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Abstract

In engineering application, a quantitative description of both fatigue and creep damage mechanisms is demanded for high-temperature alloy. This work employs the cohesive zone model to numerically simulate the crack propagation in a nickel-based super-alloy. Two separate damage evolution equations are introduced into the cohesive zone model to describe the fatigue and creep damage, respectively. Effects of temperature and loading hold period are discussed. The comparison shows a reasonable agreement between numerical results and experimental observations and confirms the potential of cohesive zone model for more complex loading conditions in engineering.

Keywords

Cohesive zone model creep damage evolution fatigue damage evolution temperature effects hold time effects

Introduction

Many mechanical parts are working under the various external loading conditions, suffering from thermal, mechanical, and environmental loads. Tong et al.¹ investigated effects of creep, fatigue, and oxidation on crack propagation in a nickel-based super-alloy at 650°C. The work concluded that the oxidation appears to be the predominant mechanism for crack growth under long dwell loading condition due to the limited creep at the crack tip. In the work of Evans et al.,² two alloys were studied under various temperatures including room temperature (RT) and hard vacuum to discuss the interactions of creep, fatigue, and environment damage. Lu et al.³ also investigated the influence of temperature and hold time on creep–fatigue crack growth behavior. Several different temperatures and hold time levels were employed in the test. The results show that temperature and hold time both have positive effects on the crack growth. The higher the temperature or hold time, the faster the crack growth. Prakash et al.⁴ studied the combined role of fatigue, creep, and oxidation, especially the time dependence in crack growth, and built up the connection between microscopic appearance of crack behavior and macroscopic variations of the loading condition. Wei et al.⁵ considered the oxygen enhancement of the nickel-based super-alloys in the crack growth. It said that some elements, such as Nb-rich carbides, can significantly enhance crack growth when they are oxidized. Roy et al.⁶ verified that even short hold time can significantly enhance the crack growth at elevated temperature. Experimental observations provide sufficient evidence for considering temperature and time effects in modeling of material failure.^7,8 The life of mechanical parts depends on both loading amplitude and loading time. Experimental data did not provide sufficient knowledge about the total life of the mechanical parts, as analyzed in Zhang et al.⁹ To give a more accurate prediction, especially for complex geometry parts, a constitutive description of the creep–fatigue crack is of importance.

In mechanical product design, the finite element method (FEM) is becoming a usable and staple numerical analysis tool. However, the damage process is not represented in most computations because it is difficult to handle numerically. Nowadays, more and more attentions are focused on the cohesive zone model (CZM). The earliest concept of the cohesive zone modeling was proposed by Barenblatt¹⁰ to investigate ideal brittle materials. Later, Dugdale¹¹ and Barenblatt¹² raised their own models to treat the unpractical infinite stresses at the crack tip. Based on these pioneer works, the CZMs have been generalized so that ductile material failure under both mixed mode loading conditions^13–15 and fatigue loading conditions^16–18 are considered in combination with the extended finite element method (XFEM).¹⁹ Modeling creep–fatigue is still a challenging task for computational mechanics of materials.

The aim of this work is to develop a CZM for predicting fatigue crack propagation of structures loaded at elevated temperature. Effects of dwell time and temperature on the crack growth in nickel-based powder metallurgy (PM) super-alloy should be integrated in the CZM. Two separate fatigue and creep damage evolution equations will be introduced based on the cohesive zone approach. Several sets of experimental data are used to determine the model parameters and to verify the computational predictions.

CZMs

Modeling of fatigue crack growth

In the CZM, the material failure is described by an additional cohesive law. As soon as the stress state in the material reaches the certain criterion, the CZM superimposes to the continuum mechanics formulation of the element. The traction-opening relation in the cohesive zone is determined by the cohesive law.

In this work, the cohesive zone law proposed by Xu and Needleman²⁰ is modified to simulate the crack propagation in the nickel-based super-alloy with a threshold value.¹⁵ Figure 1 schematically illustrates material failure under monotonic loading condition (dashed line) and under cyclic loading with one cycle (solid line), respectively. Analysis¹⁷ has confirmed that the mode I failure mechanism dominates the fatigue crack extension procedure for cracked specimens. It means the normal traction $t_{n}$ is the main driving force of fatigue crack propagation, and the shear traction $t_{s}$ can be neglected. Therefore, the normal traction $t_{n}$ is expressed as an exponential function of separation, while the shear traction $t_{s}$ is assumed in a linear relationship with the tangential separation, $δ_{t}$ , that is

\begin{array}{l} t_{n} = \frac{φ_{n}}{δ_{0}} \exp (- \frac{δ_{n}}{δ_{0}}) \\ t_{s} = G \frac{δ_{t}}{δ_{0}} \end{array}

(1)

Figure 1.

The sketch of cohesive curves under monotonic loading and cyclic loading (one cycle) conditions, respectively.

Above $δ_{n}$ and $δ_{t}$ denote the post-peak normal and tangential opening in the cohesive zone, respectively. G is the shear stiffness. $δ_{0}$ stands for the characteristic cohesive length. $φ_{n}$ is the fracture energy induced by normal separation which can be expressed as

φ_{n} = σ_{\max, 0} δ_{0}

(2)

where $σ_{\max}$ is the normal strength of the material cohesive zone.

In the CZM, the ultimate material strength is characterized by $σ_{\max}$ . To simulate time-dependent material failure, the strength has to diminish with loading cycles. As introduced in continuum damage mechanics,²¹ one may assume

σ_{\max} = (1 - D) σ_{\max, 0}

where $σ_{\max, 0}$ denotes the initial strength of the cohesive material. D stands for the material damage variable which relates to mechanical and thermal loading conditions. The cohesive law has an initial stiffness because of the non-zero separation $κ_{0}$ at the initial tensile strength $σ_{\max, 0}$ . For creep–fatigue crack problems, the material damage consists of both cyclic mechanical damage, $D_{f}$ , and creep damage, $D_{c}$ , that is

D = D_{f} + D_{c}

Considering the cyclic loading, the damage variable, $D_{f}$ , is introduced to describe the cyclic damage accumulation in the CZM. The description of $D_{f}$ is the key issue in the cyclic material damage modeling. In the past years, various suggestions were published. The first model for ductile failure was proposed by Roe and Siegmund²² in the form as

{\dot{D}}_{f} = \frac{| {\dot{κ}}_{n} |}{d_{Σ}} (\frac{t_{n}}{σ_{\max}} - \frac{σ_{f}}{σ_{\max, 0}}) H (\bar{κ} - κ_{0})

(3)

with ${\dot{D}}_{f} \geq 0$ and $\bar{κ} = \int | {\dot{κ}}_{n} | d t$ while $κ_{n}$ is the normal opening, $κ_{n} = δ_{n} + κ_{0}$ , with $κ_{0}$ as the initial normal opening related to the cohesive traction threshold value. The Heaviside function, $H (\cdot)$ , indicates that the damage will not accumulate until the separation happens in the cohesive zone, $\bar{κ}$ , exceeds the initial threshold $κ_{0}$ . $σ_{f}$ and $d_{Σ}$ represent the cohesive zone endurance limit and the accumulated cohesive length, respectively. Equation (3) demonstrated many interesting features and cannot, however, predict effects of the loading ratio, as observed in Xu and Yuan.²³ With increasing finite loading cycles, the effects of the mean stress cannot be considered. For this reason, the model has been modified by Xu²⁴ as

{\dot{D}}_{f} = \frac{| {\dot{κ}}_{n} |}{d_{Σ}} {(\frac{t_{n}}{σ_{\max, 0}} + 1)}^{m} H (σ_{e q} - f_{0})

(4)

where $σ_{e q}$ and $f_{0}$ are the equivalent principal stress around the cohesive zone tip and the material fatigue limit, respectively. Other parameters hold the same meanings as in equation (3).

Creep damage accumulation

At high temperature, the material mainly suffers from thermal attack. The mechanical stress is generally lower, so that the rate-independent plastification can be neglected. In practice, creep may occur within yield loading surface and creep behavior is decoupled with material plasticity for simplification. The creep can be divided into three phases, namely, primary, secondary, and tertiary creep regions. The creep strain rates in the primary and tertiary creep regions are not stable but time-dependent. In the secondary phase, the creep strain rate is usually considered to be independent of time and can be expressed by Norton–Bailey model as

{\dot{ε}}_{s}^{c} = A_{s} σ^{n_{s}}

(5)

where $A_{s}$ and $n_{s}$ are material constants. For engineering applications, it is the most popular model.

To consider the time effects on the crack growth under low frequency loading, especially under high temperature condition, the creep damage has to be considered separately. If the creep damage is introduced into the CZM, following the suggestion in Bouvard et al.,²⁵ the creep damage evolution in the CZM can be written as

{\dot{D}}_{c} = {〈 \frac{| | t | | - (1 - D) t_{c}}{C} 〉}^{r} {(1 - D)}^{- p}

(6)

where C, p, and r are material parameters which can be determined with experimental data. Minor modification is made on the model in Bouvard et al.²⁵ 〈·〉 denotes the Macaulay brackets with $〈 x 〉 = x$ for $x > 0$ , otherwise zero. Obviously, the model suggested by Bouvard et al.²⁵ cannot be applied in engineering since temperature dependence is not explicitly included. Should all model parameters be dependent on temperature, the whole model becomes rather complex for application. In order to explicitly consider effects of the temperature, an additional term related to the temperature ( $ϑ$ in K) is introduced into the creep damage evolution equation. The creep damage evolution (equation (6)) can be expressed as

{\dot{D}}_{c} = {〈 \frac{| | t | | - (1 - D) t_{c}}{C} 〉}^{r} {(1 - D)}^{- p} {〈 ϑ - ϑ_{0} 〉}^{ζ}

(7)

with $ζ$ as an additional model parameter for temperature influence. $ϑ_{0}$ denotes the lower limit temperature below which the creep damage disappears. $t_{c}$ is a threshold traction value which is the start point of the creep damage accumulation. $| | t | |$ is a kind of equivalent traction taking the form as

| | t | | = \sqrt{t_{n}^{2} + \frac{1}{α} t_{s}^{2}}

(8)

thus the creep–fatigue damage accumulation can be employed in the CZM.

The CZM introduced above has been implemented into the XFEM.¹⁵ The extended finite element is a handy way for considering a cohesive law containing a finite threshold value $κ_{0}$ . Extensive computations reveal that the cohesive zone should be induced only if the traction ahead of the crack tip exceeds the finite limit value $κ$ ; otherwise, stress distributions around crack tip are artificial. More detailed discussions about this point are out of the scope of this work.

Verifications

Experiments

The experiments were produced by Yang et al.^26,27 The standard compact tension (CT) specimens (Figure 2(a)) with the width $W = 50 mm$ , the initial crack length $a = 22.5 mm$ , and the thickness $B = 25 mm$ were tested at four temperature levels, that is, RT, 550°C, 650°C, and 750°C. The specimens were made of a nickel-based PM super-alloy, with the chemical composition and heat treatment summarized in Table 1. Two wave shapes of applied loads were employed in the experiment, namely, triangular and trapezoidal waves. The loading ratio was set to 0.05. Two types of cycle loads contained 1.5 s loading and 1.5 s unloading. Especially, the trapezoidal wave run with a 90 s hold time at the maximum. Subsequently, the hold time of 450 and 1500 s were involved to consider the dwell time effects on fatigue crack growth.

Figure 2.

Sketch of (a) compact tension (CT) specimen and (b) finite element mesh of crack tip.

Table 1.

Composition and heat treatment of nickel-based PM super-alloy.

The nominal chemical composition of the alloy (wt%)
C	Cr	Mo	W	Al	Ti	Co
0.02–0.06	8.0–10	3.5–4.2	5.2–5.9	4.8–5.3	1.6–2.0	15.0–16.5
Nb	Hf	Mg	Zr	B	Ce	Nickel
2.4–2.8	0.1–0.4	⩽0.02	⩽0.015	⩽0.015	⩽0.010	Balance
Heat treatment
1200°C solution treated, 8 h; furnace cooled to 1170°C; air cooled to RT; 870°C aged, 32 h, air cooled to RT

Results

In the present CZM, material failure is dominated by the damage variable D. The cohesive function (equation (1)) is indifferent for life prediction. In the known works,^25,28 the creep–fatigue can be predicted using the damage model (equation (6)); however, the dwell time was limited within very small region. No systematic experiments were considered for verifications.

For the present material, the test results shown in Figure 3 exhibit the effects of dwell time $(T_{d})$ on the crack growth rate $d a / d N$ in terms of stress intensity factor range $Δ K$ . The fatigue parameters in the CZM are identified using experimental data at high frequency loading condition $(f = 0.333 Hz)$ which is assumed as the pure fatigue region. With longer hold times, creep damage becomes intensive enough to determine the creep parameters in equation (7), given that the fatigue parameters are already known.

Figure 3.

Comparison of computational and experimental crack growth rates versus the stress intensity factor range $Δ K$ with different dwell times $(T_{d})$ at 750°C. The symbols denote experimental data and crossed lines stand for computational predictions.

In Figure 3, experiments with different dwell times are denoted using different symbols. Obviously, da/dN increases with $T_{d}$ in a power-law form. For large $T_{d}$ effects of dwell time diminishes. Computational predictions are shown in crosses with lines in the figure. The present CZM agrees roughly with experiments since both upper and lower limit cases are taken for parameter identifications. Increment of da/dN with dwell time, however, shows significant discrepancy between experiments and computations. Although the computations predict similar crack growth rate at large dwell time which is caused by the parameter identification for the CZM, the variations of the crack growth rate are very different. One may expect more significant deviations in prediction of the creep–fatigue life with larger dwell time. The results imply that the damage evolution equation does not match the experimental observation accurately.

Crack growth rates $d a / d N$ and $d a / d t$ as a function of the frequency are shown in Figure 4 for selected stress intensity factors $Δ K$ with $R = 0.05$ . Solid symbols denote experiments and the solid curves with cross are from the computational prediction. As shown in figures, the crack growth rates $d a / d N$ and $d a / d t$ increase with the rising of the stress intensity factor range $Δ K$ . The CZM can simulate the tendency well. According to the computation, the crack growth rate $d a / d N$ decreases with the reduction of the dwell time; meanwhile, $d a / d t$ increases because of the growing in cycle number per unit time. However, this simulation has some deviation with the experiment in the long dwell time zone as shown in figures. Especially when the stress intensity factor range $Δ K$ is low ( $30 MPa \sqrt{m}$ in Figure 4(a) for instance), as the dwell time reduces from 1500 s to 450 s, the crack growth rate increases. The damage evolution equation cannot predict this appearance yet.

Figure 4.

Frequency dependence of crack growth rate (a) $d a / d N$ and (b) $d a / d t$ at selected stress intensity factors $Δ K$ for nickel-based PM super-alloy tested at 750°C.

Figure 5 shows the effects of temperature on the crack growth. In the simulations, it is assumed that there exists no creep damage under RT, $ϑ_{0}$ in equation (7) was determined for optimizing the computational predictions. As shown in the figure, the crack growth rate exhibits the obvious different speeds under different temperatures, especially at low stress intensity factor range, because of creep damage dominance. As known, temperature affects creep significantly. In high stress intensity factor range, the influence of the temperature reduces and the experimental data at different temperatures gradually intermingle with each other. The crack rate is controlled mainly by the mechanical cycles. However, temperature effects at 750°C are much more severe than other temperatures. The correlation is certainly nonlinear, as assumed in equation (7). The computational results based on the CZM are shown in crossed lines in Figure 5. The figure confirms that the temperature effects are described by the present model properly. The cohesive approach provides an accurate prediction in effects of temperature and stress intensity factor range on the crack propagation behavior.

Figure 5.

Comparison of numerical and experimental crack growth rate in terms of stress intensity factor range $Δ K$ at different temperatures (T) for nickel-based PM super-alloy.

Conclusion

Creep–fatigue with long dwell times and various temperatures is generally difficult to be predicted. Using the CZM, this work has introduced a damage model for describing creep–fatigue crack propagation at elevated temperature and with long dwell time for a nickel-based PM super-alloy. The model has been implemented into the XFEM. The computational results confirm the following:

The damage evolution equation suggests a nonlinear coupling of mechanical cyclic loading with temperature.

The CZM can predict the fatigue crack growth properly in the high frequency region. In fatigue with long dwell time, the computational prediction still contains deviations.

The new creep damage evolution equation for the CZM describes temperature effects in life prediction. The existing experiments agree with computations pretty well.

The cohesive zone approach can be applied for more complex loading conditions or complex specimen geometries with great potential in numerical analysis of the creep–fatigue crack propagation.

Footnotes

Handling Editor: Michal Kuciej

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: In this paper, the research was sponsored by the National Natural Science Foundation of China (Project No. 51305025).

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Cohesive zone modeling of creep–fatigue crack propagation with dwell time

Abstract

Keywords

Introduction

CZMs

Modeling of fatigue crack growth

Creep damage accumulation

Verifications

Experiments

Results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References