On problems with the determination of the fracture resistance for materials with spatial variations of the Young’s modulus

Abstract

The paper considers the near-tip J-integral $\text {J}_{\mathrm{tip}}$, the far-field J-integral $\text {J}_{\mathrm{far}}$, and the experimental J-integral $\text {J}^{\mathrm{exp}}$ in a material with sinusoidal variation of the Young’s modulus $E$. The evaluations of $\text {J}_{\mathrm{tip}}$ and $\text {J}_{\mathrm{far}}$ are based on the concept of configurational forces; $\text {J}^{\mathrm{exp}}$ is evaluated from the area below the load point displacement curve, as prescribed by the standard testing procedures. Analytic expressions and/or approximation formulae are derived, how $\text {J}_{\mathrm{tip}} ,\,\text {J}_{\mathrm{far}}$ and $\text {J}^{\mathrm{exp}}$ depend on the system parameters of the material, i.e. wavelength and amplitude of the $E$-variation and its phase shift with respect to the crack tip position, and the global specimen dimensions. The analyses show that $\text {J}_{\mathrm{tip}}$ and $\text {J}_{\mathrm{far}}$ exhibit a strong dependency on the phase shift, but not the experimental J-integral $\text {J}^{\mathrm{exp}}$. This is the reason why the current procedures for fracture mechanics testing are not suitable to determine the true values of the fracture initiation toughness $\text {J}_\mathrm{i}$ or the crack growth resistance $R$ of a material, if the material properties exhibit a spatial variation in the direction of crack propagation. Relations are given to estimate the possible errors.

Appendix: Integrals

1.1 Integral $I_\mathrm{u}$

The integral $I_\mathrm{u} \left( {{E_\mathrm{A} }/{E_0 }, {X_0 }/\lambda , R/\lambda } \right)$, Eq. (14b), is re-formulated with $E\left( x \right)$ from Eq. (3) in the form

$$\begin{aligned} I_\mathrm{u}&= \int \limits _{-R}^R {\frac{\partial E}{\partial x}\sqrt{1-\left( {x/R} \right) ^{2}}} \hbox {d}x \nonumber \\&= 2\pi \frac{R}{\lambda }\frac{E_\mathrm{A} }{E_0 }\int \limits _{-1}^1 {\cos } \left( {2\pi \left( {\xi \frac{R}{\lambda }+\frac{X_0 }{\lambda }} \right) } \right) \sqrt{1-\xi ^{2}}\hbox {d}\xi . \end{aligned}$$

This integral can be solved both numerically and analytically, see e.g. http://www.wolfram.com/mathematica. Analytical solutions are given for $R/{\lambda =1}$ and two different values for the relative phase shift, ${X_0 }/\lambda =1/2$ (corresponding to the location of the maxima in $\text {J}_{\mathrm{tip}})$ and ${X_0 }/\lambda =0$ or 1 (corresponding to the location of the minima in $\text {J}_{\mathrm{tip}})$ as

$$\begin{aligned}&I_\mathrm{u} \left( {\frac{E_\mathrm{A} }{E_0 },\frac{1}{2},1} \right) =-\pi \frac{E_\mathrm{A} }{E_0 }\hbox {J}_1 \left( {2\pi } \right) =\pi \cdot 0.21238\frac{E_\mathrm{A} }{E_0 }, \nonumber \\&I_\mathrm{u} \left( {\frac{E_\mathrm{A} }{E_0 },0,1} \right) =\pi \frac{E_\mathrm{A} }{E_0 }\hbox {J}_1 \left( {2\pi } \right) =-\pi \cdot 0.21238\frac{E_\mathrm{A} }{E_0 }. \nonumber \\ \end{aligned}$$

(22)

The quantity $\hbox {J}_1$ symbolizes the Bessel function of first kind of order 1.

1.2 Integral $I_\mathrm{t}$

The integral $I_\mathrm{t} \left( {{E_\mathrm{A} }/{E_0 }, {X_0 }/\lambda , R/\lambda } \right)$, Eq. (14c), follows with Eq. (12b) as

$$\begin{aligned}&I_\mathrm{t} =2\int \limits _0^\pi \left( \frac{3-\nu }{4\cos \theta }+\frac{1-\nu }{2}-\frac{1-\nu }{4}\cos \theta \right) \nonumber \\&\quad \times \ln \! \left( {\frac{1\!+\!{E_\mathrm{A} }/{E_0 \cdot \sin 2\pi \left( {R/{\lambda \cdot \cos \theta \!+\!{X_0 }/\lambda }} \right) }}{1\!+\!{E_\mathrm{A} }/{E_0 \cdot \sin 2\pi {X_0 }/\lambda }}} \!\right) \!\hbox {d}\theta .\nonumber \\ \end{aligned}$$

(23a)

This integral must be solved numerically. Note that the integrand becomes an indefinite form $0/0$ for ${\theta =\pi }/2$, identified as $\left( {3-\nu } \right) /4\cdot 2\pi R/\lambda \times E_\mathrm{A} \cos \left( {2\pi {X_0 }/\lambda } \right) /E\left( {x=0} \right)$.

To avoid numerical problems ${E_\mathrm{A} }/{E_0 }$ should be kept smaller than 0.999.