J. BIOMED. MATER. RES.
VOL. 9, PP. 423439 (1975)
Slow Crack Growth in Acrylic Bone Cement P E T E R W. R . BEAUMONT and ROBERT J. YOUNG, Department of Engineering, University of Cambridge, Cambridge, Great Britain
Summary Slow crack growth in Perspex acrylic sheet (PMMA) and Simplex acrylic bone cement in air and water has been studied from a fracture mechanics viewpoint. It has been found that the crack velocity, V , for each material depends upon the intensification of stress at the tip of the crack. Experimental measurements have been made of V as a function of the stress intensity factor, K , a t the crack tip, and a derived V , K relationship has been used to predict the times-tofailure of components made from PMMA and Simplex cement. Direct measurements of time-to-failure for PMMA have shown that the predicted values give a conservative estimate of the structural lifetime of the material.
INTRODUCTION Most structural materials, perhaps all, contain defects that exist as microvoids or small cracks. The flaws may be formed during the processing of the material or can develop during service of the component. Under certain stress and environmental conditions, these imperfections can extend slowly by the coalescence of voids and microcrack progression. This occurrence of subcritical (stable) crack growth is termed “corrosion fatigue” or “stress corrosion cracking” and is the result of sustained loading (static or cyclic) combined with the action of “hostile” species in the environment. The phenomenon of slow crack growth prior t o fast (unstable) fracture leads t o a time-dependence of strength of the material. The successful exploitation of engineering materials therefore requires a detailed understanding of the time-dependent failure characteristics of the material so that accurate failure predictions can be made. Fracture involves two time-dependent processes, crack initiation and crack propagation. The structural lifetime of a component can 423 @ 1975 by John Wiley & Sons, Inc.
424
BEAUMONT AND YOUNG
therefore be considered simply as the sum of the times for completion of the two stages of failure. This means that is the time t o where #f is the time-to-failure of the component, initiate a crack of size, a,, and t,b, is the time that it takes for the crack t o propagate t o some critical size a,, a t which point fast facture occurs. If it is assumed that small cracks are already present in the structure, then the crack propagation stage of fracture can be considered dominant in the failure process. For crack propagation to occur, the surfaces a t the crack tip must be separated by a critical amount, L i t , that depends upon the material, the applied stress and the environmental conditions. To open the crack, a certain amount of work, G, must be done a t the crack tip and this is approximately given by1 3 G 1? - a 2 ~ , 6 8
where uy is the stress a t the crack tip which creates a plastic zone of size R, to accommodate the crack opening displacement, 6, as shown in Figure 1. The relationship between R,, 6, and uy is given by2 (under plane strain conditions)
where E is the Young’s modulus of the material. The size of the plastic zone, R,, is determined by the intensification of stress a t the crack tip which is described by the parameter, K , the stress intensity factor :
Combining eqs. (2), (3), and (4),then
K2
3
= -
8
T U , ~=
EG
(5)
The stress intensity factor, K , can also be described in terms of the applied stress, u,, and the crack size, ai. It can be shown that3
K =
U, 1/,*ai
(6)
ACRYLIC BONE CEMENT CRACK GROWTH
425
Fig. 1. A solid containing an internal flaw of size, 2a, showing the formation of a plastic zone, R, a t both tips, when subjected to a stress, ma, applied perpendicular to the direction of crack extension.
where a! is a constant that depends upon crack size and component geometry. For sharp cracks in infinitely large plates a equals unity. When 6 = fjcrit, then G = G , and K = K , and fast fracture occurs. The applied stress u, has reached the fracture stress, uf,and is given by
--
KIC
Uf = ___
l/ruaa,
(7)
where a, is the critical crack length. Klc is called the plane strain fracture toughness of the material.* * The subscript I denotes the crack opening mode of failure.
426
BEAUMONT AND YOUNG
Following eq. ( 5 ) , the relationship between KICand Grc (“toughness” of the material) is
KIC =
(8)
and so
assumed t o equal unity). This approach has been used4 to investigate the growth of cracks in a stainless steel used for prostheses in total hip replacements. However, the acrylic bone cement used in conjunction with the metal prosthesis is inherently much weaker and cracking of the acrylic cement has been observed in some cases.5 Under certain environmental and stress conditions, this subcritical (slow) crack growth occurs a t values of a, < us (or K1 < K1c) and the dynamics of crack propagation can be described uniquely by the stress intensity factor, K , a t the crack tip and a corresponding crack velocity, V (= da/dt under constant stress), shown schematically in Figure 2. The curve may in general be divided into 3 regions of crack growth. Regions I and I1 result from stress corrosion and depend on the diffusion of hostile species to the advancing crack tip, while region I11 represents mainly mechanical failure and is independent of environment. (ais
4
Fig. 2. A schematie representation of a crack velocity ( V ) , stress intensity factor ( K ) diagram showing three regimes of slow (subcritical)crack growth.
ACRYLIC BONE CEMENT CRACK GROWTH
427
The acrylic cements based on poly(methy1 methacrylate) (PMMA) and used in total hip and knee replacements invariably contain small pockets of trapped air or monomer and exhibit regions of incomplete mixing between polymerized methyl methacrylate (MMA) monomer and the PMMA powder (see Fig. 10a,b). These porous regions can become highly stressed due to the stress concentrating effects of the voids and may act as sites for the nucleation of small cracks. The formation of these microcracks may take place during the cure of the acrylic cement (by thermal stressing) or when the prosthesis becomes loaded in use. The growth of these small cracks can lead to fracture of the cement and loosening of the prosthesis and eventual failure of the prosthesis-cement interface. This paper describes a study into the phenomenon of slow crack growth in Simplex (surgical Simplex acrylic cement supplied by North Hill Plastics, London) and is analyzed in terms of V , K relationships and predicted times-to-failure of the cement. The effects of fabrication pressure (which controls the void content) on the fracture toughness, K I ~and , water upon the kinetic aspects of environment-sensitive cracking are also demonstrated.
EXPERIMENTAL Materials The acrylic bone cement (surgical Simplex) was supplied in the form of PMMA powder and MMA liquid monomer. The mixing operation followed the manufacturer’s instructions using a PMMA: MMA ratio of 2:l. When the resulting dough had lost its tackiness, the acrylic cement was placed between two aluminum plates and compressed to a thickness of about 6 mm. Various pressures were used between about 70 kN/m2 (10 psi) and 700 kN/m2 (100 psi) in order to produce samples with different amounts of porosity. Pressure was maintained on the samples during curing until the heat generated began to diminish and the sample started to cool down.
Specimen Design and Analysis Double-torsion (DT) specimens were machined from the slabs in the form of rectangular plates (Fig. 3). The specimens were edgenotched at one end and shallow center-grooves were machined from
428
BEAUMONT AND YOUNG P
Fig. 3. A schematic diagram of the double-torsion specimen and loading geometry. The shaded portion represents the fracture surface area.?
the notches along one face of each specimen. During the D T test the sample is supported on two parallel rollers, with the center-groove on the bottom face, and loaded using two hemispheres a t the notched end. A crack is forced t o propagate from the root of the notch and is guided by the shallow groove. It can be shown6 that the stress intensity factor,. KI a t the crack tip is independent of crack length in the DT specimen and for an elastic material is given by
Kr
=
+ ”’)“’
PW,( 3(1 WPt,
where P is the applied load, W , is the moment arm, v is Poisson’s ratio, W is the bar width, t is the bar thickness and t, is the plate thickness in the plane of the crack. If the plate is loaded and held a t a constant displacement ( d y l d t = 0), the crack growth rate, V , is given by7
where Pi,, is the initial (or final) applied load, P is the instantaneous load, ai,, is the initial (or final) crack length, and B and C are constants relating to the slope and intercept of a compliance calibration curve of the DT specimen, (y/P)f(a), respectively. The crack velocity can therefore be found directly from the rate of load relaxation, (dP/dt),, providing the initial or final values of load (Pi,,) and crack size (ai,f) are known. At any particular value of crack velocity, the stress intensity factor may be calculated
ACRYLIC BONE CEMENT CRACK GROWTH
429
from the instantaneous load, P , and specimen dimensions using eq. (10).
Procedure The test device which is illustrated schematically in Figure 3 is used in an Instron loading machine. Tests were carried out in air (relative humidity 60 f 10%) and in distilled water a t 20 f 2°C. The specimen is rapidly loaded ( d y l d t = 2 X lop4m/sec) to a preselected load ( P J and the machine is then turned off. Under conditions of constant displacement ( d y l d t = 0), the load relaxes as the crack extends. A continuous load relaxation curve is recorded as a function of time on the Instron chart recorder. When K I (or P ) has decreased t o a constant value, K O ,corresponding t o crack arrest, a second test can then be performed using the same specimen. It is possible therefore to carry out 2 or 3 experiments on the same testpiece thereby reducing the scatter of results normally found with acrylic cement. However, the viscoelastic nature of the acrylic cement can cause it to (‘creep’’ or relax under sustained loading conditions. This will then lead to erroneous determinations of the V , K diagram. The source of error can be reduced by preloading the sample to K Ofor about 30 min before carrying out the experiment. By this means, some of the molecular relaxation can be eliminated before carrying out the experiment. Nevertheless, optical measurements of crack growth rates were made regularly as a check on the validity of the analysis for these viscoelastic materials. Good agreement was found between velocities measured by the load relaxation method and those measured directly which confirmed confidence in the former method.
RESULTS AND DISCUSSION Effect of Fabrication Pressure upon KIc The plane strain fracture toughness, K I ~depends , on the fabrication pressure which in turn controls the void content of the acrylic cement. Figure 4 shows experimental KICdata obtained for Simplex cement in air using single-edge notched beam samples* and varying the fabrication pressure. I n order to isolate the effects of porosity upon subcritical crack growth from those due to the environment, all DT specimens were machined from Simplex cement fabricated using a pressure of 700 kN/m2 (100 psi).
BEAUMONT AND YOUNG
430
I
I
I
I
I
I
I
I/-[--
SIMPLEX CEMENT 16-
I
I
-
x I€
3
2
Y
“1’I,I l2
, ,
I0
0
I00
200
I 300
Fabrication Pressure
I
I
1
I
400
500
600
700
I kN
m2 1
Fig. 4. A plot of plane strain fracture toughness, KZC,of Simplex cement in air a t 20°C as a function of fabrication pressure, measured using single edgenotched beam samples loaded in slow 3-point bending.*
V, K , Measurements Slow crack growth has been observed in PMMA and Simplex cement. Tests were carried out in air and water to study the stable, subcritical crack growth under potentially hostile environments. The relationships between V and K determined for PMMA and Simplex are shown in Figures 5 and 6. All the plots are similar in shape, (the slope n = 25) but they differ in position along the K I axis. The data for Simplex in air shows fast fracture t o occur a t KIc close to 1.8 MN/rn3I2 compared with 1.4 MN/m3I2for PMMA. The value of KIc for Simplex measured in slow 3-point bending was 1.65 MN/m3/2 ( d y / d t = 2 X m/sec) which is slightly less than KIc obtained with the DT specimen (Fig. 4). It is difficult to be absolutely precise about these values because of the inherent scatter found from sample to sample, although it is important t o note that the direct measurement of crack velocity in the D T test agrees well with the values obtained using the load relaxation technique.
ACRYLIC BONE CEMENT CRACK GROWTH
431
da/dt [ m e 2 I
I
I
I
I
I
-3
I0
I.
I
I
I
I I
I
o4
I
I
d5
lo6
I I I I
I
I I
I
I
I I
I
I
I
I
I
1 I
I
I
I KIC I
[air 1
2 J3 8
1
I
-7 10
I
I
I 10
I 12 K~ [ M N G 1~
I [water] I KIC
I
1
14
16
I
Fig. 5. A V , K diagram for PMMA in air and water obtained using the DT test. The solid points represent direct optical measurements of crack velocity. (Two different thickness plates were used, 3 and 6 mm.)
It can be seen that the presence of water in general shifts the curves to the right. For PMMA, water appears t o eliminate ~ the fracture completely any slow crack growth a t K I < K I and toughness is increased to 1.7 MN/m3/2. Large “bunches” of crazes were observed a t the crack tip as soon as the PMMA was loaded in I/, K
BEAUMONT AND YOUNG
432 Imso;’
1 I
I
I
I
I 1
1
1
’
I
S I M P L E X C E M E N T 118°C1
Fig. 6. A V , K diagram for Simplex cement in air and water obtained using the D T test. The two solid curves for air were determined using two separate specimens. (The solid points represent direct optical measurements of crack velocity.)
water. This “plastic” zone ahead of the crack tip effectively blunts out the propagating crack by absorbing the stored elastic strain energy in the specimen that would have been dissipated during the slow crack growth stages of failure (eqs. (4), and (5) show that G is proportional to R,).
ACRYLIC BONE CEMENT CRACK GROWTH
433
There appears to be a small dependence of the V , K curve for PMMA upon specimen geometry but this may be due t o the variation of crack geometry from specimen to specimen. The scatter between the results for Simplex may be due to the heterogeneous nature of this material, although an order of magnitude difference in V a t a particular value of Kr is not considered significant. The accuracy of the data may be appreciated by comparing times-to-failure, #f, a t constant stress predicted using the V , K diagram with the measured times-to-failure in the next section.
Time-to-Failure Predictions The relationship between V and K can be used for purposes of fracture safe design in situations where the life-time of a component is determined by subcritical (slow) crack growth. It has been shown that in PMMA and Simplex, V is a unique function of K and therefore from the knowledge of Krc and the desired working stress, uw, (or Krz), we can calculate the time-to-failure of the component in question as a function of uw/uf (or K ~ , l K r c ) . The time-to-failure, $fl a t constant stress is simply the integral of the V , K diagram;
*,
=
Iac a.
(l/V) da
(12)
Since for finite size specimens
Kr (where Y then
=
=
Yua l/G
(13)
d?rLyand depends upon the geometry of the specimeng),
This equation is plotted for PMMA in air in Figure 7 , using a n applied stress, ua, of 2.7 MN/m2, Y equal to 4 and a KIC of 1.4 MN/m3/*. The solid points in Figure 7 are experimental time-tofailure data for tensile specimens containing edge cracks loaded t o various levels of stress intensity (Kr). The measured data shows a high degree of scatter but within experimental error it can be seen that all the points lie on or above the theoretical line. It shows that it is possible to integrate the V, K curves and obtain a conservative estimate of the lifetime of the component.
BEAUMONT AND YOUNG
434
lmin
Ihr
ldoy
lwrck
1.5
-
14
2
0
' E
-f
l 3
y"
1 2
U 0 I-
1 '
U
>
I0
IIn
z
0 9
+-W
-z
0 8
In In
w
0 7 In
I
0.6
100
10'
I
I
I
I02
lo3
TIME TO
FAILURE
$
lo4
I
to5
1 I06
[secs]
Fig. 7. Experimental values of times-to-failure for PMMA in air as a function of the initial stress intensity factor, K I , measured using single edge-notched tensile specimens. The solid curve shows the predicted times-to-failure obtained using eq. (14)where ua is 2.7 MN/m* and Y equals 4.
The scatter is probably due to the variation in crack tip geometry caused by the notching procedure. ( L Sharp" cracks were obtained by slowly pushing razor blades into the bottom of a sawcut and it was noticed that in some cases the cracks tended t o bifurcate although such samples were generally discarded. The shortest times-tofailure will be for molecularly sharp cracks. Longer times-to-failure will be obtained for cracks that have bifurcated. Having shown that times-to-failure can be obtained by the integration of the V , K diagram for PMMA, the V , K diagrams for Simplex in air and water were integrated in a similar way using eq. (14). The times-to-failure curves are plotted in Figure 8 again for u, equal to 2.7 MN/m2 and Y equal to 4. (These two values have been taken for convenience but the values of U, and Y will depend on the working stress, crack size and component geometry.)
ACRYLIC BONE CEMENT CRACK GROWTH
I
0 5 I 00
101
I 102
I I
TIME
I
o3
*.
to4
TO FAILURE
I
I
,05
,06
435
I I
o7
[I~CS]
Fig. 8. Predicted times-to-failure for Simplex cement in air and water and PMMA in air as a function of initial stress intensity factor, K I . The values of a,, and Y in eq. (14) are 2.7 MN/m2 and 4,respectively. The shaded areas represent c for materials in air and water. the scatter in measured K ~ values
The analytical derived curves shown in Figure 9 illustrate the usefulness of the stress intensity factor for fracture safe design of a Simplex cement structure being used in air and water. The timeto-failure as a function of KI, is plotted at four different stress levels. The significant effect of applied stress (or working stress) on time-tofailure is apparent a t values of KI,/Krc < 1. It can be seen that for each stress level there are three types of behavior. At values of KI close to KIc, failure only results after some incubation period which is a function of the applied stress. In the upper range (Kr 0.7 KIc), the time-to failure is slightly less sensitive to the value of KI. At low values of Kr, the failure time increases greatly with a small decrease in stress intensity level. At any particular value of KI < KIc, failure occurs after an incubation period that can either be accounted for as the time for crack initiation or as a function of the crack growth kinetics (eq. 1) as we have shown. We have therefore presented a conservative fracture safe design prediction by ignoring the possibility of a finite crack initiation time (+%= 0).
-
BEAUMONT AND YOUNG
436
I m,n
Ih0“l
-
#
-I’” a 0
z
r w
5
10
y1
w
ir y/ c
0 5 100
101
102
lo4
103 TIME
TO
FAILURE
+,
,05
lo7
I08
[SPCS]
Fig. 9. Theoretical times-to-failure curves as a function of the initial stress intensity factor Kr for Simplex cement in air and water a t 4 different stress levels, assuming Y = 4. The shaded areas represent the spread in measured K J Cvalues.
Electron Microscopy Fracture surfaces of both PMMA and Simplex were examined in a Stereoscan scanning electron microscope. The surfaczs were prepared for observation by depositing thin layers (- 100 A thick) of carbon and gold-palladium alloy on t o the fracture surfaces. Figure 10a, and b shows fracture surfaces of Simplex in air. The surface is rough and spherical particles can be seen protruding from it. There are also areas containing spherical pits where particles have been pulled out during crack propagation. I n other areas there are voids which were probably caused by pockets of trapped air or monomer. This may be contrasted with the Stereoscan micrograph of PMMA (Fig. 1Oc) a t approximately the same magnification which appears to be “perfectly” smooth. Figure 10a, b shows a mixed mode of failure in the Simplex cement with areas of transgranular and intergranular fracture. I n intergranular failure, the slowly moving crack has followed paths of weakness between spherical PMMA particles that have not completely fused together. This mode of failure may account for the rougher fracture surface appearance of the Simplex compared t o that
ACRYLIC BONE CEMENT CRACK GROWTH
437
Fig. 10. (a) and (b) Scanning electron microscopy fractographs of Simplex in air; (c) PMMA in air, showing PMMA particles and porosity in the acrylic cement and the smooth fracture surface of PMMA.
438
BEAUMONT AND YOUNG
of PMMA, although crack propagation through strongly bonded PMMA spheres also produces a fairly irregular fracture surface. Similar observations have been made by Kusy and Turner,1o for a dental material based on PMMA that failed by fast crack propagation. Paradoxically, the Simplex cement is strengthened against cracking by these weak interfaces between PMMA spheres which cause the crack front to deviate during crack progression. This is a possible explanation for the displacement of the V , K curve for Simplex cement to the right of the curve for PMMA, i.e., a t a particular value of K I < Krc, the crack moves more slowly through the acrylic cement than in the PMMA. I n terms of energy, a t a particular crack velocity, the crack driving force GI is greater in Simplex than in PMMA.
Practical Implications It is important to consider what significance the results have from a practical point of view. The results were obtained a t 20°C but it is probably more important t o consider the behavior of the material a t 37°C. It has been shown" that the crack velocity in Perspex is greater for a given K I value when the temperature is increased. This means that the times-to-failure will be less a t higher temperatures and an upper limit has been calculated here. During a, total hip replacement it is highly likely that blood will be mixed with the cement. It is also possible that the material may have folds which have not completely welded together. Both of these will act as internal flaws in the structure. If an estimate of the size of these flaws is known it is possible t o use this as the initial crack size a, in the calculation of Kri (eq. (13)). The effects of trapped blood and temperature on the time-dependence of fracture stress of Simplex acrylic bone cement is currently under investigation.
CONCLUSIONS A technique developed for measuring subcritical crack growth in brittle materials has been used successfully in determining V , K relationships for PMMA and Simplex acrylic bone cement in air and water. There is a conservative agreement between experimentally measured times-to-failure and values of #f obtained using the V , K diagram for PMMA in air and it strengthens the validity of this
ACRYLIC BONE CEMENT CRACK GROWTH
439
approach for investigating further subcritical crack growth phenomena in these kinds of materials. The V , K diagrams can also be used in the analysis of slow crack growth mechanisms and these initial results encourage their use to investigate further the effect of microstructure and environment on the fracture processes and times-to-failure in greater detail. One of the authors (P.W.R.B.) would like to acknowledge the support of a U.S. Public Health Service contract a t the University of California a t Los Angeles during part of this work, and the valuable discussions and encouragement from Professor Harlan C. Amstutz at UCLA during this period. One of the authors (R.J.Y.) would like to acknowledge the Master and Fellows of St. John’s College, Cambridge for support in the form of a Research Fellowship. P.W.R.B. wishes to thank the Royal Society of London for a travel award to the 6th Annual International Biomaterials Symposium (1974), a t Clemson University, S. Carolina, U.S.A., where this paper was presented. Both would like to thank Mr. L. Shadbolt of North Hill Plastics for supplying them with Simplex surgical cement.
References 1. A. H. Cottrell, Proc. Roy. SOC.,A276, 1 (1963). 2. F. A. McClintock and G. R. Irwin, “Fracture Toughness Testing,” ASTM STP 381, (1965). 3. P. C. Paris and G. C. Sih, ASTM STP 381 (1965). 4. K. R. Wheeler and L. A. James, J. Biomed. Mater. Res., 5 , 267 (1971). 5. Harlan C. Amstuts and K. Marchoff: private communication. 6. D. P. Williams and A. G. Evans, Mat. Res. and Standards (to be published). 7. A. G. Evans, J. Mat. Sci., 7 , 1137 (1972). 8. J. Alezska, Materials Department, UCLA; unpublished data. 9. J. E. Srawley and W. F. Brown, “Plane Strain Crack Toughness Testing,” ASTM STP 410, (1966). 10. R. B. Kusy and D. T. Turner, J. Biomed. Mater. Res., 8, 185 (1974). 11. G. P. Marshall, L. H. Coutts, and J . G. Williams, J. Mat. Sci., 9,1409 (1974).
Received September 19, 1974 Revised November 25, 1974
VOL. 9, PP. 423439 (1975)
Slow Crack Growth in Acrylic Bone Cement P E T E R W. R . BEAUMONT and ROBERT J. YOUNG, Department of Engineering, University of Cambridge, Cambridge, Great Britain
Summary Slow crack growth in Perspex acrylic sheet (PMMA) and Simplex acrylic bone cement in air and water has been studied from a fracture mechanics viewpoint. It has been found that the crack velocity, V , for each material depends upon the intensification of stress at the tip of the crack. Experimental measurements have been made of V as a function of the stress intensity factor, K , a t the crack tip, and a derived V , K relationship has been used to predict the times-tofailure of components made from PMMA and Simplex cement. Direct measurements of time-to-failure for PMMA have shown that the predicted values give a conservative estimate of the structural lifetime of the material.
INTRODUCTION Most structural materials, perhaps all, contain defects that exist as microvoids or small cracks. The flaws may be formed during the processing of the material or can develop during service of the component. Under certain stress and environmental conditions, these imperfections can extend slowly by the coalescence of voids and microcrack progression. This occurrence of subcritical (stable) crack growth is termed “corrosion fatigue” or “stress corrosion cracking” and is the result of sustained loading (static or cyclic) combined with the action of “hostile” species in the environment. The phenomenon of slow crack growth prior t o fast (unstable) fracture leads t o a time-dependence of strength of the material. The successful exploitation of engineering materials therefore requires a detailed understanding of the time-dependent failure characteristics of the material so that accurate failure predictions can be made. Fracture involves two time-dependent processes, crack initiation and crack propagation. The structural lifetime of a component can 423 @ 1975 by John Wiley & Sons, Inc.
424
BEAUMONT AND YOUNG
therefore be considered simply as the sum of the times for completion of the two stages of failure. This means that is the time t o where #f is the time-to-failure of the component, initiate a crack of size, a,, and t,b, is the time that it takes for the crack t o propagate t o some critical size a,, a t which point fast facture occurs. If it is assumed that small cracks are already present in the structure, then the crack propagation stage of fracture can be considered dominant in the failure process. For crack propagation to occur, the surfaces a t the crack tip must be separated by a critical amount, L i t , that depends upon the material, the applied stress and the environmental conditions. To open the crack, a certain amount of work, G, must be done a t the crack tip and this is approximately given by1 3 G 1? - a 2 ~ , 6 8
where uy is the stress a t the crack tip which creates a plastic zone of size R, to accommodate the crack opening displacement, 6, as shown in Figure 1. The relationship between R,, 6, and uy is given by2 (under plane strain conditions)
where E is the Young’s modulus of the material. The size of the plastic zone, R,, is determined by the intensification of stress a t the crack tip which is described by the parameter, K , the stress intensity factor :
Combining eqs. (2), (3), and (4),then
K2
3
= -
8
T U , ~=
EG
(5)
The stress intensity factor, K , can also be described in terms of the applied stress, u,, and the crack size, ai. It can be shown that3
K =
U, 1/,*ai
(6)
ACRYLIC BONE CEMENT CRACK GROWTH
425
Fig. 1. A solid containing an internal flaw of size, 2a, showing the formation of a plastic zone, R, a t both tips, when subjected to a stress, ma, applied perpendicular to the direction of crack extension.
where a! is a constant that depends upon crack size and component geometry. For sharp cracks in infinitely large plates a equals unity. When 6 = fjcrit, then G = G , and K = K , and fast fracture occurs. The applied stress u, has reached the fracture stress, uf,and is given by
--
KIC
Uf = ___
l/ruaa,
(7)
where a, is the critical crack length. Klc is called the plane strain fracture toughness of the material.* * The subscript I denotes the crack opening mode of failure.
426
BEAUMONT AND YOUNG
Following eq. ( 5 ) , the relationship between KICand Grc (“toughness” of the material) is
KIC =
(8)
and so
assumed t o equal unity). This approach has been used4 to investigate the growth of cracks in a stainless steel used for prostheses in total hip replacements. However, the acrylic bone cement used in conjunction with the metal prosthesis is inherently much weaker and cracking of the acrylic cement has been observed in some cases.5 Under certain environmental and stress conditions, this subcritical (slow) crack growth occurs a t values of a, < us (or K1 < K1c) and the dynamics of crack propagation can be described uniquely by the stress intensity factor, K , a t the crack tip and a corresponding crack velocity, V (= da/dt under constant stress), shown schematically in Figure 2. The curve may in general be divided into 3 regions of crack growth. Regions I and I1 result from stress corrosion and depend on the diffusion of hostile species to the advancing crack tip, while region I11 represents mainly mechanical failure and is independent of environment. (ais
4
Fig. 2. A schematie representation of a crack velocity ( V ) , stress intensity factor ( K ) diagram showing three regimes of slow (subcritical)crack growth.
ACRYLIC BONE CEMENT CRACK GROWTH
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The acrylic cements based on poly(methy1 methacrylate) (PMMA) and used in total hip and knee replacements invariably contain small pockets of trapped air or monomer and exhibit regions of incomplete mixing between polymerized methyl methacrylate (MMA) monomer and the PMMA powder (see Fig. 10a,b). These porous regions can become highly stressed due to the stress concentrating effects of the voids and may act as sites for the nucleation of small cracks. The formation of these microcracks may take place during the cure of the acrylic cement (by thermal stressing) or when the prosthesis becomes loaded in use. The growth of these small cracks can lead to fracture of the cement and loosening of the prosthesis and eventual failure of the prosthesis-cement interface. This paper describes a study into the phenomenon of slow crack growth in Simplex (surgical Simplex acrylic cement supplied by North Hill Plastics, London) and is analyzed in terms of V , K relationships and predicted times-to-failure of the cement. The effects of fabrication pressure (which controls the void content) on the fracture toughness, K I ~and , water upon the kinetic aspects of environment-sensitive cracking are also demonstrated.
EXPERIMENTAL Materials The acrylic bone cement (surgical Simplex) was supplied in the form of PMMA powder and MMA liquid monomer. The mixing operation followed the manufacturer’s instructions using a PMMA: MMA ratio of 2:l. When the resulting dough had lost its tackiness, the acrylic cement was placed between two aluminum plates and compressed to a thickness of about 6 mm. Various pressures were used between about 70 kN/m2 (10 psi) and 700 kN/m2 (100 psi) in order to produce samples with different amounts of porosity. Pressure was maintained on the samples during curing until the heat generated began to diminish and the sample started to cool down.
Specimen Design and Analysis Double-torsion (DT) specimens were machined from the slabs in the form of rectangular plates (Fig. 3). The specimens were edgenotched at one end and shallow center-grooves were machined from
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Fig. 3. A schematic diagram of the double-torsion specimen and loading geometry. The shaded portion represents the fracture surface area.?
the notches along one face of each specimen. During the D T test the sample is supported on two parallel rollers, with the center-groove on the bottom face, and loaded using two hemispheres a t the notched end. A crack is forced t o propagate from the root of the notch and is guided by the shallow groove. It can be shown6 that the stress intensity factor,. KI a t the crack tip is independent of crack length in the DT specimen and for an elastic material is given by
Kr
=
+ ”’)“’
PW,( 3(1 WPt,
where P is the applied load, W , is the moment arm, v is Poisson’s ratio, W is the bar width, t is the bar thickness and t, is the plate thickness in the plane of the crack. If the plate is loaded and held a t a constant displacement ( d y l d t = 0), the crack growth rate, V , is given by7
where Pi,, is the initial (or final) applied load, P is the instantaneous load, ai,, is the initial (or final) crack length, and B and C are constants relating to the slope and intercept of a compliance calibration curve of the DT specimen, (y/P)f(a), respectively. The crack velocity can therefore be found directly from the rate of load relaxation, (dP/dt),, providing the initial or final values of load (Pi,,) and crack size (ai,f) are known. At any particular value of crack velocity, the stress intensity factor may be calculated
ACRYLIC BONE CEMENT CRACK GROWTH
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from the instantaneous load, P , and specimen dimensions using eq. (10).
Procedure The test device which is illustrated schematically in Figure 3 is used in an Instron loading machine. Tests were carried out in air (relative humidity 60 f 10%) and in distilled water a t 20 f 2°C. The specimen is rapidly loaded ( d y l d t = 2 X lop4m/sec) to a preselected load ( P J and the machine is then turned off. Under conditions of constant displacement ( d y l d t = 0), the load relaxes as the crack extends. A continuous load relaxation curve is recorded as a function of time on the Instron chart recorder. When K I (or P ) has decreased t o a constant value, K O ,corresponding t o crack arrest, a second test can then be performed using the same specimen. It is possible therefore to carry out 2 or 3 experiments on the same testpiece thereby reducing the scatter of results normally found with acrylic cement. However, the viscoelastic nature of the acrylic cement can cause it to (‘creep’’ or relax under sustained loading conditions. This will then lead to erroneous determinations of the V , K diagram. The source of error can be reduced by preloading the sample to K Ofor about 30 min before carrying out the experiment. By this means, some of the molecular relaxation can be eliminated before carrying out the experiment. Nevertheless, optical measurements of crack growth rates were made regularly as a check on the validity of the analysis for these viscoelastic materials. Good agreement was found between velocities measured by the load relaxation method and those measured directly which confirmed confidence in the former method.
RESULTS AND DISCUSSION Effect of Fabrication Pressure upon KIc The plane strain fracture toughness, K I ~depends , on the fabrication pressure which in turn controls the void content of the acrylic cement. Figure 4 shows experimental KICdata obtained for Simplex cement in air using single-edge notched beam samples* and varying the fabrication pressure. I n order to isolate the effects of porosity upon subcritical crack growth from those due to the environment, all DT specimens were machined from Simplex cement fabricated using a pressure of 700 kN/m2 (100 psi).
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-
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, ,
I0
0
I00
200
I 300
Fabrication Pressure
I
I
1
I
400
500
600
700
I kN
m2 1
Fig. 4. A plot of plane strain fracture toughness, KZC,of Simplex cement in air a t 20°C as a function of fabrication pressure, measured using single edgenotched beam samples loaded in slow 3-point bending.*
V, K , Measurements Slow crack growth has been observed in PMMA and Simplex cement. Tests were carried out in air and water to study the stable, subcritical crack growth under potentially hostile environments. The relationships between V and K determined for PMMA and Simplex are shown in Figures 5 and 6. All the plots are similar in shape, (the slope n = 25) but they differ in position along the K I axis. The data for Simplex in air shows fast fracture t o occur a t KIc close to 1.8 MN/rn3I2 compared with 1.4 MN/m3I2for PMMA. The value of KIc for Simplex measured in slow 3-point bending was 1.65 MN/m3/2 ( d y / d t = 2 X m/sec) which is slightly less than KIc obtained with the DT specimen (Fig. 4). It is difficult to be absolutely precise about these values because of the inherent scatter found from sample to sample, although it is important t o note that the direct measurement of crack velocity in the D T test agrees well with the values obtained using the load relaxation technique.
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da/dt [ m e 2 I
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I [water] I KIC
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1
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Fig. 5. A V , K diagram for PMMA in air and water obtained using the DT test. The solid points represent direct optical measurements of crack velocity. (Two different thickness plates were used, 3 and 6 mm.)
It can be seen that the presence of water in general shifts the curves to the right. For PMMA, water appears t o eliminate ~ the fracture completely any slow crack growth a t K I < K I and toughness is increased to 1.7 MN/m3/2. Large “bunches” of crazes were observed a t the crack tip as soon as the PMMA was loaded in I/, K
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1 I
I
I
I
I 1
1
1
’
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S I M P L E X C E M E N T 118°C1
Fig. 6. A V , K diagram for Simplex cement in air and water obtained using the D T test. The two solid curves for air were determined using two separate specimens. (The solid points represent direct optical measurements of crack velocity.)
water. This “plastic” zone ahead of the crack tip effectively blunts out the propagating crack by absorbing the stored elastic strain energy in the specimen that would have been dissipated during the slow crack growth stages of failure (eqs. (4), and (5) show that G is proportional to R,).
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There appears to be a small dependence of the V , K curve for PMMA upon specimen geometry but this may be due t o the variation of crack geometry from specimen to specimen. The scatter between the results for Simplex may be due to the heterogeneous nature of this material, although an order of magnitude difference in V a t a particular value of Kr is not considered significant. The accuracy of the data may be appreciated by comparing times-to-failure, #f, a t constant stress predicted using the V , K diagram with the measured times-to-failure in the next section.
Time-to-Failure Predictions The relationship between V and K can be used for purposes of fracture safe design in situations where the life-time of a component is determined by subcritical (slow) crack growth. It has been shown that in PMMA and Simplex, V is a unique function of K and therefore from the knowledge of Krc and the desired working stress, uw, (or Krz), we can calculate the time-to-failure of the component in question as a function of uw/uf (or K ~ , l K r c ) . The time-to-failure, $fl a t constant stress is simply the integral of the V , K diagram;
*,
=
Iac a.
(l/V) da
(12)
Since for finite size specimens
Kr (where Y then
=
=
Yua l/G
(13)
d?rLyand depends upon the geometry of the specimeng),
This equation is plotted for PMMA in air in Figure 7 , using a n applied stress, ua, of 2.7 MN/m2, Y equal to 4 and a KIC of 1.4 MN/m3/*. The solid points in Figure 7 are experimental time-tofailure data for tensile specimens containing edge cracks loaded t o various levels of stress intensity (Kr). The measured data shows a high degree of scatter but within experimental error it can be seen that all the points lie on or above the theoretical line. It shows that it is possible to integrate the V, K curves and obtain a conservative estimate of the lifetime of the component.
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lmin
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-
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0
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I
0.6
100
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I
I
I
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FAILURE
$
lo4
I
to5
1 I06
[secs]
Fig. 7. Experimental values of times-to-failure for PMMA in air as a function of the initial stress intensity factor, K I , measured using single edge-notched tensile specimens. The solid curve shows the predicted times-to-failure obtained using eq. (14)where ua is 2.7 MN/m* and Y equals 4.
The scatter is probably due to the variation in crack tip geometry caused by the notching procedure. ( L Sharp" cracks were obtained by slowly pushing razor blades into the bottom of a sawcut and it was noticed that in some cases the cracks tended t o bifurcate although such samples were generally discarded. The shortest times-tofailure will be for molecularly sharp cracks. Longer times-to-failure will be obtained for cracks that have bifurcated. Having shown that times-to-failure can be obtained by the integration of the V , K diagram for PMMA, the V , K diagrams for Simplex in air and water were integrated in a similar way using eq. (14). The times-to-failure curves are plotted in Figure 8 again for u, equal to 2.7 MN/m2 and Y equal to 4. (These two values have been taken for convenience but the values of U, and Y will depend on the working stress, crack size and component geometry.)
ACRYLIC BONE CEMENT CRACK GROWTH
I
0 5 I 00
101
I 102
I I
TIME
I
o3
*.
to4
TO FAILURE
I
I
,05
,06
435
I I
o7
[I~CS]
Fig. 8. Predicted times-to-failure for Simplex cement in air and water and PMMA in air as a function of initial stress intensity factor, K I . The values of a,, and Y in eq. (14) are 2.7 MN/m2 and 4,respectively. The shaded areas represent c for materials in air and water. the scatter in measured K ~ values
The analytical derived curves shown in Figure 9 illustrate the usefulness of the stress intensity factor for fracture safe design of a Simplex cement structure being used in air and water. The timeto-failure as a function of KI, is plotted at four different stress levels. The significant effect of applied stress (or working stress) on time-tofailure is apparent a t values of KI,/Krc < 1. It can be seen that for each stress level there are three types of behavior. At values of KI close to KIc, failure only results after some incubation period which is a function of the applied stress. In the upper range (Kr 0.7 KIc), the time-to failure is slightly less sensitive to the value of KI. At low values of Kr, the failure time increases greatly with a small decrease in stress intensity level. At any particular value of KI < KIc, failure occurs after an incubation period that can either be accounted for as the time for crack initiation or as a function of the crack growth kinetics (eq. 1) as we have shown. We have therefore presented a conservative fracture safe design prediction by ignoring the possibility of a finite crack initiation time (+%= 0).
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436
I m,n
Ih0“l
-
#
-I’” a 0
z
r w
5
10
y1
w
ir y/ c
0 5 100
101
102
lo4
103 TIME
TO
FAILURE
+,
,05
lo7
I08
[SPCS]
Fig. 9. Theoretical times-to-failure curves as a function of the initial stress intensity factor Kr for Simplex cement in air and water a t 4 different stress levels, assuming Y = 4. The shaded areas represent the spread in measured K J Cvalues.
Electron Microscopy Fracture surfaces of both PMMA and Simplex were examined in a Stereoscan scanning electron microscope. The surfaczs were prepared for observation by depositing thin layers (- 100 A thick) of carbon and gold-palladium alloy on t o the fracture surfaces. Figure 10a, and b shows fracture surfaces of Simplex in air. The surface is rough and spherical particles can be seen protruding from it. There are also areas containing spherical pits where particles have been pulled out during crack propagation. I n other areas there are voids which were probably caused by pockets of trapped air or monomer. This may be contrasted with the Stereoscan micrograph of PMMA (Fig. 1Oc) a t approximately the same magnification which appears to be “perfectly” smooth. Figure 10a, b shows a mixed mode of failure in the Simplex cement with areas of transgranular and intergranular fracture. I n intergranular failure, the slowly moving crack has followed paths of weakness between spherical PMMA particles that have not completely fused together. This mode of failure may account for the rougher fracture surface appearance of the Simplex compared t o that
ACRYLIC BONE CEMENT CRACK GROWTH
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Fig. 10. (a) and (b) Scanning electron microscopy fractographs of Simplex in air; (c) PMMA in air, showing PMMA particles and porosity in the acrylic cement and the smooth fracture surface of PMMA.
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of PMMA, although crack propagation through strongly bonded PMMA spheres also produces a fairly irregular fracture surface. Similar observations have been made by Kusy and Turner,1o for a dental material based on PMMA that failed by fast crack propagation. Paradoxically, the Simplex cement is strengthened against cracking by these weak interfaces between PMMA spheres which cause the crack front to deviate during crack progression. This is a possible explanation for the displacement of the V , K curve for Simplex cement to the right of the curve for PMMA, i.e., a t a particular value of K I < Krc, the crack moves more slowly through the acrylic cement than in the PMMA. I n terms of energy, a t a particular crack velocity, the crack driving force GI is greater in Simplex than in PMMA.
Practical Implications It is important to consider what significance the results have from a practical point of view. The results were obtained a t 20°C but it is probably more important t o consider the behavior of the material a t 37°C. It has been shown" that the crack velocity in Perspex is greater for a given K I value when the temperature is increased. This means that the times-to-failure will be less a t higher temperatures and an upper limit has been calculated here. During a, total hip replacement it is highly likely that blood will be mixed with the cement. It is also possible that the material may have folds which have not completely welded together. Both of these will act as internal flaws in the structure. If an estimate of the size of these flaws is known it is possible t o use this as the initial crack size a, in the calculation of Kri (eq. (13)). The effects of trapped blood and temperature on the time-dependence of fracture stress of Simplex acrylic bone cement is currently under investigation.
CONCLUSIONS A technique developed for measuring subcritical crack growth in brittle materials has been used successfully in determining V , K relationships for PMMA and Simplex acrylic bone cement in air and water. There is a conservative agreement between experimentally measured times-to-failure and values of #f obtained using the V , K diagram for PMMA in air and it strengthens the validity of this
ACRYLIC BONE CEMENT CRACK GROWTH
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approach for investigating further subcritical crack growth phenomena in these kinds of materials. The V , K diagrams can also be used in the analysis of slow crack growth mechanisms and these initial results encourage their use to investigate further the effect of microstructure and environment on the fracture processes and times-to-failure in greater detail. One of the authors (P.W.R.B.) would like to acknowledge the support of a U.S. Public Health Service contract a t the University of California a t Los Angeles during part of this work, and the valuable discussions and encouragement from Professor Harlan C. Amstutz at UCLA during this period. One of the authors (R.J.Y.) would like to acknowledge the Master and Fellows of St. John’s College, Cambridge for support in the form of a Research Fellowship. P.W.R.B. wishes to thank the Royal Society of London for a travel award to the 6th Annual International Biomaterials Symposium (1974), a t Clemson University, S. Carolina, U.S.A., where this paper was presented. Both would like to thank Mr. L. Shadbolt of North Hill Plastics for supplying them with Simplex surgical cement.
References 1. A. H. Cottrell, Proc. Roy. SOC.,A276, 1 (1963). 2. F. A. McClintock and G. R. Irwin, “Fracture Toughness Testing,” ASTM STP 381, (1965). 3. P. C. Paris and G. C. Sih, ASTM STP 381 (1965). 4. K. R. Wheeler and L. A. James, J. Biomed. Mater. Res., 5 , 267 (1971). 5. Harlan C. Amstuts and K. Marchoff: private communication. 6. D. P. Williams and A. G. Evans, Mat. Res. and Standards (to be published). 7. A. G. Evans, J. Mat. Sci., 7 , 1137 (1972). 8. J. Alezska, Materials Department, UCLA; unpublished data. 9. J. E. Srawley and W. F. Brown, “Plane Strain Crack Toughness Testing,” ASTM STP 410, (1966). 10. R. B. Kusy and D. T. Turner, J. Biomed. Mater. Res., 8, 185 (1974). 11. G. P. Marshall, L. H. Coutts, and J . G. Williams, J. Mat. Sci., 9,1409 (1974).
Received September 19, 1974 Revised November 25, 1974
