A Software Tool for Mechanics of Composite Materials*

Transcription

1 Int. J. Engng Ed. Vol. 13, No., p. 433±441, X/1 $ Printed in Great Britain. # 17 TEMPUS Publications. A Software Tool for Mechanics of Composite Materials* AUTAR K. KAW and GARY WILLENBRING ENB 11, Mechanical Engineering Department, Universit of South Florida, Tampa, FL , USA The development and implementation of a software tool PROMAL' for teaching a course in Mechanics of Composite Materials is presented. The software complements the wor on length linear algebra computations for parametric studies and homewor assignments. Most important, at the end of the course it allows the students to design and analze laminated structures and appreciate their applications. PROMAL is both an instructional tool for the instructor and a guide to the student as it shows intermediate steps to the final calculations. INTRODUCTION TAUGHT first in 1, as a conventional `chal and tal' lecture course, b the first author (hereafter referred to as `I'), Mechanics of Composite Materials is now a model course in the College of Engineering at the Universit of South Florida. The course has `integrated' modern and traditional tools such as videos, slide shows, tours of companies using composite materials, lins on the World Wide Web to educational sites about composites, computerized multiple-choice question tutorials, open-ended design projects and software. This integration is based on the strong philosoph that all the tools are used onl to complement the lecture course for maing the course more interesting, and easier to understand and appreciate. A tpical course in mechanics of composites starts with a brief introduction of applications, tpes and manufacturing of composites. Then the mechanical behavior of composite materials is eplained tpicall in four main topics. Macromechanical behavior of a single lamina is eplained, including concepts about stress/strain relationship for a lamina, stiffness and strength of a lamina, and the stress/strain response due to temperature and moisture change. Mathematical models for finding mechanical and hgrothermal properties of a lamina in terms of individual properties of the constituents are developed. These properties include stiffness, strength, and coefficients of thermal and moisture epansion. Macromechanics of a single lamina are etended to the macromechanics of a laminate. Stress/ strain equations for a laminate based on individual properties of the laminae that mae the laminate are derived. Then mathematical models for stiffness and strength of a laminate, * Accepted 1 June 17. and effects of temperature and moisture on residual stresses in a laminate are developed. At the end of the course, procedures are made for failure analsis and designing of laminated composites structures. All the above four topics are augmented for instruction and understanding b two software tools. The first one is a Windows-based [1] instructional software called PROMAL [2]. The second one is the use of smbolic manipulators such as MACSYMA [3] in understanding the mechanics of composite materials described in Reference [4]. The purpose of this paper is therefore onl to discuss the development and implementation of the first software toolðpromal. For more information, please visit the WWW site at HISTORY An introductor course in mechanics of composite materials involves simple linear algebra procedures that are otherwise length [2]. So to appreciate the fundamentals, show intermediate steps in analsis, and conduct industrial strength analsis and design within a reasonable time, a suitable computer tool is essential. This has been the motivation for the development of the software tool PROMAL (acronm for PROgram for Micromechanical and Macromechanical Analsis of Lamina and Laminates). PROMAL is an in-house product developed at the Universit of South Florida. In 1, independent stud student, Steven Jourdenias and I embared on an ambitious project of developing an interactive software pacage for the mechanics of composite materials. So after eighteen months of development, PROMAL [5] (DOS version) was published nationall. Since then several students who too the mechanics of composites course continued woring with me 433

2 434 A. K. Kaw and G. Willenbring on adding other components as well as developing a graphical interface for the program. Using the Microsoft Visual Basic [] computer language, we introduced PROMAL (Windows version) in 1. The combined effort of the facult and students has generated a qualit product that meets the needs of students and the instructor. The eperiences of those who shared the authoring of the pacage with me have been rich. It has helped them not onl in better understanding of the course and in learning new programming sills, but also the have left an ecellent product for future students. Man of these students have found emploment or pursued graduate school in the field of composites. FEATURES OF PROMAL PROMAL is a Windows-based interactive software pacage and has five main programs given below. Matri algebra Throughout the course of mechanics of composite materials, the most used mathematical procedures are based on linear algebra. This feature of PROMAL allows the student to multipl matrices, invert square matrices and find the solution to a set of simultaneous linear equations. Man students now have programmable calculators and access to tools such as MATHCAD [7] to do such manipulations, and we have included this program onl for convenience. This program allows the student to concentrate on the fundamentals of the course as opposed to spending time on length matri manipulations. Lamina properties database In this program, the properties of unidirectional laminas can be added, deleted, updated and saved as shown in Fig. 1. This is useful as these properties can then be loaded in other parts of the program without repeated inputs. Macromechanical analsis of a lamina Using the properties of unidirectional laminas saved in the above database, one can find the stiffness and compliance matrices, transformed stiffness and compliance matrices, engineering constants, strength ratios based on four major failure theories, and coefficients of thermal and moisture epansion of angle laminas. These results are then presented in tetual, tabular and graphical forms. The stiffness and the compliance matrices for a unidirectional pl are found b using the following equations [2], respectivel. Fig. 1. Lamina properties database form.

3 A Software Tool for Mechanics of Composite Materials 435 Q 11 Q 12 0 QŠ ˆ 4 Q 12 Q ; 0 0 Q S 11 S 12 0 SŠ ˆ 4 S 12 S ; 1a; b 0 0 S where, S 11 ˆ 1=E 1 ; S 22 ˆ 1=E 2 ; S 12 ˆ 12 =E 1 ; S ˆ 1=G 12 ; 2a; b and Q 11 ˆ E 1 = ; Q 22 ˆ E 2 = ; 3a; b Q 12 ˆ E 1 21 = ; Q ˆ G 12 3c; d S Š ˆ QŠ 1 4 The elastic moduli for an angle lamina (Fig. 2) are given as [2]: 1=E ˆ cos 4 =E 1 1=G =E 1 sin 2 cos 2 sin 4 =E 2 5a ˆ E 12 sin 4 cos 4 =E 1 1=E 1 1=E 2 1=G 12 sin 2 cos 2 Š 5b 1=E ˆ sin 4 =E 1 1=G =E 1 sin 2 cos 2 cos 4 =E 2 5c 1=G ˆ 2 2=E 1 2=E =E 1 1=G 12 sin 2 cos 2 cos 4 sin 4 =G 12 5d m ˆ sin 2 K M cos 2 m ˆ sin 2 K M sin 2 5e 5f where K ˆ E 1 =E 2 12 E 1 = 2G 12 ; M ˆ 1 E 1 =E E 1 =G 12 The stress-strain relations in the global - aes are found b using [2]: 2 3 " 1=E =E m =E 1 7 " 4 =E 1=E m =E 1 5 ˆ m =E 1 m =E 1 1=G 7 The stresses and strains in the local 1±2 aes, and the stiffness and compliance matrices for an Fig. 2. Local and global aes of an angle lamina. angle pl are found b using the transformation matrices [2]. Micromechanics analsis of lamina Using elastic moduli, coefficients of thermal and moisture epansion, and specific gravit of fiber and matri, one can find the elastic moduli, and coefficients of thermal and moisture epansion of a unidirectional lamina. Again, the results are available in tetual, tabular and graphical forms. The above properties of a unidirectional lamina are found b using the following equations [2]. E 1 ˆ E f V f E m V m a E 2 ˆ E f E m = E f V m E m V f b G 12 ˆ G f G m = G f V m G m V f c 12 ˆ f V f m V m d 21 ˆ 12 E 2 =E 1 e P f =P c ˆ E f V f = E f V f E m 1 V f f 1 ˆ f E f V f m E m 1 V f Š E f V f E m 1 V f Š g 2 ˆ 1 f f V f 1 m m 1 V f 1 v 12 h 1 ˆ m E m c = E 1 m i 2 ˆ 1 m m c = m 1 12 j Macromechanics of a laminate Using the properties of the lamina properties from the database, one can analze laminated structures. These laminates (Fig. 3) ma be hbrid and unsmmetric. The output includes finding stiffness and compliance matrices, global and local strains, and strength ratios in response to mechanical, thermal and moisture loads. This program is used for design of laminated structures such as plates and thin pressure vessels at the end of the course. The (etensional) [A], (coupling) [B], and

4 43 A. K. Kaw and G. Willenbring Fig. 3. Schematic of a laminate. (bending) [D] stiffness matrices and the normalized [A], [B], and [D] stiffness matrices are calculated using the following equations[1]. A ij ˆ Xn Q ij z z 1 Š a ˆ1 B ij ˆ 1 X n Q ij 2 z 2 z2 1 Š ˆ1 D ij ˆ 1 X n Q ij 3 z 3 z3 1 Š ˆ1 b c The normalized stiffness matrices are given b [2]: A Š ˆ 1=t AŠ; B Š ˆ 2=t 2 BŠ; 10a c D Š ˆ 12=t 3 DŠ; Note that the above summations, and all other summations in this tet, are from ˆ 1 to n where n is the number of plies in the laminate, and is the pl number from top to bottom. Also, z is the location of the th pl relative to the midplane of the laminate. The total thicness of the laminate is t. The laminate stiffness and compliance matrices are defined as follows. $ Laminate stiffness matri ˆ A B % 11 B D " Laminate compliance matri ˆ A B # 1 B D " ˆ a b # 12 c d The compliance matrices normalized with respect to the laminate thicness are given b [2]: a Š ˆ t aš; b Š ˆ t 2 =2 bš; 13a c d Š ˆ t 3 =12 d Š; The equivalent plate properties are valid onl if there is no coupling between the forces and moments. This is the case onl for smmetric Fig. 4. Co-ordinate locations of laminae in a laminate.

5 A Software Tool for Mechanics of Composite Materials 437 laminates. These quantities are found b using the following equations [2]. Inplane longitudinal Young's modulus E 0 ˆ 1=a 11 14a Inplane transverse Young's modulus E 0 ˆ 1=a 22 14b Inplane shear modulus G 0 ˆ 1=a 14c Inplane Poisson's ratio 0 ˆ a 12 =a 11 14d Fleural longitudinal Young's modulus E f ˆ 1=d 11 14e Fleural transverse Young's modulus E f ˆ 1=d 22 14f Fleural shear modulus G f ˆ 1=d 14g Fleural Poisson's ratio f ˆ d 12 =d 11 14h The coefficients of thermal epansion for a laminate ; are found b calculating the midplane strains resulting from a unit change in temperature. Liewise, the swelling coefficients for a laminate ; are found b calculating the midplane strains caused b absorption of a unit moisture content. The effective forces and moments, and the midplane strains and curvatures are found from the stress/strain relations [2] given below. 2 3 N A 11 A 12 A 1 B 11 B 12 B 1 N A 12 A 22 A 2 B 12 B 22 B 2 N A 1 A 2 A B 1 B 2 B ˆ M B 11 B 12 B 1 D 11 D 12 D 1 M B 12 B 22 B 2 D 12 D 22 D M B 1 B 2 B D 1 D 2 D " 0 " 0 0 N N N N N N M N M N M N 15 where, N N N M M M N N N N N N M N M N M N ˆ Xn ˆ1 z ˆ Xn ˆ1 z Q ij Š z 1 z ˆ Xn ˆ1 z 1 z Q ij Š z z ˆ Xn ˆ1 z 1 " 0 " 0 0 z dz Q ij Š z z z z 1 " 0 " 0 0 z 2 dz dz z dz Q ij Š T dz Q ij Š z 1 Q ij Š z C dz7 5 1 Tz dz 3 Cz dz7 5 1 N, N, and N are stress resultants over the thicness t of the laminate; M, M, and M are the resultant moments. " 0, "0, 0,,, and are the midplane strains and curvatures; and are the transformed coefficients of thermal epansion and swelling, respectivel for each pl, T is the temperature difference, C is the moisture content, and N N, N N, N N, M N, M N, and M N are the fictitious forces and moments induced b temperature difference and moisture. The global strains and stresses are found b using the following equations [2]. " " " ˆ " 0 " 0 0 z 20

6 43 A. K. Kaw and G. Willenbring ˆ Q 11 Q 12 Q 1 4 Q 12 Q 22 Q Q 1 Q 2 Q " 4 " IMPLEMENTATION T C Each student is given a cop of the PROMAL software in the beginning of the course. Also, it is set up in all the engineering and departmental laboratories at the universit. The following are the e features of the implementation of the software. 1. Using a laptop computer and a projection sstem, PROMAL is set up for each lecture. It is used onl if a student ass a question which I can answer using PROMAL. Tpical questions relate to parametric studies or showing several other eamples of the wor shown in class. Out of three class meetings per wee, I hold one meeting in a computer laborator. The laborator has one computer for ever student and each des has a video monitor connected to the instructor's computer. This allows me to conduct numerical investigations which the student can directl see. Then the student conducts similar investigation with different inputs to reinforce the fundamentals. During this class, I conduct parametric studies and also teach how to use the computer program. Although I could teach this class on all das in the computer laborator, I have not done so because of the nature and need for efficient use of computer laboratories. Because of the monitor heights being at ee level, students have a harder time concentrating on the material unrelated to the computer implementation. Also, the demand of computer laboratories b students for other courses and general use does not mae it an efficient use of such facilities. 2. With a few eceptions, I do not allow students to use PROMAL to submit their homewor assignments. The can, however, use the Matri Algebra program of PROMAL or use mathematical tools such as MATHCAD [7] to submit their assignments. The can also verif their results as it not onl shows the results but also intermediate steps. 3. Onl nonprogrammable (but scientific) calculators are allowed in the eaminations. This is mainl done to reinforce the fact that the have to understand the fundamentals of the course as opposed to taing a `blac bo' approach to the course. 4. At the end of the course, I require that students design structures made of composite materials. The final design project carries 20% of the overall grade. Students use PROMAL and the come up with different designs as most problems are open ended. Then students discuss the various alternatives with me and see how different people approach the same problem differentl. 5. Some e features of the program which mae PROMAL different from man other similar programs are that one can switch between the sstem of units with a single clic, choose printing the outputs to a printer or a tet file which the can merge with other reports, and show each input and output clearl with used sstem of units. The emphasis on international sstem of units is critical [] with ever increasing global marets where our students will not onl be competing but also forming partnerships. ILLUSTRATIVE EXAMPLES The following eamples illustrate how PROMAL can be used in teaching the mechanics of composite materials course. These eamples show how PROMAL illustrates the fundamentals of the course and also shows the application of PROMAL to practical industrial designs. Eample for macromechanics of a lamina Structures subjected to torsion, such as a driving shaft, use angle laminas. This is because the angle plies provide more shear stiffness and strength than unidirectional lamina. For a graphite/epo unidirectional lamina with the following properties: Longitudinal Young's modulus ˆ 11 GPa Transverse Young's modulus ˆ 10.3 GPa Major Poisson's ratio ˆ 0.2 In-plane shear modulus ˆ 7.17 GPa Longitudinal tensile strength ˆ 1500 MPa Longitudinal compressive strength ˆ 1500 MPa Transverse tensile strength ˆ 40 MPa Transverse compressive strength ˆ 24 MPa In-plane shear strength ˆ MPa Find the angle of the lamina for which the shear modulus is a maimum. Also, find this value of the maimum shear modulus and compare with the shear modulus of the unidirectional lamina. Solution. Inputting the above properties in the lamina database, one can load the properties in the macromechanics of a lamina program. Using tabular or graphical output choice, one finds that the maimum shear modulus is for a 45 lamina (Fig. 5). The value of shear modulus at this angle

7 A Software Tool for Mechanics of Composite Materials 43 Fig. 5. Shear modulus as a function of the angle of lamina for a tpical graphite/epo lamina. is found as.4 GPa as compared to 7.17 GPa for a unidirectional lamina. An etension of this eample would be to stud the influence of the ratio of the longitudinal to transverse modulus of a unidirectional lamina on the shear modulus of the angle lamina. Eample for micromechanics of a lamina In space applications such as antennas, dimensional stabilit in certain directions under thermal loading is a e factor in its performance. Some tpes of graphite fibers have a negative thermal epansion coefficient in the aial direction. B combining it with a matri such as epo, the longitudinal coefficient of thermal epansion can be made zero. Find the fiber volume fraction for which the longitudinal coefficient of thermal epansion of a graphite/epo is zero if the properties of graphite and epo are given below. For sae of simplicit and since we see onl the longitudinal coefficient of thermal epansion, we assume that the graphite fiber is isotropic and longitudinal properties of an otherwise transversel isotropic fiber are used. Table 1. Longitudinal coefficient of thermal epansion of a lamina as a function of fiber volume fraction Fiber volume fraction Longitudinal CTE m/m/c Young's modulus of graphite ˆ 300 GPa Young's modulus of epo ˆ 3.5 GPa Poisson's ratio of graphite ˆ 0.20 Poisson's ratio of epo ˆ 0.35 Coefficient of thermal epansion of graphite ˆ 0:35 10 m=m=c Coefficient of thermal epansion of epo ˆ 0 10 m=m=c Solution. From the micromechanics of lamina program, after inputting the above properties, one can tabulate the coefficients of thermal epansion as given in Table 1. From Table 1, for zero longitudinal coefficient of thermal epansion, the fiber volume fraction is between 0. and 0.7. Using different values between 0. and 0.7, one finds that the required fiber volume fraction is 0.7. Eample for macromechanics of laminate Pressure vessels made of composite materials are generall manufactured b filament winding. The winding angle determines the two parametersð resistance to pressure and resistance to epansion in diameter. In some cases of clindrical pressure vessels, such as casings used for burning solid propellants, both design parameters are important. Wh []? The solid fuel burns and creates high pressure. This causes large hoop strains and epansion in the diameter of the casing. If this epansion is allowed, the propellant would crac and the rate of burning would increase and build increasing pressure. Casings made of isotropic materials can reduce the epansion b using ver thic casings which is not an acceptable solution. However, filament-wound casings made of composite materials can reduce the epansion due to the orthotropic nature of composites without resorting to large thicnesses. Determine the optimum angle for resisting maimum pressure and minimizing epansion of

8 440 A. K. Kaw and G. Willenbring Fig.. A filament-wound clindrical pressure vessel subjected to a uniform pressure. the diameter for a filament-wound clindrical pressure vessel. Assume the vessel is a four-pl smmetric laminate = Š S made of graphite/ epo. The properties of a unidirectional graphite/ epo are the same as the first eample in this section. Use the maimum strain failure theor as our failure criterion. Solution. For a thin clindrical pressure vessel under uniform internal pressure loading (Fig. ), the resultant hoop stress (-direction) is twice the resultant longitudinal stress (-direction). Maintaining this ratio between the hoop stress and the longitudinal stress resultants, assume that the normal forces perunitwidthn ˆ 1000 N=mandN ˆ 2000 N=m. Note, onl the ratio of the two values of the normal forces will affect the solution to this problem. You can now find the optimum angle for maimum pressure and minimum epansion of the diameter. First enter the properties of the graphite/epo lamina in the database as shown in Fig. 1. Then enter the laminate stacing sequence = Š S and changing the angle,, one can calculate the minimum strength ratio of the laminate as a function of the angle,. As shown in Table 2, the optimum angle is 52. Strength ratio is the ratio between the failure load and the applied load. For minimizing the epansion of the diameter, one calculates the hoop strain, that is the normal strain in the -direction as a function of the angle,. From Table 3, it shows that the optimum angle is 70. Table 2. Strength ratio as a function of winding angle Angle (degrees) Strength ratio So the two optimum angles are different. Depending on requirements such as weight, one can find a famil of filament winding angles and pressure vessel thicnesses which meet the requirements of maimum pressure and minimum epansion of the diameter. CONCLUSIONS An interactive software tool is used in teaching the mechanics of composites to reinforce the fundamentals of the course, verifing homewor assignments, conducting parametric studies and designing laminated composite structures. PROMAL increases the capabilities of the student to solve pragmatic problems and helps the instructor in using it as an instructional tool. AcnowledgmentsÐI wish to than all the students who have given feedbac and comments about the PROMAL program since 1. Special thans go to the individual students who wored on the developing and rigorous testing of the PROMAL programðsteven Jourdenais, Franc Urso, John Meers, Bruce Boucher, John Palmer, Brian Shanberg, Alan Badstuebner, David Guanio, Rafael Rodriguez. The authors also than three anonmous reviewers of the first manuscript for their helpful comments. NOMENCLATURE AŠ etensional stiffness matri A Š normalized etensional stiffness matri aš etensional compliance matri a Š normalized etensional compliance matri BŠ coupling stiffness matri Table 3. Minimum hoop strain as a function of winding angle Angle Minimum hoop strain (degrees) (m/m)

9 A Software Tool for Mechanics of Composite Materials 441 B Š normalized coupling stiffness matri bš coupling compliance matri b Š normalized coupling compliance matri C moisture content DŠ bending stiffness matri D Š normalized bending stiffness matri d Š bending compliance matri d Š normalized bending compliance matri E Young's modulus G shear modulus m coefficient of mutual influence M bending moment per unit length M N fictitious bending moment per unit length N normal force per unit length N N fictitious normal force per unit length QŠ reduced stiffness matri Q Š transformed reduced stiffness matri S Š reduced compliance matri S Š transformed reduced compliance matri t laminate thicness V volume fraction z location on z-ais from laminate midplane coefficient of thermal epansion swelling coefficient shear strain T change in temperature " normal strain fiber angle curvature Poisson's ratio aial stress shear stress Subscripts 1 along the direction of the fiber 2 perpendicular to the direction of the fiber f fiber propert m matri propert along the -ais along the -ais REFERENCES 1. Microsoft Windows, Version 3.1, Microsoft Corporation, Seattle, WA (13). 2. A. K. Kaw, Mechanics of Composite Materials, CRC Inc., Boca Raton, FL (17). 3. MACSYMA, Smbolics, Inc., USA, (15). 4. A. K. Kaw, Use of smbolic manipulators in mechanics of composites, ASEE Computers in Education Journal, 3 (13) pp. 1±5. 5. A. K. Kaw, An instructional interactive aid for mechanics of composites, International Journal of Mechanical Engineering Education, 20 (10) pp. 213±22.. Microsoft Visual Basic, Version 3.0 Professional Edition, Microsoft Corporation, Seattle, WA (13). 7. Mathcad.0, MathSoft Inc, Cambridge, Massachusetts, USA, (15).. A. K. Kaw, and M. Daniels, Should conversion to the SI sstem continue to be debated? ASCE Journal of Professional Issues in Engineering Education and Practice, 122 (1) pp. ±72.. P. C. Powell, Engineering with Fibre-Polmer Laminates, Chap. 5, Chapman & Hall, London (14). Autar K. Kaw is a Full Professor of Mechanical Engineering at the Universit of South Florida, Tampa. He received his Ph.D. degree in Engineering Mechanics from Clemson Universit, SC in 17. He is the recipient of the 11 SAE Ralph-Teetor Award, the 12 ASEE New Mechanics Educator Award, 14 State of Florida Teaching Incentive Program Award, and the 1 ASEE Southeastern Section Outstanding Contributions in Research Award. His main scholarl interests are in developing new analtical models for understanding mechanics of composite materials and developing instructional software for engineering courses. The Wright-Patterson Air Force Base and the Air Force Office of Scientific Research fund his current research. Gar Willenbring is currentl emploed b Pratt and Whitne, Palm Beach, FL. He received his BS degree in Mechanical Engineering at the Universit of South Florida, Tampa in 1. He wrote the Matri Algebra program of PROMAL as an Independent Stud Student at the Universit of South Florida.