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Load-induced stress singularities in the bending of cosserat plates

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Sommario

In questa memoria sono presentate le soluzioni di alcuni problemi relativi a piastre inflesse studiati in base ad una teoria recentemente sviluppata da Green e Naghdi, in cui si considerano distribuzioni di coppie di volume.

In ogni problema considerato le condizioni al contorno presentano una singolarità nella distribuzione di carico agente su un bordo libero di una piastra semi-indefinita. Per la risoluzione è applicato il metodo delle trasformate integrali.

In generale risulta che le singolarità nel taglio e nel momento sono dello stesso ordine di quelle date dalla teoria delle piastre di Reissner, benchè gli andamenti di queste funzioni singolari siano diversi. La teoria sviluppata porta inoltre a valutare, nella maggior parte dei casi, momenti massimi più piccoli di quelli corrispondenti ottenuti in base alla teoria di Reissner.

Si discute inoltre l'esatta relazione tra le teorie di Green-Naghdi e di Reissner.

Summary

Presented in this paper are the solutions to several plate bending problems as governed by a recent theory developed by Green and Naghdi, into which couple-stress is incorporated. Specifically, each problem considered is subjected to boundary conditions emanating from a singular load distribution acting on the free edge of a semi-infinite plate. The method of integral transforms is applied in the solutions.

In general, it is found that the singularities in the shear and moment resultants are of the same order as those given in Reissner's plate theory, however the detailed structures of these singular functions are altered. The present theory also suggests that, in most cases, the maximum magnitudes of the moment resultants are diminished as compared to the corresponding results given in Reissner's theory.

Also discussed is the exact relationship between the Green-Naghdi theory and Reissner's theory.

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Abbreviations

x i(i=1, 2, 3):

Cartesian coordinate directions

S :

undeformed Cosserat surface

s :

deformed Cosserat surface, also used as Fourier transform parameter

d :

deformed director vector

D 3,D α(α=1, 2):

components of initial director alongx i

u i :

displacement components of Cosserat surface

t :

time

M α,β (α, β=1, 2):

components of moment per unit length

N 3α :

components of shear force per unit length

p :

applied surface force per unit area

A (α,β) :

any function symmetric with respect toα andβ

A (α,β) :

any function antisymmetric with respect toα andβ

δ(α,β) :

two-dimensional Kronecker symbol

κ(α,β) :

components of curvature

δ(α) :

angles of rotation of director

a 3,α 5,α 6,α 7 :

constitutive coefficients

ν :

Poisson's ratio

E :

modulus of elasticity

D :

flexural rigidity

b :

plate thickness

ε(α,β) :

two-dimensional permutation symbol

χ, ψ, φ :

scalar functions ofx 1 andx 2

λ 2 :

a material constant defined by Eq. (13)

f(x):

arbitrary function ofx

\(\bar f\)(s):

Fourier transform off(x)

f (k)(x):

k-th derivative off(x)

A(s),B(s),C(s):

arbitrary functions ofs

a(s):

a function ofs defined by Eq. (26)

B 1---B 6 :

material constants

ℓ(s),p(s),v(s):

particular functions ofs

r :

distance from origin to (x, y)

log:

natural logarithm

U :

strain energy density function per unit area

M 0,H 0,N 0 :

concentrated bending moment, twisting moment, and shear force, respectively

O :

order of magnitude symbol

C n,C s,k 2 :

constants in the Reissner theory

References

  1. A. E. Green, P. M. Naghdi andW. L. Wainwright,A General Theory of a Cosserat Surface, Archive for Rational Mechanics and Analysis, Vol. XX, pp. 287–308, 1965.

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  2. A. E. Green andP. M. Naghdi,The Linear Theory of an Elastic Cosserat Plate, ONR Report AM-66-4, University of California, Berkeley, 1966.

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  3. R. Muki andE. Sternberg,The Influence of Couple-Stresses on Singular Stress Concentrations in Elastic Solids, Zeitschrift für Angewandte Mathematik und Physik, Vol. XVI, pp. 611–648, 1965.

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  4. M. Abramowitz, andI. A. Stegun,Handbook of Mathematical Functions, National Bureau of Standards, Washington, D. C., 1965.

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  5. R. J. Hartranft andG. C. Sih,Stress Singularities in the Plate Bending Theory of Reissner, Proceedings of the 10th Midwestern Mechanics Conference, in Press.

  6. E. Reissner,On Bending of Elastic Plates, Quarterly of Applied Mathematics, Vol. V, pp. 55–68, 1947.

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Author information

Authors and Affiliations

  1. Air Force Materials Laboratory, Dayton, Ohio

    Nicholas J. Pagano (Materials Engineer)

  2. Graduate Student, Department of Mechanics, Lehigh University, Bethlehem, Pennsylvania

    Nicholas J. Pagano (Materials Engineer)

  3. Lehigh University, Bethlehem, Pa

    George C. Sih (Professor of Mechanics)

Authors
  1. Nicholas J. Pagano
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  2. George C. Sih
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Additional information

This research is sponsored by the U. S. Navy under Contract Nonr-610(06) with the Office of Naval Research in Washington, D. C.

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Pagano, N.J., Sih, G.C. Load-induced stress singularities in the bending of cosserat plates. Meccanica 3, 34–42 (1968). https://doi.org/10.1007/BF02173991

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  • DOI: https://doi.org/10.1007/BF02173991

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