Dissipation during crack growth in a viscoelastic material from a cohesive model for a finite specimen

Abstract

In the present paper, we extend results recently given by Ciavarella et al. (J Mech Phys Solids 169:105096, 2022) to show some actual calculations of the viscoelastic dissipation in a crack propagation at constant speed in a finite size specimen. It is usually believed that the cohesive models introduced by Knauss and Schapery and the dissipation-based theories introduced by de Gennes and Persson-Brener give very similar results for steady state crack propagation in viscoelastic materials, where usually only the asymptotic singular field is used for the stress. We show however that dissipation and the energy balance never reach a steady state, despite the constant propagation crack rate and stress intensity factor. Our loading protocol permits a rigorous solution, and implies a short phase with constant specimen elongation rate, but then possibly a very long phase of constant or decreasing elongation, which differs from typical experiments. For the external work we are therefore unable to use the de Gennes and Persson-Brener theories which suggested that the increase of effective fracture energy would go up to the ratio of instantaneous to relaxed modulus, at very fast rates. We show viscoelastic dissipation is in general a transient quantity, which can vary by orders of magnitude while the stress intensity factor is kept constant, and is largely affected by dissipation in the bulk rather than at the crack tip. The total work to break a specimen apart is found also to be possibly arbitrarily large for quite a large range of intermediate crack growth rates.

1 Introduction

Crack propagation in viscoelastic materials is rate dependent, and the load to induce crack growth typically increases with the rate of crack propagation (Mueller and Knauss 1971). Gent and Lai (1994) study the rupture of elastomer sheets adhered together and show that if we vary by 9 orders of magnitude the peel rate, the apparent toughness increases by a factor of 100. Gent (1996) summarizes results for tearing, peeling and cutting of 3 elastomers and the force is found to span 5 orders of magnitude as the rate of rupture is increased, which is perhaps even higher than the increase of the elastic modulus between slow and fast rates. Master curves are typically used using the temperature-rate superposition principle, which confirms viscoelastic effects are the cause of the rate dependence of rupture propagation that is observed, although we cannot be sure if these viscoelastic effects are in the intrinsic bond scission near the crack tip, or dissipation in the bulk. Experiments of detachment of polymers from a substrate (Gent and Petrich 1969; Gent and Schultz 1972; Ondarçuhu 1997) exhibit rate phenomena similar to those observed for cohesive fracture (Mueller and Knauss 1971; Gent and Lai 1994; Gent 1996).

Kostrov and Nikitin (1970), McCartney (1977, 1980, 1981), Christensen (1979, 1980, 1981) and Christensen and McCartney (1983), Graham (1969), Barenblatt et al. (1970), Rice (1978), Goleniewski (1990), Hui et al. (2022) and others all make it clear that when there is no rupture process zone in the model for crack growth in a linear, viscoelastic material with a rate independent crack tip toughness, the applied Mode I stress intensity factor, $K_I$, is required to be equal to a critical value, $K_c$, where $K_c$ is consistent with

$$ {\Gamma } = \frac{{\left( {1 - \nu^2 } \right)K_c^2 }}{E_I } $$

(1)

where ${\Gamma }$ is the crack tip fracture toughness that is rate independent and constant, $\nu$ is Poisson’s ratio, also constant, and $E_I$ is the instantaneous Young’s modulus for the viscoelastic material. The only choice we have is to introduce a rupture process zone in the model and to use it to obtain the relationship between $V$ and $K_I$.

The first and still most general model of viscoelastic fracture, is the cohesive model (Knauss 1970; Knauss et al. 1973; Schapery 1975a, b, c; Rice 1978; Greenwood and Johnson 1981; Hui et al. 1992; Xu et al. 1992; Hui et al. 1998; Baney and Hui 1999; Rahulkumar et al. 2000; Lin and Hui 2002; Greenwood 2004; Wang et al. 2016 and others). In this model, we have a Barenblatt (1962) cohesive zone (Knauss et al. 1973), and in the simplest form that of a Dugdale (1960) cohesive zone where the cohesive strength is uniform, constant and rate independent. It is precisely this model which we shall use below. Such cohesive zone model has apparently been successful in the fitting of experiments of crack propagation in Solithane 50/50 (Knauss et al. 1973; Schapery 1975c).

The other model of fracture involves adding viscoelastic energy dissipation that occurs during crack growth, to explain the increase of the apparent toughness measured during crack growth (de Gennes 1996; Saulnier et al. 2004; Persson and Brener 2005; Persson 2017, 2021). In these models, the crack tip process zone is included via a limiting, or cut-off stress at the crack tip. This second class of models seem to generate similar conclusions that the cohesive models, at least for a semi-infinite crack with a process zone that is very small compared to all component dimensions (Ciavarella et al. 2021; Hui et al. 2022). However, the issue is more controversial in the case of finite component size effects (Ciavarella and Papangelo 2021; Persson 2017). It would seem that there is a limit to the possible size of the region of dissipation and Persson 2017 predicts that, at an intermediate crack propagation speed, the toughness will go through a maximum, and at higher rates of crack propagation the toughness will either decline or become insensitive to further increases in crack speed. This outcome has not yet been firmly established in treatments of crack growth in finite sized viscoelastic components. Indeed, in a solution for the double cantilever beam (Wang et al. 2016; Ciavarella et al. 2021), when the beams are thin in the bending direction there seems to be no reduction of the fracture load compared to the trend for a thick beam, contrary to the prediction of the de Gennes (1996) and Persson (2017) dissipation-toughening models. Also, for a finite specimen we have to consider that there are ample regions of nominal stress which can dissipate energy, which are not taken into account in the de Gennes (1996) and Persson (2017) theories. It is not clear if this dissipation should also be included in addition to the intrinsic fracture energy of the material. Hence, while it is demonstrated by Hui et al. (2022) the substantial equivalence of the Knauss et al. (1973), Schapery (1975a, b, c) and Persson and Brener (2005) models for the highly idealized semi-infinite crack under constant applied stress intensify factor, we will see that the practical realization of this model for a finite specimen is not obvious.

In the Knauss et al. (1973), Schapery (1975a, b, c) and Persson and Brener (2005) models of crack growth in viscoelastic materials, Hui et al. (2022) identify what they term the fraction of energy dissipated per unit area of crack extension, and attribute to it the role of toughening of the material during crack growth. They give it the symbol ${\Lambda }$. From their Eq. (8b), it enters the plane strain relationship

$$ {\Gamma } = \left[ {1 - {\Lambda }\left( {\frac{c}{Vt_c }} \right)} \right]\frac{{\left( {1 - \nu^2 } \right)K_I^2 }}{E_R } $$

(2)

where $c$ is the length of the cohesive zone at the crack tip, $t_c$ is the creep compliance retardation time, i.e., the time constant for viscoelastic deformation in a creep test, and $E_R$ is the Young’s modulus of the material in the relaxed state. The term outside of the square brackets on the right-hand side of Eq. (2) is the elastic energy release rate to the crack tip when the component behaves as an elastic one with Young’s modulus equal to its value in the relaxed state of viscoelastic behaviour. Indeed, it is often observed that, during fracture tests and adhesion detachment tests, the overall behaviour of viscoelastic components is consistent with the relaxed state of the material (Gent and Kinloch 1971; Mueller and Knauss 1971; Knauss et al. 1973; Maugis and Barquins 1980; Knauss (2015)).

Hui et al. (2022) assert that when the rupturing viscoelastic component overall is in the relaxed state, ${\Lambda }\left( {1 - \nu^2 } \right)K_I^2 /E_R$ is the energy per unit area of crack advance that is dissipated so that the total crack tip energy release rate, $\left( {1 - \nu^2 } \right)K_I^2 /E_R$, associated with the applied loads, is the sum of the dissipated fraction, ${\Lambda }\left( {1 - \nu^2 } \right)K_I^2 /E_R$, and the intrinsic toughness, ${\Gamma }$. However, it is not obvious to us that ${\Lambda }\left( {1 - \nu^2 } \right)K_I^2 /E_R$ is dissipated in contrast to simply not being available for energy flux to the crack tip. In addition, we are of the view that $\left( {1 - \nu^2 } \right)K_I^2 /E_R$ is not always meaningful as a measure of the crack tip energy release rate associated with the applied loads.

To explore and illustrate this situation, and to address questions that we have raised, we obtain below some exact results from linear elastic fracture mechanics converted to linear viscoelasticity under constant applied stress intensity by use of the correspondence principle (Graham 1968). Part of these results have been already illustrated in Ciavarella et al (2022a, b), but are repeated for clarity, adding some actual calculation and graphical results which help clarifying the meaning of these results. From these results, we compute the energy stored and/or dissipated while crack growth is occurring. We show that a high toughness in a viscoelastic material at a high rate of crack propagation is not due solely to viscoelastic dissipation, and, in some cases, not due at all to viscoelastic dissipation. Further, we find possibly orders of magnitude higher work rates than expected from the literature so far, occurring for realistic elastomers at intermediate speeds rather than at very fast ones. For a Maxwell material, we obtain an explicit result for the rate of energy dissipation in a strip in tension with an edge crack, and show that, per unit area of crack advance, the dissipation rate falls as the rate of crack propagation is increased. As such, these results run counter to the idea that increased dissipation leads to increased viscoelastic toughness for crack growth. Whereas Hui et al. (2022) have shown that the Persson and Brener (2005) model is effectively one in which a rupture process zone is interacting with the viscoelastic properties of the material, just as in the case of the Knauss et al. (1973) and Schapery (1975a, b, c) models, our results demonstrate that the Knauss et al. (1973) and Schapery (1975a, b, c) models, that utilize a Barenblatt (1962) cohesive zone for rupture at the crack tip, cannot be viewed as ones in which viscoelastic dissipation causes all of the increase in applied loads required to drive cracks to grow at high rates of extension.

We show that in a edge crack finite specimen, energy fluxes never reach a steady state even if we attempt to have a constant crack velocities. Only in some special geometries, like a DCB under remote moments (see Ciavarella et al. 2022a, b), or in a pure shear geometry (see Shrimali and Lopez-Pamies 2023) we could reach a true steady state after the transient. One of the results here is that dissipation can be largely occurring on the bulk of the specimen rather than at the crack tip which is why we find the peak transient work rate to be proportional to the product of width and height of the edge crack specimen. Notice that our model assumes a rate independent fracture criterion, while some experiments seem to suggest that fracture energy may be rate-dependent (see Lavoie et al. 2015). A rate dependent fracture criterion, with additional and independent time constants, would imply an even more complex transient.

2 The crack tip and the energy release rate

In the absence of a model for the process zone at a crack tip, its stress field in a linear, isotropic, viscoelastic material in infinitesimal strain theory is the standard one of linear elastic fracture mechanics: in Mode I,

$$ \sigma_{ij} = \frac{{K_I \tilde{\sigma }_{ij} \left( \theta \right)}}{{\sqrt {2\pi r} }} + \cdots $$

(3)

where $\sigma_{ij}$ is the stress tensor, $\left( {r,\theta } \right)$ is a cylindrical polar coordinate system with origin at the current crack tip and $\tilde{\sigma }_{ij} \left( \theta \right)$ is a dimensionless tensorial function (Anderson 1995). The correspondence principle (Lee 1955; Graham 1968) can be used to convert isotropic, linear elastic strain and displacement fields at the crack tip to those for isotropic linear viscoelastic materials. When a body with a Mode I crack is subjected to traction boundary conditions, we can use the correspondence principle to analyse the strains and displacements throughout the body knowing the linear elastic solution for the body if the stress field is independent of the elastic moduli, as is the case for most plane strain problems (Graham 1968).

For the Dugdale cohesive zone model (Dugdale 1960), with the length of the cohesive zone very small compared to component dimensions, the cohesive zone length, c, is given by

$$ c = \frac{\pi }{8}\left( {\frac{K_I }{{\sigma_c }}} \right)^2 $$

(4)

where $\sigma_c$ is the uniform, constant, rate independent, cohesive stress in the zone, and $K_I$ is the magnitude of the stress intensity factor in the absence of the cohesive zone. In small scale cohesion, the stresses in the region of the crack tip are finite as stress singularities are suppressed. Following Knauss et al. (1973) and Schapery (1975a), the tip opening displacement for a plane strain Mode I crack propagating steadily at the steady speed V is

$$ \delta_{tip} = \frac{{8\left( {1 - \nu^2 } \right)c\sigma_c }}{{\pi E_{eff} }} $$

(5)

where we have assumed that Poisson’s ratio, $\nu$, is constant during viscoelastic response. The parameter $E_{eff}$, the effective Young’s modulus for the energy release rate, is given by

$$ \frac{1}{{E_{eff} \left( \frac{c}{V} \right)}} = \mathop \smallint \limits_0^1 C\left( {\frac{c}{V}\left( {1 - \lambda } \right)} \right)\frac{df\left( \lambda \right)}{{d\lambda }}d\lambda $$

(6)

In the integral in Eq. (6) the creep compliance function $C\left( t \right)$ gives the strain response of the material to uniaxial stress as

$$ \varepsilon \left( t \right) = \int_{ - \infty }^t {C\left( {t - t^{\prime} } \right)\frac{{d\sigma \left( {t^{\prime} } \right)}}{{dt^{\prime} }}dt^{\prime} } $$

(7)

and the function f, the elastic cohesive zone stretch for a homogeneous system, normalized to unity at the crack tip, is (Rice 1978)

$$ f\left( \lambda \right) = \sqrt {\lambda } - \frac{1}{2}\left( {1 - \lambda } \right)\ln \frac{{1 + \sqrt {\lambda } }}{{1 - \sqrt {\lambda } }} $$

(8)

Since the crack tip energy release rate is

$$ {\rm{\mathcal{G}}} = \sigma_c \delta_{tip} $$

(9)

crack propagation occurs with $\sigma_c \delta_{tip}$ equal to ${\Gamma }$, the intrinsic crack tip toughness, thereby determining $\delta_c$, the critical value of $\delta_{tip}$, so that

$$ \sigma_c \delta_c = {\Gamma } $$

(10)

We confine ourselves to materials in which the crack tip toughness, ${\Gamma }$, has a unique value independent of the rate of crack growth. By combination of Eqs. (4) and (5) at the critical value of the crack tip opening displacement, plus Eq. (10) for the toughness, we deduce that the Irwin relationship between the crack tip toughness and the stress intensity factor is

$$ {\Gamma } = \frac{{\left( {1 - \nu^2 } \right)K_I^2 }}{{E_{eff} \left( \frac{c}{V} \right)}} $$

(11)

This relationship provides a means by which the rate of crack propagation, $V$, can be predicted from the applied stress intensity factor, $K_I$.

3 Work done by the applied loads and viscoelastic dissipation

The correspondence principle of Graham (1968) can be used for the component shown in Fig. 1, namely a strip with a single edge crack subject to Mode I, uniform tension. The applied stress at the ends of the component shown in Fig. 1 is uniform and is $\sigma_A$ and the displacement of one end of the specimen relative to the other end (more precisely, as measured on the crack centreline) is $\Delta$, while the specimen width is b, the crack length is a and the state of the component is plane strain. The cohesive zone length is assumed be very small throughout the process of crack growth, i.e., $c/a \ll 1$, $c/\left( {b - a} \right) \ll 1$ and $b/h < 1$.

Fig. 1

The single edge cracked strip subject to uniform tensile stress

3.1 Elastic results

From Tada et al. (2000), we obtain the elastic displacement of one end of the specimen relative to the other (more precisely, as measured on the crack centreline) as

$$ \Delta^e = \frac{{2\left( {1 - \nu^2 } \right)h\sigma_A }}{E} + \frac{{4\left( {1 - \nu^2 } \right)a\sigma_A }}{E}g\left( \frac{a}{b} \right) $$

(12)

where $E$ is Young’s modulus of the linear elastic material. A good approximation for $g$ is

$$ g\left( \frac{a}{b} \right) = \frac{{\left[ {0.989\frac{a}{b} - \left( {1 - \frac{a}{b}} \right)\left( {1.3 - 1.2\frac{a}{b} + 0.7\frac{a^2 }{{b^2 }}} \right)\frac{a^2 }{{b^2 }}} \right]}}{{\left( {1 - \frac{a}{b}} \right)^2 }} $$

(13)

Tada et al. (2000) state that Eq. (13) is accurate to within 1% for any $a/b$ as long as $h/b \ge 1$. Because the cohesive zone is very small, it does not perturb the result in Eq. (13) to any significant extent. Furthermore, other than the small discrepancy introduced by the cohesive zone, Eq. (12) is exact, although obtaining an exact expression for $g\left( {a/b} \right)$ instead of the approximation in Eq. (13) is not a simple task.

We note that the elastic potential energy per unit thickness is

$$ \psi = - \frac{1}{2}b\sigma_A {\Delta }^e $$

(14)

As a consequence, the elastic crack tip energy release rate is

$$ {\rm{\mathcal{G}}}^e = - \frac{{\partial \psi \left( {\sigma_A ,a} \right)}}{\partial a} = \frac{{2\left( {1 - \nu^2 } \right)b\sigma_A^2 }}{E}\left[ {g\left( \frac{a}{b} \right) + \frac{a}{b}g^{\prime} \left( \frac{a}{b} \right)} \right] $$

(15)

where $g^{\prime} \left( x \right) = dg\left( x \right)/dx$. From Eq. (15) and the elastic equivalent of Eq. (11) we deduce the stress intensity factor to be

$$ K_I = \sigma_A \sqrt {{2b\left[ {g\left( \frac{a}{b} \right) + \frac{a}{b}g^{\prime} \left( \frac{a}{b} \right)} \right]}} $$

(16)

During steady state crack propagation with the cohesive zone model, $K_I$ must be held constant. To achieve this, we treat $K_I$ as a constant during crack growth and set

$$ \sigma_A \left( a \right) = \frac{K_I }{{\sqrt {{2b\left[ {g\left( \frac{a}{b} \right) + \frac{a}{b}g^{\prime} \left( \frac{a}{b} \right)} \right]}} }} $$

(17)

This restriction may not be practical in the laboratory, but for our exploration we enforce it in our thought experiments. Therefore, Eq. (12) becomes

$$ \Delta^e = \frac{{\sqrt {2} \left( {1 - \nu^2 } \right)K_I \left[ {h + 2ag\left( \frac{a}{b} \right)} \right]}}{{E\sqrt {{b\left[ {g\left( \frac{a}{b} \right) + \frac{a}{b}g^{\prime} \left( \frac{a}{b} \right)} \right]}} }} $$

(18)

Initially, the crack length is $a_o$ and the component is free of applied load. At time zero, raise the applied stress from zero to the value in Eq. (16) with $a = a_o$. At time $t = t_o$ the crack begins to grow at speed V. Below, we will identify this viscoelastic delay in the initiation of crack growth (Knauss 1970; Schapery 1975c). For $t \ge t_o$ the crack length is given by

$$ a = a_o + V\left( {t - t_o } \right) $$

(19)

To set ideas in place, we first consider crack growth in a linear elastic component. The total work done on the specimen per unit thickness during crack growth is

$$ W\left( a \right) = W_o \left( {a_o } \right) + b\int_{a_o }^a {\sigma_A \left( {a^{\prime} } \right)\frac{{d\Delta^e \left( {a^{\prime} } \right)}}{{da^{\prime} }}da^{\prime} } $$

(20)

Since the system is elastic, the total energy per unit thickness is

$$ W = U\left( {\Delta^e ,a} \right) + {\Gamma }\left( {a - a_o } \right) $$

(21)

where $U\left( {\Delta_e ,a} \right)$ is the strain energy per unit thickness. Thus

$$ dW = b\sigma_A d{\Delta }^e - {\rm{\mathcal{G}}}^e da + {\Gamma }da = b\sigma_A d{\Delta }^e = b\sigma_A \left( a \right)\frac{{d{\Delta }^e \left( a \right)}}{da}da $$

(22)

consistent with Eq. (20) and where we have equated the crack tip energy release rate and the fracture toughness. From Eq. (21) we have also

$$ dW = dU + {\Gamma }da $$

(23)

Therefore, and as expected, the work done on the specimen per unit thickness, Eq. (23), is the sum of the change in strain energy and the increase in fracture energy.

Finally, we consider an elastic component that is unloaded to begin with, and is loaded to the critical stress intensity factor

$$ K_c = \sqrt {{\frac{{E{\Gamma }}}{1 - \nu^2 }}} $$

(24)

where ${\Gamma }$ is the rate independent, constant fracture toughness of the material. At this stage the crack is permitted to grow from its initial length, $a_o$, to a final length, $a_f$, while $K_I = K_c$ and then the component is unloaded to zero load. In this cycle of loading, crack growth and unloading, integration of Eq. (23) leads to

$$ W_{total}^F = {\Gamma }\left( {a_f - a_o } \right) $$

(25)

as the initial and final strain energy in the component is zero. Thus, we have the well-known result that, for an elastic component, the work per unit thickness required in the cycle of loading, crack growth and unloading is just the work required to grow the crack. The result in Eq. (25) is, of course, independent of the magnitude of Young’s modulus.

3.2 Viscoelastic results

3.2.1 Extremely fast loading followed by crack growth

We now consider an initially unloaded component that is rapidly loaded to a specific value of $K_I$. However, in this treatment of rapid loading we neglect mass inertia so that dynamic effects are ignored. We define

$$ \tilde{\delta }_{tip}^e = \frac{{\left( {1 - \nu^2 } \right)K_I^2 }}{\sigma_c } $$

(26)

After loading is completed, $K_I$ is held fixed, and so the result in Eq. (26) is then constant. Therefore, after rapid initial loading and during the time $0 \le t \le t_o$, the viscoelastic crack opening for the non-propagating crack, is

$$ \delta_{tip} \left( t \right) = C\left( t \right)\frac{{\left( {1 - \nu^2 } \right)K_I^2 }}{\sigma_c } $$

(27)

We note that $C\left( 0 \right) = 1/E_I$, and we choose $K_I$ such that $\delta_{tip} \left( 0 \right) \le \delta_c$. The crack will begin to propagate when $\delta_{tip} \left( t \right) = \delta_{tip} \left( {t_o } \right) = \delta_c$, thereby defining $t_o$, the delay time for crack growth. The result can also be written

$$ C\left( {t_o } \right) = \frac{{\Gamma }}{{\left( {1 - \nu^2 } \right)K_I^2 }} $$

(28)

Note that if the right-hand side of Eq. (28) is equal to $1/E_I$, then $t_o = 0$ and crack growth commences immediately after loading. We avoid loading above the level at which the right-hand side of Eq. (28) is equal to $1/E_I$ as that will lead to unstable brittle crack growth. If the right-hand side of Eq. (28) exceeds $1/E_R$ then crack growth will never commence as the critical condition $\delta_{tip} = \delta_c$ will never be reached and $\delta_{tip}$ will always remain below $\delta_c$. Therefore, we limit the applied stress intensity factor to

$$ \sqrt {{\frac{{E_R {\Gamma }}}{1 - \nu^2 }}} \le K_I \le \sqrt {{\frac{{E_I {\Gamma }}}{1 - \nu^2 }}} $$

(29)

but subject to the observation that at the lower limit $t_o = \infty$.

Note that once crack growth commences it immediately occurs at the rate determined by Eq. (11) as we are neglecting inertia and at $t = t_o$ the cohesive zone has reached the configuration it will have for steady state crack growth. We note in passing that Willis (1967) considered a crack in a viscoelastic material in antiplane shear with mass inertia accounted for.

For the next step we define

$$ \tilde{\Delta }^e \left( a \right) = \frac{{\sqrt {2} \left( {1 - \nu^2 } \right)K_I \left[ {h + 2ag\left( \frac{a}{b} \right)} \right]}}{{\sqrt {{b\left[ {g\left( \frac{a}{b} \right) + \frac{a}{b}g^{\prime} \left( \frac{a}{b} \right)} \right]}} }} $$

(30)

During the time $0 \le t \le t_o$ the crack is not propagating so that $\tilde{\Delta }^e \left( a \right) = \tilde{\Delta }^e \left( {a_o } \right)$ and is therefore constant. The viscoelastic deformation is

$$ \Delta \left( t \right) = C\left( t \right)\tilde{\Delta }^e \left( {a_o } \right) $$

(31)

After the crack begins to grow, Eq. (19) prevails, and

$$ \Delta \left( a \right) = C\left( {\frac{a - a_o }{V} + t_o } \right)\tilde{\Delta }^e \left( {a_o } \right) + \int_{a_o }^a {C\left( {\frac{{a - a^{\prime} }}{V}} \right)\frac{{d\tilde{\Delta }^e \left( {a^{\prime} } \right)}}{{da^{\prime} }}da^{\prime} } $$

(32)

This gives

$$ \frac{d\Delta }{{da}} = \frac{1}{V}C^{\prime} \left( {\frac{a - a_o }{V} + t_o } \right)\tilde{\Delta }^e \left( {a_o } \right) + C\left( 0 \right)\frac{{d\tilde{\Delta }^e \left( a \right)}}{da} + \frac{1}{V}\int_{a_o }^a {C^{\prime} \left( {\frac{{a - a^{\prime} }}{V}} \right)\frac{{d\tilde{\Delta }^e \left( {a^{\prime} } \right)}}{{da^{\prime} }}da^{\prime} } $$

(33)

where $C^{\prime} \left( t \right) = dC\left( t \right)/dt$. Therefore, the increment of work per unit area of crack advance is

$$ \begin{aligned} \frac{dW}{{da}} = & b\sigma_A \left( a \right)\frac{d\Delta }{{da}} \\ = & \frac{b\sigma_A \left( a \right)}{V}C^{\prime} \left( {\frac{a - a_o }{V} + t_o } \right)\tilde{\Delta }^e \left( {a_o } \right) + b\sigma_A \left( a \right)\frac{1}{E_I }\frac{{d\tilde{\Delta }^e \left( a \right)}}{da} + \frac{b\sigma_A \left( a \right)}{V}\int_{a_o }^a {C^{\prime} \left( {\frac{{a - a^{\prime} }}{V}} \right)\frac{{d\tilde{\Delta }^e \left( {a^{\prime} } \right)}}{{da^{\prime} }}da^{\prime} } \\ \end{aligned} $$

(34)

Analogous to Eq. (23), the increment of work, $dW$, is the sum of the work increments stored and dissipated in the material plus the increase in fracture energy, ${\Gamma }da$, due to crack growth.

3.2.2 Work done during crack growth after rapid loading

We return to Eq. (34) and observe that the left-hand side is equal to the fracture work plus viscoelastic deformation work. Therefore, we rewrite Eq. (34) as

$$ \frac{dW}{{da}} = {\Gamma } + \frac{{\delta W^{ve} }}{\delta a} = \frac{b\sigma_A \left( a \right)}{V}C^{\prime} \left( {\frac{a - a_o }{V} + t_o } \right)\tilde{\Delta }^e \left( {a_o } \right) + b\sigma_A \left( a \right)\frac{1}{E_I }\frac{{d\tilde{\Delta }^e \left( a \right)}}{da} + \frac{b\sigma_A \left( a \right)}{V}\int_{a_o }^a {C^{\prime} \left( {\frac{{a - a^{\prime} }}{V}} \right)\frac{{d\tilde{\Delta }^e \left( {a^{\prime} } \right)}}{{da^{\prime} }}da^{\prime} } $$

(35)

where $\delta W^{ve} /\delta a$ is the increment of viscoelastic work that occurs in the specimen divided by the increment of crack length. Although this term is not purely dissipative, it contains the dissipated work as well as any work stored elastically. Just after $t = t_o$ the crack begins to grow at its steady state rate. Therefore, immediately upon initiation of crack growth, Eq. (36) provides

$$ {\Gamma } + \frac{{\delta W^{ve} \left( {a_o } \right)}}{\delta a} = \frac{{b\sigma_A \left( {a_o } \right)}}{V}C^{\prime} \left( {t_o } \right)\tilde{\Delta }^e \left( {a_o } \right) + b\sigma_A \left( {a_o } \right)\frac{1}{E_I }\frac{{d\tilde{\Delta }^e \left( {a_o } \right)}}{da} $$

(36)

We note that the 2nd term on the right-hand side depends on $K_I$ but has no explicit dependence on the crack propagation speed. Furthermore, it does not depend on viscoelastic properties other than $E_I$ and thus is not a dissipative term.

To explore the result in Eq. (36), we consider the standard viscoelastic material for which

$$ C\left( t \right) = \frac{1}{E_R } + \left( {\frac{1}{E_I } - \frac{1}{E_R }} \right)e^{ - \frac{t}{t_c }} $$

(37)

and

$$ C^{\prime} \left( t \right) = \frac{1}{t_c }\left( {\frac{1}{E_R } - \frac{1}{E_I }} \right)e^{ - \frac{t}{t_c }} $$

(38)

As a result,

$$ C^{\prime} \left( {t_o } \right) = \frac{1}{t_c }\left[ {\frac{1}{E_R } - \frac{{\Gamma }}{{\left( {1 - \nu^2 } \right)K_I^2 }}} \right] $$

(39)

We now take advantage of an approximation introduced by Schapery (1975b) for the integration of Eq. (6), namely the use of $\lambda = 2/3$ in the integrand, justified by the typical relaxation spectrum exhibited by most viscoelastic materials. As a consequence, we obtain

$$ C\left( \frac{c}{3V} \right) = \frac{{\Gamma }}{{\left( {1 - \nu^2 } \right)K_I^2 }} $$

(40)

so that inversion of $C\left( {c/3V} \right)$ will allow the evaluation of the crack propagation rate, V, in term of the size of the cohesive zone and material parameters. We assume that Schapery’s (1975b) approximation is valid for the standard material and obtain

$$ V = - \frac{\pi K_I^2 }{{24\sigma_c^2 t_c \ln \left\{ {\frac{E_I }{{E_I - E_R }}\left[ {1 - \frac{{E_R {\Gamma }}}{{\left( {1 - \nu^2 } \right)K_I^2 }}} \right]} \right\}}} $$

(41)

The result in Eq. (41) is such that $V = 0$ at the lower limit of (29) and $V$ rises monotonically with $K_I^2$ until it is infinite at the upper limit of (29). When the result from Eq. (41) is substituted into Eq. (36), the term on the right-hand side of Eq. (36) that accounts for dissipation leads to

$$ \frac{{b\sigma_A \left( {a_o } \right)}}{V}C^{\prime} \left( {t_o } \right)\tilde{\Delta }^e \left( {a_o } \right) = - \frac{{24\sigma_c^2 b\sigma_A \left( {a_o } \right)\tilde{\Delta }^e \left( {a_o } \right)}}{\pi E_R K_I^2 }\left[ {1 - \frac{{E_R {\Gamma }}}{{\left( {1 - \nu^2 } \right)K_I^2 }}} \right]\ln \left\{ {\frac{E_I }{{E_I - E_R }}\left[ {1 - \frac{{E_R {\Gamma }}}{{\left( {1 - \nu^2 } \right)K_I^2 }}} \right]} \right\} $$

(42)

This result is zero at each limit in (33), indicating the absence of dissipation at each of these limits, consistent with our previous observation. As $\sigma_A \left( {a_o } \right)\tilde{\Delta }^e \left( {a_o } \right)$ is proportional to $K_I^2$, the quotient containing it is independent of $K_I$. As $K_I^2$ increases from the lower limit of (33), the right-hand side of Eq. (43) increases, goes through a maximum and falls back to zero at the upper limit of (33). Thus, as $V$ is increased from zero, the right-hand side of Eq. (37) rises from zero, goes through a maximum and then falls back to zero as $V \to \infty$. The maximum value of the right-hand side of Eq. (42) occurs at

$$ \frac{{\left( {1 - \nu^2 } \right)K_I^2 }}{{E_R {\Gamma }}} = \frac{eE_I }{{\left( {e - 1} \right)E_I + E_R }} $$

(43)

As the ratio of $E_I$ to $E_R$ is typically 2 or 3 orders of magnitude, the right-hand side of Eq. (43) is well approximated by $e/\left( {e - 1} \right) \approx 1.58$. Thus, the maximum of the right-hand side of Eq. (42) occurs at a value of $K_I$ approximately 25% greater than the lower limit of (33). As the upper limit of (33) is typically at least 10 times greater than the lower limit of (33), the results show that the right-hand side of Eq. (43) rises with increasing $K_I$ over a quite narrow range of $K_I$ at the lower end of the range in (33) and falls with increasing $K_I$ over almost all of the range in (33).

We note also that the term in square brackets in Eq. (42) is equal to the parameter ${\Lambda }$ introduced by Hui et al. (2022) and featured in our Eq. (2). As ${\Lambda }$ rises monotonically with $K_I$ within the limits of (33), the behaviour of the right-hand side of Eq. (42), rising and falling with increasing $K_I$, indicates that ${\Lambda }$ is not a good measure of the fraction of energy dissipated per unit area of crack growth. We also observe that the right-hand side of Eq. (42) depends on $h$, the half-length of the component. This reflects the fact that the rate of work done in a long specimen is greater than the rate of work done in a short specimen, simply because of the different volume of material that is being stretched viscoelastically by the applied stress. This feature does not factor out when the results are specialized to a short crack in the component shown in Fig. 1 for which the approximate result $K_I = 1.122\sigma_A \sqrt {\pi a}$ can be used. The behaviour also indicates that the dissipation rate embedded in the right-hand side of Eq. (42) accounts for more than just the propagation of the crack, but also reflects the structural behaviour of a component that would be stretching even if the crack were not growing.

We should also point out that the right-hand side of Eq. (36) provides information about the component’s viscoelastic behaviour only at the time of initiation of crack growth. It does not provide insights into the viscoelastic response of the component after some time has passed and the crack has grown beyond its initial length. Such information can be extracted from Eq. (35). As an example, we integrate Eq. (35) for a standard viscoelastic material having $E_I /E_R = 1000$ for the component in Fig. 1 with $h/a_o = b/a_o = 100$, i.e. an initially short crack. We use the approximation in Eq. (41) to evaluate the rate of crack growth, $V$, and plot results in Fig. 2 for the work rate, $dW/da$, normalized by the fracture energy Γ, from Eq. (35) for values of $K_I /\sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}} = \sqrt {1.58} , 16, 28.5$ (black, blue, red curves). To assume reasonable values for the quantities involved, we notice that the maximum dissipation at the initial instant of propagation is $1.39\left( {1 - \nu^2 } \right)\sigma_c^2 bh/\left( {E_R a_o } \right)$ which can also be written as $\frac{{1.39{\uppi }}}{8}{\Gamma }bh/\left( {c_0 a_o } \right)$, where $c_0$ is the size of the cohesive zone at very low rates of crack growth and according to Hui et al. (2022) in many elastomers that have been tested, it is a very small quantity, ranging from subnanometer to few nm. To obtain results consistent for elastomers with orders of magnitude suggested by Hui et al. (2022), we further assume Γ = 50 J/m², $E_I$ = 1GPa, $E_R$ = 1 MPa, $\sigma_c$ = $E_I$, $t_C = 0.1s$, which leads to $c_0$ = 2 · 10⁻¹¹ m (clearly, physically not corresponding to any feature of the material, as remarked by Hui et al. 2022).

Fig. 2

Absolute value of the external work per unit area of crack growth, $dW/da$, as a function of elapsed time for the edge crack in Fig. 1 for a standard material having $E_I /E_R = 1000$ and retardation time $t_c$ for values of $K_I /\sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}} = \sqrt {1.58} , 16, 28.5$ (black, blue, red curves): solid lines refer to the standard material, while dashed lines refer to the equivalent Maxwell material with viscosity appropriate to obtain the same creep function of the standard material for not too long times, namely $\eta = E_R t_c$. The load is suddenly applied at $t = 0$ and crack growth commences at $t = t_o$ where $t_o$ has a different value in each case. The work rate, $dW/da$, is normalized by the fracture energy Γ, and is positive in the first period before the first downward peak, then negative, and then positive again after the second downward peak. We assumed Γ = 50 J/m², $E_I$ = 1GPa, $E_R$ = 1 MPa, $\sigma_c$ = $E_I$, $t_C = 0.1s$, which leads to $c_0$ = 20 nm. The curves for standard and Maxwell materials become very close for fast propagation, as expected. $h/a_o = b/a_o = 100$, i.e. an initially short crack of 1 mm size

Figure 2 shows that the work rate is at times close to the initial time t₀ orders of magnitude larger than $E_I /E_R = 1000$ for all the loads chosen, which cover almost the entire range (the load for $K_I /\sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}} = 28.5$ is 0.9 of the load needed for very fast crack propagation). In particular, the maximum work rate can be several orders of magnitude (in our case, almost 9 orders) larger than the expected "steady state" increase of $E_I$/$E_R$ which in our case is a factor 1000. The reason for this important and unexpected result is that the factor (bh)/($c_0 a_0$) is easily a very large number, not only because $c_0$ is small but also because bh in principle can be arbitrarily large (regarding $a_0$ we notice that Chen et al. (2017) make remarks about how small it can be while still using meaningfully the concepts of Fracture Mechanics). We have therefore found an “ultratough” regime of crack propagation, despite this occurs only in a transient.

Looking in details to Fig. 2, it can be seen that there is an incubation period during which the crack tip is stationary. Initiation of crack growth occurs in each case at time $t = t_o$ where there is a steep rise in the work rate in Fig. 2. Thereafter, the crack grows at a constant rate in each case, given by Eq. (42), with the case plotted in red with $K_I /\sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}} = 28.5$ growing most rapidly. In this case, the crack eventually breaks the component into 2 pieces within the time period plotted with the steep rise in the work rate at approximately $t/t_c = 5000$ denoting the final fracture of the component. Similarly, the specimen breaks into two parts for the intermediate load (blue curve) at approximately $t/t_c = 20,000$ whereas for the low load (black curve), it occurs at a time later than the interval plotted in Fig. 2. Notice that Fig. 2 plots results for the standard material as solid lines and for the equivalent Maxwell material as dashed lines, which we shall discuss later. Considering both materials, it can be seen that the external work per unit area of crack growth is positive and highest immediately after initiation of crack growth, and is highest for the most lightly loaded case having $K_I /\sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}} = \sqrt {1.58}$. As this case is the one with the slowest rate of propagation of the crack, the red case with $K_I /\sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}} = 28.5$ is that having the fastest rate of crack propagation, and the curve in blue for $K_I /\sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}} = 16$ is associated with an intermediate rate of crack growth, the results in Fig. 2 show that a faster rate of crack propagation is not associated with a larger rate of external work per unit area of crack growth. This was already noted immediately after initiation of crack growth in conjunction with Eq. (42).

It can be further seen in Fig. 2 that the work done per unit area of crack growth declines rapidly after initiation of crack growth and eventually becomes negative for the standard material case (solid lines). During the stage where $dW/da < 0$ the work of fracture is being provided by release of strain energy already stored in the component, and the component does work on the applied loads, leading to the negative value of $dW/da$. This indicates that, in this stage of crack growth, the applied load is dropping sufficiently rapidly that the component is experiencing a reduction in its stretch despite crack growth and viscoelasticity causing the component’s compliance to increase, i.e., the displacement in Eq. (30) is declining and the viscoelasticity associated with Eq. (32) is insufficient to lead to increases in the displacement in Eq. (32) (these results will be clearer when discussing Fig. 3). The conclusion from these observations is that, during the stage of crack growth where $dW/da < 0$, the higher external work per unit area of crack growth anticipated to be associated with higher rates of crack growth by de Gennes (1996) and Persson and Brener (2005) does not exist because the external work per unit area of crack growth is negative.

Fig. 3

Elongation as function of time for the case of Fig. 2. The very large elongations are due to the choice of material properties

It can be seen in Fig. 2 that the external work per unit area of crack growth for the standard material becomes positive once again when the crack length approaches the width of the specimen (blue and red curves). This indicates that, when the crack tip approaches the distal free surface of the component, the viscoelastic compliance of the component increases rapidly and it stretches despite the applied load dropping as the crack continues to lengthen. As a result, the applied loads are forced once more to do work on the component.

We note further that at no time during crack growth is the external work per unit area of crack growth in steady state in Fig. 2. The value of $dW/da$ is always in transient mode once crack growth commences, with the transient changes being greatest when $dW/da > 0$. Thus, one cannot claim that there is a steady state value of a positive external work rate per unit area of crack growth associated with a constant rate of crack growth, as the proposals of de Gennes (1996) and Persson and Brener (2005) would imply. To the extent that there is any hint of a steady state value for $dW/da$ it is when $dW/da < 0$, as $dW/da$ changes relatively little in this stage, but only for the low loads regime, and not in general. However, we have trouble rationalizing a negative rate of external work per unit area of crack growth in a viscoelastic body with the concepts of de Gennes (1996) and Persson and Brener (2005). Furthermore, we used $h = b$ to compute the results in Fig. 2. A value of $h > b$ will alter the results in Fig. 2 but without changing any of the features associated with crack growth.

These results are probably clearer when we plot the elongation during time, as in Fig. 3. It can be seen that after the elastic deformation which is higher for higher loads, there is an incubation time when crack is not moving and elongation follows an exponential trend due to the creep function. Following this, there is an essentially linear phase with a speed which increases with the applied stress intensity factor; the elongation then reaches a maximum and remains there for low loads (black curve) for quite a long time, whereas it quickly decays for the highest applied loads (red curve). Elongation then finds a minimum and starts to increase again with a much higher rate when the crack is about half the width of the specimen (for the lowest load occurs for times not shown in the Figure).

In some experiments in the Literature, the stretch rate in the specimen is kept constant during the entire experiment until break (see e.g. Cristiano et al. 2011): load increases until a peak value is found, which is associated to the critical energy release rate, or fracture energy, using equations obtained for a non-linear but elastic material; a decaying load phase follows where the crack speed is estimated in an average sense from the time taken by the crack to transverse the specimen, but it is not clear how far from constant it can be. In Cristiano et al., (2011) he peak load is found to increase with elongation rate and similarly the average crack speed: hence, the fracture energy is estimated to increase monotonically with crack speed. The procedure in Cristiano et al. (2011) seems to estimate fracture energy when the crack has not yet initiated motion and could be perhaps more relevant to an “initiation” (stretch-rate dependent) fracture energy (see a recent theory proposed by Shrimali and Lopez-Pamies 2023, and commented by Persson et al. 2024). What we have learned instead from our loading protocol is that stress intensity factor and crack velocity can be made constant, but energy fluxes would be very far from constant. It is clear that the crucial difference between the two types of experiments is the phase after the constant stretch rate, which can be very long in our designed experiment.

The total work for rupturing the specimen in two parts may be also a practical quantity of interest. The total work is integrated numerically from the results of Eq. (36), and then divided by the value Γ(b − a₀), which is the pure work needed to create the two surfaces. This results in Fig. 4 where the total dimensionless work is plotted as a function of $K_I /K_{Imin}$ where $K_{Imin} = \sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}}$. We are deliberately not taking into account the initial strain energy in loading the specimen, nor the possible final strain energy dissipated at the time of fracture, which would not quantitatively change the conclusions anyway, and would be very small quantities especially considering fast loading and unloading, compared to the quantities plotted in Fig. 4. As it is clear from Fig. 4, with the reasonable set of constants we used for an elastomer, we obtain even with standard material that the total work seems to be higher than the pure work of fracture amplified by the factor $E_I /E_R = 1000$, due to the large work rate at the beginning of propagation, and the large one at the final instants. So we have effectively an ultratough interface, except for the very fast propagation limit, where the total work returns ultimately to the expected one without dissipation. Notice that the curve for the standard material case at very low $K_I /K_{Imin}$ should return also to the same very low limit, but this is not reproduced here due to numerical difficulties in integrating the equations at very large times. The result for the Maxwell material instead grows unbounded for very low speed, consistently to what we will show analytically in the next paragraph.

Fig. 4

Value of the total external work done during propagation from crack size a = a₀ up to a = b (complete breaking), for the edge crack in Fig. 1 as a function of $K_I /K_{Imin}$ where $K_{Imin} = \sqrt {{E_R {\Gamma }/\left( {1 - \nu^2 } \right)}}$. Other parameters are as in Fig. 2. Solid line is the standard material, and dashed line refer to the equivalent Maxwell material with viscosity appropriate to obtain similar creep function of the standard material for not too long times, namely $\eta = E_R t_c$

The striking result of this example reproduced in Fig. 4 is that for most of the range of applied loads, the total work to propagate the crack across the entire section of the specimen is larger than the pure work of fracture by a factor bigger than the expected factor $E_I /E_R = 1000$. This is due to the fact that dissipation occurs mostly in the bulk and not at the crack tip, as evidenced by the large difference between the Maxwell and the standard solid.

3.3 Crack growth in a Maxwell material

The standard linear solid considered so far can be represented by the parallel between a Maxwell model (a spring and a dashpot in series) in parallel with another spring. The Maxwell model could be considered to physically correspond to the standard linear solid in the limit of short time response, since the only difference is in the long time response given by the isolated spring. In other words, the Maxwell material has no limit elasticity, whereas the standard solid has a relaxed modulus. The results will reflect this physical behaviour.

To illustrate our insights more clearly, we consider a Maxwell material having creep response given by

$$ C\left( t \right) = \frac{1}{E_I } + \frac{t}{\eta } $$

(44)

where $\eta$ is the viscosity. To make a quantitative example for the case we are considering in Figs. 2, 3, we plot in Fig. 5, the example of Maxwell material which best represents the standard material considered so far, having $E_I$ = 1GPa, $E_R$ = 1 MPa (solid line), which is obtained with a viscosity $\eta = E_R t_c$ (dashed line). It is clear that the Maxwell material and the standard material’s creep functions depart significantly only for $t > t_c$, which corresponds to the fact that our results in Fig. 2 also depart at about that time for the two materials. The calculations follow the same steps as done for the standard material, and are not repeated here for brevity. However, we find for the net viscous dissipation per unit area of crack extension plus the intrinsic fracture toughness

$$ {\Gamma } + \frac{\delta W^v }{{\delta a}} = {\Gamma } + \left[ {{\Gamma } - \frac{{\left( {1 - \nu^2 } \right)K_I^2 }}{E_I }} \right]\left\{ {\frac{{24\left[ {h + 2ag\left( \frac{a}{b} \right)} \right]\sigma_c^2 }}{{\pi \left[ {g\left( \frac{a}{b} \right) + \frac{a}{b}g^{\prime}\left( \frac{a}{b} \right)} \right]K_I^2 }} - 1} \right\} $$

(45)

Fig. 5

The creep compliance for standard material considered in our examples where for the solid material $E_I$ = 1GPa, $E_R$ = 1 MPa (solid line), and for the “equivalent” Maxwell material with viscosity $\eta = E_R t_c$ (dashed line). It is clear that the Maxwell material and the standard material’s creep functions depart significantly only for $t > t_c .$

This shows that, for the Maxwell material, the net viscous dissipation per unit area of crack advance is largest when the crack propagation rate is low and diminishes as the crack speed increases. Consistent with the results of the standard material, viscous dissipation is zero when $K_I$ equals the upper limit i.e. when crack growth is fastest. Note also that the result in Eq. (45) is essentially arbitrary, as the component can be made as long or as short as is desired as long as $h \ge b$. This arbitrariness has no effect on the rate of crack propagation and makes no change to the effective toughness.

4 Discussion

Results represented as being derived from an energy balance (de Gennes 1996; Persson and Brener 2005; Persson 2017, 2021) look at the steady state, and we have shown in the present paper that viscoelastic propagation, looked from a cohesive model, can be intrinsically a transient process, despite the crack growth rate may be constant. Secondly, these models tend to neglect that strain energy could be cumulating in the bulk of the solid (and not just from the asymptotic, singular, region of stresses), and therefore in general are simplified models, which in some cases have been shown to give results compatible with the cohesive models (Ciavarella et al. 2021; Hui et al. 2022). However, the present paper has given an example of a quite more general picture of the energy fluxes during viscoelastic crack propagation which has given rise to interesting and quite unexpected results, in particular that the work rate can be much higher than what expected from the literature and happens not at large speeds but at intermediate speeds. Similarly, that the total work in propagating a crack up to a specimen failure can be also orders of magnitude higher than what expected from the very slow fracture energy limit, and being minimal only at loads extremely close to the load for very fast crack growth.

This rigorous application of the correspondence principle and the cohesive model is more general in addressing viscoelastic crack propagation problems, than the de Gennes-Persson-Brener as it shows transient processes and dissipation also in the bulk.