Elasticity problems with partially overspecified boundary conditions

ht. 1. Engng Sci. Vol.

29, No. 6, pp. 685-692, 1991

0020-7225/91 $3.00+ 0.00

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Copyright (CJ 1991 Pergamon Press plc

ELASTICITY PROBLEMS WITH PARTIALLY OVERSPECIFIED BOUNDARY CONDITIONS ZHANJUN Department

GAO?

and TOSHIO

of Civil Engineering, Northwestern

MURA

University, Evanston, IL 60208, U.S.A.

Ah&ad-Contrary to the conventional problems of elasticity, in which there is one and only one boundary condition (displacements or tractions or a combination of both) specified on every point of the boundary, the problem considered here has two boundary conditions (e.g. both displacements and tractions) imposed on a part of the boundary. For the rest part of the boundary, no boundary condition is available. A formulation of the boundary element method is presented. Applications to various problems including those with plastic deformation are discussed. We confine ourselves to 2-D cases to simplifv the notation. The results of this paper can be generalized to 3-D problems without difficulties. _

1. THE

PROBLEM

the region occupied by a 2-D elastic body and its boundary be denoted by Q and $, respectively (Fig. l), where $ = a1 + $ and & - &, =@. In a conventional problem of elasticity, the boundary condition is either specified displacement where the components of displacement Ui are prescribed on the boundary, or specified surface traction where the components of surface traction ti are assigned on the boundary. In most cases, the boundary conditions are such that over part of the boundary displacements are specified, whereas over another part the surface tractions are specified. It should be noted that in such a problem there is one and only one boundary condition (specified displacements or specified tractions) for each point of the boundary $ The problem we consider here has two boundary conditions for the points on &. Let

u? I>

on Sl,

tj = t;,

on St,

Ui

=

0) (2)

and no boundary condition on &. up and ty in the above equations are known functions. Two arbitrarily specified displacement uy and traction ty on the same & shown in equations (1) and (2) may b e inconsistent such that no elastic fields can satisfy both conditions. In practical applications, however, the consistency of equations (1) and (2) is easily achieved because one of them, say displacements, are obtained from experimental measurements. Therefore, the specified traction condition and the specified displacement condition are consistent. The solution to the problem with boundary conditions (1) and (2) is unique. Its proof follows directly from Ref. [l], which states that if tractions and displacements vanish on an arc AB of an elastic body, then the stresses and displacements vanish identically in the entire region containing the arc AB. The environment for solving such problems has been improved by the advances in mathematical methodology and computer technologies. In the following section, we establish the boundary integral equation for the above problem and employ the regularization method to obtain a stable solution.

2. THE

BOUNDARY-INTEGRAL

EQUATION

For simplicity, we consider homogeneous elastic solid with smooth boundary. The stiffness tensor of the solid is CL’,,,*By using Betti-Maxwell’s reciprocal theorem, we can easily obtain of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, U.S.A. 685

t Department

2. GAO and T. MUKA

Fig. 1. Elastic body Q with boundary S = $, + &. s is the arc length of the boundary. The value of s increases counterclockwise. The lengths of & and & are I and L, respectively.

(e-g.see PI),

-I

s, +s*

Gim(X,-Z’)ti(x)&(;z) = 0,

x’

E $1

+

$2,

(31

where Gi,(x, x’) is the solution of 2-D Kelvin problem with the unit point load at 5’. Explicit expression for Gi, can be found in Gao and Mm-a [3]. Assume that the boundary curve S = S, + & can be represented parametrically by function f(s) (see Fig. l),

$5;:$W = f(S)?

OS'SGL,

81:J(f) = i(s),

LSsCL+l.

Then, for 5’ E & and x’ E &, equation (3) is written as L

L

I0

%n(S,

r)&(s)

dS

-

rim(s~

z)ri(s)

dS

=

gm(z)7

t

E [O,

L]

(4)

i 0

and LL Xim(S, z)u~(s) do - ir Aim(s, z&i(s) do = h,(t), respectively.

II,

et al. are defined as

z E [O, ‘1,

(5)

Elasticity problems

h,(t) =

687

-IL+’ C,,G,,,,(f(~), f(;t ++(s)ul’(s) d.s r+z-. . >

+~+'G;~j!(~),~(titL))IY(~)B-Uju:(t

Equations (4) and (5) are written in a matrix form L

c

JO

!a,

t)W)

ds

=

?J E [O,

U(4,

Ll,

(74

where

C’b)

(7c) (7d) Equation (7) cannot be solved directly since it is numerically unstable. However, the following well-posed integral equation can be obtained from equation (7) by using the so-called regularization method [4],

where L Is*@,

z)

=

i0

U*(t) =

K=(T

w)K(s,

LL I$=@,w)v(w)

w)

dw,

dw

(9)

(10)

and w is a positive parameter determined from the discrepancy of measured data to the exact data. Note that IIim, Tim have singularity at s = r. The integration in equation (9) diverges even in the sense of Cauchy principal value. The definition of II,,, Tim must be modified (Appendix).

3.

APPLICATIONS

The problem shown in Fig. 1 has its counterparts in other fields of science and engineering such as nondestructive testing, inverse design and optimization, etc. For example, in the field of inverse design and optimization, shapes and locations of coolant flow passages in an internally cooled turbine blade are designed from given temperatures and heat fluxes on the surface. In some problems of elasticity, the boundary condition is partially given. A number of sensors are placed inside the solid which give us implementary information about the field variables. Such an example is shown in Fig. 2(a) in which tractions are given on S, and no boundary conditions are given on &. Sensor are placed (inside body Q along line IJ) to measure the displacements on r as complementary information. By using the principle of superposition, the problem can be expressed as the sum of problems shown in Fig. 2(b) and Fig. 2(c). The problem shown in Fig. 2(b) is a conventional elasticity problem with complete boundary

688

Z. GAO and T. MURA

t, =t;

Fig. 2. (a) A problem with sensor inside the solid to provide implementary information. (b) A conventional elasticity problem with complete boundary conditions. (c) A problem with overspecified boundary conditions on a part of the whole boundary. Problem (a) is a superposition of problems (b) and (c).

Elasticity problems

689

conditions. The problem shown in Fig. 2(c) is the overspecified boundary value problem discussed in this paper. The specified tractions tf are obtained after the problem in Fig. 2(b) is solved. The value of Ui and ti on the boundary $, as well as in the entire body, are obtained after the overspecified boundary value problem is solved. The applications of the problem shown in Fig. 1 are not restricted to elasticity problems. Let Q be a sub-region of 0. Plastic deformation occurs inside Q and the current values of surface tractions and displacements are given as ty and UP [Fig. (3a)]. Removing the material in SJ and replacing its effects by tractions and displacements on the boundary of Q, we obtain an overspecified boundary value problem [Fig. 3(b)], where Q is a hole with no material in it. The boundary of Q, i.e. aQ is taken as & boundary of the formulation shown in equation (3), while 8, the bound~y of Q, is taken as the .Sr bounda~. This problem can be solved for ti and gi on the boundary of Q. As we know, determination of field quantities of a plasticity problem requires the entire loading history. It is quite interesting to solve some characteristic quantities such as displacements and tractions on the boundary of plastic zone by knowing only the current values of displacements up and tractions ty on the surface. Another example is shown in Fig. 4(a). A body with a macro-crack is subject to surface tractions tie Micro-cracks and inhomogeneities with different sizes and orientations are localized near the tips of the macro-crack. The number of the micro-cracks and inhomogeneities, as well

Fig. 3. (a) A solid with plastic deformation inside Q. (b) The plastic zone is replaced by tractions fi and displacements ui. The resulting problem is an overspecified boundary value problem and can be solved for ui and t,.

690

Z. GAO and T. MURA

(b)

Fig. 4. (a) Micro-cracks and inhomogeneities with different sizes and orientations are localized near the tips of a macro-crack. (b) The part with micro-cracks and inhomogeneities are replaced by tractions li and displacements ui. The resulting problem is an overspecified boundary value problem.

as their sizes, orientations and locations, are generally unknown. We claim that if the displacements on the crack face AB and A’B’ are measured, the stress intensity factor of the macro-crack can be obtained. Contour r is chosen to encircle all the micro-cracks and inhomogeneities inside. This can always be done even though we do not know the exact distribution of the micro-cracks and inhomogeneities. The part of the body inside r is then replaced by surface tractions ti and displacements ui along r [Fig. 4(b)]. The elastic field of the rest of the body is obtained by solving the overspecified boundary value problem (AB and A’B’, with known displacement and traction on them, consist of $r boundary; r plus ar) is the & boundary). Therefore, the crack intensity factor of the main crack is computed by J-integral [S] J=

wdy-‘Tgds

>

Elasticity problems

691

0.0 -. . -0.1-

.

-0.2-

t P 2

-0.3-

t

-0.4-

+ -0.5. -0.6 0.0

I

I.,

0.2

.

0.4

0.6

I.

0.8

I.

1.o

1.2

Aa Fig. 5. K, the decrease

of stress intensity factor due to the existence inhomogeneities, versus Aa [see Fig. 4(a)].

of micro-cracks

and

with W = W(X, Y) = W(E) = [

Uijds,.

T is the tension vector (traction) perpendicular to r in the outside direction, u is the displacement in the crack axis direction. The numerical calculation for the stress intensity factor is shown in Fig. 5, where the decrease of stress intensity factor (normalized) of the macro-crack due to the existence of micro-cracks and inhomogeneities is plotted. In Fig. 5, l/2 P = 0.316E E;E,; dx/A , 1 UR where E$ is the fictious eigenstrain [6j introduced to simulate the existence of micro-cracks and inhomogeneities, A is the area of region in which E$ exists. E is the Young’s modulus of the material. Acknowledgement-This research was supported under U.S. Army Research Office Contract Number DAAL03-88-C0027 through a subcontract with Rockwell International Science Center.

REFERENCES (I] N. I. MUSKHELISHVILI, Some Basic Problems of the mathematical Theory of Elasticity, 3rd edn. Noordhoff, Groningen (1963). [Z] N. KINOSH~A and T. MURA, On the boundary value problem of elasticity, Res. Rept. Faculty of Engng, Meiji Univ., No. 8 (1956). [3] Z. GAO and T. MURA, J. Appt. Mech. 56, 3 (1989). [4] A. N. TIKHONOV, Dokl. Akad. Nuuk SSSR 5, 3 (1963). [5] J. R. RICE, J. Appl. Mech. 3S, 3 (1968). [6] T. MURA, Micromechanics of Defecrs in Solids, 2nd rev. edn, Martinus Nijhoff, The Hague (1987). (Received

7August

1990; accepted 27September

APPENDIX Let

u:‘(s) = u,(s) = a,,, t:(s) = z,(s) = 0,

r= 1,2,

1990)

Z. GAO and T. MURA

692

which corresponds to a rigid body motion. Then, equation (4) is reduced to L. fL(s, r) ~ = - ‘C,,k,Gk,,,(l(s)~ f(r))n, d.r. I I, I0 Choose a small positive t - e 6s C t + s, i.e.

number

E. For /S - tl < E, we define

II&,

t)

(Al) as its average

over the interval

(A2) The last equality holds due to (Al). Similarly, we define (A3) for Js - tl s E. The integral on the right-hand side of (A3) can be evaluated directly. The definitions of f&,,, r, equations (6a) and (6b) are modified as, according to equations (A2) and A3),

Cij/&km,t(f(S)rf(t))n,(s) + 0.56, ‘(t - $1, I&&,

r) d.9 -

I

lt-Sl>&,

L

h?As~ t) ds

T+E

-

j-r-’ Ci,&m,&(~)~f(4)@) d-s) It-S)‘+I&, 9

The integrand of equation (9) is, therefore,

a regular function.

in