A new basis for the application of the $J$-integral for cyclically loaded cracks in elastic–plastic materials

Abstract

Fatigue crack propagation is by far the most important failure mechanism. Often cracks under low-cycle fatigue conditions and, especially, short fatigue cracks cannot be treated with the conventional stress intensity range $\Delta K$-concept, since linear elastic fracture mechanics is not valid. For such cases, Dowling and Begley (ASTM STP 590:82–103, 1976) proposed to use the experimental cyclic $J$-integral $\Delta J^{\exp }$ for the assessment of the fatigue crack growth rate. However, severe doubts exist concerning the application of $\Delta J^{\exp }$. The reason is that, like the conventional $J$-integral, $\Delta J^{\exp }$ presumes deformation theory of plasticity and, therefore, problems appear due to the strongly non-proportional loading conditions during cyclic loading. The theory of configurational forces enables the derivation of the $J$-integral independent of the constitutive relations of the material. The $J$-integral for incremental theory of plasticity, $J^{\mathrm{ep}}$, has the physical meaning of a true driving force term and is potentially applicable for the description of cyclically loaded cracks, however, it is path dependent. The current paper aims to investigate the application of $J^{\mathrm{ep}}$ for the assessment of the crack driving force in cyclically loaded elastic–plastic materials. The properties of $J^{\mathrm{ep}}$ are worked out for a stationary crack in a compact tension specimen under cyclic Mode I loading and large-scale yielding conditions. Different load ratios, between pure tension- and tension–compression loading, are considered. The results provide a new basis for the application of the $J$-integral concept for cyclic loading conditions in cases where linear elastic fracture mechanics is not applicable. It is shown that the application of the experimental cyclic $J$-integral $\Delta J^{\exp }$ is physically appropriate, if certain conditions are observed.

Appendix: \(J\)-integral and cyclic \(J\)-integral

In this appendix, the difference between $J$-integral, Eq. (1), and cyclic $J$-integral, Eq. (3), shall be demonstrated, and Eq. (19) is derived for a simple specimen geometry.

Figure 12a shows a double cantilever beam (DCB) specimen with height $h$ and a long crack. The material is linear elastic. The specimen is clamped at the lower boundary and loaded at the upper boundary by prescribing the vertical displacement $v$. Rice (1968a) showed that the $J$-integral can be expressed as,

$$\begin{aligned} J=\phi _{\mathrm{rb}} h=\frac{1}{2}E\varepsilon _{\mathrm{appl}}^2 h. \end{aligned}$$

(25)

The term $\phi _{\mathrm{rb}}$ denotes the strain energy density at the right boundary, and the applied strain is given by $\varepsilon _{\mathrm{appl}} =v/h$. At a fixed displacement $v_{\min }$, the $J$-integral $J_{\mathrm{min}} =\phi _{\mathrm{rb,min}} h=\frac{1}{2}E\varepsilon _{\mathrm{appl,min}}^2 h$ corresponds to the area 0AE0 in Fig. 12b.

Fig. 12

a Double cantilever beam specimen with a long crack. The lower boundary is clamped. A constant displacement in $y$-direction, $v$, is prescribed on the upper boundary; the displacements $u$ in $x$-direction are free. b Load-displacement ($F$–$v$) curve and stress–strain $\left( {\sigma -\varepsilon }\right)$ curve for a linear elastic material. For cyclic loading between the displacements $v_{\min }$ and $v_{\max }$, the origin of the coordinate system is reset into point A

Now the specimen shall be loaded from the displacement $v_{\min }$ to $v_{\max }$, Fig. 12b. The increase in $J$-integral is then

$$\begin{aligned} J_{\max } -J_{\min }&= \left( {\phi _{\mathrm{rb,max}} -\phi _{\mathrm{rb,min}}} \right) h\nonumber \\&= \frac{1}{2}\;E\left( {\varepsilon _{\mathrm{appl,max}}^2 -\varepsilon _{\mathrm{appl,min}}^2} \right) h, \end{aligned}$$

(26)

corresponding to the area ABDEA in Fig. 12b. During cyclic loading between the displacements $v_{\min }$ and $v_{\max }$, the $J$-integral varies between the values of $J_{\min }$ and $J_{\mathrm{max}} =\phi _{\mathrm{rb,max}} h=\frac{1}{2}E\varepsilon _{\mathrm{appl,max}}^2 h$.

For calculating the cyclic $J$-integral $\Delta J$ as defined by Eq. (3), we should note that the cyclic stress $\Delta \sigma$ and cyclic strain $\Delta \varepsilon$ are taken from the minimum load, point A in Fig. 12b; the quantity $\phi \left( {\Delta \varepsilon }\right)$ of Eq. (4) corresponds to the area ABCA in Fig. 12b. The cyclic $J$-integral is then given by

$$\begin{aligned} \Delta J&= \phi \left( {\Delta \varepsilon } \right) h=\frac{1}{2} \;E\left( {\Delta \varepsilon }\right) ^{2}h\nonumber \\&= \frac{1}{2}\;E \left( {\varepsilon _{\mathrm{appl,max}} -\varepsilon _{\mathrm{appl,min}}}\right) ^{2}h, \end{aligned}$$

(27)

again corresponding to the area ABCA. The comparison of Eqs. (26) and (27) shows that $\Delta J\ne J_{\max } -J_{\min }$ and that

$$\begin{aligned} \Delta J=J_{\max } +J_{\min } -2\sqrt{J_{\max } J_{\min }}, \end{aligned}$$

(28)

which is in form identical to Eqs. (19) or (22).

The areas 0AEA and 0BD0 correspond to the areas $A_{\min }$ and $A_{\max }$ under the load–displacement ($F$–$v$) curve and, therefore, the magnitudes of the experimental $J$-integrals $J_{\min }^{\mathrm{exp}}$ and $J_{\max }^{\mathrm{exp}}$ can be evaluated from Eq. (2). By inserting the appropriate geometry factor $\eta$, we will find that $J_{\mathrm{min}}^{\mathrm{exp}} =J_{\mathrm{min}}$ and $J_{\mathrm{max}}^{\mathrm{exp}} =J_{\mathrm{max}}$. Similarly, the area ABCA in Fig. 12b corresponds to the area $\Delta A$ of a loading branch of the $F$–$v$-curve during a single load cycle, and we find that the experimental cyclic $J$-integral of Eq. (5) and the cyclic $J$-integral of Eq. (3) yield equal results, $\Delta J^{\mathrm{exp}}=\Delta J$.