4
Short Notes: A Uniqueness Theorem for a Nonlinear, Steadily Creeping Body Author(s): Jerome L. Sackman Source: SIAM Review, Vol. 9, No. 4 (Oct., 1967), pp. 741-743 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2028308 . Accessed: 14/06/2014 21:34 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend access to SIAM Review. http://www.jstor.org This content downloaded from 185.2.32.49 on Sat, 14 Jun 2014 21:34:58 PM All use subject to JSTOR Terms and Conditions

Short Notes: A Uniqueness Theorem for a Nonlinear, Steadily Creeping Body

Embed Size (px)

Citation preview

Page 1: Short Notes: A Uniqueness Theorem for a Nonlinear, Steadily Creeping Body

Short Notes: A Uniqueness Theorem for a Nonlinear, Steadily Creeping BodyAuthor(s): Jerome L. SackmanSource: SIAM Review, Vol. 9, No. 4 (Oct., 1967), pp. 741-743Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2028308 .

Accessed: 14/06/2014 21:34

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Review.

http://www.jstor.org

This content downloaded from 185.2.32.49 on Sat, 14 Jun 2014 21:34:58 PMAll use subject to JSTOR Terms and Conditions

Page 2: Short Notes: A Uniqueness Theorem for a Nonlinear, Steadily Creeping Body

SIAM REVIEW

Vol. 9, No. 4, October, 1967 Printed in U.S.A.

A UNIQUENESS THEOREM FOR A NONLINEAR, STEADILY CREEPING BODY*

JEROME L. SACKMANt

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

fij= CJ sij,

where Eij and sij are, respectively, the Cartesian components of the creep straill rate tensor and the deviatoric stress tensor, J is the second invariant of the deviatoric stress tensor, and C is a positive quantity inversely proportional to the coefficient of viscosity of the material. The quantities sij and J are given by

5ij = aij - aOrkk6ij

v J = 5k1lSkl

where o-ij are the Cartesian components of the stress tensor, aij is the Kronecker delta, 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) law obtained from uniaxial creep tests [2].

It is interesting to note that a uniqueness theorem for this specific nonlinear isotropic incompressible body may be simply constructed in a manner quite similar in approach to that utilized for the linear elastic body. The statement and proof of this theorem for the mixed boundary value problem follows.

THEOREM. Given a regular region of space D + B with boundary B = BI + BII and functions fi(P) (defined in D + B), gi(P) (defined on B1) and hi(P) (de- fined on B11), all in class C(), then there exists at most one set of functions o-ij(P), Eij(P) and vi(P) all in class C(1) in D + B which satisfy the following field equa- tions and boundary conditions:

(1) uij, X fi = O, P in D,

Eij = CJmsij; P in D+B

(where C > 0, J = 2sijsii, Sij = Oij - 130kkij , m _ O),

(3) Eij = I(vi,j + vj,i), P in D + B,

(4) oijnj = gi, P on BI, vi = hi, P on B,I.

In the above, vi and ni are, respectively, the Cartesian components of the velocity vector and the unit outward vector normal to the boundary surface B.

Proof. Consider two solutions, vi', eij, oij and vi , Ejij Ij , which satisfy (1)

* Received by the editors March 22, 1967, and in revised form April 26, 1967. t Department of Civil Engineering, University of California, Berkeley, California 94720.

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

741

This content downloaded from 185.2.32.49 on Sat, 14 Jun 2014 21:34:58 PMAll use subject to JSTOR Terms and Conditions

Page 3: Short Notes: A Uniqueness Theorem for a Nonlinear, Steadily Creeping Body

742 JEROME L. SACKMAN

to (4) above. The "difference solution," v* = vi- vi, etc., satisfies the linear equations (1), (3) and (4) with fi = gi = h= 0. Consider now the integral

(5) (ojvi*)nj dA - of (ujnj)vi* dA + (of*jnj)vi* dA = 0,

which is zero in view of the boundary conditions (4) satisfied by the difference solution. Utilizing the divergence theorem, the equilibrium (1) and symmetry conditions of the difference stress tensor, and the difference strain rate-velocity relations (3), we obtain

(6) f ivi ni dA = Jf Ejj dV.

Introducing (5), the definition of the deviatoric stress tensor, the condition of incompressibility (Ekk = 0 from (2)), and (2) into (6) we have

L SijEj dV = L C(s'j - 8sf3)(Jtms$j - J"msf ) dV = O

or, expanding,

(7) J C[2Jm1l - (J"m + J"m),' 'j + 2J"m+'] dV =

where J' = !si jsj j, etc. Introduce now the positive definite quadratic function

(8) = SjSij = J J -88j.

Utilizing this in (7) we have, after simplification,

(9) I C[(J'M - Jtlm)(JV - J") + J*(JVm + J Im)] dV = 0.

But (J'm - Jl) (J- J") > 0 and J*(J'm + J"lm) > 0 since m > 0 and J* J' and J" are positive definite quadratic forms. Thus it follows that

(J/m - J"m)(J I J") + J*(Jtm + Jf7) = 0,

whence

(lOa) (J - JM)(J I - J") = 0,

(lob) J*(J'n + JUm) = o.

But (lOb) implies J* = 0, and since J* is a positive definite quadratic function of S*fj (11) Sij= 0.

Thus the deviatoric stress components are uniquely determined. From the unique- ness 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

This content downloaded from 185.2.32.49 on Sat, 14 Jun 2014 21:34:58 PMAll use subject to JSTOR Terms and Conditions

Page 4: Short Notes: A Uniqueness Theorem for a Nonlinear, Steadily Creeping Body

SHORT NOTES 743

the velocity components then follows in the usual manner [3] by use of the strain rate-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 the difference solution

(12) 0j,3=O in D, oijnj=O on BI.

But from the definition of the deviatoric stress tensor and (11),

ij =- 3Okk6tij,

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

(13) 3Okkai j = Kaij = oij in D,

where K is a constant. However, oij is in class C(1) in D + B so that (12) and (13) yield

u*n =Kni=O on BI.

But ni is not identically zero on BI, hence K must be zero. In view of (13), this establishes 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 entire boundary B. In a similar manner, a uniqueness theorem can be established for the 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" f vini dA = 0, which is a statement

of the fact that no net flow of material occurs across the boundary B.) For this case uniqueness of the velocity field can be established, but it is only possible to demonstrate uniqueness of the stress tensor to within a hydrostatic state of stress. Physically, this corresponds to the fact that the addition of an arbitrary hydrostatic state of stress to an isotropic incompressible body does not affect the 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. DORN, Some fundamental experiments on high temperature creep, J. Mech. Phys. Solids, 3 (1955), pp. 85-116.

[31 I. S. SOKOLNIKOFF, Mathematical Theory of Elasticity, 2nd ed., McGraw-Hill, New York, 1956, p. 87 and ?10.

[4] B. A. BOLEY AND J. H. WEINER, Theory of Thermal Stresses, John Wiley, New York, 1960, p. 65.

This content downloaded from 185.2.32.49 on Sat, 14 Jun 2014 21:34:58 PMAll use subject to JSTOR Terms and Conditions