Research

RESEARCH STATEMENT

Professor Kaw’s main scholarly interests are in engineering education research, flipped learning, adaptive learning, open courseware development, bascule bridge design, fracture mechanics, composite materials, computational nanomechanics, and the state and future of higher education.

Professor Kaw’s research has been funded by National Science Foundation, Air Force Office of Scientific Research, Florida Department of Transportation, Research and Development Laboratories, Wright Patterson Air Force Base, and Montgomery Tank Lines.

Since 2002, under Professor Kaw’s leadership, he and his colleagues from around the nation have developed, implemented, refined and assessed online resources for open courseware in Numerical Methods. This courseware annually receives more than a million page views, 1,800,000 views of the YouTube lectures and 100,000 annual visitors to the Numerical Methods Guy blog. Professor Kaw’s has written more than 100 refereed technical papers and his opinion editorials have appeared in the St. Petersburg Times, Chronicle Vitae, and Tampa Tribune.


Educational Research

Dr. Autar Kaw is widely recognized in the field of engineering education, particularly for his research on improving instructional methodologies, enhancing student understanding, and incorporating innovative tools to bridge theoretical concepts with practical applications. As a professor of mechanical engineering, Kaw’s work emphasizes improving student engagement and comprehension, especially within the areas of numerical methods and mathematics.

A central aspect of Kaw’s research is his focus on **numerical methods**, a foundational component of engineering education. Recognizing that students often struggle with the abstract nature of these methods, Kaw has developed comprehensive, user-friendly learning modules that integrate theory with real-world applications. His approach has emphasized the importance of contextualizing numerical methods, making them accessible and relevant to engineering students. By utilizing open educational resources (OER), he has reduced barriers to learning, making high-quality resources freely available to students worldwide.

Kaw’s **implementation of digital learning tools** is another significant contribution. Through platforms like the Numerical Methods for Engineers website, YouTube video tutorials, and digital textbooks, he has created a comprehensive suite of resources that support active learning. His digital resources not only cover theoretical content but also provide step-by-step guides, example problems, and interactive simulations, all aimed at reinforcing learning through practical engagement.

A strong advocate for **student-centered learning**, Kaw incorporates pedagogical methods that engage students as active participants in their education. His research findings indicate that students learn more effectively when they are actively involved in problem-solving, analysis, and critical thinking exercises. Kaw’s methods include flipped classroom models, problem-based learning, and collaborative assignments, each designed to foster deeper understanding and long-term retention of material.

Moreover, Kaw’s work has demonstrated the **positive impacts of using open resources** on educational outcomes. His commitment to open access and OER development reflects his belief that education should be inclusive and accessible. Studies conducted by Kaw and his colleagues reveal that students who use these open resources often perform better in assessments and show greater enthusiasm for the subject matter, likely due to the ease of access and the relevance of the materials.

In summary, Dr. Autar Kaw’s contributions to engineering education have had a transformative impact on how numerical methods are taught and learned. His commitment to enhancing instructional strategies through open resources, active learning, and digital tools has set a new standard in the field, influencing educators worldwide and improving educational outcomes for students. Kaw’s work underscores the importance of accessibility, engagement, and practical application in engineering education, making complex subjects more approachable and applicable for future engineers.


Composite Materials

Dr. Autar Kaw is a renowned researcher in the field of composite materials, with significant contributions to understanding and optimizing the mechanical behavior of these materials. His research primarily focuses on the design, analysis, and applications of composites—materials made from two or more constituent materials with distinct physical or chemical properties, which, when combined, produce a material with characteristics different from the individual components.

Dr. Kaw’s work in composite materials has addressed critical issues in structural integrity, particularly in fiber-reinforced composites. These materials, often used in aerospace, automotive, and civil engineering applications, are valued for their strength-to-weight ratio, stiffness, and resistance to corrosion. However, they pose unique challenges in terms of anisotropic behavior, damage tolerance, and degradation over time, which Kaw’s research has extensively examined.

One of the key areas of Dr. Kaw’s research is the study of mechanics of composite materials, where he has contributed to the development of analytical methods to predict how these materials respond under various loads. His work has advanced computational models that help in predicting stress, strain, and failure mechanisms in composites. These models are essential for engineers to design safe and efficient structures, as they enable accurate predictions of performance and lifespan.

In addition to his research, Dr. Kaw has contributed to the educational field, authoring textbooks and developing resources that make the study of composite materials accessible to students and professionals. His textbook *Mechanics of Composite Materials* is widely regarded as an essential resource in the field, offering detailed explanations of complex concepts and practical applications. This work has become a standard reference in engineering education, supporting the next generation of engineers and researchers in composite materials.


Bridge Design Research

Dr. Autar Kaw’s research in bascule bridge design has significantly advanced the understanding and optimization of these complex movable structures. Bascule bridges, often employed in waterways to allow the passage of both vehicle and maritime traffic, require precise design to balance structural integrity, operational efficiency, and cost-effectiveness. Dr. Kaw, a prominent figure in the field of engineering mechanics and materials science, has concentrated on addressing these specific challenges through both theoretical and applied methodologies.

One of the primary focuses of Dr. Kaw’s research has been the mechanics of bascule bridge movements, particularly the dynamic and static loads that impact bridge performance. In his studies, he has analyzed how varying loads affect bridge elements such as trunnions, counterweights, and girder systems. By applying advanced computational techniques, including finite element analysis (FEA), Kaw has been able to simulate load distributions and stress concentrations that occur in real-world scenarios. These simulations contribute to the optimal placement and sizing of bridge components, reducing material requirements while maintaining structural reliability.


Sports Analytics

Autar Kaw’s research in sports analytics notably includes his development of metrics to analyze sports outcomes, particularly in college football. His work investigates season unpredictability—what he calls “topsy-turvyness”—which quantifies the variance in game outcomes to understand trends, unpredictability, and overall season dynamics. This metric sheds light on how seasons progress under unexpected conditions, providing a unique perspective for understanding competitive balance and team consistency across games.


Grants

Transforming Undergraduate Engineering Education through Adaptive Learning and Student Data Analytics”, National Science Foundation, $599,770, July 2020 – June 2024 (PI).


Collaborative Proposal: Structured Use of Metacognitive Activities in a Flipped Undergraduate Engineering Course to Enhance Learning and Professional Skill Development”, National Science Foundation, $198,253, October 2020 – September 2024 (Co-PI).


Transforming a Flipped STEM Course Through Adaptive Learning“, National Science Foundation, $299,823, September 2016 – August 2020 (PI).


Improving and Assessing Student Learning in an Inverted STEM Classroom Setting“, National Science Foundation, $598,522, September 2013-August 2016 (PI).


Collaborative Research: Development of New Prototype Tools, and Adaptation and Implementation of Current Resources for a Course in Numerical Methods, National Science Foundation, $150,000, February 2009- July 2011 (Co-PI).


Holistic Numerical Methods: Unabridged“, National Science Foundation, $500,189, January 2008-December 2012 (PI)


Holistic Numerical Methods“, National Science Foundation, $363,280, February 2004-March 2007 (PI).


“Hub Assemblies for the Next Generation of Bascule Bridges”, US Department of Transportation, $73,650, October 2004-September 2005 (Co-PI).


Holistic Numerical Methods: A Prototype“, National Science Foundation, $74,961, January 2002- May 2003 (PI).


“Hub-Girder Bolt Assembly Without An Interference Fit in Bascule Bridges”, Florida Department of Transportation, $110,000, November 2001 -August 2003 (Co-PI).


Parametric Finite Element Analysis and Full Scale Testing of Trunnion Hub Assemblies for Bascule Bridges, FDOT, $250,000, September 1998 – June 2001 (Co-PI).


Acquisition of an Axial-Torsional Loading System for Civil, Mechanical and Materials Testing“, National Science Foundation, $166,240, October 1999 – April 2001 (Co-PI).


Mechanics of Brittle Matrix Composites with Imperfect Interfaces“, AFOSR, $108,319, July 1995- August 1997 (PI).


Development of User Interfaces for Numerical Techniques for Stress Analysis of Laminated Composite Materials”, AFOSR, $50,000, September 1994 – June 1995 (PI).


“Micromechanics of Matrix Cracking in Brittle Matrix Composites with Frictional Interfaces,” Summer Research Extension Program, RDL/AFOSR, $20,000, January 1993 – December 1993 (PI).


“Mechanics of Brittle Matrix Composites with Imperfect Interfaces”, AFOSR, $93,211, September 1992 – September 1994 (PI).


“Annular Cracks In Brittle Matrix Composites,” Systran Corporation/WPAFB, Dayton, Ohio, $14,100, May 91 – August 1991 (PI).


“Design and Analysis of Stainless Steel Tanks,” Montgomery Tank Lines, Plant City, Florida, $27,653, October 89 – May 90 (Co-PI).


Journal Papers
A. Kaw, G. Lim, R. Clark, “Reflective Questions: Promoting Metacognition Through Discussion Board Prompts,” Journal of Higher Education Theory and Practice, Vol. 24 (8), 2024, Article 7238. https://articlegateway.com/index.php/JHETP/article/view/7238

A. Kaw, A. Yalcin, R.M. Clark, R.B. Gomes, L. Serrano, A. Scott, Y. Lou, “On Building and Implementing Adaptive Learning Platform Lessons for Pre-Class Learning in a Flipped Course,” ASEE Computers in Education, 2024, in print.


R. Clark, R. Guldiken, A. Kaw, O. Uynayak, “The case for Metacognition Support in a Flipped STEM Course,” International Journal of Mechanical Engineering Education, Vol 51(4), 2024,  pp. 243-269.  https://doi.org/10.1177/03064190231164719


A. Kaw, R. Clark, “Effects of Standards-Based Testing via Multiple-Chance Testing on Cognitive and Affective Outcomes in an Engineering Course,” International Journal of Engineering Education, Vol. 40 (2), 2024, pp. 303-321. https://www.ijee.ie/1atestissues/Vol40-2/09_ijee4434.pdf


A. Kaw, “Effect of Student Choice of Weights on Course Grade,” Transactions on Techniques in STEM Education, Vol. 9 (1), 2023, pp. 36-44.

Clark RM, Kaw A, Guldiken R. Metacognition instruction and repeated reflection in a fluid mechanics course: Reflective themes and student outcomes. International Journal of Mechanical Engineering Education. 2023;51(4):243-269. doi:10.1177/03064190231164719

A. Yalcin, A. Kaw, and R. Clark, On learning platform metrics as markers for student success in a course, Comput. Appl. Eng. Educ. 2023; 31: 1412–1432. https://doi.org/10.1002/cae.22653 

R.Clark, A. Kaw, Y. Lou, A, Scott, “The flipped classroom during the remote period of COVID: student perceptions compared to pre-COVID times”, International Journal of Mathematical Education in Science and Technology, (2022), 1-22. Link to Full Text (limited by subscription)


R. Clark., A. Kaw, and R. Braga Gomes, “Adaptive Learning: Helpful To The Flipped Classroom In The Online Environment Of COVID?”, Computer Applications in Engineering Education (2021), 1– 15. Link to Full Text (limited by subscription)


R. Clark, A. Kaw, “Enhancing Student Outcomes in a Blended Numerical Methods Course for Engineers: The Case for Practice and Cumulative Tests”, International Journal of Engineering Education Vol. 37, No. 3, pp. 585–593, 2021.  Link to Full Text (limited by subscription)


R. Clark, A. Kaw, “Adaptive learning in a numerical methods course for engineers: Evaluation in blended and flipped classrooms”, Computer Applications in Engineering Education, pp. 62-79, Vol. 28(1), 2020.  Link to Full Text (limited by subscription)


R. Clark, A. Kaw, “Benefits of adaptive lessons for pre-class preparation in a flipped numerical methods course”, International Journal of Mathematical Education in Science and Technology, pp. 713-729, Vol 51(5), 2020.  Link to Full Text


A. Kaw, R. Clark, E. Delgado, N. Abate, “Analyzing the Use of Adaptive Learning in a Flipped Classroom for Pre-Class Learning”, Computer Applications in Engineering Educations, Vol. 27, pp. 663‐ 678, 2019. Link to Full Text (limited by subscription)


R. Clark, A. Kaw, Y. Lou, A. Scott, M. Besterfield-Sacre “Evaluating Blended and Flipped Instruction in Numerical Methods at Multiple Engineering Schools,” International Journal for the Scholarship of Teaching and Learning: Vol. 12: No. 1, Article 11, 2018. Link to Full Text


J. Tai, A. Kaw, “Transverse Shear Modulus of Unidirectional Composites With Voids Estimated by the Multiple-Cells Model”, Composites Part A: Applied Science and Manufacturing, pp.310-320, Vol 105, 2018.



R. Clark, A. Kaw, M. Besterfield-Sacre, “Comparing the Effectiveness of Blended, Semi-Flipped, and Flipped Formats in an Engineering Numerical Methods Course”, ASEE Advances in Engineering Education, pp. 1-38, Vol. 5, 2016. Link to Full Text


G. Aden-Buie, A. Kaw, A. Yalcin, “Comparison of final examination formats in a numerical methods course”, International Journal of Engineering Education, pp. 72-82, Vol. 31, 2015.


S.H. Garapati, A.Kaw, “Analysis of Heating and Cooling Methods for Assembly of Steel Fulcra in Bascule Bridges”, Bridge Structures, pp. 121-133, Vol. 8, 2012.


A. Kaw, A. Yalcin, D. Nguyen, R. Pendyala, M. Hess, G. Lee-Thomas, G. Besterfield, J. Eison, C. Owens, “A Holistic View on History, Development, Assessment, and Future of an Open Courseware in Numerical Methods”, ASEE Computers in Education Journal, Vol 3(4), pp. 57-71, 2012.


C. Owens, A. Kaw, M. Hess, “Assessing Online Resources for an Engineering Course”, Computer Applications in Engineering Education, pp. 426-433, Vol. 20(3), 2012.


A. Kaw, A. Yalcin, “Measuring Student Learning Using Initial and Final Concept Test in a STEM Course”, International Journal of Mathematical Education in Science and Technology, pp. 435-448, Vol 43(4), 2012.


A. Yalcin, A. Kaw, “Does Grading Homework Improve Student Examination Performance?”, International Journal of Engineering Education, pp. 1333-1342, Vol 27(6), 2011.


S. Garapati, L. Snyder, A. Kaw, “Comparing Two Procedures for Assembling Steel Fulcra in Simple-Trunnion Bascule Bridges”, Bridge Structures, pp. 19-30, Vol 7(1), 2011.


S. Garapati, A. Kaw, “Effect of Geometry, Loading and Elastic Moduli on Critical Parameters in a Nanoindentation Test in Polymeric Matrix Composites with a Nonhomogeneous Interphase”, Composite Interfaces, pp. 275-294, Vol. 18, 2011.


A. Kaw, S. Garapati, “Development and Assessment of Digital Audiovisual YouTube Lectures for an Engineering Course In Numerical Methods”, ASEE Computers in Education Journal, pp. 89-97, Vol. 2 (2), 2011.


E. Gil-Herrera, A. Tsalatsanis, A. Yalcin, A. Kaw, “Predicting Academic Performance Using a Rough Set Theory Based Knowledge Discovery Methodology”, International Journal of Engineering Education, pp. 992-1002, Vol. 27 (5), 2011.


M. Rajapakshe, M. Gunaratne, A. Kaw, “Evaluation of LuGre Tire Friction Model with Measured Data on Multiple Pavement Surfaces,” Tire Science and Technology, TSTCA, Vol. 38, No. 3, pp. 213-227, 2010.


A. Kaw, A. Yalcin, “A Metric to Quantify the Topsy-Turvyness of a College Football Season”, Chance (An American Statistical Association Magazine), pp. 18-26, Vol. 22(3), 2009. (More info)


A. Kaw, A. Yalcin, B. DeMenezes, E. Allard, “Introducing and Assessing Laboratory Experience in a Numerical Methods Course for Engineers”, ASEE Computers in Education Journal, pp. 57-65, Vol. XVIII (3), 2009.


A.K. Kaw, A.Yalcin, “Problem-Centered Approach in a Course in Numerical Methods”, ASCE Journal of Professional Issues and Engineering Education, pp. 359-364, Vol. 134(4), 2008.


C. Nguyen, A.K. Kaw, J. Paul, “Sensitivity Analysis of Cooling Methods and Geometric Parameters in Assembly Procedure of Bascule Bridge Fulcrums”, The Journal of Strain Analysis for Engineering Design, pp.  337-349, Vol. 42, (2007).


P. Chalasani, A.K. Kaw, J. Daly, C. Nguyen “Effect of Geometrical and Material Parameters in Nanoindentation of Layered Materials with an Interphase”, International Journal of Solids and Structures, pp. 5380-5395, Vol. 44 (16), (2007).


A.K. Kaw, M. Hess, “Comparing Effectiveness of Instructional Delivery Modalities in an Engineering Course”, International Journal of Engineering Education, pp. 508-516, Vol. 23(3), (2007).


A.K. Kaw and S. Ho, “Introducing Approximate Methods in Theory of Elasticity”,  Computer Applications for Engineering Education, pp. 120-134, Vol. 14 (2), (2006).


N. Collier, A.K. Kaw, G.H. Besterfield, M. Rahman, “Benefits of Staged Cooling of Composite Cylinders in Shrink Fitting”, The Journal of Strain Analysis for Engineering Design, pp. 349-361, Vol. 41 (5), (2006).


A.K. Kaw, G.H. Besterfield, J. Eison, “Effectiveness of a Holistic & Customized Approach to Teaching Numerical Methods”, International Journal of Engineering Education, pp. 712-722, Vol. 21 (4), 2005.


A.K. Kaw, “Techniques Employed by Highly Effective Engineering Educators”,  ASCE Journal of Professional Issues and Engineering Education, pp. 175-177, Vol. 131 (3), (2005).


A.K. Kaw, G.H. Besterfield and S. Nichani, “Integrating a Research Problem in a Course in Applied Elasticity”, International Journal of Mechanical Engineering Education, pp. 232-242, Vol. 32 (3), (2004).


N. Collier, A.K. Kaw, “On Comparing Computational Systems – Maple, Mathcad, Matlab, and Mathematica”, ASEE Computers in Education Journal, pp. 12-24, Vol. XIV (1), (2004).


A.K. Kaw, N. Collier, M. Keteltas, J. Paul, G.H. Besterfield, “Holistic but customized resources for a course in numerical methods”, Computer Applications for Engineering Education, pp. 203-210, Vol. 11, (2003).


G.H. Besterfield, A.K. Kaw, S. Nichani, B. Ratnam, T. Cherukara, M. Denninger, “Assembly Procedures of a Trunnion-Hub-Girder for Bascule Bridges”,   Theoretical and Applied Fracture Mechanics, Vol 40, No 2, pp. 123-134, (2003).


G.H. Besterfield, S. Nichani, A.K. Kaw, T. Eason, “Full-Scale Testing of Trunnion-Hub-Girder Assemblies for Bascule Bridges”, ASCE Journal of Bridge Engineering, Vol. 8, No 4., pp. (2003).


N. Pai, A.K. Kaw and M. Weng, “Optimization of Laminate Stacking Sequence for Failure Load Maximization Using Tabu Search”, Composites Part B, Vol. 34, No. 1, pp 405-413, (2003).


J. Ye and A.K. Kaw, “Determination of Mechanical Properties of Fiber-Matrix Interface from Pushout Test”, Theoretical and Applied Fracture Mechanics, Vol. 32, No 1, pp 15-25, (1999).


A.K. Kaw and R. Rodriguez, “Synthesis of Tools in Teaching a Course in Mechanics of Composite Materials”, Mechanics, Vol. 27, No. 11, pp. 5-12, (1998).


A.K. Kaw and G. Willenbring, “A Software Tool for Mechanics of Composite Materials”, International Journal of Engineering Education, Vol. 13, No. 6, pp. 433-481, (1998).


T. Srinath, H. Madanraj, A.K. Kaw, J. Ye and G.H. Besterfield, “Effect of Extrinsic and Intrinsic Factors in Indentation Tests,” International Journal of Solids and Structures, Vol. 33, No. 24, pp. 3497-3516 (1996).


A.K. Kaw and M. Daniels, “Should We Continue to Debate over the Change to SI System?,” ASCE Journal of Professional Issues in Engineering Education and Practice, Vol. 122, No. 2, pp. 69-72 (1996).


N.J. Pagano and A.K. Kaw, “Asymptotic Stresses Around a Crack Tip at the Interface Between Planar or Cylindrical Bodies,”International Journal of Fracture, Vol. 71, No. 2, pp. 151-164, (1995).


A.K. Kaw, S. Kunchithpatham, N.J. Pagano, “Stress Field in a Composite Cylinder with a Friction Fiber-Matrix Interface,” International Journal of Solids and Structures, Vol. 32, No. 15, pp. 2127-2154, (1995).


V.T. Bechel and A.K. Kaw, “Fracture Mechanics of Composites with Nonhomogeneous Interphases and Nondilute Fiber Volume Fractions,” International Journal of Solids and Structures, Vol. 31, No. 15, pp. 2053-2070, (1994).


A.K. Kaw and J. Ye, “Comparison of Fracture Mechanics Models with Axisymmetric and Planar Assumptions,” Composites Engineering Journal, Vol. 4, No. 6, pp. 621-636, (1994).


A.K. Kaw and D. Jadhav, “Axisymmetric Elastic Response of a Composite Cylinder with a Broken Fiber”, Theoretical and Applied Fracture Mechanics, Vol. 21, pp. 197-206, (1994).


V. Bechel and A.K. Kaw, “On Using a Symbolic Manipulator in Teaching Classical Elasticity,” ASEE Computers in Education Journal, Vol. 4, No. 2, pp. 55-59, (1994).


A.K. Kaw and N.J. Pagano, “Axisymmetric Thermoelastic Response of a Composite Cylinder Containing an Annular Crack”, Journal of Composite Materials, Vol. 23, No. 6, pp. 540-571, (1993).


A.K. Kaw and K.S. Gadi, “Mechanics of Fiber-Reinforced Composites with Doubly Periodic Cracks,” Theoretical and Applied Fracture Mechanics, Vol. 19, No. 3, pp. 173-182, (1993).


A.K. Kaw, D. Jadhav, G. Sudhakar, “On Solving Crack Problems in Fiber Reinforced Composites by Symbolic Manipulations,” Advances in Engineering Software and Workstations, Vol. 16, No. 1, pp. 31-36, (1993).


A.K. Kaw and G.H. Besterfield, “Periodic Matrix Cracking in Brittle Matrix Composites,” International Journal of Solids and Structures, Vol. 29, No. 10, pp. 1193-1207, (1992).


A.K. Kaw, A. Selvarathinam and G.H. Besterfield, “Comparison of Interphase Models for a Crack in Fiber Reinforced Composites,” Theoretical and Applied Fracture Mechanics, Vol. 17, pp. 133-147, (1992).


A.K. Kaw, “Numerical Solutions of Strong Integrals in Hadamard’s Sense,” IMSL Directions, Vol. 9, No. 1, pp. 8-9, (1992).


A.K. Kaw, “An Instructional Interactive Aid for Mechanics of Composites,”  International Journal of Mechanical Engineering Education, Vol. 20, No. 3, pp. 213-229 (1992).


A.K. Kaw and G.H. Besterfield, “Effect of Interphases on Mechanical Behavior of Composites,” ASCE Journal of Engineering Mechanics, Vol. 117, No. 11, pp. 2641-2658, (1991).


A.K. Kaw and V.G. Das, “A Crack in an Imperfect Interface in Composites,”  Mechanics of Materials, Vol. 11, pp. 295-322, (1991).


A.K. Kaw and J.G. Goree, “The Semi-Infinite Strip Problem in Mechanics of Composites,” ASTM Journal of Composite Technology and Research, Vol. 13, No. 2, pp. 65-77, (1991).


A.K. Kaw, “Cauchy Singular Integral Equations with Logarithmic Singular Integrands in Solid Mechanics,” Mechanics Research Communications, Vol. 18, No.5, pp. 261-268, (1991).


A.K. Kaw, “On Evaluating Integrals with Strongly Singular Integrands,” Advances in Engineering Software and Workstations, Vol. 13, No. 2, pp. 84-103, (1991).


A.K. Kaw and J.G. Goree, “The Effect of Interleaves on Fracture of Laminated Composites: Part I – Analysis,” ASME Journal of Applied Mechanics, Vol. 57, No. 1, pp. 168-174, (1990).


A.K. Kaw and J.G. Goree, “The Effect of Interleaves on Fracture of Laminated Composites: Part II – Solution and Results,” ASME Journal of Applied Mechanics, Vol. 57, No. 1, pp. 175-181, (1990).


A.K. Kaw, “Some Simple Tools to Develop Educational Software,” The Quickbasic Journal, Vol. 1, No. 3, pp. 23-29, (1990).


A.K. Kaw and J.G. Goree, “Analysis of Notched Laminates with Interlaminar Debonding,” Journal of Engineering Fracture Mechanics, Vol. 22, No. 6, pp. 1013-1029, (1985).