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Effective fracture toughness of a marble-epoxy two phase composite
The International Journal of Cement Composites and L~ghtwelght Concrete, Volume 10, Number 3
August 1988
Effective fracture toughness of a marble-epoxy two phase composite A. K. Sachan* and C. V. S. Kameswara Rao ?
Synopsis Fracture behav=our of a two phase composite ~nwh=ch epoxy sand mortar =sone phase and marble aggregate is the second phase =s invest=gated The volume fract=ons of the phases are vaned over a w~de range, and the effectwe fracture toughness of the composite =s determined by testing notched beam specimens An attempt ~s made to correlate the effectwe fracture toughness with the fracture toughness values of the constituent phases and their volume fractions. Apphcation of the results obtained in the study to a problem in marble tile industry is indicated.
Keywords
Composite matenals, fracture properties, toughness, notch tests, aggregates, recycling, epoxy resins, mortars (matenal), fracture strength, strength of matenals
INTRODUCTION Capability to predact the effective fracture toughness of compos=tes =n wh=ch the constituent phases are randomly distnbuted is of sigmficant practical interest An example ~s ~n the technology of marble tiles. When tdes are sl~ced out of large marble blocks, constderable matenal goes as waste It =s a problem ~n the industry to dispose the waste marble An effective way of utlhsmg the waste is to cast large blocks by m=xmg wfth a suitable matnx and cutt=ng tdes out of =t The tiles cut out of th=s artificial marble are an econom=c alternative to natural marble t~les However, considerable problems are encountered ~nshcmg the tdes out of the artd~c~al marble. It was noted that whde cutting w~th a power dnven circular saw, the marble-matnx interface often gwes way and marble aggregate spalls off leaving ~rregular cut p~eces The ease w~th which the blocks can be sawn into t~les w~th sharp edges ~sdependent on the fracture characteristics of the artificial marble; the volume fract=ons of the marble and the matrix effect the fracture toughness of the compes~te
PRACTICAL OBSERVATIONS Actual observat=ons and field tnals mdtcated that ff the cutting tool can pass through both the matnx and the marble inclusions without the inclusions spalhng off, the desired neat and sharp edges are reahsed In other * Lecturer and t Professor of Cwd Engineering, Harcourt Butler Technological Institute, Kanpur-20B002. tnd=a Received 25 Januaw 1988 Accepted 29 Apn11988 (~) Longman Group UK Lid 1988 0262-5075/88/10304167/$02.00
words, the interface bond fadure dunng cutting should be avoided Several trials were conducted in the field w~th different matrices and It was observed that if the matnx and the inclusions were distinctly different with respect to their fracture behaviour, then the bond fadures were more frequent. This is quahtatively explained as follows If the matnx and inclusions have different fracture toughness values, a kind of shock phenomenon occurs as the saw blade proceeds from one phase to the other. The shock results in h~gher stresses, and interface failure ~nparticular, m cases where the bond ~srelatwely weak or not perfect. Thus there ~s need to develop composites where the constituent phases have comparable fracture toughness values and to study the fracture behawour of such composites Th~s bnef paper ts an attempt ~n th~s dwrect~on
EFFECTIVE FRACTURE TOUGHNESS S~gnlficant advances have been made ~n quantitatively predicting the effective mechamcal and physical properties of multlphase materials in terms of the properttes and volume fractions of the constituent matenals [1-3]. However, comparable knowledge does not exist to predict the effective fracture toughness of composites This ~s, to some extent, because of the complexity of the fracture process of composite mater~als Fracture being a Iocalised process, crack propagation can be either through the phases or through the interface boundaries ~n general. If the interface bond ~s adequately strong, a running crack passes through all the const~tutent phases and w~thout detounng around the inclusion phase, leading to sharp cut edges In such cases, the effective fracture toughness is an average measure of the fracture toughness of the composite [4, 5]
167
Effective fracture toughness of a marble-epoxy two phase composite
where
P
Y
c L,. J L
r
Sachan and Kameswara Rao
3 o7
+
14
_
11
+ 25 80 (~___)4
:?
er = (PL/4) (6/w 3)
-1
20crn
= 1 93 -
The values of K* obtained from equation (1) are plotted ~n Figure 3, the figure shows the vanatlon of the fracture toughness of the composite w~th the volume fractions of the constituent phases It ~s observed that the matnx, the marble and the composite have approx0mately same values of fracture toughness It is this type of composite which facd~tates easy shcmg into tdes with sharp edges It Ls interesting to examine the expenmental observations gwen in F~gure 3 with those predicted by a theoretical estimate proposed earher [4] and bnefly rewewed below
L =16cm, C : l c m , w = 4 c m Figure 1 The test specimen geometry
EXPERIMENTAL PROGRAMME
Notched beam specimens of s~ze 4cm x 4cm x 20cm were prepared The central crack length was 1 cm and was cut w=th a saw The matnx was epoxy sand mortar; the we=ght proport=ons being 150 10:1 respectwely for sand, epoxy and hardener The epoxy was Arald~te CY 212 and the hardener was HY 951 The marble phase was =n the form of l Omm aggregate =nclus=ons The volume fraction v of the marble vaned between 0 and 1, v = 0 corresponds to beam of pla=n epoxy mortar whde v = 1 corresponds to pla=n marble These tests were conducted on an Instron testing machine w~th a cross head d~splacement rate of 3 m m per m~nute. The loading arrangement and spec=men geometry =s shown m Figure 1 In total, 15 such samples were tested
2.0
v :O
matrix
v: 1
marble
1 'E Z
0
1.0
o o 0
~
o
0
0
o 0
o
EXPERIMENTAL RESULTS
The load deflect=on curves obtained from the tests are shown =n F=gure 2 If P =s the load at the onset of crack propagation, the fracture toughness K~ of the compos=te is gwen by [6] K~ = (rY V c
(1) 0
I
i
I
I
I
0.2
0.4
0.6
0.8
1.0
Volume
2.5 r
v=0
v=Q4
v=0.2
v =1.0
v= 0.6
1100% 2"0 j m a t n x
aS
0
'
2
L,
0
2
~
0
'
,
2
4
*
0
Deflection (turn) Figure 2 Typ=cal load deflect=on curves
168
Figure 3 Variation of K* with volume fraction (v)
100% marble
1.0
0
fractmn
2
4
0
i
I
2
4
Consider a two phase composite matenal shown schematically ~n F~gure 4 The matenal consists of components 1 and 2 A sample volume V of the matenal consists of V1 and V2, volumes of components 1 and 2 respectively so that V~ + V2 = V Denoting V2N = v, the hm~t~ng case of the composite consisting of component 1 only corresponds to v -- 0 and of component 2 only v = 1 The ~nterest here ~sto determsne the effectwe fracture toughness of K~ of the composste in terms of KDcand K2c, the values of fracture toughness of components 1 and 2 and v For s~mphc~ty, the method of sections [7] for analysing the stress ~ntens~ty factors ~s adopted. Consider a crack of length 2L sublected to ero at ~nfinlty as shown ~n F~gure 4 The stress ev on y = 0 ~s given by ~ry = (K*/ 2x/-E~)
Effective fracture toughness of a marble-epoxy two phase composite
Sachan and Kameswara Rao
Substituting for b = (K*/ero V ' ~ ' ~ 2 and integrating equation (4) with K* --~ K*, K1 --* K~cand K2 --* K2c at the onset of fracture one obtains
o5
K~ = (KIc - K2c)(1 - v) v2 + K2c
The imphcatlons of equation (5) are gwen in detail elsewhere [4], in particular ff K~c = K2c, as has been the case on the two phase composmte onvesttgated m thts paper
Y
,
,
~--2 L - - ~
K* = KIc = K2c
.--x
b
-
CONCLUSIONS The need for designing composites w~th effect0ve fracture toughness same as those of the constttuent phases is indicated It ts shown that ff the phases are of same fracture toughness, the resulting composite wtll have the same fracture toughness trrespectwe of the volume fractions. The study also prowdes partly an expenmental venficat~on of a theoretical model for multiphase matenals proposed earher
Figure 4 A crack subjected to uniform tension
K* being effectwe mode one stress mtensmty factor ~y asymptotically decreases to ero w~th x, the dmstance from the crack t~p If b ~sthe d~stance over whch this happens, then ~y = aro when x = b, th~s gwes b = (K*/ero (~/~--~-~))2 In the absence of the crack there would have been a stress ary = oro over a length 2L now occupied by the crack; the presence of the crack thus makes a force 2Lero absent whch ms compensated for by the increased stresses over the length b, and thus, the condmt~on of equHtbnum gwes b 2 / ~ y d x = 2 L~o
(2)
0
b / (K/ (2~/~'~x~x))dx = (~o L
(3)
0
If the medium consists of only one component, then the stress intensity factor K ~sa constant and equation (3) gaves the well known result K = ~ro X/(=L) In the composite under consmderation, =n a length b component 1 is present over an average length (I-v)b and component 2 over an average length vb In this case equaUon (3) reduces to (I-v)b
/ o
(6)
Thus the expenmental observattons support the predtcbons of equation (5).
.d
-T-
or
(5)
b [K1/ (2V'~'~x~x)]dx+ / [K2/ (2~,/(-T~x)]dx= ~oL (4) ( I-v)b
ACKNOWLEDGEMENT The authors are thankful to Mr Arun Kumar Gupta, Managing D~rector of Sermon Marbles, Naubastha, Kanpur for providing the marble specimens and marble aggregates used m the mvesttgat~ons
REFERENCES 1 Hashm, Z, 'Theory of mechanical behavlour of heterogeneous media', Applied Mechanics Reviews, Vol 17, No. 1, January 1964, pp 263-65 2 Paul, B., 'Prediction of elastic constants of multmphase matenals', Transactions of Amencan Institute of Mmmg and Metallurgical Engmeenng Vol 219, No 1, January 1960, pp. 36-41 3 HHI,R., 'Elastic properties of reinforced sohds, some theoretical principles', Journal of Mechamcs and Physics of Sohds, Vol 11, No 4, October 1963, pp 357-61 4 Kameswara Rao, C V S., 'A note on the fracture toughness of mult0phase materials', Engmeenng Fracture Mechamcs, Vol t8, No, 1, January 1983, pp 35-8. 5 Swamy, R N, 'Fracture mechanics applied to concrete, m" Developments ~n Concrete Technology', Vol 1, Apphed Sctence Pubhshers, London, 1978, pp 221-71 6 Agarwal, B D and Broutman, L J, 'Analys~s and performance of fibre composites', John WHey and Sons, New York, 1980, 300 pp 7 Patron, V Z and Morozov, E M , 'Elastic plastmc fracture mechantcs', Translated from Russman, Mtr Pubhshers, Moscow, 1978, 74 pp
169
August 1988
Effective fracture toughness of a marble-epoxy two phase composite A. K. Sachan* and C. V. S. Kameswara Rao ?
Synopsis Fracture behav=our of a two phase composite ~nwh=ch epoxy sand mortar =sone phase and marble aggregate is the second phase =s invest=gated The volume fract=ons of the phases are vaned over a w~de range, and the effectwe fracture toughness of the composite =s determined by testing notched beam specimens An attempt ~s made to correlate the effectwe fracture toughness with the fracture toughness values of the constituent phases and their volume fractions. Apphcation of the results obtained in the study to a problem in marble tile industry is indicated.
Keywords
Composite matenals, fracture properties, toughness, notch tests, aggregates, recycling, epoxy resins, mortars (matenal), fracture strength, strength of matenals
INTRODUCTION Capability to predact the effective fracture toughness of compos=tes =n wh=ch the constituent phases are randomly distnbuted is of sigmficant practical interest An example ~s ~n the technology of marble tiles. When tdes are sl~ced out of large marble blocks, constderable matenal goes as waste It =s a problem ~n the industry to dispose the waste marble An effective way of utlhsmg the waste is to cast large blocks by m=xmg wfth a suitable matnx and cutt=ng tdes out of =t The tiles cut out of th=s artificial marble are an econom=c alternative to natural marble t~les However, considerable problems are encountered ~nshcmg the tdes out of the artd~c~al marble. It was noted that whde cutting w~th a power dnven circular saw, the marble-matnx interface often gwes way and marble aggregate spalls off leaving ~rregular cut p~eces The ease w~th which the blocks can be sawn into t~les w~th sharp edges ~sdependent on the fracture characteristics of the artificial marble; the volume fract=ons of the marble and the matrix effect the fracture toughness of the compes~te
PRACTICAL OBSERVATIONS Actual observat=ons and field tnals mdtcated that ff the cutting tool can pass through both the matnx and the marble inclusions without the inclusions spalhng off, the desired neat and sharp edges are reahsed In other * Lecturer and t Professor of Cwd Engineering, Harcourt Butler Technological Institute, Kanpur-20B002. tnd=a Received 25 Januaw 1988 Accepted 29 Apn11988 (~) Longman Group UK Lid 1988 0262-5075/88/10304167/$02.00
words, the interface bond fadure dunng cutting should be avoided Several trials were conducted in the field w~th different matrices and It was observed that if the matnx and the inclusions were distinctly different with respect to their fracture behaviour, then the bond fadures were more frequent. This is quahtatively explained as follows If the matnx and inclusions have different fracture toughness values, a kind of shock phenomenon occurs as the saw blade proceeds from one phase to the other. The shock results in h~gher stresses, and interface failure ~nparticular, m cases where the bond ~srelatwely weak or not perfect. Thus there ~s need to develop composites where the constituent phases have comparable fracture toughness values and to study the fracture behawour of such composites Th~s bnef paper ts an attempt ~n th~s dwrect~on
EFFECTIVE FRACTURE TOUGHNESS S~gnlficant advances have been made ~n quantitatively predicting the effective mechamcal and physical properties of multlphase materials in terms of the properttes and volume fractions of the constituent matenals [1-3]. However, comparable knowledge does not exist to predict the effective fracture toughness of composites This ~s, to some extent, because of the complexity of the fracture process of composite mater~als Fracture being a Iocalised process, crack propagation can be either through the phases or through the interface boundaries ~n general. If the interface bond ~s adequately strong, a running crack passes through all the const~tutent phases and w~thout detounng around the inclusion phase, leading to sharp cut edges In such cases, the effective fracture toughness is an average measure of the fracture toughness of the composite [4, 5]
167
Effective fracture toughness of a marble-epoxy two phase composite
where
P
Y
c L,. J L
r
Sachan and Kameswara Rao
3 o7
+
14
_
11
+ 25 80 (~___)4
:?
er = (PL/4) (6/w 3)
-1
20crn
= 1 93 -
The values of K* obtained from equation (1) are plotted ~n Figure 3, the figure shows the vanatlon of the fracture toughness of the composite w~th the volume fractions of the constituent phases It ~s observed that the matnx, the marble and the composite have approx0mately same values of fracture toughness It is this type of composite which facd~tates easy shcmg into tdes with sharp edges It Ls interesting to examine the expenmental observations gwen in F~gure 3 with those predicted by a theoretical estimate proposed earher [4] and bnefly rewewed below
L =16cm, C : l c m , w = 4 c m Figure 1 The test specimen geometry
EXPERIMENTAL PROGRAMME
Notched beam specimens of s~ze 4cm x 4cm x 20cm were prepared The central crack length was 1 cm and was cut w=th a saw The matnx was epoxy sand mortar; the we=ght proport=ons being 150 10:1 respectwely for sand, epoxy and hardener The epoxy was Arald~te CY 212 and the hardener was HY 951 The marble phase was =n the form of l Omm aggregate =nclus=ons The volume fraction v of the marble vaned between 0 and 1, v = 0 corresponds to beam of pla=n epoxy mortar whde v = 1 corresponds to pla=n marble These tests were conducted on an Instron testing machine w~th a cross head d~splacement rate of 3 m m per m~nute. The loading arrangement and spec=men geometry =s shown m Figure 1 In total, 15 such samples were tested
2.0
v :O
matrix
v: 1
marble
1 'E Z
0
1.0
o o 0
~
o
0
0
o 0
o
EXPERIMENTAL RESULTS
The load deflect=on curves obtained from the tests are shown =n F=gure 2 If P =s the load at the onset of crack propagation, the fracture toughness K~ of the compos=te is gwen by [6] K~ = (rY V c
(1) 0
I
i
I
I
I
0.2
0.4
0.6
0.8
1.0
Volume
2.5 r
v=0
v=Q4
v=0.2
v =1.0
v= 0.6
1100% 2"0 j m a t n x
aS
0
'
2
L,
0
2
~
0
'
,
2
4
*
0
Deflection (turn) Figure 2 Typ=cal load deflect=on curves
168
Figure 3 Variation of K* with volume fraction (v)
100% marble
1.0
0
fractmn
2
4
0
i
I
2
4
Consider a two phase composite matenal shown schematically ~n F~gure 4 The matenal consists of components 1 and 2 A sample volume V of the matenal consists of V1 and V2, volumes of components 1 and 2 respectively so that V~ + V2 = V Denoting V2N = v, the hm~t~ng case of the composite consisting of component 1 only corresponds to v -- 0 and of component 2 only v = 1 The ~nterest here ~sto determsne the effectwe fracture toughness of K~ of the composste in terms of KDcand K2c, the values of fracture toughness of components 1 and 2 and v For s~mphc~ty, the method of sections [7] for analysing the stress ~ntens~ty factors ~s adopted. Consider a crack of length 2L sublected to ero at ~nfinlty as shown ~n F~gure 4 The stress ev on y = 0 ~s given by ~ry = (K*/ 2x/-E~)
Effective fracture toughness of a marble-epoxy two phase composite
Sachan and Kameswara Rao
Substituting for b = (K*/ero V ' ~ ' ~ 2 and integrating equation (4) with K* --~ K*, K1 --* K~cand K2 --* K2c at the onset of fracture one obtains
o5
K~ = (KIc - K2c)(1 - v) v2 + K2c
The imphcatlons of equation (5) are gwen in detail elsewhere [4], in particular ff K~c = K2c, as has been the case on the two phase composmte onvesttgated m thts paper
Y
,
,
~--2 L - - ~
K* = KIc = K2c
.--x
b
-
CONCLUSIONS The need for designing composites w~th effect0ve fracture toughness same as those of the constttuent phases is indicated It ts shown that ff the phases are of same fracture toughness, the resulting composite wtll have the same fracture toughness trrespectwe of the volume fractions. The study also prowdes partly an expenmental venficat~on of a theoretical model for multiphase matenals proposed earher
Figure 4 A crack subjected to uniform tension
K* being effectwe mode one stress mtensmty factor ~y asymptotically decreases to ero w~th x, the dmstance from the crack t~p If b ~sthe d~stance over whch this happens, then ~y = aro when x = b, th~s gwes b = (K*/ero (~/~--~-~))2 In the absence of the crack there would have been a stress ary = oro over a length 2L now occupied by the crack; the presence of the crack thus makes a force 2Lero absent whch ms compensated for by the increased stresses over the length b, and thus, the condmt~on of equHtbnum gwes b 2 / ~ y d x = 2 L~o
(2)
0
b / (K/ (2~/~'~x~x))dx = (~o L
(3)
0
If the medium consists of only one component, then the stress intensity factor K ~sa constant and equation (3) gaves the well known result K = ~ro X/(=L) In the composite under consmderation, =n a length b component 1 is present over an average length (I-v)b and component 2 over an average length vb In this case equaUon (3) reduces to (I-v)b
/ o
(6)
Thus the expenmental observattons support the predtcbons of equation (5).
.d
-T-
or
(5)
b [K1/ (2V'~'~x~x)]dx+ / [K2/ (2~,/(-T~x)]dx= ~oL (4) ( I-v)b
ACKNOWLEDGEMENT The authors are thankful to Mr Arun Kumar Gupta, Managing D~rector of Sermon Marbles, Naubastha, Kanpur for providing the marble specimens and marble aggregates used m the mvesttgat~ons
REFERENCES 1 Hashm, Z, 'Theory of mechanical behavlour of heterogeneous media', Applied Mechanics Reviews, Vol 17, No. 1, January 1964, pp 263-65 2 Paul, B., 'Prediction of elastic constants of multmphase matenals', Transactions of Amencan Institute of Mmmg and Metallurgical Engmeenng Vol 219, No 1, January 1960, pp. 36-41 3 HHI,R., 'Elastic properties of reinforced sohds, some theoretical principles', Journal of Mechamcs and Physics of Sohds, Vol 11, No 4, October 1963, pp 357-61 4 Kameswara Rao, C V S., 'A note on the fracture toughness of mult0phase materials', Engmeenng Fracture Mechamcs, Vol t8, No, 1, January 1983, pp 35-8. 5 Swamy, R N, 'Fracture mechanics applied to concrete, m" Developments ~n Concrete Technology', Vol 1, Apphed Sctence Pubhshers, London, 1978, pp 221-71 6 Agarwal, B D and Broutman, L J, 'Analys~s and performance of fibre composites', John WHey and Sons, New York, 1980, 300 pp 7 Patron, V Z and Morozov, E M , 'Elastic plastmc fracture mechantcs', Translated from Russman, Mtr Pubhshers, Moscow, 1978, 74 pp
169