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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.

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Notes

  1. For numerical reasons, the magnitude of \(J_{\mathrm{tip}}^{\mathrm{ep}}\) depends on the mesh size, see Kolednik et al. (2014).

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Acknowledgments

The authors would like to express their gratitude to Prof. R. Pippan and Prof. F.D. Fischer for helpful discussions. Financial support by the Austrian Federal Government and the Styrian Provincial Government within the research activities of the K2 Competence Center on “Integrated Research in Materials, Processing and Product Engineering”, under the frame of the Austrian COMET Competence Center Programme, is gratefully acknowledged (strategic projects A4.11-WP4 and A4.20-WP3).

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

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
figure 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, \(\Delta J^{\mathrm{exp}}=\Delta J\).

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Ochensberger, W., Kolednik, O. A new basis for the application of the \(J\)-integral for cyclically loaded cracks in elastic–plastic materials. Int J Fract 189, 77–101 (2014). https://doi.org/10.1007/s10704-014-9963-3

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Keywords

  • Configurational force concept
  • Crack driving force
  • Cyclic \(J\)-integral
  • Incremental theory of plasticity
  • Low-cycle fatigue