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Biaxial load effects on the crack border elastic strain energy and strain energy rate
BIAXKAL LOAD EFFECTS ON THE CRACK BORDER ELASTIC STRAIN ENERGY AND STRAIN ENERGY RATE J. EFTIS, N. SUBRAMONIANand II. LIEBOWITZ School of Engineeringand Applied Science, The George Washiion University, Washington,DC 20052, U.S.A. AMract-In a previous paper it was shown that the singular expressions for the elastic stress aad displacementsin the borderregionof a line crack are inadequate,genemlly speaking. This is nowheremore clearly demonstratedthan for the case of the infinitesheet with a flat centralcrack, biaxiiy loaded along its outer boundaries.For this particularproblem,the entireeffect of he load appliedparallelto the plane of the crack shows up in the generallydiscardedsecond (non-singular)terms of the series representationsfor the stresses and displacements.Omission of these contributionsis, in effect, equivalent to denying the presenceof the boundaryload appliedparallelto the crackand, consequently,leads to predictionof results at vmiance with expfzimentaldata. In this paper wc continuewith furtherdiscussion of the same problem,focusirg attentionon the fact that the local elastic strainenergydensity and strainenergyrate dependsigniicantiy on the biiiality of the appliedload.
1. INTRODUCTION
plane cracked-body of arbitrary size and shape, with arbitrarily applied loads, the notion that the state of plane elastic stress and displacement very close to the edge of a sharp (line) crack can be adequately and generally specified by the so-called “singular solution”, (i.e. the first term of a series representation), has more or less achieved the status of a truisim in contemporary fracture mechanics. In a previous paperlll, we have shown quite clearly that although the singular characterization of crack front stress and displacement appears on face value to be a sufficiently accurate quantitative approximation, regardless of the nature of the loads actually placed on the outer boundaries of the cracked body, as a general proposition such an approximation is unacceptable. The reasoning which favors use of a one-term characterization for the local crack tip stresses stems from the fact that the first term of the series representation has a one over the square root type of radial distance dependency, whereas the second term, with no radial dependence at all, appears therefore to be quantatively insigniicant relative to the tirst term when the radial distance from the crack tip is required to be very small. Unfortunately, when this point of view is carried over to other quantities of interest, such as in the calculation of the square of the maximum shear stress, the horizontal displacement of the edges of the crack, the’ isostats of maximum shear, the prediction of the angle of initial crack extension, etc., errors of both quantitative and qualitative nature appear. That this is actually so was shown analytically, as well as through comparisons with the results of experiment in the previously referenced paper. The simplest way to demonstrate the general inadequacy of the one-term (singular) representation of the crack tip stress and displacement field is through analysis of the infinite sheet problem with a flat line crack, with biaxial loads applied to its outer boundaries. For this problem the effects of the horizontal load (which is parallel to the plane of the crack) show up entirely in the second term of the series expansion representations for the local stress and displacements. Thus, failure to include the second contribution of the series representations, which is the general practice, in effect denies the physical presence of the load applied parallel to the crack, and misleads one into think& that load biaxiality has no significant bearing on the fracture problem whatsoever. In this paper we continue with further discussion of the same problem. It is shown that by retention of the second contribution to the series approximations for stress and displacement in calculation of the local crack border region elastic strain energy density and elastic strain energy rate, both quantities are revealed to have significant biaxial load dependency. As a matter of general interest, it is also shown that the value of the J-integral does not depend on the presence of the second term of the series expansion for stress and displacement, FOR A
154
J. EFTIS etal.
although this result could have been foreseen if the J-integral is to maintain path independence. Thus for the biaxially loaded sheet with a flat crack, Jr (like KI) is insensitive to the presence of loads applied parallel to the-plane of the crack. It should be borne in mind that the above statement does not imply that Jrc (or &), i.e. the fracture toughness, is necessarily insensitive to the presence of horizontal loads. A simple appeal to the Poisson Ratio effect caused by the presence of a horizontal load parallel to the plane of the crack demonstrates the point to be made. 2. LOCAL, ELASTIC STRAIN ENERGY DENSITY Referring to Fig. 1, when one retains the second term of the Williams eigenfunction expansion for the stress components local to the crack tip, then relative to a rectangular (x - y) coordinate system it can be shown that to O(r”*)[l],
Kz
.e
8
t~~~~rssLll2cos2cosz+~r
39
&I -cos$
I-sinfsin
y]
where the constants KI, K. and A are determined from the traction boundary conditions applied to the body, and 0 < (rfa) 4 1. Applying this point of view to the biaxially loaded infinite sheet with a flat line crack, Fii. 2, that is, when second terms of the series representation of the solution to this problem are not ignored, the local crack tip stress and displacement components correspondingly become[l]
t==-&cos:[l-sinisiny]-(1-lr)lr Kr
.e
8
e
(2.2) 38
(2.3)
PLANE CRACKED ARBITRARY SIZE AND ARBITRARILY LOADED
BODY OF SHAPE WITH BOUNDARIES
kc 0
Fii
1. General fracture
mechanics geometry.
Fii. 2.
Tendon-Compression
Plane biiially
Loading
loaded center-crack geometry.
Biaxial load effects on the
crackborderelasticstrain energy and strainenergyrate
155
where p is the elastic shear modulus, K = 3 -4v for plane strain and (3 - v)/(l + V) for generalized plane stress, v is Poisson’s Ratio, KI = ai and A = - (1 - k)u. These equations hold of course only for 0 <(da) 4 1 in the stresses and 0 5 (da) 4 1 in the displacements. For the plane problem the linear elastic strain energy density per unit volume at any point of the plane body is specified by tfi = t& j, k = 1, 2,
.
(2.4)
To calculate the elastic strain energy density in the immediate vicinity of the crack tip region for the problem of Fig. 2, use is made of eqns (2.2X2.4) together with the relations
au, ax
-=-
au,
au,sin 8
ar
ae
cos 8---
r
au, au,.
au, cos
auy
au,sin 8
-=arslne+-ae r ay au,
--case----ax ar
ae
e
r
au,_au, . au,cos 8 -;sme+-ae
aY
(2.5)
r
*
The result is
4k e) =
$ {-2sin4i+(3-.)sin’:+(r-1) (I-
k)2(K+ l>d gll .
(2.6)
It is seen that t#(r, 0) depends not only on the parameter K (or Poisson’s Ratio v), but also on the biaxial load parameter k, which of course enters as a result of the use of eqns (2) and (3), rather than the usual singular characterization of the stress and displacement components. Calculation of the local elastic strain energy density based on a singular characterization of the local stress and displacement components would produce only in the first line of eqn (2.6), thereby missing entirely the biaxial load effect on 4(r, 0). It is of some interest to examine here the relevance of eqn (2.6), that is, in particular, the effect of the second contributions to the stresses and displacements (or equivalently, for this problem, the effect of the biaxial load) in application of Sib’s hypothesis for the prediction of the direction of initial crack extension: According to this hypothesis applied to the plane crack situation, crack growth will initiate along the radial direction with orientation e, relative to the x-axis along which the local elastic strain energy density attains a minimum. In its analytical form it is required that for some very small fixed, but unspecified, radial distance r. from the crack border
(z)b=O,
($),>O
at
e=e,,
(2.7)
since at the crack border itself d(O, 60) is unbounded. Choosing r. = O.Ola as a reasonable value (which certainly satisfies 0 < (rda) 4 l), the results of plotting (0.08p/u~~(ro, 6) vs 8 using eqn (2.6) for Q(r, 0) for the plane stress case are shown in Figs. 3-8, with biaxial load factor k varying from -2 to +3, and Poisson’s ratio varying from 0.1 to 0.5. The graphs show that for unequal tension-compression loads k I - 1 and 0 5 8 I 90, the
J. EFTIS et al. 3
-I
k--2
y
2
l-
0'
d I
20
-0.1
-0.3 Y IO.6
U
I
I
40
60
I
80
I
90
eFii.3.
Variationof elastic strainenergy density with orientationat k = - 2.
3
k----l
2
8 $
1
0
I
20
I
I
I
40
60
90
e-
Fig.4. Variationof elastic strain energy density with orientationat k = - 1.
Biaxial load effects on the crack border elastic strain energy and strain energy rate
0
I
20
I
I
40 fg!?
I
so
Fii 5. Variation of e&&c strain energy &n&y with orientation at k = 0.
eFii. 6. V~i~on of elastic strain energy density with orientation at k = 1.
151
758 3
0
I
1 So
1
I
40
20
60 8-
Fig. 7. Variation
of elastic strain energy dens@ with
oxitnthn at k = 2.
k-3
I
20
i
I
60
40
1
SO
eFig, 8. Variation of elastic strain energy density with orientation at k = 3.
Biaxial load
effects on the crackbonier elastic strain encriwandstrainenergyrate
759
minimum value of 4(r0,@)occurs at 00= 0” for all values of Poisson’s ratio. However, for biaxial loads for which the biaxial load factor k ranges from k = 0 through k = 2, the minimum value of 4(r0, e), 0 s 8 s 90,canoccur either at e. = 90” or t+,= O“,depending on the value of v. For k r 3, &, e),, occurs at t+,= 90”for all values of Poisson’s Ratio. We note for purposes of comparison that for the uniaxiahy loaded case, k = 0, (for the fiat crack), application of Sib’s hypothesis in which &, 6) is calculated on the basis of using only the singular expression for the local crack tip stress components gives 4(r0, e)h at 00= 0” for all values of Poisson’s Ratio[2],which is at variance with the results shown in Fii. 5 for v = 0.1. While we make no claims either endorsing or rejecting Sib’s minimum elastic strain energy density hypothesis, we cite the results of the above calculations to demonstrate once again the explicit effect of maintaining the second term of the series expansions for the local crack tip stress and displacement components, in this instance on the local elastic strain energy density which, for the problem considered, characterizes directly the effect of biaxialloads. We see that the predictive results of the hypothesis are considerably altered by the inclusion of these second terms. This same conclusion can also be shown from a converse point of view, which leads to interesting results. In calculation of $(r, 0) using Sib’s hypothesis, Sih et al., as mentioned above, use only the singular expressions for the local elastic crack tip stress components. For the biaxially loaded flat crack problem, with the second term of the series expansion missing, the effects of the horizontally applied load parallel to the plane of the flat crack cannot appear. This, in effect, is tantamount to denying analytically the physical presence of the horizontal load. As .a consequence, when applying the hypothesis to the flat crack problem for the prediction of the angle of initial crack extension, for any system of biaxial loads one will always arrive at the result that 4(r,,, 6)ti occurs at t+,= 0” for all values of v. In other words, initial crack extension will always be predicted to be parallel to the plane of the original crack, regardless of the system of loads which are applied to produce Mode I type crack surface displacements. Suppose we calculate the second derivative of 4(r, 0), using eqn (2.6)to represent the local elastic strain energy density, and proceed to evaluate this derivative at the value 6 = 0”. With @,,/a)taken at the values 0.01 and 0.07, respectively, for purposes of illustration, the result of such a calculation will be
8woa24 ,I(3_,)+(1-k)? -a2a [ ae2 1 b_o 2
J-
$
=C+rn(l-k),
(2.8)
where for plane stress 0.07(7+ Sv)
2v >0 m_(K+ll) c_ 1+v ’ 4
2rob( a -
r&a = 0.01 1+v ’ o.l87(7+5v) rola = 0.07. 1+v ’
(2.9)
We note that for all values k < 1, it was shown in Ref. [l] that according to the maximum tensile stress criterion the angle of initial crack extension is & = o”, keeping in mind, however, that Poisson’s ratio v does not enter into the calculation related to this criterion. We also note that experimental data showing the angle of initial crack extension from flat cracks are available,to the best of our knowledge, only for tension-tension type biaxial loading[3,4]. Consider therefore values of k 2 1. The requirement that eqn (2.8) satisfy inequality (2.7)2,which for a given small value of radial distance r = ro, is the sufficient condition that t$(ro,8) be a local minimum, in this case under the a-prioti assumption that it occur at 8 = o”, necessitates that ]l-*kl>i.
(2.10)
For Poisson’s Ratio 0.1s v s 0.5, inequality (2.10)is violated for corresponding values of k as shown in Tables 1 and 2 on next page.
760
I. EFT’IS et al. Table 1. (r&z) = 0.01 V
i;aiues of k for which 19,# 0”
0.1
0.2
k z 1.38
k 2 1.70
0.3
0.4
k=2.oa
kr2.25
0.3
0.4
0.5 kzz2.48
Table 2. @do) = 0.07 V
Values of k for which 8, f 0”
0.1
0.2
krl.14
kzl.27
k z 1.38
kz1.48
0.5 kb 1.56
What this means is that, for example, at @da) = 0.07 and Y= 0.4 (see Table 2), inequality (2.9) will not be satisfied for values of k L 1.48. Hence, 4(r0, 0) will not possess a local minimum at the initially assumed value 8 = 0” and therefore, according to the Sih criterion, the angle of initial crack extension 8, will not be do= 0” under these circumstances, that is, when eqn (2.6) is used to represent the local elastic strain energy density. From the experimental data reported in Refs. [3,4] for tension-tension tests conducted on center-cracked plexiglass sheets with flat cracks (PMMA, v = 0.4) the angle of initial crack extension has already begun to turn from the initial crack plane for k = 1.3. This in qualitative agreement with the value k = 1.5 at v = 0.4 from Table 2, assuming r. = 0.07~. A shortcoming of Sih’s criterion, as for the maximum tensile stress criterion, in predicting the angle of initial crack extension is the requirement that the radial distance variable be treated as a small positive fixed but unspecified quantity ro, which presumably is left for experiment to somehow determine for any given material. Tables 1 and 2 show the effect of different choices of the ratio (r&r). Nevertheless, we repeat once again for emphasis that our discussion here is not meant to be a judgmental statement concerning the adequacy or inadequacy of the different criteria for predicting the angle of initial crack extension. The point to be made is that arbitrary omission of the second term of the series expansion for representation of the local crack-tip stress and displacement fields can lead, in general, to incorrect predictions. This has been shown in this paper, and in Ref. [ 11,through the vehicle of the biaxial load problem from an analytical point of view, with results that are in qualitative agreement with experiment. 3. IAXAL ELASTIC STRAIN ENERGY RATE The elastic strain energy per unit thickness over a circular region centered at the crack tip with radius 0 < r. 4 1 is the integral (see Fig. 9) ‘0
If
cp=
0
=
f$(r,
e)r dr de
(3.1)
--D
in which the integrand is eqn (2.6), with the result
-
16(1 k)(h
- 7)
(3.2)
15rti The rate of change of cf, with crack size is thus E=e
(2K-I)1
16(1-k)&c-7) 15&
G - . J1 a
(3.3).
The first term on the right side of eqn (3.3) is the contribution to the energy rate based on use of only the singular parts of the expressions for local crack-tip stress and displacements. The presence of the second terms of eqns (2.2) and (2.3) naturally give rise to the second contribution to the right side of (3.3), which expresses the influence of the biaxial loading on the
Biaxial load effects on the crack border elastic strain energy and strain energy rate
761
Fii. 9. Local crack tip region.
local elastic strain energy rate. The extent of this influence is shown graphically in Fiis. l&12, for (rola) values ranging from 0.01 to 0.07 and Poisson Ratio values ranging from 0.1 to 0.5, in the case of plane stress. Thus, presence of a horizontal load parallel to the plane of the crack is shown to have significant influence on the elastic strain energy rate locally, that is over a small region enclosing the crack tip, except for equal biaxial tension-tension loading, i.e. k = 1, for which case eqns (2.2) and (2.3) reduce to the usual first-term or singular approximate form. The appearance of the second contribution to eqn (3.3) for all Mode I type loading situations (with the exception of the particular case of equal biaxial tension-tension loading) appears to have additional significance in still another context. It currently seems to be the general consensus in fracture mechanics that a horizontal load applied parallel to the plane of a flat crack (in addition to a vertical load applied normal to the plane of the crack) will have, in general, no effect on the global elastic strain energy rate. An exception to this point of view is the analysis performed by Swedlowl51. However, Swedlow’s work does not seem to have gained any wide degree of acceptance. Be this as it may, the question that the presence of the
Fig. IO. Local elastic strain energy rate variation with load biaxiaiity at Y = 0.1.
J. EFMS etal.
762 5-
k=2 k=l k=O k--l
-
k=-2 k=-3
2f 0.01
b/a -
,
0.07
Fii.ll.Localelastic strain energyratevariation with load biaxiityat ~~0.3.
6-
5-
Fii.12.Localelastic strain energyratevariation withloadbiaxial@at v=O.5.
second contribution to eqn (3.3) poses ,is this: How can the locally elastic strain energy rate be dependent on the biaxial load parameter k, and yet be independent of k when evaluated globally? The addition of more terms to the series representations for local stress and displacements merely adds more terms of varying powers of radial distance to the approximation for local Q(r, e), and thus to the approximation for local a@/aa. Such additions, however, cannot cancel the second contribution to eqn (3.3) because they are terms of different
Biaxii load effectsd onthecrackborderelasticstrainenergy andstrainencrsyrate
163
powers of radial distance. Nor can this contribution vanish, since for a given r. > 0 it can never have the value zero, except for the special case when k = 1. 4 TEE J-INTEGRAL
The question of whether the J-integral value will be affected by retention of the second terms of eqns (2.2)and (2.3)can be examined here by direct calculation. For such a calculation the line integral J=
(4.1) must be evaluated along any contour enclosing the crack tip. Since the contour T is arbitrary it is easiest to choose a circular contour To centered at the crack tip with radius ro (see Fii. 9). With the help of relations (2.5), the line integral (4.1) taken about the circular contour To acquires the form
e---
au, sin 8
ae
(4.2)
r.
For the biaxial problem of Fii. 2, it remains to evaluate this integral making use of eqns (2.2), (2.3) and (2.6) for the terms appearing in the integrand, assuming r. to be sufhciently small. After making these substitutions one obtains -2sin4i+(3-r)sin2;+(w-1)
+(f-k)2a2(~+
l)ro
1%
--
KI’
I
* cos ede
cos24wfi I -2
-,r
8 8 (K- 1) X -cosB+sin2-cos8-4cos2-sin2-cost) 2 2 1 2 (K-1) 8 8 --sin2--sin’-+4cos2-sin’2 2 2 + Ml - &o ah=
8
*
8
8 2
8 de 1
+ 1)ro ~cos~{cosL~~}de
+~~!?~[cos6cos~{~+sin2~[l-4cos2~]}.d6
- (I- k)‘dK + l)ro fJP e
x sin2-
2
2 co$
sin’ 0 + cos 8 f+
xcos’
f - sin’ 8 -
SK
+
1)]d6.
cosp
1 (4.3)
164
J. EFCIS cr al.
The second, third, fifth, sixth and seventh integrals involve the biaxial load parameter k, which arise from the second term contributions to eqns (2.2) and (2.3). The third, hfth and seventh integrals vanish identically, while evaluation of the remaining integrals involving k will show that the second integral exactly cancels the sixth. This leaves only the first, fourth and eighth integrals, which are independent of k, and which are the contributions to JI due to the singular parts of eqns (2.2) and (2.3). Calculation of these remaining integrals will give the result
(4.4) where p is a numerical coefficient having one value for plane stress and one value for plane strain. Thus, the value of the J-integral fs independent of whether or not one retains the second term contributions to eqns (2.2) and (2.3). This result must necessarily be so if J is to maintain its familiar path independent property. Otherwise, from an examination of the integrals containing the biaxial load parameter k, one sees the radial dimension r. appearing which would render J path dependent, were not the sum of these integrals equal to zero. We therefore see that when applied to a biaxially loaded plane body with a flat (horizontal) crack, the J-integral is insensitive to the presence of the horizontal load &a parallel to the crack.
Acknowkdgements-The
authors wish to acknowledge support for this work in part by the NASA-Langley Research CenterthroughgrantNSG-1289,and in partby the Air Force Mice of ScientificResearchthroughgrantAPGSR-764099.
REFERENCES [l] J. Eftis, N. Subramonianand H. Liebowitx,Crackborderstress and displacementequationsrevisited.Engng Fmcturc Mech. 9, 189-210(1977). [2] G. C. Sib, A special theory of crack propagation.Mechanicsof J+ecture(Ed. G. C. Si), Vol. 1. Noordhoff,Leyden (1973). [3] I. J. KiblerandR. Roberts,lhe effect of biaxialstresseson fatigueandfracture.I. EngngIndustry ASME, 727-734, (1970). [4] P.S. Leevers,J.C.RadonandL. E. Culver,Crackgrowthinplasticpanelsunder biaxialstress.Polymer17,627-632,(1976). [S] J. L. Swedlow, Gn GritRth’stheory of fracture.J. Fmcrun Mech. 1,1%5. (Received 7 Bxember
1976)
1. INTRODUCTION
plane cracked-body of arbitrary size and shape, with arbitrarily applied loads, the notion that the state of plane elastic stress and displacement very close to the edge of a sharp (line) crack can be adequately and generally specified by the so-called “singular solution”, (i.e. the first term of a series representation), has more or less achieved the status of a truisim in contemporary fracture mechanics. In a previous paperlll, we have shown quite clearly that although the singular characterization of crack front stress and displacement appears on face value to be a sufficiently accurate quantitative approximation, regardless of the nature of the loads actually placed on the outer boundaries of the cracked body, as a general proposition such an approximation is unacceptable. The reasoning which favors use of a one-term characterization for the local crack tip stresses stems from the fact that the first term of the series representation has a one over the square root type of radial distance dependency, whereas the second term, with no radial dependence at all, appears therefore to be quantatively insigniicant relative to the tirst term when the radial distance from the crack tip is required to be very small. Unfortunately, when this point of view is carried over to other quantities of interest, such as in the calculation of the square of the maximum shear stress, the horizontal displacement of the edges of the crack, the’ isostats of maximum shear, the prediction of the angle of initial crack extension, etc., errors of both quantitative and qualitative nature appear. That this is actually so was shown analytically, as well as through comparisons with the results of experiment in the previously referenced paper. The simplest way to demonstrate the general inadequacy of the one-term (singular) representation of the crack tip stress and displacement field is through analysis of the infinite sheet problem with a flat line crack, with biaxial loads applied to its outer boundaries. For this problem the effects of the horizontal load (which is parallel to the plane of the crack) show up entirely in the second term of the series expansion representations for the local stress and displacements. Thus, failure to include the second contribution of the series representations, which is the general practice, in effect denies the physical presence of the load applied parallel to the crack, and misleads one into think& that load biaxiality has no significant bearing on the fracture problem whatsoever. In this paper we continue with further discussion of the same problem. It is shown that by retention of the second contribution to the series approximations for stress and displacement in calculation of the local crack border region elastic strain energy density and elastic strain energy rate, both quantities are revealed to have significant biaxial load dependency. As a matter of general interest, it is also shown that the value of the J-integral does not depend on the presence of the second term of the series expansion for stress and displacement, FOR A
154
J. EFTIS etal.
although this result could have been foreseen if the J-integral is to maintain path independence. Thus for the biaxially loaded sheet with a flat crack, Jr (like KI) is insensitive to the presence of loads applied parallel to the-plane of the crack. It should be borne in mind that the above statement does not imply that Jrc (or &), i.e. the fracture toughness, is necessarily insensitive to the presence of horizontal loads. A simple appeal to the Poisson Ratio effect caused by the presence of a horizontal load parallel to the plane of the crack demonstrates the point to be made. 2. LOCAL, ELASTIC STRAIN ENERGY DENSITY Referring to Fig. 1, when one retains the second term of the Williams eigenfunction expansion for the stress components local to the crack tip, then relative to a rectangular (x - y) coordinate system it can be shown that to O(r”*)[l],
Kz
.e
8
t~~~~rssLll2cos2cosz+~r
39
&I -cos$
I-sinfsin
y]
where the constants KI, K. and A are determined from the traction boundary conditions applied to the body, and 0 < (rfa) 4 1. Applying this point of view to the biaxially loaded infinite sheet with a flat line crack, Fii. 2, that is, when second terms of the series representation of the solution to this problem are not ignored, the local crack tip stress and displacement components correspondingly become[l]
t==-&cos:[l-sinisiny]-(1-lr)lr Kr
.e
8
e
(2.2) 38
(2.3)
PLANE CRACKED ARBITRARY SIZE AND ARBITRARILY LOADED
BODY OF SHAPE WITH BOUNDARIES
kc 0
Fii
1. General fracture
mechanics geometry.
Fii. 2.
Tendon-Compression
Plane biiially
Loading
loaded center-crack geometry.
Biaxial load effects on the
crackborderelasticstrain energy and strainenergyrate
155
where p is the elastic shear modulus, K = 3 -4v for plane strain and (3 - v)/(l + V) for generalized plane stress, v is Poisson’s Ratio, KI = ai and A = - (1 - k)u. These equations hold of course only for 0 <(da) 4 1 in the stresses and 0 5 (da) 4 1 in the displacements. For the plane problem the linear elastic strain energy density per unit volume at any point of the plane body is specified by tfi = t& j, k = 1, 2,
.
(2.4)
To calculate the elastic strain energy density in the immediate vicinity of the crack tip region for the problem of Fig. 2, use is made of eqns (2.2X2.4) together with the relations
au, ax
-=-
au,
au,sin 8
ar
ae
cos 8---
r
au, au,.
au, cos
auy
au,sin 8
-=arslne+-ae r ay au,
--case----ax ar
ae
e
r
au,_au, . au,cos 8 -;sme+-ae
aY
(2.5)
r
*
The result is
4k e) =
$ {-2sin4i+(3-.)sin’:+(r-1) (I-
k)2(K+ l>d gll .
(2.6)
It is seen that t#(r, 0) depends not only on the parameter K (or Poisson’s Ratio v), but also on the biaxial load parameter k, which of course enters as a result of the use of eqns (2) and (3), rather than the usual singular characterization of the stress and displacement components. Calculation of the local elastic strain energy density based on a singular characterization of the local stress and displacement components would produce only in the first line of eqn (2.6), thereby missing entirely the biaxial load effect on 4(r, 0). It is of some interest to examine here the relevance of eqn (2.6), that is, in particular, the effect of the second contributions to the stresses and displacements (or equivalently, for this problem, the effect of the biaxial load) in application of Sib’s hypothesis for the prediction of the direction of initial crack extension: According to this hypothesis applied to the plane crack situation, crack growth will initiate along the radial direction with orientation e, relative to the x-axis along which the local elastic strain energy density attains a minimum. In its analytical form it is required that for some very small fixed, but unspecified, radial distance r. from the crack border
(z)b=O,
($),>O
at
e=e,,
(2.7)
since at the crack border itself d(O, 60) is unbounded. Choosing r. = O.Ola as a reasonable value (which certainly satisfies 0 < (rda) 4 l), the results of plotting (0.08p/u~~(ro, 6) vs 8 using eqn (2.6) for Q(r, 0) for the plane stress case are shown in Figs. 3-8, with biaxial load factor k varying from -2 to +3, and Poisson’s ratio varying from 0.1 to 0.5. The graphs show that for unequal tension-compression loads k I - 1 and 0 5 8 I 90, the
J. EFTIS et al. 3
-I
k--2
y
2
l-
0'
d I
20
-0.1
-0.3 Y IO.6
U
I
I
40
60
I
80
I
90
eFii.3.
Variationof elastic strainenergy density with orientationat k = - 2.
3
k----l
2
8 $
1
0
I
20
I
I
I
40
60
90
e-
Fig.4. Variationof elastic strain energy density with orientationat k = - 1.
Biaxial load effects on the crack border elastic strain energy and strain energy rate
0
I
20
I
I
40 fg!?
I
so
Fii 5. Variation of e&&c strain energy &n&y with orientation at k = 0.
eFii. 6. V~i~on of elastic strain energy density with orientation at k = 1.
151
758 3
0
I
1 So
1
I
40
20
60 8-
Fig. 7. Variation
of elastic strain energy dens@ with
oxitnthn at k = 2.
k-3
I
20
i
I
60
40
1
SO
eFig, 8. Variation of elastic strain energy density with orientation at k = 3.
Biaxial load
effects on the crackbonier elastic strain encriwandstrainenergyrate
759
minimum value of 4(r0,@)occurs at 00= 0” for all values of Poisson’s ratio. However, for biaxial loads for which the biaxial load factor k ranges from k = 0 through k = 2, the minimum value of 4(r0, e), 0 s 8 s 90,canoccur either at e. = 90” or t+,= O“,depending on the value of v. For k r 3, &, e),, occurs at t+,= 90”for all values of Poisson’s Ratio. We note for purposes of comparison that for the uniaxiahy loaded case, k = 0, (for the fiat crack), application of Sib’s hypothesis in which &, 6) is calculated on the basis of using only the singular expression for the local crack tip stress components gives 4(r0, e)h at 00= 0” for all values of Poisson’s Ratio[2],which is at variance with the results shown in Fii. 5 for v = 0.1. While we make no claims either endorsing or rejecting Sib’s minimum elastic strain energy density hypothesis, we cite the results of the above calculations to demonstrate once again the explicit effect of maintaining the second term of the series expansions for the local crack tip stress and displacement components, in this instance on the local elastic strain energy density which, for the problem considered, characterizes directly the effect of biaxialloads. We see that the predictive results of the hypothesis are considerably altered by the inclusion of these second terms. This same conclusion can also be shown from a converse point of view, which leads to interesting results. In calculation of $(r, 0) using Sib’s hypothesis, Sih et al., as mentioned above, use only the singular expressions for the local elastic crack tip stress components. For the biaxially loaded flat crack problem, with the second term of the series expansion missing, the effects of the horizontally applied load parallel to the plane of the flat crack cannot appear. This, in effect, is tantamount to denying analytically the physical presence of the horizontal load. As .a consequence, when applying the hypothesis to the flat crack problem for the prediction of the angle of initial crack extension, for any system of biaxial loads one will always arrive at the result that 4(r,,, 6)ti occurs at t+,= 0” for all values of v. In other words, initial crack extension will always be predicted to be parallel to the plane of the original crack, regardless of the system of loads which are applied to produce Mode I type crack surface displacements. Suppose we calculate the second derivative of 4(r, 0), using eqn (2.6)to represent the local elastic strain energy density, and proceed to evaluate this derivative at the value 6 = 0”. With @,,/a)taken at the values 0.01 and 0.07, respectively, for purposes of illustration, the result of such a calculation will be
8woa24 ,I(3_,)+(1-k)? -a2a [ ae2 1 b_o 2
J-
$
=C+rn(l-k),
(2.8)
where for plane stress 0.07(7+ Sv)
2v >0 m_(K+ll) c_ 1+v ’ 4
2rob( a -
r&a = 0.01 1+v ’ o.l87(7+5v) rola = 0.07. 1+v ’
(2.9)
We note that for all values k < 1, it was shown in Ref. [l] that according to the maximum tensile stress criterion the angle of initial crack extension is & = o”, keeping in mind, however, that Poisson’s ratio v does not enter into the calculation related to this criterion. We also note that experimental data showing the angle of initial crack extension from flat cracks are available,to the best of our knowledge, only for tension-tension type biaxial loading[3,4]. Consider therefore values of k 2 1. The requirement that eqn (2.8) satisfy inequality (2.7)2,which for a given small value of radial distance r = ro, is the sufficient condition that t$(ro,8) be a local minimum, in this case under the a-prioti assumption that it occur at 8 = o”, necessitates that ]l-*kl>i.
(2.10)
For Poisson’s Ratio 0.1s v s 0.5, inequality (2.10)is violated for corresponding values of k as shown in Tables 1 and 2 on next page.
760
I. EFT’IS et al. Table 1. (r&z) = 0.01 V
i;aiues of k for which 19,# 0”
0.1
0.2
k z 1.38
k 2 1.70
0.3
0.4
k=2.oa
kr2.25
0.3
0.4
0.5 kzz2.48
Table 2. @do) = 0.07 V
Values of k for which 8, f 0”
0.1
0.2
krl.14
kzl.27
k z 1.38
kz1.48
0.5 kb 1.56
What this means is that, for example, at @da) = 0.07 and Y= 0.4 (see Table 2), inequality (2.9) will not be satisfied for values of k L 1.48. Hence, 4(r0, 0) will not possess a local minimum at the initially assumed value 8 = 0” and therefore, according to the Sih criterion, the angle of initial crack extension 8, will not be do= 0” under these circumstances, that is, when eqn (2.6) is used to represent the local elastic strain energy density. From the experimental data reported in Refs. [3,4] for tension-tension tests conducted on center-cracked plexiglass sheets with flat cracks (PMMA, v = 0.4) the angle of initial crack extension has already begun to turn from the initial crack plane for k = 1.3. This in qualitative agreement with the value k = 1.5 at v = 0.4 from Table 2, assuming r. = 0.07~. A shortcoming of Sih’s criterion, as for the maximum tensile stress criterion, in predicting the angle of initial crack extension is the requirement that the radial distance variable be treated as a small positive fixed but unspecified quantity ro, which presumably is left for experiment to somehow determine for any given material. Tables 1 and 2 show the effect of different choices of the ratio (r&r). Nevertheless, we repeat once again for emphasis that our discussion here is not meant to be a judgmental statement concerning the adequacy or inadequacy of the different criteria for predicting the angle of initial crack extension. The point to be made is that arbitrary omission of the second term of the series expansion for representation of the local crack-tip stress and displacement fields can lead, in general, to incorrect predictions. This has been shown in this paper, and in Ref. [ 11,through the vehicle of the biaxial load problem from an analytical point of view, with results that are in qualitative agreement with experiment. 3. IAXAL ELASTIC STRAIN ENERGY RATE The elastic strain energy per unit thickness over a circular region centered at the crack tip with radius 0 < r. 4 1 is the integral (see Fig. 9) ‘0
If
cp=
0
=
f$(r,
e)r dr de
(3.1)
--D
in which the integrand is eqn (2.6), with the result
-
16(1 k)(h
- 7)
(3.2)
15rti The rate of change of cf, with crack size is thus E=e
(2K-I)1
16(1-k)&c-7) 15&
G - . J1 a
(3.3).
The first term on the right side of eqn (3.3) is the contribution to the energy rate based on use of only the singular parts of the expressions for local crack-tip stress and displacements. The presence of the second terms of eqns (2.2) and (2.3) naturally give rise to the second contribution to the right side of (3.3), which expresses the influence of the biaxial loading on the
Biaxial load effects on the crack border elastic strain energy and strain energy rate
761
Fii. 9. Local crack tip region.
local elastic strain energy rate. The extent of this influence is shown graphically in Fiis. l&12, for (rola) values ranging from 0.01 to 0.07 and Poisson Ratio values ranging from 0.1 to 0.5, in the case of plane stress. Thus, presence of a horizontal load parallel to the plane of the crack is shown to have significant influence on the elastic strain energy rate locally, that is over a small region enclosing the crack tip, except for equal biaxial tension-tension loading, i.e. k = 1, for which case eqns (2.2) and (2.3) reduce to the usual first-term or singular approximate form. The appearance of the second contribution to eqn (3.3) for all Mode I type loading situations (with the exception of the particular case of equal biaxial tension-tension loading) appears to have additional significance in still another context. It currently seems to be the general consensus in fracture mechanics that a horizontal load applied parallel to the plane of a flat crack (in addition to a vertical load applied normal to the plane of the crack) will have, in general, no effect on the global elastic strain energy rate. An exception to this point of view is the analysis performed by Swedlowl51. However, Swedlow’s work does not seem to have gained any wide degree of acceptance. Be this as it may, the question that the presence of the
Fig. IO. Local elastic strain energy rate variation with load biaxiaiity at Y = 0.1.
J. EFMS etal.
762 5-
k=2 k=l k=O k--l
-
k=-2 k=-3
2f 0.01
b/a -
,
0.07
Fii.ll.Localelastic strain energyratevariation with load biaxiityat ~~0.3.
6-
5-
Fii.12.Localelastic strain energyratevariation withloadbiaxial@at v=O.5.
second contribution to eqn (3.3) poses ,is this: How can the locally elastic strain energy rate be dependent on the biaxial load parameter k, and yet be independent of k when evaluated globally? The addition of more terms to the series representations for local stress and displacements merely adds more terms of varying powers of radial distance to the approximation for local Q(r, e), and thus to the approximation for local a@/aa. Such additions, however, cannot cancel the second contribution to eqn (3.3) because they are terms of different
Biaxii load effectsd onthecrackborderelasticstrainenergy andstrainencrsyrate
163
powers of radial distance. Nor can this contribution vanish, since for a given r. > 0 it can never have the value zero, except for the special case when k = 1. 4 TEE J-INTEGRAL
The question of whether the J-integral value will be affected by retention of the second terms of eqns (2.2)and (2.3)can be examined here by direct calculation. For such a calculation the line integral J=
(4.1) must be evaluated along any contour enclosing the crack tip. Since the contour T is arbitrary it is easiest to choose a circular contour To centered at the crack tip with radius ro (see Fii. 9). With the help of relations (2.5), the line integral (4.1) taken about the circular contour To acquires the form
e---
au, sin 8
ae
(4.2)
r.
For the biaxial problem of Fii. 2, it remains to evaluate this integral making use of eqns (2.2), (2.3) and (2.6) for the terms appearing in the integrand, assuming r. to be sufhciently small. After making these substitutions one obtains -2sin4i+(3-r)sin2;+(w-1)
+(f-k)2a2(~+
l)ro
1%
--
KI’
I
* cos ede
cos24wfi I -2
-,r
8 8 (K- 1) X -cosB+sin2-cos8-4cos2-sin2-cost) 2 2 1 2 (K-1) 8 8 --sin2--sin’-+4cos2-sin’2 2 2 + Ml - &o ah=
8
*
8
8 2
8 de 1
+ 1)ro ~cos~{cosL~~}de
+~~!?~[cos6cos~{~+sin2~[l-4cos2~]}.d6
- (I- k)‘dK + l)ro fJP e
x sin2-
2
2 co$
sin’ 0 + cos 8 f+
xcos’
f - sin’ 8 -
SK
+
1)]d6.
cosp
1 (4.3)
164
J. EFCIS cr al.
The second, third, fifth, sixth and seventh integrals involve the biaxial load parameter k, which arise from the second term contributions to eqns (2.2) and (2.3). The third, hfth and seventh integrals vanish identically, while evaluation of the remaining integrals involving k will show that the second integral exactly cancels the sixth. This leaves only the first, fourth and eighth integrals, which are independent of k, and which are the contributions to JI due to the singular parts of eqns (2.2) and (2.3). Calculation of these remaining integrals will give the result
(4.4) where p is a numerical coefficient having one value for plane stress and one value for plane strain. Thus, the value of the J-integral fs independent of whether or not one retains the second term contributions to eqns (2.2) and (2.3). This result must necessarily be so if J is to maintain its familiar path independent property. Otherwise, from an examination of the integrals containing the biaxial load parameter k, one sees the radial dimension r. appearing which would render J path dependent, were not the sum of these integrals equal to zero. We therefore see that when applied to a biaxially loaded plane body with a flat (horizontal) crack, the J-integral is insensitive to the presence of the horizontal load &a parallel to the crack.
Acknowkdgements-The
authors wish to acknowledge support for this work in part by the NASA-Langley Research CenterthroughgrantNSG-1289,and in partby the Air Force Mice of ScientificResearchthroughgrantAPGSR-764099.
REFERENCES [l] J. Eftis, N. Subramonianand H. Liebowitx,Crackborderstress and displacementequationsrevisited.Engng Fmcturc Mech. 9, 189-210(1977). [2] G. C. Sib, A special theory of crack propagation.Mechanicsof J+ecture(Ed. G. C. Si), Vol. 1. Noordhoff,Leyden (1973). [3] I. J. KiblerandR. Roberts,lhe effect of biaxialstresseson fatigueandfracture.I. EngngIndustry ASME, 727-734, (1970). [4] P.S. Leevers,J.C.RadonandL. E. Culver,Crackgrowthinplasticpanelsunder biaxialstress.Polymer17,627-632,(1976). [S] J. L. Swedlow, Gn GritRth’stheory of fracture.J. Fmcrun Mech. 1,1%5. (Received 7 Bxember
1976)