Physically appropriate characterization of fatigue crack propagation rate in elastic–plastic materials using the $J$-integral concept

Abstract

The current paper discusses the physically correct evaluation of the driving force for fatigue crack propagation in elastic–plastic materials using the $J$-integral concept. This is important for low-cycle fatigue and for short fatigue cracks, where the conventional stress intensity range ($\Delta K$) concept cannot be applied. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008) , have derived the $J$-integral for elastic–plastic materials with incremental theory of plasticity, $J^{\mathrm{ep}}$, which is applicable for cyclic loading and/or for growing cracks, in contrast to the conventional $J$-integral. The variation of this incremental plasticity $J$-integral $J^{\mathrm{ep}}$ is studied in numerical investigations conducted on two-dimensional C(T)-specimens with long cracks under cyclic Mode I loading. The crack propagates by an increment after each load cycle. The maximum load is varied so that small- and large-scale yielding conditions prevail. Three different load ratios are considered, from pure tension to tension-compression loading. By theoretical considerations and comparisons with the variation of the crack tip opening displacement $\delta _{\mathrm{t}}$, it is demonstrated that the cyclic, incremental plasticity $J$-integral $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, which is computed for a contour around the active plastic zone of the growing crack, is physically appropriate to characterize the growth rate of fatigue cracks. The validity of the experimental cyclic $J$-integral, $\Delta J^{\mathrm{exp}}$, proposed by Dowling and Begley (ASTM STP 590:82–103, 1976), is also investigated. The results show that $\Delta J^{\mathrm{exp}}$ is correct for the first load cycle, however, not fully appropriate for a growing fatigue crack.

1 Introduction

This paper deals with the physically correct evaluation of the driving force of cyclically loaded, growing cracks in elastic–plastic materials for cases where linear elastic fracture mechanics is not applicable.

The conventional $J$-integral $J^{\mathrm{conv}}$ (Rice 1968a, b), which is commonly applied in the regime of nonlinear fracture mechanics, relies on deformation theory of plasticity, i.e. the elastic–plastic material is treated as being nonlinear elastic. For this reason, $J^{\mathrm{conv}}$ suffers from two fundamental problems when it is applied to elastic–plastic materials: (i) $J^{\mathrm{conv}}$ is formally not applicable for non-proportional loading conditions (Rice 1968a, b; Anderson 1995), (ii) $J^{\mathrm{conv}}$ does not describe the real driving force of a crack in elastic–plastic materials (Rice 1968a, b). In spite of these problems, Dowling and Begley (1976) proposed the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ as a parameter characterizing the growth rate ${\hbox {d}a}/{\hbox {d}N}$ of fatigue cracks for cases where the stress intensity range $\Delta K$ is not applicable. Although supported for some materials by experimental data (e.g. Dowling and Begley 1976; Dowling 1976; Lambert et al. 1988; Banks-Sills and Volpert 1991), the applicability of $\Delta J^{\exp }$ has remained doubtful due to the lack of its theoretical basis (Suresh 1998).

New insight into this problem has been gained by adopting the concept of configurational forces, which enables the derivation of the $J$-integral for elastic–plastic materials with incremental theory of plasticity (Simha et al. 2008). This incremental plasticity $J$-integral $J^{\mathrm{ep}}$ has the physical meaning of a real driving force term of a crack in an elastic–plastic material even under strongly non-proportional loading conditions, however it is path dependent (Simha et al. 2008). Kolednik et al. (2014) studied this path dependence and demonstrated the usefulness of $J^{\mathrm{ep}}$ for stationary and growing cracks under monotonic loading conditions. In a very recent study, Ochensberger and Kolednik (2014) have investigated the application of $J^{\mathrm{ep}}$ for stationary cracks in elastic–plastic materials that are cyclically loaded, and it has been shown that the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ is, in principle, correct, if certain conditions are observed.

The current paper complements the study of Ochensberger and Kolednik (2014) by considering growing cracks in elastic–plastic, cyclically loaded materials. It will be demonstrated that important differences occur compared to the case of a stationary crack. The paper shall provide a new basis for the application of the $J$-integral concept for characterizing the crack growth rate in fatigue.

The next section briefly reviews the incremental plasticity $J$-integral $J^{\mathrm{ep}}$ and the findings of the papers by Kolednik et al. (2014) and Ochensberger and Kolednik (2014) that are necessary for the understanding of the current paper. Readers who are already familiar with the topic may continue reading at the last paragraph in Sect. 2.

2 Incremental plasticity $J$-integral $J^{\mathrm{ep}}$ and crack driving force

Configurational forces are thermodynamic driving forces on defects in materials (Maugin 1995; Gurtin 1995, 2000; Kienzler and Herrmann 2000). The application of the configurational force concept for studying fracture mechanics problems has gained increasing research interest, see e.g. Simha et al. (2003; 2005), Nguyen et al. (2005), Özenç et al. (2014), Sistaninia and Kolednik (2014), Kolednik et al. (2014) and the according references in Sect. 2 therein. The concept rests on the notion of the second-rank configurational stress tensor, which is defined in the form, $\mathbf{C}=\phi \mathbf{I}-\mathbf{F}^{\mathrm{T}}\mathbf{S}$ (Eshelby 1951, 1970).¹ The parameter $\phi$ denotes the strain energy density, I the identity tensor, $\mathbf{F}^{\mathrm{T}}$ the transposed of the deformation gradient tensor F, and S the $1^{\mathrm{st}}$ Piola-Kirchhoff stress. A configurational force f in a body is associated with the divergence of the configurational stress tensor,

$$\begin{aligned} \mathbf{f}=-\nabla \cdot \mathbf{C}=-\nabla \cdot \left( {\phi \mathbf{I}-\mathbf{F}^{\mathrm{T}}\mathbf{S}}\right) . \end{aligned}$$

(1)

The vector f gives the magnitude and direction of the thermodynamic driving force acting on the defect.

2.1 Configurational forces and $J$-integrals for elastic–plastic materials

A literature review on the application of configurational force concept for the prediction of the behavior of cracks has been given in Kolednik et al. (2014) and shall not be repeated here. Figure 1a shows a sketch of a homogeneous body ${\mathcal {B}}$ containing a crack with length $a_{0}$ and a unit vector in the nominal crack growth direction e. The configurational force concept allows the derivation of the $J$-integral. The scalar, near-tip $J$-integral $J_{\mathrm{tip}}$ is related to the configurational force vector emanating from the crack tip, $\mathbf{f}_{\mathrm{tip}}$, in the form (see e.g. Simha et al. 2003),

$$\begin{aligned} J_{\mathrm{tip}} =\mathbf{e}\cdot \mathbf{J}_{\mathrm{tip}} =\mathbf{e}\cdot \left( {-\mathbf{f}_{\mathrm{tip}}}\right) =\mathbf{e}\cdot \lim _{r\rightarrow 0} \int \limits _{\Gamma _r} {\mathbf{C}\,\mathbf{n}\,\hbox {d}s}, \end{aligned}$$

(2)

where $\Gamma _r$ is a contour drawn from the lower to the upper crack surface in counterclockwise direction at a distance $r$ around the crack tip; n is the outward unit normal vector to the contour $\Gamma$, and d$s$ is an increment of the integration path. The scalar $J$-integral $J_{\mathrm{tip}}$ of Eq. (2) is the projection of the near-tip $J$-integral vector $\mathbf{J}_{\mathrm{tip}}$ into the nominal crack growth direction e. The $J$-integral along an arbitrary contour $\Gamma$ can be evaluated from the relation,

$$\begin{aligned} J&= \mathbf{e}\cdot \mathbf{J}=\mathbf{e}\cdot \int \limits _\Gamma {\mathbf{C}\,\mathbf{n}\,\hbox {d}s} =\mathbf{e}\cdot \int \limits _\Gamma {\left( {\phi \mathbf{I}-\mathbf{F}^{\mathrm{T}}\mathbf{S}}\right) \mathbf{n}\,\hbox {d}s}\nonumber \\&= J_{\mathrm{tip}} - \mathbf{e}\cdot \int \limits _{\mathcal {D}} {\mathbf{f}\,\hbox {d}A} , \end{aligned}$$

(3)

where ${\mathcal {D}}$ denotes the area bounded by $\Gamma$, but excluding the crack tip. Note that Eqs. (2) and (3) do not rely on constitutive equations of the material.

Fig. 1

a Homogeneous elastic–plastic body ${\mathcal {B}}$ with a long, stationary crack. The contour $\Gamma _{\mathrm{PZ}}$ encloses the crack tip plastic zone. b Body ${\mathcal {B}}$ after crack extension $\Delta a$. The contour $\Gamma _{\mathrm{actPZ}}$ encloses the active plastic zone of the current crack tip, whereas the contour $\Gamma _{\mathrm{PZ}}$ encloses the entire crack tip plastic zone, including the plastic wake. c Stress–strain $(\sigma - \varepsilon )$ curve for point P in the plastic zone. Only the elastic part of the strain energy density $\phi _\mathrm{e}$ is reversible. The total strain energy density $\phi =\phi _\mathrm{e} +\phi _\mathrm{p}$ would be recoverable in a comparable nonlinear elastic material with the identical $\sigma -\varepsilon$-curve

We assume that body ${\mathcal {B}}$ is homogeneous and consists of elastic–plastic material. Then we can distinguish between two different types of $J$-integral, the conventional $J$-integral $J^{\mathrm{conv}}$, which presumes deformation theory of plasticity (see e.g. Simha et al. 2003), and the incremental plasticity $J$-integral $J^{\mathrm{ep}}$ for materials with incremental theory of plasticity (Simha et al. 2008),

$$\begin{aligned} J^{\mathrm{conv}}&= J_{\mathrm{tip}}^{\mathrm{conv}} + \mathbf{e}\cdot \int \limits _{\mathcal {D}} {\nabla \cdot \left( {\phi \mathbf{I}-\mathbf{F}^{\mathrm{T}}\mathbf{S}}\right) \hbox {d}A}\nonumber \\&= J_{\mathrm{tip}}^{\mathrm{conv}} - \mathbf{e}\cdot \int \limits _{\mathcal {D}} {\mathbf{f}^{\mathrm{def.pl}} \hbox {d}A} ,\end{aligned}$$

(4)

$$\begin{aligned} J^{\mathrm{ep}}&= J_{\mathrm{tip}}^{\mathrm{ep}} + \mathbf{e}\cdot \int \limits _{\mathcal {D}} {\nabla \cdot \left( {\phi _\mathrm{e} \mathbf{I}-\mathbf{F}^{\mathrm{T}}\mathbf{S}}\right) \hbox {d}A} \nonumber \\&= J_{\mathrm{tip}}^{\mathrm{ep}} - \mathbf{e}\cdot \int \limits _{\mathcal {D}} {\mathbf{f}^{\mathrm{ep}} \hbox {d}A} . \end{aligned}$$

(5)

The difference between $J^{\mathrm{conv}}$ and $J^{\mathrm{ep}}$ appears in the substitution of the strain energy density $\phi$. For deformation plasticity, the total strain energy density $\phi$ is inserted as for a nonlinear elastic material, whereas for incremental plasticity only the elastic part of the strain energy density $\phi _\mathrm{e}$ is inserted, see Fig. 1c. From Eqs. (4) and (5), or by inserting either $\phi$ or $\phi _\mathrm{e}$ into Eq. (1), it is clear that also two different types of configurational force exist, the deformation plasticity configurational force $\mathbf{f}^{\mathrm{def.pl}}$ and the elastic–plastic configurational force $\mathbf{f}^{\mathrm{ep}}$.

If the body ${\mathcal {B}}$ deforms only elastically, there exists no difference between $\mathbf{f}^{\mathrm{def.pl}}$ and $\mathbf{f}^{\mathrm{ep}}$, or between $J^{\mathrm{conv}}$ and $J^{\mathrm{ep}}$. The bulk configurational force f vanishes, and configurational forces appear only at the crack tip, $\mathbf{f}_{\mathrm{tip}}$, and at the external boundary $\partial {\mathcal {B}}$ (Fischer et al. 2012). Therefore, the $J$-integral is path-independent, see the right-hand side extension of Eq. (4).

If body ${\mathcal {B}}$ is also plastically deformed, bulk configurational forces $\mathbf{f}^{\mathrm{ep}}$ are induced in the plastically deformed regions of the body and, according to the right-hand side extension of Eq. (5), the incremental plasticity $J$-integral $J^{\mathrm{ep}}$ becomes path-dependent. The bulk configurational force $\mathbf{f}^{\mathrm{ep}}$ at a material point in the elastic–plastic body evolves proportional to the gradient of the plastic component of the deformation gradient (Simha et al. 2008),

$$\begin{aligned} \mathbf{f}^{\mathrm{ep}}=\left( {\mathbf{F}^{\mathrm{e}}}\right) ^{\mathrm{T}}\mathbf{S}:\frac{\partial \mathbf{F}^{\mathrm{p}}}{\partial \mathbf{X}}. \end{aligned}$$

(6)

In Eq. (6), $\mathbf{F}^{\mathrm{e}}$ and $\mathbf{F}^{\mathrm{p}}$ are the elastic and plastic components of the deformation gradient tensor F, and ${\partial \mathbf{F}^{\mathrm{p}}}/{\partial \mathbf{X}}$ is the gradient of $\mathbf{F}^{\mathrm{p}}$ with respect to the unloaded reference coordinate system.

Performing a numerical cyclic tensile test, Kolednik et al. (2014) have demonstrated the problem of idealizing elastic–plastic materials with deformation theory of plasticity: artificial bulk configurational forces $\mathbf{f}^{\mathrm{def.pl}}$ emerge on positions with a gradient in plastic strain as soon as non-proportional loading occurs. These bulk configurational forces do not have a physical background and lead to an artificial path dependence of $J^{\mathrm{conv}}$, compare Eq. (4); see also Brocks et al. (2003) or, e.g., in Kuna (2008).

A big advantage of the incremental plasticity $J$-integral $J^{\mathrm{ep}}$ is that it has the physical meaning of a true driving force term in elastic–plastic materials even for non-proportional loading conditions, such as a growing crack or a cyclically loaded crack, while, on the contrary, the conventional $J$-integral $J^{\mathrm{conv}}$ possesses the well-known restrictions outlined in the Introduction.

2.2 Path dependence of $J^{\mathrm{ep}}$ and driving force for cracks under monotonic loading

The path dependence of the incremental plasticity $J$-integral $J^{\mathrm{ep}}$ for stationary and growing cracks in monotonically loaded elastic–plastic materials has been investigated in Kolednik et al. (2014). The investigation showed that for a stationary crack, the $J$-integral for a contour enclosing the entire crack tip plastic zone $J_{\mathrm{PZ}}^{\mathrm{ep}}$, Fig. 1a, should be taken as parameter characterizing the driving force. Crack extension occurs, if $J_{\mathrm{PZ}}^{\mathrm{ep}}$ is equal or larger than the crack growth resistance.

The incremental plasticity $J$-integral for a contour enclosing the crack tip plastic zone $J_{\mathrm{PZ}}^{\mathrm{ep}}$ has the physical meaning of the driving force for the combined movement of the crack tip and the crack tip plastic zone. From a comparison of Fig. 1a, b it becomes clear that it is impossible for a crack to grow in an elastic–plastic material without the simultaneous movement of the surrounding plastic zone. Thus, the near-tip $J$-integral $J_{\mathrm{tip}}^{\mathrm{ep}}$, which is the driving force for the translational movement of the crack tip alone, is meaningless for the assessment of crack extension. The numerical results in Kolednik et al. (2014) suggest that the magnitude of the incremental plasticity near-tip $J$-integral is zero, $J_{\mathrm{tip}}^{\mathrm{ep}} =0$, for both a stationary and a growing crack under monotonic loading. As noted already in classical papers, e.g. Rice and Johnson (1970) or Rice (1979), also the conventional, deformation plasticity near-tip $J$-integral is zero, $J_{\mathrm{tip}}^{\mathrm{conv}} =0$, for stationary and growing cracks under monotonic loading.

For a contour around the crack tip plastic zone, the conventional $J$-integral $J_{\mathrm{PZ}}^{\mathrm{conv}}$ is identical to the incremental plasticity $J$-integral $J_{\mathrm{PZ}}^{\mathrm{ep}}$,

$$\begin{aligned} J_{\mathrm{PZ}}^{\mathrm{ep}} =J_{\mathrm{PZ}}^{\mathrm{conv}}. \end{aligned}$$

(7)

The requirement for this equality is that the crack tip plastic zone is completely surrounded by material that is only elastically deformed. In this case there is no difference in the formulation between deformation- and incremental plasticity, see middle term in Eq. (3).

Notice that a relation similar to Eq. (7) for the far-field $J$-integrals, $J_{\mathrm{far}}^{\mathrm{ep}} =J_{\mathrm{far}}^{\mathrm{conv}}$, only exists, if no part of the outer boundary of the specimen is plastically deformed. A back-face plasticity region appears in case of large-scale yielding (lsy) conditions, compare Fig. 4b. Configurational forces $\mathbf{f}^{\mathrm{ep}}$ with a positive component in $x$-direction are induced in the back-face plasticity region so that $J^{\mathrm{ep}}$ decreases and $J_{\mathrm{far}}^{\mathrm{ep}} <J_{\mathrm{PZ}}^{\mathrm{ep}}$, compare Eq. (5). On the contrary, the conventional $J$-integral remains constant, $J_{\mathrm{far}}^{\mathrm{conv}} =J_{\mathrm{PZ}}^{\mathrm{conv}}$, since proportional loading conditions prevail. It is well known that for a stationary crack the conventional $J$-integral $J^{\mathrm{conv}}$ equals the experimental $J$-integral $J^{\mathrm{exp}}$, which is determined from the load–displacement curve (Rice et al. 1973; Kolednik 1991). Therefore, the incremental plasticity $J$-integral for a contour around the crack tip plastic zone $J_{\mathrm{PZ}}^{\mathrm{ep}}$ equals the experimental $J$-integral for a stationary crack, $J_{\mathrm{PZ}}^{\mathrm{ep}} =J_{\mathrm{PZ}}^{\mathrm{conv}} =J_{\mathrm{far}}^{\mathrm{conv}} =J^{\mathrm{exp}}$. This means that both the conventional and the experimental $J$-integral give the correct driving force for the initiation of crack growth.

For a continuously growing crack at constant load, the $J$-integral $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ for a contour $\Gamma _{\mathrm{actPZ}}$ around the active plastic zone of the moving crack tip, Fig. 1b, is the physically correct crack driving force parameter (Kolednik et al. 2014). Here, the contour $\Gamma _{\mathrm{PZ}}$ encloses the initial plastic zone of the stationary crack, the plastically deformed regions along the crack flanks (plastic wake), and the active plastic zone around the current crack tip. Since the contour $\Gamma _{\mathrm{PZ}}$ goes only through elastic deformed regions, Eq. (7) is still valid. However, note that $J_{\mathrm{actPZ}}^{\mathrm{ep}} \ne J_{\mathrm{actPZ}}^{\mathrm{conv}}$, since the contour $\Gamma _{\mathrm{actPZ}}$ crosses the plastic wake along the crack flanks (Fig. 1b). It should be mentioned that the experimental $J$-integral $J^{\mathrm{exp}}$ does, in general, not reflect the driving force for a growing crack, $J_{\mathrm{actPZ}}^{\mathrm{ep}} \ne J^{\mathrm{exp}}$; see Kolednik (1991; 1993) and Turner and Kolednik (1994). This means that neither the conventional nor the experimental $J$-integral give the correct driving force for a continuously growing crack under monotonic loading.

2.3 Driving force for cyclically loaded, stationary cracks in elastic–plastic materials

The characteristic properties of the incremental plasticity $J$-integral $J^{\mathrm{ep}}$ under cyclic loading conditions, but for a stationary crack, have recently been presented in Ochensberger and Kolednik (2014). The path-dependence of $J^{\mathrm{ep}}$ was investigated for various, positive and negative, load ratios $R={F_{\min }}/{F_{\max }}$ under lsy-conditions. It has been shown that negative $J^{\mathrm{ep}}$-values can appear during the unloading stages on contours close to the crack tip; they originate from compressive residual stresses caused by reverse plasticity within the crack tip plastic zone.

The incremental plasticity $J$-integral for a contour enclosing the crack tip plastic zone $J_{\mathrm{PZ}}^{\mathrm{ep}}$ also characterizes the driving force of a crack in an elastic–plastic material, which is monotonically loaded after a cyclic pre-deformation. It is clear that $J_{\mathrm{PZ}}^{\mathrm{ep}} =J_{\mathrm{PZ}}^{\mathrm{conv}}$, since Eq. (7) is valid also for cyclic loading conditions, as long as the crack tip plastic zone is surrounded by elastically deformed material.

Fig. 2

To the determination of the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ for a C(T) or deeply notched bend specimen from the area $\Delta A$ below a single loading branch of the load–displacement $(F-v)$ curve, a for load ratio $R > 0$ without crack closure, b for $R < 0$. The load $F_{\mathrm{op}}$ denotes the point of crack opening

In the fatigue of metals and alloys, the magnitude of the crack driving force at the maximum load $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ is considerably smaller than the crack growth resistance, so that the crack cannot extend at $F_{\mathrm{max}} =\hbox {constant}$. Driving force terms for fatigue crack growth have been introduced, which should allow the prediction of the crack propagation rate of a fatigue crack (e.g. see Suresh 1998). Note that these terms are not necessarily real driving force terms in the thermodynamic sense. The stress intensity range $\Delta K$ (Paris et al. 1961; Paris and Erdogan 1963) or the effective stress intensity range $\Delta K_{\mathrm{eff}}$ (Elber 1970, 1971) are used, if linear elastic fracture mechanics is applicable. For the regime of elastic–plastic fracture mechanics, Dowling and Begley (1976) proposed the application of the experimental cyclic $J$-integral $\Delta J^{\exp }$, which is determined from a single loading or unloading branch of the load–displacement ($F-v$) curve, similar to $J^{\exp }$ for monotonic loading. For deeply notched bend- and C(T)-specimens, $\Delta J^{\exp }$ is given by the relation

$$\begin{aligned} \Delta J^{\mathrm{exp}}=\frac{\eta \Delta A}{b B}, \end{aligned}$$

(8)

where $\Delta A$ is the area below a single loading branch of the $F-v$-curve (Fig. 2a), $b=W-a$ is the ligament length, with $W$ as the specimen width and $a$ as the crack length, and $B$ is the specimen thickness. The geometry factor $\eta (a/W)$ depends on the specimen type (see also ESIS P2-92 1992 or ASTM E1820 2005). In spite of empirical results showing that $\Delta J^{\exp }$ correlates to ${\hbox {d}a}/{\hbox {d}N}$ for specific materials under certain cyclic loading conditions (e.g. Dowling and Begley 1976; Dowling 1976; Lambert et al. 1988; Banks-Sills and Volpert 1991), the general applicability of $\Delta J^{\exp }$ remained doubtful due to the lack of theoretical basis (Anderson 1995; Suresh 1998).

Ochensberger and Kolednik (2014) have shown that the incremental plasticity cyclic $J$-integral for a contour around the crack tip plastic zone, $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$, is a physically appropriate parameter for characterizing the driving force for a cyclically loaded, stationary crack. The parameter $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ should be evaluated by the relation,

$$\begin{aligned} \Delta J_{\mathrm{PZ}}^{\mathrm{ep}} =J_{\mathrm{PZ,max}}^{\mathrm{ep}} +J_{\mathrm{PZ,}\min }^{\mathrm{ep}} -2\sqrt{J_{\mathrm{PZ,}\max }^{\mathrm{ep}} J_{\mathrm{PZ,}\min }^{\mathrm{ep}}}. \end{aligned}$$

(9)

Here, $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ,min}}^{\mathrm{ep}}$ denote the maximum and minimum $J_{\mathrm{PZ}}^{\mathrm{ep}}$-values achieved in a single load cycle. It should be mentioned that the magnitude of $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ is, due to the square root term in Eq. (9), very sensitive to small values of $J_{\mathrm{PZ,min}}^{\mathrm{ep}}$. This fact is especially important for small positive load ratios. For a stationary crack, negative load ratios gave minimum $J_{\mathrm{PZ}}^{\mathrm{ep}}$-values of exactly zero, $J_{\mathrm{PZ,min}}^{\mathrm{ep}} =0$, leading to $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}} =J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ (Ochensberger and Kolednik 2014).

The application of Eq. (9) for the correct calculation of the cyclic $J$-integral can be demonstrated by comparison with the cyclic crack tip opening displacement $\Delta \delta _{\mathrm{t}}$ (Ochensberger and Kolednik 2014). In a very recent study, Metzger et al. (2014) demonstrated the correlation between the conventional cyclic $J$-integral (Lamba 1975; Wüthrich 1982; Tanaka 1983) and $\Delta \delta _{\mathrm{t}}$. Note that the conventional cyclic $J$-integral used in Metzger et al. (2014) is, in principle, equal to the expression in Eq. (9), see Appendix in Ochensberger and Kolednik (2014) for details.

If the crack tip plastic zone is completely surrounded by elastically deformed material, Eq. (7) applies for both the $J^{\mathrm{ep}}$-values at maximum and minimum load. Therefore, the incremental plasticity and deformation plasticity cyclic $J$-integrals for a contour around the crack tip plastic zone must be equal,

$$\begin{aligned} \Delta J_{\mathrm{PZ}}^{\mathrm{ep}} =\Delta J_{\mathrm{PZ}}^{\mathrm{conv}} =\Delta J^{\mathrm{exp}}. \end{aligned}$$

(10)

The right-hand side extension of Eq. (10) results from Ochensberger and Kolednik (2014), who have shown that the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ reflects the magnitude of $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$, provided that in cases of crack closure a correct procedure is applied for the determination of the area $\Delta A$ in Eq. (8). The conventionally applied procedure via the determination of the closure- or opening loads, $F_{\mathrm{cl}}$ and $F_{\mathrm{op}}$ (Fig. 2b), as proposed by Dowling and Begley (1976), can lead to inaccurate results, see also Sect. 5.2 below.

The results of Ochensberger and Kolednik (2014) demonstrate that the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ proposed by Dowling and Begley (1976) is, in principle, a physically appropriate driving force parameter for a cyclically loaded specimen. It should be noted, however, that this investigation has been conducted for a stationary crack. Therefore, the question remains to be solved whether these findings are applicable also in the case where the crack grows under cyclic loading, as it occurs in fatigue crack growth.

We consider in the current paper cyclic loading with crack extension. It will be demonstrated in this paper that important differences appear between a stationary and a growing crack, although the crack growth rate per load cycle is small.

3 Numerical modeling and computational aspects

For the numerical investigations the same procedure is applied as in Ochensberger and Kolednik (2014). The simulations are performed using the finite element (FE) program ABAQUS (see http://www.simulia.com/products/abaqus_fea.html). A two-dimensional C(T)-specimen (ASTM E1820, 2005) is modeled with a straight crack in horizontal $x$-direction, Fig. 3a. The specimen dimensions are: width $W=50\,\hbox {mm}$, height $H = 60\,\hbox {mm}$, nominal thickness $B = 25\,\hbox {mm}$, and initial crack length $a_0= 25\,\hbox {mm}$. The specimen is subjected to cyclic Mode I loading by prescribing the load $F$ at the load application point. Plane strain conditions are assumed.

Fig. 3

a Model of the C(T)-specimen with boundary conditions. b Applied load $F$ versus time $t$ for a load ratio $R = 0$; the increment of crack extension per load cycle $\Delta (\Delta a)$ is two element lengths. Apart from the cases with constant loads, an additional study is conducted for a single tensile overload $F_{\mathrm{OL}}$, applied during the fourth load cycle

The specimen consists of homogeneous, isotropic, elastic–ideally plastic material with Young’s modulus $E = 200\,\hbox {GPa}$, Poisson’s ratio $\nu = 0.3$, and yield strength $\upsigma _{\mathrm{y}} = 270\,\hbox {MPa}$. A small strip near the left boundary of the specimen is adopted as linear elastic, with Young’s modulus $E = 200\,\hbox {GPa}$ and Poisson’s ratio $\nu = 0.3$. This is done to prevent large plastic deformation at the load application point. Note that this does not cause any problems, since the plastic zone does not approach the elastic region.

Half of the specimen is discretized (Fig. 3a). The mesh consists of bilinear 4-node continuum elements. The inner region A, where the crack propagates, has a dimension of $13.5\times 3.0\,\hbox {mm}^{2}$; it is filled with elements of equal size. If not specified otherwise, the element size is $m=0.1\;\hbox {mm}$. Geometric nonlinearity is applied to consider large deformations around the crack tip. Crack flank contact without friction is modeled; a rigid body serves as counterpart to the upper half of the specimen. The nodes on the plane $y = 0$, except the nodes on the crack flank, are locked in $y$-direction, but unlocked in $x$-direction.

The loading steps are shown in Fig. 3b. Each load cycle $N$ can be divided into three steps. In the first step, finished at time $t = 3N - 2$ with $N \in {\mathbb {N}}$, the specimen is loaded to maximum load $F_{\max }$. In the second step, finished at $t = 3N - 1$, the specimen is unloaded to minimum load $F_{\min }$. The crack length is held fixed during the loading and unloading stages. In the third step, finished at $t = 3N$, the crack extends by an increment at the minimum load $F_{\min }$. We refer to Solanki et al. (2004) regarding the preferred load level for incremental crack extension. Incremental crack extension is modeled by adopting the node release technique (Ohji et al. 1975; Newman 1976). Hereby initially bonded nodes on the plane $y = 0$ are released according to a pre-defined crack length function of time (see also Kuna 2008). Two elements are chosen as crack extension increment per load cycle, $\Delta (\Delta a)=2m$. The number of applied load cycles is $N = 24$; the total crack advance is $\Delta a=\sum _{i=1}^N {\Delta (\Delta a)_i} =4.8\;\hbox {mm}$.

The maximum load is varied so that we get small-scale yielding (ssy) and large-scale yielding (lsy) conditions at $F_{\mathrm{max}} =12.5\,\hbox {kN}$ and $F_{\mathrm{max}} =27\,\hbox {kN}$, respectively (Fig. 4a, b). Large-scale yielding is assumed to start with the onset of plasticity at the back face of the specimen (Fig. 4b). Three load ratios are considered, $R = 0$ (zero-tension loading, Fig. 3b), $R = 0.5$ (pure tension), and $R = -1$ (tension-compression).

Fig. 4

Maps of the equivalent plastic strain $\upvarepsilon _{\mathrm{eq}}^\mathrm{p}$ at an applied load of a $F_{\mathrm{max}} =12.5\,\hbox {kN}$ (small-scale yielding conditions), b $F_{\mathrm{max}} =27\,\hbox {kN}$ (large-scale yielding, i.e. after onset of back face plasticity), and c at $F_{\mathrm{max}} =27\,\hbox {kN}$ and $\Delta a=4.6\,\hbox {mm}$ (general yielding). In a the configuration after $\Delta a=1\,\hbox {mm}$ crack extension is depicted, when the active plastic zone has completely left the monotonic plastic zone of the initial crack tip; the plastic zone shapes are marked with dashed lines. $\Gamma _{\mathrm{PZ}}$ and $\Gamma _{\mathrm{actPZ}}$ denote the contours around the entire and the active plastic zone, respectively. Directions of the bulk configurational force $\mathbf{f}^{\mathrm{ep}}$ are schematically indicated

The FE stress and strain analyses are in all cases performed using the incremental plasticity model provided by ABAQUS. Subsequently, the configurational forces are evaluated by a self-written post-processing routine, which is based on the papers by Müller et al. (2002; 2004) and Denzer et al. (2003). Deformation plasticity and incremental plasticity are alternatively applied for this post-processing in order to calculate at each node the deformation- and incremental plasticity configurational force, $\mathbf{f}^{\mathrm{def.pl}}$ and $\mathbf{f}^{\mathrm{ep}}$, by inserting either $\phi$ and $\phi _{\mathrm{e}}$ into Eq. (1).

Fig. 5

Incremental plasticity $J$-integrals around the crack tip plastic zone, $J_{\mathrm{PZ}}^{\mathrm{ep}}$, and around the active plastic zone, $J_{\mathrm{actPZ}}^{\mathrm{ep}}$, versus time $t$ for a small-scale yielding; c large-scale yielding, turning to general yielding after show $t = 69, \Delta a=4.6\,\hbox {mm}$. b, d show a detail of the $11^{\mathrm{th}}$ load cycle. The values reached at maximum load, $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ and $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$, are marked with full and open dots, respectively. The vertical dashed lines indicate when the active plastic zones leave the initial crack tip plastic zones

The scalar $J$-integral for an arbitrary contour $\Gamma$ is calculated by a summation of the configurational forces over all nodes within the area $\mathcal {D}$ bounded by $\Gamma$, including the crack tip node, compare Eqs. (4) and (5). The $J$-integrals for deformation plasticity and incremental plasticity, $J^{\mathrm{conv}}$ and $J^{\mathrm{ep}}$, are given by

$$\begin{aligned} J_\Gamma ^{\mathrm{conv}}&= \sum _{n \in {\mathcal {D}}\cup {\mathrm{tip}}} {-\left( {\mathbf{e}\cdot \mathbf{f}^{\mathrm{def.pl}}}\right) \Delta } A_n ,\end{aligned}$$

(11)

$$\begin{aligned} J_\Gamma ^{\mathrm{ep}}&= \sum _{n \in {\mathcal {D}}\cup {\mathrm{tip}}} {-\left( {\mathbf{e}\cdot \mathbf{f}^{\mathrm{ep}}}\right) \Delta } A_n . \end{aligned}$$

(12)

The parameter $\Delta A_n$ denotes the element area corresponding to a specific node $n$.

Table 1 Values of the incremental plasticity $J$-integrals around the crack tip plastic zone, $J_{\mathrm{PZ}}^{\mathrm{ep}}$, and around the active plastic zone, $J_{\mathrm{actPZ}}^{\mathrm{ep}}$, for small-scale yielding conditions

Table 2 Values of the incremental plasticity $J$-integrals around the crack tip plastic zone, $J_{\mathrm{PZ}}^{\mathrm{ep}}$, and around the active plastic zone, $J_{\mathrm{actPZ}}^{\mathrm{ep}}$, for large-scale yielding conditions

Important are the computation of $J$-integrals around the crack tip plastic zone $J_{\mathrm{PZ}}$ and around the active plastic zone $J_{\mathrm{actPZ}}$, see Fig. 1b. The shape of the active plastic zone is obtained by observation of currently yielding integration points, i.e. when the plastic strain $\varvec{\upvarepsilon }^{\mathrm{p}}$ changes during a loading sequence. An example is presented in Fig. 4a; currently yielding integration points are marked by x-symbols. The integration path $\Gamma _{\mathrm{actPZ}}$ is chosen so that it includes all marked integration points at maximum load $F_{\mathrm{max}}$. It should be noted that the magnitude of the integration path is held fixed for a specific loading and unloading sequence, i.e. $\Gamma _{\mathrm{actPZ}}$ does not vary with increasing or decreasing load.

In addition to the $J$-integrals derived from configurational forces, $J^{\mathrm{conv}}$ and $J^{\mathrm{ep}}$, the conventional computational $J$-integral of ABAQUS $J^{\mathrm{VCE}}$, which is based on the virtual crack extension method developed by Parks (1977), is also computed. Note that $J^{\mathrm{VCE}}$ implicitly relies on deformation plasticity, when applied to elastic–plastic materials.

4 Incremental plasticity $J$-integral $J^{\mathrm{ep}}$ for crack extension under cyclic loading

4.1 Characteristic incremental plasticity $J$-integral terms, $J_{\mathrm{PZ}}^{\mathrm{ep}}$ and $J_{\mathrm{actPZ}}^{\mathrm{ep}}$

Figure 5a, c present, for a load ratio $R=0$, the variations of the incremental plasticity $J$-integrals around the crack tip plastic zone, $J_{\mathrm{PZ}}^{\mathrm{ep}}$, and around the active plastic zone, $J_{\mathrm{actPZ}}^{\mathrm{ep}}$, under ssy- and lsy-conditions, i.e. $F_{\mathrm{max}} =12.5\,\hbox {kN}$ and $F_{\mathrm{max}} =27\,\hbox {kN}$, respectively. Shown are $N=24$ load cycles; Fig. 5b, d show in more detail the $11^{\mathrm{th}}$ load cycle. The $J^{\mathrm{ep}}$-values reached at maximum load $F_{\mathrm{max}}$ are marked with full (for $J_{\mathrm{PZ,max}}^{\mathrm{ep}})$ and open (for $J_{\mathrm{actPZ,max}}^{\mathrm{ep}})$ dots.

It is seen from Fig. 5 that, due to the crack extension after each load cycle, the $J^{\mathrm{ep}}$-values at maximum load, $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ and $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$, continuously increase, whereas the values taken at minimum load, $J_{\mathrm{PZ,min}}^{\mathrm{ep}}$ and $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$, remain approximately constant. While $J_{\mathrm{PZ}}^{\mathrm{ep}}$ shows a regular behavior, with values very close to zero at $F_{\mathrm{min}}$, the $J_{\mathrm{actPZ}}^{\mathrm{ep}}$-curve exhibits some irregularities during the first load cycles. The $J_{\mathrm{actPZ}}^{\mathrm{ep}}$-curve appears shifted downwards compared to the $J_{\mathrm{PZ}}^{\mathrm{ep}}$-curve.

The values of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ}}^{\mathrm{ep}}$ are equal for the first loading and unloading cycle, since crack extension has not occurred yet. With the first crack extension step, time $t>2$, the active plastic zone starts to leave the plastic zone of the initial crack tip, and $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ differs from $J_{\mathrm{PZ}}^{\mathrm{ep}}$. The active plastic zone has completely left the initial plastic zone after a crack extension of $\Delta a=0.6$ and 2 mm, $N=4$ and 11, for ssy- and lsy-conditions, respectively. This is indicated by a vertical dashed line in Fig. 5a, c. It is seen that the irregularities of the $J_{\mathrm{actPZ}}^{\mathrm{ep}}$-curve occur before this line; e.g. especially high $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$-values appear during $N=2$, followed by especially low $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$-values. This behavior will be discussed in more detail in the next section.

The $J$-integral values are collected in Tables 1 and 2. The values of $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ lie approximately 6 % lower than the values of $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$; the difference decreases to approximately 4 % with increasing crack extension. The difference almost disappears, if the active plastic zone merges with the back-face plasticity region so that general yielding (gy) conditions prevail. This happens for the values in Table 2 after $\Delta a=4.6\,\hbox {mm}$, see also Figs. 4c and 5c. The values of $J_{\mathrm{PZ,min}}^{\mathrm{ep}}$ are very close to zero for ssy-conditions (Table 1), whereas they reach several tenths of $\hbox {kJ}/\hbox {m}^{2}$ for lsy-conditions (Table 2), probably caused by irreversible elastic strain energy stored around the crack tip (Atkins and Mai 1986). The $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$-values are, in general, negative.

The incremental plasticity far-field $J$-integral $J_{\mathrm{far}}^{\mathrm{ep}}$ equals $J_{\mathrm{PZ}}^{\mathrm{ep}}$ for ssy-conditions. Due to the appearance of back-face plasticity, $J_{\mathrm{far}}^{\mathrm{ep}}$ is lower than $J_{\mathrm{PZ}}^{\mathrm{ep}}$ for lsy- and gy-conditions. With increasing crack length the difference between $J_{\mathrm{far,max}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ increases from 0.5 % up to 33 %, due to the increase of the back-face plasticity region.

Fig. 6

Distribution of incremental plasticity bulk configurational force $\mathbf{f}^{\mathrm{ep}}$ in the entire crack tip plastic zone at a maximum load, $F_{\mathrm{max}} =12.5\,\hbox {kN}$, after a crack extension of $\Delta a=1\,\hbox {mm}$, compare Fig. 4a. Regions on the boundaries of the plastic zone and the bunting region of the initial crack tip are enlarged in detailed views; a finer mesh size of $m = 0.033\,\hbox {mm}$ is used for a better visualization of the $\mathbf{f}^{\mathrm{ep}}$-vectors

In all cases the $J$-integral for deformation plasticity $J^{\mathrm{conv}}$ equals the computational $J$-integral of ABAQUS, $J^{\mathrm{VCE}}$. It is clear from Eq. (7) that $J_{\mathrm{PZ}}^{\mathrm{conv}} =J_{\mathrm{PZ}}^{\mathrm{ep}}$, unless gy-conditions appear. At the maximum load of every load cycle the deformation plasticity far-field $J$-integral comes close to the value around the plastic zone, $J_{\mathrm{far,max}}^{\mathrm{conv}} \approx J_{\mathrm{PZ,max}}^{\mathrm{conv}} =J_{\mathrm{PZ,max}}^{\mathrm{ep}}$, whereas at the minimum load of each load cycle, $J_{\mathrm{far,min}}^{\mathrm{conv}}$ strongly differs from the values of $J_{\mathrm{PZ,min}}^{\mathrm{conv}} =J_{\mathrm{PZ,min}}^{\mathrm{ep}}$. The reason has been already explained in detail in Ochensberger and Kolednik (2014), Sect. 4.2 therein: Artificial bulk configurational forces are induced in the back-face plasticity region since the conditions of proportional loading are violated during unloading. The deformation plasticity $J$-integral around the active plastic zone $J_{\mathrm{actPZ}}^{\mathrm{conv}}$ shows an oscillating curve that continuously decreases with every load cycle, so that after a certain number of load cycles the values are negative even at maximum load.

Tables 1 and 2 collect also additional results of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ}}^{\mathrm{ep}}$ for load ratios $R=0.5$ and $-$1 under ssy- and lsy-conditions. Especially for $R = 0.5$ and lsy-conditions, the difference between the values at maximum load, $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$, can be significant.

4.2 Bulk configurational forces in the crack tip plastic zone

In order to explore the differences that appear in Fig. 5 between $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ}}^{\mathrm{ep}}$, we analyze in this section the distribution of the bulk configurational force $\mathbf{f}^{\mathrm{ep}}$ in the crack tip plastic zone. For easier understanding, it is useful to transform Eq. (6) into small strain plasticity (Simha et al. 2008),

$$\begin{aligned} \mathbf{f}^{\mathrm{ep}}=\varvec{\upsigma }:\frac{\partial \varvec{\upvarepsilon }^{\mathrm{p}}}{\partial \mathbf{x}}, \end{aligned}$$

(13)

where $\varvec{\upsigma }$ denotes the Cauchy stress tensor and $\varvec{\upvarepsilon }^{\mathrm{p}}$ the plastic part of the linear strain tensor $\varvec{\upvarepsilon }$. Note that for crack and crack growth in $x$-direction, only the $x$-component of the configurational force $\mathbf{f}^{\mathrm{ep}}$-vector,

$$\begin{aligned} f_x^{\mathrm{ep}} =\sigma _{xx} \frac{\partial \varepsilon _{xx}^\mathrm{p}}{\partial x}+2\sigma _{xy} \frac{\partial \varepsilon _{xy}^\mathrm{p}}{\partial x}+\sigma _{yy} \frac{\partial \varepsilon _{yy}^\mathrm{p}}{\partial x}, \end{aligned}$$

(14)

contributes to the scalar $J^{\mathrm{ep}}$-integral, see Eq. (5).

Figure 6 presents, for a load ratio $R = 0$, the distribution of the bulk configurational force $\mathbf{f}^{\mathrm{ep}}$ in the total crack tip plastic zone at maximum load $F_{\mathrm{max}} =12.5\,\hbox {kN}$ during the $6^{\mathrm{th}}$ load cycle, after 1 mm of crack growth. Important regions are enlarged in the detailed views where a finer mesh of $m = 0.033\,\hbox {mm}$ is used to obtain a better visualization of the $\mathbf{f}^{\mathrm{ep}}$-vectors.²

On the left boundary of the crack tip plastic zone, $\mathbf{f}^{\mathrm{ep}}$-vectors appear with a positive $f_x^{\mathrm{ep}}$-component, while $f_x^{\mathrm{ep}}$ is negative for $\mathbf{f}^{\mathrm{ep}}$-vectors on the right boundary. Along the upper boundary, $\mathbf{f}^{\mathrm{ep}}$-vectors emerge with $x$-components of almost zero; they only have a negative $y$-component $f_y^{\mathrm{ep}}$. It is seen that in all cases the direction of $\mathbf{f}^{\mathrm{ep}}$-vectors clearly follows the gradient of the plastic strain, corresponding to Eq. (13); Fig. 4a includes schematically the directions of these $\mathbf{f}^{\mathrm{ep}}$-vectors. In the blunting region of the initial crack tip, the $\mathbf{f}^{\mathrm{ep}}$-vectors point to either direction, but the resulting configurational force, i.e. the sum of the $x$-components of all $\mathbf{f}^{\mathrm{ep}}$-vectors lying within this region, points into the negative $x$-direction.

The largest $\mathbf{f}^{\mathrm{ep}}$-vectors appear around the current crack tip, since there both stress and gradient of plastic strain are largest, Eq. (13). Due to symmetry, each $\mathbf{f}^{\mathrm{ep}}$-vector that emerges from a node directly along the ligament in front of the current crack tip does have a “companion” $\mathbf{f}^{\mathrm{ep}}$-vector from the lower specimen half. The resulting $\mathbf{f}^{\mathrm{ep}}$-vector of both specimen halves has only a component in $x$-direction. This applies also for $\mathbf{f}_{\mathrm{tip}}^{\mathrm{ep}}$ which emerges from the current crack tip. For numerical reasons, $J_{\mathrm{tip}}^{\mathrm{ep}} =\mathbf{e}\cdot \left( {-\mathbf{f}_{\mathrm{tip}}^{\mathrm{ep}}}\right)$ has a finite value which depends on the FE-mesh size, see Kolednik et al. (2014). A decrease of the mesh size leads to a reduction of the magnitude of $J_{\mathrm{tip}}^{\mathrm{ep}}$ and yields finally $J_{\mathrm{tip}}^{\mathrm{ep}} = 0$. However, the values of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ (and $J_{\mathrm{PZ}}^{\mathrm{ep}})$ are not affected by a mesh refinement; Sect. 5.1 provides an example.

The variation of the difference between $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ shown in Fig. 5a can be understood from Fig. 6 and Eq. (12): More and more bulk configurational forces $\mathbf{f}^{\mathrm{ep}}$ of the initial plastic zone become excluded from the integration contour $\Gamma _{\mathrm{actPZ}}$, when it is shifted to the right during crack extension. First, the $\mathbf{f}^{\mathrm{ep}}$-vectors with positive $f_x^{\mathrm{ep}}$-components from the left boundary are excluded, leading to the peak value of $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ observed at $N=2$. Next, the integration contour $\Gamma _{\mathrm{actPZ}}$ excludes the blunting region of the initial crack tip. Since the resulting $\mathbf{f}^{\mathrm{ep}}$ in this blunting region has a negative $x$-component, the value of $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ drops below the value of $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$. With further crack extension the difference between $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ remains equal, since the excluded $\mathbf{f}^{\mathrm{ep}}$-vectors of the plastic wake have no $x$-component. It should be mentioned that a similar variation of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ is reported in Section 5.3 of Kolednik et al. (2014) for a growing crack under constant load.

For lsy-conditions, the difference between the values of $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ,max}}^{\mathrm{ep}}$ decreases with increasing crack extension, because the right boundary of the active plastic zone, with $\mathbf{f}^{\mathrm{ep}}$-vectors in negative $x$-direction, increases. Finally, the difference nearly vanishes under gy-conditions.

The differences between $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ,min}}^{\mathrm{ep}}$ at minimum load $F_{\mathrm{min}}$ can be explained analogously. Note that, due to the appearance of compressive stresses during unloading, the $\mathbf{f}^{\mathrm{ep}}$-vectors at $F_{\mathrm{min}}$ point into the opposite $x$-direction compared to the $F_{\mathrm{max}}$-stage. This has been explained already in Ochensberger and Kolednik (2014), see also Sect. 2.3. It can be shown that negative $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$-values originate for $R \le 0$ from these negative stresses; the relevance of these negative $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$-values will be discussed in the next section.

5 Driving force for fatigue crack growth

For a cyclically loaded, stationary crack, Ochensberger and Kolednik (2014) proposed the cyclic, incremental plasticity $J$-integral around the crack tip plastic zone, $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$, as physically appropriate driving force parameter. Consequently, for a growing fatigue crack it is reasonable to adopt the cyclic, incremental plasticity $J$-integral around the active crack tip plastic zone, $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, as the appropriate driving force parameter for fatigue crack growth. One argument is that only the active plastic zone travels with the crack tip during crack extension; the initial plastic zone and the plastic wake do not move. Another reason is that plasticity far from the current crack tip cannot be responsible for fatigue crack propagation that occurs due to cyclic plastic deformation at the current tip.

It seems obvious that $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ should be evaluated, analogously to Eq. (9), by the relation,

$$\begin{aligned} \Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}&= J_{\mathrm{actPZ,max}}^{\mathrm{ep}} +J_{\mathrm{actPZ,min}}^{\mathrm{ep}} \nonumber \\&-\,2\sqrt{J_{\mathrm{actPZ,max}}^{\mathrm{ep}} J_{\mathrm{actPZ,min}}^{\mathrm{ep}}}. \end{aligned}$$

(15)

At this point we arrive at a dilemma when the incremental plasticity $J$-integral at minimum load, $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$, is negative, as shown in Tables 1 and 2 for load ratios $R=0$ and $R=-1$, since this leads to a complex square root term. Therefore, it should be clarified in which form the negative values of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ in a load cycle deliver a contribution to the driving force for fatigue crack growth. We do this by studying the relation between the incremental plasticity $J$-integral $J^{\mathrm{ep}}$ and the crack tip opening displacement $\delta _{\mathrm{t}}$.

5.1 Incremental plasticity $J$-integral $J^{\mathrm{ep}}$ and crack tip opening displacement

Fatigue crack growth in ductile metals and alloys is driven by cyclic plasticity at the current crack tip (Laird 1967, 1979). The crucial role of crack tip plasticity was confirmed by experiments, e.g. Tanaka (1989), Krupp et al. (2002), and Pippan et al. (2010), where it was shown that the crack growth rate of a fatigue crack ${\hbox {d}a}/{\hbox {d}N}$ is a function of the cyclic crack tip opening displacement at the current crack tip, ${\hbox {d}a}/{\hbox {d}N}\propto \Delta \delta _{\mathrm{t}}$ with $\Delta \delta _{\mathrm{t}} =\delta _{\mathrm{t,max}} -\delta _{\mathrm{t,min}}$.

A correct numerical evaluation of the crack tip opening displacement $\delta _{\mathrm{t}}$ during fatigue crack propagation is difficult, since it requires a very fine FE-mesh. Experimental investigations using stereophotogrammetric measurements show that the crack tip opening displacement should be determined a distance behind the blunted crack tip, which corresponds to the width of the stretched zone (Kolednik and Stüwe 1985; Kolednik and Kutlesa 1989; Siegmund et al. 1990); a good approximation of this distance is for many materials given by $0.4\;\delta _{\mathrm{t}}$, see Kolednik and Stüwe (1987), Heerens et al. (1988). Therefore, $\delta _{\mathrm{t}}$ should not be determined at a fixed distance behind the crack tip, since measurements at too large distances lead to an overestimate of $\delta _{\mathrm{t}}$, and vice versa. However, it is difficult to fulfill these requirements in practice. Dougherty et al. (1997) give a minimum element size $m$ in order to determine accurate $\delta _{\mathrm{t}}$-values, $m/{r_{\mathrm{pl}}}\le 0.1$, where $r_{\mathrm{pl}}$ denotes the radius of the crack tip plastic zone. According to Solanki et al. (2003, 2004) the mesh size shall be further reduced by a factor $3 \div 4$. The relation $r_{\mathrm{pl}} =\beta {JE}/{\upsigma _\mathrm{y}^2}$, with $\beta \approx 0.1$, yields for the maximum load of the 1$^\mathrm{st}$ load cycle $r_{\mathrm{pl}} \approx 0.59\;\hbox {mm}$ and $r_{\mathrm{pl}} \approx 2.90\;\hbox {mm}$ for ssy- and lsy-conditions, respectively; this is roughly in agreement with Fig. 4a, b. The relation $J=k \upsigma _{\mathrm{y}} \delta _{\mathrm{t}}$, with $J=J_{\mathrm{PZ}}^{\mathrm{ep}}$ and $k\approx 2$, yields values of the crack tip opening displacement, $\delta _{\mathrm{t}} \approx 3.95\;\upmu \hbox {m}$ and $\delta _{\mathrm{t}} \approx 19.6\;\upmu \hbox {m}$, for ssy- and lsy-conditions, respectively. According to the criterion by Solanki et al. (1997), a mesh size of approximately $m \approx 4\delta _{\mathrm{t}}$ would be sufficient for gaining accurate $\delta _{\mathrm{t}}$-values, which appears somewhat doubtful to us.

However, the computation of accurate $\delta _{\mathrm{t}}$-values is not very important for our purposes; our interest lies only in the correct reflection of crack tip opening and -closure behavior during the loading and unloading stages. Therefore, a mesh size of $m = 0.1\,\hbox {mm}$ should be sufficient for lsy-conditions; additional computations with smaller mesh size, $m=10\,\upmu \hbox {m}$, are conducted for ssy-conditions. The crack tip opening displacement is taken one element behind the current crack tip, as e.g. in Solanki et al. (2004).

Fig. 7

Incremental plasticity $J$-integrals, $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ}}^{\mathrm{ep}}$, plotted against the crack tip opening displacement $\delta _{\mathrm{t}}$ during re-loading after $\Delta a=1\,\hbox {mm}$ crack extension. Values are depicted for a small-scale yielding, mesh sizes $m = 0.1\,\hbox {mm}$ and $10\,\upmu \hbox {m}$; c large scale yielding, $m=0.1\,\hbox {mm}$. b, d Enlarged views of the beginnings of the loading sequences. The results suggest that loading stages where $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ is negative do not contribute to fatigue crack propagation, since $\delta _{\mathrm{t}}$ is zero

Figure 7 shows the curves $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ versus $\delta _{\mathrm{t}}$ during loading from $F_{\mathrm{min}}$ to $F_{\mathrm{max}}$ during the 6th load cycle, i.e. after a crack extension of $\Delta a=1\,\hbox {mm}$. The values for $J_{\mathrm{PZ}}^{\mathrm{ep}}$ are included for comparison. The curves for ssy-conditions are plotted in Fig. 7a; Fig. 7b gives a detailed view of the region near the minimum load. Figure 7c, d present the curves for lsy-conditions. It is seen that $\delta _{\mathrm{t}} =0$ during the very initial loading stages. The curves suggest that the crack tip opening displacement $\delta _{\mathrm{t}}$ does not start to increase before $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ becomes positive.

Thus, we state that negative values of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ do not deliver a contribution to the fatigue crack growth rate, since the crack tip is closed during this stage. Moreover, from a thermodynamic view, a negative $x$-component of the $\mathbf{J}_{\mathrm{actPZ}}^{\mathrm{ep}}$-vector means that the crack feels a driving force for shortening its length. Therefore, the stage where the crack driving force is negative cannot deliver a contribution to the driving force for fatigue crack growth. For these reasons, we conclude that the negative values of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ are not relevant for calculating the driving force for fatigue crack growth, and $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}} =J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ for $R=0$.

It should be mentioned that the values of $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$ (and $J_{\mathrm{PZ,max}}^{\mathrm{ep}})$ are not affected by the decrease in mesh size $m$ by a factor ten for ssy-conditions. For example, at maximum load we get $J_{\mathrm{actPZ,max}}^{\mathrm{ep}} =2.279\;\hbox {kJ/m}^{2}$ for $m=10\;\upmu \hbox {m}$ and $J_{\mathrm{actPZ,max}}^{\mathrm{ep}} =2.280\, \hbox {kJ/m}^{2}$ for $m=0.1\;\hbox {mm}$. On the contrary, the values of the crack tip opening displacement show a big variation: at maximum load we get $\delta _{\mathrm{t}} =3.11\;\upmu \hbox {m}$ for $m=10\;\upmu \hbox {m}$ and $\delta _{\mathrm{t}} =4.65\;\upmu \hbox {m}$ for $m=0.1\;\hbox {mm}$. This clearly demonstrates the advantage of the $J$-integral concept.

For negative load ratios $R < 0$ it can be also shown that negative $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$-values do not give a contribution to the cyclic $J$-integral $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$. Therefore, the driving force for fatigue crack growth is equal to the incremental plasticity $J$-integral around the active plastic zone at maximum load for zero-tension and tension-compression loading,

$$\begin{aligned} \Delta J_{\mathrm{actPZ}}^{\mathrm{ep}} =J_{\mathrm{actPZ,max}}^{\mathrm{ep}} \quad \hbox {for}\quad R\le 0. \end{aligned}$$

(16)

Note that the exact upper boundary for the validity of Eq. (16) has not been determined. For positive load ratios $R > 0$, the curves $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $J_{\mathrm{PZ}}^{\mathrm{ep}}$ versus $\delta _{\mathrm{t}}$ are almost linear and do not have a vertical part near $F_{\min }$. Since both the $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$- and $\delta _{\mathrm{t,min}}$-values are positive at minimum load, Eq. (15) must be used to evaluate $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$.

Table 1 lists $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$-values with increasing load cycle number $N$ and crack extension $\Delta a$ for load ratios, $R = 0, 0.5$ and $-1$, under ssy-conditions. The results for $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$, Eq. (9), are collected for comparison. The difference between $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ is of the order of 2 % for $R\le 0$; on the contrary, $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ can even exceed $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ by 13 % for $R=0.5$. Table 2

lists corresponding values for lsy-conditions. Here, the difference between $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ can become large for all load ratios. For $R = 0.5$, the values of $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$ are distinctively smaller than the $J_{\mathrm{PZ,min}}^{\mathrm{ep}}$-values which cause lower square root terms in Eq. (15) and, thus, significantly higher values of $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ compared to $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$. Figure 8 presents, for a load ratio $R = 0$, the variations of $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ with increasing crack extension. The variations of the cyclic crack tip opening displacement, $\Delta \delta _{\mathrm{t}} =\delta _{\mathrm{t,max}}-\delta _{\mathrm{t,min}}$, with crack extension are shown for comparison.

Fig. 8

Development of the cyclic, incremental plasticity cyclic $J$-integrals around the crack tip plastic zone, $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$, and around the active plastic zone, $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, with crack extension $\Delta a$. Values of the cyclic crack tip opening displacement $\Delta \delta _{\mathrm{t}}$ and the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ are drawn for comparison. a Small-scale yielding, b large-scale yielding conditions. The computations are made for a FE-mesh size of $m = 0.1\,\hbox {mm}$; the first crack growth periods are re-calculated for a finer mesh size, $m = 10\,\upmu \hbox {m}$, see dotted curves

The conclusion of this section is that the loading stages where $J_{\mathrm{actPZ,min}}^{\mathrm{ep}}$ becomes negative do not play a role for the calculation of the driving force for fatigue crack propagation. In the following section, the validity of the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ proposed by Dowling and Begley (1976) shall be checked.

5.2 Comparison to the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$

The experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ (Dowling and Begley 1976) is computed from the area $\Delta A$ below a single loading branch of the load–displacement $(F-v)$ curve, Eq. (8). The $\Delta J^{\mathrm{exp}}$-values for $R = 0$ are drawn into Fig. 8 and listed in Tables 1 and 2. It is seen that $\Delta J^{\mathrm{exp}}$ fits very well to the values of the cyclic, incremental plasticity $J$-integral $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$. This corresponds to the findings reported in Sects. 2.2 and 2.3, and to Eq. (10). The experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ does not fit so well to $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, although the error remains small for ssy-conditions. However, the error can reach 20 % for lsy-conditions. The values listed for $\Delta _{\Delta J\mathrm{PZ}}^{\Delta J\mathrm{actPZ}}$ in Tables 1 and 2 are approximately equal to the relative difference between $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $\Delta J^{\mathrm{exp}}$. Note that a small error of about 2 % has to be taken into account in the evaluation of $\Delta J^{\mathrm{exp}}$, which seems to result from an inaccuracy of the geometry factor $\eta$ taken from ASTM E1820 (2005), compare the values listed for $N = 1$ in Tables 1 and 2.

Tables 1 and 2 collect also the $\Delta J^{\mathrm{exp}}$-values for load ratios $R = 0.5$ and $-1$. For load ratios $R<0$, in presence of crack closure, the values of $\Delta J^{\mathrm{exp}}$ depend on the correct determination of $\Delta A$. According to Dowling and Begley (1976), the opening load $F_{\mathrm{op}}$ shall be determined from the compliance change that is visible as kink in the $F-v$-curve, Fig. 2b. Ochensberger and Kolednik (2014) showed for stationary cracks that this procedure tends to overestimate the driving force. Instead, $\Delta A$ should be determined as the area above the load $F_{J_{\mathrm{PZ}}^{\mathrm{ep}} = 0}$, where the incremental plasticity $J$-integral around the crack tip plastic zone reaches its minimum value, $J_{\mathrm{PZ,min}}^{\mathrm{ep}} =0$. The same procedure is used for the evaluation of the $\Delta J^{\mathrm{exp}}$-values listed in Tables 1 and 2 for $R=-1$. The results show that $\Delta J^{\mathrm{exp}}$ fits very well to$\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$, and Eq. (10) remains valid, unless gy-conditions prevail.

Crack closure does not occur for $R>0$. It is seen that the misfit between $\Delta J^{\mathrm{exp}}$ and $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ is small for various load ratios under ssy- and lsy-conditions, see Tables 1 and 2. In both cases the misfit slightly increases with increasing crack extension.

The validity of Eq. (10) becomes clear from the following: the condition $J_{\mathrm{PZ}}^{\mathrm{ep}} =J_{\mathrm{PZ}}^{\mathrm{conv}}$, Eq. (7), is fulfilled for cyclic loading conditions as long as the crack tip plastic zone is surrounded by elastically deformed material, see Sects. 2.2 and 2.3. The requirement for the validity of the condition $J_{\mathrm{PZ}}^{\mathrm{conv}} =J^{\mathrm{exp}}$ is that a single loading cycle of a growing fatigue crack can be treated like a stationary crack under monotonic loading; only then, the conventional $J$-integral and the experimental $J$-integral $J^{\mathrm{exp}}$ lead to the same results, see Rice et al. (1973), Kolednik (1991). This requirement is fulfilled during loading in each load cycle.

Table 1 shows that the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ overestimates the driving force for fatigue crack growth $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ by approximately 2 % for ssy-conditions and $R=0$ and $R=-1$. On the contrary, $\Delta J^{\mathrm{exp}}$ considerably underestimates $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ for $R=0.5$, whereby the misfit decreases up to 12 % with increasing crack extension. For lsy-conditions, Table 2, $\Delta J^{\mathrm{exp}}$ underestimates $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ for all load ratios. The misfit increases with crack extension to approximately 20 % for $R=0$ and to 5 % for $R=-1$. For $R=0.5$, the misfit of about 15 % is rather independent of crack extension. The reason for the high misfit under $R = 0.5$, even for ssy-conditions, is currently not fully understood by the authors, but it might be caused by the fact that proportional loading is fulfilled for each maximum load of a load cycle, which is required for the validity of $J_{\mathrm{PZ}}^{\mathrm{ep}} =J^{\mathrm{exp}}$, whereas this is not the case for the minimum load.

We can conclude this section by stating that the experimental cyclic $J$-integral does not exactly reflect the driving force for growing fatigue cracks in elastic–plastic materials, since it corresponds to the cyclic, incremental plasticity $J$-integral around the crack tip plastic zone, and not around the active plastic zone, $\Delta J^{\mathrm{exp}} \approx \Delta J_{\mathrm{PZ}}^{\mathrm{ep}} \ne \Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$. The difference between $\Delta J^{\mathrm{exp}}$ and $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ can reach approximately 20 % for lsy-conditions.

6 The effect of a single tensile overload

In this section, a further test shall be conducted in order to check whether the cyclic, incremental plasticity $J$-integral for a contour around the active plastic zone, $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, is the appropriate parameter for the characterization of the driving force for fatigue crack propagation, and not the cyclic $J$-integral for a contour around the plastic zone, $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$. It is investigated whether $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ is able to reflect the well-known overload effect: Crack growth retardation occurs, after a single tensile overload has been superimposed to cyclic loading with constant load amplitude. With further crack extension the crack growth rate gradually increases, reaching again the crack growth rate pertaining to the constant fatigue load somewhat after the active plastic zone has escaped from the plastic zone produced by the overload (Schijve 1960; Christensen 1959; von Euw et al. 1972; Suresh 1983; Fleck 1988; Skorupa 1998; Sadananda et al. 1999).

The fatigue crack growth rate ${\hbox {d}a}/{\hbox {d}N}$ in ductile metals and alloys is proportional to the cyclic crack tip opening displacement $\Delta \delta _{\mathrm{t}}$; this is so also after applying an overload. This fact was confirmed by experiments, e.g. Pippan et al. (2005) and Bichler and Pippan (1999; 2007), and by numerical studies, e.g. Tvergaard (2005). For example, Bichler and Pippan (2007) conducted $\Delta K$-controlled fatigue tests under ssy-conditions with load ratio $R = 0.05$, using C(T)-specimens fabricated of ductile austenitic CrNi steel. The constant $\Delta K$-fatigue history, with $\Delta K = 70\,\hbox {MPa}\surd \hbox {m}$, was interrupted by a single tensile overload with various overload ratios, $R_{\mathrm{OL}} {=F_{\mathrm{OL}}}/{F_{\max }}=1.1\div 2$. By analyzing fatigue striations (Zappfe and Worden 1951), which usually correlate with ${\hbox {d}a}/{\hbox {d}N}$ (Forsyth and Ryder 1960), and by conducting stereophotogrammetric measurements (Kolednik 1981) of the crack tip opening displacement $\delta _{\mathrm{t}}$, Bichler and Pippan (2007) demonstrated that the relation, ${\hbox {d}a}/{\hbox {d}N}\propto \Delta \delta _{\mathrm{t}}$, is valid also in the post-overload regime.

Our numerical test is conducted for zero-tension cyclic loading under ssy-conditions with applied maximum load, $F_{\mathrm{max}} =12.5\,\hbox {kN}$. A tensile overload, $F_{\mathrm{OL}} =25\,\hbox {kN}$, is applied in the fourth load cycle after $\Delta a = 0.6\,\hbox {mm}$ crack extension (Fig. 3b); the overload ratio is ${R_{\mathrm{OL}} =F_{\mathrm{OL}}}/{F_{\max }}=2$. Note that ssy-conditions still prevail during application of the overload, i.e. no back-face plasticity occurs. For our purpose, a FE-mesh size of $m = 0.1\,\hbox {mm}$ was seen to be sufficient.

Fig. 9

Effect of a single tensile overload, applied during the $4^{\mathrm{th}}$ load cycle at $\Delta a = 0.6\,\hbox {mm}$. The cyclic, incremental plasticity $J$-integrals around the crack tip plastic zone, $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$, and around the active plastic zone, $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, are plotted against crack extension $\Delta a$. Values of the cyclic crack tip opening displacement $\Delta \delta _{\mathrm{t}}$ and the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ are drawn for comparison. The parameter $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ correlates perfectly with $\Delta \delta _{\mathrm{t}}$, exhibiting a crack growth retardation effect, whereas this is not the case for $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ and $\Delta J^{\mathrm{exp}}$

Figure 9 presents the variations of $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}, \Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$, the cyclic experimental $J$-integral $\Delta J^{\mathrm{exp}}$, and $\Delta \delta _{\mathrm{t}}$ as functions of the crack extension $\Delta a$. It is seen that $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ varies analogously to $\Delta \delta _{\mathrm{t}}$ and clearly shows the retardation effect, whereas this is not the case for $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ and $\Delta J^{\mathrm{exp}}$.

The minimum value of $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}} =0.269{\hbox {kJ}}/{\hbox {m}^{2}}$ is reached after 0.4 mm of crack growth following the overload cycle, i.e. $\Delta a=1\;\hbox {mm}$ total crack extension. This gives a reduction of about 88 % compared to the $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$-value at $\Delta a=1\,\hbox {mm}$ for a constant fatigue load, see Fig. 8a and Table 1. This value is in excellent agreement with the experimental results obtained by Bichler and Pippan (2007) from $25\;\hbox {mm}$ thick C(T)-specimens (plane strain dominance), subjected to $R_{\mathrm{OL}} =2$, where a maximal reduction in ${\hbox {d}a}/{\hbox {d}N}$ of about 84 % was observed after approximately 0.5 mm crack growth following the overload.

The overload case study has confirmed that the cyclic, incremental plasticity $J$-integral for a contour around the active plastic zone, $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, is a physically appropriate driving force parameter for assessing the fatigue crack growth rate.

7 Computational aspects in the evaluation of $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$

In the following, a few issues regarding the computation of $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ shall be pointed out, which might be important for the practical application of the concept presented in the current paper.

The values of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ are not affected of the used FE-mesh size, especially, in comparison to the expensive numerical effort that is necessary to obtain accurate results for the crack tip opening displacement $\delta _{\mathrm{t}}$, see Fig. 7a.

Provided that the active plastic zone has left the plastic zone of the initial crack tip, the magnitude of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ is not significantly influenced by the magnitude of the integration contour $\Gamma _{\mathrm{actPZ}}$. The reason is that the configurational forces in the plastic wake do not deliver a contribution to the value of $J_{\mathrm{actPZ}}^{\mathrm{ep}}$, see Sect. 4.2. Therefore, it is not important to find the exact shape of the active plastic zone (Fig. 4a).

However, one has to be careful during the initial stages of crack extension, especially, before the active plastic zone excludes the blunting region of the initial crack tip. In Sect. 4 we have shown that $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ at maximum load exhibits a peak value after the onset of crack extension; compare the $J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$-values listed for $N = 2$ in Tables 1 and 2. Note that both position and magnitude of this peak value depend on the FE-mesh size $m$: A smaller mesh size gives a higher peak at smaller crack growth distance. The effect is shown in Fig. 8, for a reduction of $m$ by a factor ten and a crack extension increment per load cycle of $\Delta (\Delta a)=2m$. In this way, the finer FE-mesh is connected to a smaller crack extension increment per load cycle. Nevertheless, graphs similar to Fig. 7 can be also drawn for the first load cycles, showing that the crack tip is closed during the stages where $J_{\mathrm{actPZ}}^{\mathrm{ep}}$ is negative and, thus, $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}} =J_{\mathrm{actPZ,max}}^{\mathrm{ep}}$, Eq. (16), is still valid and that $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ correlates to $\Delta \delta _{\mathrm{t}}$.

The dependence of this peak value on the mesh size $m$ is caused by the inhomogeneity of the plastic strain field around the crack tip, especially, near the blunted tip. A coarse mesh leads to a smoothing of stress and strain peaks and causes a reduction of the magnitude of the configurational force $\mathbf{f}^{\mathrm{ep}}$, see Eq. (13).

Furthermore, it should be noted that additional FE-analyses are conducted where the crack extension in each load cycle occurs at maximum load. Basically, the results do not change compared to the procedure with crack extension at minimum load, and the same conclusions can be drawn as presented above.

8 Summary

The current paper discusses the physically correct evaluation of the driving force for fatigue crack propagation in elastic–plastic materials using the $J$-integral concept. Numerical investigations are conducted for a two-dimensional compact tension specimen with a long crack under cyclic Mode I loading. The crack extends by an increment after each load cycle at the minimum load. The maximum load is varied so that small- and large-scale yielding conditions prevail. Three different load ratios are considered, from pure tension to tension-compression loading. In general, maximum and minimum load are held constant during crack extension; in addition, the effect of a single tensile overload is also studied.

The results of the analyses show that the cyclic, incremental plasticity $J$-integral $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, which is computed for a contour around the active plastic zone of the growing crack, is physically appropriate to characterize the growth rate of fatigue cracks in elastic–plastic materials.

The experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$, proposed by Dowling and Begley (1976), measures the cyclic, incremental plasticity $J$-integral for a contour around the total plastic zone, $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$; the contour includes also the plastic zone around the initial crack tip and the plastic wake. The experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ reflects the driving force of a stationary crack, i.e. it is valid for the first load cycle. After crack extension, the incremental plasticity $J$-integral around the total plastic zone $J_{\mathrm{PZ}}^{\mathrm{ep}}$ differs from that around the active plastic zone $J_{\mathrm{actPZ}}^{\mathrm{ep}}$. Therefore, $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}} \ne \Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, and the experimental cyclic $J$-integral $\Delta J^{\mathrm{exp}}$ is not fully appropriate to reflect the driving force for a growing fatigue crack.

The difference between $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ and $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ is most clearly seen in the overload case, where $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$ is able to correctly reflect the well-known crack growth retardation effect, whereas $\Delta J_{\mathrm{PZ}}^{\mathrm{ep}}$ and $\Delta J^{\mathrm{exp}}$ would predict a constant crack growth rate.