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Controllable infinitesimal deformations in homogeneously deformed compressible elastic materials

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Abstract

A controllable static deformation is a deformation that may be maintained in all materials of a given class under the action of surface forces alone. For compressible, homogeneous, isotropic elastic materials the only controllable deformations are homogeneous. However, it is known that there are solutions of the static equations of finite elasticity, linearized about a finite homogeneous deformation, which do not correspond to homogeneous deformations. These approximate solutions are investigated here. Three cases arise, depending on whether none, two, or three of the basic principal stretches are equal.

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Abbreviations

A :

arbitrary vector potential

a 1, a 2, a 3 :

bounding coordinates of body

B, B ij :

left Cauchy-Green tensor

C, C ijpq :

elasticity tensor

c, c 1, c 2, c 3 :

arbitrary constants

N 0, N 1, N 2 :

elastic response functions

n :

vector normal to surface of body

T 1, T 2, T 3 :

surface tractions

t 1, t 2, t 3 :

surface tractions

t, t ij :

Cauchy stress tensor

t 0, t 0 ij :

Cauchy stress corresponding to basic homogeneous deformation

u, u i :

infinitesimal displacement from basic homogeneous deformation

X, X i :

position vector in reference state

x, x i :

position vector

α :

arbitrary function

δ ij :

Kronecker delta

λ, λ 1, λ 2, λ 3 :

principal stretches

φ :

arbitrary function

ψ :

arbitrary function

ϰ :

arbitrary function

I, II, III:

principal invariants of B

References

  1. Singh, M. and A. C. Pipkin, Arch. Rational Mech. Anal. 21 (1966) 169.

    Google Scholar 

  2. Ericksen, J. L., Z. angew. Math. Phys. 5 (1954) 466.

    Google Scholar 

  3. Ericksen, J. L., J. Math. Phys. 34 (1955) 126.

    Google Scholar 

  4. Klingbeil, W. W. and R. T. Shield, Z. angew. Math. Phys. 17 (1966) 489.

    Google Scholar 

  5. Singh, M. and A. C. Pipkin, Z. angew. Math. Phys. 16 (1965) 706.

    Google Scholar 

  6. Rivlin, R. S. and D. W. Saunders, Phil. Trans. Roy. Soc. Lond. A 243 (1951) 251.

    Google Scholar 

  7. Green, A. E. and R. T. Shield, Phil. Trans. Roy. Soc. Lond. A 244 (1951) 47.

    Google Scholar 

  8. Truesdell, C. and W. Noll, The non-linear field theories of mechanics. Handbuch der Physik III/3, Springer-Verlag, Berlin-Heidelberg-New York 1965.

    Google Scholar 

  9. Green, A. E. and W. Zerna, Theoretical elasticity, Clarendon Press, Oxford 1954.

    Google Scholar 

  10. Weatherburn, C. E., Advanced vector analysis, Bell, London 1924.

  11. Woo, T. C. and R. T. Shield, Arch. Rational Mech. Anal. 9 (1962) 196.

    Google Scholar 

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Authors and Affiliations

  1. Dept. of Mathematics, Simon Fraser University, Burnaby 2, B.C., Canada

    P. Currie

Authors
  1. P. Currie
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Currie, P. Controllable infinitesimal deformations in homogeneously deformed compressible elastic materials. Appl. Sci. Res. 23, 212–220 (1971). https://doi.org/10.1007/BF00413199

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  • DOI: https://doi.org/10.1007/BF00413199

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