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 ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAZCAYAAAAxFw7TAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAACiSURBVEhL7ZNLDYAwEEQrAAMYwAAGUIACHOAAC1hAAyZwgRmY19JLDz20e4CEl0zSkjLdT9f9fI9G6hNV0UmndD3aJBMwm8KynkHCkPRNWKU9LG0wTZemYNj6nQGLRJeLSd8ZZpgWMUqkR5oQ98XpYkBERDlLmPEthY4fYenPZRvGAX7gqeTGLNaV6KvHEbiU0piYAZlQDrPnhCEj+fN+nLsB5Fgba6vF1sUAAAAASUVORK5CYII=)
Fig.9 shows the
nominal stress-strain data at
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAAVCAYAAAA0GqweAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAEpSURBVEhL7ZQBDQIxDEVPAAYwgAEMoAAFOMABFrCABkzgAjPQd+QnS9ON7kIICXtJc3ej237/OqbBYPB9zhYPi8389WPsLI4WK4srA5+GqlkYB+4We4uItQU5CkQBz+3rtSmQ+XKaIJe9GWtysWAyaAFc8ZwscMmTEUiOitJeiMOQ2/xVgYU1QTAJ0SXkyDHPO4ESF50MYxTeBZt42+Us1fqNWj1IYcyriaCwWktViUQARyI3fAE4zri/xSosao1FUHGzJwxEsGmmcvJqfbkInPAuROCk79MIBHb3WA0cyYgDeicr0LfDIhDX06w4mMnneGs9iBm6/U0icQcLLcqz/J33jHugfqWvJUbrpY6eRBbwUV4ULg7/jYzzxL0eEKlbrjW8IYN/YpqelytKlr/7fYcAAAAASUVORK5CYII=)
for all three materials from (RSb). These
quantities are the Piola axial stress
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAPCAYAAADtc08vAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAB1SURBVDhPYxjeQBKIjXFgvIAHiOcA8X8cGCSHE4A0HwZikCKQC0AARIPEiALZQHwLwkQB64A4BsLED0BOxKYQ5AKQ4QQByAD0QAJ5AZs4VgBS6ABhwgHI+d0QJmEA0gwKAz8oBgUmUU5HBqCYICq+hwVgYAAAz2cX5jwPZN8AAAAASUVORK5CYII=)
and the strain
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAPCAYAAADUFP50AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAB2SURBVDhPYxgewBiIs6HYDyRADFgHxDEQJoMDEP8HYpBBeAHI9MMQJhwQ1AQCakAMspEHiInSAAMgP1UBMciJINtBBhAE3UAMsg0ZgFxAEIACAWQjDIAMQTcIKwAFCgiD/DYHiEGaiLIRBEAKQRolwbyhDhgYAHa3D2r2+/YOAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
. 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 GN 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
![](data:image/png;base64,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)
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.