The near-crack-tip stresses in any planar coupon of arbitrary geometry subjected to mode 1 loading may be equated to those in an infinite center-cracked panel subjected to the appropriate equivalent remote biaxial stresses (ERBS). Since this process can be done for all such mode 1 coupons, attention may be focused on the behavior of the equivalent infinite cracked panel. To calculate the ERBS, the constant term in the series expansion of the crack-tip stress must be retained. It is proposed that the ERBS may be used quantitatively to explain different fracture phenomena such as crack branching

Goree, James G.

Richardson, David E.

English

NASA Technical Reports Server

/Micro-Mechanical Analysis of Damage
Growth and Fracture in Discontinuous
Fiber Reinforced Metal Matrix Composites
NASA-Grant NAG-I-971
Semi-Annual Report
by
James G. Goree
Centennial Professor
Department of Mechanical Engineering
Clemson University
Clemson, South Carolina 29634
and
David E. Richardson
Du Pont Fellow in Composite Materials
(Ph.D. Student in Engineering Mechanics)
Department of Mechanical Engineering
Clemson University
Clemson, South Carolina 29634
(NASA-CR- | L_7020) MICRO-MECHANICAL ANALYSIS N91-I0120
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ABSTRACT
The near-crack-tip stresses in any planar coupon of arbitrary
geometry subjected to mode I loading may be equated to those in an
infinite center-cracked panel subjected to the appropriate
equivalent remote biaxial stresses _ERBS). Since this process can
be done for all such mode I coupons, researchers may focus their
attention on the behavior of the equivalent infinite cracked panel.
To calculate the ERBS, the constant term in the series expansion of
the crack-tip stress must be retained. It is proposed that the
ERBS may be used quantitatively to explain different fracture
phenomena such as crack branching.
Equivalent Remote Biaxial Stresses _ERBS)
for Planar Cracked Body Analyses
Introduction
In general, the fracture behavior of arbitrary cracked bodies
subjected to various loadings is dependent on the near-crack-tip
stress state. Exact solutions, however, can be very complicated,
and in many cases almost impossible to obtain, even though the
general form of the stress state is known. For this reason
numerical methods (finite element, boundary element, etc.) have
been used to determine local stress states. One of the few exact
solutions available is that of an infinitely large center-cracked
panel. Authors such as Inglis [i], Griffith [2], Muskhelishvili
[3], Lekhnitskii [4], Savin [5], and others have considered this
problem for both isotropic and orthotropic materials. In this
paper we will show how the solution for the infinite cracked plate
may be used in analyzing the stress states in arbitrary cracked
coupons subjected to mode I loading.
To start, consider the near-crack-tip stress state. Using a
form similar to that of Savin [5], it can be shown that the series
expansions of the stresses around the crack tip in an arbitrary
anisotropic body subjected to mode I loading (see Figure la) are:
a fz ae( SlS2 [ s2- sl ]}+Az+O(V7), (i)
x (%-%) V
o- fz ae{ i [ Sl _S_xub_2_2 ]}+O(V7), and (2 )
f_ R_ SlS2 [ I i ]}+O(VT) (3}
v_ (%-%) _
where K I and A I are constants. In an infinite center-cracked panel
of crack length c" (see Figure ib),
Kz-o_ , and Az-O'yRe{SiS2}+U'x,
giving:
. - _-&" _ , SlS2 S2 Sl
a oy_xe_ (s_-s=) [ ] }+o'_e{s_s_}+o'=+o(V-E),
(4)
oy-O'v_c" Re{ I [ st_ s2 ]}+o(v?),
(s_-s_) #-CT_vrt?
and (5)
" Vrx--C"Re{ $IS2 1 1 ])+O(v/}-) , (6)
where CI and C2 are defined by,
Ci-cos(B)+Sisin(8),
and S I and S2 are the appropriate complex roots of the polynomial,
AIIS4_2(AI2+_6S2)_2A26S+A22_O. (7)
The coefficients Aij are the classical anisotropic elastic
compliances.
F,?. /
P
(a) (b)
G"x
Equivalent Remote Biaxial Stresse_
The near-crack-tip stresses (retaining only the singular and
constant terms) of any arbitrary coupon subjected to mode I loading
can be equated to those in the infinite biaxially loaded center-
cracked panel (see Figure I). This is done by equating the
transverse stresses of equations (i) and (4), and the longitudinal
stresses series of equations (2) and (5) and determining the
Equivalent Remote Biaxial Stresses (ERBS i.e. ux®, a;) which, when
applied to the infinite cracked panel, make the near-crack-tip
stresses equal to those in the arbitrary body. Because the stress
distributions in both coupons have the same r and 8 dependence,
ERBS are independent of r and 8. To be specific the infinite
cracked plate can be assumed to have a fixed crack length
(c*=constant). The ERBS are then as follows:
u'x- -Kz Re{SIS2}+Ax,
_/_c"
and (8)
From these equations, the ERBS ratio (B) is defined as:
For an orthotropic material, equation (7) reduces to:
A11Sa+ (2A12+2_6) $2+A22-0,
(9)
(10)
(11)
where,
All" _ ' _2"_ A12- v12 and _6- 1
' E11
In order to find S I and S 2 a simplification may be made,
G12"-_1+%÷2v1_%
(12)
(An approximation suggested in [6]). Using this relation along
with equations (12), equation (ii) reduces to:
S4+ (I÷E_)S2+EI-0.m_
The roots of this equation are:
S-±i, and ±i._.
_ E2
Of these roots, only the two positive roots are the appropriate
roots to be used in equations (8-10), giving:
c---7 _ +Az"
(13)
a_- Kz , and (14)
i (is)
To calculate the ERBS it is necessary to determine the near-
crack-tip stresses in the arbitrary coupon (either analytically or
numerically) to find K i and AI, and to choose the fixed crack length
c*. Since these expressions (8-10, and 13-15) for calculating ERBS
are independent of r and 8, the stress state at any point near the
crack-tip in the arbitrary coupon is sufficient to be able to
calculate the constants K I and A I. In most cases ERBS will be used
to analyze the near-crack-tip stresses in the infinite panel,
therefore c* must be chosen to be much larger than the radius at
which the analysis will be made (c*>>r). This restriction will
ensure that the first two terms in the series expansion of the
stresses around the crack-tip in the infinite panel dominate the
stress field. Possible choices for c would be 1.0 in., or 25 mm..
Some important observations can be noted from the expressions
(8-10, or 13-15). First, if the near-crack-tip stress state is the
same, then it can be expected that the materials response to
loading would be similar in both the arbitrary coupon and the
infinite panel. For example, this implies that both configurations
would respond in a similar manner under fatigue loading. Under
quasi-static loading both coupons would have the same fracture
characteristics. Thus, if crack branching occurs either under
fatigue or during fracture in the infinite panel, one could expect
branching to occur in the arbitrary coupon and visa versa. Because
of this expected loading response similarity, various near-crack-
tip phenomena (such as crack branching) may be modeled
mathematically by studying the biaxially loaded infinite center-
cracked panel (with the fixed crack length of 2c'), and then
relating coupons of arbitrary geometries to the infinite solution.
We are currently studying the use of ERBS in predicting fracture in
advanced materials (such as composites) and in non-plane strain
coupons [7-8]. Inconsistencies in both values for fracture
toughness and crack growth direction (during both fracture and
fatigue) have been noted in these materials.
It is important to note that the constant term in the series
expansions of the transverse stress is not neglected in the
formulation, and the elements of expressions (8-10, and 13-15)
which come from the constant term are not insignificant when
compared with the other terms in the equations. The constant term
has been shown by many researchers [9-12] to be significant when
considering fracture behavior of materials.
Another interesting feature of the model is that the ERBS show
considerable sensitivity to changes in geometry and crack length.
This can be seen in Figures 2 and 3, which are numerical results
for an edge-notched coupon (geometry seen in Figure 4 compared with
that of a standard compact tension coupon) of a material with
El/E2=0.8. Figures 2 depicts the change in the ERBS ratio, B, with
change in the length of the edge-notched coupon (c=19.05mm).
Figure 3 gives the change in B with crack length in an edge-notched
coupon (L=30.5mm). The numerical work was done using a finite
element alternating method developed by Raju and Fichter [13].
Reedy [15] noted that branching occurred in compact tension
(CT) coupons, but did not occur in center-cracked (CCT) panels of
continuous boron fiber reinforced aluminum. Qualitatively, this
can be explained by considering the pattern depicted in figure 3.
For a material with El/E2=0.8, the ERBS ratio (B) in a CCT coupon
would be nearly zero and in a CT coupon B would be close to two.
This would lead one to expect a higher tendency for crack branching
in a CT coupon. The effect of the biaxial stress ratio on crack
branching in infinite center-cracked panels subjected is discussed
in [16-17]. The occurrence of splitting cannot be explained using
the stress intensity factor approach of LEFM. As indicated
earlier, the authors are currently investigating the use of ERBS to
quanitativly explain this branching behavior.
The purpose of this paper has been to present in detail the
mathematical derivation of ERBS, and to discuss qualitatively the
uses of ERBS. In future papers we will present the results of a
detailed experimental program to investigate the use of ERBS in
predicting the fracture behavior of cracked bodies of various
geometries.
m1.9
1.7
1.5
1.3
1.1
25
! Length of the Standard
! ! I
35 45 55
L (mm)
!
65
9
P2.2
2.0
1.8
1.6
1.4
8
_\ Typical Region For
i I I " I I "
10 12 14 16 18 20
c (ram)
a--
lO
L=30.5 mm
31.8 mm
I 25.4 mm
d --
1- '
J._8.3 mm
_J6.4 mm DIA.
Side View
12.7 mm
L=30.5 mm
L
31.8 mm
I-'- 25.4 mm --
!._6.48.3 mm
mm DIA.
I
C --!
= : a__._I
I
I
I
I
I
!
2
--1
(a) (b)
J
j-
ll
CONCLUSIONS
lo
•
The near-crack-tip stress state in an arbitrary planar coupon
subjected to mode I loading may be equated to those in an
infinite cracked panel of the same material subjected to the
appropriate equivalent remote biaxial stresses (ERBS).
We suggest that researchers focus their mathematical
analyses on biaxially loaded infinite cracked panels and
relate arbitrary coupon geometries to such studies through
the use of ERBS.
•
ERBS may be used to qualitatively explain crack branching
seen by Reedy [15].
•
The constant term in the series expansion of the near-crack-
tip stresses is neither neglected, nor insignificant in the
development of ERBS.
12
••
•
•
.
.
•
•
•
i0.
Ii.
12.
13.
14.
15.
16.
17.
REFERENCES
C. E. Inglis, Transactions of the Institution of Naval
Architects 55 (1973) 219-230•
A. A. Griffith, Philosophical Transactions of The Royal
Society, A221 (1920) 163-197.
N. Muskhelishvili, Some Basic Problems of the Mathematical
Theory of Elasticity, English Translation, Noordhoff (1953).
S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic
Bo__, Mir Publishers, Moscow (1981)•
G. N. Savin, Stress Concentrations Around Holes, Pergamon
Press, London (1961).
V. E. Saouma, M. L. Ayari, and D. A. Leavell, Enqineerinq
Fracture Mechanics, 27 (1987) 171-183•
J. G. Goree, Nasa Grant NAG-I-971, Nasa Langley Research
Center (1990).
J. G. Goree, E. I. Du Pont De Nemours and Company, Graduate
Student Fellowship and Research Contract, Pioneering
Research Laboratory, Wilmington, Delaware (1990)•
A. Piva and E. Viola, Enqineerinq Fracture Mechanics, 13
(1980) 143-174.
J. Eftis, N. Subramonian, and H. Liebowitz, Enqineering
Fracture Mechanics, 9 (1977) 189-210.
H. Liebowitz, J. D. Lee, and J. Eftis, Enqineerinq Fracture
Mechanics, i0 (1978) 315-335.
B. Cotterell, International Journal of Fracture Mechanics,
13 (1980) 189-192.
I. S. Raju, and W. B. Fichter, Engineerinq Fracture
Mechanics, 33, No. 4 (1989) 525-540.
S. Mostovoy, P. B. Crosley, and E. J. Ripling, Journal of
Materials, 2, No. 3 (1967) 661-681•
E. D. Reedy, Journal of Composite Materials Supplement, 14
(1980) 118-131.
J. J. Kibler, and R. Roberts, Journal of Enqineerinq for
Industry, (1970) 727-734•
J. C. Radon, P. S. Leevers, and L. E. Culver, L. E.,
Fracture, 3 (1977) 1113-1118.
13
List of Captions
Figure i: Comparison of: a) an edge-notched coupon of arbitrary
geometry with that of b) an infinite center-cracked
panel.
Figure 2: B as a function of coupon length (L) for an edge-
notched coupon with crack length c=19.05mm.
Figure 3: B as a function of crack length (c) for an edge-
notched coupon with coupon length L=30.5mm.
Figure 4: Comparison of: a) the standard compact tension coupon
geometry with that of b) the edge-notched coupon used
in the numerical analysis.
14


Micro-mechanical analysis of damage growth and fracture in discontinuous fiber reinforced metal matrix composites

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