## Abstract

Shrimali and Lopez-Pamies (2023, “The ‘Pure-Shear’ Fracture Test for Viscoelastic Elastomers and Its Revelation on Griffth Fracture,” Extreme Mech. Lett., 58, p. 101944) have recently shown that the Griffith criticality condition that governs crack growth in viscoelastic elastomers can be reduced to a fundamental form that involves exclusively the intrinsic fracture energy G_{c} of the elastomer, and, in so doing, they have brought resolution to the complete description of the historically elusive notion of critical tearing energy T_{c}. The purpose of this article—which can be viewed as the third installment of a series—is to make use of this fundamental form to explain one of the most popular fracture tests for probing the growth of cracks in viscoelastic elastomers, the trousers test.

## 1 Introduction

_{0}, while the right-hand side

*T*

_{c}stands for the critical tearing energy,

*a characteristic property of the elastomer that depends on the loading history*. In Eq. (2), on the other hand, $\u2212\u2202WEq/\u2202\Gamma 0$ denotes the change in equilibrium elastic energy $WEq$ stored in the bulk with respect to an added surface area to the pre-existing crack Γ

_{0}, while

*G*

_{c}stands for the intrinsic fracture energy,

*a material constant of the elastomer*. Experiments have shown that its value is typically in the same relatively narrow range

*not*knowing how the critical tearing energy

*T*

_{c}depends on the applied loading history for a given elastomer and given geometry of the body of interest. The work of Shrimali and Lopez-Pamies [1] has brought resolution to this decades-old problem by first recognizing that the total deformation energy $W$ admits the partition

^{2}and then establishing from experiments that

From a fundamental point of view, the criticality condition (2) is a strikingly simple and intuitive condition as it states that whether an elastomer simply deforms or, on the other hand, creates new surface from a pre-existing crack is dictated by a competition between its stored equilibrium elastic energy and its intrinsic fracture energy, irrespective of its viscosity.

From a practical point of view, the criticality condition (2) is also conveniently simple. This is because it is based on two properties of the elastomer that can be measured experimentally once and for all by means of conventional tests: (i) its viscoelastic behavior, from which the storage of equilibrium elastic energy can be identified, and (ii) its intrinsic fracture energy.

As a first effort to gain precise and quantitative insight into the fracture behavior of viscoelastic elastomers, Shrimali and Lopez-Pamies [1,8] have made use of the newly minted fundamental form (2) of the Griffith criticality condition to explain two types of popular fracture tests for viscoelastic elastomers, the so-called pure-shear fracture test and the delayed fracture test. The objective of this article—which can be viewed as the third instalment of the series—is to deploy the criticality condition (2) to explain yet another popular fracture test: the trousers test.

This article is organized as follows. In the same spirit of the pioneering global (elastic) analysis provided by Rivlin and Thomas [2], we begin in the first part of Sec. 2 by presenting a global analysis of fracture nucleation in the trousers test for viscoelastic elastomers. In the same spirit of the global analysis provided by Greensmith and Thomas [3], we also present in the latter part of Sec. 2 a global analysis of fracture propagation. In Sec. 3, in preparation to carry out corresponding full-field analyses, we formulate the initial boundary value problem of the trousers test *per se*. In Sec. 4, we present and discuss full-field solutions for the canonical case of a viscoelastic elastomer with Gaussian elasticity and constant viscosity. These results serve to lay bare the key features of the trousers test in the simplest of settings. We conclude in Sec. 5 by summarizing the main findings of this work and by recording a number of closing remarks.

## 2 Global Analysis of the Trousers Test

### 2.1 Fracture Nucleation.

Consider the trousers test schematically depicted in Fig. 1. The specimen, which is taken to be made of an isotropic incompressible viscoelastic elastomer, is such that its thickness is much smaller than its height (*B* ≪ *H*), its height is smaller than its length (*H* < *L*), and the initial length of the pre-existing crack is larger than the height of the specimen but, obviously, smaller than its length (*H* < *A* < *L*). In experiments, one typically encounters the ranges of ratios *L*/*H* ∈ [2, 4], *A*/*H* ∈ [1, 2], and *H*/*B* ∈ [20, 40].

The two legs of the trousers are bent opposite to one another and brought to lie in the same plane, their bottom ends are held firmly by stiff grips, and these are then pulled apart. In experiments, the pulling is done either by applying a force *P*(*t*) or by applying a deformation *l*(*t*) over a time interval *t* ∈ [0, *T*]; see, e.g., Fig. 2 in Ref. [3]. For the former case, a force is typically ramped up over an initial time interval [0, *t*_{0}] and then held constant. When a deformation is applied, on the other hand, the grips are typically separated at a constant rate $l\u02d90$ so that the current distance between the grips is given by the relation $l(t)=l0+l\u02d90t$, where *l*_{0} ≈ 2*A* since *B* ≪ *A*.

Because of the special geometry of the specimen, at any given time *t* ∈ (0, *T*), there are four different regions of deformation; see Fig. 1. In keeping with the same type of region labeling used by Rivlin and Thomas [2]—see Fig. 7 in their work—the region $D$ is substantially undeformed, the crack front region $C$ and the grip region $A$ are in a complex state of deformation (highly nonuniform in space), while region $B$ is substantially in a state of spatially uniform uniaxial tension, and this provided that the deformation *l*(*t*) between the grips is not exceedingly large. Large deformations *l*(*t*) lead to significant twisting of the specimen, which in turn results in region $B$ also exhibiting a state of deformation that is *not* uniform. In the remaining of this section, we tacitly assume that *l*(*t*) is small enough and hence that the state of deformation in region $B$ is one of uniform uniaxial tension.

*P*(

*t*), consider an increase in the crack surface of amount dΓ

_{0}=

*B*d

*A*at time

*t*. This increase in crack surface does

*not*alter the complex states of deformation in $A$ and $C$. Instead, it simply shifts the region $C$ in the direction of the added crack, resulting in the growth of region $B$ at the expense of region $D$. In other words, an added crack dΓ

_{0}at constant applied force

*P*(

*t*) results in the transferring of a volume

*H*dΓ

_{0}of the specimen from an undeformed state to a state of uniaxial tension under the same force

*P*(

*t*). Making use of this observation, we can immediately deduce that

*P*denotes differentiation at fixed force

*P*(

*t*),

*λ*(

*t*) is the stretch that results by subjecting the elastomer to uniform uniaxial tension with a force

*P*(

*t*), and $\psi utEq$ stands for the equilibrium elastic energy density stored at that state of deformation.

*l*(

*t*), and

*not*at fixed force

*P*(

*t*). It follows from a result of Shrimali and Lopez-Pamies [8] that these two derivatives are related to one another according to the equality

*l*in the derivative $\u2212\u2202WEq/$∂Γ

_{0}, since there is no longer risk of confusion.

This last relation is the result that we are after. It reveals that the computation of the energy release rate $\u2212\u2202WEq/\u2202\Gamma 0$ in the Griffith criticality condition (2) for a trousers fracture test—regardless of whether the test is carried out by prescribing a force *P*(*t*) or a deformation *l*(*t*)—amounts to determining two quantities:

the stretch

*λ*(*t*) that results in the elastomer by subjecting it to uniaxial tension with the same force*P*(*t*) that is prescribed or measured, if*l*(*t*) is prescribed, in the test andthe equilibrium elastic force $PEq$, as defined by Eq. (8).

While the stretch *λ*(*t*) can be readily determined from a separate uniaxial tension test, the determination of $PEq$ would appear to require, in principle, having access to the local deformation field in the trousers specimen at hand, which is only possible by solving in full the pertinent initial boundary value problem making use of an appropriate viscoelastic model for the elastomer. Fortunately, in practice, the equilibrium elastic force $PEq$ can be determined directly in terms of a global measurement from the trousers test itself. This is because of two distinguishing properties of $PEq$, which we outline next.

*is the dominant term in Eq. (9)*. In practice, as already noticed by Rivlin and Thomas [2] from their own experimental results, fracture in trousers specimens with sufficiently large height

*H*nucleates when the stretch in region $B$ is negligible, that is, when

*λ*(

*t*) ≈ 1. This allows to simplify relation (9) to

*is substantially only a function of the global stretch*

*l*(

*t*)/

*l*

_{0}

*between the grips*. For typical trousers specimens, for which relation (10) applies, the full-field analysis presented in Sec. 4.3 reveals that—rather remarkably—the equilibrium elastic force $PEq$ is de facto only a function of the current value of the global stretch

*l*(

*t*)/

*l*

_{0}between the grips, and hence, in particular, independent of the length

*A*of the pre-existing crack and of the loading rate. With a slight abuse of notation, we write

*Ergo*, given an elastomer of interest, in order to determine $PEq$ in practice, it would suffice to carry out a trousers test at a slow enough rate that viscous dissipation is negligible, measure both

*P*(

*t*) and

*l*(

*t*)/

*l*

_{0}, and then use these measurements to determine the function (11), since $PEq=P(t)$ in the absence of viscous dissipation.▪

*A unique critical global stretch l*-. When combined with the Griffith criticality condition (2), the facts that the energy release rate $\u2212\u2202WEq/\u2202\Gamma 0$ is given exclusively in terms of $PEq$ and that $PEq$ is only a function of the global stretch

_{c}/l_{0}*l*(

*t*)/

*l*

_{0}imply that, for a given elastomer of interest, there is a unique critical global stretch

Interestingly, the aforementioned is the same type of unique criticality result that occurs in “pure-shear” fracture tests [1].

Regrettably, virtually none of the experimental studies that have been reported in the literature for trousers fracture tests include measurements of the critical deformation *l*_{c} at which fracture nucleation takes place. The sole exception that we are aware of is the original data of Rivlin and Thomas [2], which pertains to natural rubber specimens that contained pre-existing cracks of various different lengths *A*, but that were loaded at the same (slow) rate; see Figs. 2(ii) and 4(ii) in their work. Their results indicate (to within experimental error) that fracture nucleation occurs at a critical stretch *l*_{c}/*l*_{0} that is indeed independent of the length of the pre-existing crack.

### 2.2 Fracture Propagation.

The analysis presented in the preceding subsection pertains to the *nucleation* of fracture, that is, the first instance at which new surface is created from the pre-existing crack. As first recognized by Greensmith and Thomas [3], trousers fracture tests are also particularly useful to study fracture *propagation* in viscoelastic elastomers. This is because under the often satisfied twofold premise,

*λ*(*t*) ≈ 1 andcrack propagation is a smooth process in time,

*t*)/d

*t*is given in terms of the rate d

*l*(

*t*)/d

*t*of separation between the grips by the simple relation:

*t*) =

*B*d

*a*(

*t*), by the relation

*a*(

*t*) of the crack.

*The experimentally prominent case of applied deformation*. For the case when the test is carried out by separating the grips at a constant rate $l\u02d90$, so that, again, $l(t)=l0+l\u02d90t$, it follows immediately from (12) that

*l*(t) at a constant rate*t*> 0, where we have made critical use of the inequality

*B*≪

*A*.

The string of equalities (14) reveals that the propagation (13) of the crack in a trousers fracture test carried out by pulling the grips apart at a constant rate of deformation is such that the current global stretch *l*(*t*)/2 *a*(*t*) between the grips is always at the critical global stretch *l*_{c}/*l*_{0}. In other words, the Griffith criticality condition (2) is constantly satisfied. It is for this reason that what is typically observed in experiments—whenever the crack propagation happens to be smooth in time—is that the crack length evolves according to Eq. (13) and that it does so at a constant force *P*(*t*), of different values for different applied deformation rates $l\u02d90$; see, e.g., Refs. [3,9,10].

*Lack of smoothness/continuity of crack propagation in time*. Already in their early pioneering experiments, Greensmith and Thomas [3] noticed that crack propagation in trousers tests may not be smooth or even continuous in time. They referred to such a type of propagation as “stick-slip”; see Fig. 3(*a*) in their work. For noncrystallizable elastomers—in particular, for various types of styrene-butadiene rubber (SBR)—they observed that crack propagation is mostly smooth in time, except possibly at high rates of deformation and low temperatures. For natural rubber, the most prominent crystallizable elastomer, on the other hand, they observed that crack propagation is mostly discontinuous in time, irrespective of the loading rate and temperature.▪

## 3 Formulation of the Initial Boundary Value Problem for the Trousers Test

Having analyzed the trousers fracture test from a global perspective, we now turn to its full-field analysis.

### 3.1 Initial Configuration.

*L*= 150 mm and height

*H*= 40 mm in the

**e**

_{3}and

**e**

_{1}directions and constant thickness

*B*= 1 mm in the

**e**

_{2}direction. The specimens contain a pre-existing edge crack of five different lengths:

**e**

_{3}direction. These specific values for

*L*,

*H*,

*B*, and

*A*are chosen here because they are representative of those typically used in experiments; see, in particular, the classical experiments of Greensmith and Thomas [3]. Here, {

**e**

_{i}} stands for the laboratory frame of reference. We place its origin at the specimens’ midplane along the bottom edge so that, in their $i$nitial configuration, the specimens occupy the domain

### 3.2 Reference Configuration and Kinematics.

As already noted in the preceding global analysis of the problem, in an actual trousers fracture test, the specimen is mounted in the testing machine by bending in opposite directions the two legs of the trousers until they are brought to lie in the same plane, at which point their bottom ends are firmly gripped and the specimen is ready to be loaded. For this reason, it proves convenient not to use the initial configuration as the reference configuration to carry out the analysis, but to use, instead, the configuration as initially mounted in the testing machine.

*Residual stresses in the reference configuration.* In the reference configuration identified earlier, there are residual stresses due to the bending of the legs of the trousers. Numerical simulations show that these have no significant impact on the response of the specimen when the grips are pulled apart and hence that they can be neglected altogether; this is hardly surprising, since the specimens are very thin (*H*/*B* = 40).▪

*t*= 0, in their reference configuration, we consider that the specimens occupy the domain

*The fillet in the reference configuration*. In the reference configuration defined by the aforementioned domains, the pre-existing crack has been assumed to feature a circular fillet of inner radius *B* = 1 mm, and so its initial inner length is *not* simply *A* but *A* + *πB*. Numerical simulations show that the specifics of the fillet have no significant impact on the response of the specimens. Again, this is because *B* ≪ *H* < *A*. For notational simplicity, in the sequel, we will continue referring to *A* as the initial length of the pre-existing crack with the understanding that its actual initial length contains a correction of order *B*.▪

*t*∈ [0,

*T*], in response to the applied boundary conditions described below, the position vector

**X**of a material point in the specimens will move to a new position specified by

**y**is a mapping from Ω

_{0}to the current configuration Ω(

*t*). We consider only invertible deformations, and write the deformation gradient field at

**X**and

*t*as follows:

### 3.3 Constitutive Behavior of the Elastomer.

**S**at any material point

**X**∈ Ω

_{0}and time

*t*∈ [0,

*T*] is given by the following relation:

**C**

^{v}is defined implicitly as the solution of the evolution equation:

*p*stands for the arbitrary hydrostatic pressure associated with the incompressibility constraint $J=detF=1$,

**C**=

**F**

^{T}

**F**denotes the right Cauchy-Green deformation tensor, the “dot” notation stands for the Lagrangian time derivative (i.e., with

**X**held fixed), and

*μ*≥ 0,

*ν*≥ 0, and

*η*≥ 0 are three material constants. Specifically,

*μ*denotes the initial shear modulus associated with the Gaussian elasticity of the elastomer at states of thermodynamic equilibrium,

*ν*denotes the initial shear modulus associated with its additional Gaussian elasticity at nonequilibrium states, and

*η*stands for its viscosity.

*I*

_{1}= tr

**C**and $I1e=tr(CCv\u22121)$, and dissipates energy according to the dissipation potential

### 3.4 Initial and Boundary Conditions.

**y**(

**X**,

*t*), the pressure field

*p*(

**X**,

*t*), and the internal variable

**C**

^{v}(

**X**,

*t*).

_{0}of the specimens is traction free. The left and right grip boundaries are separated in the

**e**

_{2}direction at the constant rate $l\u02d90$ so that, as a function of time

*t*∈ [0,

*T*], the current separation between the grips is given by the relation $l(t)=l0+l\u02d90t$, where

*l*

_{0}= 2

*A*+ 3

*B*. Precisely, by making use of the notation

**s**(

**X**,

*t*) =

**SN**, we have that the boundary conditions in full read

**N**stands for the outward unit normal to the boundary ∂Ω

_{0}.

### 3.5 Governing Equations.

_{1,2}and (19), where $S(X,t)=\mu \u2207y+\nu \u2207yCv\u22121\u2212p\u2207y\u2212T$, coupled with the evolution equation:

_{3}, for the deformation field

**y**(

**X**,

*t*), the pressure field

*p*(

**X**,

*t*), and the internal variable

**C**

^{v}(

**X**,

*t*).

In general, the initial boundary value problems (20) and (21) with (18) and (19) do not admit analytical solutions and hence must be solved numerically. All the results that we present below are generated by a variant of the numerical scheme introduced by Ghosh et al. [12], which is based on a Crouzeix-Raviart finite-element discretization of space and a high-order explicit Runge-Kutta discretization of time.

## 4 Results for a Canonical Elastomer With Gaussian Elasticity and Constant Viscosity

*τ*=

*η*/

*ν*= 0.5, 1, 2.5, 5, 10, 12.5, 20, 50 s.

### 4.1 The Force–Deformation Response.

Figure 3 presents results for the total force *P*(*t*) required to deform the specimens with nonequilibrium shear modulus *ν* = 2 MPa, viscosity *η* = 5 MPa s, and pre-existing cracks of length *A* = 49, 50, 51 mm at three constant deformation rates $l\u02d90$. The results are shown for *P*(*t*) as a function of the applied deformation *l*(*t*) for $l\u02d90=2\xd710\u22123mm/s\u22121$ in Fig. 3(a), $l\u02d90=2\xd710\u22121mm/s\u22121$ in Fig. 3(b), and $l\u02d90=100mm/s\u22121$ in Fig. 3(c).

There are three observations worth pointing out. First, specimens with larger cracks require smaller forces to reach the same deformation. Second, larger forces are required to reach a given deformation applied at a higher deformation rate. Finally, all force–deformation responses exhibit some nonlinearity at the beginning of the loading process, whereas they are mostly linear at larger deformations.

### 4.2 The Total Deformation Energy $W$ and Its Partition Into $WEq$, $WNEq$, and $Wv$.

Figure 4 shows results for $WEq$, $WNEq$, and $Wv$—as computed from expressions (22)–(24) and the pertinent numerical solutions for the deformation field **y**(**X**, *t*) and internal variable **C**^{v}(**X**, *t*)—at the deformation *l*(*t*) = 105.743 mm, plotted as functions of the initial crack surface Γ_{0} = *A* × *B* and the deformation rate $l\u02d90$. The results at other fixed values of the deformation *l*(*t*) are not fundamentally different from those shown in Fig. 4 for *l*(*t*) = 105.743 mm, which therefore can be viewed as representative of those at any deformation *l*(*t*).

The following comments are in order. With the sole exception of the dependence of the equilibrium elastic energy $WEq$ on the rate $l\u02d90$ of the applied deformation, all three parts of the deformation energy appear to depend nonlinearly on both the crack surface Γ_{0} and $l\u02d90$. Distinctly, with respect to $l\u02d90$, both the nonequilibrium energy $WNEq$ and the viscous dissipated energy $Wv$ appear to be bounded, but whereas $WNEq$ increases monotonically with increasing $l\u02d90$, $Wv$ exhibits a Γ_{0}-dependent local maximum away from which $Wv$ becomes vanishingly small.

### 4.3 The Derivative $\u2212\u2202WEq/\u2202\Gamma 0$.

From the type of the 3D plot presented in Fig. 4(a), we can compute numerically the energy release rate $\u2212\u2202WEq/\u2202\Gamma 0$ entering the Griffith criticality condition (2). Figure 5 reports such a computation of $\u2212\u2202WEq/\u2202\Gamma 0$ in terms of the applied deformation rate $l\u02d90$ for specimens with nonequilibrium shear modulus *ν* = 2 MPa, viscosity *η* = 5 MPa s, and pre-existing cracks of length *A* = 49, 50, 51 mm at the same fixed deformation *l*(*t*) = 105.743 mm considered in Fig. 4(a). For direct comparison, the results produced by the approximate formula (10) are also included in the figure (dotted lines).

We remark that, much like the results presented in Fig. 4(a) are representative of any fixed value of *l*(*t*), the results for $\u2212\u2202WEq/\u2202\Gamma 0$ at other fixed values of the deformation *l*(*t*) are qualitatively the same as those shown in Fig. 5 for *l*(*t*) = 105.743 mm.

A key observation from Fig. 5 is that the energy release rate $\u2212\u2202WEq/\u2202\Gamma 0$ is essentially independent of the rate $l\u02d90$ at which the grips are separated. On the other hand, $\u2212\u2202WEq/\u2202\Gamma 0$ does depend on the length *A* of the pre-existing crack, in particular, it increases with decreasing *A*. Another key observation from Fig. 5 is that Eq. (10) provides a good approximation for $\u2212\u2202WEq/\u2202\Gamma 0$.

To gain further insight into the energy release rate $\u2212\u2202WEq/\u2202\Gamma 0$ and its approximate representation (10), Fig. 6 presents the results for the equilibrium elastic force *P*^{Eq} as a function of the global stretch *l*(*t*)/*l*_{0} between the grips for trousers specimens with nonequilibrium shear modulus *ν* = 2 MPa, viscosity *η* = 5 MPa s, and pre-existing cracks of length *A* = 49, 50, 51 mm. The results are shown for (*a*) $l\u02d90=2\xd710\u22123mm/s\u22121$ and (*b*) $l\u02d90=100mm/s\u22121$, that is, the slowest and fastest deformation rates $l\u02d90$ considered in Fig. 5.

Foremost, as announced in Remark 2, the results in Fig. 6 show that the equilibrium elastic force *P*^{Eq}—and hence, as per Eq. (10), the energy release rate $\u2212\u2202WEq/\u2202\Gamma 0$—is de facto only a function of the global stretch *l*(*t*)/*l*_{0} and hence, in particular, independent of the length *A* of the pre-existing crack as well as of the rate $l\u02d90$ at which the deformation is applied.

*A unique critical global stretch l*. When combined with the Griffith criticality condition (2)—consistent with the conclusion established from the global analysis in Sec. 2—the full-field results presented in Figs. 5 and 6 for a canonical elastomer imply that, indeed, there is a unique critical global stretch

_{c}/l_{0}To see this via an example, consider that the viscoelastic behavior of the elastomer being tested can be described by the canonical behavior assumed in this section, with equilibrium shear modulus *μ* = 1 MPa, nonequilibrium shear modulus *ν* = 2 MPa, and viscosity *η* = 5 MPa s. Consistent with the standard range (3) for common elastomers, consider further that the intrinsic fracture energy of the elastomer being tested is *G*_{c} = 100 N/m. Then, according to the Griffith criticality condition (2) and Eq. (10), fracture nucleation will occur whenever *P*^{Eq} = *B G*_{c}/2 = (10^{−3} m) × (100 N/m)/2 = 0.05 N. In turn, according to the results shown in Fig. 6, fracture nucleation will occur at the critical global stretch *l*_{c}/*l*_{0} = 1.0071, irrespective of the length *A* of the pre-existing crack and of the deformation rate $l\u02d90$ used to carry out the test.

### 4.4 The Critical Tearing Energy *T*_{c}.

*T*

_{c}as a function of the rate of propagation $a\u02d9(t)$ of the current length of the crack

*a*(

*t*). In truth, it is a plot of

*T*

_{c}= 2

*P*(

*t*)/

*B*and $a\u02d9(t)=l\u02d90/2$. In the sequel, we present such a plot for the canonical elastomer under investigation here.

As a first step, note that from the type of 3D plots presented in Figs. 4(b) and 4(c), we can compute numerically the derivatives $\u2212\u2202WNEq/\u2202\Gamma 0$ and $\u2212\u2202Wv/\u2202\Gamma 0$ entering Eq. (5) for the critical tearing energy *T*_{c}. Figure 7 presents the results obtained from this computation in terms of the applied deformation rate $l\u02d90$ for a specimen with nonequilibrium shear modulus *ν* = 2 MPa, viscosity *η* = 5 MPa s, and a pre-existing crack of length *A* = 51 mm at the same fixed deformation *l*(*t*) = 105.743 mm considered in Figs. 4(b) and 4(c).

As expected on physical grounds, contrary to the derivative $\u2212\u2202WEq/\u2202\Gamma 0$ of the equilibrium elastic energy, note that both derivatives $\u2212\u2202WNEq/\u2202\Gamma 0$ and $\u2212\u2202Wv/\u2202\Gamma 0$ depend strongly on $l\u02d90$. In particular, $\u2212\u2202WNEq/\u2202\Gamma 0$ is bounded from below (by zero) and from above and increases monotonically with increasing $l\u02d90$. By contrast, $\u2212\u2202Wv/\u2202\Gamma 0$ is also bounded from below (by zero) and from above, but is not monotonically increasing in $l\u02d90$, instead it exhibits a single local maximum at some value of $l\u02d90$ (in the present case, around $l\u02d90=2\xd710\u22121mm/s$).

Now, making direct use of the type of results presented in Fig. 7, we can readily determine the critical tearing energy (5).

To see this via an example, consider once more that the viscoelastic behavior of the elastomer being tested can be described by the canonical behavior assumed in this section, with equilibrium shear modulus *μ* = 1 MPa, nonequilibrium shear modulus *ν* = 2 MPa, and viscosity *η* = 5 MPa s. Consider as well that its intrinsic fracture energy is *G*_{c} = 100 N/m. For this choice of material constants, as established earlier, the Griffith criticality condition (2) is satisfied when the global stretch between the grips is *l*_{c}/*l*_{0} = 1.0071. This is precisely the global stretch (*l*(*t*)/*l*_{0} = 105.743 mm/105 mm = 1.0071) that the results in Fig. 7 pertain to. Then, according to Eq. (5), the computation of *T*_{c} in this case simply amounts to summing the constant *G*_{c} = 100 N/m to the results in Figs. 7(a) and 7(b). Figure 8 reports such a computation of *T*_{c} in terms of the rate of crack propagation $a\u02d9(t)$, as defined by $a\u02d9(t)=l\u02d90/2$.

An immediate observation from Fig. 8 is that the critical tearing energy *T*_{c} exhibits the “S” shape that is the hallmark of trousers fracture tests for viscoelastic elastomers; see, e.g., Fig. 6 in Ref. [3], Fig. 2 in Ref. [9], and Fig. 6 in Ref. [10]. Specifically, as $a\u02d9\u21980$, for sufficiently slow crack propagation rates, $Tc\u2198Gc$. As $a\u02d9$ increases, so does *T*_{c} monotonically. As $a\u02d9\u2197+\u221e$, for sufficiently fast crack propagation rates, *T*_{c} approaches an asymptotic maximum, *T*_{max} say. The transition of *T*_{c} from its minimum value *G*_{c} to its maximum value *T*_{max} is controlled by both the nonequilibrium elasticity of the elastomer and its viscosity.

*The effect of the nonequilibrium elasticity.* Specifically, for the canonical elastomer under investigation here, the nonequilibrium shear modulus *ν* controls the maximum value *T*_{max} of *T*_{c} and, by the same token, how fast *T*_{c} increases from *G*_{c} to *T*_{max}. Figure 9 illustrates this effect by presenting results of *T*_{c} as a function of $a\u02d9(t)$ for three different nonequilibrium shear moduli, *ν* = 2, 5, 10 MPa. Save for the value *η* = 25 MPa s of the viscosity, the values of the remaining material constants are the same as shows in Fig. 8.

*The effect of the viscosity*. On the other hand, for the canonical elastomer under investigation here, the viscosity *η* controls the range of crack propagation rates $a\u02d9(t)$ over which *T*_{c} increases from *G*_{c} to *T*_{max}. Figure 10 illustrates this effect by presenting results of *T*_{c} as a function of $a\u02d9(t)$ for three different viscosities, *η* = 5, 25, 100 MPa s. The values of the remaining material constants are the same as shown in Fig. 8.

The results in Figs. 9 and 10 make it plain that the critical tearing energy *T*_{c}, as measured from trousers fracture tests carried out at constant deformation rates $l\u02d90$, is primarily a direct manifestation of the viscoelastic behavior—and *not* of the fracture behavior, as commonly portrayed in the literature—of the elastomer at hand. More specifically, it is a manifestation of the nonequilibrium elasticity and the viscosity of the elastomer.

### 4.5 The Local Fields in the Regions $A$, $B$, $C$, and $D$.

For completeness, we close this section by reporting in Fig. 11 a representative contour plot of the equilibrium elastic part *ψ*^{Eq}(*I*_{1}) of the free energy (17) in a trousers specimen deformed at *l*(*t*) = 105.743 mm at a deformation rate $l\u02d90=100mm/s$. The result pertains to an elastomer with nonequilibrium shear modulus *ν* = 2 MPa, viscosity *η* = 5 MPa s, pre-existing crack of length *A* = 51 mm, and is shown over the deformed configuration.

The plot allows to identify the precise locations of the so-called regions $A$, $B$, $C$, and $D$ in the global analysis of the problem; see Fig. 1. In particular, the plot shows that regions $A$, $B$, and $D$ are substantially undeformed, while the crack front region $C$ concentrates all the deformation around the crack front.

## 5 Summary and Final Comments

Since the celebrated works of Rivlin and Thomas [2] and Greensmith and Thomas [3], experimental studies of nucleation and propagation of fracture from large pre-existing cracks in elastomers subjected to quasi-static mechanical loads have been centered on three types of tests:

the “pure-shear” fracture test,

the delayed fracture test, and

the trousers fracture test.

One of three main results that we have established in this work is that there is a critical global stretch *l*_{c}/*l*_{0}—that is, a critical separation between the grips normalized by their initial separation—at which fracture nucleates from the pre-existing crack in a trousers test, irrespective of the length of the pre-existing crack and of the loading rate at which the test is carried out. The existence of such a critical global stretch appears to have gone unnoticed until now.

Since the early pioneering experiments of Greensmith and Thomas [3], it has been well documented that in a trousers test carried out at a constant deformation rate $l\u02d90$, whenever the crack propagates steadily, the resulting force *P*(*t*) at the grips is constant. As a second main result, we have established that this behavior is nothing more than a manifestation of the fact noted earlier that the Griffith criticality condition (2) in a trousers fracture test happens to be satisfied at a critical global stretch *l*_{c}/*l*_{0} that is independent of the crack length and of the loading rate.

As a third main result, we have provided quantitative insight into the effects that the nonequilibrium elasticity and the viscosity of the elastomer have on the critical tearing energy *T*_{c} obtained from a trousers fracture test carried out at a constant deformation rate. This result makes it clear that *T*_{c} is essentially a measure of the capability of the elastomer to dissipate energy through viscous deformation and *not* a measure of its fracture properties. We hope that this result will encourage future experimental studies centered on trousers fracture tests to include separate measurements of the finite viscoelastic behavior of the elastomer being investigated. Regrettably, virtually none of the plethora of experimental studies that have been reported to date in the literature include such measurements.

In conclusion, when viewed collectively, the results presented in this work, together with those presented by Shrimali and Lopez-Pamies [1,8] for the “pure-shear” and delayed fracture tests, provide broad evidence that the Griffith criticality condition (2) may indeed be the universal condition that governs crack growth in elastomers undergoing finite deformations in response to quasi-static mechanical loads.

Accordingly, as suggested in Ref. [8], given the “seamless” mathematical generalization that the Griffith criticality condition (2) provides of the classical Griffith criticality for elastic brittle materials [13], the next sensible step would be to follow in the footstep of Francfort and Marigo [14] in order to turn the Griffith criticality condition (2) into a complete mathematical description of the growth of cracks in viscoelastic elastomers.

## Footnote

The interested reader is referred to Ref. [7] for a detailed account on the viscoelasticy of elastomers. Here, we merely recall that rheological representations of elastomers provide a helpful visualization of the energy partition (4). For instance, in the Zener-type rheological representation depicted in Fig. 2, $WEq$ and $WNEq$ correspond to the elastic energy stored in the equilibrium and nonequilibrium springs, whereas $Wv$ corresponds to the viscous energy dissipated by the dashpot.

## Funding Data

• The National Science Foundation (Grant Nos. CMMI–1901583 and CMMI–2132528).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.