size effect tests and fracture characteristics of aluminum - Civil ...
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Engineaing Fracture Mechanics Vol. 26, No. I. pp. 45-57.1987 Printed in Great BrItain
SIZE SIZE EFFECT EFFECT TESTS TESTS AND AND FRACTURE FRACTURE CHARACTERISTICS OF CHARACTERISTICS OF ALUMINUM ALUMINUM ZDENEK P. BAZANT,t SOO-GON LEEt A. ZDENl?K BAgANT,? SOO-GON LEES and and PHILLIP PHILLlP A. PFEIFFER§ PFEIFFERP Center Northwestern University, Center for for Concrete Concrete and and Geomaterials, Geomaterials, Northwestern University, Evanston, Evanston, IL IL 60201, 60201, U.S.A. U.S.A. Abstract-The recently size Abstract-The recently established established approximate approximate size effect effect law law for for blunt blunt fracture fracture is shown shown to to agree agree with with three-point bent fracture with test test data data for for aluminum. aluminum. The The data data are are obtained obtained with three-point bent fracture specimens specimens of of depths Non-linear fracture parameters can be obtained by a linear depths ranging ranging from from 0.25 0.25 to to 4 in. Non-linear fracture parameters can be obtained by linear regression based based on be detennined using only regression on the the size size effect effect law. law. Thus, Thus, the the fracture fracture energy energy can can be determined using only the the maximum values from be obtained maximum load load values from fracture fracture tests, tests, and and the the R-curve R-curve can can be obtained as as an an envelope envelope of of fracture fracture equilibrium No measurements of equilibrium curves. curves. No measurements of crack crack length length or or specimen specimen compliance compliance are are needed. needed.
INTRODUCTION INTRODUCTION FRACTURE generally FRACTURE of of aluminum aluminum generally does does not not follow follow linear linear elastic elastic fracture fracture mechanics. mechanics. This This means means of the non-linearity that a non-linear fracture model must be used. One conspicuous consequence that non-linear fracture model must be used. One conspicuous consequence of the non-linearity of properties is the of fracture fracture properties the apparent apparent variation variation of of the the energy energy required required for for crack crack growth, growth, R, with with the the length of the crack. Irwin[l] and Krafft et al.[2] proposed that the dependence of R on the length of the crack. Irwin[l] and Krafft al.[2] proposed that the dependence of on the crack crack length, be considered be approximately the length, c, may may be considered to to be approximately the same same for for various various fracture fracture geometries. geometries. This This means use for a unique unique curve, means that that one one can can use for fracture fracture calculations calculations curve, R(c), R(c), called called the the R-curve R-curve (resistance (resistance curve), be a material property. curve), which which is considered considered to to be material property. The the by an The existing existing methods methods for for determining determining the R-curve, R-curve, specified specified by an ASTM ASTM standard standard [3-6], [3-61, rely rely 2 2 2 Ip a/(Ecb d ) in on on the the relation relation R = kk,P2a/(Ecb2d2) m which which ee,.c = = Young's Young’s elastic elastic modulus, modulus, b = specimen specimen thickness, thickness, d = characteristic dimension plus notch, characteristic dimension of of the the specimen, specimen, a = = length length of of the the crack crack plus notch, P = load load at at which which the the crack crack extends extends and and k k,I = a coefficient coefficient determined determined for for the the given given specimen specimen geometry geometry according according to to linear linear elastic elastic fracture fracture mechanics. mechanics. A series series of of R-values R-values is measured, measured, either either on on a single single specimen specimen for to for various various crack crack lengths lengths corresponding corresponding to various various loads, loads, P, or or on on a series series of of specimens specimens from from the the critical crack be critical values values of of P and and the the corresponding corresponding crack lengths. lengths. In In each each case, case, the the crack crack lengths lengths must must be determined, either by direct or by measurements of determined, either by direct measurements measurements or indirectly indirectly by measurements of the the specimen's specimen’s comcompliance at unloading or obstacle, pliance at unloading or reloading. reloading. The The need need to to measure measure the the crack crack length length is a considerable considerable obstacle, first because the ~ip is not because the first because the location location of of the the crack crack tip not easy easy to to define, define, and and second second because the visible visible crack crack length for use of length is not not the the most most relevant relevant information information for the the use of the the R-curve R-curve in in structural structural analysis. analysis. For For that purpose, the that purpose, the crack crack length length should should ideally ideally represent represent the the length length of of a certain certain equivalent equivalent crack crack that that yields yields the the correct correct remote remote elastic elastic stress stress field, field, rather rather than than the the actual actual crack crack length length in in the the non-linear non-linear material. the material. Another Another drawback drawback of of determining determining the R-curve R-curve on on a single single specimen specimen (or (or specimens specimens of of the the same point on to same kind) kind) is the the fact fact that that only only one one point on the the R-curve R-curve actually actually corresponds corresponds to a critical critical state state of of the under load the specimen specimen (i.e. (i.e. to to failure failure under load control). control). If be used used for predicting failure If the the R-curve R-curve should should be for predicting failure loads loads (maximum (maximum loads) loads) of of structures, structures, it is certainly use data certainly more more desirable, desirable, as as well well as simpler, simpler, to to use data on on the the failure failure loads loads alone. alone. This This is made made possible by by using using specimens possible specimens of of different different sizes sizes for for which which the the failure failure occurs occurs at at different different crack crack lengths. lengths. A method previously developed be proposed proposed method of of this this type, type, similar similar to to a method method previously developed for for concrete[7], concrete[7], will will be in paper. It of in this this paper. It will will make make use use of of the the size size effect effect and and the the salient salient characteristics characteristics of fracture fracture mechanics. mechanics. The been investigated before The effect effect of of specimen specimen size size on on the the apparent apparent fracture fracture toughness toughness values values has has been investigated before [8, 9]; been exploited the properties. 91; however, however, it has has not not been exploited for for determining determining the non-linear non-linear fracture fracture properties. SIZE SIZE EFFECT EFFECT LAW LAW The size size effect effect can can be exploited because an approximate approximate but quite general general size size effect effect law, law, which which The be exploited because an but quite can be used for for smoothing smoothing and extrapolating extrapolating the data data from from specimens specimens of of various various sizes, sizes, has has can be used and the
t7 Professor Professor of of Civil Civil Engineering Engineering and and Director. Director. Visiting Scholar. Scholar. Prof. Prof. on on leave leave from from Department Department of Architectural Architectural Engineering, Chonnam National University, t$ Visiting of Engineering, Chonnam National University, Kwangju, Kwangju, Chonnam, Chonnam, Korea. Korea. aGraduate Research Assistant: Assistant; presently Resident Student Student Associate Associate at at Argonne Argonne National Laboratory. Argonne, IL, IL, ~Graduate Research presently Resident National Laboratory. Argonne, U.S.A. U.S.A.
45
Z. P. BAZANT et al. BAiANT al.
46 46
been recently been applied been recently established[10]. established[lO]. It It has has already already been applied with with success success for for concrete concrete and and reinforced reinforced concrete[7, concrete[7, 11]. 111.However, However, this law is not not restricted restricted to concrete concrete alone alone and and should should hold hold as an approxiapproximation blunting in general[lO]. mation for for non-linear non-linear fracture fracture with with crack crack blunting general[lO]. Consider Consider geometrically geometrically similar similar specimens specimens or or structures structures with with geometrically geometrically similar similar notches, notches, having purpose of having the the same same thickness, thickness, b, and and made made of of the the same same material. material. For For the purpose of determining determining the size effect, effect, one one may may introduce introduce the the following following approximate approximate hypothesis. hypothesis. The The total total energy, energy, W, dissipated dissipated due process due to fracture fracture depends depends on: on : (1) the the length length a of of the crack crack and and (2) the the size of of the fracture fracture process zone, zone, dr, d,, which which is assumed assumed to be constant constant at failure failure state. state. parameters aid Consequently, Consequently, W must must be a function function of of two non-dimensional non-dimensional parameters a/d and and dr/d, df/d, and dimensional dimensional analysis analysis with with similitude similitude arguments arguments then then leads[l2] leads[ 121to the the following following size effect effect law[lO]: law[ IO] :
oN
Bfy
=
( ‘T2 &
1+
3
d
(1) (1)
j” = --
4’
A0
in which P/bd == nominal which (IN oN == Pjbd nominal stress stress at failure, failure, P == maximum maximum load load (i.e. failure failure load), load), d = = characteristic beam's depth), characteristic dimension dimension of of the the specimen specimen or or structure structure (e.g. the the beam’s depth), /y f, = = uniaxial uniaxial yield yield stress, stress, A = relative relative structure structure size and and B, Ao ilo = = two two empirical empirical constants constants which which depend depend on the geometrical but not geometrical shape shape of of the the specimen specimen or or structure structure but not on its size. In the graph graph of of log (IN B,,,VS vs log d, eqn plotted in Fig. 1. eqn (1) is plotted If parentheses in eqn If the structure structure is very very small small (A (A --> -+ 0), then then the the second second term term in the the parentheses eqn (1) is negligible B/y = negligible compared compared to 1, 1, therefore therefore (IN ON = = Bfy = const. const. This This limit limit case, case, in which which there there is no size effect, effect, represents represents the the yield yield criterion criterion of of failure failure and and corresponds corresponds to a horizontal horizontal line in Fig. 1. 1. If If the the parentheses structure structure is very very large large (A (A --> -+ 00), co), then then 1 is negligible negligible compared compared to the second second term term in the the parentheses of ~ const/jd. of eqn eqn (1), (l), and and then then (IN (TVN const/,,&. This This is the the size effect effect typical typical oflinear of linear elastic elastic fracture fracture mechanics; mechanics ; it corresponds corresponds to the the inclined inclined straight straight line in Fig. I, 1, having having the the slope slope - 1/2. l/2. In general, general, the the size effect effect law according represents a gradual according to eqn eqn (1) represents gradual transition transition from from the the yield yield criterion criterion for for very very small small structures structures to the the energy energy criterion criterion of of linear linear elastic elastic fracture fracture mechanics mechanics for for very very large large structures. structures. For For smoothing smoothing and and extrapolation extrapolation of of the the measured measured raw raw data data as well as their their statistical statistical analysis, analysis, it is important plot important that that the the size effect effect law from from eqn eqn (1) can can be transformed transformed to a linear linear regression regression plot
01 Ol.----------------,~--------~
(a)
0.0 00
I-~-.;....:...-.....;..----~_..,._--_l
,_
9 b’
-01
k? _g
o-N=Bfy /vl+d/)\o0 -02 -0 2
Bf, =3291
PSI
Aao; =2.362m -03~---~-----~----~------~ -0 3
-08 -08
-04 -04
00 00 log log d(ln) dh)
04 04
08
3.---------------------------~ (b)
., Average load 0 Average iood from from 3 3 tests tests
Y=AX+C A=3.904 x 10- 8 C=9.220xlo- 8
NZ
b
'-.
C
o
2
4
Depth Depth d (In) (II?)
Fig. with Fig. 1. The The size size effect effect law, law, its its linear linear regression regression and and comparison comparison with test test data data for for aluminum. aluminum.
Size effect tests and and fracture fracture characteristics characteristics
47
Y=AX+C Y= AX+C
(2) (2)
of of the the form: form :
in in which which X=d, X = d,
Y y=‘= = ~ = (bd)2 F2, p' 4(T~
0
c A =/2&
1 “=m’
(3) (3)
A is the the slope slope of of the the regression regression line line [Fig. [Fig. 1(b)] (b)] and and C is the the Y-intercept. Y-intercept. Plotting Plotting the the measured measured failure failure data similar plot, one data for for geometrically geometrically similar specimens specimens of of different different sizes sizes in in this this regression regression plot, one can can obtain obtain easily by least-square by linear easily A and and C by least-square regression. regression. After After determining determining A and and C by linear regression, regression, we have have B = ..)/C//; \,k;,f( .., )'0 2,, = Cj(Ad CI(Ad,).r)·
DETERMINA TION OF ENERGY DETERMINATION OF FRACTURE FRACTURE ENERGY FROM FROM SIZE SIZE EFFECT EFFECT MEASUREMENTS MEASUREMENTS
Let Let c represent represent the the length length of of the the crack crack from from the the tip tip of of the the notch notch and and ao a0 represent represent the the length length of of the the notch; notch ; let let a a = = ao+c. a,, + c. For For a larger larger specimen, specimen, the the crack crack length length c at at failure failure is normally normally longer. longer. However, the bounded by by a certain process However, the crack crack length length at at failure failure is bounded certain ratio ratio to to the the size size of of the the fracture fracture process zone, zone, and and therefore, therefore, for for very very large large specimens, specimens, the the relative relative crack crack length length at at failure, failure, a CI = = ciao, c/a,, is essentially ~ 00, ~ 00, GO, for for which which ao a, + co, the the value value of of the the relative relative crack crack essentially constant. constant. Thus, Thus, in in the the limit limit d + length length a CIapproaches approaches ao CI~ = = aod. sod. At At the the same same time, time, for for very very large large specimens, specimens, the the size size of of the the fracture fracture process zone becomes negligible negligible compared process zone becomes compared to to the the size size of of the the specimen, specimen, and and so linear linear elastic elastic fracture fracture mechanics mechanics must must apply apply in in the the limit. limit. Thus, Thus, for for very very large large specimens: specimens :
(4) (4)
in g(ao) is the energy in which which g(ao) the non-dimensional non-dimensional energy release release rate rate according according to to linear linear elastic elastic fracture fracture mechmechanics g(ao) is known anics for for a crack crack of of length length ao. a,. The The value value of of g(a,) known for for many many typical typical specimen specimen geometries geometries and g(ao) can be determined by linear and is found found in textbooks textbooks and and handbooks[13J. handbooks[ 131. The The value value of of g(a,J can always always be determined by linear finite finite element element analysis. analysis. According According to to the the size size effect effect law law [eqn [eqn (I)], (I)], for for very very large large d we we have have
(5) (5)
Now, equating Now, equating expressions expressions for for (TN bN [eqns [eqns (4) and and (5)] and and solving solving for for fracture fracture energy energy G[, Gf, we obtain obtain the basic result[7] the basic result[7] : g(ao) _ S(%> Grf-- AE AE . G
=
(6)
Thus we see that that the the fracture fracture energy energy is inversely inversely proportional to the the slope slope of of the the size size effect effect regression regression Thus proportional to line )]. This provides a convenient of line [Fig. [Fig. I1(b (b)]. This provides convenient method method of of determination determination of fracture fracture energy. energy.
DETERMINATION OF R-CURVE R-CURVE FROM SIZE SIZE EFFECT EFFECT MEASUREMENTS MEASUREMENTS DETERMINATION OF FROM The size size effect effect law law is also also helpful helpful for for determining determining the R-curve[7]. R-curve[7]. In In terms terms of of the the R-curve R-curve R(c), R(c), The the which describes describes the the dependence dependence of of the the energy energy required required for for crack crack growth growth on on the the crack crack length length measured measured which from notch tip, balance condition requires that from the the notch tip, the the energy energy balance condition requires that
G(a) G(a) = = R(c) R(c)
(a=ao+c), (a = a,+c),
(7) (7)
48
Z. P. BAZANT BAZANT et et al. nl.
in which which G(a) G(a) is the the energy energy release release rate rate according according to an equivalent equivalent linear linear elastic elastic analysis. analysis. The The foregoing relation may may foregoing relation may be regarded regarded as an equilibrium equilibrium relation relation for for fracture. fracture. The The equilibrium equilibria may be stable, critical critical or or unstable. The critical critical state state is the the state state at maximum maximum load load P,,,. The condition condition of of stable, unstable. The Pmax' The stability stability of of fracture fracture is known known to have have the form form : dR(e) dG(a) df2(c) dG(a) .... ----~ 0 (stable) - -dn- > (stable) de da dr
.
(8)
(critical). = 0 (critical).
The The energy energy release release rate rate according according to linear linear elastic elastic fracture fracture mechanics mechanics has has always always the the form form G(a) G(a) = = p2g(a)/(Eb 2d), and Pzg(cc)/(Eb2d), and so for for the the maximum maximum measured measured loads loads we have have (9)
in which which g(a) g(a) is a function function available available in handbooks handbooks for for typical typical specimen specimen geometries[13] geometries[ 131and and obtainable obtainable in general by linear general by linear finite finite element element analysis. analysis. According According to eqns eqns (7)-(8), (7)-(8), failure failure occurs occurs when when the the curve curve G(a) G(a) has has the same same value value and and the the same same slope plotted slope as the the curve curve R(c). R(c). Consequently, Consequently, the the R-curve R-curve represents represents the envelope envelope of of all curves curves G{a), G(a), plotted according according to to eqn eqn (9) for for various various measured measured values values of of maximum maximum load, load, P P,,,. max • Thus, plotted for Thus, the the R-curve 1P-curve may may be obtained obtained from from size effect effect tests if if the the curve curve G(a) G(a) is plotted for each each specimen practice, a specimen size and and the the corresponding corresponding maximum maximum load, load, using using the the known known function function g(a). g(a). In practice, problem arises problem arises however however for for two two reasons: reasons : (l) (1) the test test data data exhibit exhibit statistical statistical scatter scatter and and (2) the the range range of of measured measured data data is usually usually insufficient insufficient for for determining determining the the complete complete envelope. envelope. To To avoid avoid these these difficulties, di~culties, we must must take take recourse recourse to the the size effect effect law [eqn [eqn (l)].If (l)].If the the raw, raw, scattered values of P,,, scattered values of are used used to plot the curves curves G(a) G(u) for various various specimen specimen sizes, no common common Pmax are plot the envelope envelope can can be obtained obtained [Fig. 2(b)]. Thus, Thus, smoothing smoothing of of the the measured measured data data according according to the size effect effect law is required required before the R-curve R-curve can can be determined. determined. Furthermore, Furthermore, if if the the curves curves G(a) G(a) are are before the plotted only plotted only for for a limited limited range range of of measured measured Pm,,Pmax-values, of values, determination dete~ination of the the R-curve R-curve is ambiguambiguous, as illustrated illustrated in Fig. 2(c). It It is necessary necessary to plot the curves curves G(a) G(a) for for P P,,,-values within a range range ous, plot the max-values within of of I: 1: 100 or or more, more, and and such such a range range can can hardly hardly be obtained obtained except except by generating generating the the Pmax-values P,,,-values from parameters Band been determined by regression from the the size effect effect law [eqn (1)] (l)] after after parameters B and ).0 & have have been determined by regression of of the the test test data. data. The principle, according The R-curve R-curve can can be also constructed, constructed, in principle, according to the the failure failure load load for for different different notch notch lengths lengths on on specimens specimens of of the the same same size. However, However, this this method method fails for for the the same same reason reason as mentioned; mentioned ; the the range range of of the the curve curve G(a) G(a) is insufficient, insufficient, like in Fig. Fig. 2(c). Z(c). The The R-curve R-curve can can be obtained obtained from from such such data data only only if if additional additional properties, such as the crack crack length length or or the the specimen specimen compliance, compliance, properties, such are measured. particular the measured. However, However, this has has the the drawback drawback already already stated, stated, in particular the fact fact that that load load values values which do which do not not correspond correspond to to failure failure states states must must be also also included included in the the evaluation. evaluation. It been shown[7] It has has been shown[;l that that the the R-curve R-curve obtained obtained from from test test data data smoothed smoothed by the the size effect effect law iaw in eqn eqn (1) is very very closely closely approximated approximated by the the formula formula R(e)=G,[l-!l-~~i
for
cc,,.
(10)
in which parameters em which G Grr is already already known known in advance advance from from eqn eqn (6) and and parameters c, and and n are are to be identified identified by fitting fitting of of the the envelope. envelope. Usually, Usually, as a good good approximation, approximation, n = = 3.6. It It must must be kept kept in mind mind that that the the R-curve A-curve is an approximate approximate concept. concept. In theory, theory, the the R-curves R-curves for however, experience has shown for different different specimen specimen geometries geometries should should not not be the the same; same; however, experience has shown that that in many situations the R-curves are nearly the same, while in others they are not[7]. many situations the R-curves are nearly same, while in others they are not[7]. FRACTURE TESTS TESTS OF OF ALUMINUM ALUMINUM FRACTURE
After procedure outlined been applied After initial initial success success with with concrete[7], concrete[7], the the new test test procedure outlined here here has has been applied to bent fracture been aluminum product of aluminum A16061-T651 Al 6061-T651 (a product of Alcoa Alcoa Co.). Co.). Three-point Three-point bent fracture specimens specimens have have been
49
Size effect tests tests and and fracture fracture characteristics characteristics 1000
(a)
'" Ci'" ~
(\J
Ci ~
800
/"
~
d·1l
"0 d, =3.990 In. A=1586xIO- B C=6.329xIO- B G,=70941b/in
-02 -02-'::-0-:-4-----:0"::O-----::O"::4----~O~ -04
00
04
Or‘
log log dOn) d(in 1
Fig. plot of proportional thicknesses. Fig. 6. Size Size effect effect plot of test test results results of of specimens specimens with with proportional thicknesses.
the boundary conditions, it possibly have been better better the specimen specimen does does not not match match the the surface surface boundary conditions, it might might possibly have been for of proportional to beam depth. for comparability comparability of specimens specimens of of different different sizes sizes to to use use thicknesses thicknesses proportional to beam depth. However, was not not done because the the elastic Nevertheless, in However, this this was done because elastic effects effects seem seem less less important. important. Nevertheless, in a preliminary test beam of been tested, preliminary test one one beam of different different thickness, thickness, b = = 0.75 0.75 in., in., has has been tested, and and the the results results (row (row H in based on in Table Table 1) are are still still essentially essentially in in agreement agreement with with the the values values of of aN (TVfor for the the size size effect effect law, law, based on A-E. rows rows A-E. A been also A few few tests tests have have been also carried carried out out for for a fixed fixed specimen specimen size size (d (d = = 1 in.) in.) and and different different notch notch lengths, Fin Table lengths, aid a/d = = 0.25,0.5 0.25, 0.5 and and 0.75 0.75 (rows (rows G, A and and Fin Table 1). The The fracture fracture equilibrium equilibrium curves curves for for the plotted in the measured measured maximum maximum loads loads (Table (Table 1) are are plotted in Fig. Fig. 2(c), 2(c), and and it it is clear clear that that determination determination of for as already of the the R-curve R-curve as as an an envelope envelope is totally totally ambiguous ambiguous for such such a set set of of measurements, measurements, already emphasized. be used just the emphasized. So So a single single specimen specimen size size cannot cannot be used unless unless more more than than just the maximum maximum load load is measured, measured, as is well well known. known. The been carried Northwestern University. The foregoing foregoing tests tests have have been carried out out at at Northwestern University. A further further test test series series has has National University, to been carried been carried out out at at Chonnam Chonnam National University, to check check the the effect effect of of scaling scaling in in which which the the thickness proportion to beam depth thickness is varied varied in in proportion to the the beam depth and and length. length. The The results results of of these these tests, tests, carried carried out with an next out with an Instron Instron machine, machine, are are given given in in Figs Figs 6 and and 7 where where the the maximum maximum loads loads are are indicated indicated next to points of by smoothing to the the points of aN' oN. The The fracture fracture energy energy value value obtained obtained by smoothing with with the the size size effect effect law law is for for these previous tests these tests tests G Grr = = 709 Ib/in., lb/in., which which is only only 6.0% 6.0% 'larger ‘larger than than in in the the previous tests with with constant constant be scaled specimen specimen thickness. thickness. From From this this it seems seems that that the the question question of of how how the the thickness thickness should should be scaled is not not very very important. important. CRACK MODEL CRACK BAND BAND FINITE FINITE ELEMENT ELEMENT MODEL
Fracture behavior which be closely Fracture behavior which follows follows the the size size effect effect according according to to eqn eqn (1) can can be closely modeled modeled in in finite using the band theory[15, 16]. finite element element analysis analysis using the crack crack band theory[l5, 161. The The fact fact that that this this size size effect effect law law is obtained process zone obtained from from the the hypothesis hypothesis that that the the fracture fracture process zone has has a certain certain fixed fixed effective effective width[lO] width[ IO] means be modeled band of progressively cracking means that that the the fracture fracture can can be modeled as a band of progressively cracking elements elements of of the the same same width, relationship with width, h. The The material material in in the the finite finite elements elements is assumed assumed to to exhibit exhibit a stress-strain stress-strain relationship with gradual relation gradual strain strain softening, softening, for for example example as shown shown in Fig. Fig. 8. A A simple simple triaxial triaxial constitutive constitutive relation which diagram been formulated[16]. which yields yields the the stress-strain stress-strain diagram with with strain strain softening softening shown shown in in Fig. Fig. 8 has has been formulated[lh]. The process zone The size size of of the the finite finite element, element, which which represents represents the the effective effective width width of of the the fracture fracture process zone at at the be expressed the fracture fracture front, front, may may be expressed as
h = GlWo,
h[ -__
Fig. band model Fig. 8. Finite Finite element element crack crack band model and and tensile tensile stress--strain stress--strain diagram diagram with with yield yield and and strain strain softening. softening. EF~1
26: 1-D
((12) 12)
Z. P. BAZANT BATANT et al.
54
in under the diagram in which which Wo W, represents represents the the area area under the complete complete tensile tensile stress-strain stress-strain diagram with with strain strain softening softening (Fig. plateau and slope E t in (Fig. 8). The The length length of of the the yield yield plateau and the the strain-softening strain-softening slope Et in Fig. Fig. 8 govern govern chiefly chiefly the the length length of of the the fracture fracture process process zone zone ahead ahead of of the the crack crack tip, tip, and and E, E t can can be be determined determined if if this this length length is known. known. It is important to precise element by eqn important to realize realize that that a certain certain precise element size. size, as as given given by eqn (12), (12). is required. required. an excessively large number of elements, However, in situations where this would lead to However, in situations where this would lead to an excessively large number of elements, larger larger finite elements may be used, but then curve, finite elements may be used, but then the the slope slope of of the the declining declining segment segment of of the the stress-strain stress-strain curve, and and possibly possibly also also the the peak peak stress stress value, value, must must be be adjusted adjusted so that that the the product product of of the the area area under under the the complete curve complete tensile tensile stress-strain stress-strain curve with with the the element element width width used used would would give give the the correct correct value value of of fracture the fracture energy energy Gr. Then Then the the results results are are approximately approximately the same same as as those those for for the the correct correct finite finite element element size by eqn size given given by eqn (12). (12). It It has has been been demonstrated[16] demonstrated[ 161 that that a finite finite element element model model of of this this type type gives gives the the correct correct size size effect effect by eqn curve as as shown shown in in Fig. Fig. 1 and and described described by eqn (1), (l), and and generally generally agrees agrees with with a wide wide range range of of data data curve on on non-linear non-linear fracture fracture of of various various materials, materials, including including the the effect effect of of the the notch notch length length on on the the maximum maximum load load and and on on the the variation variation of of energy energy required required for for crack crack growth growth as as a function function of of the the length length of of the the crack crack from from the the notch notch tip, tip, i.e. i.e. the the R-curve. R-curve. Note that R-curve is obtained Note that the the correct correct R-curve obtained from from the the finite finite element element model model even even though though only only one one constant constant value value of of fracture fracture energy, energy, G Gr,r, is used used in in the the model. model. This This value value of of fracture fracture energy energy coincides coincides with with the the energy energy release release rate rate only only in in those those situations situations where where the the ligament ligament cross-section cross-section is much much larger larger than process zone, process zone than the the fracture fracture process zone, so that that the the fracture fracture process zone can can fully fully develop develop and and does does not not interfere boundaries of interfere with with the the boundaries of the the structure. structure. If If the the structure structure is smaller, smaller, then then the the full full size size of of the the fracture process zone boundaries occurs, fracture process zone cannot cannot develop, develop, and and interference interference with with the the boundaries occurs. e.g. e.g. when when finite finite elements undergo strain elements that that undergo strain softening softening are are close close to to the the notch notch tip tip or or to to the the end end of of the the ligament. ligament. This This is why the why the the model, model, with with its its fixed fixed fracture fracture energy energy value, value, is capable capable of of representing representing the R-curve. R-curve. It be noted relation It may may be noted that that the the limiting limiting case case in in which which the the tensile tensile stress-strain stress-strain relation is considered considered as as elastic, elastic, followed followed by by a sudden sudden stress stress drop drop to to zero zero as as the the strength strength limit limit is reached, reached, yields yields solutions solutions which in provided there which are are approximately approximately in agreement agreement with with linear linear elastic elastic fracture fracture mechanics mechanics provided there are are about because, for about 15 or or more more finite finite elements elements across across the the ligament. ligament. This This is because, for a sudden sudden stress stress drop, drop, the the fracture process zone undergo cracking) fracture process zone (elements (elements that that undergo cracking) is always always limited limited to to a single single element element width, width, so that process zone possible within that the the fracture fracture process zone is localized localized to to the the maximum maximum possible within the the finite finite element element mesh. mesh. This This fact fact also also shows shows that that it is not not necessary necessary to to model model fracture fracture in in finite finite element element codes codes as as a line line interelement crack band and interelement crack with with a sharp sharp tip. tip. In In fact, fact, solutions solutions with with a crack crack band and a sudden sudden stress stress which which drops drops to to zero zero come come as as close close to to the the linear linear elastic elastic fracture fracture mechanics mechanics solutions solutions as as do do the the solutions solutions with crack. band approach be more with a sharp sharp interelement interelement crack. The The crack crack band approach to to fracture fracture modeling modeling seems seems to to be more flexible, program and permits modeling band of flexible, easier easier to to program and also also permits modeling fracture fracture in in any any direction direction as as a zigzag zigzag band of crack elements. So far this technique has been used widely only for concrete and geomaterials; crack elements. So far this technique has been used widely only for concrete and geomaterials; however, potential for fracture however, it it may may have have a potential for non-linear non-linear fracture modeling modeling in in metals. metals.
GENERALIZED SIZE LAW GENERALIZED SIZE EFFECT EFFECT LAW Equation (1) possible approximation to fracture. Equation (1) is the the simplest simplest possible approximation to the the size size effect effect for for non-linear non-linear fracture. The be written written in The most most general general size size effect effect law law may may be in the the form form of of asymptotic asymptotic expansion[16]: expansion[l6] :
(IN
A)~r + I + (A) ..f + (") ~2 + (A)4 / +...
= Bfy [( )~
J. 1/2r ,
(13) (13j
B, r, )"0, A10 AZ, A2. )03, obtained by where are material material constants. constants. The The first first order order approximation, approximation, obtained by where ff,, ILo, A,, Ax, ... . . are y, B, AI = A, A2 == A3 A3 == ... setting setting A1 . . = = 0, i.e. i.e. (IN
= B/y
[1 + (~)11:2r
(14) (14)
is considerably more possible shapes considerably more flexible flexible than than eqn eqn (1) for for describing describing various various possible shapes of of the the curve curve of of log be obtained by finite In log (IN dN vs log log IAwhich which may may be obtained by finite element element calculations. calculations. In fact, fact, finite finite element element experience experience indicates[17] that indicates[l7] that more more than than the the first first two two terms terms from from eqn eqn (13) (13) are are never never really really needed. needed.
Size Size effect effect tests tests and and fracture fracture characteristics characteristics
55
In the fitting fitting of of available available test data, data, however, however, coefficient coefficient r in eqn eqn (14) cannot cannot be determined determined unambiguously because because of unambiguously of the random random scatter scatter of of test test results results and and the usual usual limitations· limitations’ofof the size range. broad size range Band range. Tests Tests over over a very very broad range would would be required required to determine determine r, r, in addition addition to B and Ao. Nevertheless, Nevertheless, if used to determine 2,. if sufficient sufficient data data are given, given, then then eqn eqn (14) may may be also easily easily used determine the parameters of the fracture fracture parameters of the material. material. Either Either a non-linear non-linear optimization optimization subroutine subroutine such such as the Marquardt-Levenberg algorithm Ao, and Marquardt-Levenberg algorithm may may be used used to determine determine B, &, and r simultaneously, simultaneously, or or eqn eqn (14) may may be algebraically algebraically rearranged rearranged to the linear linear relation relation in eqn eqn (2) in which, which, instead instead of of eqn eqn (3), X=d’, X= d',
Y Y= = O"N 0;-2, 2r,,
A = C&d,)-‘,
(15) (1%
C = (BF,)- “.
A set of of r-values, r-values, e.g. 1, 1, 0.8, 0.6, 0.4, may may be selected selected and, and, for for each, each, one one finds A and and C as well as the sum of regression. Considering of squared squared deviations deviations from from the test test data data by linear linear regression. Considering the dependence dependence of 1/2,//y, lo Ao = of this sum on r, one one may may identify identify the the r-value r-value that that minimizes minimizes this sum. Then Then B = cC-ii2r/fy, = (C/A)l/'/d (C/A) “‘/df.r . For For dfd d/d‘+ co, eqn (14) reduces reduces again again to eqn (5). Thus, Thus, the fracture fracture energy energy is again again given by eqn r --+ 00, (6), (6) although although for a different different rr a different different G Grr will be obtained. obtained.
COMPARISON COMPARISON WITH WITH R-CURVES R-CURVES BASED BASED ON ON ASTM ASTM STANDARD STANDARD
According According to ASTM ASTM Standard Standard Recommendation Recommendation E561-81, E561-8 1, the R-curve R-curve is determined determined from from compliance compliance measurements measurements on on a single compact compact tension tension specimen specimen of of the type type shown shown in Fig. 9. These These tests have been also carried have been carried out out using using three three groups groups of of specimens specimens of of different different sizes (specified (specified in the the Table Table in Fig. 1). 1). The The crack crack opening opening displacement, displacement, COD, COD, was measured measured by a clip gage with an accuracy base length under w/4, mounted accuracy of of 10-IO--66 m and and base length under mounted at distance distance (0.1576 ± + O.0006)w 0.0006)~ ahead ahead of of
.s "-
.0
oel
o Thickness b,1 in. 0
02 02
"C3
b,0.75 in. 0
02 02
b'O.5 in. 0
0.4 04
02 02
0.6 06
Fig. with R-curves obtained by ASTM Fig. 9. Comparison Comparison with R-curves obtained by ASTM procedure procedure based based on on compact compact tension tension specimens specimens (solid R-curve from present method). (solid curve curve = R-curve from present method). Dimensions Dimensions (in., (in., except except s, mm) mm) Specimen Specimen
b
d d
e
ff
i
AI,A2 Al, A2 BI, Bl, B2 82 Cl, Cl, C2, c2, C3 c3
1.00 1.00 0.75 0.75 0.50 0.50
2.4 2.4 1.8 1.8 1.2 1.2
0.550 0.550 0.413 0.413 0.275 0.275
0.18 0.18 0.118 0.118 0.098 0.098
0314 0.314 0.236 0.236 0-157 0.157
2.500 2.500 1.875 1.875 1.250 1.250
Al Al
A2 A2
BI Bl
B2
CI Cl
C2 c2
C3 c3
0.998 0.998 2.004 2.004 0.819 0.819 0.13 0.13
0.994 0.994 2.004 2.004 0.803 0.803 0.13 0.13
0_748 0.748 1.496 1.496 0.628 0.628 0.13 0.13
0.744 0.744 1.496 1.496 0.628 0.628 0.13 0.13
0.498 0.498 0.976 0.976 0386 0.386 030 0.30
0.498 0.498 0.992 0.992 0.407 0.407 0.13 0.13
0.498 0.498 0_996 0.996 0.406 0.406 0.30mm 0.30 mm
t IV w
ao a0 ss
m 112
p P
qY
u
v
zz
1.2 0.9 0.9 0.6 0.6
0.650 0.650 0.488 0.488 0.325 0.325
0.20 0.20 0.15 0.15 0-10 0.10
1.2 0.9 0.9 0.6 0.6
0.500 0.500 0.375 0.375 0.250 0.250
56
Z. P. BAZANT BAZANT et et al.
the axis connecting pins (for connecting the the centers centers of of the loading loading pins (for w see Fig. 9). The The initial initial notch notch was cut cut with with a saw and, producing a fatigue and, instead instead of of producing fatigue crack, crack, a slit crack crack extending extending from from the notch notch tip was machined by a diamond paper of machined by diamond disc paper of thickness thickness 0.13 or or 0.30 mm mm (see the the table table in Fig. 1). The The crosscrosshead head speed speed of of the the machine machine was 0.2 mmjmin. mm/min. The points obtained The R-curve R-curve points obtained from from these these tests are are shown shown in Fig,. Fig,. 1. 1. They They were were obtained obtained by the the following procedure. Take representative representative points the recorded recorded graph graph of of load load P vs 2vI 221, = = COD. COD. following procedure. Take points on the Calculate Calculate the the values values of of 2v 11 Ebj Eb/PP and and find the the equivalent equivalent crack crack length length a a,e from from the the compliance compliance table table (ASTME561-SI, Kcr = Ph-1w (ASTM E561-81, Table Table 2). ThencalculateR(c) Then calculate R(c) = = K:rjE(forplane K$/E (for plane stress) stress) where where Kc, = Ph‘w 1'2(0 “‘f(t) [13, IS, ~ = ae/H'. 18, 19] 191where where < CI,.M’. f(~) = (0.866+4.64((0.S66+4.64~-13.32~2+ 14.72~3_5.6~4)[(2+~)j(1_~)3!2]. f(t) 13.32t2+ 14.72~3-5.6~4)[(2+~)/(l -
Engineaing Fracture Mechanics Vol. 26, No. I. pp. 45-57.1987 Printed in Great BrItain
SIZE SIZE EFFECT EFFECT TESTS TESTS AND AND FRACTURE FRACTURE CHARACTERISTICS OF CHARACTERISTICS OF ALUMINUM ALUMINUM ZDENEK P. BAZANT,t SOO-GON LEEt A. ZDENl?K BAgANT,? SOO-GON LEES and and PHILLIP PHILLlP A. PFEIFFER§ PFEIFFERP Center Northwestern University, Center for for Concrete Concrete and and Geomaterials, Geomaterials, Northwestern University, Evanston, Evanston, IL IL 60201, 60201, U.S.A. U.S.A. Abstract-The recently size Abstract-The recently established established approximate approximate size effect effect law law for for blunt blunt fracture fracture is shown shown to to agree agree with with three-point bent fracture with test test data data for for aluminum. aluminum. The The data data are are obtained obtained with three-point bent fracture specimens specimens of of depths Non-linear fracture parameters can be obtained by a linear depths ranging ranging from from 0.25 0.25 to to 4 in. Non-linear fracture parameters can be obtained by linear regression based based on be detennined using only regression on the the size size effect effect law. law. Thus, Thus, the the fracture fracture energy energy can can be determined using only the the maximum values from be obtained maximum load load values from fracture fracture tests, tests, and and the the R-curve R-curve can can be obtained as as an an envelope envelope of of fracture fracture equilibrium No measurements of equilibrium curves. curves. No measurements of crack crack length length or or specimen specimen compliance compliance are are needed. needed.
INTRODUCTION INTRODUCTION FRACTURE generally FRACTURE of of aluminum aluminum generally does does not not follow follow linear linear elastic elastic fracture fracture mechanics. mechanics. This This means means of the non-linearity that a non-linear fracture model must be used. One conspicuous consequence that non-linear fracture model must be used. One conspicuous consequence of the non-linearity of properties is the of fracture fracture properties the apparent apparent variation variation of of the the energy energy required required for for crack crack growth, growth, R, with with the the length of the crack. Irwin[l] and Krafft et al.[2] proposed that the dependence of R on the length of the crack. Irwin[l] and Krafft al.[2] proposed that the dependence of on the crack crack length, be considered be approximately the length, c, may may be considered to to be approximately the same same for for various various fracture fracture geometries. geometries. This This means use for a unique unique curve, means that that one one can can use for fracture fracture calculations calculations curve, R(c), R(c), called called the the R-curve R-curve (resistance (resistance curve), be a material property. curve), which which is considered considered to to be material property. The the by an The existing existing methods methods for for determining determining the R-curve, R-curve, specified specified by an ASTM ASTM standard standard [3-6], [3-61, rely rely 2 2 2 Ip a/(Ecb d ) in on on the the relation relation R = kk,P2a/(Ecb2d2) m which which ee,.c = = Young's Young’s elastic elastic modulus, modulus, b = specimen specimen thickness, thickness, d = characteristic dimension plus notch, characteristic dimension of of the the specimen, specimen, a = = length length of of the the crack crack plus notch, P = load load at at which which the the crack crack extends extends and and k k,I = a coefficient coefficient determined determined for for the the given given specimen specimen geometry geometry according according to to linear linear elastic elastic fracture fracture mechanics. mechanics. A series series of of R-values R-values is measured, measured, either either on on a single single specimen specimen for to for various various crack crack lengths lengths corresponding corresponding to various various loads, loads, P, or or on on a series series of of specimens specimens from from the the critical crack be critical values values of of P and and the the corresponding corresponding crack lengths. lengths. In In each each case, case, the the crack crack lengths lengths must must be determined, either by direct or by measurements of determined, either by direct measurements measurements or indirectly indirectly by measurements of the the specimen's specimen’s comcompliance at unloading or obstacle, pliance at unloading or reloading. reloading. The The need need to to measure measure the the crack crack length length is a considerable considerable obstacle, first because the ~ip is not because the first because the location location of of the the crack crack tip not easy easy to to define, define, and and second second because the visible visible crack crack length for use of length is not not the the most most relevant relevant information information for the the use of the the R-curve R-curve in in structural structural analysis. analysis. For For that purpose, the that purpose, the crack crack length length should should ideally ideally represent represent the the length length of of a certain certain equivalent equivalent crack crack that that yields yields the the correct correct remote remote elastic elastic stress stress field, field, rather rather than than the the actual actual crack crack length length in in the the non-linear non-linear material. the material. Another Another drawback drawback of of determining determining the R-curve R-curve on on a single single specimen specimen (or (or specimens specimens of of the the same point on to same kind) kind) is the the fact fact that that only only one one point on the the R-curve R-curve actually actually corresponds corresponds to a critical critical state state of of the under load the specimen specimen (i.e. (i.e. to to failure failure under load control). control). If be used used for predicting failure If the the R-curve R-curve should should be for predicting failure loads loads (maximum (maximum loads) loads) of of structures, structures, it is certainly use data certainly more more desirable, desirable, as as well well as simpler, simpler, to to use data on on the the failure failure loads loads alone. alone. This This is made made possible by by using using specimens possible specimens of of different different sizes sizes for for which which the the failure failure occurs occurs at at different different crack crack lengths. lengths. A method previously developed be proposed proposed method of of this this type, type, similar similar to to a method method previously developed for for concrete[7], concrete[7], will will be in paper. It of in this this paper. It will will make make use use of of the the size size effect effect and and the the salient salient characteristics characteristics of fracture fracture mechanics. mechanics. The been investigated before The effect effect of of specimen specimen size size on on the the apparent apparent fracture fracture toughness toughness values values has has been investigated before [8, 9]; been exploited the properties. 91; however, however, it has has not not been exploited for for determining determining the non-linear non-linear fracture fracture properties. SIZE SIZE EFFECT EFFECT LAW LAW The size size effect effect can can be exploited because an approximate approximate but quite general general size size effect effect law, law, which which The be exploited because an but quite can be used for for smoothing smoothing and extrapolating extrapolating the data data from from specimens specimens of of various various sizes, sizes, has has can be used and the
t7 Professor Professor of of Civil Civil Engineering Engineering and and Director. Director. Visiting Scholar. Scholar. Prof. Prof. on on leave leave from from Department Department of Architectural Architectural Engineering, Chonnam National University, t$ Visiting of Engineering, Chonnam National University, Kwangju, Kwangju, Chonnam, Chonnam, Korea. Korea. aGraduate Research Assistant: Assistant; presently Resident Student Student Associate Associate at at Argonne Argonne National Laboratory. Argonne, IL, IL, ~Graduate Research presently Resident National Laboratory. Argonne, U.S.A. U.S.A.
45
Z. P. BAZANT et al. BAiANT al.
46 46
been recently been applied been recently established[10]. established[lO]. It It has has already already been applied with with success success for for concrete concrete and and reinforced reinforced concrete[7, concrete[7, 11]. 111.However, However, this law is not not restricted restricted to concrete concrete alone alone and and should should hold hold as an approxiapproximation blunting in general[lO]. mation for for non-linear non-linear fracture fracture with with crack crack blunting general[lO]. Consider Consider geometrically geometrically similar similar specimens specimens or or structures structures with with geometrically geometrically similar similar notches, notches, having purpose of having the the same same thickness, thickness, b, and and made made of of the the same same material. material. For For the purpose of determining determining the size effect, effect, one one may may introduce introduce the the following following approximate approximate hypothesis. hypothesis. The The total total energy, energy, W, dissipated dissipated due process due to fracture fracture depends depends on: on : (1) the the length length a of of the crack crack and and (2) the the size of of the fracture fracture process zone, zone, dr, d,, which which is assumed assumed to be constant constant at failure failure state. state. parameters aid Consequently, Consequently, W must must be a function function of of two non-dimensional non-dimensional parameters a/d and and dr/d, df/d, and dimensional dimensional analysis analysis with with similitude similitude arguments arguments then then leads[l2] leads[ 121to the the following following size effect effect law[lO]: law[ IO] :
oN
Bfy
=
( ‘T2 &
1+
3
d
(1) (1)
j” = --
4’
A0
in which P/bd == nominal which (IN oN == Pjbd nominal stress stress at failure, failure, P == maximum maximum load load (i.e. failure failure load), load), d = = characteristic beam's depth), characteristic dimension dimension of of the the specimen specimen or or structure structure (e.g. the the beam’s depth), /y f, = = uniaxial uniaxial yield yield stress, stress, A = relative relative structure structure size and and B, Ao ilo = = two two empirical empirical constants constants which which depend depend on the geometrical but not geometrical shape shape of of the the specimen specimen or or structure structure but not on its size. In the graph graph of of log (IN B,,,VS vs log d, eqn plotted in Fig. 1. eqn (1) is plotted If parentheses in eqn If the structure structure is very very small small (A (A --> -+ 0), then then the the second second term term in the the parentheses eqn (1) is negligible B/y = negligible compared compared to 1, 1, therefore therefore (IN ON = = Bfy = const. const. This This limit limit case, case, in which which there there is no size effect, effect, represents represents the the yield yield criterion criterion of of failure failure and and corresponds corresponds to a horizontal horizontal line in Fig. 1. 1. If If the the parentheses structure structure is very very large large (A (A --> -+ 00), co), then then 1 is negligible negligible compared compared to the second second term term in the the parentheses of ~ const/jd. of eqn eqn (1), (l), and and then then (IN (TVN const/,,&. This This is the the size effect effect typical typical oflinear of linear elastic elastic fracture fracture mechanics; mechanics ; it corresponds corresponds to the the inclined inclined straight straight line in Fig. I, 1, having having the the slope slope - 1/2. l/2. In general, general, the the size effect effect law according represents a gradual according to eqn eqn (1) represents gradual transition transition from from the the yield yield criterion criterion for for very very small small structures structures to the the energy energy criterion criterion of of linear linear elastic elastic fracture fracture mechanics mechanics for for very very large large structures. structures. For For smoothing smoothing and and extrapolation extrapolation of of the the measured measured raw raw data data as well as their their statistical statistical analysis, analysis, it is important plot important that that the the size effect effect law from from eqn eqn (1) can can be transformed transformed to a linear linear regression regression plot
01 Ol.----------------,~--------~
(a)
0.0 00
I-~-.;....:...-.....;..----~_..,._--_l
,_
9 b’
-01
k? _g
o-N=Bfy /vl+d/)\o0 -02 -0 2
Bf, =3291
PSI
Aao; =2.362m -03~---~-----~----~------~ -0 3
-08 -08
-04 -04
00 00 log log d(ln) dh)
04 04
08
3.---------------------------~ (b)
., Average load 0 Average iood from from 3 3 tests tests
Y=AX+C A=3.904 x 10- 8 C=9.220xlo- 8
NZ
b
'-.
C
o
2
4
Depth Depth d (In) (II?)
Fig. with Fig. 1. The The size size effect effect law, law, its its linear linear regression regression and and comparison comparison with test test data data for for aluminum. aluminum.
Size effect tests and and fracture fracture characteristics characteristics
47
Y=AX+C Y= AX+C
(2) (2)
of of the the form: form :
in in which which X=d, X = d,
Y y=‘= = ~ = (bd)2 F2, p' 4(T~
0
c A =/2&
1 “=m’
(3) (3)
A is the the slope slope of of the the regression regression line line [Fig. [Fig. 1(b)] (b)] and and C is the the Y-intercept. Y-intercept. Plotting Plotting the the measured measured failure failure data similar plot, one data for for geometrically geometrically similar specimens specimens of of different different sizes sizes in in this this regression regression plot, one can can obtain obtain easily by least-square by linear easily A and and C by least-square regression. regression. After After determining determining A and and C by linear regression, regression, we have have B = ..)/C//; \,k;,f( .., )'0 2,, = Cj(Ad CI(Ad,).r)·
DETERMINA TION OF ENERGY DETERMINATION OF FRACTURE FRACTURE ENERGY FROM FROM SIZE SIZE EFFECT EFFECT MEASUREMENTS MEASUREMENTS
Let Let c represent represent the the length length of of the the crack crack from from the the tip tip of of the the notch notch and and ao a0 represent represent the the length length of of the the notch; notch ; let let a a = = ao+c. a,, + c. For For a larger larger specimen, specimen, the the crack crack length length c at at failure failure is normally normally longer. longer. However, the bounded by by a certain process However, the crack crack length length at at failure failure is bounded certain ratio ratio to to the the size size of of the the fracture fracture process zone, zone, and and therefore, therefore, for for very very large large specimens, specimens, the the relative relative crack crack length length at at failure, failure, a CI = = ciao, c/a,, is essentially ~ 00, ~ 00, GO, for for which which ao a, + co, the the value value of of the the relative relative crack crack essentially constant. constant. Thus, Thus, in in the the limit limit d + length length a CIapproaches approaches ao CI~ = = aod. sod. At At the the same same time, time, for for very very large large specimens, specimens, the the size size of of the the fracture fracture process zone becomes negligible negligible compared process zone becomes compared to to the the size size of of the the specimen, specimen, and and so linear linear elastic elastic fracture fracture mechanics mechanics must must apply apply in in the the limit. limit. Thus, Thus, for for very very large large specimens: specimens :
(4) (4)
in g(ao) is the energy in which which g(ao) the non-dimensional non-dimensional energy release release rate rate according according to to linear linear elastic elastic fracture fracture mechmechanics g(ao) is known anics for for a crack crack of of length length ao. a,. The The value value of of g(a,) known for for many many typical typical specimen specimen geometries geometries and g(ao) can be determined by linear and is found found in textbooks textbooks and and handbooks[13J. handbooks[ 131. The The value value of of g(a,J can always always be determined by linear finite finite element element analysis. analysis. According According to to the the size size effect effect law law [eqn [eqn (I)], (I)], for for very very large large d we we have have
(5) (5)
Now, equating Now, equating expressions expressions for for (TN bN [eqns [eqns (4) and and (5)] and and solving solving for for fracture fracture energy energy G[, Gf, we obtain obtain the basic result[7] the basic result[7] : g(ao) _ S(%> Grf-- AE AE . G
=
(6)
Thus we see that that the the fracture fracture energy energy is inversely inversely proportional to the the slope slope of of the the size size effect effect regression regression Thus proportional to line )]. This provides a convenient of line [Fig. [Fig. I1(b (b)]. This provides convenient method method of of determination determination of fracture fracture energy. energy.
DETERMINATION OF R-CURVE R-CURVE FROM SIZE SIZE EFFECT EFFECT MEASUREMENTS MEASUREMENTS DETERMINATION OF FROM The size size effect effect law law is also also helpful helpful for for determining determining the R-curve[7]. R-curve[7]. In In terms terms of of the the R-curve R-curve R(c), R(c), The the which describes describes the the dependence dependence of of the the energy energy required required for for crack crack growth growth on on the the crack crack length length measured measured which from notch tip, balance condition requires that from the the notch tip, the the energy energy balance condition requires that
G(a) G(a) = = R(c) R(c)
(a=ao+c), (a = a,+c),
(7) (7)
48
Z. P. BAZANT BAZANT et et al. nl.
in which which G(a) G(a) is the the energy energy release release rate rate according according to an equivalent equivalent linear linear elastic elastic analysis. analysis. The The foregoing relation may may foregoing relation may be regarded regarded as an equilibrium equilibrium relation relation for for fracture. fracture. The The equilibrium equilibria may be stable, critical critical or or unstable. The critical critical state state is the the state state at maximum maximum load load P,,,. The condition condition of of stable, unstable. The Pmax' The stability stability of of fracture fracture is known known to have have the form form : dR(e) dG(a) df2(c) dG(a) .... ----~ 0 (stable) - -dn- > (stable) de da dr
.
(8)
(critical). = 0 (critical).
The The energy energy release release rate rate according according to linear linear elastic elastic fracture fracture mechanics mechanics has has always always the the form form G(a) G(a) = = p2g(a)/(Eb 2d), and Pzg(cc)/(Eb2d), and so for for the the maximum maximum measured measured loads loads we have have (9)
in which which g(a) g(a) is a function function available available in handbooks handbooks for for typical typical specimen specimen geometries[13] geometries[ 131and and obtainable obtainable in general by linear general by linear finite finite element element analysis. analysis. According According to eqns eqns (7)-(8), (7)-(8), failure failure occurs occurs when when the the curve curve G(a) G(a) has has the same same value value and and the the same same slope plotted slope as the the curve curve R(c). R(c). Consequently, Consequently, the the R-curve R-curve represents represents the envelope envelope of of all curves curves G{a), G(a), plotted according according to to eqn eqn (9) for for various various measured measured values values of of maximum maximum load, load, P P,,,. max • Thus, plotted for Thus, the the R-curve 1P-curve may may be obtained obtained from from size effect effect tests if if the the curve curve G(a) G(a) is plotted for each each specimen practice, a specimen size and and the the corresponding corresponding maximum maximum load, load, using using the the known known function function g(a). g(a). In practice, problem arises problem arises however however for for two two reasons: reasons : (l) (1) the test test data data exhibit exhibit statistical statistical scatter scatter and and (2) the the range range of of measured measured data data is usually usually insufficient insufficient for for determining determining the the complete complete envelope. envelope. To To avoid avoid these these difficulties, di~culties, we must must take take recourse recourse to the the size effect effect law [eqn [eqn (l)].If (l)].If the the raw, raw, scattered values of P,,, scattered values of are used used to plot the curves curves G(a) G(u) for various various specimen specimen sizes, no common common Pmax are plot the envelope envelope can can be obtained obtained [Fig. 2(b)]. Thus, Thus, smoothing smoothing of of the the measured measured data data according according to the size effect effect law is required required before the R-curve R-curve can can be determined. determined. Furthermore, Furthermore, if if the the curves curves G(a) G(a) are are before the plotted only plotted only for for a limited limited range range of of measured measured Pm,,Pmax-values, of values, determination dete~ination of the the R-curve R-curve is ambiguambiguous, as illustrated illustrated in Fig. 2(c). It It is necessary necessary to plot the curves curves G(a) G(a) for for P P,,,-values within a range range ous, plot the max-values within of of I: 1: 100 or or more, more, and and such such a range range can can hardly hardly be obtained obtained except except by generating generating the the Pmax-values P,,,-values from parameters Band been determined by regression from the the size effect effect law [eqn (1)] (l)] after after parameters B and ).0 & have have been determined by regression of of the the test test data. data. The principle, according The R-curve R-curve can can be also constructed, constructed, in principle, according to the the failure failure load load for for different different notch notch lengths lengths on on specimens specimens of of the the same same size. However, However, this this method method fails for for the the same same reason reason as mentioned; mentioned ; the the range range of of the the curve curve G(a) G(a) is insufficient, insufficient, like in Fig. Fig. 2(c). Z(c). The The R-curve R-curve can can be obtained obtained from from such such data data only only if if additional additional properties, such as the crack crack length length or or the the specimen specimen compliance, compliance, properties, such are measured. particular the measured. However, However, this has has the the drawback drawback already already stated, stated, in particular the fact fact that that load load values values which do which do not not correspond correspond to to failure failure states states must must be also also included included in the the evaluation. evaluation. It been shown[7] It has has been shown[;l that that the the R-curve R-curve obtained obtained from from test test data data smoothed smoothed by the the size effect effect law iaw in eqn eqn (1) is very very closely closely approximated approximated by the the formula formula R(e)=G,[l-!l-~~i
for
cc,,.
(10)
in which parameters em which G Grr is already already known known in advance advance from from eqn eqn (6) and and parameters c, and and n are are to be identified identified by fitting fitting of of the the envelope. envelope. Usually, Usually, as a good good approximation, approximation, n = = 3.6. It It must must be kept kept in mind mind that that the the R-curve A-curve is an approximate approximate concept. concept. In theory, theory, the the R-curves R-curves for however, experience has shown for different different specimen specimen geometries geometries should should not not be the the same; same; however, experience has shown that that in many situations the R-curves are nearly the same, while in others they are not[7]. many situations the R-curves are nearly same, while in others they are not[7]. FRACTURE TESTS TESTS OF OF ALUMINUM ALUMINUM FRACTURE
After procedure outlined been applied After initial initial success success with with concrete[7], concrete[7], the the new test test procedure outlined here here has has been applied to bent fracture been aluminum product of aluminum A16061-T651 Al 6061-T651 (a product of Alcoa Alcoa Co.). Co.). Three-point Three-point bent fracture specimens specimens have have been
49
Size effect tests tests and and fracture fracture characteristics characteristics 1000
(a)
'" Ci'" ~
(\J
Ci ~
800
/"
~
d·1l
"0 d, =3.990 In. A=1586xIO- B C=6.329xIO- B G,=70941b/in
-02 -02-'::-0-:-4-----:0"::O-----::O"::4----~O~ -04
00
04
Or‘
log log dOn) d(in 1
Fig. plot of proportional thicknesses. Fig. 6. Size Size effect effect plot of test test results results of of specimens specimens with with proportional thicknesses.
the boundary conditions, it possibly have been better better the specimen specimen does does not not match match the the surface surface boundary conditions, it might might possibly have been for of proportional to beam depth. for comparability comparability of specimens specimens of of different different sizes sizes to to use use thicknesses thicknesses proportional to beam depth. However, was not not done because the the elastic Nevertheless, in However, this this was done because elastic effects effects seem seem less less important. important. Nevertheless, in a preliminary test beam of been tested, preliminary test one one beam of different different thickness, thickness, b = = 0.75 0.75 in., in., has has been tested, and and the the results results (row (row H in based on in Table Table 1) are are still still essentially essentially in in agreement agreement with with the the values values of of aN (TVfor for the the size size effect effect law, law, based on A-E. rows rows A-E. A been also A few few tests tests have have been also carried carried out out for for a fixed fixed specimen specimen size size (d (d = = 1 in.) in.) and and different different notch notch lengths, Fin Table lengths, aid a/d = = 0.25,0.5 0.25, 0.5 and and 0.75 0.75 (rows (rows G, A and and Fin Table 1). The The fracture fracture equilibrium equilibrium curves curves for for the plotted in the measured measured maximum maximum loads loads (Table (Table 1) are are plotted in Fig. Fig. 2(c), 2(c), and and it it is clear clear that that determination determination of for as already of the the R-curve R-curve as as an an envelope envelope is totally totally ambiguous ambiguous for such such a set set of of measurements, measurements, already emphasized. be used just the emphasized. So So a single single specimen specimen size size cannot cannot be used unless unless more more than than just the maximum maximum load load is measured, measured, as is well well known. known. The been carried Northwestern University. The foregoing foregoing tests tests have have been carried out out at at Northwestern University. A further further test test series series has has National University, to been carried been carried out out at at Chonnam Chonnam National University, to check check the the effect effect of of scaling scaling in in which which the the thickness proportion to beam depth thickness is varied varied in in proportion to the the beam depth and and length. length. The The results results of of these these tests, tests, carried carried out with an next out with an Instron Instron machine, machine, are are given given in in Figs Figs 6 and and 7 where where the the maximum maximum loads loads are are indicated indicated next to points of by smoothing to the the points of aN' oN. The The fracture fracture energy energy value value obtained obtained by smoothing with with the the size size effect effect law law is for for these previous tests these tests tests G Grr = = 709 Ib/in., lb/in., which which is only only 6.0% 6.0% 'larger ‘larger than than in in the the previous tests with with constant constant be scaled specimen specimen thickness. thickness. From From this this it seems seems that that the the question question of of how how the the thickness thickness should should be scaled is not not very very important. important. CRACK MODEL CRACK BAND BAND FINITE FINITE ELEMENT ELEMENT MODEL
Fracture behavior which be closely Fracture behavior which follows follows the the size size effect effect according according to to eqn eqn (1) can can be closely modeled modeled in in finite using the band theory[15, 16]. finite element element analysis analysis using the crack crack band theory[l5, 161. The The fact fact that that this this size size effect effect law law is obtained process zone obtained from from the the hypothesis hypothesis that that the the fracture fracture process zone has has a certain certain fixed fixed effective effective width[lO] width[ IO] means be modeled band of progressively cracking means that that the the fracture fracture can can be modeled as a band of progressively cracking elements elements of of the the same same width, relationship with width, h. The The material material in in the the finite finite elements elements is assumed assumed to to exhibit exhibit a stress-strain stress-strain relationship with gradual relation gradual strain strain softening, softening, for for example example as shown shown in Fig. Fig. 8. A A simple simple triaxial triaxial constitutive constitutive relation which diagram been formulated[16]. which yields yields the the stress-strain stress-strain diagram with with strain strain softening softening shown shown in in Fig. Fig. 8 has has been formulated[lh]. The process zone The size size of of the the finite finite element, element, which which represents represents the the effective effective width width of of the the fracture fracture process zone at at the be expressed the fracture fracture front, front, may may be expressed as
h = GlWo,
h[ -__
Fig. band model Fig. 8. Finite Finite element element crack crack band model and and tensile tensile stress--strain stress--strain diagram diagram with with yield yield and and strain strain softening. softening. EF~1
26: 1-D
((12) 12)
Z. P. BAZANT BATANT et al.
54
in under the diagram in which which Wo W, represents represents the the area area under the complete complete tensile tensile stress-strain stress-strain diagram with with strain strain softening softening (Fig. plateau and slope E t in (Fig. 8). The The length length of of the the yield yield plateau and the the strain-softening strain-softening slope Et in Fig. Fig. 8 govern govern chiefly chiefly the the length length of of the the fracture fracture process process zone zone ahead ahead of of the the crack crack tip, tip, and and E, E t can can be be determined determined if if this this length length is known. known. It is important to precise element by eqn important to realize realize that that a certain certain precise element size. size, as as given given by eqn (12), (12). is required. required. an excessively large number of elements, However, in situations where this would lead to However, in situations where this would lead to an excessively large number of elements, larger larger finite elements may be used, but then curve, finite elements may be used, but then the the slope slope of of the the declining declining segment segment of of the the stress-strain stress-strain curve, and and possibly possibly also also the the peak peak stress stress value, value, must must be be adjusted adjusted so that that the the product product of of the the area area under under the the complete curve complete tensile tensile stress-strain stress-strain curve with with the the element element width width used used would would give give the the correct correct value value of of fracture the fracture energy energy Gr. Then Then the the results results are are approximately approximately the same same as as those those for for the the correct correct finite finite element element size by eqn size given given by eqn (12). (12). It It has has been been demonstrated[16] demonstrated[ 161 that that a finite finite element element model model of of this this type type gives gives the the correct correct size size effect effect by eqn curve as as shown shown in in Fig. Fig. 1 and and described described by eqn (1), (l), and and generally generally agrees agrees with with a wide wide range range of of data data curve on on non-linear non-linear fracture fracture of of various various materials, materials, including including the the effect effect of of the the notch notch length length on on the the maximum maximum load load and and on on the the variation variation of of energy energy required required for for crack crack growth growth as as a function function of of the the length length of of the the crack crack from from the the notch notch tip, tip, i.e. i.e. the the R-curve. R-curve. Note that R-curve is obtained Note that the the correct correct R-curve obtained from from the the finite finite element element model model even even though though only only one one constant constant value value of of fracture fracture energy, energy, G Gr,r, is used used in in the the model. model. This This value value of of fracture fracture energy energy coincides coincides with with the the energy energy release release rate rate only only in in those those situations situations where where the the ligament ligament cross-section cross-section is much much larger larger than process zone, process zone than the the fracture fracture process zone, so that that the the fracture fracture process zone can can fully fully develop develop and and does does not not interfere boundaries of interfere with with the the boundaries of the the structure. structure. If If the the structure structure is smaller, smaller, then then the the full full size size of of the the fracture process zone boundaries occurs, fracture process zone cannot cannot develop, develop, and and interference interference with with the the boundaries occurs. e.g. e.g. when when finite finite elements undergo strain elements that that undergo strain softening softening are are close close to to the the notch notch tip tip or or to to the the end end of of the the ligament. ligament. This This is why the why the the model, model, with with its its fixed fixed fracture fracture energy energy value, value, is capable capable of of representing representing the R-curve. R-curve. It be noted relation It may may be noted that that the the limiting limiting case case in in which which the the tensile tensile stress-strain stress-strain relation is considered considered as as elastic, elastic, followed followed by by a sudden sudden stress stress drop drop to to zero zero as as the the strength strength limit limit is reached, reached, yields yields solutions solutions which in provided there which are are approximately approximately in agreement agreement with with linear linear elastic elastic fracture fracture mechanics mechanics provided there are are about because, for about 15 or or more more finite finite elements elements across across the the ligament. ligament. This This is because, for a sudden sudden stress stress drop, drop, the the fracture process zone undergo cracking) fracture process zone (elements (elements that that undergo cracking) is always always limited limited to to a single single element element width, width, so that process zone possible within that the the fracture fracture process zone is localized localized to to the the maximum maximum possible within the the finite finite element element mesh. mesh. This This fact fact also also shows shows that that it is not not necessary necessary to to model model fracture fracture in in finite finite element element codes codes as as a line line interelement crack band and interelement crack with with a sharp sharp tip. tip. In In fact, fact, solutions solutions with with a crack crack band and a sudden sudden stress stress which which drops drops to to zero zero come come as as close close to to the the linear linear elastic elastic fracture fracture mechanics mechanics solutions solutions as as do do the the solutions solutions with crack. band approach be more with a sharp sharp interelement interelement crack. The The crack crack band approach to to fracture fracture modeling modeling seems seems to to be more flexible, program and permits modeling band of flexible, easier easier to to program and also also permits modeling fracture fracture in in any any direction direction as as a zigzag zigzag band of crack elements. So far this technique has been used widely only for concrete and geomaterials; crack elements. So far this technique has been used widely only for concrete and geomaterials; however, potential for fracture however, it it may may have have a potential for non-linear non-linear fracture modeling modeling in in metals. metals.
GENERALIZED SIZE LAW GENERALIZED SIZE EFFECT EFFECT LAW Equation (1) possible approximation to fracture. Equation (1) is the the simplest simplest possible approximation to the the size size effect effect for for non-linear non-linear fracture. The be written written in The most most general general size size effect effect law law may may be in the the form form of of asymptotic asymptotic expansion[16]: expansion[l6] :
(IN
A)~r + I + (A) ..f + (") ~2 + (A)4 / +...
= Bfy [( )~
J. 1/2r ,
(13) (13j
B, r, )"0, A10 AZ, A2. )03, obtained by where are material material constants. constants. The The first first order order approximation, approximation, obtained by where ff,, ILo, A,, Ax, ... . . are y, B, AI = A, A2 == A3 A3 == ... setting setting A1 . . = = 0, i.e. i.e. (IN
= B/y
[1 + (~)11:2r
(14) (14)
is considerably more possible shapes considerably more flexible flexible than than eqn eqn (1) for for describing describing various various possible shapes of of the the curve curve of of log be obtained by finite In log (IN dN vs log log IAwhich which may may be obtained by finite element element calculations. calculations. In fact, fact, finite finite element element experience experience indicates[17] that indicates[l7] that more more than than the the first first two two terms terms from from eqn eqn (13) (13) are are never never really really needed. needed.
Size Size effect effect tests tests and and fracture fracture characteristics characteristics
55
In the fitting fitting of of available available test data, data, however, however, coefficient coefficient r in eqn eqn (14) cannot cannot be determined determined unambiguously because because of unambiguously of the random random scatter scatter of of test test results results and and the usual usual limitations· limitations’ofof the size range. broad size range Band range. Tests Tests over over a very very broad range would would be required required to determine determine r, r, in addition addition to B and Ao. Nevertheless, Nevertheless, if used to determine 2,. if sufficient sufficient data data are given, given, then then eqn eqn (14) may may be also easily easily used determine the parameters of the fracture fracture parameters of the material. material. Either Either a non-linear non-linear optimization optimization subroutine subroutine such such as the Marquardt-Levenberg algorithm Ao, and Marquardt-Levenberg algorithm may may be used used to determine determine B, &, and r simultaneously, simultaneously, or or eqn eqn (14) may may be algebraically algebraically rearranged rearranged to the linear linear relation relation in eqn eqn (2) in which, which, instead instead of of eqn eqn (3), X=d’, X= d',
Y Y= = O"N 0;-2, 2r,,
A = C&d,)-‘,
(15) (1%
C = (BF,)- “.
A set of of r-values, r-values, e.g. 1, 1, 0.8, 0.6, 0.4, may may be selected selected and, and, for for each, each, one one finds A and and C as well as the sum of regression. Considering of squared squared deviations deviations from from the test test data data by linear linear regression. Considering the dependence dependence of 1/2,//y, lo Ao = of this sum on r, one one may may identify identify the the r-value r-value that that minimizes minimizes this sum. Then Then B = cC-ii2r/fy, = (C/A)l/'/d (C/A) “‘/df.r . For For dfd d/d‘+ co, eqn (14) reduces reduces again again to eqn (5). Thus, Thus, the fracture fracture energy energy is again again given by eqn r --+ 00, (6), (6) although although for a different different rr a different different G Grr will be obtained. obtained.
COMPARISON COMPARISON WITH WITH R-CURVES R-CURVES BASED BASED ON ON ASTM ASTM STANDARD STANDARD
According According to ASTM ASTM Standard Standard Recommendation Recommendation E561-81, E561-8 1, the R-curve R-curve is determined determined from from compliance compliance measurements measurements on on a single compact compact tension tension specimen specimen of of the type type shown shown in Fig. 9. These These tests have been also carried have been carried out out using using three three groups groups of of specimens specimens of of different different sizes (specified (specified in the the Table Table in Fig. 1). 1). The The crack crack opening opening displacement, displacement, COD, COD, was measured measured by a clip gage with an accuracy base length under w/4, mounted accuracy of of 10-IO--66 m and and base length under mounted at distance distance (0.1576 ± + O.0006)w 0.0006)~ ahead ahead of of
.s "-
.0
oel
o Thickness b,1 in. 0
02 02
"C3
b,0.75 in. 0
02 02
b'O.5 in. 0
0.4 04
02 02
0.6 06
Fig. with R-curves obtained by ASTM Fig. 9. Comparison Comparison with R-curves obtained by ASTM procedure procedure based based on on compact compact tension tension specimens specimens (solid R-curve from present method). (solid curve curve = R-curve from present method). Dimensions Dimensions (in., (in., except except s, mm) mm) Specimen Specimen
b
d d
e
ff
i
AI,A2 Al, A2 BI, Bl, B2 82 Cl, Cl, C2, c2, C3 c3
1.00 1.00 0.75 0.75 0.50 0.50
2.4 2.4 1.8 1.8 1.2 1.2
0.550 0.550 0.413 0.413 0.275 0.275
0.18 0.18 0.118 0.118 0.098 0.098
0314 0.314 0.236 0.236 0-157 0.157
2.500 2.500 1.875 1.875 1.250 1.250
Al Al
A2 A2
BI Bl
B2
CI Cl
C2 c2
C3 c3
0.998 0.998 2.004 2.004 0.819 0.819 0.13 0.13
0.994 0.994 2.004 2.004 0.803 0.803 0.13 0.13
0_748 0.748 1.496 1.496 0.628 0.628 0.13 0.13
0.744 0.744 1.496 1.496 0.628 0.628 0.13 0.13
0.498 0.498 0.976 0.976 0386 0.386 030 0.30
0.498 0.498 0.992 0.992 0.407 0.407 0.13 0.13
0.498 0.498 0_996 0.996 0.406 0.406 0.30mm 0.30 mm
t IV w
ao a0 ss
m 112
p P
qY
u
v
zz
1.2 0.9 0.9 0.6 0.6
0.650 0.650 0.488 0.488 0.325 0.325
0.20 0.20 0.15 0.15 0-10 0.10
1.2 0.9 0.9 0.6 0.6
0.500 0.500 0.375 0.375 0.250 0.250
56
Z. P. BAZANT BAZANT et et al.
the axis connecting pins (for connecting the the centers centers of of the loading loading pins (for w see Fig. 9). The The initial initial notch notch was cut cut with with a saw and, producing a fatigue and, instead instead of of producing fatigue crack, crack, a slit crack crack extending extending from from the notch notch tip was machined by a diamond paper of machined by diamond disc paper of thickness thickness 0.13 or or 0.30 mm mm (see the the table table in Fig. 1). The The crosscrosshead head speed speed of of the the machine machine was 0.2 mmjmin. mm/min. The points obtained The R-curve R-curve points obtained from from these these tests are are shown shown in Fig,. Fig,. 1. 1. They They were were obtained obtained by the the following procedure. Take representative representative points the recorded recorded graph graph of of load load P vs 2vI 221, = = COD. COD. following procedure. Take points on the Calculate Calculate the the values values of of 2v 11 Ebj Eb/PP and and find the the equivalent equivalent crack crack length length a a,e from from the the compliance compliance table table (ASTME561-SI, Kcr = Ph-1w (ASTM E561-81, Table Table 2). ThencalculateR(c) Then calculate R(c) = = K:rjE(forplane K$/E (for plane stress) stress) where where Kc, = Ph‘w 1'2(0 “‘f(t) [13, IS, ~ = ae/H'. 18, 19] 191where where < CI,.M’. f(~) = (0.866+4.64((0.S66+4.64~-13.32~2+ 14.72~3_5.6~4)[(2+~)j(1_~)3!2]. f(t) 13.32t2+ 14.72~3-5.6~4)[(2+~)/(l -
