《复合材料 Composites》课程教学资源（学习资料）第一章 复合材料基础_复合材料力学 Mechanics Research Communications

Mechanics Research Communications 41(2012)1-20 Contents lists available at SciVerse Science Direct MECHANICS Mechanics research communications ELSEVIER journalhomepagewww.elsevier.com/locate/mechrescom Review Mechanics of composites: A historical review CarI T. Herakovich University of virginia, USA ARTICLE INFO A BSTRACT Available online 21 January 2012 This review is concerned with mechanics of continuous fiber composites. The earliest and most important advancements in the field are emphasized. No doubt the coverage is limited to some extent by the interests and experiences of the writer as well as time and space considerations. The advancements in mechanics of composites have been influenced to a great extent by the development of advanced composites through materials science. No attempt is made to discuss these developments. This reviev emphasizes the use of theoretical and applied mechanics in the development of theories, confirmed by mentation, to predict the response of composite materials and structures. Citations have been given for many published works, but certainly not all Apologies to those not listed; numerous additional references can be found in the works cited o 2012 Elsevier Ltd. All rights reserved Contents 1. Since the beginning 2. The early year 3. Anisotropic, elastic constitutive equations 4. Micromechanics 4. 1. Micromechanics model comparisons 5. Lamination theory 2334668 6. Environmental effects 6.1. Thermal effects Moisture effects 7 8. Unsymmetric laminates 9. Tubes 10. Plate 0B44 11. Failure 11.2. failure criteria 13.1 556 13. 2. Compression 133. Shea 34. Off-axis tensile test 14. Nanocomposites 15. University and government program 16. Closure 17. Books on mechanics of composites 777778 knowledgement 3 This section was written by Mike Hyer, with a very few modifications provided by the author. 6413/s-see fi ro 2012 Elsevier Ltd. All rights reserved. 1016/.meche 201201.006

Mechanics Research Communications 41 (2012) 1–20 Contents lists available at SciVerse ScienceDirect Mechanics Research Communications jou rnal homepage: www.elsevier.com/locate/mechrescom Review Mechanics of composites: A historical review Carl T. Herakovich University of Virginia, USA a r t i c l e i n f o Available online 21 January 2012 a b s t r a c t This review is concerned with mechanics of continuous fiber composites. The earliest and mostimportant advancements in the field are emphasized. No doubt the coverage is limited to some extent by the interests and experiences of the writer as well as time and space considerations. The advancements in mechanics of composites have been influenced to a great extent by the development of advanced composites through materials science. No attempt is made to discuss these developments. This review emphasizes the use of theoretical and applied mechanics in the development of theories, confirmed by experimentation, to predict the response of composite materials and structures. Citations have been given for many published works, but certainly not all. Apologies to those not listed; numerous additional references can be found in the works cited. © 2012 Elsevier Ltd. All rights reserved. Contents 1. Since the beginning .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. The early years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Anisotropic, elastic constitutive equations .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Micromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.1. Micromechanics model comparisons .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5. Lamination theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6. Environmental effects .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6.1. Thermal effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6.2. Moisture effects .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 7. Interlaminar stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 8. Unsymmetric laminates3 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 9. Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 10. Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 11. Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 11.1. Unidirectional lamina. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 11.2. Quadratic failure criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 12. Damage mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 13. Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 13.1. Tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 13.2. Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 13.3. Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 13.4. Off-axis tensile test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 14. Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 15. University and government programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 16. Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17. Books on mechanics of composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 E-mail address: herak@virginia.edu 3 This section was written by Mike Hyer, with a very few modifications provided by the author. 0093-6413/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2012.01.006

CT. Herakovich/ Mechanics Research Communications 41(2012)1-20 "More important than any one new application is the new Egyptianscirca4000 B C. They laid up strips from the fibrous materials concept itself. It marks a shift from concern with papyrus plant in two layers with one layer at right angles to the substances to concern with structures, a shift from artisan to other In present day mechanics terminology, such a fabric would cientist as mans artificer, a shift from chemistry to physics as be called an unsymmetric, cross-ply laminate. While it is possible the basic discipline, and a shift, above all, from the concrete that the Egyptians used a symmetric laminate to eliminate curva- experience of the workshop to abstract mathematics, a shift ture this author has found no indication that such was the case from starting with what nature provides to what man wants The development of papyrus paper was so important to the egyp- to accomplish tians that they guarded the secret of how it was produced thereby Peter F Drucker, The Age of Discontinuity, 1969 creating a monopoly. Papyrus paper revolutionized the way peo- ple saved valuable information. As a result, it was ancient Egypt greatest export for many centuries. 1. Since the beginning Cuttings from the papyrus plant also were used in bundles by early Egyptians to make boats, sails, baskets and ropes. Fig. 1 shows A historical review of the mechanics of composites must first an image of a passage from the New Testament written on papyrus consider the question"what initiated the study of thes geneous, anisotropic materials"? Composite materials m around the beginning of the 3rd century, some 1800 years ago. It is referred to as the bodmer Papr can(2007) oresent since the beginning of time. Many objects such Fig. 2 is an example of present-day egyptian artwork on papyrus nd animals are fibrous composite systems. This is very trees and their leaves. in the wings of birds and the fins Another early, but totally different, application of a man-made human body is the most complex fibrous composite system. On one fibrous composite was the use of straw to strengthen bricks made level it consists of a musculoskeletal system of bones, muscles and from mud. According to the Book of Exodus ("do not give them tendons. On a microscopic level these objects themselves are com- straw for their bricks, make them find their own straw"), this prac- posite systems consisting of a variety of components that give ris tice was used as early as 1300 B.C.; it is still in use today. Fig. 3 is a to heterogeneous, anisotropic materials. picture of such a brick that was taken in the Middle East in the mid The first production and man-made use of a fibrous com- 20th cent ry Present-day mechanics would classify these bricks posite material appears to be the papyrus paper made by the randomly reinforced, short fiber composites According to Hartman et al. (1996), ancient egyptians also made containers of coarse fibers drawn from heat softened glass, and the French scientist Reaumur considered the potential of forming fine glass fibers as early as the 18th century. It was not until 1939 that "MLE NoN eEeTTA)EAT continuous glass fibers were produced commercially( Knox, 1982). These glass fibers were produced mainly for high temperature electrical applications. Two more decades passed before the crA AAUHTMAYT-YEITENAR called"advanced fibers were produced, boron(Talley, 1959)carbon NAC 的)Fx) TAINT 入 Fig. 1. New testament on papyrus. Fig. 2. Artwork on papyrus

2 C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 “More important than any one new application is the new ‘materials’ concept itself. It marks a shift from concern with substances to concern with structures, a shift from artisan to scientist as man’s artificer, a shift from chemistry to physics as the basic discipline, and a shift, above all, from the concrete experience of the workshop to abstract mathematics, a shift from starting with what nature provides to what man wants to accomplish”. Peter F. Drucker, The Age of Discontinuity, 1969 1. Since the beginning A historical review of the mechanics of composites must first consider the question “what initiated the study of these hetero￾geneous, anisotropic materials”? Composite materials have been present since the beginning of time. Many objects such as plants and animals are fibrous composite systems. This is very evident in trees and their leaves, in the wings of birds and the fins of fish. The human body is the most complex fibrous composite system. On one level it consists of a musculoskeletal system of bones, muscles and tendons. On a microscopic level these objects themselves are com￾posite systems consisting of a variety of components that give rise to heterogeneous, anisotropic materials. The first production and man-made use of a fibrous com￾posite material appears to be the papyrus paper made by the Fig. 1. New testament on papyrus. Egyptianscirca4000 B.C. They laid up strips from the fibrous papyrus plant in two layers with one layer at right angles to the other. In present day mechanics terminology, such a fabric would be called an unsymmetric, cross-ply laminate. While it is possible that the Egyptians used a symmetric laminate to eliminate curva￾ture, this author has found no indication that such was the case. The development of papyrus paper was so important to the Egyp￾tians that they guarded the secret of how it was produced, thereby creating a monopoly. Papyrus paper revolutionized the way peo￾ple saved valuable information. As a result, it was ancient Egypt’s greatest export for many centuries. Cuttings from the papyrus plant also were used in bundles by early Egyptians to make boats, sails, baskets and ropes. Fig. 1 shows an image of a passage from the New Testament written on papyrus around the beginning of the 3rd century, some 1800 years ago. It is referred to as the Bodmer Papyrus XIV-XV in the Vatican (2007). Fig. 2 is an example of present-day Egyptian artwork on papyrus paper. Another early, but totally different, application of a man-made fibrous composite was the use of straw to strengthen bricks made from mud. According to the Book of Exodus (“do not give them straw for their bricks, make them find their own straw”), this prac￾tice was used as early as1300 B.C.; it is still in use today. Fig. 3 is a picture of such a brick that was taken in the Middle East in the mid- 20th century. Present-day mechanics would classify these bricks as randomly reinforced, short fiber composites. According to Hartman et al. (1996), ancient Egyptians also made containers of coarse fibers drawn from heat softened glass, and the French scientist Reaumur considered the potential of forming fine glass fibers as early as the 18th century. It was not until 1939 that continuous glass fibers were produced commercially (Knox, 1982). These glass fibers were produced mainly for high temperature electrical applications. Two more decades passed before the “so￾called” advanced fibers were produced, boron (Talley, 1959) carbon Fig. 2. Artwork on papyrus

CT. Herakovich/Mechanics Research Communications 41(2012)1-20 Table Early activities, contributions and accomplishments in mechanics of composites. Papyrus paper developed pc Egypt 1780s Youngs modulus defined Great Britain general equations of elasticity aude-Louis Navier Anisotropic equations of elasticity ugustin-Louis Cauchy Strain energy density defined-21 elastic constant 887/1889 Treatise on the Mathematical Theory of Elasticity AEH. Love Great Britain Uniform stress modulus prediction A. Reuss Fiberglass fabrics available to market Owns-Corning and H. goldsmith Mathematical Theory of elasticity ISA opic Elastic Body G. Lekhnitshkii Russia Japan Society of Reinforced Plastics formed bricated glass reinforced plastic glider of Unidirectional lamina A. Kelly and G Davies Great Britain L Broutman and R.H. Kro Journal of Composite Materials Vol 1 No. 1 Composite Materials Workshop Tsai, Halpin and pagano USA The Analysis of Laminated Composite Structures ee calcite mer on Composite Materials: Analysis shton, Halpin and Petit Ashton and J.M. whitney eory of Anisotropic plates SA Ambartsumya Russia Mechanics of composite Materials R.M. Jones Mechanics of composite Materia R. M. Christensen USA (Soltes, 1961)and aramid( Kwolek, 1964 ). The development of the 2. The early years advanced fibers in the late 1950s and early 1960s spurred great interest in the development of theoretical and applied mechanics Table 1 summarizes the early activities, contributions and or applications to fibrous composite materials and structures accomplishments related to advances in the mechanics of fibrous rom theearliest applications of fibrous composites by the Egyp- composites. The remainder of this paper is organized according tians to the introduction of advanced composites in the second half to subject matter. Topics covered include constitutive equations, of the 20th Century, roughly 6000 years have passed. The progress micromechanics, laminates, thermal and moisture effects, damage in the use of fibrous composites in the most recent fifty years and failure, experimental methods, interlaminar stresses, tubes, and years. The Egyptians were artisans in that they undoubtedly This leaves many related subjects still to be reviewe t programs was much greater than that during the preceding nearly six thou- plates, nanocomposites, and university and governm developed their products through trial and error. during the past fifty years, theoretical and applied mechanics has been employed 3. Anisotropic, elastic constitutive equations in order to exploit the vast potential of man-made fibrous com- posites. These advancements are exemplified dramatically by the Discussions on the advances in the development of constitu- application of advanced fibrous composites in SpaceShipOne and its tive equations for elastic materials can be found in Love's work unchvehiclewhIteKnight(fiG.4,seehttp://www.scaled.com/).(1892-1927),Sokolnikoff(1946/1956)andTimoshenko'sHistoryof Space ShipOne is an all-composite, suborbital spaceplane launched Strength of Materials(1953). The development of constitutive equa tions for homogeneous, elastic materials began with the work of Hooke(1678)who stated that for an elastic body there is propor- tionality between stress and strain Navier(1821) generalized this Fig 3. Brick with straw fibers Fig 4. White knight and Space shipone

C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 3 Table 1 Early activities, contributions and accomplishments in mechanics of composites. Year Activity People Country 4000 BC Papyrus paper developed Egyptians Egypt 1660 Hooke’s Law Robert Hooke Great Britain 1780s Young’s modulus defined Thomas Young Great Britain 1821 Formulation of general equations of elasticity Claude-Louis Navier France 1822 Anisotropic equations of elasticity Augustin-Louis Cauchy France 1837 Strain energy density defined – 21 elastic constants George Green Great Britain 1887/1889 Uniform strain modulus prediction W. Voigt Germany 1892 Treatise on the Mathematical Theory of Elasticity A.E.H. Love Great Britain 1929 Uniform stress modulus prediction A. Reuss Germany 1935 Papers on anisotropic bodies S.G. Lekhnitshkii Russia 1938 Owens-Corning developed fiberglass Owens-Corning USA 1941 Air Force Materials Lab initiated composites activity Robert T. Schwartz USA 1941 Fiberglass fabrics available to market Owns-Corning and H. Goldsmith USA 1946 Mathematical Theory of Elasticity I.S. Sokolnikoff USA 1947 Anisotropic Plates S.G. Lekhnitshkii Russia 1950 Theory of Elasticity of an Anisotropic Elastic Body S.G. Lekhnitshkii Russia 1954 Japan Society of Reinforced Plastics formed Tsuyoshi Hayashi Japan 1954 Fabricated glass reinforced plastic glider Tsuyoshi Hayashi Japan 1961 Theory of Anisotropic Shells S.A. Ambartsumyan Russia 1965 Strength of Unidirectional Lamina A. Kelly and G.J. Davies Great Britain 1967 Modern Composite Materials L.J. Broutman and R.H. Krock USA 1967 Journal of Composite Materials Vol. 1 No. 1 Stephen W. Tsai USA 1968 Composite Materials Workshop Tsai, Halpin and Pagano USA 1969 The Analysis of Laminated Composite Structures Lee Calcote USA 1969 Primer on Composite Materials: Analysis Ashton, Halpin and Petit USA 1970 Theory of Laminated Plates J.E. Ashton and J.M. Whitney USA 1970 Theory of Anisotropic Plates S.A. Ambartsumyan Russia 1972 Theory of Fiber Reinforced Materials Zvi Hashin USA 1975 Mechanics of Composite Materials R.M. Jones USA 1979 Mechanics of Composite Materials R.M. Christensen USA (Soltes, 1961) and aramid (Kwolek, 1964). The development of the advanced fibers in the late 1950s and early 1960s spurred great interest in the development of theoretical and applied mechanics for applications to fibrous composite materials and structures. Fromthe earliest applications offibrous composites by the Egyp￾tians to the introduction of advanced composites in the second half of the 20th Century, roughly 6000 years have passed. The progress in the use of fibrous composites in the most recent fifty years was much greater than that during the preceding nearly six thou￾sand years. The Egyptians were artisans in that they undoubtedly developed their products through trial and error. During the past fifty years, theoretical and applied mechanics has been employed in order to exploit the vast potential of man-made fibrous com￾posites. These advancements are exemplified dramatically by the applicationof advancedfibrous composites inSpaceShipOne andits launch vehicle White Knight, (Fig. 4, see http://www.scaled.com/). SpaceShipOne is an all-composite, suborbital spaceplane launched in 2003 by Scaled Composites. Fig. 3. Brick with straw fibers. 2. The early years Table 1 summarizes the early activities, contributions and accomplishments related to advances in the mechanics of fibrous composites. The remainder of this paper is organized according to subject matter. Topics covered include constitutive equations, micromechanics, laminates, thermal and moisture effects, damage and failure, experimental methods, interlaminar stresses, tubes, plates, nanocomposites, and university and government programs. This leaves many related subjects still to be reviewed. 3. Anisotropic, elastic constitutive equations Discussions on the advances in the development of constitu￾tive equations for elastic materials can be found in Love’s work (1892–1927), Sokolnikoff (1946/1956) and Timoshenko’s History of Strength of Materials (1953). The development of constitutive equa￾tions for homogeneous, elastic materials began with the work of Hooke (1678) who stated that for an elastic body there is propor￾tionality between stress and strain. Navier (1821) generalized this Fig. 4. White knight and SpaceShipOne

CT. Herakovich/ Mechanics Research Communications 41(2012)1-20 idea to arrive at differential equations describing elastic response: directions 1, 2, 3, Eq (2)takes the form(3)when written in terms however, his equations included only one elastic constant Cauchy of engineering constants and reduced notation (1822), building on the work of Navier, developed equations of elasticity for isotropic materials with two elastic constants. I Cauchy (1828)generalized his results for anisotropic (or 一v1-2 olotropic)materials and found that there are at most, twenty one independent constants. However, he believed that only fifteen 000 of these constants were"elastics constants". Like Cauchy, Poisson (1829) believed that there were only fifteen elastic constants These beliefs were based upon a molecular structure of solids and The symmetry of the compliance matrix provides additional intermolecular forces. During this period, there were two com- relationships between the moduli and poisson ratios peting theories as to the number of elastic constants, either 21 The earliest publications employing anisotropic constitutive or 15. The question was resolved when Green(1839)introduced equations for the solution of real problems appear to be those of the concept of strain energy and arrived at equations of elastic Lekhnitskii. During the years 1935-1942, he published on plane problems, cylindrical anisotropy, torsion and bending Wood ity from the principal of virtual work. The equations developed was the primary material considered in his work. See the bibli using Greens approach have 21 independent, elastic constants. ography in Lekhnitskii,s Theory of elasticity of an Anisotropic Body Lord Kelvin(Thomson, 1855)used the First and Second Laws(1950) for additional references. It is possible that Navier and co- energy function. The existence of the strain energy function and th presence of 21 independent elastic constants in the most general on wood structures, but no references have been found by this nisotropic case is now the accepted theory author The most general form of the anisotropic constitutive equation Cu G CC gs C 4 C15C25C35C45C55C56yx|8s C16 C26 C36 C46 C56 Ce here aj and t are normal and shear components of stress, respectively, Ey and yi are the normal and shear components of strain,respectively, and C is the symmetric stiffness matrix with 4. Micromechanics 21independent, elastic constants(or stiffness coefficients). (The n of stress and strain in (1)is common practice The study of composite materials at the fiber and matrix level for analysis of composite materials. These constitutive equations is referred to as micromechanics. It is desired to predict the overall the same form in Lekhnitskii(1947)Anisotropic Plates effective(or average)elastic properties and inelastic response of book(p 10, Eq (2.1). It is noted that the preface to the 1947 edition the composite based upon the known properties, arrangement and actually was written in May 1944 volume fraction of the constituent phases. Examples of compos- Lekhnitskii also shows that a monoclinic material (one plane of ites at the fiber and matrix level are shown in Fig. 5. Fig 5a shows symmetry)has 13 independent constants, an orthotropic material carbon fibers in an epoxy matrix and Fig 5b is a photomicrograph (three planes of symmetry) has 9 independent constants, a trans- of ceramic fiber(silicon carbide)in a titanium matrix. The silicon ersely isotropic material (isotropic properties in one of the planes carbide fiber has a tungsten core that is clearly visible in the figure. of symmetry) has five independent constants, and an isotropic The carbon fibers are actually a collection(called tows)of numer material(properties independent of direction) has two indepen- ous carbon filaments(2000-30,000 or more). As indicated in these dent constants. He also discussed the case of a material with figures, ceramic fibers typically have a much larger diameter than cylindrical anisotropy. carbon fibers. The distribution of fibers is quite uniform in metal Inversion of (1) gives expressions for the strains in terms of matrix composites, but is variable in resin matrix composites. This stresses and compliance coefficients Si significant difference in the distribution of fibers requires that a E S1S12S13S14S15S16(ox larger region (number of fibers)be considered as the representative volume element(RVE) for micromechanics studies when the fiber S12S22523S24525526ay distribution is nonuniform. When the fibers are uniformly spaced as SI (2) in Fig 5b, it is reasonable to consider a single fiber and surrounding matrix material as the rve. in this latte symmetry arguments often can be used to reduce the region under consideration even The constitutive Eq(2)can be written in terms of the en%ar astic properties of composites have been offered(Table 2).The A wide variety of methods for predicting the effective thermo- moduli Gi As an example, for an orthotropic material with principal earliest works are those of Voigt(1889)and Reuss(1929). while these early studies were concerned primarily with polycrystals, the theories can be applied to fibrous composites. Voigt assumed that the strains were constant throughout the material under load. In contrast, Reuss assumed that the stresses were constant through out the material. Hill (1952) showed that the voigt assumption of the results presented in this article are described in detail in the authors text results in upper bounds on effective elastic properties and the Reuss (Herakovich, 1998). assumption results in lower bounds

4 C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 idea to arrive at differential equations describing elastic response; however, his equations included only one elastic constant. Cauchy (1822), building on the work of Navier, developed equations of elasticity for isotropic materials with two elastic constants.1 Cauchy (1828) generalized his results for anisotropic (or aeolotropic) materials and found that there are, at most, twenty￾one independent constants. However, he believed that only fifteen of these constants were “elastics constants”. Like Cauchy, Poisson (1829) believed that there were only fifteen elastic constants. These beliefs were based upon a molecular structure of solids and intermolecular forces. During this period, there were two com￾peting theories as to the number of elastic constants, either 21 or 15. The question was resolved when Green (1839) introduced the concept of strain energy and arrived at equations of elastic￾ity from the principal of virtual work. The equations developed using Green’s approach have 21 independent, elastic constants. Lord Kelvin (Thomson, 1855) used the First and Second Laws of Thermodynamics to argue for the existence of Green’s strain energy function. The existence ofthe strain energy function and the presence of 21 independent elastic constants in the most general anisotropic case is now the accepted theory. The most general form of the anisotropic constitutive equations for homogeneous, elastic, composite materials can be written2: ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 1 2 3 4 5 6 ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ ≡ ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ xx yy zz yz zx xy ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ C11 C12 C13 C14 C15 C16 C12 C22 C23 C24 C25 C26 C13 C23 C33 C34 C35 C36 C14 C24 C34 C44 C45 C46 C15 C25 C35 C45 C55 C56 C16 C26 C36 C46 C56 C66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ εxx εyy εzz yz zx xy ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ ≡ ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ ε1 ε2 ε3 ε4 ε5 ε6 ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ (1) where ij and ij are normal and shear components of stress, respectively, εij and ij are the normal and shear components of strain, respectively, and Cij is the symmetric stiffness matrix with 21independent, elastic constants (or stiffness coefficients).(The sin￾gle subscript notation of stress and strain in (1) is common practice for analysis of composite materials.) These constitutive equations appear in the same form in Lekhnitskii (1947) Anisotropic Plates book (p. 10, Eq.(2.1)). Itis noted thatthe preface to the 1947 edition actually was written in May 1944. Lekhnitskii also shows that a monoclinic material (one plane of symmetry) has 13 independent constants, an orthotropic material (three planes of symmetry) has 9 independent constants, a trans￾versely isotropic material (isotropic properties in one of the planes of symmetry) has five independent constants, and an isotropic material (properties independent of direction) has two indepen￾dent constants. He also discussed the case of a material with cylindrical anisotropy. Inversion of (1) gives expressions for the strains in terms of stresses and compliance coefficients Sij: ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ εxx εyy εzz yz zx xy ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ S11 S12 S13 S14 S15 S16 S12 S22 S23 S24 S25 S26 S13 S23 S33 S34 S35 S36 S14 S24 S34 S44 S45 S46 S15 S25 S35 S45 S55 S56 S16 S26 S36 S46 S56 S66 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ xx yy zz yz zx xy ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ (2) The constitutive Eq. (2) can be written in terms of the engi￾neering constants, elastic moduli Ei, Poisson ratios ij and shear moduliGij.As anexample,for anorthotropicmaterial withprincipal 1 Dates given for Navier and Cauchy correspond to when they read their paper to the Paris Academy. Publication, if any, was at a later date. 2 Commonly used notations for composite mechanics and developments of many of the results presented in this article are described in detail in the author’s text (Herakovich, 1998). directions 1, 2, 3, Eq. (2) takes the form (3) when written in terms of engineering constants and reduced notation: ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ ε1 ε2 ε3 ε4 ε5 ε6 ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1⁄E1 −21⁄E2 −31⁄E3 0 0 0 −12⁄E1 1⁄E2 −32⁄E3 0 0 0 −13⁄E1 −23⁄E2 1⁄E3 0 0 0 0 0 0 1⁄G23 0 0 0 0 00 1⁄G31 0 0 0 0 00 1⁄G12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 1 2 3 4 5 6 ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ (3) The symmetry of the compliance matrix provides additional relationships between the moduli and Poisson ratios. The earliest publications employing anisotropic constitutive equations for the solution of real problems appear to be those of Lekhnitskii. During the years 1935–1942, he published papers on plane problems, cylindrical anisotropy, torsion and bending. Wood was the primary material considered in his work. See the bibli￾ography in Lekhnitskii’s Theory of Elasticity of an Anisotropic Body (1950) for additional references. It is possible that Navier and co￾workers used the equations that they had developed in their work on wood structures, but no references have been found by this author. 4. Micromechanics The study of composite materials at the fiber and matrix level is referred to as micromechanics. It is desired to predict the overall effective (or average) elastic properties and inelastic response of the composite based upon the known properties, arrangement and volume fraction of the constituent phases. Examples of compos￾ites at the fiber and matrix level are shown in Fig. 5. Fig. 5a shows carbon fibers in an epoxy matrix and Fig. 5b is a photomicrograph of ceramic fiber (silicon carbide) in a titanium matrix. The silicon carbide fiber has a tungsten core that is clearly visible in the figure. The carbon fibers are actually a collection (called tows) of numer￾ous carbon filaments (2000–30,000 or more). As indicated in these figures, ceramic fibers typically have a much larger diameter than carbon fibers. The distribution of fibers is quite uniform in metal matrix composites, but is variable in resin matrix composites. This significant difference in the distribution of fibers requires that a larger region (number of fibers) be considered as the representative volume element (RVE) for micromechanics studies when the fiber distributionisnonuniform.Whenthefibers are uniformly spaced as in Fig. 5b, it is reasonable to consider a single fiber and surrounding matrix material as the RVE. In this latter case, symmetry arguments often can be used to reduce the region under consideration even further. A wide variety of methods for predicting the effective thermo￾elastic properties of composites have been offered (Table 2). The earliest works are those of Voigt (1889) and Reuss (1929). While these early studies were concerned primarily with polycrystals,the theories can be applied to fibrous composites. Voigt assumed that the strains were constant throughout the material under load. In contrast, Reuss assumed that the stresses were constant through￾out the material. Hill (1952) showed that the Voigt assumption results in upper bounds on effective elastic properties and the Reuss assumption results in lower bounds.

CT. Herakovich/Mechanics Research Communications 41(2012)1-20 Table 2 Developments in micromechanics. Author(s) ountry 837 Strain energy density defined -21 elastic constants Great Britain Germany Determination of the elastic field of an ellipsoidal inclusion and related problems rediction of Elastic Constants of Multiphase Materials lashin Variational Approach to the Theory of The Elastic Behavior of Multiphase Materials ashin and shtrikman USA and israel heory of Mechanical Behavior of Heterogeneous Media USA trengthened Materials-L. Elastic Behaviour The el uli of Fiber-Reinforced Materials Hashin and Rosen Reinforcement of metals Kelly and Davies Great Britain Tsai, Halpin and Pagano Theory of Fiber Reinforced Materials On the Effective Moduli of Composite Materials: Slender rigid Inclusions at Dilute Concentrations Russel and acrivos 975 A Theory of Elasticity with Microstructure for Directionally Reinforced Composites Fiber Composites w ropic Constituents Mechanics of Composite Materials: A Unified Micromechanical Approach Micromechanics: overall properties of heterogeneous materials Nemat-Nasser and hori USA The development of micromechanics models for predicting the of a fiber core surrounded by a matrix annulus, such that the effective properties of composites experienced a flurry of activity size of the cylinders varies as needed to fill the entire volume of eginning in the late 1950s and extending through the 1960s and material. The ratio of fiber radius to cylinder radius is held constant to the 1970s. The earlier works were concerned with the pre- throughout, thereby maintaining a constant fiber volume fraction diction of effective properties for materials(both solids and fluids) in each cylinder Four of the necessary five effective properties for a consisting of inclusions in a carrier material (see Hashin, 1964). transversely isotropic composite can be determined using the CCa Paul (1960) and Hill (1963)used energy approaches to obtain model. upper and lower bounds on elastic moduli of heterogeneous mate- In the following from Christensen(1979), with subscripts f and rials consisting of inclusions in a matrix. In general, the inclusions m indicating fiber and matrix respectively, V fiber volume fraction, were of arbitrary shape but both authors did make reference to E axial modulus, v Poissons ratio, u shear modulus, k bulk modulus, fiber-like inclusions. It is noteworthy that they both showed that and the fiber and matrix are taken to be transversely isotropic, the the voigt and reuss approximations are upper and lower bounds CCa model provides expressions for four of the effective properties on moduli Hashin and Shtrikman(1963)presented a variational in terms of the phase properties and fiber volume fraction. approach to derive upper and lower bounds for the effective elas- Effective axial modulus. tic moduli of quasi-isotropic and quasi-homogeneous multiphase materials of arbitrary phase geometry. Hill(1964)addressed the v(1-v Xu-Vm) elastic mechanical properties of fiber-strengthened materials In E1 (1972)Russel and Acrivos considered the effective modulus of com- osites with slender rigid inclusions at dilute concentrations. Effective axial Poisson's ratio: Hashin and Rosen(1964)employed a co cylinder assem- blage(CCA)model to develop upper and lower bounds as well V12=(1-v)Vm+ Vry as specific expressions for(some of) the effective elastic mod- uli of transversely isotropic composites. Their model consists of v(1-VXy-m)(m/(km+m/3)-m(k+H/3) an assemblage of concentric cylinders, each cylinder consisting +(-V)m)/(k+/3)+Vm/(km+m/3)+1 (5) a)Carbon/Epoxy b)Silicon Carbide/Titanium Fig. 5. Fiber and matrix photomicrographs

C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 5 Table 2 Developments in micromechanics. Year Development Author(s) Country 1837 Strain energy density defined – 21 elastic constants George Green Great Britain 1887/1889 Uniform strain modulus prediction W. Voigt Germany 1929 Uniform stress modulus prediction A. Reuss Germany 1957 Determination of the elastic field of an ellipsoidal inclusion and related problems Eshelby Great Britain 1960 Prediction of Elastic Constants of Multiphase Materials Paul USA 1962 The Elastic Moduli of Heterogeneous Materials Hashin USA 1963 Elastic Properties of Reinforced solids: some theoretical principles Hill Great Britain 1963 Variational Approach to the Theory of The Elastic Behavior of Multiphase Materials Hashin and Shtrikman USA and Israel 1964 Theory of Mechanical Behavior of Heterogeneous Media Hashin USA 1964 Theory of Mechanical Properties of Fiber-Strengthened Materials – I. Elastic Behaviour Hill USA 1964 The Elastic Moduli of Fiber-Reinforced Materials Hashin and Rosen USA 1965 The Principle of the Fiber Reinforcement of Metals Kelly and Davies Great Britain 1967 Modern Composite Materials Broutman and Krock USA 1968 Composite Materials Workshop Tsai, Halpin and Pagano USA 1972 Theory of Fiber Reinforced Materials Hashin USA 1972 On the Effective Moduli of Composite Materials: Slender Rigid Inclusions at Dilute Concentrations Russel and Acrivos USA 1975 A Theory of Elasticity with Microstructure for Directionally Reinforced Composites Achenbach USA 1979 Analysis of Properties of Fiber Composites with Anisotropic Constituents Hashin Israel 1991 Mechanics of Composite Materials: A Unified Micromechanical Approach Aboudi Israel 1993 Micromechanics: overall properties of heterogeneous materials Nemat-Nasser and Hori USA The development of micromechanics models for predicting the effective properties of composites experienced a flurry of activity beginning in the late 1950s and extending through the 1960s and into the 1970s. The earlier works were concerned with the pre￾diction of effective properties for materials (both solids and fluids) consisting of inclusions in a carrier material (see Hashin, 1964). Paul (1960) and Hill (1963) used energy approaches to obtain upper and lower bounds on elastic moduli of heterogeneous mate￾rials consisting of inclusions in a matrix. In general, the inclusions were of arbitrary shape, but both authors did make reference to fiber-like inclusions. It is noteworthy that they both showed that the Voigt and Reuss approximations are upper and lower bounds on moduli. Hashin and Shtrikman (1963) presented a variational approach to derive upper and lower bounds for the effective elas￾tic moduli of quasi-isotropic and quasi-homogeneous multiphase materials of arbitrary phase geometry. Hill (1964) addressed the elastic mechanical properties of fiber-strengthened materials. In (1972) Russel andAcrivos considered the effective modulus of com￾posites with slender rigid inclusions at dilute concentrations. HashinandRosen(1964) employeda concentric cylinder assem￾blage (CCA) model to develop upper and lower bounds as well as specific expressions for (some of) the effective elastic mod￾uli of transversely isotropic composites. Their model consists of an assemblage of concentric cylinders, each cylinder consisting of a fiber core surrounded by a matrix annulus, such that the size of the cylinders varies as needed to fill the entire volume of material. The ratio of fiber radius to cylinder radius is held constant throughout, thereby maintaining a constant fiber volume fraction in each cylinder. Four of the necessary five effective properties for a transversely isotropic composite can be determined using the CCA model. In the following from Christensen (1979), with subscripts f and m indicating fiber and matrix respectively, V fiber volume fraction, E axial modulus,  Poisson’s ratio,  shear modulus, k bulk modulus, and the fiber and matrix are taken to be transversely isotropic, the CCA model provides expressions for four of the effective properties in terms of the phase properties and fiber volume fraction. Effective axial modulus: E∗ 1 = Vf Ef + (1 − Vf )Em + 4Vf (1 − Vf )(vf − vm) 2m ((1 − Vf )m)/(kf + f /3) + Vf m (4) Effective axial Poisson’s ratio: 12 = (1 − Vf )m + Vf f + Vf (1 − Vf )(f − m)  m/(km + m/3) − m/(kf + f /3) ((1 − Vf )m)/(kf + f /3) + Vf m/(km + m/3) + 1 (5) Fig. 5. Fiber and matrix photomicrographs

CT. Herakovich/ Mechanics Research Communications 41(2012)1-20 Effective plane strain bulk modulus MOCT Mori-Tanaka 172. 1/-km+(-pm)/3]+(1-v)(km+4m/3) Effective axial shear modulus. 12f(1+V)+pm(1-v) 68.9 It is evident from eqs. (4)and(5 that the first two terms cor- respond to a rule of mixtures. The last term is typically very small for most composites in use today. Thus, the rule of mixtures (i.e. Voigt upper bound) is a very good predictor for the effective axial 0.25005000.750 modulus and effective axial Poisson s ratio. this cannot be said for he other two properties. Fig. 6. Axial modulus predictions for carbon/epoxy Chamis and Sendeckyj(1968)presented an extensive critique of the theories known at the time for predicting the thermoelastic properties of fibrous composites. The theories reviewed were clas- 4.1. Micromechanics model comparisons sified as: netting analysis, mechanics of materials, self-consistent model, variational, exact, statistical, discrete element, semi empir- Figs 6-9 show comparisons of micromechanics predictions for the effective properties E1, E2, V12, and Gu of unidirectional ca ical methods, and theories accounting for microstructure. They bonepoxy(Lissenden and Herakovich, 1992)as a function of the included comparisons of predictions by different theories for uni- directional glass-epoxy, boron-epoxy and graphite-epoxy. fiber volume fraction. The methods compared include: Voigt, Reuss, Hashin(1972) gave an extensive theoretical treatment of concentric cylinder assemblage, self-consistent, method of cells hermoelastic properties, thermal and electrical conduction, and Several important features are evident from these comparisons. electrostatics and magnetostatics behavior. For the effective axial modulus, E:(Fig. essentially all models give Achenbach(1974) and Achenbach(1975)considered wave the same prediction, with the lower bound reuse model being the The composite with microstructure is distinguished from a com- provides excellent predictions for the effective axial modulus posite that is modeled as a homogeneous, anisotropic continuum Schapery(1967)has shown that the results for linear elastic using effective properties. The point is made that for dynamic materials can be extended to linear viscoelastic materials in a sim- response such as wave propagation, the characteristic lengths of ple and accurate manne the deformation be small and the effective modulus theory ay not suffice. The proposed theory showed good comparison 5. Lamination theory with ultrasonic data for fibrous composites and finite element pre dictions. Lectures on this subject were given at the International Possibly the most fundamental result for the application of Centre for Mechanical Sciences(CISM)in Udine, Italy, in July 1973 fibrous composites in structural and devices is Classical lamination for composite materials and provided an in-depth analysis of the The theory considers an assemblage of layers bonded together Method of Cells for thermo-elastic, viscoelastic, nonlinear behav- to form a laminate. The individual layers are taken to be homoge ior of resin matrix composites, initial yield surfaces and inelastic neous with properties that can range from isotropic to anisotropic. behavior of metal matrix composites, and composites with imper- Typically, the layers are unidirectional fibrous composites with the fect bonding. The method of cells consists of a periodic square array of rectangular subcells, one representing the fiber and three 2.50 172 similar subcells representing the matrix. This model provides a -I MOC-Tl computationally efficient method for predicting inelastic response 丰 The effects of different types of fiber orthotropy on the effective roperties of composites were considered by Knott and Herakovich 拿εCN (1991a). Nemat-Nasser and Hori (1993)presented a treatise on the mechanics of solids with microdefects such as cavities cracks, and inclusions, including elastic composites 68 Discrete element methods such as the finite element method have been used to predict effective properties of unidirectional to 0500 3.4 be that of Foye who studied effective elastic properties, inelastic response, and stress distributions in unidirectional boron/epoxy. Finite element studies can be valuable when the fiber distribution 0.250 0.500 100 as shown for the ceramic fiber in a titanium matrix less so for random fiber distributions such as the arbon/epoxy of Fig 5a. Fig. 7. Transverse modulus predictions for carbon/epoxy

6 C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 Effective plane strain bulk modulus: K∗ 23 = km + m 3 + Vf 1/[kf − km + (f − m)/3] + (1 − Vf )/(km + 4m/3) (6) Effective axial shear modulus: ∗ 12 m = f (1 + Vf ) + m(1 − Vf ) f (1 − Vf ) + m(1 + Vf ) (7) It is evident from Eqs. (4) and (5) that the first two terms cor￾respond to a rule of mixtures. The last term is typically very small for most composites in use today. Thus, the rule of mixtures (i.e. Voigt upper bound) is a very good predictor for the effective axial modulus and effective axial Poisson’s ratio. This cannot be said for the other two properties. Chamis and Sendeckyj (1968) presented an extensive critique of the theories known at the time for predicting the thermoelastic properties of fibrous composites. The theories reviewed were clas￾sified as: netting analysis, mechanics of materials, self-consistent model, variational, exact, statistical, discrete element, semi empir￾ical methods, and theories accounting for microstructure. They included comparisons of predictions by different theories for uni￾directional glass-epoxy, boron-epoxy and graphite-epoxy. Hashin (1972) gave an extensive theoretical treatment of micromechanics. He considered effective elastic, viscoelastic and thermoelastic properties, thermal and electrical conduction, and electrostatics and magnetostatics behavior. Achenbach (1974) and Achenbach (1975) considered wave propagation in fiber-reinforced composites with microstructure. The composite with microstructure is distinguished from a com￾posite that is modeled as a homogeneous, anisotropic continuum using effective properties. The point is made that for dynamic response such as wave propagation, the characteristic lengths of the deformations may be small and the effective modulus theory may not suffice. The proposed theory showed good comparison with ultrasonic data for fibrous composites and finite element pre￾dictions. Lectures on this subject were given at the International Centre for Mechanical Sciences (CISM) in Udine, Italy, in July 1973 with publication of the (expanded) monograph in 1975. Aboudi (1991) presented micromechanical analysis methods for composite materials and provided an in-depth analysis of the Method of Cells for thermo-elastic, viscoelastic, nonlinear behav￾ior of resin matrix composites, initial yield surfaces and inelastic behavior of metal matrix composites, and composites with imper￾fect bonding. The method of cells consists of a periodic square array of rectangular subcells, one representing the fiber and three similar subcells representing the matrix. This model provides a computationally efficient method for predicting inelastic response of composites. The effects of different types of fiber orthotropy on the effective properties of composites were considered by Knott and Herakovich (1991a). Nemat-Nasser and Hori (1993) presented a treatise on the mechanics of solids with microdefects such as cavities, cracks, and inclusions, including elastic composites. Discrete element methods such as the finite element method have been used to predict effective properties of unidirectional composites. The earliest work using finite elements appears to be that of Foye who studied effective elastic properties, inelastic response, and stress distributions in unidirectional boron/epoxy. Finite element studies can be valuable when the fiber distribution is very regular as shown for the ceramic fiber in a titanium matrix of Fig. 5b, but less so for random fiber distributions such as the carbon/epoxy of Fig. 5a. Fig. 6. Axial modulus predictions for carbon/epoxy. 4.1. Micromechanics model comparisons Figs. 6–9 show comparisons of micromechanics predictions for the effective properties E1, E2, 12, and G12 of unidirectional car￾bon/epoxy (Lissenden and Herakovich, 1992) as a function of the fiber volume fraction. The methods compared include:Voigt, Reuss, concentric cylinder assemblage, self-consistent, method of cells, Mori-Tanaka and strength of materials. Several important features are evident from these comparisons. For the effective axial modulus, E∗ 1 (Fig. 6) essentially all models give the same prediction, with the lower bound Reuse model being the exception. Thus, a simple rule of mixtures (the Voigt upper bound), provides excellent predictions for the effective axial modulus. Schapery (1967) has shown that the results for linear elastic materials can be extended to linear viscoelastic materials in a sim￾ple and accurate manner. 5. Lamination theory Possibly the most fundamental result for the application of fibrous composites in structural and devices is Classical Lamination Theory. The theory follows the original works of Pister and Dong (1959), Reissner and Stavsky (1961) and Dong et al. (1962) The theory considers an assemblage of layers bonded together to form a laminate. The individual layers are taken to be homoge￾neous with properties that can range from isotropic to anisotropic. Typically, the layers are unidirectional fibrous composites with the Fig. 7. Transverse modulus predictions for carbon/epoxy

CT. Herakovich/Mechanics Research Communications 41(2012)1-20 4.0 MOC-Tl L St Matls Fig 10. Composite laminate. 0,25非 0750 Poisson ratio: Ey Fig 8. Shear modulus predictions for carbon/epoxy Vxy 8x (11) fibers in the kth layer oriented at an angle ek from a global x-axis s depicted in Fig. 10. ix (12) Analysis results in the fundamental equation relating the inplane forces N and moments ( M acting on the laminate to Coefficient of mutual influence the midplane strains E) and curvatures( x) through coefficients [A] [B]and ( D] that are functions of the material properties, layers nxy xei ai1 thickness and stacking sequence of the layers N A B The coefficient of mutual influence(13)quantifies the shear ( 8) strain associated with normal strain; it is non-zero when the lam- inate compliance t The effective engineering properties of symmetric laminates can Specific examples of the range of engineering properties thatcan be predicted from Eq ( 8)through a series of thought experimen be affected through the choice of material and stacking sequence here the laminate is subjected to a series of specified loadings. are presented in Figs 11-13. These figures show the variation in With the laminate compliance defined: axial modulus, Poisson ratio and shear modulus for T300/5208 car- bon/epoxy. la]=2HIAJ-I (9) These three figures show that the effective engineering prop- for the engineering properties of the laminate. Examples are. ons erties of angle-ply laminates are higher than those of the Axial modulus: inates can exhibit values greater than 1.0, and the shear modulus of angle-ply laminates is largest at 45. Ex Another most interesting result for laminated composites (Fig. 14)is the fact that the through-the-thickness P 20.0 0.400 0.200 MOC-TI 0.100 .Mori-Tanaka 含 St matls 0.0 0.00 0.0 0.250 0.750 20.00 8000 Fig. 11. Axial modulus -unidirectional and angle-ply laminates

C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 7 Fig. 8. Shear modulus predictions for carbon/epoxy. fibers in the kth layer oriented at an angle k from a global x-axis as depicted in Fig. 10. Analysis results in the fundamental equation relating the inplane forces {N} and moments {M} acting on the laminate to the midplane strains {ε◦} and curvatures { } through coefficients [A], [B] and [D] that are functions of the material properties, layers thickness and stacking sequence of the layers.  N M  =  A B B D  ε◦  (8) The effective engineering properties of symmetric laminates can be predicted from Eq. (8) through a series of thought experiments where the laminate is subjected to a series of specified loadings. With the laminate compliance defined: [a∗] ≡ 2H[A] −1 (9) The results of these thought experiments provide expressions for the engineering properties of the laminate. Examples are: Axial modulus: Ex = ¯ x ε◦ x = 1 a∗ 11 (10) Fig. 9. Poisson’s ratio predictions for carbon/epoxy. Fig. 10. Composite laminate. Poisson ratio: xy = −ε◦ y ε◦ x = −a∗ 12 a∗ 11 (11) Shear modulus: Gxy = ¯xy ◦ xy = 1 a∗ 66 (12) Coefficient of mutual influence: xy,x = ◦ xy ε◦ x = a∗ 16 a∗ 11 (13) The coefficient of mutual influence (13) quantifies the shear strain associated with normal strain; it is non-zero when the lam￾inate compliance term a∗ 16 is non-zero. Specific examples ofthe range of engineering properties that can be affected through the choice of material and stacking sequence are presented in Figs. 11–13. These figures show the variation in axial modulus, Poisson ratio and shear modulus for T300/5208 car￾bon/epoxy. These three figures show that the effective engineering prop￾erties of angle-ply laminates are higher than those of the corresponding laminae. Further, Poisson’s ratio of angle-play lam￾inates can exhibit values greater than 1.0, and the shear modulus of angle-ply laminates is largest at 45◦. Another most interesting result for laminated composites (Fig. 14) is the fact that the through-the-thickness Poisson’s ratio Fig. 11. Axial modulus – unidirectional and angle-ply laminates.

CT. Herakovich/ Mechanics Research Communications 41(2012)1-20 0600 120 0.450 00 0.800 0.150 H儿L 0.150 0.300 0015003000450060.007500 30.04560.0 75,0 Fig. 12. Poisson s ratio- unidirectional and angle-ply laminates Fig. 14. Through-the-thickness poissons ratio. Vxz is negative over a significant range of fber orientations for some Composites often are the material of choice where thermal stresses ngle-ply laminates( Herakovich, 1984). xpansion are important. The coefficient of thermal Another interesting feature of laminates is that, depending expansion in the fiber direction of unidirectional composites is n the stacking sequence of the layers, they can exhibit cou- often near zero and can be slightly negative. This has huge pling between inplane and bending effects. Laminates that are consequences when designing laminates for low,or matching, coef- unsymmetric about the laminate midplane have a non-zero [b] ficients of thermal expansion. Thermal stresses can be extremely matrix resulting in coupling between inplane and out-of-plane important for the application of fibrous composite materials as responses( see Eq(8) Unsymmetric laminates exhibit curvature essentially all composite materials are fabricated at an elevate hen subjected to pure inplane loading. Likewise, unsymmetric temperature. The constituent phases become bonded at anelevated laminates exhibit inplane strains when subjected to pure bending temperature resulting in residual thermal stresses in the composite moments. More on unsymmetric laminates is provided in a later after it has cooled to room temperature section Fundamental problems at the micromechanics level are predic tion of the residual stresses al 6. Environmental effects unidirectional composites. At the laminate level, it is necessary to predict the residual stresses and the laminate effective coefficient 6. 1. Thermal effects of thermal expansion(CTE). This latter property is very important as it is one of the unique aspects of laminated composite materi- ronmental effects often play a critical role in the choice als: composite laminates can exhibit ChE values over a wide range erial for many applications in devices and structures. including zero, positive and negative. The earliest papers dealing with thermal effects in anisotropic materials appear to be those by Ambartsumyan (1952)who consid- ered thermal stresses in anisotropic, laminated plates, and hayashi (1956)who considered thermal stresses in orthotropic plates. The earliest works at the micromechanics level appears to be that of Van Fo Fy(1965) who considered thermal effects in com- posites consisting of periodic arrays of continuous, circular glass fibers. He used stress analysis to determine exact thermal coef- ficients for specific phase geometries. Levin(1967) presented an approach for determining the effective coefficients of thermal expansion for two phase composites with isotropic phases. The 300 work used an extension Hill,s approach and included bounds on the expansion coefficients of transversely isotropic, undirect 200 fiber-reinforced composites. Rosen(1968)investigated th expansion coefficients for composite materials. Much of this is incorporated in the later paper by Rosen and Hashin (1970)on 100 expansion coefficients. Schapery(1968)derived upper and lower bounds as well as specific approximations for thermal expansion coefficients of lin- ear elastic and viscoelastic composite materials. He extended the 000 6000 00 previous work of Levin and Van Fo Fy for an arbitrary number e of constituents and phase geometries, for isotropic phases. The approach provided upper and lower bounds using the principles Fig 13. Shear modulu of complementary and potential energy. Approximate expressions

8 C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 Fig. 12. Poisson’s ratio – unidirectional and angle-ply laminates. xz is negative over a significant range of fiber orientations for some angle-ply laminates (Herakovich, 1984). Another interesting feature of laminates is that, depending on the stacking sequence of the layers, they can exhibit cou￾pling between inplane and bending effects. Laminates that are unsymmetric about the laminate midplane have a non-zero [B] matrix resulting in coupling between inplane and out-of-plane responses (see Eq. (8)). Unsymmetric laminates exhibit curvature when subjected to pure inplane loading. Likewise, unsymmetric laminates exhibit inplane strains when subjected to pure bending moments. More on unsymmetric laminates is provided in a later section. 6. Environmental effects 6.1. Thermal effects Environmental effects often play a critical role in the choice of material for many applications in devices and structures. Fig. 13. Shear modulus – unidirectional and angle-ply laminates. Fig. 14. Through-the-thickness Poisson’s ratio. Composites often are the material of choice where thermal stresses or thermal expansion are important. The coefficient of thermal expansion in the fiber direction of unidirectional composites is often near zero and can be slightly negative. This has huge consequences when designing laminates for low, or matching, coef- ficients of thermal expansion. Thermal stresses can be extremely important for the application of fibrous composite materials as essentially all composite materials are fabricated at an elevated temperature. The constituent phases become bonded at an elevated temperature resulting in residualthermal stresses in the composite after it has cooled to room temperature. Fundamental problems at the micromechanics level are predic￾tion of the residual stresses and the effective thermal properties of unidirectional composites. At the laminate level, it is necessary to predict the residual stresses and the laminate effective coefficient of thermal expansion (CTE). This latter property is very important as it is one of the unique aspects of laminated composite materi￾als: composite laminates can exhibit CTE values over a wide range including zero, positive and negative. The earliest papers dealing with thermal effects in anisotropic materials appear to be those by Ambartsumyan (1952) who consid￾ered thermal stresses in anisotropic, laminated plates, and Hayashi (1956) who considered thermal stresses in orthotropic plates. The earliest works at the micromechanics level appears to be that of Van Fo Fy (1965) who considered thermal effects in com￾posites consisting of periodic arrays of continuous, circular glass fibers. He used stress analysis to determine exact thermal coef- ficients for specific phase geometries. Levin (1967) presented an approach for determining the effective coefficients of thermal expansion for two phase composites with isotropic phases. The work used an extension Hill’s approach and included bounds on the expansion coefficients of transversely isotropic, unidirectional, fiber-reinforced composites. Rosen (1968) investigated thermal expansion coefficients for composite materials. Much of this work is incorporated in the later paper by Rosen and Hashin (1970) on expansion coefficients. Schapery (1968) derived upper and lower bounds as well as specific approximations for thermal expansion coefficients of lin￾ear elastic and viscoelastic composite materials. He extended the previous work of Levin and Van Fo Fy for an arbitrary number of constituents and phase geometries, for isotropic phases. The approach provided upper and lower bounds using the principles of complementary and potential energy. Approximate expressions

CT. Herakovich/Mechanics Research Communications 41(2012)1-20 x axI(±)l nterface 12.0 21,6 16.2 Generic Cross- Section I08 Generic Quarter-Section 0.0150 45.060.075.0 Fig. 16. Finite width coupon under axial load. Fig 15. CTE-unidirectional and angle-ply laminates. 6. 2. Moisture effec or the axial and transverse thermal coefficients of expansion for The analysis of moisture effects in organic matrix composites is unidirectional, fiber-reinforced composites were presented. Hashin analogous to that for thermal effects at both the micromechanics (1979)extended Schapery's elastic results for composites with and laminate levels. Much of this work is detailed in three volumes transversely isotropic phases. The final forms of the predictions for edited by Springer(1981), Springer(1984), and Springer(1988a) the axial and transverse thermo-elastic coefficients of expansion Volume 3, Chapter 1(Springer, 1988b) provides a broad review of presented by daniel and Ishai, 1994)are the effects of temperature and moisture on organic matrix com posites. In general, moisture effects are not nearly as significant as Era V+EmamVm (Ea) (14) thermal effects. 7. Interlaminar stresses for the coefficient in the fiber direction and The first publication concerned with interlaminar stresses in laminated composites ap be that of Hayashi(1967)who 02=a2f +1 investigated interlaminar shear stresses in an idealized lam consisting of orthotropic layers separated by isotropic shear (U12 V+v12m Vm) Ear)1 (15) ers. Other important early works include those by Bogy(1968 who investigated the singular behavior of stresses at the inte section of a boundary and bonded dissimilar isotropic materials he coefficient in the transverse direction. In the above, f andand the first three-dimensional(numerical)analysis of inter- fer to fiber and matrix, respectively, V is volume fraction, E laminar stresses in laminated Is modulus, a is coefficient of thermal expansion and v is Pois-(1970) son's ratio,(Ea)1=Ea Vf+ Emam Vm and En is the rule of mixture Pipes and Pagano provided the first complete analysis of the mposite modulus in the fiber direction. problem of an axially loaded, laminated coupon with free edges Additional works on thermal effects in composites include the(Fig. 16). They formulated a reduced system of elasticity equa view article by Tauchert(1986)and that by Herakovich and tions governing the laminate behavior by assuming independence Aboudi(1999). of the stress and strain state on the axial coordinate and then The first presentation of the thermal-elastic formulation for solved the system of equations using the finite difference method posite laminates was by Tsai(1968). An early textbook presen- Their results showed the existence of all three interlaminar stress tation of the formulation is that by Calcote(1969). Amostimportant components in the boundary layer regions along the free edges result of the formulation is an expression for the effective coeffi- of finite width laminated coupons under inplane tensile load ent of thermal expansion (@) for a symmetric N-layered laminate, ing. They presented results for a variety of fiber orientations and laminate stacking sequences and showed that the width of the boundary layer is approximately equal to the thickness of the lam- l2=A∑广at nate, that the interlaminar normal stress oz and the interlaminar (16) shear stress tz can exhibit singular behavior as the free edge is approached, at the sign and magnitude of the interlami nar stresses are functions of the laminate configuration including Ashton et al. 1969 presented results for the varia- material type, fiber orientations, layer thicknesses and stacking tion of thermal strains as a function of fber orientation sequence. The free edge problem has been studied on a continuing basis shows that rather large, negative coefficients of ther- ever since the original work in the late 1960s. The finite dif- are possible for a typical carbon/epoxy ference solution of Pipes and Pagano was followed quickly by a naterial(T300/5208 in Fig. 15)over a range of fiber orientations three-dimensional finite element solution by Rybicki( 1971). Later. for angel-ply laminates it was recognized that the tensile coupon problem also could be

C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 9 Fig. 15. CTE – unidirectional and angle-ply laminates. for the axial and transverse thermal coefficients of expansion for unidirectional,fiber-reinforcedcomposites werepresented.Hashin (1979) extended Schapery’s elastic results for composites with transversely isotropic phases. The final forms of the predictions for the axial and transverse thermo-elastic coefficients of expansion (as presented by Daniel and Ishai, 1994) are: ˛1 = Ef ˛f Vf + Em˛mVm Ef Vf + EmVm = (E˛)1 E1 (14) for the coefficient in the fiber direction, and ˛2 = ˛2f Vf  1 + 12f ˛1f ˛2f  + ˛2mVm  1 + 12m ˛1m ˛2m  − (12f Vf + 12mVm) (E˛)1 E1 (15) for the coefficient in the transverse direction. In the above, f and m refer to fiber and matrix, respectively, V is volume fraction, E is modulus, ˛ is coefficient of thermal expansion and  is Pois￾son’s ratio, (E˛)1 = Ef˛fVf + Em˛mVm and E1 is the rule of mixture composite modulus in the fiber direction. Additional works on thermal effects in composites include the review article by Tauchert (1986) and that by Herakovich and Aboudi (1999). The first presentation of the thermal-elastic formulation for composite laminates was by Tsai (1968). An early textbook presen￾tationofthe formulationis that by Calcote (1969).Amostimportant result of the formulation is an expression for the effective coeffi- cient of thermal expansion {˛¯ } for a symmetric N-layered laminate, namely: {˛¯ } = [A] −1 N k=1 [Q¯ ] k {˛} ktk (16) Ashton et al., 1969 presented results for the varia￾tion of thermal strains as a function of fiber orientation for unidirectional and angle play laminates. Fig. 15 shows that rather large, negative coefficients of ther￾mal expansion are possible for a typical carbon/epoxy material (T300/5208 in Fig. 15) over a range of fiber orientations for angel-ply laminates. Fig. 16. Finite width coupon under axial load. 6.2. Moisture effects The analysis of moisture effects in organic matrix composites is analogous to that for thermal effects at both the micromechanics and laminate levels. Much of this work is detailed in three volumes edited by Springer (1981), Springer (1984), and Springer (1988a). Volume 3, Chapter 1 (Springer, 1988b) provides a broad review of the effects of temperature and moisture on organic matrix com￾posites. In general, moisture effects are not nearly as significant as thermal effects. 7. Interlaminar stresses The first publication concerned with interlaminar stresses in laminated composites appears to be that of Hayashi (1967) who investigated interlaminar shear stresses in an idealized laminate consisting of orthotropic layers separated by isotropic shear lay￾ers. Other important early works include those by Bogy (1968) who investigated the singular behavior of stresses at the inter￾section of a boundary and bonded dissimilar isotropic materials, and the first three-dimensional (numerical) analysis of inter￾laminar stresses in laminated composites by Pipes and Pagano (1970). Pipes and Pagano provided the first complete analysis of the problem of an axially loaded, laminated coupon with free edges (Fig. 16). They formulated a reduced system of elasticity equa￾tions governing the laminate behavior by assuming independence of the stress and strain state on the axial coordinate and then solved the system of equations using the finite difference method. Their results showed the existence of all three interlaminar stress components in the boundary layer regions along the free edges of finite width laminated coupons under inplane tensile load￾ing. They presented results for a variety of fiber orientations and laminate stacking sequences and showed that the width of the boundary layer is approximately equal to the thickness of the lam￾inate, that the interlaminar normal stress z and the interlaminar shear stress zx can exhibit singular behavior as the free edge is approached, and that the sign and magnitude of the interlami￾nar stresses are functions of the laminate configuration including material type, fiber orientations, layer thicknesses and stacking sequence. The free edge problem has been studied on a continuing basis ever since the original work in the late 1960s. The finite dif￾ference solution of Pipes and Pagano was followed quickly by a three-dimensional finite element solution by Rybicki (1971). Later, it was recognized that the tensile coupon problem also could be

CT. Herakovich/ Mechanics Research Communications 41(2012)1-20 Undeformed Boundary formulated as a two-dimensional finite element problem because of the independence of the stress and strain states on the axial coor- dinate. The finite element formulation for cross-ply laminates as a two-dimensional problem was presented by Foye and Baker(1971). and the two-dimensional finite element formulation for laminates including off-axis layers, thermal stresses and non-linear response was presented by Herakovich et al. (1976) Noteworthy approximate analytical solutions include a pertur bation solution by Tang(1975), a variational approach by Pagano (1978), a solution employing complex stress potentials and eigen function series by Wang and choi(1982), and solutions based upon 45245452 statically admissible stress states( Kassapoglou and lagace, 1986 Rose and Herakovich, 1993). The free edge problem has also been nvestigated experimentally, e.g., Pipes and Daniel (1971). Oplinge et al. (1974), and Herakovich et al. (1984). The experimental inves- tigations provided physical evidence of a boundary layer with large strain gradients near free edges. All of the above studies have clearly shown interlaminar stresses are the result of the mismatch in pois- son's ratios and coefficients of mutual influence and the presence of a stress free boundary. The laminate stacking sequence plays n important role in the magnitude and sign of the interlaminar stresses [30230430 es of the deformations of two-dimensional finite element grids of the generic cross-section near the free edge f carbon epoxy laminates under axial loading( Buczek et al, 1983) Results are presented for two different stacking sequences of quas isotropic laminates with the +45 always adjacent to one another It is evident from this figure that the displacements(and related stresses and strains)are a strong function of the stacking sequence with the interlaminar normal stresses being positive or negative depending upon the stacking sequence. Figs. 17 and 18 combined provide a complete picture of the possible edge effects on the three generic planes (top face, free edge and transverse cross-section)of Fig 18(Herakovich et al, 1984) shows Moire fringe patterns for Fig. 18. Moire fringe patterns-angle-ply laminates the axial displacements on the face and free edge of angle-ply, car- bon/epoxy laminates subjected to axial loading On the coupon face it is evident that the width of the edge effect is approximatelyequal 8. Unsymmetric laminates to the thickness of the laminate On the edge, the shear strains y2 are proportional to the gradient of the fringe lines and are max As noted previously in the section on lamination theory, ima at the interfaces between the layers: the 10 and 30 laminates unsymmetric laminates exhibit coupling between inplane and exhibit much higher shear strain(and stress)than the 450 lami- out-of-plane responses. Hyer(1988)reviewed many features of nate. Analytical studies are in agreement with the fiber orientation dependence which is directly related to the mismatch in layer prop erties. It is also evident from the figure that the displacements on 3 This section was written by Mike Hyer, with a very few modifications provided the face and on the edge are uniform along the length by the author

10 C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 Fig. 17. Free edge deformations – quasi-isotropic laminates. formulated as a two-dimensional finite element problem because ofthe independence ofthe stress and strain states on the axial coor￾dinate. The finite element formulation for cross-ply laminates as a two-dimensional problemwas presented by Foye and Baker (1971), and the two-dimensional finite element formulation for laminates including off-axis layers, thermal stresses and non-linear response was presented by Herakovich et al. (1976). Noteworthy approximate analytical solutions include a pertur￾bation solution by Tang (1975), a variational approach by Pagano (1978), a solution employing complex stress potentials and eigen￾function series by Wang and Choi (1982), and solutions based upon statically admissible stress states (Kassapoglou and Lagace, 1986; Rose and Herakovich, 1993). The free edge problem has also been investigated experimentally, e.g., Pipes and Daniel(1971), Oplinger et al. (1974), and Herakovich et al. (1984). The experimental inves￾tigations provided physical evidence of a boundary layer with large strain gradients near free edges.All ofthe above studies have clearly shown interlaminar stresses are the result of the mismatch in Pois￾son’s ratios and coefficients of mutual influence and the presence of a stress free boundary. The laminate stacking sequence plays an important role in the magnitude and sign of the interlaminar stresses. Fig. 17 shows examples ofthe deformations oftwo-dimensional finite element grids of the generic cross-section near the free edge of carbon/epoxy laminates under axial loading (Buczek et al., 1983). Results are presented for two different stacking sequences of quasi￾isotropic laminates with the ±45◦ always adjacent to one another. It is evident from this figure that the displacements (and related stresses and strains) are a strong function of the stacking sequence with the interlaminar normal stresses being positive or negative depending upon the stacking sequence. Figs. 17 and 18 combined provide a complete picture of the possible edge effects on the three generic planes (top face, free edge and transverse cross-section) of a finite width coupon under axial load. Fig. 18 (Herakovich et al., 1984) shows Moiré fringe patterns for the axial displacements on the face and free edge of angle-ply, car￾bon/epoxy laminates subjected to axial loading. On the coupon face itis evidentthatthe width ofthe edge effectis approximately equal to the thickness of the laminate. On the edge, the shear strains zx are proportional to the gradient of the fringe lines and are max￾ima at the interfaces between the layers; the 10◦ and 30◦ laminates exhibit much higher shear strain (and stress) than the 45◦ lami￾nate. Analytical studies are in agreement with the fiber orientation dependence which is directly related to the mismatch in layer prop￾erties. It is also evident from the figure that the displacements on the face and on the edge are uniform along the length Fig. 18. Moiré fringe patterns – angle-ply laminates. 8. Unsymmetric laminates3 As noted previously in the section on lamination theory, unsymmetric laminates exhibit coupling between inplane and out-of-plane responses. Hyer (1988) reviewed many features of 3 This section was written by Mike Hyer, with a very few modifications provided by the author