Raymond D. Mindlin and Applied Mechanics

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

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 1

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Springer-Verlag, New York, Berlin, 1989

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

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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

Germain-Lagrange’s Equation of
Plates (1803-1812)

∇2 (D∇2w) + ρh∂2w / ∂t2 = q(x, y, t)

w-Transverse Motion of a Plate in xy plane

h-plate thickness, p-mass density

D ≡ EI 2) = 2GI , I ≡ h3
(1−ν (1−ν ) 12

D-Flexural Rigidityc

E-Young’s modulus

G-shear modulus

v-Poisson’s ratio

Mindlin’s Equation of Plates (1951)

(∇2 − ρ ∂2 )(D∇2 − ρI ∂2 )w + ρh ∂2w * “Influence of Rotatory Inertia and Shear on Flexural Motions
κ 2G ∂t 2 ∂t 2 ∂t 2 of Isotropic, Elastic Plates” in Journal of Applied Mechanics,
1951
= (1 − D∇2 + ρI ∂2 )q( x, y, t)
κ 2Gh κ 2Gh ∂t 2 18

κ2-shear coefficient

Eqs. of Elasticity and Eqs. of Plate Plate displacement uα (x,t) = zψα (x, y,t),α = x, y

ui-displacement component in xi direction (i=1,2,3) u3(x,t) ≈ w(x, y,t)

σij-stress component, （ ∂ ） Plate stress h
∂xi
∂i ≡ ∫[ M x , M y , M x y ] ≡ 2 h z [σ xx ,σ yy , σ xy ]d z
2
−

εij-strain component, ( 2εij = (∂iu j + ∂ jui ) ) h
y
∫[Qx ,Qy ] ≡ 2 h [σ xz ,σ yz ]dz
2
−

zh 1. Plate Stress Equation of Motion, ∂x ≡ ∂
∂x
∂ M x + ∂ y M yx Qx ρ Iψx
w ψy x ∂ x M yx + ∂ y M y − Qy = ρ Iψy
ψx x − =
1. Stress-Equation of Motion
∂xQx + ∂ yQy + q = ρhw

∂ xσ ∂ yσ ∂ zσ xz ρux 2. Plate Stress and Plate-Displacement Relation
 ∂ yσ ∂ zσ yz ρuy
∂ xσ xx + xy + = Mα = 2GI µ (∂βψ β ) + 2GI ∂ψ α , α , β = x, y ∂βψ β = ∂ψ x + ∂ψ y
yx + yy + =  1− µ ∂α ∂x ∂y

∂xσ zx + ∂ yσ zy + ∂ zσ zz = ρuz Mαβ = GI (∂ψα + ∂ψ β ), α, β = x, y α ≠ β
 ∂β ∂α
2. Stress-Displacement Relation
 Qα = ∂w +ψα , κ 2 − shear coefficient
 κ 2Gh ∂α
σ ij = λε δkk ij + 2Gε ij 

= 2µG ∂ k uk δ ij + G(∂iu j + ∂ jui ) 3. Plate Displacement Equation of Motion (Mindlin, 1951)
1− 2µ

3. Displacement- Equation of Motion (D 2)[(1− µ)∇2ψα + (1+ µ)∂α (∂βψ β )] − κ 2Gh(ψα + ∂α w) = ρ Iψα
(Navier-Cauchy, 1822-28) κ 2Gh(∇2w + ∂βψ β ) + q = ρhw

(λ + G)∂i (∂ ju j ) + G∂ j∂ jui + ρ fi = ρui (∇2 − ρ ∂2 )(D∇2 − ρI ∂2 )w + ρh ∂2w = (1 − D∇2 + ρI ∂2 )q( x, y,t)
κ 2G ∂t 2 ∂t 2 ∂t 2 κ 2Gh κ 2Gh ∂t 2

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Cauchy’s Eq. of Motion(1828) Higher Order Theories of Plates

∂ jσ ji + ρ fi = ρui ui ( x1 , x2 , x3 , t) = u(0) ( x1 , x2 , t) + x3ui(1) ( x1 , x2 ,t) + x32 +"
i

u(0) ≡ w, u (1) = x3ψ 1,2
3 1,2

Generalized Hooke’s Law for Crystalline Solids Zeroth-Order (Extensional Motion)

σ ij = εcijkl kl = cijkl∂k ul uu12((00 ) = u(x1, x2 , t)
) = v(x1, x2 , t)
Equation of Motion for Crystals
First Order (Thickness Shear and
∂ j (cijkl∂kul ) + ρ fi = ρui
Flexural Motion)

uuα3((10)) = x3ψα (x1, x2 ,t),α = 1, 2
= w(x1, x2 , t)

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

f = ω / 2π = 1.664MHz

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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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Epilogue
The Canterbury Tales, by Geoffrey Chaucer (1340-1400)

“…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. - 28 -
Deresiewicz