In this section we use the superscript *R* on the displacements and its gradients, recognizing that all of Sections 3 and 4 apply here when this modification is made. The stress and \({J_v}\)integral notation is unchanged.

A very important result from the elasticity analysis is that \({u_x}^{R} \approx 0.\)This means that the local deformation consists primarily of only normal strain and shear strain in the *y*-direction, as depicted in Fig. 6. The rotation in Fig. 3b is *not* relevant because the *entire* deformation field can be essentially accounted for using only the two gradients \(\partial {u_y}^{R}/\partial x\)and \(\partial {u_y}^{R}/\partial y\) in the strain energy; thus, Green’s strains are not needed to eliminate the effect of rotation. *This finding helps to show why the theory and experiment in (*RSb) *are in such good agreement with very large strains.*

In one comparison in (RSb), the *shape* of the COD in the singularity, was predicted and found to agree with experiment for rubber filled with two different amounts of carbon black (CB), 0.09% and 0.17%.

However, the COD itself was not predicted because the relationship between \({J_v}\)and \(CO{D^R}\)was not available. However, it is now in Eq. (39). Here we shall provide details on the prediction of the \(CO{D^R}\) for the 0.09% and 0.17% CB rubber .The \(CODs\) will then be compared to experimental values.

Additionally, the effect of viscoelasticity on the deformation field will be shown.

## 5.1 Linear viscoelastic creep compliance.

The normalized *master curves* of shear creep compliances are in Figs. 7 and 8 using logarithmic coordinates. They are plotted for a reference temperature of \({25^0}C\), which is the temperature used in the pertinent experiments. The equilibrium compliances are

$$C{s_e}=0.88{\text{ }}{(MPa)^{ - 1}}{\text{for }}0.0{\text{9}}\% {\text{CB and }}C{s_e}=0.{\text{6}}0{\text{ for }}0.{\text{17}}\% {\text{CB}}.$$

49

These compliances were developed from the master curves of the real part of the dynamic shear modulus by using a cubic spline fit to the data. Each master curve in (MTU) has a data spread of approximately 12%, while the fit was to its centerline.

## 5.2 Stress-strain data and determination of

Fig.9 shows the

*nominal* stress-strain data at

for all three materials from (RSb). These

* *quantities are the Piola axial stress

and the strain

. In (RSb) it was assumed that the material was essentially in an equilibrium (elastic) state; this was believed to be acceptable because of the low rate of stress relaxation during the test period.

The stress-strain curves (from a so-called pure shear test) were generated using the same strain rate as in the crack growth study (Urayama 2022). Specifically, specimens were loaded at a displacement rate of 1 mm/s, with a specimen height of 20 mm. In crack growth tests, when the desired constant displacement was reached, it took only 1–2 s while an edge crack was cut, followed immediately by crack growth (Urayama 2022). The time scale of each test was not long enough to neglect viscoelastic effects in the stress-strain behavior, especially in predicting COD, as done here.

This condition necessitates a modification of the data reduction, this time using *pseudo displacements* instead of *displacements*. This correction is needed because, according to theory, the value of \({J_v}\) is equal to the energy release rate based on pseudo displacement, *not* displacement itself.

This correction is done by first modifying the stress-strain curve using pseudo strain for the abscissa. According to Eq. (2),

$${\varepsilon ^R} \equiv \int\limits_{{{0^ - }}}^{t} {\bar {G}(t - \tau )} \frac{{\partial \varepsilon }}{{\partial \tau }}d\tau$$

50

which requires the relaxation modulus. This modulus may be easily constructed from the creep compliance using the well-known equation (Ferry J 1980),

$$G(Lt)=\frac{{\sin (n\pi )}}{{n\pi }}\frac{1}{{{C_S}(Lt)}}$$

51

where

$$n=n(Lt)=\frac{{d\log ({C_S})}}{{\partial \log (Lt)}}$$

52

is the local log-log slope. It is very accurate for functions spread over many decades (Schapery and Park 1999). The relaxation moduli are drawn in Figs. 7 and 8. For the purpose of numerical integration they are fit using a cubic polynomial.

The factor *G*N is also needed. The strain energy Eq.(14), for the test conditions

$${\lambda _1}^{2} \gg {\text{1, }}{\lambda _2}^{2}=1,{\text{ }}{\lambda _3}^{2} \ll 1,{\text{ }}\varepsilon {\text{=}}\frac{{\partial {u_y}}}{{\partial y}},{\text{ thickness=1mm}}$$

53

becomes, in terms of pseudo strains,

$$W={G_N}{({\lambda _1}^{R})^{N+1}}={G_N}{\left( {{\varepsilon ^R}+1} \right)^{N+1}} \approx {G_N}{({\varepsilon ^R})^{N+1}}$$

54

Thus,

$$\sigma =(N+1){G_N}{({\varepsilon ^R})^N}$$

55

The numerical values of various quantities for CB = 0.09 and CB = 0.17 will have 9 and 7 subscripts, respectively.

With *time* as the common parameter, the stress-strain curve is converted to stress-pseudo strain. When \(\varepsilon =3\), the pseudo strains are \({\varepsilon _9}^{R}=4.90\) and \({\varepsilon _7}^{R}={\text{6}}{\text{.47}}\). The complete stress-pseudo strain curves are the solid lines in Fig. 10. Matching Eq. (55) to them at the \(\varepsilon =3\) state gives, in MPa-mm(= kJ/m^2**)** units,

$${G_{9N}}={\text{0}}{\text{.0979, }}{G_{7N}}={\text{0}}{\text{.134}}$$

56

These values are used in Eq. (55) to draw the dotted lines in Figs. 10 and 11.

The solid lines in Fig. 11 are the stress-pseudo strain curves having a maximum value at the pseudo strain used to photograph the COD. That the end point touches the power law means that the far-field strain is high enough to be in the power law regime, assuring that *N* = 1.8 for all strains in the singularity.

For the pure shear test, the value of \({J_v}\) is equal to the specimen height (20 mm) multiplied by the area under the solid lines in Fig. 11. This yields, in MPa-mm units,

$${J_{9v}}=122,{\text{ }}{J_{7v}}=183$$

57

which are the \({J_{{v_{}}}}\)values when the CODs were photographed.

## 5.3 Prediction of the elastic COD

The elastic COD in the singularity is given by Eq. (39),

$$CO{D^R}={( - x)^{\frac{N}{{N+1}}}}{\left[ {\frac{{{J_v}}}{{{G_N}}}} \right]^{\frac{1}{{N+1}}}}{C_J}$$

58

From Eqs. (56) -(58),

$$\begin{gathered} CO{D_9}^{R}=20.0{( - x)^{\frac{N}{{N+1}}}}{\text{ }} \hfill \\ CO{D_7}^{R}=20.7{( - x)^{\frac{N}{{N+1}}}} \hfill \\ \end{gathered}$$

59

which provides the values at *x* = 1,

$${u^R}_{{19}}=10.0,{\text{ }}{u^R}_{{17}}=10.3{\text{ }}$$

60

to be used in the elastic displacements, Eq. (23), which are now designated as \({u_r}^{R}\)and \({u_\theta }^{R}\).

## 5.4 Prediction of the viscoelastic COD

Equation (3) provides the viscoelastic COD,

$$COD=\int\limits_{0}^{t} {\bar {C}(t - \tau )\frac{{\partial CO{D^R}}}{{\partial \tau }}d\tau }$$

61

Assuming crack speed \(\dot {a}\) is constant, at least during the time it takes for the crack to propagate the length of the singularity, and selecting *X* = 0, without loss in actual generality, *COD* becomes

$$COD=\int\limits_{0}^{{\bar {x}}} {\bar {C}(\frac{{\bar {x} - \xi }}{{\dot {a}}})} \frac{{\partial CO{D^R}}}{{\partial \xi }}d\xi$$

62

where

$$\bar {x} \equiv - x$$

63

As done in (Schapery 1975), we change the integration variable to

$$z=\log \left( {1 - \frac{\xi }{{\bar {x}}}} \right)$$

64

and find

$$COD=CO{D^R}\int\limits_{{LL}}^{0} {\bar {C}\left( {\log \left( {\frac{{\bar {x}}}{{\dot {a}}}} \right)+z} \right)} w(z,p)dz$$

65

where the lower limit is \(LL= - \infty\) and a weight function *w*(z, *p*) is defined,

$$w(z,p) \equiv p\ln (10){10^z}{(1 - {10^z})^{p - 1}}$$

66

Also

Now, Eq. (65) does not converge numerically unless the lower limit is finite. The factor \({10^z}\)in the weight function enables the lower limit to be changed to -3 with negligible error. Additionally, the narrow width of the weight function leads to (RASb),

$$COD=CO{D^R}\bar {C}\left( {\frac{{\bar {x}}}{{{s_f}\dot {a}}}} \right)$$

68

for \(\bar {x} \geqslant 0\).The factor \({s_f}\)is a quantity that depends on *N* and the local log-log slope of the creep compliance; for the materials used here \({s_f} \approx 2\)for the relevant range of \(\bar {x}/\dot {a}\).

We can now predict the COD for the two materials. The applicable set of parameters is

$$\begin{gathered} {{\dot {a}}_9}=2510{\text{ }}\frac{{mm}}{s}{\text{, }}{J_{9v}}{\text{=}}122{\text{ }}\frac{{kJ}}{{{m^2}}}{\text{, }}{G_{9N}}{\text{=0}}{\text{.0979 }} \hfill \\ {{\dot {a}}_7}=531{\text{ }}\frac{{mm}}{s}{\text{, }}{J_{7v}}{\text{=}}181{\text{ }}\frac{{kJ}}{{{m^2}}}{\text{, }}{G_{7N}}{\text{=0}}{\text{.134 }} \hfill \\ \end{gathered}$$

69

and, for both materials,

$${C_J}=1.57,{\text{ }}N=1.8,{\text{ }}{s_f}=2,$$

70

With these numbers, we find at the end points \({x_9}= - 6\)and \({x_7}= - 4\), using Eqs. (59) and (65),

\(CO{D_9}=17.2,{\text{ }}CO{D_7}=\) 9.16 (71)

while approximate Eq. (68) yields

\(CO{D_9}=17.4,{\text{ }}CO{D_7}=\) 9.21 (72)

The experimental CODs are,

$$xCO{D_9}=15.9,{\text{ }}xCO{D_7}=9.59$$

73

Thus, the theoretical COD is 8% greater than the experimental value for CB = 0.09%, while it is 4% less than the measured value for CB = 0.17%. Recognizing that there are specimen-to-specimen differences in stress-strain behavior (Urayama 2022), such as the 12% mentioned earlier for the linear viscoelastic modulus, it may be concluded that the theory satisfactorily agrees with the data.

At the end points,

$${\bar {C}_9}=0.274,{\text{ }}{\bar {C}_7}=0.183$$

74

showing that viscoelasticity greatly suppresses the COD compared to the pseudo value (which uses the equilibrium value, \(\bar {C}=1\)).

Moreover, this strong effect of viscoelasticity means that the neglect of first order terms in Green’s pseudo strains produces considerably less error than the actual strains in the singularity.

## 5.5 Effect of viscoelasticity on \({J_v}\).

In (RSb) the\({J_v}\)integral was calculated directly from the data in Fig. 9, using

$${J_v}=\int\limits_{0}^{{\hat {\varepsilon }}} {\sigma (\varepsilon } )d\varepsilon$$

75

where \(\hat {\varepsilon }\) is the applied strain used for the COD pictures. In this case,

$${J_{v9}}=77,{\text{ }}{J_{v7}}=87$$

76

which are considerably less than the values in Eq. (57).

However, the data analysis of crack *growth* in (RSb) made use of *only* the ratio \(\bar {J}={J_v}/\Gamma\), where \(\Gamma\)is the intrinsic fracture energy, as found by shifting data. Consequently, the fracture energy values reported in (RSb) should be increased by the factors

$${f_{\Gamma 9}}=\frac{{122}}{{77}}=1.58{\text{, }}{f_{\Gamma 7}}=\frac{{181}}{{87}}=2.09$$

77

These factors have been found to be essentially the same (within 5%) for all crack speed data points in Fig. 7b,c in (RSb); if this 5% variation were accounted for, there would be no observable effect on the log-log plots. Thus, the only correction needed to (RSb) is the value of Γ (and, of course, the associated change in value of the failure zone parameters, \({\sigma _m}\)and\({v_0}\), such that \({\sigma _m}{v_0}=\Gamma\).

When the stress-strain data are *not* corrected for viscoelasticity, as in (RSb), the COD predictions are,

$$CO{D_9}=11.0,{\text{ }}CO{D_7}{\text{= 4}}{\text{.31}}$$

78

which are much smaller than the experimental values, Eq. (73).

## 5.6 Prediction of the viscoelastic deformation field

Given a constant crack speed for all time we may easily predict the viscoelastic displacement and gradient fields. In contrast to the COD, it is necessary to consider all past time; thus,

\({u_y}=\int\limits_{{ - \infty }}^{t} {\bar {C}(t - \tau )} \frac{{\partial {u_y}^{R}}}{{\partial \tau }}d\tau\) (79)

Use \(x=\dot {a}t\), and make a further change of integration variable, like Eq. (64), but now for positive and negative *x*,

$$z=\log \left( {1+\frac{\xi }{x}} \right)$$

80

With these two changes Eq. (79) becomes

$${u_y}= - \left| x \right|\ln (10)\int\limits_{{ - LL}}^{{UL}} {\bar {C}\left( {\log \left( {\frac{{\left| x \right|}}{{\dot {a}}}} \right)+z} \right)} {w_u}(z,p)dz$$

81

in which lower *LL* and upper *UU* limits must be large, but finite for convergence of the numerical integration. The weight function is

$$\begin{gathered} {\text{for x<}}0, \hfill \\ {w_u}=\frac{{\partial {u_y}^{R}}}{{\partial x}}\left[ {(1 - {{10}^z})x,y} \right]{10^z} \hfill \\ {\text{for x>}}0, \hfill \\ {w_u}=\frac{{\partial {u_y}^{R}}}{{\partial x}}\left[ {(1+{{10}^z})x,y} \right]{10^z} \hfill \\ \end{gathered}$$

82

In the following examples \({u_1}^{R}=10\)and \(\dot {a}=2510\) are used, as before. The viscoelastic displacement and gradients are drawn in Figs. 12–14. The gradients \({w_{ux}}\) and \({w_{uy}}\)are predicted by simply replacing \({u_y}^{R}\)with \(\partial {u_y}^{R}/\partial x{\text{ }}\)and \(\partial {u_y}^{R}/\partial y\), respectively, in Eq. (82)

The weight functions are drawn in Figs. 12–14 for selected (*x*, *y*); but they are similar for all (*x*, *y*). We can use *LL =* 3 and *UL* = 6.6 because there is no significant contribution to the integrals beyond these limits. The numerical integration for \(\partial {u_y}/\partial y\)does not converge for *UL* > 6.6. The farther the location from the crack tip, the larger the limits must be, especially *UL*.

These weight functions are much broader than that for the COD, Eq. (66), which is approximately one decade. As a result, approximations such as Eq. (68) will not be nearly as accurate.