Simple crack propagation model of pressure-sensitive adhesives

The cavity volume is governed by the cavity dynamics and it is described by the Rayleigh–Plesset equation [3]

$$\begin{equation} \frac{{{\rm{d}}R_{\! j} }}{{{\rm{d}}t}} = \frac{{R_{\! j} }}{{2\eta }}(P_{{\rm{cav}},j} - P_{\! j} ) - \frac{\gamma }{{2\eta }}, \end{equation} \tag{ 1 }$$

where R_j and P_cav,j are radius and pressure of the cavity in the jth block, and P_j is the pressure of the jth block, γ is the surface energy per unit area and η is the viscosity of the adhesive material.

Let us focus our attention on the jth block. Initially the block has width W₀ = L₀ /N_block and height H₀. When the adhesive layer is stretched by a factor λ in z-direction, the height and the width of this block are, assuming uniform deformation, changed to H = λH₀ and W = W₀ /λ, respectively (figure 2). Here, (X,Z) are the coordinates of the center of gravity, and X_s is the coordinate of the central position of the surface.

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In Yamaguchi's model, the block is affected by two kinds of deformation, i.e. the elongational deformation and the shear deformation. The elongational deformation is described by a parameter λ, while the shear deformation is described by another parameter C, which characterizes the parabolic deformation like the Poiseuille flow of the block. This parameter C is proportional to the curvature of the side surface of the block.

Usually, the adhesive materials have highly nonlinear viscoelasticity and have multiple relaxation times. However, for simplicity, we adopt a simple model, called the upper-convected Maxwell fluid model with a single relaxation time [4, 5]

$$\begin{eqnarray} \frac{{\partial \sigma _{ij} }}{{\partial t}} &=& - v_k \frac{{\partial \sigma _{ij} }}{{\partial x_k }} - \frac{1}{\tau }\sigma _{ij} + \left( {\frac{{\partial v_i }}{{\partial x_k }}} \right)^{\rm{T}} \sigma _{kj} + \sigma _{ik} \left( {\frac{{\partial v_j }}{{\partial x_k }}} \right)\nonumber\\ && + G\left[ {\frac{{\partial v_i }}{{\partial x_j }} + \left( {\frac{{\partial v_i }}{{\partial x_j }}} \right)^{\rm{T}} } \right], \end{eqnarray} \tag{ 2 }$$

where σ_ij is the stress tensor, ∂v_i /∂x_j is the velocity gradient tensor and G is its elastic modulus.

In this case, the relaxation time is determined by the following relation:

$$\begin{equation} \tau = \frac{\eta }{G}, \end{equation} \tag{ 3 }$$

where η is the viscosity of the PSA.

On peel tack test, one side of the substrates, which are near the boundary where the external strain is imposed, deform the flat layer in initial state to the curved one in deformation state (see figure 1(b)). Using the terminology of differential geometry, radius of surface curvature of the layer is changed from infinite to finite value near the external strain imposed boundary. We assume that dissipation of energy required for the surface deformation is proportional to the curvature at that point:

$$\begin{equation} \eta \frac{{{\rm{d}}\lambda _j }}{{{\rm{d}}t}} = \gamma \frac{{{\rm{d}}^2 \lambda _j }}{{{\rm{d}}x^2 }}. \end{equation}\noindent \tag{ 4 }$$

The rhs of this equation expresses the curvature elasticity [6].

To describe rupture process, we introduce a condition for the rupture formation given by the following equation:

$$\begin{equation} \lambda _j > \lambda _{{\rm{limit}}} , \end{equation} \tag{ 5 }$$

where we assume that this block ruptures into two pieces with the same size if the elongation ratio of this block λ_j is larger than the artificial threshold value λ_limit.

After this rupture process, the lower surface of this block, which is newly created through the rupture process, shows the same motion as the free surface according to the surface tension, as was described above.

To consider the effect of cavity, we compare two cases, i.e. CASE1 with a cavity and CASE2 without cavity. Figure 4 shows the stress–strain (S–S) curves for CASE1 and CASE2. We can understand that the cavity has a very important effect on the rupture process by weakening the stress of the adhesive layer.

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Figure 5 shows S–S curves for various values of the thickness (table 2) of the adhesive layer. We observed the following properties:

• 1.  
  the initial stage is strongly affected by the thickness of the adhesives,
• 2.  
  the peak value of the stress decreases as the thickness of the adhesives increases,
• 3.  
  the peak position is dependent on the film thickness.

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From a detailed calculation of upper convected Maxwell model (equation (2)), we get the following result:

$$\begin{eqnarray*}&&\frac{{\partial \sigma _{{\rm{s}},j} }}{{\partial t}} = - \frac{{\sigma _{{\rm{s}},j} }}{\tau } + \frac{{4\skew3\dot C}}{{H_0 \lambda }}(\sigma _{zz,j} + G), \end{eqnarray*}$$

where σ_s,j is shear stress at the surface of the j-the block. As the initial state is stress free (σ_s,j = 0 ), we then obtain

$$\begin{equation} \frac{{\partial \sigma _{{\rm{s}},j} }}{{\partial t}} \cong \frac{{4\skew3\dot C}}{{H_0 \lambda }}(\sigma _{zz,j} + G). \end{equation} \tag{ 6 }$$

Therefore, the thinner the thickness of the adhesives H₀, the larger the rate of stress increases.

Figure 6 shows S–S curves for various values of separation speed (table 3) of the substrate. We observe that the peak value of the stress increases as the separation speed $\dot \lambda$ increases.

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Table 3. Used parameters for various friction separation speeds.

CASE	$\dot \lambda$
CASE 1	0.1
CASE 5	0.5
CASE 6	0.02

Higher separation rate causes larger shear stress and larger resistance for the cavity expansion, leading to a larger negative pressure. To understand this effect, the Rayleigh–Plesset equation (equation (1)) is rewritten in the following form:

$$\begin{eqnarray*}&&P_j = P_{{\rm{cav}},j} - \left( {\frac{{2\eta \dot R_{\! j} }}{{R_{\! j} }} + \frac{\gamma }{{R_j }}} \right). \end{eqnarray*}$$

Among the two terms in the parentheses on the rhs, the first term describes the dissipation phenomena and the second term expresses the surface tension. In this case, R_j is small for a long time because of the high separation speed. These two terms lead to a large negative pressure.

Figure 7 shows S–S curves for various values of the friction coefficient (table 4) at the interface between adhesives layer and the substrate. The shear stress acting on the interface is proportional to the slip velocity with a coefficient μ:

$$\begin{equation} \sigma _{{\rm{si}}} = \mu \dot X_{{\rm{si}}}. \end{equation}\noindent \tag{ 7 }$$

Comparing equation (6) with equation (7), we understand that the stress increasing rate is proportional to the friction coefficient:

$$\begin{eqnarray*} &&\dot \sigma _{{\rm{si}}} \propto \mu. \end{eqnarray*}\noindent$$

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