Raymond D. Mindlin and Applied Mechanics

Transcription

1 Raymond D. Mindlin and Applied Mechanics Yih-Hsing Pao, Ph.D. (Columbia University, 1959) Professor of College of Civil Engineering and Architecture, Zhejiang University Professor Emeritus, Institute of Applied Mechanics, National Taiwan University Joseph C. Ford Professor Emeritus, Theoretical and Applied Mechanics, Cornell University In collaboration with C.-S. Lam, Ph.D. (Princeton University, 1986) Mindlin Centennial Symposium 15th U.S. National Congress of Theoretical and Applied Mechanics Boulder, Colorado, June 5-30,

2 Raymond D. Mindlin, Ph.D. Sept. 17, 1906 ~ Nov., Contents Abstract and Diagrams I. Biographical Sketch (p. 1-7) II. R. D. Mindlin and Applied Mechanics (p. 8-17) III. Theories of Plates by R. D. Mindlin (p. 18-3) Epilogue 0 4

3 Raymond D. Mindlin-His Life ( ) Born in New York City, New York, in 1906; educated in private elementary and preparatory schools. College Education at Columbia University in the City of New York (194-36): Bachelor of Arts in 198 (B.A.); Bachelor of Science in 1931 (B.S.); Civil Engineer in 193 (C.E.); Doctor of Philosophy in 1936 (Ph.D). Teaching and Research Career in the School of Civil Engineering, Columbia University ( ): Assistant ( ); Instructor ( ); Assistant Professor ( ); Associate Professor ( ); Professor ( ); James Kip Finch Professor of Applied Science ( ); Retired in World War II Service (194-46) at the Johns Hopkins University Applied Physics Laboratory: Awarded the Presidential Medal for Merit in 1946 by US Navy (for developing the proximity fuse of the anti-aircraft gun shell). Research Contributions: to the fields of applied mechanics, structural engineering, geotechnical engineering, applied physics (acoustics, photoelasticity, piezoelectricity). Honors and Awards: member of the National Academy of Engineering (1966), member of the National Academy of Sciences (1973). von Karman Medal of the ASCE in 1961, Timoshenko Medal of the ASME in 1964, C.B. Sawyer Award of the Army Electronics Command in 1967, Honorary Member in 1969 and ASME Medal in 1976 of ASME (American Society of Mechanical Engineering). National Medal of Science (a Presidential Award) in Died in Hanover, New Hampshire in

4 Springer-Verlag, New York, Berlin, 1989 Edited by H. Deresiewicz, M. P. Bieniek, F. L. DiMaggio 3 Mindlin s Publications ( ) 4

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11 17 Theories of Plates by R. D. Mindlin Germain-Lagrange s Equation of Plates ( ) ( D w) + ρh w/ t = q( x, y, t) w-transverse Motion of a Plate in xy plane h-plate thickness, p-mass density EI 3 D GI, I h (1 ν ) (1 ν) 1 D-Flexural Rigidityc E-Young s modulus G-shear modulus v-poisson s ratio Mindlin s Equation of Plates (1951) ρ w ( )( D ρi ) w+ ρh κ G t t t D ρi = (1 + ) qxyt (,, ) κ Gh κ Gh t * Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates in Journal of Applied Mechanics, 1951 κ -shear coefficient 18

12 Eqs. of Elasticity and Eqs. of Plate u i -displacement component in x i direction (i=1,,3) σ ij -stress component, ( i ) ε ij -strain component, ( ε ij = ( iuj + jui ) ) y z ψ y w x ψ x 1. Stress-Equation of Motion y xσ xx + yσxy + zσxz xσ yx + yσ yy + zσ yz = ρu x = ρu xσ zx + yσzy + zσzz = ρu z. Stress-Displacement Relation x i h Plate displacement Plate stress M + M Q = ρi ψ + = xqx + yqy + q = ρhw u ( x, t) = zψ ( x, y, t), α = x, y 1. Plate Stress Equation of Motion, α x x y yx x x M M Q ρi ψ y x yx y y y. Plate Stress and Plate-Displacement Relation GIµ ψα ψ ψ x y Mα = ( βψβ) + GI, α, β = x, y βψβ = + 1 µ α x y ψ ψ α β Mαβ = GI( + ), αβ, = x, y α β β α Qα w = + ψ, shear coefficient α κ κ Gh α 3. Plate Displacement Equation of Motion (Mindlin, 1951) α u ( x, t) w( x, y, t) 3 h [ M x, M y, M x y ] h z [ σ x x, σ y y, σ x y ] d z h [ Qx, Qy] h[ σxz, σ yz] dz x x 3. Displacement- Equation of Motion (Navier-Cauchy, 18-8) ( λ + G) ( u ) + G u + ρ f = ρu i j j j j i i i = κ Gh( w + βψβ) + q = ρhw ( D )[(1 µ ) ψ α (1 µ ) α( βψβ)] κ Gh( ψα αw) ρiψα ρ w D ρi ( )( D ρi ) w+ ρh = (1 + ) q( x, y, t) κ G t t t κ Gh κ Gh t 19 Cauchy s Eq. of Motion(188) σ + ρ f = ρu j ji i i Higher Order Theories of Plates u ( x, x, x, t) = u ( x, x, t) + x u ( x, x, t) + x + (0) (1) i i 3 i u w, u = xψ (0) (1) 3 1, 3 1, Generalized Hooke s Law for Crystalline Solids k σij εkl = = cijkl cijkl ul Equation of Motion for Crystals ( c u ) + ρ f = ρu j ijkl k l i i Zeroth-Order (Extensional Motion) u1 = u( x1, x, t) (0) u = v( x1, x, t) (0) First Order (Thickness Shear and Flexural Motion) uα = x3ψα( x1, x, t), α = 1, (1) u = w( x1, x, t) (0) 3 0

13 * Thickness-Shear and Flexural Vibrations of Crystal Plates in Journal of Applied Physics, 1951 f = ω/ π = 1.664MHz 1 An Introduction to the Mathematical Theory of Vibrations of Elastic Plates

14 *Mechanics Research Communications, 13, (1986) 3 Epilogue The Canterbury Tales, by Geoffrey Chaucer ( ) Of study took he utmost care and heed Not one word spoke he more than was his need And that was said in fullest reverence And short and quick and full of high good sense Pregnant of moral virtue was his speech And gladly would he learn and gladly teach. *Excerpt from Raymond David Mindlin, A Biographical Sketch, 1989 by H. Deresiewicz -8-