Toughening and Instability Phenomena in Quantized Fracture Process: Euclidean and Fractal Cracks

Abstract

Basic concept underlying Griffith’s theory of fracture of solids was that, similar to liquids, solids possess surface energy and, in order to propagate a crack by increasing its surface area, the corresponding surface energy must be compensated through the externally added or internally released energy. This assumption works well for brittle solids, but is not sufficient for quasi-brittle and ductile solids.

Here some new forms of energy components must be incorporated into the energy balance equation, from which the input of energy needed to propagate the crack and subsequently the stress at the onset of fracture can be determined. The additional energy that significantly dominates over the surface energy is the irreversible energy dissipated by the way of the plastic strains that precede the leading edge of a moving crack. For stationary cracks, the additional terms within the energy balance equation were introduced by Irwin and Orowan. An extension of these concepts is found in the experimental work of Panin, who showed that the irreversible deformation is primarily confined to the pre-fracture zones associated with a stationary or a slowly growing crack.

The present study is based on the structured cohesive crack model equipped with the “unit step growth” or “fracture quantum.” This model is capable to encompass all the essential issues such as stability of subcritical cracks, quantization of the fracture process, and fractal geometry of crack surfaces and incorporate them into one consistent theoretical representation.

Appendices

Appendix A

Let us recall Eq. 43

$$ \frac{d}{ d X}\left[ p\left(\alpha \right){\left[\frac{2}{\pi}\sqrt{\frac{2 Y}{X}}\right]}^{\frac{1-2\alpha}{\alpha}} Y\right]={M}_f-\frac{1}{2}-\frac{1}{2} \ln \left(4\rho p\left(\alpha \right){\left[\frac{2}{\pi}\sqrt{\frac{2 Y}{X}}\right]}^{\frac{1-2\alpha}{\alpha}} Y\right) $$

(51)

When the product of ${\left(\frac{2}{\pi}\right)}^{\frac{1-2\alpha}{\alpha}}$ and p(α) is denoted by f(α), then the left-hand side (LHS) of Eq. 51 can be written as $f\left(\alpha \right)\frac{d}{ d X}\left[{\left(\frac{2 Y}{X}\right)}^{\frac{1-2\alpha}{2\alpha}} Y\right]$. Now one proceeds with the differentiation

$$ LHS= f\left(\alpha \right)\left[{\left(\frac{2 Y}{X}\right)}^{\frac{1-2\alpha}{2\alpha}}\frac{dY}{dX}+\left(\frac{1}{2\alpha}-1\right) Y{\left(\frac{2 Y}{X}\right)}^{\frac{1}{2\alpha}-2}2\frac{X\frac{dY}{dX}- Y}{X^2}\right] $$

(52)

hence

$$ LHS= f\left(\alpha \right)\left[{\left(\frac{2 Y}{X}\right)}^{\frac{1}{2\alpha}-1}\left(1+\frac{1}{2\alpha}-1\right)\frac{dY}{dX}\right]- f\left(\alpha \right)\left(\frac{1}{\alpha}-2\right){\left(\frac{Y}{X}\right)}^2{\left(\frac{2 Y}{X}\right)}^{\frac{1}{2\alpha}-2} $$

(53)

With G denoting the RHS of Eq. 51, one has

$$ f\left(\alpha \right){\left(\frac{2 Y}{X}\right)}^{\frac{1}{2\alpha}-1}\left[\frac{1}{2\alpha}\frac{dY}{dX}\right]- f\left(\alpha \right)\left(\frac{1}{\alpha}-2\right){\left(\frac{Y}{X}\right)}^2{\left(\frac{2 Y}{X}\right)}^{\frac{1}{2\alpha}-2}= G\left( X, Y,\alpha \right) $$

(54)

This reduces to

$$ \frac{dY}{dX}=\frac{2\alpha G}{f\left(\alpha \right){\left(\frac{2 Y}{X}\right)}^{\frac{1}{2\alpha}-1}}+2\alpha \left(\frac{1}{\alpha}-2\right){\left(\frac{Y}{X}\right)}^2\frac{X}{2 Y} $$

(55)

With ${\left(\frac{2}{\pi}\right)}^{\frac{1-2\alpha}{\alpha}} p\left(\alpha \right)= f\left(\alpha \right) n,$ this equation becomes identical with Eq. 45.

Appendix B

Fracture in an ideally brittle solid (and for the fractal exponent α = ½) occurs when the ductility index ρ = R/Δ → 1. It would be worthwhile to prove that in this case the differential equation governing motion of the subcritical crack in Eq. 23 predicts no stable crack growth and that the δCOD criterion reduces then to the classic case of Griffith. In order to prove this point, let us write the governing equation derived from the δCOD criterion, Eq. 23, in this form:

$$ \begin{array}{l}\frac{dR}{da}= M-\frac{R}{\Delta}+\frac{R}{\Delta} F\left(\Delta / R\right)\\ {} M=\frac{\pi E}{4{\sigma}_Y}\left(\frac{\widehat{u}}{\Delta}\right)\end{array} $$

(56)

where M is the tearing modulus and the function F is defined as follows:

$$ F\left(\Delta / R\right)=\sqrt{1-\frac{\Delta}{R}}-\frac{\Delta}{2 R} \ln \frac{1+\sqrt{1-\frac{\Delta}{R}}}{1-\sqrt{1-\frac{\Delta}{R}}} $$

(57)

For ductile solids, Δ is much smaller than R, and thus ρ≫1. Under this condition, the function F reduces as follows:

$$ F{\left(\frac{\Delta}{R}\right)}_{\rho \gg 1}=1-\frac{\Delta}{2 R}+\frac{\Delta}{2 R} \ln \left(\frac{\Delta}{4 R}\right) $$

(58)

This form leads to the differential in Eq. 24 considered in the last section. To obtain the ideally brittle limit, one needs to expand the function F into a power series for ρ approaching one. The results is

$$ F{\left(\frac{\Delta}{R}\right)}_{\rho \to 1}=-\frac{2}{3}{\left(1-\frac{\Delta}{R}\right)}^{3/2} $$

(59)

When this is substituted into Eq. 56, one obtains the differential equation governing an R-curve for quasi-brittle solids, namely,

$$ \frac{dR}{da}= M-\frac{R}{\Delta}-\frac{2}{3}\frac{R}{\Delta}{\left(1-\frac{\Delta}{R}\right)}^{3/2} $$

(60)

For the ideally brittle solid, two things happen, first, one has R = Δ and, second, the slope of the R-curve defined by Eq. 60 equals zero (the R-curve reduces now to a horizontal line drawn at the level R = R _ini ). Therefore, Eq. 60 reduces to

$$ \frac{dR}{da}=0\kern0.75em \mathrm{or},\kern0.5emM=1 $$

(61)

It is also known that for an ideally brittle solid, the size of the Neuber particle Δ can be identified with the length of the cohesive zone

$$ \Delta = R=\frac{\pi E}{8{\sigma}_Y}\mathrm{CTOD} $$

(62)

Here symbol CTOD stands for the crack tip opening displacement. The final stretch $\widehat{u}$ is now equal half of the CTOD, namely,

$$ \widehat{u}=\frac{1}{2}\mathrm{CTOD} $$

(63)

When Eqs. 62 and 63 are substituted into the definition of the tearing modulus shown in Eq. 56, one gets

$$ M=\frac{\pi E}{4{\sigma}_Y}\left(\frac{\widehat{u}}{\Delta}\right)=\frac{\pi E}{4{\sigma}_Y}\left(\frac{\left(1/2\right)\mathrm{CTOD}}{\left(\pi E/8{\sigma}_Y\right)\mathrm{CTOD}}\right)=1 $$

(64)

In this way it is confirmed that the requirement of zero slope of the R-curve in the limiting case of an ideally brittle solid, expressed by Eq. 61, is satisfied when $\widehat{u}\equiv \left(1/2\right)\mathrm{CTOD}$ and Δ ≡ R. In other words, the δCOD criterion for the onset of fracture reduces to the CTOD criterion of Wells or – equivalently – to the J-integral criterion of Rice. The latter is in full accord with the Irwin driving force criterion G = G _c , and this yields the result identical to the ubiquitous Griffith expression for the critical stress

$$ {\sigma}_G=\sqrt{\frac{2 E\gamma}{\pi a}}=\sqrt{\frac{G_c E}{\pi a}}=\frac{K_c}{\sqrt{\pi a}} $$

(65)

Similar conclusion may be obtained directly from the fact that the R-curve is given as a horizontal line (of zero slope) drawn at the level of R _ini . Setting the equilibrium length of the cohesive zone R equal to its critical value R _c leads to Eq. 65, as expected. To complete this consideration, one is reminded that the quantities R and K_I are related as follows:

$$ R=\frac{\pi}{8}{\left(\frac{K_I}{\sigma_Y}\right)}^2={R}_c=\frac{\pi}{8}{\left(\frac{K_c}{\sigma_Y}\right)}^2\kern0.75em \mathrm{or},\kern0.5em {K}_I={K}_c\kern0.5em \mathrm{or},\kern0.5em {\sigma}_{\mathrm{crit}}={\sigma}_G $$

(66)

Therefore, it has been demonstrated that the nonlinear theory described in the preceding sections encompasses the classic theory of fracture, which becomes now a special case of a more general mathematical representation.