AUTOMATIC TWO-DIMENSIONAL QUASI-STATIC AND FATIGUE CRACK PROPAGATION USING THE BOUNDARY ELEMENT METHOD

A Thesis Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment for the Degree of Doctor of Philosophy

by George Elbert Blandford January, 1981

ABSTRACT

The objective of this thesis research is to develop a boundary element code which automatically generates the modified data required to perform two-dimensional linear elastic quasi-static and fatigue crack propagation.

The work begins with the derivation of the direct boundary integral equation for the analysis of elastostatic boundary value problems in Chapter II. The numerical implementation of the boundary integral equation, referred to as the boundary element method, uses isoparametric linear and/or quadratic polynomial approximation of the boundary geometry, displacements and tractions. Gaussian quadrature is used to evaluate the discretized boundary integrals. Because of the singularities which exist in the boundary integrals, special singularity actorization schemes are presented so that the singular integral scan be evaluated accurately. A multidomain boundary element formulation is also presented. Several example problems are presented verifying the accuracy, efficiency and flexibility of the present boundary element method implementation.

Chapter III deals with the subject of linear elastic fracture mechanics. The analytically dominant displacement and stress variations in the vicinity of the crack tip are derived. Special crack elements are developed which include the dominant r displacement variation and the dominant 1/ r singularity exhibited by the stresses. Displacement correlation techniques are presented for calculating the mode I and mixed-mode stress intensity factors which equate the analytical displacement variation to the numerical solution of the special crack tip element, i.e., the traction singular quarter-point boundary element. several example problems of mode I and mixed-mode fracture are presented. These results show that the use of traction singular quarter-point boundary elements yields accurate mode I and mixed-mode stress intensity factors for crack tip element lengths, l, to crack lengths, a, of 0.10 ý l/a ý0.20.

The multidomain boundary element formulation of Chapter II is shown to be an effective strategy for analyzing mixed-mode fracture problems without resorting to complex variable transformations. The accuracy of the computed stress intensity factors is shown to be sufficient for the analysis of quasi-static and fatigue crack propagation problems.

The analysis of mode I and mixed-mode quasi-static crack propagation problems is presented in Chapter IV. The maximum circumferential tensile stress, the minimum strain energy density and the maximum energy release rate mixed-mode fracture initiation theories are discussed. Four closed-form fracture initiation theories, i.e., the ~Bmax~ 5~8~min' G(e)max and ~r(e)max' are discussed at length and implemented in the boundary element code of this thesis. The numerical implementation of the stable primary crack algorithm for both mode I and mixed-mode fracture problems is presented. The mode I quasi-static boundary element crack propagation code is verified by analyzing a tapered double cantilever beam. A good correlation with experimental and analytical results is shown. The mixed mode crack propagation code is used to analyze an angled-notch rock fracture problem loaded in compression. Good correlation of the boundary element method solutions is shown with available experimental data and finite element analysis of the same problem.

Chapter v deals with the analysis of mode I and mixed-mode linear elastic constant-amplitude cyclic-load fatigue crack propagation problems. The Paris law is used to model the fatigue crack growth behavior. The boundary element formulation for the analysis of linear elastic fatigue crack propagation is given. The boundary element method results are shown to agree with published experimental and numerical data.

SIMULATION AND VISUALIZATION OF HYDRAULIC FRACTURE PROPAGATION IN POROELASTIC ROCK

A Dissertation presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

by Thomas James Boone May, 1989

ABSTRACT

This thesis is a detailed investigation of hydraulically-driven fracture propagation in poroelastic rock. Biot's theory of poroelasticity is used to study coupling between rock deformation and fluid flow within its mass. The topic is developed as follows: (1) a nonlinear fracture mechanics model is adapted for a poroelastic continuum, (2) poroelastic concepts and effects are illustrated through application to the 1-D, PKN fracture model, (3) a 2-D, plane strain, numerical model for hydraulic fracture in poroelastic materials is described and verified, (4) a specialized code for 3-D visualization of coupled processes, using computer graphics, is presented, and (5) the plane strain model and visualization capabilities are used to further investigate a variety of poroelastic effects.

The plane strain numerical model solves three sets of fully-coupled equations representing (1) equilibrium of the rock mass, (2) conservation of fluid mass within the rock matrix, and (3) conservation of fluid mass in the fracture. The first two sets of equations are derived from a finite element approximation to Biot's theory of poroelasticity, while the third set is approximated by a finite difference model. An equilibrium fracture model is incorporated in a manner that produces a crack length that is a natural product of the solution procedure, and allows modeling of fracture initiation from a borehole. An iterative, staggered solution procedure, which implicitly advances the solution at each time step, has been designed to take advantage of vector processing on a mini-supercomputer. Images from a specially developed, workstation-based, 3D visualization tool are found to be an effective means of communicating the results and physics of coupled processes.

The primary application of this work is hydraulic fracturing in oil or gas bearing rock. Poroelastic effects on the PKN and CGDD models are investigated, along with effects on mini-frac tests which can be used to determine a variety of relevant material and fracture parameters. These latter studies include detailed simulations of fracture closure which can occur after flow into the fracture is shut-in. The results may also be applicable to dredging, drilling and cutting of fluid saturated rock.

THREE-DIMENSIONAL SIMULATION OF NEAR-WELLBORE PHENOMENA RELATED TO HYDRAULIC FRACTURING FROM A PERFORATED WELLBORE

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

by Jose Luiz Antunes de oliveira e Sousa May 1992

ABSTRACT

This thesis addresses the problem of stimulation of hydrocarbon reservoirs from cased wellbores with multiple perforations. Developments in related areas are integrated towards a realistic simulation of near-wellbore phenomena in hydraulic fracturing from an arbitrarily oriented wellbore. In the early stages of hydraulic fracturing under such conditions, fractures starting from different perforations propagate and coalesce with each other, changing orientation to an optimum direction, normal to the minimum in situ stress.

New developments in this thesis include (1) a general approach for computation of stress intensity factors in three dimensions, capable of handling results from both boundary and finite element techniques, (2) a numerical, energy-based approach for three-dimensional propagation of arbitrary fractures, (3) an integrated system for simulation of hydraulic fracturing processes, capable of handling complex rock and fracture geometries, implemented in a state-of-the-art computational environment. An object oriented Programming discipline is followed in the implementation of the hydraulic fracturing system, efficiently combining recent developments in (1) computer graphics for interactive modeling and visualization of results, (2) structural modeling of solids containing fractures, (3) structural analysis of fractured solids by the boundary element technique, (4) three-dimensional simulation of arbitrary fracture propagation, and (5) finite element modeling of fluid flow between the fracture walls, coupled with the rock structural behavior. The architecture of the resulting simulation system facilitates expansions to account for more sophisticated theories and incorporation of new developments in the related areas. It is asserted that such a system architecture provides an efficient environment for the investigation of the several related phenomena in the simulation of hydraulic fracturing processes involving highly complex geometries.

The developed system is applied to the simulation of experiments conducted by Schlumberger Perforating & Testing Center and Dowell Schlumberger Incorporated, aiming at insight into the three dimensional geometrical evolution of fluid-induced fractures initiating from a cased and perforated wellbore. The simulated fracture geometry was in good agreement with experimental observation.

Difficulties were found in correlating the history of crack growth to the pressure log from the experiment. However, further enhancements to the computational environment, and correction of imperfections detected in the experiments, can lead to a realistic simulation.

COMPUTER SIMULATION OF LINEAR AND NONLINEAR CRACK PROPAGATION IN CEMENTITIOUS MATERIALS

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

by Tulio Nogueira Bittencourt May 1993

This thesis deals with the computer simulation of crack propagation in cementitious materials. Both linear and nonlinear aspects of crack propagation are addressed. The problems addressed gradually increase in complexity, starting from LEFM (Linear Elastic Fracture Mechanics) for two dimensional models, and going up to NLFM (NonLinear Fracture Mechanics) for three dimensional problems. The goal is not only to model the physics involved, but also to provide tools for visualizing the crack propagation process. Size effects are also investigated and related to the crack propagation process.

First, the use of LEFM concepts for two dimensional models is considered, and a strategy to model crack propagation with minimum user interaction is presented. In a following step, the application of the fictitious cohesive crack model for two-dimensional problems is explored. A new, integrated, arbitrary, cohesive crack propagation strategy is proposed. This strategy is based on: interactive, effective total crack (true crack plus fracture process zone) length control, a criterion for propagation based on fictitious crack tip parameters (opening profile, tip stress, or tip singularity), a local principal-stress-based criterion for direction of propagation, a dynamic relaxation solver for determining propagation length, and automatic remeshing to accommodate arbitrary growth. Finally, a new method to solve the cohesive crack problem in three dimensions is proposed. This method is capable of modeling the propagation of both the true crack and process zone on a pre-defined crack path for different specimen geometries and absolute sizes. The influence of specimen size on the determination of fracture toughness is investigated by simulating the short-rod specimen response for concrete.

The use of computer graphics is stressed not only to control the crack propagation process, but also to allow fast and comprehensive interpretation of the results.

A SOFTWARE FRAMEWORK FOR SIMULATING CURVILINEAR CRACK GROWTH IN PRESSURIZED THIN SHELLS

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

by David Oskar Potyondy, Cornell University 1993

A methodology for simulating crack growth in pressurized, stiffened, thin shell structures is described. Crack trajectories are allowed to be arbitrary and are computed as part of the simulation. The methodology is implemented within a software framework that consists of a fracture simulation code and a separate analysis code. The fracture simulation code embodies a representational model that supports a high-level problem description that is independent of the discretization. The representational model --consisting of a constrained hierarchy of topology-based, boundary representation, geometric models with which simulation attributes may be associated --is well-suited for tracking the geometry changes resulting from crack growth. Cracks are represented discretely in the mesh and crack growth results in localized mesh deletion. The deleted regions can be remeshed automatically using a newly developed arbitrary region, all quadrilateral element surface meshing algorithm.

Structural response of pressurized, stiffened, thin shells is computed using a geometrically nonlinear shell finite element analysis procedure installed in the STAGS (STructural Analysis of General Shells) code. Crack growth is characterized by four stress intensity factors that model the membrane behavior using two dimensional, plane stress elasticity and the bending behavior using Kirchhoff plate theory. The four stress intensity factors for mixed-mode problems are computed from the results of a-finite element shell analysis using an extension of the modified crack closure integral method. Crack trajectory is determined by applying the maximum tangential stress criterion to a point on the shell mid-surface.

The effectiveness of the methodology is demonstrated by simulating crack growth in a typical narrow-body pressurized aircraft fuselage idealized as a stiffened, thin shell structure. A hierarchy of structural models, ranging from a relatively coarse global shell model to a highly refined local shell model, provides the kinematic boundary conditions for the 2x2 bay stiffened panel model represented within the fracture simulation code. Crack trajectory and stress intensity factor variation as a function of total crack length are computed and a fatigue life prediction is made. The predicted crack trajectory and life compare well with measurements of these same quantities from a full-scale pressurized panel test.

THREE-DIMENSIONAL FINITE ELEMENT ANALYSIS OF CYCLIC FATIGUE CRACK GROWTH OF MULTIPLE SURFACE FLAWS

A Thesis Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment for the Degree of Doctor of Philosophy

Corneliu Manu May, 1980

The objective of the thesis was to develop a numerical tool capable of dealing with interacting multiple surface flaws in a plate subjected to cyclic loading.

The work starts with the presentation of a consistent method for computing stress-intensity factors from three-dimensional quarter-point element nodal displacements. The method is generalized to permit a functional evaluation of stress-intensity factors along the crack front. Embedded and surface crack problems are solved using the proposed technique. Results are compared to previous finite element and boundary element solutions. The comparison shows that use of the functional evaluation technique allows a dramatic decrease in problem size while still maintaining engineering accuracy. A three-dimensional stress-intensity factor calibration of the ASTM E399-74 standard three-point bend configuration is presented, revealing the fact that the optimal quarter-point element length is dependent on loading type and Poisson ratio of the material.

The above problems involved only crack initiation; crack propagation problems benefit even further from the proposed technique. Since fatigue crack growth rate is highest along the region of the crack front experiencing the highest stress-intensity factor range, accurate evaluation of stress-intensity along the front is essential to accurate predictions of, for example, direction of fastest growth or flaw interaction effects.

A computer program has been developed to evaluate material para-meters that govern fatigue propagation of surface flaws subjected to cyclic loading. A complete listing of the program, a sample input and output are presented.

A plate containing a single surface flaw is analyzed at several stages of crack growth, computing stress-intensity factors with the proposed functional technique. The crack growth rates are derived from experimental measurements made by other workers using the incremental polynomial method presently recommended by ASTM for through-thickness cracks. The crack growth rates are correlated with the computed cyclic stress-intensity factors through a regression analysis to obtain the material parameters that characterize the cyclic crack growth properties of 304 stainless steel under the tested conditions for the Paris model and the Walker model. These models give similar results for this problem, so all subsequent work is done using the Paris model only. A fairly constant value was obtained for the exponent of the Paris model along the crack front. A large variation was obtained for the coefficient of the Paris model along the crack front. Two new techniques to simulate a fairly constant value for the coefficient of the Paris model are presented and discussed.

After obtaining a fit for these material parameters, propagation of two interacting surface flaws in a plate subjected to cyclic loading is studied. The breakthrough life of the plate is predicted with very good agreement compared to experimental results when finite element analyses are used to evaluate stress-intensity factors. Use of the ASME Code to predict fatigue growth of the interacting cracks (including the Code's charts to evaluate stress-intensity factors) led to extremely Conservative results for the case of loading in direct tension.

It was concluded in the experimental work that interacting multiple surface flaws in a plate subjected to cyclic loading accelerate the crack growth, leading to earlier breakthrough of the plate than in the case of a single surface flaw. The good agreement between the numerical result of the present work with the experimental result shows that through efficient three-dimensional finite element analysis a good pre-diction of breakthrough life of a plate with interacting surface flaws subjected to cyclic loading can be obtained.

TOPOLOGICAL AND GEOMETRICAL MODELING APPROACH TO NUMERICAL DISCRETIZATION AND ARBITRARY FRACTURE SIMULATION IN THREE-DIMENSIONS

A Dissertation presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

by Luiz Fernando Campos Ramos Martha August 1989

This thesis proposes and implements modeling features which enable practical, arbitrary, and realistic three-dimensional fracture simulation in solid structures. The scenario is the development of a software system which is specifically designed to simulate problems with evolutionary geometry. The system takes full advantage of the latest advances in computer hardware, data structures, numerical analysis, interactive graphics, and scientific visualization. The components of the system addressed in this thesis are the internal data representation, numerical discretization, and the development of a new boundary element numerical technique for fracture analysis which avoids multi-domain decomposition.

The system relies on a sophisticated topology-based data structure with explicit model geometry representation and on a common user interface. simulation is performed using solid modeling concepts with the true geometric representation. Both boundary and finite element models are represented in a consistent fashion, allowing for an easy coupling of these techniques, although this is not implemented. Semi-automatic, user-driven meshing is based on a technique of progressive hierarchical refinement, which relies on the underlying topological representation to implement both mapped-meshing schemes and general, arbitrary triangulation algorithms. Because loads, support conditions, material properties, and other physical attributes are attached to geometric and topological entities, they need not be reassigned during meshing and remeshing and during the course of a fracture simulation. In addition, the topology-based data structure supports accurate prediction and tracking of geometry changes. This provides the ability to specify flaws of arbitrary shape (including non-planar flaws), size, and orientation at arbitrary locations in the geometric model. The flaw is specified, using interactive graphics techniques, at the desired location in the actual geometric representation, rather than at a location in the mesh. Automatic local remeshing is used to simulate flaw growth. Any change is fast, with minimum user interaction, and affects only locally the existing mesh.

AN INTEGRATED BOUNDARY ELEMENT ANALYSIS SYSTEM WITH INTERACTIVE COMPUTER GRAPHICS FOR THREE-DIMENSIONAL LINEAR-ELASTIC FRACTURE MECHANICS

Thesis Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

by Renato Perucchio January 1984

The objective of the present research is the creation of a numerical tool for the solution of general problems in three-dimensional linear-elastic fracture mechanics. This is achieved with the integration of boundary element analysis and interactive computer graphic techniques.

A library of quadratic, isoparametric surface elements is implemented into a linear-elastic boundary element analysis code designed for virtual memory mini-computer operations. The program incorporates a multidomain analysis capability designed to allow the treatment of domains built up as a sequence of adjacent regions, each having different material properties.

An interactive computer graphic system is developed for generating and editing three-dimensional meshes for boundary-integral element analysis. The surf ace generation algorithm used for geometric modelling is based on the assumption that the three-dimensional region can be subdivided in a series of contiguous subdomains, which, in turn, can be separately described by a set of planar cross sections. Discrete transfinite mapping and cubic spline blending algorithms are used, respectively, to create plane cross-sectional meshes, and to generate the three-dimensional geometry by interpolating between the cross sections. A digitizing tablet and a refresh vector scope provide the communication between analyst and machine.

In the application of the boundary element analysis code to linear-elastic fracture mechanics, two issues are addressed: crack surface modeling and singularity representation. The multidomain option is used to separate numerically the two crack surfaces by including the crack plane within the interface of two adjoining subdomains. Quadratic, isoparametric, elements are made to embody the correct crack tip displacement variation by repositioning side nodes at the quarter-point. The inclusion of the correct singular traction term is accomplished by multiplying the traction shape functions by an "ad hoc" shape function.

The combined strength of boundary element analysis and interactive preprocessing is demonstrated by the application of the integrated system to a variety of three-dimensional fracture mechanics problems. The geometrical modeling of solid domains containing cracks is easily accommodated within the structure of the preprocessor and the determination of stress-intensity factors is performed accurately and efficiently by the boundary element code.

DISCRETE MODELING OF CRACK PROPAGATION: THEORETICAL ASPECTS AND IMPLEMENTATION ISSUES IN TWO AND THREE DIMENSIONS

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Paul Andrew Wawrzynek August 1991

Abstract

This thesis is a detailed investigation of the numerical simulation of discrete crack propagation for two-and three-dimensional problems. Discrete modeling of crack propagation deals with the analysis of fracture processes in which the geometries of individual cracks are considered explicitly as integral parts of the analysis. Because of the difficulty of obtaining solutions, this can be done analytically for only a few simplified cases, usually involving symmetry conditions. Most realistic problems are much more complex, and a numerical simulation is required to obtain a solution.

This thesis presents an approach to the simulation of arbitrary crack growth that relies on the integration of geometrical modeling, numerical stress analysis, and principles of fracture mechanics. Computational topology is used as an agent to integrate these, and to support the interactive and incremental nature of these simulations. Specific topics addressed include: the evaluation of crack-tip parameters, -crack stability, crack propagation, initial stresses, interfacial cracks, automatic meshing, computational topology, and computer graphics for computational mechanics.

The approach presented here has been implemented in programs for two-and three-dimensional crack propagation analyses. Numerous verification and example problems are presented that illustrate the effectiveness of this approach for a broad range of practical engineering problems. The three-dimensional program is particularly significant because it represents the first practical capability for performing realistic (i.e., planar and non-planar) three-dimensional discrete crack propagation simulations.

NUMERICAL METHODS FOR HYPERSINGULAR AND LINEAR-SINGULAR BOUNDARY INTEGRALS IN FRACTURE MECHANICS

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Earlin David Lutz, Ph.D. Cornell University 1991

The boundary integral equation is one of several equivalent forms of governing equations that can be used to compute approximate solutions to boundary value problems in elasticity and potential flow analvsis. Since it determines the entire solution in terms of values only on the boundary. there are possible order of-magnitude advantages in solution time and geometric complexity over better known domain-based methods such as finite elements and finite differences.

In practice, it has been hard to capitalize on these advantages. Many of the difficulties center around inability to perform certain numerical integrations. This thesis presents (a) a systematic 'modal' method of converting singular integrals to easier integrals over 'far' surfaces (b) an optimal quadrature method for the 'nearly singular' integration problem. An existence proof is given to show that all surface integrals arising from the 3D boundary integral can be converted to easier contour integrals if basis functions are constructed in a Cartesian sense, rather than the common parametric formulations. Stokes vectors needed to make this result useful are demonstrated for the Laplace equation and for some cases of elasticity.

Comparison to analytic benchmark cases shows that the method produces ac- curate stress intensity factors for 3-dimensional fracture analysis.

MODELING MIXED-MODE DYNAMIC CRACK PROPAGATION USING FINITE ELEMENTS

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Daniel Vernon Swenson, Ph.D. Cornell University 1986

This thesis describes a consistent approach to modeling mixed mode dynamic crack propagation using the finite element method. Applying linear elasto-dynamic fracture mechanics concepts, discrete cracks are allowed to propagate through the mesh in arbitrary directions.

The solution scheme is implemented with explicit time integration and quadratic triangular elements. To preserve uniform element size, local remeshing is automatically performed when the crack tip elements become excessively distorted. The use of interactive computer graphics makes the program a "finite element processor" that unifies the traditionally separate tasks of preprocessing, analysis, and post-processing. The insight obtained from on-line viewing of results greatly aids in understanding the physical behavior in an analysis.

The asymptotic solutions for displacement, velocity, and stress around a steady state moving crack are derived. These solutions are used to calculate stress intensities by displacement correlation with the finite element results. Given the current state at the crack tip and a KID(V) relation, the crack velocity is updated by assuming the change in velocity is proportional to the difference between the current stress intensity and the critical value. The maximum circumferential stress criterion is used to predict crack propagation direction.

Verification problems include an infinite body problem for which analytic solutions can be obtained using Freund's method. The rapid change in stress intensity due to an instantaneous change in velocity is captured by the model.

Applications include analyses of double cantilever beam experiments and compact tension experiments. These calculations demonstrate the difficulty of using a combined experiment/analysis approach to generate a KID(V) curve. Relatively minor changes in the specified crack velocity result in significantly different KID(V) curves. Therefore, extreme caution should be exercised when using this type of generated data.

Biaxial loading and stress wave interaction with a moving crack are two problems used to examine curving cracks. The use of a maximum circumferential stress criterion for finding the crack propagation direction and the use of a KID(V) curve for critical stress intensity give good correlation with experimental data.

CRACK GROWTH SIMULATION AND RESIDUAL STRENGTH PREDICTION IN THIN SHELL STRUCTURES

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Chuin-Shan Chen January 1999

Abstract

The dissertation mainly deals with self-similar and non-self-similar crack growth simulations in thin-shell metallic structures. An analysis methodology and a software program for predicting the structural integrity and residual strength of pressurized, thin-shell, built-up structures are developed.

The first part of the dissertation discusses the crack tip opening angle (CTOA) fracture criterion obtained and correlated from coupon tests to predict fracture behavior and residual strength of built-up aircraft fuselages. Geometrically nonlinear, elastic-plastic, thin shell finite element analyses are used to simulate stable crack growth and to predict residual strength. Both measured and predicted results of laboratory flat panel tests and full-scale fuselage panel tests show substantial reduction of residual strength due to the occurrence of multi-site damage (MSD). Detailed comparisons of stress distributions, stable crack growth history, and residual strength between the predicted and experimental results are used to assess the feasibility and validity of the analysis methodology.

The second part of the dissertation discusses issues related to crack trajectory prediction in thin shells; an evolving methodology uses the crack turning phenomenon to improve the structural integrity of aircraft structures. A directional criterion is developed based on the maximum tangential stress theory, but taking into account the effect of T-Stress and fracture toughness orthotropy. Possible extensions of the current crack growth directional criterion to handle geometrically and materially nonlinear problems are discussed. The path independent contour integral method for T-Stress evaluation is derived and its accuracy is assessed using a p- and hp-version adaptive finite element method. Curvilinear crack growth is simulated in coupon tests and in full-scale fuselage panel tests. Both T-Stress and fracture toughness orthotropy are found to be essential to predict the observed crack paths.

The analysis methodology and software program developed herein will allow engineers to maintain aging aircraft economically while insuring continuous airworthiness. Consequently, it will improve the technology to support the safe operation of the current aircraft fleet as well as the design of more damage-tolerant aircraft for the next-generation fleet.

VIRTUAL CRACK EXTENSION METHOD FOR CALCULATING RATES OF ENERGY RELEASE RATE AND NUMERICAL SIMULATION OF CRACK GROWTH IN TWO AND THREE DIMENSIONS

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Changyu Hwang May 1999

Abstract

This thesis develops an analytical virtual crack extension method for calculating rates of energy release rate and provides a numerical procedure for simulating a growth of multiple crack systems in two and three dimensions.

First, the thesis generalizes the analytical virtual crack extension method presented by Lin and Abel by providing the higher order derivatives of energy release rate due to crack extension for multiply cracked bodies in two and three dimensions. It provides derivations and verifications of the following: extension to the general case of multiple crack systems in two and three dimensions, extension to axisymmetric case, inclusion of crack-face and thermal loading, and evaluation of the second derivative of energy release rate. The salient feature of this method is that the energy release rate and its higher order derivatives for multiple crack systems are computed in a single analysis. Maximum errors for the mesh density used in the examples are about 0.2 % for energy release rate, 2-4 % for its first derivatives, and 5-10 % for its second derivative.

Second, this thesis proposes crack-growth model and numerical procedure for simulation of a growth of planar cracks in two and three dimensions, using the first derivative of the energy release rate provided by the present virtual crack extension method. The model is based on the concept of maximizing the total energy released as a crack propagates, which results in the problem of constrained optimization. The main advantages of this approach are threefold: (a) the present approach provides crucial information about the stability of a propagating crack; (b) the interaction between crack extensions at different points along the crack front is considered in the shape prediction; (c) the energy release rates and their derivatives at all points along the crack front can be accurately calculated by the present virtual crack extension method in a single analysis.

Third, this thesis provides an approximate numerical procedure for simulating a growth of non-straight cracks in two dimensions. In the approach, the potential energy variation during the next kink extension is approximated as a quadratic polynomial function of the kink extension in the preferred direction of propagation. The energy release rate and its derivative variations, G(l) and , during the kink extension are approximated as linear and constant functions of the kink extension, by using the energy release rate and its derivative at the half-way point of the next kink extension range, respectively. The present approach provides an excellent quadratic polynomial approximation for potential energy variation during various kink extension ranges, with differences of less than 1 % from the actual variation of potential energy obtained by finite element analysis. This research demonstrates that, through numerical simulation of inclined central cracks subjected to wedge force on the crack surface, the present approach can predict a reasonable crack-growth pattern and stability consistent with predictions made under this study.

INTERFACE MODELING OF COMPOSITE MATERIAL DEGRADATION

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Tong-Seok Han, Ph.D., Cornell University, May 2001

Abstract

In this dissertation, feasibility studies are performed on applying interface elements with cohesive constitutive models (cohesive element models) to composite structural members to simulate material degradation. Two types of composite structural members are investigated, a honeycomb composite panel, and a reinforced concrete frame building.

The cohesive element approach is proposed as a tool for simulating delamination propagation between a face sheet and a core in a honeycomb core composite panel. To determine the critical energy release rate (Gc) of the cohesive model, Double Cantilever Beam (DCB) tests were performed. The DCB coupon test is simulated using the measured fracture parameters, and sensitivity studies to the parameters for the cohesive model are performed. The cohesive model determined from DCB tests is then applied to a full-scale, 36x36 in. debond panel subjected to edge compression loading, and results are compared with an experiment. It is concluded that the cohesive element approach can predict the delamination propagation of a honeycomb panel with reasonable accuracy.

Seismic analyses of reinforced concrete frame structures are performed using the finite element method to propose a simulation modeling strategy which is efficient, accurate, and relatively simple in developing fragility curves for structural performance. Shake table tests of a lightly reinforced concrete three story frame buildings are simulated. The effects of modeling boundary conditions, and of considering the initial micro-cracking of concrete on changes in natural frequency are investigated. These parameters are used to calibrate finite element models to experimental results. For efficient analysis, an approach, wherein nonlinear material response is concentrated into possible failure sections while other portions of the simulation model are kept elastic, is proposed. The method is applied to a frame structure by inserting cohesive elements at failure sections, and by calibrating the constitutive model for the interface elements. It is found that seismic analyses of the calibrated models predict the experimental results of two different types of shake table test with reasonable accuracy. It is concluded that with further calibration such an interface model will be able to predict the seismic behavior of reinforced concrete frame structures accurately and efficiently to develop fragility information.

MODELING AND SIMULATION OF FATIGUE CRACK GROWTH IN METALS USING LEFM AND A DAMAGE-BASED COHESIVE MODEL

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Ani Ural, Ph.D., Cornell University, May 2004

Abstract

Failure of structures due to cyclic loading is an important concern for designers. Fracture mechanics has been widely used in the prediction and assessment of crack growth due to fatigue over the last few decades. In this thesis, two different approaches using fracture mechanics concepts are developed for predicting fatigue crack growth in metals.

Part I of this thesis summarizes new results for predicting crack shape and fatigue life for a spiral bevel pinion gear using computational fracture mechanics. The predictions are based on linear elastic fracture mechanics theories combined with the finite element method, and incorporating plasticity-induced fatigue crack closure and moving loads. It is shown that arbitrarily shaped fatigue crack growth in a spiral bevel gear can be simulated more efficiently and with much higher resolution than with a previous boundary-element-based approach using the finite element method along with a better representation of moving loads. Another very significant improvement is the decrease in solution time of the problem by employing a parallel PC-cluster, an approach that is becoming more common in both research and practice. This reduces the computation time for a complete simulation from days to a few hours. Finally, the effect of change in the flexibility of the cracking tooth on the location and magnitude of the contact loads and also on stress intensity factors and fatigue life is investigated.

DECOHESION OF GRAIN BOUNDARIES IN THREE-DIMENSIONAL STATISTICAL REPRESENTATIONS OF ALUMINUM POLYCRYSTALS

A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by

Erin Iesulauro, Ph.D., Cornell University, August 2006

Abstract

Since the 1950's, researchers have studied fatigue crack propagation utilizing fracture mechanics. Such work has provided advances in calculating stress inten- sity factors, determining elastic-plastic crack tip parameters, and investigating the e®ects of crack closure. Predictions of fatigue life have been made using crack growth rate models. Over the years, this work has served to in°uence structural maintenance and damage tolerance philosophies; however, understanding, predict- ing, and simulating fatigue crack growth is still based on experimental curve ¯tting and phenomenological rate \laws." The work discussed in this thesis is a step toward understanding fatigue crack incubation, nucleation and microstructurally small crack growth from a ¯rst prin- ciples approach. To this end, capabilities have been created and assembled to generate, mesh, analyze, and post-process 3D statistical representations of metal- lic polycrystals with cohesive grain boundaries. A component-based framework facilitates °exibility, growth, and multiscale modeling. Components are accessed and connected through Web service interfaces. The Polycrystal Generator accesses the components for generating, meshing, and assigning properties and boundary conditions to a 3D polycrystal sample. It also provides an interface to a molecular dynamics component to facilitate loosely coupled multi-scale analyses. Analyses are conducted utilizing a parallel solution software package, PETSc, and in-house ¯nite element library, FemLib. The large samples and resulting data is managed using Microsoft SQL Server 2000, an o®-the-shelf relation database. Finally, sam- ple geometries, mesh models, and results are visualized using PView, a real-time visualization tool created using OpenDX, Python, and SQL. The assembled framework is used to conduct a parametric study of 3D statisti- cal polycrystals under monotonic loading. The samples are analyzed with variation introduced in geometry, grain constitutive model and parameter values, cohesive grain boundary parameter values, and boundary conditions. This parametric study gives insight into how each variation in°uences when and where cracks nucleate. Finally, the results from the parametric study are utilized to conduct simula- tions under cyclic loading. These analyses give insight into the ability to accurately capture grain boundary decohesion leading to fatigue crack nucleation.

Souma

Gerstle

Pettit

Hanson

Riddell