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Existence and uniqueness theorems for the solution of boundary integral equations for linearized elastostatics
Compurers d Strucrures Vol. 15. So. Pnntcd I” Great Bnram.
3, pp. 365-369.
1997
EXISTENCE AND UNIQUENESS THEOREMS FOR THE SOLUTION OF BOUNDARY INTEGRAL EQUATIONS FOR LINEARIZED ELASTOSTATICS XIXGFA
WA%
Department of Materials and Mechanics. Beijing Institute of Technology.
Beijing. China
(Receiwd I6 June 1986) Abstract-The paper consists of both theory and application. The existence and uniqueness theorems for the solution of boundary integral equations for linearized elastostatics are proposed. and computational models for direct formulation of boundary integral equations are presented. By using the existence and uniqueness theorems of this paper, the existence and uniqueness of the solution of the boundary integral equations are determined, showing that the method used here is successful. The computational models presented are suited to the solution of mixed boundary value problems and linear elasticity problems in geomechanics.
1. INTRODUCTION
fields including mining engineering. rapid tunnelling. explosive rock breakage, stability analysis of massive hydraulic structures, as well as underground excavation, experimental rock mechanics and so on. The rapid rock fracture process can be viewed mainly as a result of crack growth. The ability of fracture mechanics to account for the behaviour of a crack makes it a suitable tool for dealing with rapid rock fracture problems. But this problem differs from the usual linear elastic fracture mechanics problem which is stated in terms of a given global geometry, flaw geometry and auxiliary (boundary and initial) conditions. The problem is to determine conditions for the incipient growth and propagation of this flaw. The unknown location of initiation as well as the growth of the damage region requires investigation of the stress field for which the BIEM is particularly well suited. In this paper, the existence and uniqueness theorems of solution of boundary integral equations for linearized elastostatics are proposed and the computational models of direct formulation of boundary integral equations are presented. By using the existence and uniqueness theorems presented here, the existence and uniqueness of the solution of the boundary integral equations in this paper are determined.
The boundary integral equation method (BIEM) is based on an integral equation formulation of a given problem and is used for the solution of the boundary vahte problems. The boundary integral equation is rooted in classical integral equation theory. Rizzo [I] and Cruse [Z] first proposed the discretization of the partial differential equation into the boundary integr. I equation. The formulations first developed are preliminary, but the accuracy of this method has been and improved by Cruse[3] and extended Ricardella [4]. Recently BIEM has been widely used in many branches of science and engineering [5-l 3]. BIEM has several advantages over the widely used finite-element method (FEM). In BIEM the number of unknowns in the resultant algebraic systems depends on the boundary (or surface) discretization rather than on the discretization of the entire domain of the body as in FEM. The maximum principle may be often used for problems of error estimation in BIEM. With regard to accuracy and computational efficiency, BIEM is better than FEM in many appliobtained by cations. The physical quantities differentiation of the primary variables, e.g. stresses, are determined pointwise inside and on the body in BIEM, so that discontinuities in these variables across element boundaries cannot arise. This is a very 2. THEOREY important advantage for solutions of non-linear The classical result for uniqueness in elasticity problems in geomechanics. A third important advantheory is given by Kirchhoff[l4]. It states that the tage is that problems in infinite regions can be solved as easily as those in finite regions. This makes the standard mixed boundary vaIue problem for a homogeneous, isotropic, linear, efastic material in equimethod very suitabfe for handling geome~hani~s problems. librium and ~~upying a bounded three-dimensional The present paper consists of theory and applicaregion of space possesses at most one solution in the tion of BIEM to linear elasticity problems in geo- classical sense, provided the Lame i. and shear moduli mechanics. Such problems are of interest in many G obey the inequalities (3;. + 2G) > 0 and G > 0. In 365
366
XlNGF.4
linear elastodynamics an analogous result was given by Neumann [IS]. Subsequently the existence and uniqueness of various solutions m the classicaf sense for elasticity theory were developed. But the analytical solution in a classical sense often cannot be obtained in many engineering problems. Recently in a large number of applications boundary integral equations with some kind of singular kernel have arisen. If the singularity is strong enough, the principal value of the integral must be defined. These problems are closed concerning the existence and the uniqueness of the weak sofution of boundary integral equations with a singular kernel in the principal value sense.
WANG
The traction vector field 9. E c’B, has points MOE B and M E ZB. Theorem 1. Given arbitrary vector fields of U,(M) E H[ZB] and P,(M) +zH[ciB] at each point of ZB, respectively. there is a unique solution U,(M,) E B which assumes these boundary vector fields. 2.3. The uniqueness theoremfor the solution ofbortndnry integrnf eqz~fftions with the Cal~e~z~kernel for finearked efastostatics Regularized boundary integral equations with the Cauchy kernel may be written as the following form of the Fredholm integral equation of the second kind
*h cb(X) - j. k(% .x)4(s) d.s =f(;t) (1) 2. t The existence theorem of sohtion of b~z~n~~r~ J. integral equation with the Caatchy kernel for linearized where the parameter i of eqn (I) is an arbitrary elastostarics complex variable. unknown function 4(x) and funcA function d, is continuous within a twotion f(x) are square integrable functions over [a, b], dimensional domain B (or a three-dimensional doand the kernel i;(x ‘sf of (1) is a square integrable main), bounded by a closed piecewise smoothing function over [a. 6; a, b]. Function (b is continuous curve SB (or a closed piecewise smoothing surface). within a two-dimensional domain B (or a threeTwo distinct points P, M E ZB. and SB = SB, ti I?&. dimensional domain), bounded by a closed piecewisc Theroem 1. Given arbitrary values of smoothing curve ?B (or a closed piecewise smoothing 6(M) E H[L?B,] and ?#(M)jSn c H[?B,] at each surface). Two distinct points p, :M EZB, and point of dB, and SB,, respectively, there exist the as = f%, u dB,. weak solutions 4(p) E SB2 and &$ ( p)/i?n E B, which Theorem 5. Given values of arbitrary assume these boundary values, respectively. where c;b(M) E H[aB,] and &$(.V)/Sn E H[SB,] at each a$/& denotes the outward normal derivative of 4 at point of dB, and as ZB,, respectively, and given i, as arbitrary point on boundary SB. a regular value. there is at most one weak solution to A displacement vector field Vi is continuous within 4(p) E ii& and E#( p)/8n E SB, which assumes these a two-dimensional domain B (or a three-dimensional boundary values. respectively, where ~~~~n denotes domain), bounded by a closed piecewise smoothing the outward normal derivative of (b at an arbitrary curve 2B (or a closed piecewise smoothing surface). point on the boundary. The traction vector field P,E dB has two distinct A displacement vector field U, is continuous with points p, A4 E SB, and 2B = SB, U 2Bz. a two-dimensional domain B (or a three-dimensional Theorem 2. Given arbitrary vector fields of domain), bounded by a closed piecewise smoothing Uj(M) E H[aB,] and P,(M) E N[S&] at each point of curve dB {or a closed piecewise smoothing surface). 8B, and d&, respectiveiy, there exist the weak soluThe traction vector field P,edB has two distinct tions U,(p) E 8B2 and P,(p) E 8B, which assume these points p, A4 E 8B. and (38 = aB, U dB:. boundary vector fields, respectively. Theorem 6. Given arbitrary vector fields of u,(M) E H[dB,] and P,(M) E H[8B2] at each point of 2.2. The existence-uniqueness theorem for the solution of boundary integral equations for linearized dB, and aBz, respectively, and given i. being a regular value, there is at most one weak solution to elastostatics uj( p) E ML! and P,(p) E c’B, which assumes these A function 4 is continuous within a twoboundary vector fields, respectively. dimensional domain B (or a three-dimensional domain), bounded by a closed piecewise smoothing 3. PROOF curve 2B (or a closed piecewise smoothing surface). Point MOE B, and point M E c?B. The singular integral equation with the Cauchy Theorem 3. Given arbitrary values of Q)(M)6 kernel may be expressed in the form H[dB] and dq5(M)/En E H[aB] at each point of dB V = A(P)?(P) respectively, there is a unique solution of ~(.~*) E B which assumes these boundary values, where &/an denotes the outward normal derivation of $J at arbitrary point on boundary SB. A displacement vector Ui is continuous within a where II denotes the singular operator, 8B the smooth curve; two distinct points M, p E dB; known functwo-dimensional domain B (or a three-dimensional tions ff p), A(p) and K( p, M)f HfZBf, and undomain), bounded by a closed piecewise smoothing known function (3(M) E H(dB). curve ZB (or a closed piecewise smoothing surface).
Solution of boundary integral equations for linearized elastostatics The existence of solutions to the Fredholm integral equation was first rigorously demonstrated by Fredholm [ 161. Hilbert and Poincarb subsequently deduced the same form as eqn (2) and established the theory of a singular integral equation with the Cauthy kernel. The class of boundary integraf equations with the Cauchy kernel for linearized elastostatics may be expressed in the form of eqn (2). By using singular integral equation theory and Fredholm integral equation theory [ 17-201, the above-mentioned theorems may be proved in view of (2). (Detailed processes of the proof are omitted here. )
4. COMPUTATIONAL
MODEL
Based on the above-mentioned theory, we may now develop the computational models for the direct formulation of boundary integral equations for linearized elastostatics. The fundamental solution satisfying the H-condition is first given in terms of the ~-function. Consider an m-order differential equation with constant coe~cient
u = E*/
with points IW~E B, and M E C’Band where r is the distance between two distinct points MO and M. If 4(M) and its normal derivative ?&/an are known on boundary dB, then (p(M,) at the point M, in the region B may be determined in terms of eqn (8).
Consider a biharmonic equation of linear elasticity v4p = 0
(9)
where p is the stress function. The fundamenta1 solution of eqn (9) is r2 In r. By using the abovementioned procedure, we transfer the biha~oni~ equation into one set of the boundary integral equation as follows
(3)
If p(Z)E = 6, E is called the fundamental solution of p(a)u = 0, where 6 is Dirac’s delta function. Since E is the fundamental solution of p(a)u = 0, then, p(d)u =J
lation, we can obtain the foIlowing two-dimensional boundary integral equation of the Dirichlet and Neumann problems (detailed deductive processes are omitted here):
4.2. Biharntonic equnrion
4.1. The Dirichfet and the Neumann problems
p(d)u = c Il;,d%J= 0.
367
p(M)$A(r’Inr) a&M)
- A(r* In r) T
(4)
a
+4(M)-(r’lnr) an
where E*_f= Consider function equation
1 [$(M)~ hr b,Kd=&~ 1 - (r2 In r) v
+ZZE(x -u)f(v)dv. 1 --c 4 to be satisfied by the Laplace
dS,W
EB
W(M)
--Inr
an
dS,%.
4.3. Three-dimensional
Luplace equation and the Navier-type displacement equilibrium eqwrion
where &#/&I denotes the outward normal derivative of #I at arbitrary point on boundary 83, the boundary functionfor g, defined over a bounded region B with boundary dB, may be unknown. These are the wellknown Dirichlet and Neumann problems. By using the d-function and the first Green’s formulation, we may obtain the fundamental solution cp of eqn (5) cp = -1nr. Substituting
(7)
eqn (7) into the second Green’s formu-
By using the same above-mentioned procedure, we can obtain the boundary integral equation of the three-dimensional Laplace equation as follows
.?
-d(M);( ;)Ida,
where ): is the boundary dimensional body.
surface
(11)
of a three-
XINGFA WASG
368
Similarly the Navier-type Tao-dimensiona displacement equilibrium equation may be transformed into the following boundary integral equation
U,(iM*)
U,jf,(iCf)
=
- T,,u,(M)
1
(14)
dS,
(i,j = 1,2).
(12)
If the boundary vector fields of the displacement u,(M) and the traction P,(M) at each point on boundary aB are known, the displacement vector field ui(M,,) at any point MO in the body may be determined in terms of (12). Uij and qj in (12) are the second-order tensors: Uij = c, (Si,cz In r - rirsj) 7;$
r
-
[
d2r an
Csijc4
+
dS,
ar,ir,,f
U,(P) = 2
drjni
-
uijpjtM)
r,,n,)
1
- T;,u,(M)
in which -1
“=87tG(1 Cl = 3 - 4jl,
-Jf)’
(15)
-1 cJ= 4n(l -,U) c, = 1 - 2p
where n, and nj are the unit direction cosines of the normal at a point on boundary ZB, G is the shear modulus, p is Poisson’s ratio of materials, and rqi, rJ denote the partial derivative of r with respect to orthogonal coordinates .r,, .xj. 4.4. The computational model of the boundary integral equations with the Cauchy kernel
In general the boundary values of function 4(M) and 24(M)/& or the boundary vector fields of traction P,(M) and displacement u,(M) at each point on boundary dB (or surface C) are not simultaneously known. Since the fundamental solutions in the paper satisfy the ~-conditions (8), (10-12) are transfo~ed, respectively, into the following form
(13)
1
d.S,,
(lj = 1,2)
where boundary curve ?B (or surface X) must be piecewise smoothing, and rp,Mis the distance between two distinct points P and M on boundary 8B (or surface Z). All unknown boundary factors in (S), (10-12) may now be determined by using (13-16) then unknown functions and the displacement vector field at any point M,, in the body B are obtained by using (81, (lo- 12) respectively. 5.
APPLICATION
By using the existence and uniqueness theorems of the paper the existence and uniqueness of solution of the above-mentioned boundary integral equations may now be determined.
(1) Based on Theorem 1, if the boundary value of
ME H[aI3,] and Z&(M)/an E H[aB,] at each point of c?B, and aBz are given respectively, then for the boundary integral eqns (13-I 5) there exist the solutions #(p) E aB2 and a~~~)/an EdB,. (2) Based on Theorem 2, if the boundary vector fields of u,(M) E H(c^B,J and P,(N) E H[aB?] at each point of dB, and ZB, are given respectively, then for the boundary integral equation of (16) there exists the sotutions U,(P)E ZBz and P,(P)E~&. (3) Based on Theorem 3, if the boundary values of ME H[aB] and ZQ,(M)/an E H[dB] at each point of dB are given, then the boundary integral equations of (8), (10) (11) have a unique solution ~(~~)~B. (4) Based on Theorem 4, if the boundary vector fields of uj(M)e H[cB] and p,(M)e H[dB] at
Solution of boundary integral equations for linearized elastostatics
each point of ?B are given respectively, then the boundary integral equation of (12) has a unique solution u,(MO) E B. (5) Based on Theorem 5, if the boundary values of 4c.M) E H[SB,] and @(M)/Zn E H[ZB:] at each point of SB, and dB, are given respectively and a regular value i. is chosen, then the boundary integral equations of (13-15) have at most one solution 4(p) E ZBz and Z&(p)/dn E SB,. (6) Based on Theorem 6, if the boundary vector fields of u,(M) E H[dB,] and p,(M) E H[?B,] at each point of ?B, and aBz are given respectively and a regular value E.is chosen, then the boundary integral equation of (16) has at most one solution u,(p) E LIBz as well as p,(p) E ?B, If the singularity is strong enough, (13-16) of the boundary integral equations with the Cauchy kernel are to be interpreted in the Cauchy principal value sense and their solutions have the wholly definite sense. 6. DISCUSSION
The solving ability of a numberical method can only be judged by obtaining numerical solutions to specific problems. The existence-uniqueness of the numerical solutions to these problems must first be taken into consideration. Boundary integral equations with various singular kernels are now used for solving a large number of problems in science and engineering, e.g. wave propagation, contact problems, fluid-structure interaction, viscoplasticity, creep, fracture, and complicated non-linear problems. This numerical method is evidently of considerable engineering significance, but its theory has not yet been well developed. An important limitation of the developed boundary integkl equation method is the determination of the singular kernels of integral equations. The kernels of the integral equations have been successfully constructed by means of these effective mathematical tools and the method can be definitely used in a wide range of science and engineering applications. These problems are closed concerning the developed existence and uniqueness of the solution of this method. Acknoi~ledgemenrs-I wish to express my sincere gratitude to professors at the Institute of Geology, University of Ruhr, in particular Prof. Dr-Ing. H. K. Kutter for his support and help during the writing of this paper. The
369
author sincerely appreciates the financial support afforded by the Heinrich-Hertz-Stiftung. REFERENCES
I.
2.
3.
4.
5.
F. J. Rizzo. An integral equation approach to boundary value problems of classical elastostatics. Q. ._~ppl.Ma/h. 25, 83-98 (1967). T. A. Cruse. Numerical solutions in three-dimensional elastostatics. In,. J. Solids Struct. 5, 1259-1274 (1969). T. A. Cruse, An improved boundary integral equation method for three-dimensional elastic stress analysis. Compur. Srrucr. 4, 741-757 (1974). P. C. Ricardella, An improved boundary integral equation method for two-dimensional elasticity problems. Carneeie-Mellon Universitv (1972). T. A. &use and F. J. Rizzd (Eds). Boundary lnregral
Equation Method: Computational Applications in Applied Mechanics. ASME, New York (1975). 6. S. L. Crouch and A. M. Starfield, B0undar.v Elemenr Methods in Solid Mechanics. George Allen & Unwin.
London (1983). 7. C. A. Brebbia (Ed.), Progress in B0undar.v Elemenf Methods, Vol. 2. Pentech Press, London (1983). 8. R. Butterfield, The application of integral equation methods to continuum problems in soil mechanics. Proc. Roscoe Memorial Symp. (Edited by R. H. G. Parry). 573-587. Henley-on-Thames (1971). 9. P. K. Banerjee and R. Butterfield (Eds), Derelopmenr in Boundary Elemeni Methods, Vol. I. Applied Science, London (1979). 10. P. K. Banerjee and R. P. Shaw (Eds), Developments in Boundary Element Merhodr. Vol. 2. Applied Science, London (1982). II. C. A. Brebbia and P. Geogiou, Combination of boundary and finit elements in elastostatics. Appl. Math. Modell. 3, 212-220 (1979).
12 T. A. Cruse, Boundary-integral equation method for three-dimensional elastic fracture mechanics analysis. Air Force Office of Scientific research-TR-75-0813. Accession No. ADA 011660 (1975). 13. S. Mukherjee, Boundary Elemenr Methods in Creep and Fracrure. Applied Science, London (1982). 14. G. Kirchhoff. Vorlesuwen iiber mafhematische fhvsik ’ Mechanik, Vol. 3. Teubuner, Leipzig (1877). IS. F. Neumann, Vorlesungen tiber die Theorie der Elasriziriir der fesren K&per und des Lichrdrhers, Vols 3. 8. Teuber, Leipzig (I 885). 16. I. Fredholm, Sur une class d’equations fonctionelles. Acra Mar. 27, 365-390 (1903).
17. S. G. Mikhlin, Multidimensional singular Integrals and lnregral Equations. Fizmatigiz, Moscow (1962). 18. N. I. Muskhelishvili, Singular lnfegral Equalions. Noordhoff, Groningen (1972). 19. N. I. Muskhelishvili, Some Basic Problems of the Mafhemarical Theory of Elasticity. Noordhoff, Groningen (1953). 20. G. Hoheisel, lnregral Equations. Nelson. London (1967).
3, pp. 365-369.
1997
EXISTENCE AND UNIQUENESS THEOREMS FOR THE SOLUTION OF BOUNDARY INTEGRAL EQUATIONS FOR LINEARIZED ELASTOSTATICS XIXGFA
WA%
Department of Materials and Mechanics. Beijing Institute of Technology.
Beijing. China
(Receiwd I6 June 1986) Abstract-The paper consists of both theory and application. The existence and uniqueness theorems for the solution of boundary integral equations for linearized elastostatics are proposed. and computational models for direct formulation of boundary integral equations are presented. By using the existence and uniqueness theorems of this paper, the existence and uniqueness of the solution of the boundary integral equations are determined, showing that the method used here is successful. The computational models presented are suited to the solution of mixed boundary value problems and linear elasticity problems in geomechanics.
1. INTRODUCTION
fields including mining engineering. rapid tunnelling. explosive rock breakage, stability analysis of massive hydraulic structures, as well as underground excavation, experimental rock mechanics and so on. The rapid rock fracture process can be viewed mainly as a result of crack growth. The ability of fracture mechanics to account for the behaviour of a crack makes it a suitable tool for dealing with rapid rock fracture problems. But this problem differs from the usual linear elastic fracture mechanics problem which is stated in terms of a given global geometry, flaw geometry and auxiliary (boundary and initial) conditions. The problem is to determine conditions for the incipient growth and propagation of this flaw. The unknown location of initiation as well as the growth of the damage region requires investigation of the stress field for which the BIEM is particularly well suited. In this paper, the existence and uniqueness theorems of solution of boundary integral equations for linearized elastostatics are proposed and the computational models of direct formulation of boundary integral equations are presented. By using the existence and uniqueness theorems presented here, the existence and uniqueness of the solution of the boundary integral equations in this paper are determined.
The boundary integral equation method (BIEM) is based on an integral equation formulation of a given problem and is used for the solution of the boundary vahte problems. The boundary integral equation is rooted in classical integral equation theory. Rizzo [I] and Cruse [Z] first proposed the discretization of the partial differential equation into the boundary integr. I equation. The formulations first developed are preliminary, but the accuracy of this method has been and improved by Cruse[3] and extended Ricardella [4]. Recently BIEM has been widely used in many branches of science and engineering [5-l 3]. BIEM has several advantages over the widely used finite-element method (FEM). In BIEM the number of unknowns in the resultant algebraic systems depends on the boundary (or surface) discretization rather than on the discretization of the entire domain of the body as in FEM. The maximum principle may be often used for problems of error estimation in BIEM. With regard to accuracy and computational efficiency, BIEM is better than FEM in many appliobtained by cations. The physical quantities differentiation of the primary variables, e.g. stresses, are determined pointwise inside and on the body in BIEM, so that discontinuities in these variables across element boundaries cannot arise. This is a very 2. THEOREY important advantage for solutions of non-linear The classical result for uniqueness in elasticity problems in geomechanics. A third important advantheory is given by Kirchhoff[l4]. It states that the tage is that problems in infinite regions can be solved as easily as those in finite regions. This makes the standard mixed boundary vaIue problem for a homogeneous, isotropic, linear, efastic material in equimethod very suitabfe for handling geome~hani~s problems. librium and ~~upying a bounded three-dimensional The present paper consists of theory and applicaregion of space possesses at most one solution in the tion of BIEM to linear elasticity problems in geo- classical sense, provided the Lame i. and shear moduli mechanics. Such problems are of interest in many G obey the inequalities (3;. + 2G) > 0 and G > 0. In 365
366
XlNGF.4
linear elastodynamics an analogous result was given by Neumann [IS]. Subsequently the existence and uniqueness of various solutions m the classicaf sense for elasticity theory were developed. But the analytical solution in a classical sense often cannot be obtained in many engineering problems. Recently in a large number of applications boundary integral equations with some kind of singular kernel have arisen. If the singularity is strong enough, the principal value of the integral must be defined. These problems are closed concerning the existence and the uniqueness of the weak sofution of boundary integral equations with a singular kernel in the principal value sense.
WANG
The traction vector field 9. E c’B, has points MOE B and M E ZB. Theorem 1. Given arbitrary vector fields of U,(M) E H[ZB] and P,(M) +zH[ciB] at each point of ZB, respectively. there is a unique solution U,(M,) E B which assumes these boundary vector fields. 2.3. The uniqueness theoremfor the solution ofbortndnry integrnf eqz~fftions with the Cal~e~z~kernel for finearked efastostatics Regularized boundary integral equations with the Cauchy kernel may be written as the following form of the Fredholm integral equation of the second kind
*h cb(X) - j. k(% .x)4(s) d.s =f(;t) (1) 2. t The existence theorem of sohtion of b~z~n~~r~ J. integral equation with the Caatchy kernel for linearized where the parameter i of eqn (I) is an arbitrary elastostarics complex variable. unknown function 4(x) and funcA function d, is continuous within a twotion f(x) are square integrable functions over [a, b], dimensional domain B (or a three-dimensional doand the kernel i;(x ‘sf of (1) is a square integrable main), bounded by a closed piecewise smoothing function over [a. 6; a, b]. Function (b is continuous curve SB (or a closed piecewise smoothing surface). within a two-dimensional domain B (or a threeTwo distinct points P, M E ZB. and SB = SB, ti I?&. dimensional domain), bounded by a closed piecewisc Theroem 1. Given arbitrary values of smoothing curve ?B (or a closed piecewise smoothing 6(M) E H[L?B,] and ?#(M)jSn c H[?B,] at each surface). Two distinct points p, :M EZB, and point of dB, and SB,, respectively, there exist the as = f%, u dB,. weak solutions 4(p) E SB2 and &$ ( p)/i?n E B, which Theorem 5. Given values of arbitrary assume these boundary values, respectively. where c;b(M) E H[aB,] and &$(.V)/Sn E H[SB,] at each a$/& denotes the outward normal derivative of 4 at point of dB, and as ZB,, respectively, and given i, as arbitrary point on boundary SB. a regular value. there is at most one weak solution to A displacement vector field Vi is continuous within 4(p) E ii& and E#( p)/8n E SB, which assumes these a two-dimensional domain B (or a three-dimensional boundary values. respectively, where ~~~~n denotes domain), bounded by a closed piecewise smoothing the outward normal derivative of (b at an arbitrary curve 2B (or a closed piecewise smoothing surface). point on the boundary. The traction vector field P,E dB has two distinct A displacement vector field U, is continuous with points p, A4 E SB, and 2B = SB, U 2Bz. a two-dimensional domain B (or a three-dimensional Theorem 2. Given arbitrary vector fields of domain), bounded by a closed piecewise smoothing Uj(M) E H[aB,] and P,(M) E N[S&] at each point of curve dB {or a closed piecewise smoothing surface). 8B, and d&, respectiveiy, there exist the weak soluThe traction vector field P,edB has two distinct tions U,(p) E 8B2 and P,(p) E 8B, which assume these points p, A4 E 8B. and (38 = aB, U dB:. boundary vector fields, respectively. Theorem 6. Given arbitrary vector fields of u,(M) E H[dB,] and P,(M) E H[8B2] at each point of 2.2. The existence-uniqueness theorem for the solution of boundary integral equations for linearized dB, and aBz, respectively, and given i. being a regular value, there is at most one weak solution to elastostatics uj( p) E ML! and P,(p) E c’B, which assumes these A function 4 is continuous within a twoboundary vector fields, respectively. dimensional domain B (or a three-dimensional domain), bounded by a closed piecewise smoothing 3. PROOF curve 2B (or a closed piecewise smoothing surface). Point MOE B, and point M E c?B. The singular integral equation with the Cauchy Theorem 3. Given arbitrary values of Q)(M)6 kernel may be expressed in the form H[dB] and dq5(M)/En E H[aB] at each point of dB V = A(P)?(P) respectively, there is a unique solution of ~(.~*) E B which assumes these boundary values, where &/an denotes the outward normal derivation of $J at arbitrary point on boundary SB. A displacement vector Ui is continuous within a where II denotes the singular operator, 8B the smooth curve; two distinct points M, p E dB; known functwo-dimensional domain B (or a three-dimensional tions ff p), A(p) and K( p, M)f HfZBf, and undomain), bounded by a closed piecewise smoothing known function (3(M) E H(dB). curve ZB (or a closed piecewise smoothing surface).
Solution of boundary integral equations for linearized elastostatics The existence of solutions to the Fredholm integral equation was first rigorously demonstrated by Fredholm [ 161. Hilbert and Poincarb subsequently deduced the same form as eqn (2) and established the theory of a singular integral equation with the Cauthy kernel. The class of boundary integraf equations with the Cauchy kernel for linearized elastostatics may be expressed in the form of eqn (2). By using singular integral equation theory and Fredholm integral equation theory [ 17-201, the above-mentioned theorems may be proved in view of (2). (Detailed processes of the proof are omitted here. )
4. COMPUTATIONAL
MODEL
Based on the above-mentioned theory, we may now develop the computational models for the direct formulation of boundary integral equations for linearized elastostatics. The fundamental solution satisfying the H-condition is first given in terms of the ~-function. Consider an m-order differential equation with constant coe~cient
u = E*/
with points IW~E B, and M E C’Band where r is the distance between two distinct points MO and M. If 4(M) and its normal derivative ?&/an are known on boundary dB, then (p(M,) at the point M, in the region B may be determined in terms of eqn (8).
Consider a biharmonic equation of linear elasticity v4p = 0
(9)
where p is the stress function. The fundamenta1 solution of eqn (9) is r2 In r. By using the abovementioned procedure, we transfer the biha~oni~ equation into one set of the boundary integral equation as follows
(3)
If p(Z)E = 6, E is called the fundamental solution of p(a)u = 0, where 6 is Dirac’s delta function. Since E is the fundamental solution of p(a)u = 0, then, p(d)u =J
lation, we can obtain the foIlowing two-dimensional boundary integral equation of the Dirichlet and Neumann problems (detailed deductive processes are omitted here):
4.2. Biharntonic equnrion
4.1. The Dirichfet and the Neumann problems
p(d)u = c Il;,d%J= 0.
367
p(M)$A(r’Inr) a&M)
- A(r* In r) T
(4)
a
+4(M)-(r’lnr) an
where E*_f= Consider function equation
1 [$(M)~ hr b,Kd=&~ 1 - (r2 In r) v
+ZZE(x -u)f(v)dv. 1 --c 4 to be satisfied by the Laplace
dS,W
EB
W(M)
--Inr
an
dS,%.
4.3. Three-dimensional
Luplace equation and the Navier-type displacement equilibrium eqwrion
where &#/&I denotes the outward normal derivative of #I at arbitrary point on boundary 83, the boundary functionfor g, defined over a bounded region B with boundary dB, may be unknown. These are the wellknown Dirichlet and Neumann problems. By using the d-function and the first Green’s formulation, we may obtain the fundamental solution cp of eqn (5) cp = -1nr. Substituting
(7)
eqn (7) into the second Green’s formu-
By using the same above-mentioned procedure, we can obtain the boundary integral equation of the three-dimensional Laplace equation as follows
.?
-d(M);( ;)Ida,
where ): is the boundary dimensional body.
surface
(11)
of a three-
XINGFA WASG
368
Similarly the Navier-type Tao-dimensiona displacement equilibrium equation may be transformed into the following boundary integral equation
U,(iM*)
U,jf,(iCf)
=
- T,,u,(M)
1
(14)
dS,
(i,j = 1,2).
(12)
If the boundary vector fields of the displacement u,(M) and the traction P,(M) at each point on boundary aB are known, the displacement vector field ui(M,,) at any point MO in the body may be determined in terms of (12). Uij and qj in (12) are the second-order tensors: Uij = c, (Si,cz In r - rirsj) 7;$
r
-
[
d2r an
Csijc4
+
dS,
ar,ir,,f
U,(P) = 2
drjni
-
uijpjtM)
r,,n,)
1
- T;,u,(M)
in which -1
“=87tG(1 Cl = 3 - 4jl,
-Jf)’
(15)
-1 cJ= 4n(l -,U) c, = 1 - 2p
where n, and nj are the unit direction cosines of the normal at a point on boundary ZB, G is the shear modulus, p is Poisson’s ratio of materials, and rqi, rJ denote the partial derivative of r with respect to orthogonal coordinates .r,, .xj. 4.4. The computational model of the boundary integral equations with the Cauchy kernel
In general the boundary values of function 4(M) and 24(M)/& or the boundary vector fields of traction P,(M) and displacement u,(M) at each point on boundary dB (or surface C) are not simultaneously known. Since the fundamental solutions in the paper satisfy the ~-conditions (8), (10-12) are transfo~ed, respectively, into the following form
(13)
1
d.S,,
(lj = 1,2)
where boundary curve ?B (or surface X) must be piecewise smoothing, and rp,Mis the distance between two distinct points P and M on boundary 8B (or surface Z). All unknown boundary factors in (S), (10-12) may now be determined by using (13-16) then unknown functions and the displacement vector field at any point M,, in the body B are obtained by using (81, (lo- 12) respectively. 5.
APPLICATION
By using the existence and uniqueness theorems of the paper the existence and uniqueness of solution of the above-mentioned boundary integral equations may now be determined.
(1) Based on Theorem 1, if the boundary value of
ME H[aI3,] and Z&(M)/an E H[aB,] at each point of c?B, and aBz are given respectively, then for the boundary integral eqns (13-I 5) there exist the solutions #(p) E aB2 and a~~~)/an EdB,. (2) Based on Theorem 2, if the boundary vector fields of u,(M) E H(c^B,J and P,(N) E H[aB?] at each point of dB, and ZB, are given respectively, then for the boundary integral equation of (16) there exists the sotutions U,(P)E ZBz and P,(P)E~&. (3) Based on Theorem 3, if the boundary values of ME H[aB] and ZQ,(M)/an E H[dB] at each point of dB are given, then the boundary integral equations of (8), (10) (11) have a unique solution ~(~~)~B. (4) Based on Theorem 4, if the boundary vector fields of uj(M)e H[cB] and p,(M)e H[dB] at
Solution of boundary integral equations for linearized elastostatics
each point of ?B are given respectively, then the boundary integral equation of (12) has a unique solution u,(MO) E B. (5) Based on Theorem 5, if the boundary values of 4c.M) E H[SB,] and @(M)/Zn E H[ZB:] at each point of SB, and dB, are given respectively and a regular value i. is chosen, then the boundary integral equations of (13-15) have at most one solution 4(p) E ZBz and Z&(p)/dn E SB,. (6) Based on Theorem 6, if the boundary vector fields of u,(M) E H[dB,] and p,(M) E H[?B,] at each point of ?B, and aBz are given respectively and a regular value E.is chosen, then the boundary integral equation of (16) has at most one solution u,(p) E LIBz as well as p,(p) E ?B, If the singularity is strong enough, (13-16) of the boundary integral equations with the Cauchy kernel are to be interpreted in the Cauchy principal value sense and their solutions have the wholly definite sense. 6. DISCUSSION
The solving ability of a numberical method can only be judged by obtaining numerical solutions to specific problems. The existence-uniqueness of the numerical solutions to these problems must first be taken into consideration. Boundary integral equations with various singular kernels are now used for solving a large number of problems in science and engineering, e.g. wave propagation, contact problems, fluid-structure interaction, viscoplasticity, creep, fracture, and complicated non-linear problems. This numerical method is evidently of considerable engineering significance, but its theory has not yet been well developed. An important limitation of the developed boundary integkl equation method is the determination of the singular kernels of integral equations. The kernels of the integral equations have been successfully constructed by means of these effective mathematical tools and the method can be definitely used in a wide range of science and engineering applications. These problems are closed concerning the developed existence and uniqueness of the solution of this method. Acknoi~ledgemenrs-I wish to express my sincere gratitude to professors at the Institute of Geology, University of Ruhr, in particular Prof. Dr-Ing. H. K. Kutter for his support and help during the writing of this paper. The
369
author sincerely appreciates the financial support afforded by the Heinrich-Hertz-Stiftung. REFERENCES
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