Advances in Materials Science and Engineering

Volume 2018, Article ID 5728174, 16 pages

https://doi.org/10.1155/2018/5728174

## Effect of Mean Stress on the Fatigue Life Prediction of Notched Fiber-Reinforced 2060 Al-Li Alloy Laminates under Spectrum Loading

^{1}Institute of Modern Design and Analysis, College of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819, China^{2}Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of China, Northeastern University, Shenyang 110819, China^{3}School of Mechanical Engineering, Shenyang Jianzhu University, Shenyang 110168, China^{4}Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX 76019, USA^{5}College of Science, Shenyang Jianzhu University, Shenyang 110168, China

Correspondence should be addressed to Weiying Meng; moc.361@520gniyiewgnem

Received 24 December 2017; Revised 17 March 2018; Accepted 22 March 2018; Published 6 June 2018

Academic Editor: Frederic Dumur

Copyright © 2018 Weiying Meng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a study on the fatigue life prediction of notched fiber-reinforced 2060 Al-Li alloy laminates under spectrum loading by applying the constant life diagram. Firstly, a review on the state of the art of constant life diagram models for the life prediction of composite materials is given, which highlights the effect on the forecast accuracy. Then, the fatigue life of notched fiber-reinforced Al-Li alloy laminates (2/1 laminates and 3/2 laminates) is tested under cyclic stress, which has different stress cycle characteristics (constant amplitude loading and Mini-Twist spectrum loading). The introduced models are successfully realized based on the available experimental data of examined laminates. In the case of Mini-Twist spectrum loading, the effect of the constant life diagram on the life prediction accuracy of examined laminates is studied based on the rainflow-counting method and Miner damage criteria. The results show that the simple Goodman model and piecewise linear model have certain advantages compared to other complex models for the life prediction of notched fiber metal laminates with different structures under Mini-Twist loading. From the engineering perspective, the S-N curve prediction based on the piecewise linear model is most applicable and accurate among all the models.

#### 1. Introduction

Fiber metal laminates (Arall and Glare et al.), as the new generation of aircraft structural materials, are the best alternatives for aluminum materials [1]. These laminates are composed of the metal layers and the fiber-reinforced resin layer, and these fibers can be placed in different directions [2]. Compared with conventional materials (metals or fiber-reinforced epoxy resin composites), fiber metal laminates could enhance the mechanical property by combining the metal layer and fiber layer, which exhibit excellent fatigue and impact properties and allow for flexible structural designs. Currently, fiber metal laminates have been used in the fuselage, leading edge, and other parts of the Airbus [3–6]. In view of the abovementioned advantages, fiber metal laminates have the potential wide application in aerospace equipment. Thus, understanding their fatigue performance is very important.

In general, fatigue strength is often referring to the structural strength under constant amplitude fatigue loading. However, constant amplitude loading is too ideal for engineering structures in practice. Therefore, variable amplitude cyclic stress should be considered [7]. In addition, not only the smooth material is used in actual engineering, but also notched materials are used more frequently. Since the failure mechanism of the smooth material is different from that of the notched material, the study on the fatigue properties of the notched material is also necessary. In the past few decades, the fatigue life prediction of composites under variable amplitude cyclic stress has received increasing interests. At the same time, the fatigue properties of the notched composite material are also widely concerned.

The life prediction problem of unnotched composite materials under variable amplitude loading is mainly based on the theory of fatigue damage cumulative and progressive fatigue damage. Some researchers have made some achievements in the application of fatigue damage cumulative theory to predict the fatigue life. A fatigue damage cumulative model for the fiber-reinforced plastics (FRPs) under variable amplitude loading was presented by Yao and Himmel [8], which was assumed that the damage state of laminates could be described phenomenologically by residual strength. A review on damage accumulation rules and residual strength approaches for FRP materials subjected to spectrum loading was made by Post [9], and the prediction accuracy of these introduced models was compared and analyzed. The effects of the cycle-counting method, constant life diagram formulation, and damage accumulation theory on the life prediction problem for composite materials under spectrum loading were discussed by Passipoularidis and Philippidis [10]. The fatigue properties of carbon fiber-reinforced plastic laminates under variable amplitude loading were studied by Hosoi et al. [7], who found that damage cumulative under variable amplitude loading could be better described by considering residual strength. Some researchers have also made some progress in the study of progressive fatigue damage theory for life prediction. In order to simulate the damage process of composite laminates under fatigue loading, a progressive fatigue damage model was first proposed by Shokrieh and Lessard [11]. Based on the fatigue test of the unidirectional laminates, the fatigue life prediction system of composite laminates was constructed by the three-part cyclical process of stress analysis, failure analysis, and material performance degradation. A progressive damage model applied in fatigue life prediction for composites under block and spectrum loading was discussed by Passipoularidis et al. [12], who used classical lamination theory to perform stress analysis, applied the failure criterion of Puck to implement adequate stiffness discount tactics, and merged residual strength into fatigue damage accumulation criteria. Fatigue properties of notched composites under variable amplitude loading have also been studied by some scholars. In order to increase the life prediction accuracy of notched composite materials under spectrum loading, an improved progressive damage model was proposed by Hu et al. [13]. In the model, a stiffness-strength degradation coupled model which was related to the damage of composites was developed, and a damage equivalent was made by the continuity of stiffness degradation so that it could be effectively applied to fatigue life prediction under complex loading. The fatigue crack growth model and delamination growth behavior of notched fiber metal laminates were studied by Khan et al. [14–17], who considered the effect of variable amplitude loading. A mechanism-based life prediction model of FMLs under constant amplitude loading was developed by Wu and Guo [18]. And the fatigue behaviors of GLARE laminates under single overloads were investigated experimentally and were predicted by an equivalent crack closure model. The fatigue crack growth and delamination extension behaviors of FMLs under single tensile overload were studied by Huang et al. [19], who found that fatigue crack growth and delamination behaviors could be influenced by the stress intensity factor of the crack tip in the metal layer when the overload is applied.

Studying on the potential damage mechanism of composite laminates and interpreting the relationship of feedback between materials and loading are a new research trend for the fatigue model of composite laminates [20]. The method based on the actual damage mechanism involves the properties of the fiber and matrix and the fatigue behavior of the laminate structure. However, the studies on fatigue life prediction of composites using the actual damage mechanism are restricted due to the complexity of the phenomenon [9]. As a result, the phenomenological approach has been more prevailing in the life prediction of composite materials under variable amplitude loading. It is an experienced method based on macroscopic mechanical properties of composites, which avoids the independent hypothesis units of component materials and obtains the model parameters by curve fitting. In order to verify the effectiveness of this method, composite materials with different compositions and those with different layer structures under various loading conditions have been examined by various researchers [10, 21–33].

The phenomenological approach has the same applicability for the unnotched and notched laminates, and the approach can phenomenally reflect the material damage. This is because the approach is a phenomenological empirical method that does not take into account the specific damage situation, and the model parameters can be obtained by curve fitting. The phenomenological approach for fatigue life prediction of composite materials under variable amplitude has three processes: (1) loading cycle counting, (2) the constant life diagram (CLD) model, and (3) the damage criterion. In order to determine an effective life prediction method under practical spectrum loading, some researchers have attempted to compare different models for each process [10]. Among them, the rainflow-counting method and Palmgren–Miner (PM) damage criteria are, respectively, the most widely used methods in the process of loading cycle counting and damage criterion. These CLD models are derived from different mathematical principles and based on the fatigue data of materials under constant amplitude loading. For the CLD-modeling process, it has been shown in recent studies that different models have a great impact on the fatigue life prediction accuracy of composite laminates under spectrum loading [21].

Unlike fiber-reinforced resin laminates, the fiber-reinforced metal laminates have a metallic composite structure and a nonmetallic composite structure, which leads to a different fatigue damage mechanism. A large number of documents showed that the research of this material was mainly focused on fatigue crack propagation and delamination behavior. Generally, it is very difficult to study the fatigue life prediction under complex spectrum loading based on fatigue crack propagation and delamination behavior, and there is very few research on the fatigue life prediction for notched fiber metal laminates based on the phenomenological approach.

In this paper, different CLD models in the life prediction of composite materials are introduced and compared. The effect of CLD models on the life prediction of notched fiber-reinforced Al-Li alloy laminates under spectrum loading is the focus of this study. The applicability of the model, the requirement of the test data, and the prediction accuracy are treated as important assessment parameters. The 2/1 laminates and 3/2 laminates of the notched fiber-reinforced Al-Li alloy are tested under three kinds of cyclic stress (constant amplitude loading with the stress ratio , that with the stress ratio , and Mini-Twist spectrum loading). Taking the fatigue life prediction under Mini-Twist spectrum loading for two kinds of the material as an example, the effect of CLD models on life prediction under Mini-Twist spectrum loading is studied through the use of the rainflow-counting method and Miner damage criteria.

#### 2. Life Prediction Theory for Composite Material

##### 2.1. Composite Life Prediction

At present, the research work on fatigue life prediction of composite materials under variable amplitude mainly focuses on the phenomenological approach. For life prediction under the complex stress cycle, it is a simple and effective method. The basic mechanical properties of the examined material used as input in the calculation are the compression static strength, tension static strength, and at least one S-N curve. The implementation process of the phenomenological approach is described as follows: Firstly, the number of cycles for the applied spectrum or block loading will be counted in order to predict the fatigue life under the complex stress cycle. The rainflow method is regarded as the most common cycle-counting method used in the irregular loading spectrum and is therefore adopted in this study. Secondly, the fatigue life at desired is evaluated by using the aforementioned material properties. An empirical model is required by most of the damage accumulation approaches to determine the total failure cycles under a constant amplitude stress which is equivalent to the current applied stress. Hence, the total failure cycles can be determined directly by the constant life diagram for a given stress cycle. The method will be later discussed in detail. Finally, the damage accumulation will be calculated. In this study, the Miner damage rule is adopted. It is a well-known and simple linear empirical rule.

##### 2.2. Constant Life Diagram Models

Constant life diagrams are treated as a predictive tool to estimate the material fatigue life under loading patterns. It can reflect the combined effect of material properties and mean stress on the fatigue life [22]. The main parameters used to build CLD models are the cyclic stress amplitude , the mean cyclic stress , and the stress ratio . The is defined as the ratio of the minimum cyclic stress to the maximum cyclic stress, that is, .

A number of CLD models have been proposed in the past to describe the characteristics of different composite materials. Based on symmetric linear Goodman and nonlinear Gerber equations, a variety of correction models are derived to simulate the behavior characteristics of composite materials [21]. The Goodman diagram is the simplest and most basic CLD and often used to verify the validity of other models. Without any assumptions, the piecewise linear (Pwlinear) interpolation was used between different S-N curves based on modified Goodman diagram concepts [23–25]. According to this idea, the analytical expressions for any desired S-N curve were proposed [26]. The piecewise nonlinear (PNL) model was developed by Anastasios, in which simple phenomenological equations were established based on the relationship between the stress amplitude and the stress ratio [22]. In addition, the PNL model was a mathematical representation of the material’s properties on the plane, and the equations were derived without any assumptions. The Bell model was derived by Harris et al., and it was based on the semiempirical formulations [27–29]. The result of this model was a continuous bell-shaped line from compressive strength to tensile strength, which was obtained from a nonlinear equation by fitting the experimental data. Based on the theory of the Gerber line, Boerstra deduced another CLD formulation [30]. In this model, the theory of variable slopes with various mean stresses for the S-N lines was introduced, and the exponent 2 of the Gerber equation was modified as a variable. This model provided a simple method for the fatigue life prediction and avoided fatigue data classification based on values, when the laminate material was subjected to cyclic stress with continuously varying mean stress. An alleged heteromorphic CLD was derived by Kawai, and the model was established by introducing the “critical” S-N curve concept [31, 32]. The ratio of ultimate compression strength (UCS) to ultimate tensile strength (UTS) was treated as the value of the critical S-N curve for the examined material. Based on the assumption that fatigue failure probability and static failure probability are equivalent, Kassapoglou recently developed a new CLD model [33]. Only the compressive and tensile static strength data were required to obtain the desired S-N curve in the Kassapoglou model. However, this model had an obvious drawback: it ignored different damage mechanisms resulted from fatigue stress and led to erroneous results in some cases. The models mentioned above will be described in detail below.

###### 2.2.1. Goodman Diagram

For any cycle stress with the stress amplitude and mean stress , the equivalent stress amplitude at is expressed as follows:

The equivalent stress corresponding to any cycle stress can be obtained by using the above equations. It will be used as an input into the S-N curve at in order to determine the allowable number of cycles for the current cycle stress. The model can be used directly based on a power S-N curve or an exponential S-N curve.

###### 2.2.2. Piecewise Linear CLD Model

The piecewise linear CLD model is deduced by linear interpolation theory between different S-N curves on the (-) plane [23–26]. This CLD model is constructed based on the compression static strength data and tension static strength data as well as a limited number of known S-N curves (generally at , , and ). The S-N curves at any values can be obtained by linear interpolation theory between experimentally determined static strength and fatigue strength data.

In this study, two experimentally determined S-N curves and static strength data are available. The unknown S-N curves can be obtained by the linear interpolation theory. The diagram of the piecewise linear model is shown in Figure 1, and the derivation process of the model is as follows. The values of two determined S-N curves are defined as and , respectively, and the value of desired S-N curves is defined as . Firstly, the mean stress and stress ratio of the current cycle are determined. Secondly, the values of and () for known S-N curves are calculated based on the slope *a*_{1} and intercept *b*_{1}, and the slope a_{3} and intercept *b*_{3}. The position of and on the (-) plane is described by and . The position of on the constant life plot can only be described by and , so the exact location cannot be determined at present. Thirdly, an initial fatigue life *N* is assumed. When (), for the assumed life *N*, the corresponding formulas of mean stress and amplitude stress are shown as follows:and when (), then