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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 UniversityIn collaboration with

C.-S. Lam, Ph.D. (Princeton University, 1986)

Mindlin Centennial Symposium

15th U.S. National Congress of Theoretical and Applied MechanicsBoulder, Colorado, June 25-30, 2006

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Raymond D. Mindlin, Ph.D.

Sept. 17, 1906 ~ Nov. 22, 1987

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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. 18-23)

Epilogue

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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 (1924-36): Bachelor of Arts in 1928 (B.A.); Bachelor of Science in 1931 (B.S.); Civil Engineer in 1932 (C.E.); Doctor of Philosophy in 1936 (Ph.D).

Teaching and Research Career in the School of Civil Engineering, Columbia University (1933-1975):Assistant (1933-38); Instructor (1938-39); Assistant Professor (1940-44); Associate Professor (1945-46); Professor (1947-67); James Kip Finch Professor of Applied Science (1967-75); Retired in 1975.

World War II Service (1942-46) 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 1979.

Died in Hanover, New Hampshire in 1987

Raymond D. Mindlin-His Life (1906-1987)

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Springer-Verlag, New York, Berlin, 1989

3Edited by H. Deresiewicz, M. P. Bieniek, F. L. DiMaggio

Mindlin’s Publications (1934-1950)

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Mindlin’s Publications (1950-1962)

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Mindlin’s Publications (1962-1972)

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Mindlin’s Publications (1972-1986)

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Theories of Plates by R. D. Mindlin

* “Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates” in Journal of Applied Mechanics, 1951

Mindlin’s Equation of Plates (1951)2 2 2

2 22 2 2 2

2 2

2 2 2

( )( )

(1 ) ( , , )

wD I w hG t t t

D I q x y tGh Gh t

ρ ρ ρκ

ρκ κ

∂ ∂ ∂∇ − ∇ − +

∂ ∂ ∂∇ ∂

= − +∂

Germain-Lagrange’s Equation of Plates (1803-1812)2 2 2 2( ) / ( , , )D w h w t q x y tρ∇ ∇ + ∂ ∂ =

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2

2 ,(1 ) (1 ) 12

EI GI hD Iν ν

≡ = ≡− −

w-Transverse Motion of a Plate in xy plane

h-plate thickness, p-mass density

D-Flexural Rigidityc

E-Young’s modulus

G-shear modulus

v-Poisson’s ratio

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κ2-shear coefficient

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2

[ , , ] [ , , ]h

hx y x y x x y y x yM M M z d zσ σ σ−

≡ ∫

2

2 ( )1 2

ij kk ij ij

k k ij i j j i

GG u G u u

σ λε δ ε

µ δµ

= +

= ∂ + ∂ + ∂−

( , ) ( , , ), ,u t z x y t x yα αψ α= =x

2 ( )ij i j j iu uε = ∂ + ∂

x x y yx x x

x yx y y y y

x x y y

M M Q IM M Q I

Q Q q hw

ρ ψ

ρ ψ

ρ

∂ + ∂ − =∂ + ∂ − =∂ + ∂ + =

1. Plate Stress Equation of Motion,

2. Plate Stress and Plate-Displacement Relation

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2 ( ) 2 , , ,1

( ), , ,

,

yxGIM GI x yx y

M GI x y

Q w shear coefficientGh

αα β β β β

βααβ

αα

ψψ ψµ ψ α β ψµ α

ψψα β α β

β α

ψ κκ α

∂ ∂ ∂= ∂ + = ∂ = + − ∂ ∂ ∂

∂∂ = + = ≠ ∂ ∂ ∂

= + −∂

x xx y xy z xz x

x yx y yy z yz y

x zx y zy z zz z

uuu

σ σ σ ρ

σ σ σ ρ

σ σ σ ρ

∂ + ∂ + ∂ =∂ + ∂ + ∂ =∂ + ∂ + ∂ =

Eqs. of Elasticity and Eqs. of Plate

ui-displacement component in xi direction (i=1,2,3)

σij-stress component, （ ）

εij-strain component, ( )

Plate displacement

2. Stress-Displacement Relation

3. Plate Displacement Equation of Motion (Mindlin, 1951)2 2

2 2

( 2)[(1 ) (1 ) ( )] ( )

( )

D Gh w I

Gh w q hwα α β β α α α

β β

µ ψ µ ψ κ ψ ρ ψ

κ ψ ρ

− ∇ + + ∂ ∂ − + ∂ =

∇ + ∂ + =

Plate stress

x x∂

∂ ≡∂

3. Displacement- Equation of Motion (Navier-Cauchy, 1822-28)

( ) ( )i j j j j i i iG u G u f uλ ρ ρ+ ∂ ∂ + ∂ ∂ + =

1. Stress-Equation of Motion

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∂∂ ≡

∂

h

ψx

ψyw x

z

y

3( , ) ( , , )u x t w x y t≈

2

2

[ , ] [ , ]h

hx y xz yzQ Q dzσ σ−

≡ ∫

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2 2 2 2 22 2

2 2 2 2 2 2 2( )( ) (1 ) ( , , )w D ID I w h q x y tG t t t Gh Gh tρ ρρ ρ

κ κ κ∂ ∂ ∂ ∇ ∂

∇ − ∇ − + = − +∂ ∂ ∂ ∂

ij ijkl kl ijkl k lc c uσ ε= = ∂

(0) (1) 21 2 3 1 2 3 1 2 3( , , , ) ( , , ) ( , , )i i iu x x x t u x x t x u x x t x= + + +

j ji i if uσ ρ ρ∂ + =

Cauchy’s Eq. of Motion(1828)

Generalized Hooke’s Law for Crystalline Solids

Equation of Motion for Crystals( )j ijkl k l i ic u f uρ ρ∂ ∂ + =

Higher Order Theories of Plates

(0) (1)3 1,2 3 1,2,u w u xψ≡ =

Zeroth-Order (Extensional Motion)(0)1 1 2(0)2 1 2

( , , )

( , , )

u u x x tu v x x t

=

=

First Order (Thickness Shear and Flexural Motion)

(1)3 1 2

(0)3 1 2

( , , ), 1, 2

( , , )

u x x x tu w x x tα αψ α = =

=

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*“Thickness-Shear and Flexural Vibrations of Crystal Plates” in Journal of Applied Physics, 1951

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/ 2 1.664f MHzω π= =

An Introduction to the Mathematical Theory of Vibrations of Elastic Plates

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*Mechanics Research Communications, 13, 349-357 (1986)

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“…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

EpilogueThe Canterbury Tales, by Geoffrey Chaucer (1340-1400)