Efficient Risk Assessment Using Probability of Fracture Nomographs

Transcription

1 Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2011 Efficient Risk Assessment Using Probability of Fracture Nomographs Venkateswaran Shanmugam Wright State University Follow this and additional works at: Part of the Engineering Commons Repository Citation Shanmugam, Venkateswaran, "Efficient Risk Assessment Using Probability of Fracture Nomographs" (2011). Browse all Theses and Dissertations. Paper 512. This Dissertation is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact

2 Efficient Risk Assessment Using Probability of Fracture Nomographs A dissertation submitted in partial fulfillment of the requirements for the degree of the Doctor of Philosophy by Venkateswaran Shanmugam B.E., Bharathiar University, Wright State University

3 Wright State University SCHOOL OF GRADUATE STUDIES October 26, 2011 I HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER MY SUPERVISION BY Venkateswaran Shanmugam ENTITLED Efficient Risk Assessment Using Probability of Fracture Nomographs BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy. Ravi C. Penmetsa, Ph.D. Dissertation Director Ramana V. Grandhi, Ph.D. Director, Ph.D. in Engineering Program Committee on Final Examination Andrew Hsu, Ph.D. Dean of Graduate Studies Ravi C. Penmetsa, Ph.D. Ramana V. Grandhi, Ph.D. Nathan W. Klingbeil, Ph.D. Eric J. Tuegel, Ph.D. Robert A. Brockman, Ph.D. Stephen B. Clay, Ph.D.

4 ABSTRACT Shanmugam, Venkateswaran. Ph.D., Engineering Ph.D. Program, Wright State University, Efficient Risk Assessment Using Probability of Fracture Nomographs. The traditional risk-based design processes involve designing the structure based on risk estimates obtained during several iterations of an optimization routine. This approach is computationally expensive for large-scale aircraft structural systems because of the iterative nature of the risk assessment methods. Therefore, this research introduces the concept of risk-based design plots that can be used for both structural sizing and risk assessment of stiffened plates when maximum allowable crack length information is available. These plots are obtained using normalized probability density functions of load and material properties and are applicable for any arbitrary load and strength magnitudes that follow similar scatter. Risk-based design plots serve as a tool for failure probability assessment given geometry and applied load, or they can determine geometric constraints to be used in sizing, given allowable failure probability. This approach would transform a reliability-based optimization problem into a deterministic optimization problem with geometric constraints that implicitly incorporates risk into the design. Moreover, these plots provide a unique graphical tool to visualize the sensitivity of risk to geometric changes and loading conditions. In situations where crack length is defined as a probability distribution, the presented approach can only be applied for various percentiles of crack lengths. To demonstrate the methodology outlined in this research, a cracked flat and stiffened plate configurations of Aluminum 2024-T3 are investigated using both a Stress Intensity Factor and Cohesive Zone Model approach. This research also presents a material property calibration process for the probabilistic cohesive zone model for Aluminum 2024-T3. In order to demonstrate the robustness of the calibration process, it was also applied to a composite (IM7/977-3) double cantilever beam peel test to capture the scatter in experimental measurements of delamination strength. iii

5 List of Abbreviations AF RL ASIP CDF CZM DCB DCM F ALST AF F F EA F EM F F T LEF M LHS MCS P DF P I P of P P R P SO RBD RSM SIF Air Force Research Laboratory Aircraft Structural Integrity Program Cumulative Distribution Function Cohesive Zone Model Double Cantilever Beam Displacement Compatibility Method Fighter Aircraft Loading STAndard For Fatigue Finite Element Analysis Finite Element Model Fast Fourier Transformation Linear Elastic Fracture Mechanics Latin Hypercube Sampling Monte Carlo Simulation Probability Density Function Probability Index Probability of Fracture Park-Paulino-Roseler Particle Swarm Optimization Risk-Based Design Response Surface Model Stress Intensity Factor iv

6 Contents 1 Introduction Vision Motivation Dissertation Organization Fracture Nomographs using Linear Elastic Fracture Mechanics Probability Density Function Modeling Failure Criteria for Flat and Stiffened Plate Failure Criteria for Flat Plate Failure Criteria for a Stiffened Plate Probability of Fracture Nomograph Creation of Plot Creation of Plot Creation of Plot Using the Probability of Fracture Nomograph Example Plate with a Hole Displacement Compatibility Method (DCM) to Determine the Geometric Effect Stiffened Plate with Z and C Stiffeners J-Integral Evaluation J-Analytical Finite Element Model Results and Discussion Summary Fracture Nomographs using Cohesive Zone Modeling Introduction PPR Traction-Separation Law Probability Distribution Identification Example Geometry v

7 3.4.2 Traction-Separation Law Conventional Material Properties and Initial CZM Parameter Estimates Finite Element Model Response Surface Model Results Uncorrelated Parameters Correlated Parameters Material Strength Scatter Probability of Fracture Nomograph Nomograph for a Flat Plate Nomograph for a Stiffened Plate Procedure for Calibration Summary Scientific Contributions and Limitations of Nomograph List of Scientific Contributions Limitations of PoF Nomograph Summary and Future Work Summary Future Work A Appendix A: Probabilistic Cohesive Zone Model for Fiber Bridging 90 A.1 Introduction A.2 Cohesive Bridging Law Definition A.3 DCB Geometry and Finite Element Model A.4 Probabilistic Cohesive Zone Model A.4.1 Estimation of Steady State Energy Release Rate, Load and Crack Length A.4.2 Upper and Lower Bounds A.4.3 Response Surface Model A.4.4 Monte Carlo Simulation A.5 Results A.5.1 Calibration of Probabilistic CZM Parameters A.5.2 Prediction of Variation in Steady State Energy A.6 Summary Bibliography 117 vi

8 List of Figures 1.1 Vision for aircraft risk assessment PoF using traditional reliability analysis PoF using nomograph approach Research outline Normalized limit load (LL) Limit load (actual = psi) Normalized fracture strength Mean fracture strength, psi Cracked flat plate Two-bay skin crack with broken center stiffener Plot capturing the geometric variations for a stiffened plate with rectangular stiffeners (Stiffener spacing, b = 4 and 8 ) Applied stress-fracture strength plot Plot capturing the stress at limit load and mean critical fracture toughness variations Probability index versus probability of fracture Fracture toughness, K C Sample PDF discretization Probability density for limit state Z Applied stress and fracture strength PDF Normalized final Z PDF Probability of fracture estimate for other load distribution Probability of fracture nomograph Cracked plate with a center hole Plot 1 of the nomograph Plot 2 of the nomograph Probability of fracture nomograph for flat plate with a hole Applied stress and reaction forces in a cracked stiffened plate[65] Different Z-stiffener configuration (dimensions, inch) Z and rectangular stiffener results C, Z and rectangular stiffener results ABAQUS Z-stiffener model vii

9 2.27 Collapsed element representation J-contour integral values J-contour locations (red color) J-contour picked set (3D), A-top, B-middle, C-bottom J-contour value (3D) Linear traction separation law Exponential traction separation law Traction separation law based on PPR potential model Spectrum of curves generated using the traction separation law in Equation Flow chart for determining the CZM parameters and their PDFs Cracked plate analyzed using a cohesive zone model Bilinear traction-separation law PPR traction-separation law Residual strength CDF comparison of PPR model Residual strength CDF comparison, PPR, Bilinear, and test data Realizations of PPR model (a) No correlation, (b) Positive correlation, (c) Negative correlation Lower tail region comparison for fracture strength PDF PDFs of fracture strength for different crack lengths PDFs of normalized fracture strength for different crack lengths Mean fracture strength for different crack lengths (Plot 1) Mean fracture strength with probability index (Plot 2) Probability of fracture with probability index Probability of fracture nomograph for a flat plate Probability of fracture nomograph for a stiffened panel, µ= A.1 Fiber bridging [104] A.2 DCB specimen A.3 Crack tip of a DCB specimen A.4 Typical bridging law A.5 Adjusted bridging law A.6 Load vs load line displacement of 15 test results A.7 Steady state energy release rate vs crack length of 15 test results A.8 FEA model A.9 Flow chart for determining the CZM parameters and their PDFs A.10 Energy release rate, J vs crack length, a (test data 4) A.11 Steady state energy, test vs optimized result A.12 Steady state load, test vs optimized result A.13 Steady state crack, crack length vs optimized result A.14 Steady state energy release rate vs crack length - 51 and 76.2 mm A.15 Steady state energy release rate, test vs predicted result A.16 Steady state load, test vs predicted result A.17 Steady state crack, test vs predicted result viii

10 List of Tables 2.1 FALSTAFF spectrum with normalized stress exceedance values J-integral value comparison Material property Latin hypercube samples for bilinear model Latin hypercube samples for PPR model Bilinear model-initial and optimal distribution results (Uncorrelated) PPR model-initial and optimal distribution results (Uncorrelated) PPR model-initial and optimal distribution results (Correlated) Entropy results Bhattacharyya distance results A.1 Material properties of IM7/977-3 composite A.2 Steady state energy rate, load and crack length from 15 tests A.3 Upper and lower bounds A.4 Coefficients for load and crack length RSM A.5 Bridging law distribution results A.6 Coefficients for load and crack Length RSM mm A mm - Steady state energy rate, load and crack length from 9 tests ix

11 Acknowledgment First and foremost, I would like to thank my advisor Dr. Penmetsa for the continuous guidance and support he provided for me in order to achieve this challenging task. I would also like to thank Dr. Grandhi for providing admission to the incredible Ph.D. Program at Wright State University (WSU). I am grateful to all the members of my dissertation committee, Dr. Grandhi, Dr. Klingbeil, Dr. Tuegel, Dr. Brockman, and Dr. Clay, for taking time out of their busy schedules to provide their reviews and valuable comments on my research work. I would also like to thank Dr. Tuegel from the Air Force Research Laboratory (AFRL), the Midwest Structural Sciences Center (MSSC), the Dayton Area Graduate Studies Institute (DAGSI), and the WSU Ph.D. in Engineering Program Office for providing funding to complete my research. In addition, I would like to thank the WSU and University of Dayton professors for challenging coursework to maintain the utmost standards and expectations of the graduate school. I also appreciate Ms. Alysoun Taylor-Hall, Ph.D. Program Coordinator, for monitoring and advising the progress of my research work. I greatly appreciate the WSU University Center for International Education (UCIE) for providing this wonderful opportunity to upgrade my technical knowledge to meet today s global requirements. I appreciate all my Wright State and family friends who provided their suggestions and help over these years. I look forward to continuing relationships with each of them in the coming years. Finally, I would like to thank the two most important people in my life: my wife, Sailakshmi Venkateswaran and our son, Arvind Kumar Venkateswaran. Without them, it would have been impossible to achieve this dream and professional goal. Their support and sacrifice over the years has been enormous, and saying thank you will never be sufficient. x

12 Dedicated to My loving and caring wife Sailakshmi and our wonderful son Arvind Kumar xi

13 Chapter 1 Introduction Advances in computational tools and capabilities have made it possible for the U. S. Air Force to plan development of a digital twin for every aircraft platform. This digital twin will be subjected to the same load spectrum as the physical system. The digital twin represents the state of the physical structure in a probabilistic sense, with statistical definitions representing manufacturing flaws, geometric tolerances, and damage history. This information can be used to determine the Probability Density Function (PDF) of the damage state at any given time. The damage state PDF can then be used to predict the risk of failure of future missions whose load profiles are obtained from a load spectrum forecasting model. The digital twin can provide risk of failure and damage state information to an operational model that performs risk benefit analysis and determines whether the structure needs to be repaired before deployment. The operational model would have the damage states and mission capability information for all the aircraft in the fleet. Therefore, it will be capable of automatically prioritizing maintenance of these systems and scheduling future depot visits while maximizing their availability. This type of an environment would minimize operational costs to the U. S. Air Force while ensuring maximum availability and mission capability for its fleet. These type of systems will become a necessity for the U. S. Air Force because of aging fleets and increased volatility in the global arena. 1

14 1.1 Vision The ideal scenario where structures and computational tools communicate to inform various depots when they are due for a specific maintenance task that need to be performed on an individual vehicle/system, will revolutionize the condition-based maintenance program. This is not a far-fetched idea and can be achieved through technological advancements in various aspects of this scenario. One aspect of the project is a load spectrum forecasting model that uses past usage history to predict future mission profiles and loading sequences. This information, combined with the computational damage models, would enable probabilistic life prediction capabilities of aerospace components. These estimates can directly impact crucial decisions about aircraft usage for any given mission. Figure 1.1 illustrates the U.S. Air Force vision for a probabilistic damage prediction process. The failure mode effects and criticality analysis of structural systems presented in Figure 1.1 have already been developed and published by Penmetsa et. al. in Ref. [1]. As shown in Figure 1.1, damage states are modeled using the cohesive traction separation law, and the degraded model parameters represent the evolution of damage within the material. The technological challenges relating to the calibration of probabilistic cohesive zone models and efficient propagation of material strength scatter through the structure are the focus of this research. These developments address analysis of structures, which is a one-time evaluation of the state of the component. However, in a design environment these analyses need to be performed multiple times to evaluate competing design ideas. Therefore, along with a new analysis capability, this research also explores various strategies to enable rapid risk assessment in a design environment. With the advances in computing speed in recent years, multidisciplinary optimization of large structural systems has become increasingly common in various industries. Nuclear and offshore industries [2, 3] have even introduced formal risk-based design practices to minimize risk of failure of various components. The Literature suggests that structures 2

15 Figure 1.1: Vision for aircraft risk assessment designed for minimum risk not only have lower failure rates, but they also result in reduced operating costs over the life of the component. Despite this, the aircraft industry has traditionally relied on factor-of-safety based techniques based techniques to design structures. This approach has proved capable of producing safe structures as long as they are used for the intended purpose as outlined in the original design requirements. However, many platforms in the Air Force fleet have exceeded their design life and are maintained using ASIP guidelines to manage risk. These guidelines rely on propagating uncertainties through the structural system and determining the probability of failure of a structural component during a given flight. There are several risk assessment algorithms available in the literature [8]-[14] to propagate uncertain input information through the structural system 3

16 to determine its probability of failure. These techniques have already been used in several multidisciplinary optimization algorithms [15]-[24] to size structures for minimum weight and risk. In the past, several researchers [25]-[34] have developed risk-based design algorithms that use surrogate models of the response to improve the efficiency of the risk assessment process. While all these past advancements have made risk-based optimization [35]-[44] practical for large-scale structures, they have still relied heavily on iterative risk analysis algorithms. These risk analysis algorithms also require the user to be familiar with risk assessment methods in order to be able to integrate them into the optimization process. For a risk-based design algorithm to be successfully and efficiently transitioned into the aircraft industry, it should not place requirements for additional expertise on structural design engineers. 1.2 Motivation There are three major inputs to the structural safety assessment of a structure. One is the geometry, the second is the load that the structure experiences during service, and the third is the strength or resistance of the material to withstand the applied load. Since aircraft structure predominantly fail by fracture, traditional risk assessment of the aircraft structure is based on Probability Of Fracture (PoF). Typically, all the aircraft are periodically inspected by maintenance personnel, and during the inspection the crack sizes at critical locations are analyzed to verify structural integrity. Using these crack sizes, material strength, and mission profiles, reliability engineers can estimate the PoF using reliability analysis methods. Based on the PoF value, reliability engineers can give their recommendations regarding the operational capacity of the aircraft. For example, as per FAA guidelines, if the PoF is less than 1 x 10 7, the aircraft is safe to fly, otherwise the aircraft has to be repaired before its next flight. 4

17 Reliability engineers often use software, such as PROF (PRobability Of Fracture [45]) or DARWIN (Design Assessment of Reliability With INspection [46]), to estimate PoF using a reliability index or safety index, β RI. A reliability index is a normalized measure of the number of standard deviations from the failure surface. Typically, the reliability index is determined iteratively through an optimization algorithm, which searches the design space to identify the most probable failure point (MPP) on the failure surface (also called the limit state). The flow chart in Figure 1.2 shows the different steps involved in calculating the reliability index. There are different reliability methods, [47] such as FORM (First Order Reliability Method) and SORM (Second Order Reliability Method), that are implemented based on the nature of the limit state to find the reliability index. Also, based on the type of limit state function (linear or nonlinear), different first order algorithms, such as MVFOSM (Mean Value First Order Second Moment), HL (Hasofer Lind [48]), TANA (Two-point Adaptive Nonlinear Approximation [55, 56]), and second order algorithms developed by Breitung [49], Tvedt [50, 51], Hohenbichler and Rockwitz [52], Koyluoglu and Nielsen [53], Cai and Elishakoff [54], and Wang and Grandhi [56], are available in the literature to determine the most probable point and reliability index accurately. While each of the reliability methods have their own strengths and weaknesses, one aspect that is common among the methods is that the optimization process to obtain MPP has to be restarted if there are any changes made to the geometry, loading, or material properties. Moreover, in an optimization framework where the structure is being designed to meet certain reliability metrics, the traditional reliability methods require repeated analysis in each optimization iteration as the structural design space is being explored. Therefore, the concept of risk-based design plots was developed during this research to present risk information to the designers in a simplified format that also overcomes some of the drawbacks of the iterative risk assessment methods. These plots directly provide the risk of failure estimates for a component using simple scaling factors that are similar to margin of safety calculations. Once digitized, these plots represent a tool that eliminates 5

18 Limit State Set initial design point Compute initial, β RI Compute new design point Compute safety index, β RI (Most Probable Point-MPP) β RI converged? Final safety index, β RI and PoF Figure 1.2: PoF using traditional reliability analysis the expensive iterative risk analysis algorithms that are typically used in most optimization routines. These design plots, also called nomographs (Figure 1.3), were developed in such a way that they not only capture the geometric variation, but also capture material and loading variation. The nomograph developed in this research serves two purposes: one is to determine an allowable crack length given acceptable failure probability and the other is to determine failure probability given geometry and loading conditions of the structural component. A unique feature of this nomograph is that when the geometry of interest is changed from a stiffened plate to a plate with a hole, only the first plot that represents geometry needs to be modified. The other two plots remain unaltered as long as the material and scatter information are the same for both the geometries. All three plots indicated in Figure 1.3 can be created independently, and only the geometry plot requires computational effort. Once these nomographs are developed by reliability engineers, maintenance engineers can efficiently access the risk of the aircraft structure rapidly and make a GO / NO-GO decision at the maintenance depot. Maintenance engineers do not need any reliability knowledge or 6

19 Probability of Fracture - PoF optimization skills to use the nomograph. K Stiffener / applied Load applied / K C = 0.4 applied / K C = 0.6 applied / K C = 0.8 applied / K C = Probability Index - PI Geometry =0.2, b=8 =0.4, b=8 =0.6, b= a/b FALSTAFF PoF Probability Index-PI Figure 1.3: PoF using nomograph approach 1.3 Dissertation Organization The focus of this work is to develop a framework for rapid risk assessment using nomographs to estimate probability of fracture. While fatigue was not addressed in this research, contributions of this work can be used to further develop a capability for probabilistic fatigue damage progression. Nomograph-based fracture risk assessment is a novel contribution of this work that is demonstrated on two fracture theories. One is based on the Stress Intensity Factor (SIF), and the other is an approach based on the Cohesive Zone Model (CZM), as shown in Figure1.4. CZM is introduced in this research because of its capability to handle both linear and nonlinear fracture mechanics. Also as compared to Linear Elastic Fracture Mechanics (LEFM), CZM does not require an initial crack to start with; 7

20 one can insert CZM at critical locations in the structure, and based on the loading conditions, a crack can nucleate and propagate. This is because the failure or fracture criteria for that material is assigned into the cohesive elements. The following paragraphs outline how various chapters in this document are organized. Rapid Risk Assessment Metallic (Aluminum) Stress Intensity Factor Cohesive Zone Modeling CZM Parameter Calibration Cracked Flat and Stiffened Plate Probability of Fracture Nomograph Figure 1.4: Research outline Chapter 2 explains the development of nomographs to determine the probability of fracture in Al 2024-T3 using a Stress Intensity Factor (SIF) approach. This methodology is demonstrated using cracked flat and stiffened plate models. To determine the distribution of fracture strength, a fracture toughness distribution is used from existing test data of an unknown aluminum alloy, which is then normalized for use with Al 2024-T3. Chapter 3 investigates the applicability of Cohesive Zone Modeling (CZM) for the development of nomographs. The difficulty of using CZM stems from the lack of experimental data for determining the CZM parameters and their distributions. Using assumptions 8

21 about PDFs for the CZM parameters will introduce uncertainty into PoF estimates. Since material fracture toughness data is available from experimental tests, this research is aimed at using that information to determine appropriate Probability Density Functions (PDF) of the cohesive zone parameters that can simulate experimentally observed fracture strength scatter. A finite width cracked plate is used as a test case to demonstrate the process. This research investigates the possibility that the material scatter can be isolated from the geometric effects to determine a normalized PDF of fracture strength for a given material. This normalized PDF can then be scaled, using mean fracture strength, to any crack configuration. Chapter 4 presents scientific contributions and limitations of probability of fracture nomographs. In this dissertation the primary focus is to investigate probabilistic fracture in metallic materials. However, in order to verify the validity of the CZM calibration process developed in Chapter 3, a composite delamination criterion is considered in this research. Currently, the Air Force Research Laboratory (AFRL), located at Wright-Patterson Air Force Base, Dayton, OH, is testing unidirectional composite (IM7/977-3) material using double cantilever beam (DCB) specimens to estimate Mode-I fracture toughness. During testing, the energy plot of DCB specimens showed an increase in energy release rate with increase in crack length. Also, it was found during measurement that there is a variation of about 20 to 200 percent in the energy release rate between samples. Therefore, the goal of this study is to calibrate the cohesive zone model for the composite material and capture steady state energy release rate variations of DCB specimens. The details of the cohesive zone model, the calibration process, and results are included in Appendix-A. 9

22 Chapter 2 Fracture Nomographs using Linear Elastic Fracture Mechanics This chapter presents information on how to generate Probability of Fracture (PoF) nomographs using a Stress Intensity Factor (SIF) based approach. A cracked flat plate and a cracked stiffened plate are selected as examples to demonstrate the development and use of risk integrated design plots. The following sections provide details on Probability Density Function (PDF) modeling and normalization, failure criteria considered for flat plate and stiffened plate, and risk integrated design plots. Also, details about a parametric study performed on Z and C type stiffeners are presented in this chapter. 2.1 Probability Density Function Modeling In this research, applied load or stress (L or σ), yield strength (σ y ), and fracture toughness were modeled as random variables whose distribution functions are determined using a weighted distribution-fitting scheme presented in Ref. [57]. The FALSTAFF (Fighter Aircraft Loading STAndard For Fatigue) load spectrum [58], Mil-HDBK information [59], and 10

23 lot release data were used to determine load, material yield strength, and fracture toughness PDFs, respectively. These distributions were then normalized to common reference values. The load distribution was normalized such that 1.0 on the abscissa represented the Limit Load (LL) condition where P r[load > LL] = The PDFs for normalized limit load and actual limit load of psi are shown in Figure (2.1 & 2.2). DOD Joint Services Specification Guide of Aircraft structures JSSG-2006 [60], in section , suggests selecting a value for frequency of occurrence of limit load. In this study, 10 7 was selected. Using this value, the limit load is defined as load whose frequency of occurrence is less than or equal to This value can be selected based on the design requirements of various aircraft platforms and the PDFs can be updated accordingly by using PDF shifting. For normalized strength distribution, 1.0 corresponded to either A or B basis strength allowables. A-basis values for material strength are those values that will be exceeded 99 percent of the time with a 95 percent confidence interval. B-basis values are exceeded 90 percent of the time with a 95 percent confidence interval. Since mean values for fracture toughness are typically available in the Mil-Handbooks, the PDF was normalized such that 1.0 corresponds to mean K C (fracture toughness of the material). The normalized fracture strength and psi mean fracture strength (corresponding to a 2 in half crack length Aluminum center crack specimen) PDFs are shown in Figure (2.3 & 2.4 ) 11

24 PDF PDF Limit Load (LL) = Normalized applied stress (PSI) Figure 2.1: Normalized limit load (LL) x Limit Load (LL) = psi Applied stress (PSI) (psi) x 10 4 Figure 2.2: Limit load (actual = psi) 12

25 PDF PDF Normalized fracture strength (PSI) Figure 2.3: Normalized fracture strength 1.2 x Fracture strength (PSI) (psi) x 10 4 Figure 2.4: Mean fracture strength, psi 13

26 2.2 Failure Criteria for Flat and Stiffened Plate Failure Criteria for Flat Plate Two failure criteria were considered in this study for design and analysis of a flat plate under uniaxial tension (Mode-I, opening). Failure here has been defined as stress exceeding the fracture strength of the structure. The following equations show the residual strength for a cracked plate. R.S NetSection = σ y t(w 2a) (2.1) 2w R.S F racture = K C t πα cos(πα 2 ) (2.2) Figure 2.5: Cracked flat plate Equation 2.1 represents residual strength before net-section yielding and Equation 14

27 2.2 represents residual strength before fracture. As shown in Figure 2.5, σ is the applied uniaxial stress, a is the half crack length, t is the plate thickness, w is the plate width, α = 2a, σ w y is the yield strength of the material, and K C is the material fracture toughness. Based on these definitions of residual strength, the probability of failure for this flat plate subject to random load and material properties can be assessed using the following equations. Probability of net-section yielding of a cracked plate: P fnetsection = P [(1 2a w )σ y σ < 0] (2.3) Probability of fracture of a cracked plate: 2w P ff racture = P [K C t πα cos(πα 2 ) L < 0] (2.4) In Equation 2.3 if m = (1 2a w ), X = σ y, Y = σ then Equation 2.3 becomes P fnetsection = P [mx Y < 0] (2.5) where m represents geometry, X and Y are two random variables. A similar expression can be obtained for Equation 2.4 P ff racture = P [na B < 0] (2.6) 2w where n = t cos( πα), A = K πα 2 C, B = L. To efficiently perform the integration of these equations, a FFT (Fast Fourier Transform)-based numerical convolution technique [61] was selected to determine the failure probability of the individual failure modes. This FFT method is capable of estimating the failure probability for highly nonlinear problems with less computational effort compared to Monte Carlo Simulation (MCS) and can be 15

28 implemented to non-traditional PDFs without creating any approximations Failure Criteria for a Stiffened Plate One of the design criteria to ensure damage tolerance of aircraft structures is the capability to sustain a two bay skin crack with broken central stiffener as shown in Figure 2.6. The displacement compatibility method (DCM) based solution approach [62] was used in this study to analyze the stiffened panel to determine the geometric factors required for residual strength calculations. The failure criteria used to determine the probability of fracture were as follows: P fstiffened = P [ResidualStrength < AppliedStress] K C P fstiffened = P [ < σ] πaβ(a) K C P fstiffened = P [ σ < 0] (2.7) πaβ(a) where K C is the critical stress intensity factor, a is the half crack length, and β(a) is the geometric effect on stress intensity. The geometric effect due to stiffeners was estimated as follows: β(a) = K Stiffened σ πa (2.8) where K Stiffened is determined using DCM (section 2.5) and the denominator is the stress intensity factor for an unstiffened flat plate. Equation 2.7 can be rewritten as P fstiffened = P [cr S < 0] (2.9) 16

29 where c = 1 πaβ(a), R = K C, S = σ. This is clearly in the same form as Eqs. 2.5 and 2.6, and therefore, can be easily integrated using FFT based convolution technique. Figure 2.6: Two-bay skin crack with broken center stiffener 2.3 Probability of Fracture Nomograph This research introduces a novel concept of risk-based design plots, called PoF nomograph, which is a series of plots that share a common axes that can be used for both design and analysis of stiffened panels. The PoF nomograph contains three plots, one for geometric variations, one for failure criterion, and one for representing scatters in load and material yield strength. This section contains details of the process that will lead to the generation of nomograph. 17

30 2.3.1 Creation of Plot 1 The first plot, Figure 2.7, represents geometric variations by using normalized quantities that capture different configurations of a stiffened plate with rectangular stiffeners. In this case, a stiffness ratio shown in Equation 2.10 was selected to represent changes in stiffener thickness, stiffener width, plate thickness, and elastic modulus. While the stiffness ratio captures most of the geometric changes it is not suitable for stiffener spacing variations. Figure 2.7 shows the geometric variations of 4 and 8 stiffener spacings. For rectangular stiffener configurations, stress intensity factor K Stiffened indicated in Equation 2.8 were determined using DCM, which will be discussed in Section 2.5. Even though DCM was used in this work, any other technique that will enable determination of stress intensity factor of a cracked plate for different crack lengths can be used [63]. The y-axis of plot 1 was selected as β(a) πa, which is also K Stiffened σ applied and plotted for various ratios of a/b (crack length / stiffener spacing). Here, β(a) πa, was selected as axis that will be common with the next plot. These common axes between the plots relate all the necessary variables in order to analyze or design a stiffened plate. µ = Stiffener Stiffness P late Stiffness = w s t s E s w s t s E s + bte (2.10) Since geometric changes were modeled in the first plot, failure criterion and material and load scatter need to be captured in the next two plots Creation of Plot 2 Tuegel and Penmetsa [57] previously demonstrated that probability of failure can be plotted with respect to an index, like the margin of safety, which measures the distance between applied stress (stress at limit load) and fracture strength (mean strength to fracture). This index is called Probability Index (PI) and it is shown in Figure 2.8. This plot will be the 18

31 K Stiffener / applied =0.2, b=8 =0.4, b=8 =0.6, b=8 =0.2, b=4 =0.4, b=4 =0.6, b= a/b, (Crack Length / Stiffener Spacing) Figure 2.7: Plot capturing the geometric variations for a stiffened plate with rectangular stiffeners (Stiffener spacing, b = 4 and 8 ) third plot and it eliminates the need for probability integration as long as the index is known. Probability Index represented by Equation 2.11, serves as a link between β(a) πa from the first plot and the probability of fracture from the third plot. Therefore, through this the geometry, material strength, and applied stress were all related to determine the probability of fracture for a given crack length. (or) P I = K C β(a) πa σ applied σ applied (2.11) P I = 1 σ applied K C β(a) πa 1 where β(a) πa = K stiffened σ applied With the two axes identified for the second plot, it now represents a contour of the ratio of applied stress at reference load condition to the mean critical fracture toughness, 19

32 PDF Applied Stress Fracture Strength Limit Load Mean Strength to Fracture Probability Index Applied Stress or Fracture Strength (psi) 1 Figure 2.8: Applied stress-fracture strength plot ( σ applied K C ). In order to generate plot 2, several values of PI are generated using Equation 2.11 by providing β(a) πa and σ applied K C values. Figure 2.9 shows the contour plot for different values of σ applied K C, where K C is 100 ksi in and applied stresses are 100 ksi, 80 ksi, 60 ksi, and 40 ksi Creation of Plot 3 Finally, the third plot represents the relationship between the probability index and probability of fracture. Based on the crack size, the fracture strength of the structure varies. That variation changes the distance between the fracture strength and applied stress PDFs and thereby changes the probability of fracture. In this research, the probability of fracture, represented by Equation 2.7 (also called the limit state), was determined using Fast Fourier Transformation (FFT) based integration. The following outlines the steps that were taken 20

33 Probability of Fracture - PoF K Stiffener / applied / K = 0.4 applied C / K = 0.6 applied C applied / K C = 0.8 / K = 1.0 applied C Probability Index - PI Figure 2.9: Plot capturing the stress at limit load and mean critical fracture toughness variations FALSTAFF Probability Index-PI Figure 2.10: Probability index versus probability of fracture 21

34 to perform FFT integration: 1. To demonstrate the FFT procedure, the probability of fracture represented by Equation 2.7 was simplified as shown below: g(x) = F racture Strength(σ F ) Applied Stress(σ applied ) and P of = P [F racture Strength(σ F ) Applied Stress(σ applied ) < 0] or P of = P [g(x) < 0] Where the notation g(x) < 0 indicates failure region. 2. In this study the applied stress PDF, represented by Equation 2.12, was derived from the FALSTAFF spectrum [58]. The FALSTAFF spectrum is a standard load sequence considered representative of the load-time history in the lower wing root of a fighter aircraft. Table 2.1 shows the exceedance data from the FALSTAFF spectrum, where 1.0 corresponds to the maximum load experienced during the data collection period. This spectrum is usually available as a normalized exceedance plot and can be scaled to the desired level. 22

35 σ applied P DF = ( LogN( , ) N(0.2849, ) N( , ) N( , ) N( , )). (2.12) Table 2.1: FALSTAFF spectrum with normalized stress exceedance values Normalized Stress Exceedances The fracture strength PDF ( σ F = K C β(a) πa ) was determined from the fracture toughness PDF represented by Equation The fracture toughness PDF was derived from 74 fatigue test samples as shown in Figure To determine the fracture 23

36 da/dn PDF toughness (K C ) of this alloy, a region was selected where the fatigue curves transition to a vertical line, representing fracture (10 2 on the y-axis). Then the fracture toughness information from the above 74 samples was normalized to represent the mean fracture toughness of 1.0 as shown in Figure K C P DF = 0.34 LogN[1.004, 0.013] N[0.999, 0.037] (2.13) Del K Normalized fracture toughness a) Fracture toughness test data b) Normalized fracture toughness Figure 2.11: Fracture toughness, K C 4. Once the fracture strength (σ F ) and applied stress (σ applied ) PDFs became available, their means and standard deviations (SD) were used to define upper and lower bounds of integration as follows: Upper σapplied = Mean + 8 * SD Lower σapplied = Mean - 8 * SD 24

37 PDF Upper σf = Mean + 8 * SD Lower σf = Mean - 8 * SD 5. Next the above PDFs were discretized using a common discretization factor (see Figure 2.12) represented by the following equation. The most efficient FFT integration process was achieved when the number of terms in the discretized vector were a power of 2. Therefore, 2 18 was selected as the size of the vector in this example. Discretization F actor = Minimum[Upper σ applied Lower σapplied, Upper σf Lower σf ] x Discretization Factor F or applied (PSI) (psi) Figure 2.12: Sample PDF discretization 6. The limit state, g(x), was converted into a linear combination of variables. In this case g(x) = Z. Z = Z 1 + Z 2 25

38 Where Z 1 = σ F and Z 2 = -σ applied. 7. The PDF of Z 1 and Z 2 was determined by the chain rule as follows: f Z1 = dσ F dz 1 f σf = 1 f σf (Z 1 ) = f σf (σ F ) (2.14) where f Z1 is the PDF of Z 1 and f σf is the PDF of σ F. f Z2 = dσ applied dz 2 f σapplied = 1 f σapplied ( Z 2 ) = f σapplied (σ applied ) (2.15) Where f Z2 is the PDF of Z 2 and f σapplied is the PDF of σ applied. 8. Based on the above chain rule, the range and PDFs of Z 1 and Z 2 were found to be same as σ F and σ applied, respectively. Since Z 2 = - σ applied, it is the opposite sign of σ applied PDF. 9. The upper and lower bounds of the final Z PDF were calculated using the individual bounds calculated previously: Upper Z = Lower σf - Upper σapplied Lower Z = Upper σf - Lower σapplied 10. The final range vector size of the Z PDF was calculated using the following equation: Z V ector size = Upper Z Lower Z Discretization F actor 11. In order to apply the discrete FFT algorithm, the sizes of the vectors must be equal. Therefore, Z 1 and Z 2 PDF vectors were padded with zeros based on the final range vector size of the Z PDF found in the previous step. 26

39 12. The final Z PDF, represented by Z = Z 1 + Z 2, was determined by using FFT. The FFT algorithm converts an expensive convolution in the physical domain to an inexpensive product of two signals (PDF in this case) in the frequency domain. Therefore, the PDF of Z, the convolution of the individual PDFs of Z 1 and Z 2, can be written as: f Z = f Z1 f Z2 13. After applying FFT to both sides of f Z = f Z1 f Z2 the following relationship was obtained: FFT [f Z ] = FFT[f Z1 ]* FFT [f Z2 ] 14. The final f Z PDF of the limit state Z was then obtained by taking the inverse FFT. 15. The final f Z PDF was then normalized by dividing by its area under the curve. 16. Next the shaded area (failure region) shown in Figure 2.13 was integrated from - to 0 using Equation 2.16 to determine the PoF of the limit state, Z. f z Z = 0 Z < 0 Failure Region Z > 0 Safe Region PoF Z Figure 2.13: Probability density for limit state Z 27

40 PDF P of = 0 f Z dz (2.16) Where f Z is the PDF of Z. 17. Finally to determine PoF in this study, a discretization factor of 0.5 was used and 8 standard deviation was used to define upper and lower bounds of fracture strength (σ F ). Since the applied stress PDF (σ applied ) was not symmetrical, the lowest and limit loads were considered as the lower and upper bounds, respectively. Figure 2.14 shows the applied stress and fracture strength PDFs, where Figure 2.15 shows normalized f Z PDF for a certain case. 6 x Fracture Strength ( F ) Applied Stress ( ) Applied Stress and Fracture strength PDF x 10 4 Figure 2.14: Applied stress and fracture strength PDF The PI discretization used in Figure 2.9 was also used in plot 3 (Figure 2.10) to determine probability of fracture values. This plot was generated for different values of PI by 28

41 PDF 14 x Final Z PDF Normalized Z PDF x 10 4 Figure 2.15: Normalized final Z PDF integrating the convolution integral based on the applied stress PDF and fracture strength PDF. The PDFs for the input variables were modeled using the approach outlined in Ref. [57]. Depending on the PDFs of stress and fracture strength, various curves were plotted to represent different load spectrum at different locations of the aircraft and different materials. This is the only plot that uses the PDF information. However, this plot does not require any analysis of the structure, thus resulting in minimal computational effort. Only plot 1, shown in Figure 2.7, requires structural analysis to determine the SIF, and therefore it requires significant computational effort to represent numerous structural configurations. Therefore, Figs. 2.9 and 2.10, can be plotted independently of the first plot and require minimal computational effort. Only the first plot will need to be replaced to represent changes in geometry, such as stiffened plate with hat-stiffeners, or a plate with a hole, etc., as long as the material properties and loading conditions are similar for the new component. Moreover, if a new loading spectrum must be explored for the same rectangular stiffener problem, only the third plot needs to be changed. Figure 2.16 shows probability of frac- 29

42 ture estimation for other load distributions. These three plots represent all the information required for determining probability of fracture for a specified crack size. Figure 2.16: Probability of fracture estimate for other load distribution Using the Probability of Fracture Nomograph A stiffened plate with rectangular stiffeners shown in Figure 2.17 was selected as an example to demonstrate the nomograph. The three plots (Figure 2.7 with b = 8 stiffener spacing, 2.9 and 2.10) can be attached to each other, as shown in Figure 2.17, based on the common axes they share. This results in a nomograph that can be used to draw a series of vertical and horizontal lines to determine the failure probability. As seen in Figure 2.17 (plot 1), an 8 crack in a plate with µ = 0.4 and stiffener spacing of 8 resulted in β(a) πa = For a situation where the applied stress was equal to 60 ksi and the fracture toughness was 100 ksi in, σ applied K C = 0.6. As shown in Figure 2.17 (plot 2), a horizontal line is drawn from β(a) πa = to the contour that represents σ applied K C = 0.6. This horizontal line gives the location parameter to draw a vertical line that determines the 30

43 Probability of Fracture - PoF probability index (PI = in this case). For this PI, the failure probability is PoF = 1.0 x 10 4 as shown in figure 2.17 (plot 3) / K = 0.4 applied C / K = 0.6 applied C applied / K C = 0.8 / K = 1.0 applied C =0.2, b=8 =0.4, b=8 =0.6, b=8 K Stiffener / applied Probability Index - PI a/b FALSTAFF Probability Index-PI Figure 2.17: Probability of fracture nomograph 2.4 Example Plate with a Hole As an example, a plate with a hole, as shown in Figure 2.18, was considered, for which stress intensity factors were obtained from the closed-form Equation 2.17 [64]. From Equation 2.17, K P late σ applied was plotted for different hole radii and plate widths. Two plate widths, 3 in and 2 in, and two radii for each of these widths were selected. Other cases can easily 31

44 be plotted based on the requirements. Figure 2.19 shows plot 1 of the nomograph. Similar to stiffener plate configuration, here also y-axis maintained as K P late σ applied. K P late = σ Applied πa ΦF (2.17) where Φ = λ λ λ λ 4 λ = (1 + a r ) 1, where r is radius of the hole and a is the crack length F = Sec[(π/2) (2r+a) ] Sec[(π/2)(2r/w)] W a K plate σ Applied = πa ΦF Figure 2.18: Cracked plate with a center hole Plot 2 (Figure 2.20) was regenerated using Equation 2.11 considering the K P late σ applied values obtained from plot 1. Plot 3 does not require any change since there was no change in material system. Therefore combining all the plots (Figures 2.19, 2.20 and 2.10), Figure 2.21 shows the nomograph that was obtained for a flat plate with a hole. 32

45 K Plate / applied r = 0.25", W = 2" r = 0.125", W = 2" r = 0.5, W = 3" r = 0.25, W = 3" Half crack length / Width (a / W) Figure 2.19: Plot 1 of the nomograph K Stiffener / appied applied / K C = 0.6 applied / K C = 0.8 applied / K C = Probability Index - PI Figure 2.20: Plot 2 of the nomograph 2.5 Displacement Compatibility Method (DCM) to Determine the Geometric Effect The Displacement Compatibility Method uses superposition of elasticity solutions to approximate the displacement and strain fields in a stiffened plate. It is a semi-analytical 33

46 K Stiffener / appied Probability of Fracture - PoF / K = 0.6 applied C / K = 0.8 applied C applied / K C = 1.0 K Plate / applied r = 0.25", W = 2" r = 0.125", W = 2" r = 0.5, W = 3" r = 0.25, W = 3" Half crack length / Width (a / W) FALSTAFF Probability Index-PI Figure 2.21: Probability of fracture nomograph for flat plate with a hole method where a set of equations are derived using analytical expressions of displacements that are solved numerically using matrix algebra. The basic concept of the displacement compatibility method is that the displacements on the stiffener and the plate are compatible at all the fastener locations. Figure 2.22 below, taken from Ref. [65], clearly shows the applied stress in the plate (S), stiffener (S*(E S /E)), and the reaction forces at the rivet locations (Q i ). In this model, applied stress is distributed among the plate and the stiffener based on their relative stiffness. In order to develop the displacement compatibility equations, free body diagrams, shown in Figure 2.22, of the stiffener and the plate were constructed. Based on these free body diagrams, displacement of the plate at any one of the rivet locations is a function of the far-field applied stress (S), force at each of the rivets (Q i ), and its location with respect to the center of the crack. In this model, the crack in the plate was considered centered about a stiffener and symmetrical on each side of the stiffener. The fastener forces along this stiffener are numbered from 1 to n and they are symmetric about the x-axis. This 34

47 numbering continues through other fasteners as shown in the above Figure Due to symmetry of the problem only a quarter plate needs to be analyzed. Figure 2.22: Applied stress and reaction forces in a cracked stiffened plate[65] Displacement at any of the fasteners can be obtained using superposition of displacements due to far-field stress and displacements due to rivet forces. These equations were derived by C. C. Poe and published in Ref. [65]. They have been included below for completeness. v i = j A ij Q j + B i S (2.18) where v i is the displacement at the i th rivet, A ij is the displacement at the i th rivet due to unit force at the j th rivet, and B i is the displacement at the i th rivet due to unit far-field applied stress. Similarly, the displacement at the i th rivet in the stiffener can be determined using the following equation. 35

48 v S i = j A S ijq j + B S i E S E S (2.19) where the superscript and subscript S represents the stiffener quantities. Based on these two equations the displacement compatibility equation becomes: j (A ij + A S ij)q j (B i E S E BS i )S = 0 (2.20) Expressions for the individual terms in Equation 2.20 have been implemented using equations from Ref. [63]. Once the analytical expressions were implemented, the solution of the unknown rivet forces was determined by using matrix algebra. These rivet forces were used to determine the stress intensity factor for the stiffened flat plate using Equation 2.21, which is a combination of stress intensity factor due to far-field stress and individual rivet forces. K Stiffened = S πa + j ˆK j Q j (2.21) ˆK = πa πt [(3 + ν)φ 1 (1 + ν)φ 2 ] Using this stress intensity factor the geometric factor was plotted for different crack lengths and stiffened plate configurations. 2.6 Stiffened Plate with Z and C Stiffeners In the previous section the nomograph for stiffened plate was developed using only a rectangular stiffener. Since DCM is applicable only to rectangular stiffener, the stress intensity factor for Z or C stiffeners must be obtained from finite element analysis. This section explores the need for any modification in the stiffness ratio (Equation 2.10) to generate 36

49 nomographs for Z or C type stiffeners. The stiffness ratio used for rectangular stiffener is shown below: µ = Stiffener Stiffness P late Stiffness = w s t s E s w s t s E s + bte A parametric study was conducted to determine whether the same stiffness ratio could be used for Z or C type stiffeners. For this study, a flat plate (30 in x 30 in with in thick) [62] with one broken center stiffener and four intact outer stiffeners was used. The stiffness ratio Equation 2.10 was modified to the following form to include Z- stiffener dimensions, µ = (w 1 t 1 + w 2 t 2 + w 3 t 3 )E s (w 1 t 1 + w 2 t 2 + w 3 t 3 )E s + bte where w 1, t 1, w 2, t 2, w 3, t 3 are the dimensions of a Z stiffener as shown in Figure 2.23 (a) After introducing the following assumptions, w 1 = w 3, t 1 = t 3, w 2 = 1.3 w 1 and t 2 = 0.6 t 1, the stiffness ratio was reduced to µ = (2.78w 1 t 1 )E s (2.78w 1 t 1 )E s + bte and by further introducing E s = E the equation was reduced to µ = (2.78w 1 t 1 ) (2.78w 1 t 1 ) + bt (2.22) Four different stiffener configurations were analyzed, including three Z stiffeners shown in Figure 2.23 and a rectangular stiffener, by fixing the stiffener area as in 2, µ = , b = 6 in and t = in. Case 1) w 1 = in, t 1 = in as shown in Figure 2.23(b) Case 2) w 1 = in, t 1 = in as shown in Figure 2.23(c) 37

50 0.938 w1 t t2 w t3 w3 (a) (b) (c) (d) Figure 2.23: Different Z-stiffener configuration (dimensions, inch) Case 3) w 1 = 0.7 in, t 1 = 0.1 in as shown in Figure 2.23(d) and Case 4) w 1 = in, t 1 = in rectangular stiffener The finite element results of the four cases are shown in Figure Similarly, a C-type stiffener was analyzed and compared with Z and rectangular stiffener. The results are shown in Figure Based on the Z, C, and rectangular stiffener parametric study for a stiffness ratio of µ = , the stress intensity ratio remained equal and did not change for any change in stiffener type. As a result the stiffener ratio µ for any configuration was simply represented by µ = (Area of the stiffener)e s (Area of the stiffener)e s + (Area of the plate)e 38

51 µ = z63_bcs_area z83_bcs_area 0.189_shape changed z10_bcs_area0.189_shape changed FP938_bcs_0.189_rectang 1.80E E+00 K stiffenr /K un-stiffenr 1.40E E E E E E E E Half Crack Length Figure 2.24: Z and rectangular stiffener results Also the half crack length vs. β(a) πa was maintained as X and Y-axis to remain consistent with the rectangular stiffener plot developed in previous section, see Figure 2.7. The finite element model shown in Figure 2.26 was created using ABAQUS [66]. The plate and stiffener were modeled with C3D20R element (20-node quadratic brick, reduced integration), and around the crack-tip collapsible type elements were used that consist of a ring of triangles with concentric layers. This modeling approach was tested for convergence of J-integral values for the various contours and used for fracture strength evaluation. The following section presents details about the implementation of this process for a flat plate configuration. 39

52 z63_bcs_area C63_bcs_ area FP938_bcs_area K stiffenr /K un-stiffenr Half Crack Length Figure 2.25: C, Z and rectangular stiffener results 2.7 J-Integral Evaluation To evaluate the J-integral in ABAQUS [66], the following model was created both in 2D and 3D to compare the simulation results with analytical values of fracture strength. A flat plate of size 508 mm x 1016 mm x 1 mm and a center crack of size mm (2a) was modeled. For this model, Aluminum material properties (E = MP a and ν = 0.3) were selected and a load of 200 MP a was applied in Mode-I J-Analytical The following formula [67] was used to calculate the Mode I stress intensity factor, 40

53 Intact Outer Stiffener Center Broken Stiffener Intact Outer Stiffener Figure 2.26: ABAQUS Z-stiffener model K I = P B f(a/w) (2.23) W where B is the specimen thickness, 2W is the width of the specimen. The above equation can also be written in the following form using applied stress, σ = P, instead of load P, 2W B K I = 2σW W f(a/w) (2.24) where f(a/w) is f(a/w) = πa πa sec 4W 2W [ ( a ) 2 ( a ) ] W W (2.25) 41

54 and the J-integral can found by the following equation for the plane stress condition. J = K I 2 E (2.26) Finite Element Model A 2D finite element model was meshed with 8 node plane stress elements (CPS8R). To model the crack tip the crack front and q-vector (crack extension direction) were specified and the initial crack was modeled as sharp crack using the ABAQUS seam feature. For the contour integral evaluation, a midside node parameter value of 0.25 (quarter point) was used to create collapsed elements as shown in Figure 2.27 with the collapsed element side, single node option to maintain 1 r singularity at the crack tip. Since the collapsible element type was available in ABAQUS, it was used in this study with a coarser mesh. This same effect could also be obtained by using a fine mesh around the crack tip for contour integral evaluation. For Z and C stiffener configuration, Mode-I loading was considered assuming the crack initiation and propagation is parallel to crack path. The J-integral was evaluated for contours and the mesh was refined until the convergence of the J-Integral value was obtained. The first few contour values were ignored and the remaining contour values were considered for the J integral convergence. The J- integral values (Figure 2.28) and the contour locations (Figure 2.29) are shown below. For the 3D finite element model, the part was meshed with 20 node quadratic brick, reduced integration elements (C3D20R). Similar to the 2D case, collapsed elements were inserted after creating wedges around the crack tip. The J-integral evaluation in 3D was based on the average value across the thickness of the model. If the number of elements across the thickness is one then there were three sets (called picked sets) of J-integral values which were available for each of the contours (Figure 2.31). Then the J-integral value for 42

55 Typical Finite element Figure 2.27: Collapsed element representation J-integral, KJ/m J-contour integral 2D J integral (2D) = KJ/m Number of contours Figure 2.28: J-contour integral values 43

56 Contour 1 Contour 2 Contour 3 Figure 2.29: J-contour locations (red color) the 3D geometry was calculated using the following averaging method (Equation 2.27) J shell = J A + 4J B + J C 6 (2.27) Results and Discussion The comparison of J-contour integral results for 2D and 3D finite element models are listed in Table 2.2. Based on the type of problem (plane stress or plane strain), the above J- integral value was compared to the critical J-integral value of the material to determine if a crack was initiated or not. For crack initiation, the model should satisfy J = G C condition. Table 2.2: J-integral value comparison Description Analytical J-integral(2D) J-integral(3D) J-integral (KJ/m 2 )

57 C B A Figure 2.30: J-contour picked set (3D), A-top, B-middle, C-bottom 2.8 Summary In this chapter, risk-based design plots were presented for flat plate and a stiffened panel. The risk-based design plots can easily be extended to include several other complexities that can be added to a stiffened panel, like riveted stiffeners, welded stiffener, cracked stiffener, different material systems, etc. Using these plots a designer could obtain information about the failure probability and its relation to all the structural changes for a given allowable crack length. The nomograph presented in this chapter also captures the entire design space of a particular component and requires no further simulations to analyze changes in material mean fracture strength and load magnitude. This now becomes a communication tool between the designers, who will be using these plots, and the reliability engineers, who would develop these plots. The concept of nomographs presented in this chapter has been published as a journal article [68] and presented in conferences [69, 70]. The PoF nomograph can be applied to complete wing based on stiffener configuration provided the section 45

58 J-Integral, KJ/m J-Contour integral (A) J-Contour integral (B) J-Contour integral (C) B J-integral (avg) = KJ/m 2 A & C Number of contours Figure 2.31: J-contour value (3D) is subjected to tension along the stiffener direction. If the stiffener arrangement varies along the wing span, then typical sections can be considered for generating the nomograph. In Reference [71], a structural risk assessment of lower wing stringers was performed on The Royal Australian Air Force (RAAF) B707 aircraft to evaluate the stringer damages. For aging aircraft one could develop a nomograph, as outlined in this chapter, to assess the risk of damage of the wing before clearing the aircraft for its next mission. 46

59 Chapter 3 Fracture Nomographs using Cohesive Zone Modeling In the previous chapter, to estimate PoF (Equation 2.6 or 2.7) the fracture strength distribution was directly derived from fracture toughness test data. However, in a Cohesive Zone Model (CZM) based approach there is no closed-form equation available to determine the fracture strength and it can be obtained only from Finite Element Analysis (FEA). Also, CZM requires more than one parameter to define the fracture process zone and there are no standardized tests available to determine distributions of these CZM parameters. Therefore, in order to use CZM for probabilistic study, first fracture strength must be estimated before developing a PoF nomograph. This chapter presents details regarding a process to estimate and calibrate CZM parameters to determine fracture strength distribution and what modifications must be adopted to develop a PoF nomograph. 47

60 3.1 Introduction While linear elastic fracture mechanics-based approaches have been sufficient for life predictions for existing platforms and design practices, they may not be adequate for future platforms that are built with new material systems and experience extreme environments. A CZM on the other hand can reproduce the effects of a linear elastic fracture mechanics based approach and it also has the capability to handle local non-linear behavior of the fracture process zone. This CZM can represent both linear and nonlinear traction behavior, can be made temperature dependent, and can also be used for automatic fatigue crack growth by using appropriate strength degradation models. Despite many benefits, there are several technological challenges that need to be overcome before a CZM can be used for life prediction of a hypersonic aircraft. One such challenge is the identification of the appropriate model itself including its parameters and their random scatter. Once an appropriate fracture model is obtained, various testing procedures can be developed to repeat this process at different temperature levels to calibrate a temperature dependent CZM. Models that are developed using fracture criteria can be combined with an appropriate degradation model to determine the probability distribution of the fatigue life of a component. Identification of the degradation model is still an open research problem that was not part of this dissertation. While cohesive zone model techniques have been around for decades, it has only recently became practical due to today s computational capabilities. In the late 50 s, Barenblatt [75] pioneered the idea of the representation of fracture as a material separation across a crack surface. Based on this modeling approach, cracks and other material discontinuities can be represented using a zero thickness process zone in a finite element framework. As the load increases this process zone separates according to a pre-determined law while applying finite tractions until a critical separation is reached. These tractions disappear when the separation exceeds a critical value and a new cracked surface is created. Several researchers have proposed models for this traction separation that shared the following fea- 48

61 tures: traction increases until it reaches a maximum value and then it approaches zero as the separation increases. While maintaining these two key aspects of traction separation, they all differed in the actual functional form used to represent the decay. Only a few of the types of models available in the literature are discussed here, but these models clearly demonstrate the wide spectrum of shapes used for the cohesive zone. Some researchers [76, 77, 78] have used a linear softening relation and selected peak traction, fracture energy, and critical separation parameters for the model that reproduce the test results with minimal error. In this model, only three parameters are needed to define the triangular traction separation law. Equation 3.1 and Figure 3.1 show the form of the traction separation law. In this equation T n is the traction, δ is the normal separation, δ c is the critical separation, and σ c is the peak traction. The area under the triangle is the fracture energy, the peak of the triangle is the peak traction, and the traction reaches zero at the critical separation. σ c Traction,T n K Fracture Energy, φ n δ c Separation, δ δ Figure 3.1: Linear traction separation law 49

62 T n = σ c δ c δ (δ eakt raction ) σ c δ ( 1 δ 1 δ c )δ (δ > eakt raction ) (3.1) T n = β η β δβ 1 exp( ( δ η )β ) (3.2) T n = exp(1)σ c δ δ c exp( ( δ δ c ) (3.3) σ c Traction,T n Fracture Energy δ c Separation, δ Figure 3.2: Exponential traction separation law Bhate and Subbarayan [74] used a Weibull density function based CZM shown in Equation 3.2 (Figure 3.2). They selected this model because the shape and scale factors of the Weibull distribution can be used to explore a wide range of traction separation laws that 50

63 would best suit the application of interest. Moreover, the analytical expression for the area under the curve, which in a probability sense is a cumulative distribution, now represents critical fracture energy. The two distribution parameters η and β are adjusted based on the test data to be replicated using this model. An exponential traction separation law, represented by Equation 3.3, was used by researchers [79, 72, 81] to model nonlinear behavior. Nonlinearity in the exponential model can be adjusted by introducing a variable into the exponential power that controls the decay of the traction separation relation. This model is also similar to the Weibull distribution function type model because both the functional forms are dependent on an exponential function. The exponential law is most commonly used in the literature. Chen et al. [82] have proposed a polynomial model shown in Equation 3.4 to represent the traction separation law. The C s in the equation are the unknown coefficients that are used to fit test data. Volokh [86] has compared the response of two rigid blocks joined by a cohesive layer by using bilinear, parabolic, sinusoidal, and exponential models to represent the traction separation law. These models were selected to demonstrate the shape sensitivity of the results and to emphasize the need to select the correct shape that represents the fracture process. Even though all these models have differed in their shapes they were all used as a surrogate model for the material response. Similar to the traditional regression model where a linear, quadratic, nonlinear, etc. can be selected to represent the response, the CZM has various shapes that can be used to represent the material response. Therefore, the higher flexibility to control the shape of the model is desirable. T n = Cσ c δ δ c [1 + c 1 δ δ c + c 2 ( δ δ c ) 2 ] (δ eakt raction ) 0 (δ > eakt raction ) (3.4) While all the above-described models were developed and implemented for Mode I fracture, Park, et. al. [87] presented a potential function based CZM that is derived based on physical field equations. They have presented a Park-Paulino-Roesler (PPR) potential 51

64 based model that can be used for mixed mode fracture (normal and tangential modes). The gradients of the PPR potential directly lead to a mixed mode traction separation law that can be calibrated based on four parameters in each mode. These four parameters provide greater control of the shape of the CZM than the other models presented previously. Using the four model parameters, several shapes explored to identify the best shape that represents the fracture process in the material selected. Along with the shapes of the CZM, randomness in the parameters that define these models also must be identified through a formal process. The PPR based CZM model was selected in this research to represent the fracture process, and its details are explained in Section 3.2. Since the main objective was to calibrate a CZM from material test data to determine fracture strength distribution, the steps involved to determine the statistical characteristics of the CZM parameters are given below, and a detailed step-by-step procedure is also provided at the end of the chapter. 1. First, the fracture strength distribution was estimated based on Mode-I fracture toughness test data. 2. The initial mean values of each CZM parameters (refer to section 3.4.3: Conventional Material Properties and Initial CZM Parameter Estimates) were then estimated from material properties. 3. A Response Surface Model (RSM) using Latin Hypercube Sampling (LHS) technique was developed to represent the fracture strength distribution in terms of CZM parameters. 4. Monte Carlo Simulation (MCS) was applied on the RSM to determine the statistical characteristics (mean, std. deviation, skewness, kurtosis and correlation coefficients) of the CZM parameters by fitting the response surface distribution with fracture strength distribution obtained from test. 5. Nomographs (risk-based design plots) were developed using the calibrated CZM for efficient design decisions. 52

65 The overall idea behind the estimation of fracture strength PDF and algorithm for determining the statistical characteristics of the CZM parameters is explained in Section 3.3. In Section 3.4, a simple center crack model is used as an example to demonstrate the calibration process with two different types of traction-separation laws and in Section 3.5 the optimization results are presented for uncorrelated and correlated CZM parameters. Finally in Section 3.6 probability of fracture nomographs for flat and stiffened plate configurations are presented. 3.2 PPR Traction-Separation Law In this research, only mode I fracture (tension) was considered in order to develop an algorithm that can be used to determine an appropriate CZM and its statistical characteristics associated with scatter. This approach can be implemented for any material system of interest based on fracture strength tests. The Mode I traction separation law based on the PPR potential is given in Equation 3.5. The parameters of the PPR model are energy, Φ n, cohesive or peak traction, σ c, shape factor, α, and ratio, λ n. The four parameters must be determined for each material system. Since Φ n is the Mode I fracture energy, an estimate of this value is available for most materials. The values of the other three parameters are not directly available and are obtained through the proposed probabilistic parameter identification algorithm in this work. Figure 3.3 shows how the shape parameter, α, influences the traction separation law. In order to generate the three curves in Figure 3.3, m was set to 2 and Φ n was set to in.lb/in 2. T n (δ) = Γ n δ f [m(1 δ δ f ) α ( m α + δ δ f ) m 1 α(1 δ δ f ) α 1 ( m α + δ δ f ) m ] (3.5) Γ n = Φ n ( α m )m (3.6) 53

66 m = α(α 1)λ2 n (1 αλ 2 n) (3.7) λ n = δ c δ f (3.8) δ f = Φ n σ c αλ n (1 λ n ) α 1 ( α m + 1)( α m λ n + 1) m 1 (3.9) Traction, T n /T Peak < 2 = 2 > Separation, / c Figure 3.3: Traction separation law based on PPR potential model Figure 3.4 shows how the parameters of the model can be changed to obtain a wide range of shapes with constant fracture energy (area under the traction separation curve). All the curves in Figure 3.4 are generated by changing the non-dimensional parameter m and 54

67 Traction, T n /T Peak Separation, / f Figure 3.4: Spectrum of curves generated using the traction separation law in Equation 3.5 the shape parameter α while keeping the Mode I fracture energy constant. The proposed optimization algorithm explored these shapes to determine a shape that would be necessary to model fracture behavior of a cracked plate. Once the shape was identified then the PDFs of the CZM parameters were determined such that these models would result in the same scatter for fracture strength as that observed in the fracture toughness tests. 3.3 Probability Distribution Identification Most of the material properties used in structural design, like the yield strength, ultimate strength, shear strength, fracture toughness, etc., are obtained from test data and typically have scatter models developed from the test data. This information has also been used to 55

68 determine statistical strength allowables in various military handbooks. However, when moving from a traditional failure analysis method to the new CZM based method there are no tests specifically designed to calibrate the model parameters to determine their scatter information. Most of the model identification methods have been based on matching one of the existing strength metrics from the test to the CZM based method. Therefore, in this CZM calibration, a similar approach is followed, but from a probabilistic perspective. For a cracked flat plate, fracture strength can be determined using Linear Elastic Fracture Mechanics (LEFM) principles as shown in Equation Since fracture toughness is a probabilistic quantity, strength to fracture is also random and its PDF can be determined using simple chain rule as shown in Equation 3.11 for any given crack length. In this research, crack length was assumed to be a deterministic quantity. σ F racture = K IC β πa (3.10) f σ (σ) = β πaf KIC (σβ πa) (3.11) Also, Al 2024-T3 was used because it is a common material of choice for aircraft structures. There is no fracture toughness distribution data publicly available for Al 2024-T3, so lot release data from 74 samples of an unknown aluminum alloy, provided by Air Force Research Laboratory (AFRL), were used to determine the fracture toughness PDF f KIC for Al 2024-T3. Since the actual alloy details were not available, the fracture information from the above 74 samples was normalized to represent mean fracture toughness of 1.0. This process provided scatter information for an unknown aluminum alloy which was then applied to Al 2024-T3 by using the mean fracture strength of Al 2024-T3 as the scaling parameter. As indicated earlier, the main objective is to calibrate the CZM parameters, the fracture information from the 74 samples was used to develop the calibration framework. One can use this calibration process when any material data of interest becomes available. 56

69 When the assumptions of LEFM are satisfied, the fracture strength PDF in Equation 3.11 is accurate. Therefore, when using CZM based method to analyze structures that satisfy LEFM assumptions, the fracture strength PDF should be exactly same as that obtained from LEFM. When conditions are similar, both the methods predict the same fracture strength and its PDF. Based on this principle, an optimization algorithm will be implemented to determine the PDFs of the CZM parameters that would result in the same fracture strength PDF as the one obtained from Equation Here, the PDFs of the CZM parameters were modeled using Pearson system of distributions. By selecting a Pearson system, only four statistical characteristics are required for each PDF to explore a wide range of possible distributions. This is possible because Pearson system embeds seven basic types of distributions, such as normal and t distributions, simple transformations of shifted, scaled beta distribution, the inverse gamma distribution, and one Type IV distribution. The system automatically determines an appropriate PDF for the variable based on four statistical characteristics. These characteristics are mean, standard deviation, skewness, and kurtosis. These four statistical characteristics were chosen in an optimization algorithm to match the Cumulative Distribution Function (CDF) of fracture strength obtained from a CZM based approach to that of LEFM fracture strength CDF. While LEFM fracture strength CDF is readily available as a closed form equation from the fracture toughness CDF, obtaining a CZM fracture strength PDF is not a trivial task. Fracture strength evaluation using CZM involves non-linear finite element analysis for each realization of the various parameters that define the CZM; therefore, surrogate modeling techniques were used to determine the CDF. The algorithm shown in Figure 3.5 was used to determine the PDFs of the CZM parameters defined using the Pearson system of distributions. While there are 16 parameters for the PPR model that define the PDFs the algorithm only used the four mean values in the sampling scheme to construct the surrogate model. The algorithm begins with a sampling scheme that generates uniformly distributed realizations of the CZM parameters. At 57

70 each of these realizations a non-linear finite element analysis was performed to determine fracture strength of the flat plate. Using these samples a surrogate model described in the following section was constructed to represent the fracture strength of the cracked plate. This surrogate model was then used to determine the joint PDF of fracture strength, which was then integrated to determine the CDF of fracture strength. Fracture strength obtained was compared with the fracture strength of the LEFM approach using an error model as shown in Equation In this algorithm the shape of the CZM and the PDF information of the parameters controlling the shape were altered in each iteration until the final optimum was obtained. Since gradient information is not available for this problem a Particle Swarm Optimization (PSO) algorithm was implemented on the surrogate model to determine the optimum configuration. Error = Σ(CDF LEF M CDF CZM ) 2 (3.12) Latin Hypercube Sampling Realizations of the mean of the CZM parameters Surrogate Model Construction Second order non-linear regression model Initialize PDF Parameters Sampling from Pearson Distributions Realizations of Mean, Standard Deviation, Skewness and Kurtosis Update parameters Not converged Monte Carlo Simulation Determine the CDF of fracture strength from CZM Compare CDF of CZM and LEFM Minimize error between the CDF s If converged CZM Parameter Estimates and PDF Figure 3.5: Flow chart for determining the CZM parameters and their PDFs 58

71 Surrogate Model A second order non-linear regression model [89] was used as a surrogate model that represents the finite element solution for different realizations of the CZM parameters. Equation 3.13 shows the form of the surrogate model used in this research. In this surrogate model β 0, β i and β ii are regression coefficients, P i are the non-linear indices and X i are explanatory or coded variables for each CZM parameters and it is represented by X i = X X L X U X L, where X U and X L are the upper and lower bounds of the CZM parameters. The coefficients and the non-linearity indices of the surrogate model were determined using a least squares algorithm to minimize the residual. While no single surrogate model can guarantee complete representation of the underlying non-linear response, it is assumed that the nonlinearity indices along with the coefficients can be calibrated to minimize the error in the final CDF in regions that are critical to the probability of fracture calculations. The CDF of the fracture strength was determined by performing MCS on the surrogate model using the Pearson family distributions for the CZM parameters. The entire process was repeated by adding more sample points until the parameters and their PDF information had converged. Ỹ = β 0 + Σβ i X P i i + Σβ ii X 2P i i (3.13) 3.4 Example Geometry In this section, a finite width cracked flat plate of size 20 in x 20 in x 0.04 in thickness with 4 in center crack (2a), shown in Figure 3.6, was used to implement the above algorithm to determine the CZM parameter values and their PDFs. To reduce the computational time, a symmetry model was created based on the assumption that two crack tips have identical material properties. ABAQUS was used to perform the nonlinear analysis on the plate 59

72 model. A thin plate of thickness 0.04 in (1 mm) was selected here to demonstrate how probabilistic models can be developed for real life structures like aircraft wing and fuselage skin where thin sheet of metals are widely used. Figure 3.6: Cracked plate analyzed using a cohesive zone model Traction-Separation Law In order to demonstrate the CZM parameter calibration process described in the previous section, two different models, (1) Bilinear traction-separation law, and (2) PPR traction separation law were used. The traction-separation response shown in Figure 3.7 represents a bilinear traction-separation law, which is defined using the stiffness parameter, K, that represents the loading before reaching the peak traction. This stiffness parameter was determined using the peak traction value, σ c, and the damage initiation value, δ c. The rate of damage evolution was controlled through the descending slope of the traction separation law. The slope of damage evolution was determined using the value of critical energy release rate, Φ n, for the material and the critical normal separation, δ f. ABAQUS damage 60

73 modeling using a bilinear model requires peak traction, σ c, critical fracture energy, Φ n, and the stiffness, K. Therefore, these three parameters were used to construct a response surface model. In the PPR traction separation law, shown in Figure 3.8, there are four parameters that are energy, Φ n, cohesive traction, σ c, shape factor, α, and ratio, λ n. Here the shape factor was used to capture material softening response, such as brittle, quasi-brittle, etc., and ratio, λ n, was defined as critical displacement at which damage initiates to final element displacement where the element has completely failed. The response surface model developed for the PPR model uses these four parameters. σ c Damage initiation Linear Damage evolution Fracture Energy, Ф n (area) Traction,T n K δ c Separation, δ δ f Figure 3.7: Bilinear traction-separation law The failure mechanism in the traction-separation law involves a three step process (i) Damage initiation, (ii) Damage evolution, and (iii) Element removal [66]. Damage initiation indicates when the degradation of the element begins, which is after it reaches the peak traction. Damage evolution represents the rate of element stiffness degradation after reaching the damage initiation criteria. Finally element removal was performed after the traction approaches zero at a critical normal separation value. 61

74 Damage initiation σ c Damage evolution Traction,T n Fracture Energy, Ф n (area) Shape factor, α δ c Separation, δ δ f Figure 3.8: PPR traction-separation law Conventional Material Properties and Initial CZM Parameter Estimates To estimate the mean initial values for CZM parameters to begin the optimization algorithm, material properties [80] shown in Table 3.1 was used for the example. Two different unit systems are shown in the material property table as well as in the initial parameter calculation for clarity, but the English system of units were used in this example. Table 3.1: Material property Material E, psi (MP a) ν σ 0, ksi (MP a) K IC, ksi in(mp a m) Al2024-T E7 (71300) (345) (35) Based on the experimental data and analyses by Roy et al. [79] and Weizhou Li et.al. [80], the peak traction (cohesive strength), σ c, was approximated by Equation 3.14 and the stiffness, K, is calculated by Equation 3.15 based on the slope of the damage initiation line indicated in Figure 3.7. One can determine δ c by assuming an appropriate value for λ n 62

75 ( δc δ f ), which generally ranges between 0 and 1. Based on λ n, δ c was determined by selecting a value for δ f. The starting value of δ f can be approximated by 0.05 times the cohesive element size [66, 95]. Based on the suggestions from the reference [80], a value of δ c = in ( mm) was selected. Then using energy (area under the curve) and equation of a triangle relationship calculated δ f = in ( mm) for the bilinear model. The ratio came as λ n = in (0.184 mm). σ c 2σ 0 = psi (690 MP a) (3.14) K = ( σ c δ c ) = 2.77E8 psi/in ( MP a/mm) (3.15) To calculate cohesive energy, Weizhou Li et. al. [80] considered a plain strain fracture toughness value of K IC = psi in (35 MP a m) to be a true crack initiation value for Al 2024-T3 based on experimental results determined using high-resolution detection of crack tip damage and crack initiation in thin aluminum sheet specimens [83, 84]. Also by considering the relationship J IC = ( K IC 2 ), and the ratio between the I/III mixed E mode initiation toughness, and the pure mode I toughness, they estimated that the cohesive energy, φ n, for crack initiation and crack propagation have the same value of Φ n = in.lb/in 2 (17KJ/m 2 ) for this material. In this example, based on their recommendation, the cohesive energy value Φ n = in.lb/in 2 (17KJ/m 2 ) was used to determine the static fracture strength. The selected geometry with a 2 in initial half crack length satisfied the necessary LEFM-small scale yielding conditions (i.e., SSY < crack length and uncracked ligament). SSY was determined by the plastic zone size which is 2 r y = in (3.27 mm), where r y is calculated by Equation 3.16 [67], and the same is verified with a finite element model using elastic-perfectly plastic conditions. r y = 1 2π ( KIC σ 0 ) 2 for plane stress (3.16) 63

76 The PPR model has four CZM parameters, of which three of the parameters are the same as the bilinear model. The three parameters values of ratio, λ n, peak traction, σ c, and cohesive energy, Φ n, were kept identical to the bilinear model, and α = 3 was determined based on bilinear and PPR model fracture strength validation. While the above methods are given here as guidelines to find out the initial mean values for the CZM parameters, and one could also assume these values for creating the response surface model and to initialize the optimization starting points. Also in this research, other statistical characteristics including standard deviation, skewness and kurtosis were assumed as 10 percent of mean, 0, and 3 respectively as initial values for the optimizer. Parameters in Table 3.1 were determined using a deterministic value of K IC and a deterministic fracture strength value. Therefore, the parameters that control the CZM are deterministic. However, in practical situations the fracture strength is non-deterministic and requires the presented approach to determine the scatter in CZM parameters that represents experimental scatter observed for fracture strength Finite Element Model The traction-separation law was applied to the cohesive elements, COH2D4 (4 node bilinear two dimensional cohesive element), which were defined ahead of the crack tip and were surrounded by continuum elements, CPS4R (4-node bilinear plane stress element). Since the thickness of the plate is 0.04 in, a plane stress condition was used for the analysis. Siegmund and Brocks [85] also suggested plane stress conditions for their crack growth simulations that they have calibrated using extensive tests with 1 mm thick, Al 2024-T3 panels. Based on the mesh convergence studies it was determined that cohesive zone element sizes should be less than 50 δ c. This is consistent with the findings published by Roy et. al. [79]. This length scale is consistent with the findings published by Tomar et. al. [90] for element size requirements for a cohesive finite element model. In this calibration pro- 64

77 cess, only one crack tip was considered in the finite element model, as the CZM parameter characteristics were determined against static fracture strength (crack initiation) not based on fatigue crack growth. In case of crack growth one must degrade the material properties of the cohesive zone model after crack initiation and calibrate the CZM parameters accordingly. Also for fatigue crack growth, the cohesive zone ahead of crack tip must be created along the entire width of the finite element model. Because static fracture strength was adopted as a failure criteria, it was found that a large cohesive zone ahead of crack tip is not necessary. To come to that conclusion, several analysis were performed and it was determined that a minimum half of half crack length size (example, if a = 2 in (50.8 mm), then CZ length = 1 in (25.4 mm)) of cohesive zone was required to obtain accurate results for any traction separation law selected in the optimization process. This CZ length was sufficient because for a load controlled static fracture strength simulation, once the first element fails the crack length is increased, which results in a lower allowable fracture strength for the damaged structure than the subsequent load increments. Also as indicated in the beginning of the geometry section, the crack tip material property was assumed to be identical in either side of the center crack model and based on that assumption it was decided to create a symmetry model to reduce the computational time of the finite element simulation. Using this modeling approach, load to fracture for different configurations of the CZM was explored Response Surface Model To create a surrogate model a nonlinear Response Surface Model (RSM) was constructed for strength to fracture the plate using 15 Latin Hypercube Sampling (LHS) points to represent various combinations of the CZM parameter settings. Table 3.2 and 3.3 shows latin hypercube samples for each CZM parameters and fracture strength obtained from ABAQUS for bilinear and PPR model. Equation 3.17 shows the model that was used for the bilinear traction separation law based fracture strength, where Ỹ represents strength to fracture 65

78 and X1, X2, X3 are the coded (normalized) variables for σ c, Φ n and K. Equation 3.18 represents the equation for the PPR model based fracture strength and X1, X2, X3, and X4 are Φ n, σ c, α, and the ratio λ n. In this research, even though there are 16 random variables (mean, standard deviation, kurtosis, skewness) representing the PDFs of the four CZM parameters, only 15 sample points captured the response accurately because the nonlinear regression model was only in terms of the four CZM parameters. Moreover, the optimization routine yielded several local minima which warranted no further design space exploration because of any one of those minima provided a good fit for fracture strength CDF. Table 3.2: 15 Latin hypercube samples for bilinear model LHS σ c (psi) Φ n (psi.in) K (psi/in) Fracture Strength (psi) LHS E LHS E LHS E LHS E LHS E LHS E LHS E LHS E LHS E LHS E LHS E LHS E LHS E LHS E LHS E Ỹ = X X X X X X (3.17) 66

79 Table 3.3: 15 Latin hypercube samples for PPR model LHS Φ n (psi.in) σ c (psi) α λ n Fracture Strength (psi) LHS LHS LHS LHS LHS LHS LHS LHS LHS LHS LHS LHS LHS LHS LHS Ỹ = X X X X X X X X (3.18) 3.5 Results Uncorrelated Parameters Monte Carlo Simulation (MCS) was performed on the surrogate models (3.17, 3.18) using Pearson family distributions to determine the CDF of fracture strength using the CZM parameters for both the models. There are a total of 12 uncorrelated variables for the bilinear model (Equation 3.19) and 16 uncorrelated variables for the PPR model (Equation 3.20). 67

80 12 variables > (P eak T raction Mean, SD, Skewness and Kurtosis) (Energy M ean, SD, Skewness and Kurtosis) (Stiffness Mean, SD, Skewness and Kurtosis) (3.19) 16 variables > (P eak T raction Mean, SD, Skewness and Kurtosis) (Energy M ean, SD, Skewness and Kurtosis) (Shapef actor M ean, SD, Skewness and Kurtosis) (Ratio M ean, SD, Skewness and Kurtosis) (3.20) The initial settings for the CZM parameter PDFs were randomly assigned based on expected bounds for the material of interest and then optimized using PSO to minimize the error between the CZM parameter based fracture strength and test data based fracture strength CDFs. Each of the PSO iterations used 24 particles and on an average PSO took approximately iterations to find the optimum solution. After the optimization process, PSO determines mean, std. deviation, skewness, and kurtosis for each of the four CZM parameters. Table 3.4 shows the initial and optimal values for the bilinear model and Table 3.5 shows the optimal values for the PPR model. In both of the tables, the third column indicates the initial values, where the mean values of the CZM parameters were estimated using the guidelines indicated in Section and all the other statistical characteristics including standard deviation, skewness and kurtosis were assumed as normal distributions with 10 percent of mean, 0 and 3 respectively as initial starting values. In order to check the consistency of the optimum solution, the optimization (PSO) was repeated 68

81 several times. This process showed that the optimal solution did vary, but the error between the CDFs and the final shape of the CDF did not show significant changes. Figure 3.9 shows the fracture strength CDF for three different optimal solutions obtained from optimization of PPR model. This indicates that the optimization uses all 12 or 16 variables to fit the test data CDF and there are multiple optimum solutions that meet that criterion. The occurrence of multiple solutions resulted due to the nonlinearity of the problem and it was not entirely due to the presence of 12 or 16 variables. At the same time, this is not a weakness of the proposed method and one can reduce the occurrence of multiple solutions by introducing a constraint or simply by selecting one of the multiple solutions if any test data becomes available for the CZM parameters to compare with. Since no test data are available for the CZM parameters at the moment, all the solutions obtained can be considered as equivalent good probabilistic CZM characteristics as they were determined by comparing with fracture strength CDF obtained from test data. Another point to be noted here is that the multiple optimum designs have the same error model (Equation 3.12), which is a sum of squares of error at selected points on the CDF. Therefore, in order to use CZM for fracture simulation in a probabilistic sense, this is the only formal process to determine the distributions. The other observation was that the optimized mean energy value in all these cases showed very small change and one can see this even in bilinear and PPR models (Tables 3.4 and 3.5). This shows that for a particular specimen geometry and material system, the fracture energy seems to be same. The results shown in Tables 3.4 and 3.5 represents the solution that results in the least error among all the local minimums. Figure (3.10) shows the final CDF s of fracture strength determined using the PPR and the bilinear model. The error in these models was significantly lower compared to the confidence bounds for the CDF obtained from test data. This is due to the low number of tests, 74, used to develop the CDF. All the parameters in this study were initially assumed to be normal distributions with 10 percent coefficient of variation and the optimizer determined the final distributions for all the parameters. Figure 3.11 shows 100 realizations of the PPR model based on the 69

82 F( ) distribution information obtained from the optimal values Optimal Solution 1 ( Critical using PPR ) Optimal Solution 2 ( Critical using PPR ) Optimal Solution 3 ( Critical using PPR ) PSI psi x 10 4 Figure 3.9: Residual strength CDF comparison of PPR model Table 3.4: Bilinear model-initial and optimal distribution results (Uncorrelated) CZM Parameters Pearson Distribution Param. Initial Optimal Peak Traction, σ c Mean (psi) Std.Dev (psi) Skewness Kurtosis Fracture Energy, Φ n Mean (in.lb/in 2 ) Std.Dev (in.lb/in 2 ) Skewness Kurtosis Stiffness, K Mean (psi/in) 2.77E8 2.17E8 Std.Dev 2.77E E7 Skewness Kurtosis

83 F( ) Table 3.5: PPR model-initial and optimal distribution results (Uncorrelated) CZM Parameters Pearson Distribution Param. Initial Optimal Peak Traction, σ c Mean (psi) Std.Dev (psi) Skewness Kurtosis Fracture Energy, Φ n Mean (in.lb/in 2 ) Std.Dev (in.lb/in 2 ) Skewness Kurtosis Shape Factor, α Mean Std.Dev Skewness Kurtosis Ratio,λ n Mean Std.Dev Skewness Kurtosis Critical using PPR Critical using KIC Upper bound using KIC Lower bound using KIC Critical using Bilinear PSI psi x 10 4 Figure 3.10: Residual strength CDF comparison, PPR, Bilinear, and test data 71

84 Traction, Tn 14 x Optimal values 100 Realizations Separation, x Correlated Parameters Figure 3.11: Realizations of PPR model Once the process for optimizing the parameters to match the scatter from material test data was automated, the correlations between the variables were introduced to add more flexibility to control the final PDF. The correlations among the variables have improved the accuracy of the fracture strength PDF in the lower and upper tails. Since the lower tail region is typically integrated with the applied stress PDF to determine the fracture probability, its accuracy is critical to the risk assessment process. Once the correlations of the variables were determined from the optimization process, MCS was implemented using the Iman and Conover [91, 92] algorithm to generate correlated random samples. Figure 3.12 shows the implementation of the Iman and Conover method using 500 samples, where positive and negative correlations are shown for the two design variables. By controlling the correlation coefficient between these variables during the optimization process an optimal 72

85 Variable X2 Variable X2 Variable X2 fit for the fracture strength PDF was determined. For this exercise, correlations shown in Equation 3.21 were introduced between each of the CZM parameters on the PPR model, which results in six additional variables to the previous 16 variables for the optimization Variable X1 (a) No correlation with correlation with correlation Variable X1 (b) Variable X1 (c) Figure 3.12: (a) No correlation, (b) Positive correlation, (c) Negative correlation 6 correlation > (Energy and P eak T raction) (Energy and Alpha) (Energy and Ratio) (P eak T raction and Alpha) (P eak T raction and Ratio) (Ratio and Alpha). (3.21) 73

86 F( ) Table 3.6 shows the final optimized settings of the PPR model and Equation 3.22 shows the coefficients between each CZM s parameters obtained through the optimization process. Figure 3.13 clearly shows the improvement of the lower tail of the CDF with respect to test data after adding the correlations between the PPR model parameters Correlated Critical using PPR Critical using KIC 0.04 Un correlated Critical using PPR 0.03 Improved lower tail for correlated variables PSI psi x 10 4 Figure 3.13: Lower tail region comparison for fracture strength PDF C = (3.22) Material Strength Scatter While mean fracture strength for a given geometry would be a function of the crack length, it will be shown in this section that the random scatter information associated with each 74

87 Table 3.6: PPR model-initial and optimal distribution results (Correlated) CZM Parameters Pearson Dist. Param. PPR Un-Correlated PPR Correlated Peak Traction, σ c Mean (psi) 10,910 89,568 Std.Dev (psi) 14,276 11,689 Skewness Kurtosis Energy, Φ n Mean (in.lb/in 2 ) Std.Dev (in.lb/in 2 ) Skewness Kurtosis Shape Factor, α Mean Std.Dev Skewness Kurtosis Ratio,λ n Mean Std.Dev Skewness Kurtosis of the fracture strength distributions for different crack lengths can be determined using a normalized scatter distribution. The material strength scatter distribution represents the randomness in the fracture process due to uncertainty in the material characteristics at the crack tip. Figure 3.14 shows the fracture strength PDFs for 2 in and 1 in half crack lengths. These PDFs were determined using the correlated PPR model presented in the previous section. Both of these distributions were normalized using their respective mean fracture strength values, thereby, resulting in the PDFs shown in Figure These 2 in and 1 in normalized PDFs are statistically identical, demonstrating that there exists a normalized scatter PDF that can be scaled to any fracture strength PDF using only the mean fracture strength. Analytical comparison of the two PDFs also proved that they represent the same scatter information. Two different analytical tools were used to understand scatter. One was the Bhattacharya distance measure and the other was based on entropy and cross entropy of the distributions. The Bhattacharya distance [93] measure as shown in Equation 3.23, uses 75

88 an integration scheme where values between 0.9 and 1.1 represent close agreement between the distributions and a value closer to 1.0 represents a very good match. The entropy [94] and cross entropy measures shown in Equation 3.24 require the cross entropy to match with the entropy to represent similarity between the PDFs. The closer the entropy values the better the match between the distributions. Table 3.7 and 3.8 show these measures for the two distributions checked for five repetitions with 100,000 samples each and they all confirm that these two normalized distributions represent same level of uncertainty. Therefore, the normalized distribution becomes the scatter distribution that represents the randomness in the fracture process for a given material. The normalized fracture strength distribution de-couples the material randomness from the geometric effects. By de-coupling, only one probabilistic analysis, based on a single crack configuration, is necessary to determine the normalized fracture strength PDF. The normalized PDF can then be scaled using mean fracture strength, obtained from a deterministic fracture simulation, for all other realizations of crack lengths. The normalized fracture strength PDF improves the computational efficiency of a probabilistic fracture simulation and enables its integration into large-scale structural problems. Bhattcharya Distance = n fa (x) f b (x) (3.23) i=1 n Entropy = f a (x) logf a (x) (3.24) i=1 n CrossEntropy = f a (x) logf b (x) i=1 76

89 F( ) Table 3.7: Entropy results Description Entropy(2in) Entropy(1in) Encross(2in on 1in) Table 3.8: Bhattacharyya distance results Description Bhattacharyya distance x inch HCL 1 inch HCL Fracture Strength (, PSI) psi x 10 4 Figure 3.14: PDFs of fracture strength for different crack lengths 3.6 Probability of Fracture Nomograph In the previous chapter, a PoF nomograph was developed based on SIF approach to assess the risk of a damaged structure efficiently without the need for repeated reliability analyses. In this section, a similar nomograph is developed for both flat and stiffened plates using the calibrated probabilistic CZM parameters. 77

90 F( ) inch HCL-Normalized 1 inch HCL-Normalized Normalized Fracture Strength ( ) Figure 3.15: PDFs of normalized fracture strength for different crack lengths Nomograph for a Flat Plate Similar to PoF nomograph developed based on Stress Intensity Factor, a PoF nomograph for CZM also has three plots. The first plot represents the geometry effects, the second plot represents the equation of the failure criterion, and the third plot represents the probability of fracture obtained by integrating normalized stress and strength distributions. Creation of Plot 1 In the previous section it was demonstrated that using a deterministic scale factor, the PDF of fracture strength for any crack length can be determined based on the normalized fracture strength PDF with mean of 1.0, Fig This deterministic scale factor is the mean fracture strength obtained by using the mean values of the correlated PPR model. This mean fracture strength is plotted for various crack lengths in plot 1 of the nomograph, Figure The main difference between the SIF and CZM approaches in plot 1 is their y-axis. In SIF, the y-axis was β(a) πa whereas in CZM it is Mean Fracture Strength because 78

91 Mean Fracture Strength (PSI) (psi) there is no closed form equation available for CZM, and fracture strength can be obtained only from FEA. 3.5 x Half Crack Length (in), a Figure 3.16: Mean fracture strength for different crack lengths (Plot 1) Creation of Plot 2 As indicated in plot 1 the y-axis of the plot 2, Figure 3.17 was also maintained as Mean Fracture Strength. Using Equation 3.25 and mean fracture strength ranging from to psi, generated contour plots for four applied stress values 15000, 25000, and psi. Otherwise, the procedure for generating plot 2 is identical to same as plot 2 generated in SIF approach. P I = F racturestrength, σ F σ applied σ applied (3.25) 79

92 (psi) Mean Fracture Strength (PSI), F 3.5 x PSI PSI PSI PSI Probability Index - PI Figure 3.17: Mean fracture strength with probability index (Plot 2) Creation of Plot 3 The third plot integrates the normalized stress and strength distributions to determine the PoF. Since the same failure criteria is also used in CZM approach, the PoF plot represented by Equation 3.26 is identical as Figure 3.18 (plot 3) generated in the SIF approach. P of = P [F racturestrength < AppliedStress] or P of = P [F racturestrength AppliedStress < 0] (3.26) With all the three plots arranged as shown in Figure 3.19, the Probability of Fracture can be determined for any crack length without the need for any reliability analysis. The nomograph for a flat plate is explained using an example of a structure that has a half 80

93 Probability of Fracture - PoF FALSTAFF Probability Index-PI Figure 3.18: Probability of fracture with probability index crack length, a = 0.5 in and applied stress, σ applied = 35, 000 psi. From plot 1, using interpolation, the fracture strength,σ F was found as 2.5e4 psi. From the second plot using the information σ F = 2.5e4 psi and σ applied = 35, 000 psi, the probability index, PI, was Using P I = 0.29, the PoF was found to be with the help of plot 3 of the nomograph Nomograph for a Stiffened Plate Since it was determined the cohesive zone model parameters were independent of geometry, the optimized parameters obtained previously can be applied to any configuration. From the flat plate nomograph, it can be clearly seen that plot 2 and plot 3 will not change for situations with new geometry. For a stiffened plate, plot 1 must be generated for multiple crack lengths using deterministic analyses with mean values of the correlated PPR model. In the previous chapter the stiffness ratio [65] was used to represent various geometric 81

94 Probability of Fracture - PoF (psi) Mean Fracture Strength (PSI), F Mean Fracture Strength (PSI) 3.5 x PSI PSI PSI PSI 3.5 x Probability Index - PI Half Crack Length (in), a a Center crack Probability Index-PI Figure 3.19: Probability of fracture nomograph for a flat plate configuration and is indicated again for reference. µ = Stiffner Stiffness P late Stiffness = w s t s E s w s t s E s + bte (3.27) Here t is the thickness of the plate, w s and t s are the width and thickness of the stiffener, E and E s are the Young s modulus of the plate and stiffener and b is the stiffener spacing. A plate of size 20 in x 20 in x 0.04 in thickness with four stiffeners of size w s = 0.8 in, t s = 0.2 in and b = 4 in (µ = 0.5) was modeled with rivet spacing p = 0.67 in and rivet diameter d = in to maintain p/b = 1/6 and d/p = 1/4 specification. In this model, all the stiffeners are intact and there were no rivets interfering with the crack propagation. Similar to the flat plate considered above, the fracture strength of the stiffened plate 82

95 Probability of Fracture - PoF Mean Fracture Strength (PSI), (psi) F 3.5 x PSI PSI PSI PSI Probability Index - PI 3.5 x Stiffened panel Half Crack Length (in), a Probability Index-PI Figure 3.20: Probability of fracture nomograph for a stiffened panel, µ=0.5 was obtained using FEA for different crack lengths and plotted as a function of half crack length. The nomograph of a stiffened panel is shown in Figure The nomograph for stiffened panel is explained using an example of a structure that has a half crack length, a = 0.5 in and σ applied = 35, 000 psi. From the plot 1, using interpolation, the fracture strength,σ F was found as 2.61e4 psi. From the second plot using the information σ F = 2.61e4 psi and applied load of psi, the probability index, PI was Using P I = 0.257, the PoF was found to be from plot 3. 83

96 3.7 Procedure for Calibration Previously the calibration process was demonstrated in detail with two examples, in this section, a step-by-step process is presented for calibrating the CZM parameters: 1. Create a Mode-I model with appropriate crack size (which satisfy small scale yielding conditions) for which the fracture strength distributions are to be determined. 2. Obtain material properties and fracture toughness test data for the material of interest. 3. Estimate fracture strength distribution using Equation 3.10 and represent the fracture strength as a CDF. 4. Identify the number of CZM parameters required to define the traction-separation law. 5. Using the material properties estimate the mean values of each CZM parameter (refer to Section Conventional Material Properties and Initial CZM Parameter Estimates). 6. Based on the mean values, generate several LHS points by providing acceptable upper and lower limits for each of the CZM parameters. 7. Use the above LHS points and determine the fracture strength for each of those sample points using ABAQUS software. 8. Develop a RSM by using the fracture strength results obtained from the previous step. 9. For each CZM parameter in the RSM, apply a Pearson distribution and generate the fracture strength CDF using MCS. Also, correlation between each CZM parameter can be incorporated by using Iman and Conover method. 84

97 10. Use PSO to minimize the error between the response surface generated CDF fracture strength CDF obtained from test data. 11. After minimizing the error, PSO provides the optimized values including mean, standard deviation, skewness and kurtosis for each of the CZM parameters. 12. Use the response surface with optimized CZM parameters to generate the fracture strength PDF. 3.8 Summary Typically when performing reliability analysis, PDFs of the input parameters are selected either based on experience or experimental data. For a cohesive zone model there are no formal tests in the literature to determine the scatter in the data, to assign PDFs to CZM parameters. Assuming PDFs without any justification introduces uncertainty into the probability of fracture estimates. Therefore, in this chapter demonstrated a formal process to identify the distributions for the cohesive zone model parameters based on existing test data from LEFM. It was assumed that for situations where LEFM assumptions are not violated, both LEFM and CZM based methods predict the same probability of fracture. In situations where LEFM assumptions are violated, nonlinearity in the CZM parameters captured the material behavior in the fracture process zone. Once the distributions were identified using coupon test models they were used to predict crack growth and probability of fracture in complex structures. Therefore, the demonstrated process is a tool to develop material behavior models that can be embedded into the structural system to assess initiation and propagation of damage. Finally, the chapter also presents a nomograph that uses a cohesive zone model to assess probability of fracture without the need for repeated reliability analyses. The calibration process and development of nomograph presented in this chapter have been published [96, 97]. 85

98 Chapter 4 Scientific Contributions and Limitations of Nomograph 4.1 List of Scientific Contributions 1. Developed of Probability of Fracture nomograph concept for efficient risk assessment using stress intensity factor. There are several methods and software available in the literature to calculate probability of fracture estimates, but all these methods require repeated risk analysis for every change in geometry or loading condition. However, the nomograph represents risk information in a graphical form and it captures geometric variations, mean material strength, and reference stress magnitude changes. This nomograph eliminates the need for repeated risk assessment and also presents risk sensitivity information in graphical form. 2. This research also developed a formal process for calibration of the probabilistic cohesive zone model for both metallic and composite materials. This development enables assigning realistic probability distributions to CZM parameters that represent 86

99 the fracture strength scatter noticed as it appears in experimental test data. 3. In addition to the above contributions, this research also developed probabilistic CZM to predict steady state energy release rate variations of the unidirectional composite DCB specimens. This demonstrated the possibility of extending the probabilistic model calibration process to non-metallic material systems. 4.2 Limitations of PoF Nomograph 1. This research developed the nomograph for fracture where the geometry, material strength, and loading are separable. However, the applicability of this concept to other failure criteria where its constituents are not separable still must be investigated. 2. If crack length was available as a probability distribution, the current form of the nomograph does not have a means to perform the double integral of probability of fracture over all possible crack lengths. 3. Applicability and practicality of a nomograph in a real engineering situation has not been investigated. 87

100 Chapter 5 Summary and Future Work 5.1 Summary In this research, a PoF nomograph was developed by integrating information from three independently generated plots that represent geometry, failure criterion, and random scatter. These plots enable rapid PoF assessment without the need for repeated reliability analysis to accommodate for changes in geometry. The nomograph developed in this research serves two purposes: one is to determine an allowable crack length given acceptable failure probability, and the other is to determine failure probability given geometry and loading conditions of the structural component. A unique feature of this nomograph is that when the geometry of interest is changed from a flat plate to a stiffened plate only the first plot that represents geometry would need to be modified. The other two plots remain unaltered as long as the material and scatter information are the same for both the geometries. In this research, the nomograph was demonstrated for Al 2024-T3 material using stress intensity factor and cohesive zone model approach for flat and stiffened plates. While developing nomograph using SIF approach, the fracture strength distribution was directly derived from fracture toughness test data. But in CZM approach, a material 88

101 calibration process was required before developing PoF nomograph. This calibration process was developed in this research to identify the distributions of the CZM parameters to estimate the fracture strength distribution. Also in this research, the calibration process was extended to a composite material and a probabilistic CZM was developed to predict steady state energy release rate variations in unidirectional composite DCB specimens. 5.2 Future Work 1. In this research, only fracture was considered as a failure criterion to develop the nomograph. Further research is needed to determine if this concept can be extended to other failure criteria including fatigue, buckling, etc. While a nomograph is a visual tool to assess reliability, its strength lies in its ability to use separable normalized quantities to enable rapid assessment. 2. PoF nomograph has been developed for a generic flat plate and a stiffened plate and it needs to be extended to real aircraft structural components. In reference [71], the DSTO (Defense Science and Technology Organization) conducted structural risk assessment on four B707 aging aircraft to study the reliability of the four lower wing stringers and their bolt holes. A nomograph can be developed for this structure by performing appropriate stress intensity factor plots based on the scenarios of interest. This nomograph would then become a tool for rapid reliability assessment given knowledge of crack sizes and loading conditions. 89

102 Appendix A Appendix A: Probabilistic Cohesive Zone Model for Fiber Bridging Delamination between plies is the most common failure mode in composite laminates that can cause fiber breakage and reduction in life of the composite. Currently, Mode-I interlaminar fracture toughness or critical energy release rate of a composite is measured using double cantilever beam (DCB) with unidirectional composites (ASTM D5528). Unlike metals, the energy plot of a DCB specimen shows increase in energy release rate with increase in crack length. Also, during testing the steady state energy release rate from each of the samples of the same batch shows a lot of variation due to fiber cross-over bridging that occurs only in unidirectional composites. In this study, a probabilistic Cohesive Zone Model (CZM) is developed to capture steady state energy release rate variations of 51 mm crack size DCB specimens based on unidirectional composite (IM7/977-3) test data. Then using the probabilistic CZM, the energy release rate variations of 76.2 mm crack size DCB model are predicted and compared to the test results. The predictions showed good agreement with the experiments suggesting a probabilistic CZM is capable of simulating the strength scatter during the delamination process in unidirectional composites that were of 90

103 interest in this research. Nomenclature P 2h b a a 0 I Load Height or Thickness of DCB Width of DCB Crack Length Initial Crack Length Second Moment of Inertia λ, ρ Elastic Anisotropic Measures J 0 J SS Crack Initiation Fracture Energy Steady State Fracture Energy J SS Difference in Fracture Energy (J SS J 0 ) δ u 0 u 2 α End-Opening of the Bridging Zone at the Notch Root Steady State Crack Opening Displacement at the Notch Root Crack Initiation Opening Displacement at the Notch Root Power-Law Factor A.1 Introduction Fiber-reinforced composite material offers high strength to weight ratios when compared to metals. Military and commercial aircraft industries are now using composites to reduce structural weight while meeting the necessary strength requirements. From the introduction of boron fiber-epoxy as a skin material for an F-14 aircraft in 1969, the use of composite materials has been increased drastically over the past few decades [98]. In addition to 91

104 its light weight properties, composites exhibit high corrosive resistance and high fatigue strength properties. In general, a composite consists of two materials, one is called the matrix and the other is called the fiber. The combination of these two systems gives the desired material strength. The matrix holds the fibers in place and forms the bulk of the material. The matrix material can be thermoset polymer, thermoplastic polymer, metallic, or ceramic, while the fiber material can include carbon, glass, aramid, boron, or ceramic, which are impregnated into the matrix and provide the strength to the composites. Fiberreinforced composite material has high strength and stiffness in the fiber direction when compared to transverse direction. A laminated composite has more than one layer of fiberreinforcement, called a ply, and in each ply, fibers may be oriented in the same or different directions. Based on the type of loading, the failure mechanisms of a fiber-reinforced composites can be fiber pull-out, fiber bridging, fiber/matrix debonding, fiber failure, matrix cracking, delamination, microbuckling and buckling delamination [67]. Delamination between plies is the most common failure mode in composite laminates and it occurs at free edges, such as holes and notches in a laminate. This interply delamination can cause fiber breakage and reduce the life of the composite. Generally, Mode-I interlaminar fracture toughness or critical energy release rate of a composite is measured using double cantilever beam (DCB) with unidirectional composites (ASTM D5528). Unlike metals, the energy plot of a DCB specimen shows increase in energy release rate with increase in crack length [100]. This is due to fiber cross-over bridging at the interface between the top and bottom arm of DCB. During fiber bridging, shown in Figure A.1, a crack jumps from one fiber/matrix interface to another without breaking the fiber. This occurs only in unidirectional composites where the fibers migrate during the curing process of composite preparation. The increase in energy release rate with respect to crack length is called the R-curve effect [67]. Sorensen and Jacobsen [101] indicated that the behavior of R-curves depend on specimen geometry and therefore they cannot be considered as material property when large-scale bridging phenomenon occurs. Large-scale bridging occurs 92

105 whenever the length of the crack at fiber bridging is equal to or greater than the geometry size. During DCB testing, energy required for crack initiation and crack propagation are measured with respect to crack length of each DCB specimens. From the measurement it was found that the variations in energy for crack initiation is between 5-10 percent, whereas variation in energy for steady state crack propagation was between percent for each of the samples in one batch. Even though fiber-bridging is not desirable, researchers and designers like to characterize the behavior of the structure when fiber-bridging occurs during structural failure and also to understand the relationship between matrix fracture toughness and composite fracture toughness [105]. In order to understand energy variation during steady-state crack propagation, researchers test various crack sizes of DCB specimens. This research primarily concentrates on developing a methodology to capture and predict the steady state energy variations of the DCB specimens during the crack propagation stage. In order to accomplish that goal, a CZM with appropriate bridging law (or traction-separation law) is introduced ahead of the initial crack. As the load increases the cohesive zone starts separating based on the bridging law until it reaches a critical limit. The bridging law definition, its formulation, and advantages are described in Section A.2. In Section A.3 details about the DCB geometry, finite element construction and simulation using ABAQUS are explained and in Section A.4 the calibration process for the CZM parameters using the test data is presented. In Section A.5, results of 51 mm crack size are presented along with the correlations between predictions and test results for a 76.2 mm initial crack size specimen. This sections also presents a process for implementing the probabilistic CZM based steady state crack propagation energy prediction. 93

106 Figure A.1: Fiber bridging [104] A.2 Cohesive Bridging Law Definition To determine fiber bridging characteristics of unidirectional fiber composites of a double cantilever beam, Bao [99], Spearing and Evans [100], and Sorensen and Jacobsen [101] measured bridging law parameters based on the path independent J integral approach [102]. The global J integral evaluated along the external boundaries of a DCB specimen under Mode-I loading (Figure A.2) can be determined analytically [100] using Equation A.1. J = P 2 a 2 be 11 I [1 + ɛ(ρ)λ1/4 ( h a )]2 (A.1) where P is the load, a is the crack length, h is half of specimen thickness, b is the specimen width, I is the second moment of area of the DCB model represented by, 94

107 P Fiber Direction Fiber bridging X 2 LLD δ* X 1 2h P a 0 a Figure A.2: DCB specimen I = bh3 12 λ and ρ are elastic anisotropy measures and calculated by λ = E 22 E 11 ρ = (E 11E 22 ) 1 2 2G 12 (ν 12 ν 21 ) 1 2 where E 22 and E 11 are the transverse and longitudinal Young s Modulus, G 12 is the in plane shear modulus and ν 12, ν 21 are the major and minor Poisson s ratio. The coefficient ɛ is denoted by 95

108 ɛ = (ρ 1) (ρ 1) (ρ 1) 3 The DCB specimen (Figure A.2) exhibits fiber bridging between the top and bottom of the crack faces during loading. Sorensen and Jacobsen [101] indicated that the closure traction or bridging stress, σ, depends only on local crack separation, δ, and therefore assumed the bridging law, σ = σ(δ), is identical at every point along the length of the bridging zone. The bridging law is defined as a relationship between local crack opening displacement, δ, and local bridging stress, σ. The bridging stress disappears once a maximum crack opening displacement, δ 0 is reached. The J integral evaluated locally (Figure A.3) along the bridging zone and around the crack tip [102] can be represented using Equation A.2. Fiber bridging σ δ* J tip a 0 Г loc Figure A.3: Crack tip of a DCB specimen δ J = σ(δ) dδ + J 0 0 (A.2) where the first term is the energy dissipation in the bridging zone and the second term, J 0, 96

109 is energy release rate at the crack tip. By continuously measuring the end-opening, δ, at the notch root along with the crack growth resistance (R-curve), the bridging law, σ = σ(δ), can be determined by σ(δ ) = J R δ where J R is the energy release rate during crack growth, initially J R (A.3) = J 0 when crack initiates and then it follows as per Equation A.2. When δ = δ 0, J R reaches steady state crack growth resistance, J SS (R-curve). Based on crack growth resistance versus crack opening data for carbon fiber/epoxy, glass-fiber/epoxy, and glass-fiber/polyester the following analytical equation was found to fit all the experimental data curves [103, 101]. J R (δ ) = J 0 + J ss ( δ δ 0 ) 1 2 (A.4) where J SS = (J SS J 0 ) is the crack growth resistance due to fiber bridging and δ 0 is the crack opening displacement at which bridging stress disappears. Differentiating the above equation as per Equation A.3 yields the following, σ(δ ) = J ss 2δ 0 ( δ δ 0 ) 1 2 = J ss 2δ 0 ( ) 1 δ0 2 J ss = δ 2 δ 0 δ 0 < δ < δ 0 (A.5) The bridging law for a typical material system is shown in Figure A.4 and it is independent of specimen geometry and can be considered a material property [101]. Adjusted Bridging Law [103] To improve the agreement between the experiment and numerical data of the R-curve and also to avoid a trial and error iterative process to obtain the bridging law parameters, Feih 97

110 Bridging Stress, σ 1 2 ( ) 2 J SS 0 0 Crack Opening, δ δ 0 Figure A.4: Typical bridging law [103] adjusted the bridging law as shown in Figure A.5. They also developed cohesive elements using ABAQUS user element to introduce the adjusted bridging law for composite structures. The non-linear bridging law shown in Figure A.4 is adjusted for two reasons, first the bridging stress goes to infinity when the crack opening displacement δ = 0 and second after differentiation the bridging law no longer has the crack initiation J 0 term. For the adjusted bridging law, J 0 (crack initiation) and J SS (steady state fracture energy) are separated and a K fac factor (when u > u 2 ) is introduced to avoid over prediction of experimental results. The advantage of the adjusted bridging law is that, with the exception of u 1, all the parameters (J 0, J SS, u 0, and u 2 ) can be determined from the experiment and all have physical meaning associated with the fiber bridging process. The power-law factor, α, is introduced so that the tangent at u = u 2 of the bridging law is maintained close to zero before changing to negative slope. At α = 100, both initial linear increase and power law increase of the bridging law provide a constant stress value. Both u 0 and u 2 can be determined from crack growth resistance versus crack opening curve, where u 0 is the steady state crack opening value and u 2 is crack opening value at initial fracture energy, J 0. Similarly, J 0 and J SS can be measured from crack growth resistance versus crack extension, a curve (R-curve). 98

111 Bridging Stress, σ u ( 2 u K u 2 u) 1 fac J SS u u1 1 u J SS u0 u1 J 0 ΔJ SS Δu 2 Δu 1 Crack Opening, Δu Δu 0 Figure A.5: Adjusted bridging law After determining J 0, J SS, u 2 and u 0 from the experiment, original peak stress, σ 2, required peak stress, σ 0, and K fac can be calculated using the following equations: σ 2 = 1.5 J ss u 1 u 0 u 1 (A.6) σ 0 = J 0 u 2 ( ) α + 1 α (A.7) K fac = Required peak stress, σ 2 Original peak stress, σ 0 (A.8) In this section only relevant information from Feih [103] was included to describe the adjusted bridging law. The following section presents details about DCB modeling and simulation in ABAQUS. 99

112 A.3 DCB Geometry and Finite Element Model The Air Force Research Laboratory (AFRL) tested unidirectional carbon/epoxy (IM7/977-3) DCB composite specimens to determine Mode-I interlaminar fracture toughness. The specimens were 254 mm long, 25 mm wide with two 6 mm thick arms. Also each specimen had an initial crack of 51 mm, which was introduced by using a Teflon strip. Table A.1 shows the material properties for the above mentioned composite material. During each test, load, load line displacement, and crack length were measured. Using load and crack length measurement, energy release rate J was calculated using Equation A.1. The test results of load vs. load line displacement and energy release rate (J) vs. crack length (a) are shown in Figure A.6 and A.7 for 15 test samples. In this study, DCB specimens were modeled using a 2D finite element model. Each Finite Element Model (FEM) shown in Figure A.8 consists of 8 node 2D plane strain elements (CPE8) with an element size of 0.5 mm and 6 node zero-thickness cohesive elements. These cohesive elements were placed ahead of the crack tip, between the top and bottom arm of the FE model. To represent fiber bridging, these cohesive elements were assigned the adjusted bridging law described in the previous section. Table A.1: Material properties of IM7/977-3 composite Description Batch-2 E 11 (GPa) 146 ν E 22 (GPa) ν G 12 (GPa) 4.53 J 0 Initiation (J/m 2 )

113 Energy (J/m 2 ) Load (Newton) Displacement (m) data1 data2 data3 data4 data5 data6 data7 data8 data9 data10 data11 data12 data13 data14 data15 x 10-3 Figure A.6: Load vs load line displacement of 15 test results Crack Length, a (m) data1 data2 data3 data4 data5 data6 data7 data8 data9 data10 data11 data12 data13 data14 data15 Figure A.7: Steady state energy release rate vs crack length of 15 test results 101

114 Cohesive element (user subroutine) Figure A.8: FEA model A.4 Probabilistic Cohesive Zone Model Most of the material properties used in structural design, including yield strength, ultimate strength, shear strength, fracture toughness, etc., are obtained from test data and typically have scatter models developed from the test data. However, when moving from a traditional failure analysis method to the new CZM based method, there are no tests specifically designed to calibrate the CZM bridging parameters to determine their scatter information. In this work, to capture the steady state energy release rate (J SS ) variation, the following procedure was developed: 1. Estimate steady state energy release rate, load and crack length distribution from test data and represent energy in CDF (Cumulative Distribution Function) format. 2. Use test data to estimate the upper and lower bounds of each of the adjusted bridging law parameters. 102