Introduction

Rubbers, or elastomers, have exotic mechanical properties, such as entropic elasticity, viscosity and incompressibility. Due to such properties rubbers find various industrial applications, e.g., tyres, dampers, etc. From the viewpoint of industry, it is important to understand the fracture mechanism of rubbers for better design of reliable rubber products. The relationship between the velocity of the crack propagation and the energy release rate has been vigorously investigated as a fundamental property to evaluate how easily rubber products are damaged under external loading. A large number of studies have been carried out to investigate the dynamic crack propagation in rubber materials, including experiments1,2,3,4,5,6, theoretical studies7,8,9,10,11 and numerical analyses12,13,14,15,16,17,18.

Some experiments for the crack propagation in rubbers reported an interesting phenomenon: The crack velocity shows a discontinuous jump as energy release rate increases, which is called a “mode transition”5,6. There are two mode regions, namely the “slow mode” with low crack velocity at low energy release rates and the “fast mode” with high crack velocity at high energy release rates. The two propagation modes show distinct features such as roughness on the crack surface, which implies highly complicated physics behind the mode transition. The mode transition phenomenon is industrially important because it is empirically known that the property of the mode transition is related to lifetime of rubber products; i.e., the higher the transition energy (energy release rate at the mode transition) becomes, the higher durability the rubber has. There is another kind of transition of the crack propagation in rubber materials that occurs at the vicinity of the sound speed (subsonic-supersonic transition) and has been investigated intensively16,19,20,21,22. The mechanism of the subsonic-supersonic transition is theoretically well explained16,20 and this phenomenon has been reproduced successfully by numerical simulations16,22. On the other hand, the mechanism of the slow-fast mode transition, which is found around the crack velocity much lower than the sound speed in the material, has not been revealed because of the lack of a simple model to reproduce the phenomenon. While Carbone and Persson11 have theoretically explained the discontinuity of the mode transition through the effect of the locally high temperature (“hot crack”), their explanation stands only when the crack grows very fast and, in reality, the mode transition can be observed at a low crack velocity. In addition, the hot crack model presupposes a quite high temperature on the crack tip. Thus, there is still room for discussion about the mechanism of the slow-fast mode transition on the dynamic crack propagation in rubbers.

Furthermore, no numerical simulation has successfully explained or reproduced the mode transition phenomenon thus far while some simulations have been performed for the dynamic crack propagation in rubber materials12. It is also unknown what relationship exists between the transition energy and the mechanical properties. It is of great importance to reveal the mechanism of the mode transition for industry because it would lead to new designs for rubber materials with improved durability.

In this study, we aim at revealing the nature of the mode transition phenomenon by performing numerical simulation based on the finite element method (FEM). This paper demonstrates that, with proper descriptions of hyperelastic and viscoelastic properties, it is possible to realise the mode transition phenomenon within the framework of FEM. It is observed how the elements on the crack path act during the crack propagation and the viscoelastic behaviours are compared between the slow and fast modes. Then we discuss the relationship between the mode transition phenomenon and the mechanical response at the crack tip.

Results

Crack velocity

Figure 1(a) shows the relationship between the crack propagation velocity ν and the energy release rate G obtained by the FEM simulation, compared with an experiment6 (Fig. 1(b)). It was observed that the crack velocity changes discontinuously around the energy release rate of G ≈ 42 kJ/m2, which is referred to as the transition energy Gtrans. This jump corresponds to the slow-fast mode transition observed in experiments5,6. The crack velocity changes at the transition by nearly 2 orders of magnitude in this simulation. To the best of our knowledge, this is the first FEM result ever that has succeeded in explicitly reproducing the mode transition in the crack propagation.

Figure 1
figure 1

Crack velocity as a function of energy release rate obtained by (a) FEM simulation in this study and (b) experiment by ref. 6. The transition energy Gtrans indicates the energy release rate at the mode transition. Note that the figures are plotted on a different scale.

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On the whole, the simulation result is not in a quantitative agreement with the corresponding experiment. The crack velocity and the transition energy are overestimated in the simulation, e.g. Gtrans ≈ 11 kJ/m2 in the experiment6. Note that our FEM results inevitably include a quantitative deviation from experiments because of the finiteness of elements, especially in the vicinity of the crack tip. Moreover, the criterion of fracture, which we assume is stipulated by the critical principal stress, may be too simple to achieve a quantitative agreement with the experiments. In this study, we basically focus on the qualitative aspect of the simulation results because, as is explained in the Discussion section, the qualitative aspect is helpful enough to understand the basic mechanism of the mode transition.

Crack profile

It was observed that the crack propagated along a straight path in the x-axis without any oscillation or roughness regardless of the mode type while an experiment reported the formation of rough crack surfaces in the slow mode and smooth crack surfaces in the fast mode5. This disagreement is basically likely to be attributed to the finiteness of the elements (aligned along the crack path) and the size of elements, which is not fine enough to describe rough crack surfaces.

The shape of the crack tip was observed to become sharper as a larger strain is applied. Figure 2 shows the shapes of the crack tip under two different energy release rates, i.e., different applied strains εload. While the crack shape is blunt at the lower energy release rate (Fig. 2(a)), the shape becomes sharp at the higher energy release rate (Fig. 2(b)). This result is in a qualitative agreement with experiments6. It may be worth noting that the change of the crack shape is not discontinuous at the mode transition point in contrast to the clear discontinuity in the crack velocity.

Figure 2: Shape of propagating crack tip at two different energy release rates, G.
figure 2

(a) G = 27 kJ/m2 and (b) G = 56 kJ/m2.

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In the slow mode, it was observed that the crack propagates repeating the cycle of two states, namely the state with considerable energy dissipation right after element deletion and the state of a blunt crack after energy dissipation. This observation indicates that fracture in the slow mode occurs basically after energy dissipation nearly vanishes. In other words, elasticity mainly contributes to stress on the crack-tip elements at the moment of fracture rather than viscosity does. On the other hand, in the fast mode, crack propagates steadily, which suggests that both elasticity and viscosity contribute to stress on the crack-tip element at the moment of fracture.

Mechanical response of crack-tip element

Figures 3(a,b) show the maximum principal stress exerted on the elements on the crack path in the slow and fast modes, respectively. In both modes, the principal stress on the element increases suddenly when the element faces the crack tip (shown as t = t0 in Fig. 3). After a certain time, the principal stress reaches the critical stress σc and then drops instantly to naught because the element is deleted. It is important to note the following characteristic difference between the two modes: In the slow mode, stress on a crack-tip element rapidly increases and reaches a local maximum point. Then the stress exhibits a slight decrease and turns to increase again until fracture. In the fast mode, in contrast, the stress at the crack tip exhibits a monotonic increase until fracture. Moreover, the two cases of the slow mode presented in Fig. 3(a) exhibit temporal development of the stress similar to each other. The difference in the mechanical response around the crack tip plays an essential role in the mode transition.

Figure 3: Maximum principal stress on an element on the crack path as a function of time.
figure 3

(a) Slow mode (G = 27 and 41 kJ/m2). (b) Fast mode (G = 42 and 56 kJ/m2). The element faces the carck tip at time t = t0. Note that the magnitude of absissae in figures are different by about 2 orders.

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Discussion

Let us consider the mechanical response of the element at the crack tip to understand the mechanism of the mode transition in this simulation. Here, we suppose the following process:

  1. 1

    Two elements, A and B, on the crack path are considered. The element B locates on the crack tip and the element A sites next to the element B at the initial state (Fig. 4(a)). The elements A and B have been fully relaxed at time t < t0. The element A stores stress σ = σ0 at this state.

    Figure 4: Schematic illustration of elements in the vicinity of crack tip.
    figure 4

    (a) The element A, next to the crack, holds stress σ = σ0 at t = t0. (b) The crack tip element B is deleted at t = t0. (c) At t > t0, the structural relaxation proceeds to the new equilibrium state, where the element A holds σ = σ1.

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  2. 2

    The element B is broken (deleted) at time t = t0 (Fig. 4(b)). The structure has not yet been relaxed at this moment.

  3. 3

    The element A, which newly faces the crack tip, gets relaxed to the new equilibrium state with stress σ = σ1 (Fig. 4(c)).

Furthermore, the stress and strain are regarded as scalar values for simplification, which should be reasonable since the load is uniaxial. As shown in Fig. 3(a), the element A is expected to undergo a local maximum of σ(t) at t > t0.

If σ does not reach the fracture criterion (σ1 < σc), no more crack propagation occurs (Fig. 5(a)). As increasing strain εload, σ reaches σc after a certain time Δt passes (Fig. 5(b)); i.e., Δt gives the time necessary for fracture of one element. Then the crack velocity is obtained as le/Δt, where le denotes the element length along the crack path. Under a certain value of strain, the local maximum reaches σc, where the time necessary for fracture jumps from Δt to Δt′ (Fig. 5(c)). Around this transition point, both Δt and Δt′ are theoretically possible and either of them actually (in reality) occurs according to various factors of perturbation such as inhomogeneity of a specimen. Thus, the crack propagation mode can be easily switched from the slow mode to the fast mode and vice versa. This is presumably the reason why the stick-slip behaviour is observed in the experiments. Indeed, an experiment provided an evidence to regard the stick-slip behaviour as a continual alternation between the slow and fast modes5. Under higher strains, the fracture is expected to occur at the local maximum, as shown in Fig. 5(d), leading to the fast mode crack propagation. Interestingly, the basic mechanical response at the crack tip is likely to be common to both the modes; the difference between them is only whether fracture occurs after or before reaching the local maximum in σ(t), shown in Fig. 5.

Figure 5: Schematic illustration of the mechanical response of the element on the crack tip.
figure 5

The characteristics of the curve shape are exaggerated for clarity. (a) No fracture occurs under very low tensile load εload. (b) Fracture occurs after undergoing local maximum point (slow mode). (c) Local maximum in the stress curve reaches the critical stress and the mode transition occurs. (d) Fracture occurs within local maximum point (fast mode). Fracture time per element is shown as Δt and Δt′.

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The discussion above is basically based on simple hypotheses such as hyperelasticity, linear viscosity and a fracture criterion based on the principal stress. Thus, those factors are regarded as the sufficient condition for the mode transition phenomenon to occur. The probable least condition required is the distribution of the relaxation time in the representation of viscosity. If there exists a unique relaxation time that dominates all the relaxation behaviour of stress and strain, then the local maximum point is unlikely to appear.

It is found that the observed transition phenomenon is distinguished from the subsonic-supersonic transition through some striking differences between the phenomena: The observed transition is characterised by the discontinuous change of the crack velocity as explained above. Furthermore, in the present analysis the crack velocity at the transition point is sufficiently slow compared with the sound speed. For example, the lowest crack velocity at the fast mode measures 14 m/s (Fig. 1(a)), while the sound speed cs of rubber materials is typically about 50 m/s (according to the evaluation from the first-order shear modulus μ(1) and mass density ρ, m/s for the compound under consideration6). In contrast to our FEM simulation results, the subsonic-supersonic transition is a continuous change of the crack velocity at the vicinity of the sound speed, as shown in experiments and numerical analyses (see e.g. Fig. 3 in ref. 22).

There are some non-trivial factors that were omitted in the discussion above; e.g., energy dissipation at the process zone, inertia effect, finite deformation, etc., which are implicitly included in the FEM simulation. Obviously, the factors can quantitatively contribute to the crack propagation behaviour. Nevertheless, they can be regarded as marginal factors when we focus on the essential mechanisms of the mode transition. It is important to note that the mode transition can be successfully explained through the characteristic mechanical response on the crack-tip element despite the omission of those factors. Besides the factors above, the present FEM simulations do not treat other possible factors, such as temperature distribution, plastic deformation and the Mullins effect23,24. It will be necessary to include those effects in order to perform more realistic simulations.

While the mode transition phenomenon was clearly explained through the characteristic mechanical response at the crack tip, it is necessary to thoroughly reveal the origin of such mechanical response. In addition, there is room for investigation in quantitative aspect of the mode transition, e.g., evaluation of the transition energy. These subjects will be our future work.

Method

Constitutive model

The constitutive model of rubber material in this study consists of hyperelasticity and linear viscosity. The hyperelastic and viscous terms represent the static and rate-dependent responses, respectively. The total stress tensor is given by the sum of both contributions (referred to below as and ). Hyperelasticity independent of rate effect is described by the Ogden model25. The Ogden model gives the strain energy density W as follows:

where K and J denote the bulk modulus and the relative volume, respectively. The bulk modulus K was determined through the Poisson’s ratio, which was set to 0.495 in this study. The series of denotes the principal stretch divided by J. Material parameters μi and αi are empirically determined. In this study, the number of terms for the Ogden model, n, was set to 3. The strain density function gives the second Piola-Kirchhoff stress tensor of hyperelasticity, , as follows:

where Eij denotes the Green-St. Venant strain tensor. The effect of viscosity on stress, , is given by the following convolution integral;

where g(t) denotes the relaxation function. The Einstein’s summation convention is applied for indices k and l. We adopted the Prony series for g(t), defined as

where series of , and βi are material parameters. The number of terms for the Prony series, m, was set to 12, in this study. The Prony series description is equivalent to the Maxwell model.

The material parameters in the constitutive equations were determined in such a way that they reproduce the uniaxial stress-strain curve at a low strain rate and the stress relaxation curve obtained by experiments for a cross-linked acrylonitrile-butadiene rubber (35% acrylonitrile content) with carbon black filler (0.05 volume fraction) shown in ref. 6. The determined material parameters are listed in Supplementary Table S1. It was confirmed that both the uniaxial stress-strain relationship and the stress relaxation function were reproduced well by the constitutive model in the wide range of strain and time (see Supplementary Fig. S1).

The fracture criterion was given by the maximum principal stress. If an element holds a principal stress as high as the given critical stress, σc, then the element is deleted to realise fracture. The critical principal stress was estimated through the experimental result of uniaxial tensile strength under the quasi-static condition. Since the experimental strength was measured as the nominal stress, it was converted into the true stress on the supposition that the volume is constant under the quasi-static uniaxial tension.

In our model, the Mullins effect23,24, which is known as the hysteresis effect of mechanical properties observed in rubber materials, was not considered for simplification.

Simulation setup

We carried out FEM simulations of dynamic crack propagation mimicking the pure shear test, where the crack is expected to propagate at a constant velocity under a given strain. We adopted the FEM simulation package LS-DYNA (Livermore Software Technology Corporation).

According to Rivlin and Thomas1,26, the energy release rate in the pure shear test, G, is given as the product of the strain energy density W and the width of the unstrained specimen, h0;

Equation 5 indicates that G is simply given as a function of applied strain, εload. Here, W is calculated with the Ogden model (1) under the pure shear condition; i.e., .

The length (in the x-axis), the width (in the y-axis) and the thickness (in the z-axis) of the specimen were set to 180 mm, 20 mm and 1 mm, respectively. The dimension of the elements on which the crack was expected to propagate was set to 0.50 mm, 0.19 mm and 0.20 mm in the x-, y- and z-axes, respectively. As explained in the previous section, the elements were to be deleted when the maximum principal stress reaches the critical stress, σc = 56 MPa, to describe the crack propagation.

Figure 6 schematically shows the procedure of simulation. Tensile strain εload was loaded in the y-direction by controlling the displacement on the top xz-plane at a constant strain rate s−1. The degrees of freedom regarding the displacement along all the directions on the bottom xz-plane and those along the x- and z-directions on the top xz-plane were fixed during the whole simulation. The degree of freedom along the y-direction on the top plane was also fixed after the tensile strain control.

Figure 6: Schematic illustration of simulation procedure.
figure 6

A specimen is set thin along the z-axis to realise the pure shear condition. Firstly, a specimen is stretched along the y-axis to a predetermined strain. Secondly, initial crack is introduced by removing elements. Then crack starts propagating along the x-axis and the velocity of crack propagation is measured. The relationship between the crack velocity and the applied strain by repeating this procedure with various strains.

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Right after reaching the predetermined strain, the initial crack was introduced with a length of 10 mm in the x-direction by removing elements. When the crack started propagating spontaneously and steadily, the rate of propagation (crack velocity) was measured. Since the energy release rate G can be calculated with the applied strain εload, we obtained the relationship between the energy release rate and the crack velocity.

Additional Information

How to cite this article: Kubo, A. and Umeno, Y. Velocity mode transition of dynamic crack propagation in hyperviscoelastic materials: A continuum model study. Sci. Rep. 7, 42305; doi: 10.1038/srep42305 (2017).

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