A Uniqueness Theorem for a Nonlinear, Steadily Creeping Body

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  • SIAM REVIEWVol. 9, No. 4, October, 1967

    Printed in U.S.A.

    A UNIQUENESS THEOREM FOR A NONLINEAR,STEADILY CREEPING BODY*

    JEROME L. SACKMANf

    A stress-strain rate relation often used in analytic investigations of high tern-.perature, quasi-static, infinitesimal, steady creep of isotropic incompressiblematerials is [1]

    ej CJmsiwhere ei and si are, respectively, the Cartesian components of the creep strahrate tensor and the deviatoric stress tensor, J is the second invariant of thedeviatoric stress tensor, and C is a positive quantity inversely proportional tothe coefficient of viscosity of the material. The quantities si and J are given by

    J8i] ffij Okki] 8klSkl

    where are the Cartesian components of the stress tensor, s is the Kroneckerdelta, and the usual convention, of summation over repeated indices is adopted..This steady creep law (which is nonlinear when m > 0) represents a simple,generalization to three-dimensional states of stress of the familiar (power) lawobtained from uniaxial creep tests [2].

    It is interesting to note that a uniqueness theorem for this specific nonlinearisotropic incompressible body may be simply constructed in a manner quitesimilar in approach to that utilized for the linear elastic body. The statement andproof of this theorem for the xed boundary value problem follows.THEOREM. Given a regular region of space D + B with boundary B B B

    and functis f(P) (defined in D + B), g(P) (defined on B) and hi(P) (de-fined on Bx), all in class C( then there exists at most one set of functions as(P)e.is(P) and v(P) all in class C(1) in D + B which satisfy the following field equa-tions and boundary conditions:

    (1) s,s + f O, P in D,ei CJss, P in D + B

    (2)(where C > O, J sisi si a&i, m => 0),

    (3) es (v.s + vs.), P in D + B,(4) asns= g, P on Bz, v= h, P on Bx.In the above, v and n are, respectively, the Cartesian components of the velocityvector and the unit outward vector normal to the boundary surface B.

    If II

    Proof. Consider two olutions, v, , nd v , , which stisfy (1)* Received by the editors March 22, 1967, and in revised form April 26, 1967.Department of Civil Engineering, University of California, Berkeley, California 94720.

    This research was supported in part by the United States Army Research Office (Durham).741D

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  • 742 JEROME L. SACKMAN

    to (4) above. The "difference solution," vi* vi v etc., satisfies the linearequations (1), (3) and (4) withf g h 0. Consider now the integral

    (5) f.( * * f. dA / f, dA O,ovi )n dA * * *o-n)v* on)vwhich is zero in view of the boundary conditions (4) satisfied by the differencesolution. Utilizing the divergence theorem, the equilibrium (1) and symmetryconditions of the difference stress tensor, and the difference strain rate-velocityrelations (3), we obtain

    * * fD * *(6) v n dA dV.(Tij.ijIntroducing (5), the definition of the deviatoric stress tensor, the condition ofincompressibility ek 0 from (2)), and (2) into (6) we have

    fo * * fos dV C(s s) s s) dV 0or, expanding,

    (7) fo C[2J+ (J" + ]")s[s + 2J"+] dV O,where J -SijSij etc.

    Introduce now the positive definite quadratic function

    8 J*Utilizing this in (7) we have, after simplification,

    (9) fo C[(j, g,,,n)(j, j,,) + j.(g,. + j.,)] dV O.But (J" J")(J J") 0 and J*(J" + J) 0 since m >= 0 andJ* J" and are positive definite quadratic forms. Thus it follows that

    (j,m j..)(j, j,,) + j.(j,. + jn.) O,whence

    (lOa)

    (lOb)

    (J"- J")(J- J") O,J*( J" + J") O.

    But (10b) implies J* 0, and since J* is a positive definite quadratic functionof si*.(11) s*- 0.Thus the deviatoric stress components are uniquely determined. From the unique-hess of the deviatoric stress components, the uniqueness of the strain rate com-ponents follows by use of the stress-strain rate relation (2). The uniqueness of

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  • SHORT :NOTES 743

    the velocity components then follows in the usual manner [3] by use of the strainrate-velocity relation (3) and the velocity boundary conditions of (4).To establish uniqueness of the stress tensor, we again employ the equilibrium

    equation (1) and the traction boundary conditions (4), written in terms of thedifference solution

    (12) aj.= 0 in D, ajn 0 on BI.But from the definition of the deviatoric stress tensor and (11),

    Oij -Okkij

    whence it follows from (12) that *ak. 0 in D, or

    (13) 1 *a Ki a*. in D,*where K is a constant. However, a is in class in D - B so that (12) and(13) yield Kn-- 0 on BI.

    But n is not identically zero on B, hence K must be zero. In view of (13), thisestub]ishes the uniqueness of the stress tensor and concludes the proof.This theorem may be extended to the case of "mixed-mixed" [4] boundary

    conditions, and to the case where the tractions are specified over the entireboundary B. In u similar munner, a uniqueness theorem can be established forthe case where velocities are prescribed over the entire boundary B. (The ve-locities cannot be specified arbitrarily throughout B since the condition of

    incompressibility implies the "constraint" j: v. dA 0, which is a statementof the fact that no net flow of material occurs across the boundary B.) For thiscase uniqueness of the velocity field can be established, but it is only possibleto demonstrate uniqueness of the stress tensor to within a hydrostatic state ofstress. Physically, this corresponds to the fact that the addition of an arbitraryhydrostatic state of stress to an isotropic incompressible body does not affectthe deformation of that body.

    REFERENCES

    [1] B. VENKATRAMAN, Solutions of some problems in steady creep, PIBAL Rep. 402, Poly-technic Institute of Brooklyn, Brooklyn, New York, 1957.

    [2] J. E. DoN, Some fundamental experiments on high temperature creep, J. Mech. Phys.Solids, 3 (1955), pp. 85-116.

    [3] I. S. SooNxo, Mathematical Theory of Elasticity, 2nd ed., McGraw-Hill, New York,1956, p. 87 and 10.

    [4] B. A. BOLE: ND J. H. WEANER, Theory of Thermal Stresses, John Wiley, New York,1960, p. 65.

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