decrease in \({G(\varepsilon)}\) at low strains (Fig. 8a) and to a continuing delamination above the yield point. However, the overall effect is still compensated by the faster increase in surface area (Fig. 8b).

*W*(*ɛ*) as a function of layer thickness

*G*is qualitatively similar. Figure 10 shows \({W(\varepsilon)}\) up to the strain at yielding. \({W(\varepsilon)}\) equal to zero indicates that the interface becomes metastable to fracture. Clearly, \({{U_E(\varepsilon)}/{A_{\rm nom}(\varepsilon)}}\) dominates \({W(\varepsilon)}\). For the 10 μm coating \({W(\varepsilon)}\) decreases to a value of about 0.6 at the yield strain, indicating that this coating is stable against delamination. \({\frac{U_E(\varepsilon)}{A_{\rm nom}(\varepsilon)}}\) increases linearly with

*d*, and for the coatings of 30 μm and of 60 μm the energy \({W(\varepsilon)}\) turns out to be smaller than zero and therefore these situations are metastable.

## Discussion

Another interesting aspect is the relation to analytical results relating \({G(\varepsilon)}\) to the roughness of interfaces. In the literature analytical studies have been reported for situations [15, 16] in which a flat elastic body is brought into perfect contact with a rigid rough body (see appendix A for a brief overview of these treatments).

*increase*of the change of interface energy upon contact. On the other hand for complete contact to occur elastic energy is stored in the material and this contributes to a

*decrease*in interface energy. Depending on the properties of interface, substrate and polymer (geometric as well as elastic) one of these two effects dominates and in fact a critical modulus

*E*

_{c}can be defined for the polymer layer that separates these two regimes [15]. For

*E*

_{c}it is found that

*r*

_{min}represents the smallest length scale in the system (in our case 1 μm). In the case of PET and steel and for a substrate geometry typical for the ones discussed here we find \({E_{\rm c}\sim 70}\) MPa which means that

*E*

_{PET}>>

*E*

_{c}. So, based on the analytical approach the interfaces are expected to show a

*decrease*in interface energy for increasing roughness. An expression for \({G(\varepsilon)/G_0}\) is given in [15] (see Appendix A for a derivation and a description of the assumptions):

*E*that reflects the elastic properties of both the cohesive zone and the polymer coating. In the graph a value of 118 MPa is used.

The figure shows that the analytic result (indicated by a drawn line) predicts a monotonous linear decrease of \({G(\varepsilon)/G_0}\) as a function of \({w(\varepsilon)^2}\). We note that the existence of an enveloping curve (indicated by a dash-dot line in the figure) may be inferred from the numerical simulations (also shown in Fig. 11). This shows that as long as the PET is in the elastic regime the interface energy depends only on \({w(\varepsilon)^2}\) which is in qualitative accordance with the analytical results. A difference in this respect is the occurrence of a non-linear regime at low strains for the numerical solutions which is due to the description of the interface with a stress-separation law with a certain working distance. Decreasing the interaction distance for the cohesive zone will lead to a closer correspondence with the analytical result.

Deviation from the enveloping curve occurs for all numerical calculations as soon as the average strain in the PET reaches the yield strain. The value of \({w(\varepsilon)^2}\) at which this occurs depends on \({w_{\rm sat}}\). It can be seen that for small \({w_{\rm sat}}\) the deviations from the envelope curve occur for very low \({w(\varepsilon)^2}\). The softening that occurs in the PET above the yield strain leads to an increase or partial recovery of \({G(\varepsilon)/G_0}\), a behavior that differs drastically from what is expected from the analytical result.

## Conclusion

The following generic picture emerges that describes the energetics of a strained ductile glassy polymer layer with a roughening interface:

At the interface local delamination leading to a decrease in adhered area competes with roughening that leads to an increase in adhered area.

A decrease of the interface energy occurs in the regime where the polymer deforms elastically, a (partial) recovery occurs during the softening phase of the polymer followed by a renewed decrease during the hardening phase of the polymer coating.

For layers of practical thickness the elastic energy stored in the polymer coating by straining at the yield stress dominates the work of adhesion and the stability of the interface against delamination.

## Notes

### Acknowledgements

This work was financially supported by the Netherlands Technology Foundation STW (project number GTF.4901).

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