Elsevier

International Journal of Fatigue

Volume 27, Issues 10–12, October–December 2005, Pages 1297-1306
International Journal of Fatigue

Time derivative equations for mode I fatigue crack growth in metals

https://doi.org/10.1016/j.ijfatigue.2005.06.034Get rights and content

Abstract

Predicting fatigue crack growth in metals remains a difficult task since the available models based on the Paris law are cycle-derivative equations (da/dN), while service loads are often far from being cyclic. This imposes a cycle-reconstruction of the load sequence, which significantly modifies the load history in the signal. The main objective of this paper is therefore to propose a set of time-derivative equations for fatigue crack growth in order to avoid any cycle reconstruction. The model is based on the thermodynamics of dissipative processes. Its main originality lies in the introduction of a supplementary state variable for the crack, which allows describing continuously the state of the crack throughout any complex load sequence. The state of the crack is considered to be fully characterized at the global scale by its length a, its plastic blunting ρ, and its elastic opening. In the equations, special attention is paid to the elastic energy stored inside the crack tip plastic zone, since, in practice, residual stresses at the crack tip are known to considerably influence fatigue crack growth. The model consists finally in two laws: a crack propagation law, which is a relationship between dρ/dt and da/dt and which observes the inequality stemming from the inequality of Clausius Duhem, and an elastic–plastic constitutive behaviour for the cracked structure, which provides dρ/dt versus load and which stems from the energy balance equation. The model was implemented and tested. It successfully reproduces the main features of fatigue crack growth as reported in the literature, such as the Paris law, the stress ratio effect, and the overload retardation effect.

Introduction

Predicting fatigue crack growth in metals under variable amplitude fatigue remains difficult since the available models based on the Paris law provide expressions of the cyclic crack growth rate ‘da/dN=f(Kmax, Kop, etc…)’, while loads during operating conditions are often far from being cyclic. Engineering practice consists of converting original load sequences into reconstructed sequences of fatigue cycles. However, the cycle reconstruction usually significantly modifies the load history in the signal despite the fact that is has been repeatedly shown in the past 30 years that fatigue crack growth is very sensitive to load history.

The load history effect is usually taken into account in the modelling through the variations of the crack opening stress intensity factor, Kop [1]. The effective part of a fatigue cycle is calculated as ΔKeff=KmaxKop and the cyclic crack growth rate is usually calculated as da/dN=CΔKeffm. The determination of the evolution of Kop with load history can be achieved using finite element method or semi-analytical models. Since the sensitivity to load history of Kop includes the sensitivity of the material itself to load history [2], the finite element method is preferred over semi-analytical models because it allows the introduction of suitable material constitutive behaviours.

However, in engineering practice, finite element methods are not employed for this purpose. First of all, it is not realistic to compute millions of elastic–plastic cycles as applied to a cracked structure and even more unrealistic if a complex elastic–plastic constitutive behaviour is needed for the material. Secondly, elaborate and time-consuming methods for computing the evolution of Kop with load history are essentially irrelevant since a significant part of load-history is lost during the cycle-reconstruction operation.

The aim of this paper is therefore to discuss the basis and to show the versatility of a mode I fatigue crack growth model expressed as a set of time derivative equations (da/dt). The present model remains within the framework of fracture mechanics and is based on the thermodynamics of dissipative processes. Its main originality lies in the introduction of a supplementary state variable for the crack which allows a continuous description of the behaviour of the crack versus time. Once this new variable is defined, the principle of virtual power, the energy balance equation, and the inequality of Clausius-Duhem are written and employed to set up the model [3]

Section snippets

Continuous characterization of the state of the crack

In practice it is common to encounter fatigue lives that stretch to tens of millions of ‘cycles’. This leads to a preference for a global approach rather than a local one. The global approach is aimed at characterizing the state of the crack through as few global variables as possible. The first state variable is the crack length a. Then, for a given crack length, the displacement field around the crack tip during a load sequence in mode I is analysed in order to find how it could be

Energy balance equation: yield criterion for the cracked structure.

The problem is assumed to be quasi-static and isothermal. In such a case, the energy balance equation is as follows:Pext+Q=dUdt=dψdt+TdSdtPext is the power of the external forces, Q is the heat received by the cracked structure, and U its internal energy. The variation of the internal energy is the sum of the variation of the free energy dψ and of the variation of the stored energy Us=T dS. Eq. (1) can be alternatively expressed as follows, putting on one side the reversible terms and on the

Unzooming by the FEM

In the previous sections, a new state variable ρ was chosen for characterizing the state of a crack continuously and using the framework of the dissipative processes, the expression of its conjugate force ϕ was determined. The expression of ϕ is now known as a function of the applied remote stress and ρ is calculated as explained before using the displacement profiles of the crack faces as calculated using the FEM. It is now possible to employ the FEM in order to study the evolution of ϕ versus

Application of the model

The equations were implemented initially with an explicit integration scheme [3] and at present with the use of an implicit integration scheme.

The material parameters of the cyclic constitutive behaviour of the cracked structure are determined automatically using the finite element method. No information is required from fatigue crack growth experiments. The only experiment necessary is a cyclic push–pull test on a bar in order to determine the parameters of the cyclic constitutive behaviour of

Conclusions

In this paper, the basis of a fatigue crack growth model constituted by a set of time-derivative equations were discussed. Recognizing that service loads during operating conditions are often far from being cyclic, this new approach was developed to produce predictive capability. The classical cycle-derivative models for fatigue crack growth, based on the Paris law, imply that cycles should be extracted from the real signal. This operation usually modifies the load history in the signal.

Prospects

Since we have developed an automated identification protocol, it is reasonable to consider providing the material parameters of the equations for the cyclic-plasticity of the cracked structure directly as a function of the material parameters, such as Young's modulus, Poisson's ratio, yield strength and rate and amount of kinematics hardening.

The addition of time-dependent and temperature dependent material properties may allow this theoretical framework to be expanded to the problem of

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