# Microstructure-based Multistage Fatigue Modeling of Aluminum Alloy 7075-T651

## Abstract

The multistage fatigue model for high cycle fatigue of a cast aluminum alloy developed by McDowell et al. is modiﬁed to consider the structure–property relations for cyclic damage and fatigue life of a high strength aluminum alloy 7075-T651 for aircraft structural applications. The multistage model was developed as a physically-based framework to evaluate sen¬sitivity of fatigue response to various microstructural features to support materials process design and component-speciﬁc tailoring of fatigue resistant materials. In this work, the model is ﬁrst generalized to evaluate both the high cycle fatigue (HCF) and low cycle fatigue (LCF) regimes for multiaxial loading conditions, with appropriate modiﬁcations introduced for wrought materials. The particular microstructural features of relevance to fatigue in aluminum alloy 7075-T651 include micron-scale Fe-rich intermetallic particles and rolling textures. The model speciﬁcally addresses the role of local con¬strained cyclic microplasticity at fractured inclusions in fatigue crack incubation and microstructurally small crack growth, including the eﬀect of crystallographic orientation on crack tip displacement as the driving force. The model is able to pre¬dict lower and upper bounds of the fatigue life based on measured inclusion sizes.

## Introduction

Figure 1 (a) Fractured intermetallic particles in interrupted fatigue experiments observed under scanning electron microscope; (b) geometry of finite element analysis (FEA); and (c) response of the elastic–plastic FEA of a fractured inclusion.

Recent prognosis programs for aging military aircraft have revived research on high strength 7075-T6 series Al alloys that have been used as structural materials for airframes. The prediction of fatigue crack formation and crack growth rate at sub-mm scale is very important for safety evaluation and strategicdecision-making. Moreover, crack formation and growth under variable loading is relevant to the selection of nondestructive testing procedures with appropriate resolution and structural designs for crack arrest based on crack growth mechanisms. The microstructure-based multistage fatigue (MSF)model of McDowell et al. [1][1] is an appealing model for prognosis applications, particularly in the HCF regime, since it captures fatigue behavior in the initial stages of crack incubation (formation and growth within the zone of inﬂuence of inclusion or defect where formed) and small crack growth, and explicitly addresses the role of microstructure. This model was originally devel¬oped to characterize the constant amplitude loading high cycle fatigue of cast Al–Mg–Si alloys with a partic¬ular goal of characterizing the potency of a hierarchy of microstructure features ranging from micron-scale interdendritic inclusions to dendrite cells to casting pores and trapped oxides. Interactions of both crack for¬mation and small crack growth processes with microstructure were considered. The model addresses the role of constrained microplasticity at debonded particles or gas pores in forming and growing microstructurally small fatigue cracks, using the cyclic crack tip displacement as the crack driving force rather than DK of linear elastic fracture mechanics (LEFM). For LEFM to be valid, the scale of the cyclic plastic zone at the crack tip must be small relative to crack length, as must the scale of the damage process zone. Furthermore, validity of the homogeneous material solutions for the mode I stress intensity factor of LEFM requires that the cyclic plastic zone must enclose a suﬃcient number of lower scale inclusions that control the rate of crack advance; this is typically much more demanding than the LEFM requirements on cyclic plastic zone size relative to crack length. The inhomogeneity issue remains even if the elastic–plastic fracture mechanics (EPFM) is employed to more adequately capture elastic–plastic crack tip conditions. A successful fracture mechanics approach for small crack growth in Ref. [2][2]was based on extensive experiments to eﬀectively homogenize the microstructure features using the equivalent initial ﬂaw size concept. However, the model requires extensive experiments in the threshold crack growth regime and small crack growth rate evaluations prior to application. In this paper, we extend the MSF approach [1] that decomposes fatigue life into four consecutive stages based on the microstructural details of fatigue crack growth as follows:

where $N_Total$ is the total fatigue life. Here, $N_inc$ is the number of cycles to incubate a crack at a micronotch that includes the nucleation of crack-like damage and early crack propagation through the zone of the micronotch root inﬂuence; NMSC is the number of cycles required for propagation of a microstructurally small crack with the crack length, ai < a < k MS, with MS deﬁned as a characteristic lengthscale of interaction with microstruc¬tural (MS) features and k as a multiplier in the range between one and three; NPSC is the number of cycles required for propagation of a physically small crack (PSC), k MS < a <O(10 MS), during the transition from MSC status to that of a dominant long crack (LC). Depending on the microstructural inclusion morphology and texture of the matrix, the PSC regime may extend to 300–800 $\mu\,$m. In Eq. (1)

In this research, the multistage fatigue model [1] is further generalized to consider high strength wrought 7075¬T651 Al alloy in both HCF and LCF regimes. The local plastic accumulation around inclusions is estimated using micromechanical simulations of fractured intermetallic particles. The incubation life of crack-like dam¬age is evaluated using an extended Coﬃn–Manson plastic strain-life relation at the microscale and is further extended to evaluate multiaxial loading cases. Certain parameters in this model are estimated using macro¬scopic material properties. The cyclic crack tip displacement ($\Delta$CTD) is used as the crack driving force for MSC/PSC growth and the grain orientation eﬀect on crack growth based on rolling textures is introduced in the model. The model parameter correlation is conducted using experimental strain-life relations. Finally, the model is validated using constant amplitude experiments following an overload event. Following calibra¬tion, the model is applied to predict lower and upper bounds of the fatigue life based on observed inclusion sizes which agree well with experimental results.

## Multistage Fatigue Model

Crack incubation

To characterize the damage incubation life as a function of micronotch root cyclic plastic deformation, a modified Coffin–Manson law was implemented based on the nonlocal maximum plastic shear strain for uniaxial

where $\beta\,$ is the nonlocal maximum plastic shear strain amplitude around the inclusion, and Cinc and $\alpha\,$ are coefﬁcient and exponent, respectively, in the modiﬁed Coﬃn–Manson law linking microplasticity to incubation life. The exponent $\alpha\,$ is chosen identical to that of the macroscopic Coﬃn–Manson (C–M) law [1];

Microstructurally small crack growth

Crack growth in the MSC/PSC regime is governed by the range of the crack tip displacement, $\Delta\,$CTD

where $\chi\,$ is a material constant that reflects crack tip irreversibility, typically in the range of 0.333 to 0.5; it is taken as 0.35 for aluminum alloys [1]. We assign the threshold value for crack tip displacement according to the Burger’s vector for the Al rich matrix, $\Delta$CTDth = b = 2.85 E-04 $\mu\,$m for pure FCC Al. The crack tip displacement is related to the remote loading according to [1]

where C1, CII, and all exponents are material constants to be determined based on fatigue crack growth experiments in the MSC regime. The effect of texture on MSC growth was captured using the ratio of grain orientation to the typical rolling texture orientation {011}<100> for FCC Al,$(GO/GO0)^\xi$ . The ratio of grain size to the reference grain size in the alloy $(GS/GS0)^\varpi$ describes the effect of grain size distribution on small crack growth.

Long crack growth

Finally, the LC growth regime is primarily mode I opening fatigue crack growth and is modeled using linear elastic fracture mechanics (LEFM), i.e.,

## Results

Figure 5 Strain-life data and model correlation for 7075-T651 Al alloy for completely reversed strain-controlled loading, with the fatigue cracks formed at 8 lm fractured particles. Note the higher fraction of incubation life to total fatigue life in the high cycle fatigue regime.
Figure 8 Predicted upper and lower bounds of total fatigue life for completely reversed, strain-controlled fatigue of 7075-T651 Al alloy(D = particle inclusion size).

Uniaxial fatigue experiments under constant amplitude loading were conducted on smooth cylindrical fatigue specimens at room temperature (approx. 68 F) in laboratory air with relative humidity near 45–60 RH.completely reversed remote strain amplitude. Failure was defined as 50% drop of the maximum stress. The strain-life experimental results, Re = -1, are shown in Fig. 5.

Assuming an extremely large particle diameter of 20 lm as the site of crack formation, a lower bound of fatigue life of 7075-T651 is predicted. Assuming a small pore size of 4 lm as the site of damage incubation, an upper bound for fatigue life was predicted. As shown in Fig. 8, the capability of predicting accurate upper and low bounds is an important attribute of the multistage fatigue model. In addition, the model can conceptually predict the variability due to grain size or texture modification, another key aspect, which is another key aspect.

## References

1. McDowell DL, Gall K, Horstemeyer MF, Fan J. Microstructure-based fatigue modeling of cast A356-T6 alloy. Engng Fract Mech 2003;70:49–80.
2. Newman Jr JC, Wu XR, Venneri SL, Li CG. Small-crack eﬀects in high-strength aluminum alloys, NASA reference publication, 1309, 1994

Citation: Microstructure-based multistage fatigue modeling of aluminum alloy 7075-T651, Y. Xue a, D.L. McDowell, M.F. Horstemeyer, M.H. Dale, J.B. Jordon, Engineering Fracture Mechanics 74 (2007) 2810–2823