1、NASA CONTRACTOR REPORT STRENGTH CHARACTERISTICS OF COMPOSITE MATERIALS by Stephen W. Tsm Prepared under Contract No. NAS 7-215 by PHILCO CORPORATION Newport Beach, Calif. f OY NATIONAL AERONAUTICS AND SPACE ADMINISTRATION l WASHINGTON, D. C. l APRIL 1965 Provided by IHSNot for ResaleNo reproduction
2、or networking permitted without license from IHS-,-,-NASA CR-224 STRENGTH CHARACTERISTICS OF COMPOSITE MATERIALS By Stephen W. Tsai Distribution of this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it.
3、 Prepared under Contract No. NAS 7-215 by PHILCO CORPORATION Newport Beach, Calif. for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $4.00 II. Provided by IHSNot for ResaleNo reproducti
4、on or networking permitted without license from IHS-,-,-I I I., a-.,. . - . -m- .- - -.-. -. ._- Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FOREWORD This is an annual report of the work done under the National Aeronautics and Space Administratio
5、n Contract NAS 7-215, “Structural Behavior of Composite Materials, ” for the period January 1964 to January 1965. The program is monitored by Mr. Norman J. Mayer, Chief, Advanced Structures and Materials Application, Office of Advanced Research and Technology. The author wishes to acknowledge the co
6、ntributions of his colleague Dr. Victor D. Azzi, and his consultants Dr. George S. Springer of the Massachusetts Institute of Technology, and Dr. Albert B. Schultz of the University of Delaware. Mr. Rodney L. Thomas contribution in the experimental work, and Mr. Douglas R. Doner and Miss Alena Fongs
7、 contributions in the numerical analysis and computation are also acknowledged. . . . 111 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ABSTRACT The strength characteristics of quasi-homogeneous, nonisotropic materials are derived from a generalize
8、d distortional work criterion. For unidirectional composites, the strength is governed by the axial, transverse, and shear strengths, and the angle of fiber orientation. The strength of a laminated composite consisting of layers of uni- directional composites depends on the strength, thickness, and
9、orientation of each constituent layer and the temperature at which the laminate is cured. In the process of lamination, thermal and mechanical interactions are induced which affect the residual stress and the subsequent stress distribution under external load. A method of strength analysis of lamina
10、ted composites is delineated using glass-epoxy composites as examples. The validity of the method is demonstrated by appropriate experiments. Commonly encountered material constants and coefficients for stress and strength analyses for glass-epoxy composites are listed in the Appendix. iv Provided b
11、y IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CONTENTS SECTION PAGE 1 INTRODUCTION 2 STRENGTH OF ANISOTROPIC MATERIALS Mathematical Theory . . . . . . Quasi-homogeneous Composites . . . Experimental Results . . . . . . 3 STRENGTH OF LAMINATED COMPOSITES Math
12、ematical Theory . . . . . . Cross-ply Composites . . . . . . Angle-ply Composites . . . . . . Structural Behavior of Composite Materials . Scope of Present Investigation . . . 4 CONCLUSIONS . . . . . . . . . . REFERENCES . . . . . . . . . . . . APPENDIX. . . . . . . . . . . . . . . . 1 2 5 7 13 19 2
13、8 43 53 57 59 V Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ILLUSTRATIONS Figure 1. Comparative Yield Surfaces . . . . . . . . . . . . 8 Figure 2. Coordinate Transformation of Stress 9 Figure 3. Tensile Test Specimens . 14 Figure 4. Strength of U
14、nidirectional Composites . 16 Figure 5. Strength of a Typical Cross-ply Composite . 37 Figure 6. Strength of Cross-ply Composites 39 Figure 7. Thermal Warping of a Two-layer Composite 40 Figure 8. Strength of Angle-ply Composites 51 vi Provided by IHSNot for ResaleNo reproduction or networking permi
15、tted without license from IHS-,-,-NOMENCLATURE A = A 1J = In-plane stiffness matrix, lb/in. :C A: lj = A: = Intermediate in-plane matrix, in. /lb A. . = A 1J = In-plane compliance matrix, in. /lb B = B 1J = Stiffness coupling matrix, lb g: B = B iJ = Intermediate coupling matrix, in. B!. = B 1J = Co
16、mpliance coupling matrix, 1 /lb C 13 = Anisotropic stiffness matrix, psi D = D = Flexural stiffness matrix, lb-in. iJ Df:. = D”: = Intermediate flexural matrix, lb-in. iJ D!. = D 1J = Flexural compliance matrix, l/lb-in. E = Youngs modulus, psi E11 = Axial stiffness, psi 9: H. = Hz” 1J = Intermediat
17、e coupling matrix, in. h = Plate thickness, in. M. =M M; = Distributed bending (and twisting) moments, lb = MT = Thermal moment, lb Kri =m = Effective moment = Mi + MT m = cos 0, or = cross-ply ratio (total thickness of odd layers over that of even layers) vii Provided by IHSNot for ResaleNo reprodu
18、ction or networking permitted without license from IHS-,-,-NOMENCLATURE (Continued) N. = N N; = Stress resultant, lb/in. = NT = Thermal forces, lb/in. FT. =fi 1 = Effective stress resultant = Ni + NT n 1 sin 8, or = total number of layers P = Ratio of normal stresses = a2/ 01 9 = Ratio of shear stre
19、ss = as/ 01 r = Ratio of normal strengths = X/Y S = Shear strength of unidirectional composite, psi S = Shear strength ratio = X/s ij = Anisotropic compliance matrix, l/psi T = Temperature, degree F T+ = Coordinate transformation with positive rotation T- = Coordinate transformation with negative ro
20、tation X = Axial strength of unidirectional composite, psi Y = Transverse strength of unidirectional composite, psi a. 1 = Thermal expansion matrix, in. /in. /degree F f. 1 = Strain component, in. /in. 0 F . 1 = In-plane strain, component, in. /in. viii Provided by IHSNot for ResaleNo reproduction o
21、r networking permitted without license from IHS-,-,-8 K. 1 x u. 1 r iJ u u 12 v21 NOMENCLATURE (Continued) = Fiber orientation or lamination angle, degree = Curvature, I/in. = 1 - V12 v21 = Stress components, psi = Shear stress, psi = Poissons ratio = Major Poissons ratio = Minor Poissons ratio SUPE
22、RSCRIPTS t = Positive rotation or tensile property = Negative rotation or compressive property k = k-th layer in a laminated composite -1 = Inverse matrix SUBSCRIPTS 1, J = I, 2, . . . 6 or x, y, z in 3-dimensional space, or = 1, 2, 6 or x, y, s in 2-dimensional space ix Provided by IHSNot for Resal
23、eNo reproduction or networking permitted without license from IHS-,-,-SECTION 1 INTRODUCTION Structural Behavior of Composite Materials The purpose of the present investigation is to establish a rational basis of the designs of composite materials for structural applications. Ultimately, materials d
24、esign can be integrated into structural design as an added dimension. Higher performance and lower cost in materials and structures applications can therefore be expected. Following the research method outlined previously, 1* the present program combines two traditional areas of research - materials
25、 and structures. These two areas are linked by a mechanical constitutive equa- tion, the simplest form of which is the generalized Hookes law. The mate- rials research is concerned with the influences of the constituent materials on the coefficients of the constitutive equation, which in this case,
26、are the elastic moduli. The structures research, on the other hand, is concerned with the gross behavior of an anisotropic medium. An integrated structural design takes into account, in addition to the traditional variations in thick- nesses and shapes, the controllable magnitude and direction of ma
27、terial properties through the selection of proper constituent materials and their geometric arrangement. “References are listed at the end of this report. 1 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Following the framework just described, the e
28、lastic moduli of aniso- tropic laminated composites were reported previously. 2, 3 The appropriate constitutive equation was: This equation, of course, included the quasi-homogeneous orthotropic com- posite, which represented a unidirectional composite, as a special case. The material coefficients A
29、, B, and D were expressed in terms of material and geometric parameters associated with the constituent materials and the method of lamination. This information provided a rational basis for the design of elastic stiffnesses of an anisotropic laminated composite. Thus, the investigation reported in
30、References 2 and 3 involved both structures research, in the establishment of Equation (1) as an appropriate constitutive equation, and materials research, in the establishment of the parameters that govern the material coefficients of Equation (1). The present report covers the strength characteris
31、tic of anisotropic laminated composites, which again includes the quasi-homogeneous com- posite, as a special case. Unlike the case of the elastic moduli, the present report covers only the structures aspect of strengths; the materials aspect is to be investigated in the future. The appropriate cons
32、titutive equation for the strength characteristics is established in this report. Only when this information is available, can the area of research from the materials stand- point be delineated. Guidelines for the design of composites from the strength consideration can be derived. Scope of Present
33、Investigation The present investigation is concerned with the structures aspect of the strength characteristics of composite materials. The strength of a quasi-homogeneous anisotropic composite is first established. Then the strength of a laminated composite consisting of layers of quasi-homogeneous
34、 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-composites bonded together is investigated. The validity of the theoretical predictions is demonstrated by using glass-epoxy resin composites as test specimens. The main result of this investigation is
35、 that a more realistic method of strength analysis than the prevailing netting analysis is obtained. The structural behavior of composite materials is now better understood, and one can use these materials with higher precision and greater confidence. A stride is made toward the rational design of c
36、omposite materials. Although much more analyses and data generation still remain to be done, the present knowledge of stiffnesses and strengths of composite materials, as reported in References 2 and 3, and in this report, respectively, is approaching the level of knowledge presently available in th
37、e use of isotropic homogeneous materials. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SECTION 2 STRENGTH OF ANISOTROPIC MATERIALS Mathematical Theor
38、y Several strength theories of anisotropic materials are frequently encountered in the study of composite materials. Hill postulated a theory in 1q484 and later repeated it in his plasticity book. 5 Using his notation, it is assumed that the yield condition is a quadratic function of the stress comp
39、onents 2f ( oij) = F ( uy - oz) t G ( oz - ox)2 f H(Ox - 7) 2 (2) t2L r 2 t 2M r 2 YZ Z;t2Nr =1 XY where F, G, H, L, M, N are material coefficients characteristic of the state of anisotropy, and x, y, z are the axes of material symmetry which are assumed to exist. This yield condition is a generaliz
40、ation of von Mises condition proposed in 1913 for isotropic materials. Note that Equation (2) reduces to von Mises condition when the material coefficients are equal. Beyond this necessary condition, there seems to be no further rationale. Nevertheless, this yield condition has the advantages of bei
41、ng reasonable and readily usable in a mathematical theory of strength because it is a con- tinuous function in the stress space. For identification purposes, this con- dition will be called the distortional energy condition. Provided by IHSNot for ResaleNo reproduction or networking permitted withou
42、t license from IHS-,-,-Marin proposed a strength theory equivalent to Equation (2), except the principal stress components were used instead of the general stress components. The use of principal stresses is, in fact, more difficult to apply to anisotropic materials, since the axes of material symme
43、try, the principal stress, and the principal strain are, in general, not coincident. Thus, principal stresses per se do not have much physical significance. Another strength theory of anisotropic material is called the “inter- action formula, ” as described by a series of reports by the Forest Produ
44、cts Laboratory 7, 8, 9 and apparently independently by Ashkenazi. 10 The interaction formula in Hills notation” takes the following form: ux 2 ( ) - - X “,- + (Y$)2t(+Y)2=1 plates, ( ) 2 3. - Y *y; + ($)2 t (.+,” = 1 uz ( 2 - - Z y$ + (rz)2t (LE) = 1 (3) Since the composite material of interests now
45、 is in the form of thin a state of plane stress is assumed. Then Equations (2) and (3) can be reduced, respectively: z i. e., the strength characteristics as a function of the orientation of the symmetry axes, 8. For uniaxial tension, p = q = 0, the failure condition is m4+ (s 2 - 1) m2n2 f r2n4 = (
46、X/ alI2 or C 112 7 = X/ m4+ (S2 - 1) m2n2 + r2n4 1 (9) (10) 10 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Thus, by performing uniaxial tension tests on specimens with different orientations of the material symmetry axes; i. e., different values
47、of 0, one finds directly the transformation property of strength. What is equally important is that the strength characteristics of a quasi-homogeneous aniso- tropic material under combined stresses are simultaneously verified. By a simple substitution of Equation (6) into (q), while maintaining p =
48、 q = 0, one recovers, as expected, the original yield condition shown in Equation (4). Equation (8) can be reduced to other simple cases in a straight- forward manner. For example, the case of hydrostatic pressure requires p = 1, q = 0, from which one can show that the maximum pressure is equal to the transverse strength, Y, and is independent of the orientation, 8. The case of an internally pressuri
