An algorithm is presented which can be used to develop compliance matrices for cracked bodies. The method relies on the numerical solution of singular integral equations with Cauchy-type kernels and provides an efficient and accurate procedure for relating applied loadings to crack opening displacements. The algorithm should be of interest to those performing repetitive calculations in the analysis of experimental results obtained from fracture specimens

Ballarini, R.

English

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NASA Contractor Report 179478
Compliance Matrices for
Cracked Bodies
,
(NASA-CB-179478) COHFLIftNCE MfiTBICES POB N86-30236
CBACKED BODIES Final Beport (Cleveland
State Dniv.)/ 8 p CSCL 20K
Unclas
63/39 43562
R. Ballarini
Cleveland State University
Cleveland, Ohio
July 1986
Prepared for
Lewis Research Center
Under Grant NCC 3-46
NASA
National Aeronautics and
Space Administration
https://ntrs.nasa.gov/search.jsp?R=19860020764 2020-03-20T14:38:42+00:00Z
COMPLIANCE MATRICES FOR CRACKED BODIES
R. BallaMnl
Cleveland State University
Department of C1v1l Engineering
Cleveland, Ohio 44115
SUMMARY
An algorithm 1s presented which can be used to develop compliance matrices
for cracked bodies. The method relies on the numerical solution of singular
Integral equations with Cauchy-type kernels and provides an efficient and accu-
rate procedure for relating applied loadings to crack opening displacements.
The algorithm should be of Interest to those performing repetitive calculations
1n the analysis of experimental results obtained from fracture specimens.
INTRODUCTION
Two dimensional problems 1n linear elastic fracture mechanics are often
reduced to a singular Integral equation (or system of Integral equations) of
the form
f ft±Mr * f K(x,t) f(t) dt = g(x) -1 < x < 1 (1)J1 T - x J1
where f(t) 1s a function to be determined, and K(x,t) and g(x) are known
functions related to the geometry of the cracked body and the loading on the
crack surface(s), respectively. The function f(t), which 1s called the dis-
location density, 1s the slope of the crack profile, and, 1f the crack 1s
closed at Its tips, satisfies the consistency condition
/] f(t) dt = 0 (2)
In this paper an algorithm 1s presented which can be used to develop a compli-
ance matrix for a cracked solid when such a formulation 1s used. This matrix
relates the stresses at the collocation points arising from the applied loads
(Including those applied to the crack surfaces) to the values of the opening
of the crack at the Integration points. The method relies on the numerical
procedure developed by Gerasoulls (ref. 1), which 1s used to reduce
equation (1) to a system of algebraic equations for unknown values of f(t)
at discrete points 1n the Interval [-1,1].
FORMULATION
The algorithm 1s best explained through a simple example. For the cracked
plate'shown 1n figure 1 the governing equations for the dislocation density are
2u f f(t) dt
 = a i < x < 1 (3)
«(* + 1) I, t - x = a» -1 X ] (3)
/[ f(t) dt .-0 (4)
where y 1s the shear modulus, K = 3 - 4w for plane strain, and v 1s
Polsson's ratio.
The crack opening displacement 1s given by
u* - u" =
 x
]
 f(t) dt (5)
The exact solution to this problem 1s
Following [1], (after nond1mens1ona!1z1ng by setting 2v/ira09(ic + 1) equal
to unity), f(t) 1s expressed by
f(t) = o(t) (1 - t*)-"< (8)
and <|>(t) 1s approximated as plecewlse quadratic 1n [-1,1]. The result 1s
that equations (3) and (4) are reduced to a system of algebraic equations
through quadrature formulas. The details of the quadrature can be found 1n
reference 1 and are omitted here. The results are
2N+1
£ w.(x. ) 4>(t.) =1 k = 1.2N (9)
1=1 1 K n
2N+1
Z v^t^ = 0 (10)
We note that the unknown values <t> are those at the Integration points t^,
while the stresses on the right hand side of equation (9) are at the colloca-
tion points Xfc.
For Illustration we take five points for the quadrature, and equations (9)
and (10) become
-3.236 1.527 0.699 0.683 0.328
-.948 -2.311 1.626 1.190 .443
-.443 -1.190 -1.626 2.311 .948
-.328 -.683 -.699 -1.527 3.236
.571 .858 .283 .858 .571
Jj
S
1
1
1
1
0
(11)
1k "1
In equation (11), matrix G 1s characteristic of the geometry of the prob-
lem and matrix L represents the loading and crack closure condition. What 1s
usually of Interest 1h such a problem 1s the stress Intensity factor, and since
the stress Intensity factor 1s proportional to the value of 4* at the end-
points (ref. 1), equation (11) 1s solved for the unknown vector <t>.
Instead of doing this, 1n this paper the Inverse of matrix G 1s obtained,
and the product of this matrix and (1-t2)"1/2 1s Integrated term by term
to obtain a matrix C which 1s called the compliance matrix for this particu-
lar geometry. The Integration of each term 1s performed using the weights for
the Lagrange Interpolation polynomials, since the function <t> 1s approxi-
mated 1n this manner. The results of the Integration lead to the following
(12)
0.000 0.000 0.000 0.000
.087 .121 .049 .019
.029 .130 .130 .029
.019 .049 .121 .087
.000 .000 .000 .000
C1k x
1
1
1
1 -
"X,3
(u* - u")1
(u* - u-)2
(u* - u-)
<u* - u-)4
(u* - u-)5
ru+ - tr)f
We note that matrix C 1s not a square matrix. This 1s because the number of
Integration points 1s one more than the number of collocation points. Matrix
C^k relates the displacement of the crack faces at the point t^ to the
stress at point X|<.
For the present problem the applied loading vector 1s unity, and premultl-
plylng Vt by the compliance matrix leads to
(u*
(u*
4.(u*
(u*
4-(u*
- u"
- u~
- u~
- u
- u~
h
' 2
*3
>4
J5
= 0
= 0
= 0
= 0
= 0
.0
.276
.318
.276
.0
(13)
The above displacements agree with the exact solution.
The usefulness of the compliance matrix becomes evident when one 1s Inter-
ested 1n Investigating the effects of many loading cases, and 1n particular,
1f the stresses along the crack surfaces depend on the crack opening displace-
ments. Such loadings are used 1n models for fiber reinforced concrete, rocks,
ceramics, and other materials where mlcrocracklng, fiber bridging, and other
nonlinear effects are modeled as nonlinear springs along the crack surfaces
(refs. 2 to 5). For these models equation (1) becomes nonlinear, and an Iter-
ative solution 1s needed. With the use of the compliance matrix, this Iterat-
ion procedure 1s efficient and fast.
As an example, let us assume that the crack surfaces are bridged by fibers,
and that the stresses transmitted by the fibers to the crack depend on the
opening of the crack. The displacements along the crack will be governed by
(u* - u"). = C..x o (applied loads) + a (u+ - u")
t1 1K |_ xk xk (14)
where now the stresses are decomposed Into those arising from the applied
loads, and those due to the fiber bridging. The function o(u+ - u-) 1s
determined from experiments (ref. 6). For the first Iteration the stresses due
to fiber bridging are assumed to be equal to zero. Premult1ply1ng the stresses
arising from the applied loading by the compliance matrix results 1n the first
approximation to the crack opening displacements. From these displacements the
first approximation to the fiber bridging stresses are determined, and these
are applied to the crack surfaces. The procedure 1s repeated until convergence
1s reached. This procedure was used 1n reference 4 (where experiments per-
formed on concrete and fiber reinforced concrete were analyzed) and convergence
was observed to be very fast (only several Iterations were needed for three and
four point bending specimens).
CONCLUSIONS
An algorithm has been presented which can be used to develop compliance
matrices for cracked bodies. The usefulness of the matrices becomes evident
when one Is Interested 1n performing parameter studies to Investigate the
effects of various loadings on crack opening displacements. Even though the
example presented 1n this paper Involved Mode-I loading, the method can be
extended to Include mixed mode problems (Including three-dimensional problems).
REFERENCES
1. Gerasoulls, A.: The Use of P1ecew1se Quadratic Polynomials for the
Solution of Singular Integral Equations of Cauchy Type. Comput. Math.
Appl., vol. 8, no. 1, 1982, pp. 15-22.
2. Keer, L.H.; and Heade, K.P.: A Note on a No-SUp Interface Crack,
J. Appl. Mech., vol. 49, no. 2, June 1982, pp. 454-455.
3. Wecharatana, M.; and Shah, S.P.: A Model for Predicting Fracture
Resistance of Fiber Reinforced Concrete. Cem. Concr. Res., vol. 13,
no. 6, Nov. 1983, pp. 819-829.
4. Ballar1n1, R.; Shah, S.P.; and Keer, L.M.: Crack Growth 1n Cement Based
Composites, Engineering Fracture Mechanics, vol. 20, no. 3, 1984,
pp. 433-445.
5. Catalano, D.M.; and Ingraffea, A.R.: Concrete Fracture: A Linear Elastic
Fracture Mechanics Approach, Report No. 82-1, Dept. of Structural
Engineering, Cornell University, 1982.
6. Shah, S.P.; Stroeven, P.; Dalhulsen, D.; van Stekelenberg, P.: Complete
Stress-Strain Curves for Steel Fiber Reinforced Concrete 1n Un1ax1al
Tension and Compression. Proc. Int. Symp., RILEM-ACI-ASTM, Sheffield,
1978, pp. 339-408.
KJl
-1 -.5
iL
0 +5 +1
-.75 -.25 +.25 +.75
FIGURE 1.- CRACKED PLATE SHOWING INTEGRATION AND COLLOCATION
POINTS FOR FIVE POINT QUADRATURE.
1. Report No.
NASA CR-179478
2. Government Accession No. 3. Recipient's Catalog No.
4. Title and Subtitle
Compliance Matrices for Cracked Bodies
5. Report Date
July 1986
6. Performing Organization Code
7. Authors)
R. Ballar1n1
8. Performing Organization Report No.
None
10. Work Unit No.
9. Performing Organization Name and Address
Cleveland State University
Department of C1v1l Engineering
Cleveland, Ohio 44115
11. Contract or Grant No.
NCC 3-46
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, O.C. 20546
13. Type of Report and Period Covered
Contractor Report
14. Sponsoring Agency Code
506-42-11 (E-3158)
15. Supplementary Notes
Final Report. Project Manager, Anthony M. Calomlno, Structures Division, NASA
Lewis Research Center, Cleveland, Ohio 44135.
16. Abstract
An algorithm 1s presented which can be used to develop compliance matrices for
cracked bodies. The method relies on the numerical solution of singular Integral
equations with Cauchy-type kernels and provides an efficient and accurate proce-
dure for relating applied loadings to crack opening displacements. The algorithm
should be of Interest to those performing repetitive calculations 1n the analysis
of experimental results obtained from fracture specimens.
17. Key Words (Suggested by Authors))
Integral equations
Compliance matrix
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Unclassified - unlimited
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Unclassified
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Unclassified
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