Biorheology, 1975, Vol. 12, pp. 39-55. Pergamon Press. Printed in Great Britain
ANALYSIS OF SUCTION EXPERIMENTS ON RED BLOOD CELLS ELSPETH RICHARDSON*
Bioengineering Unit, University of Strathclyde, Glasgow, Scotland (Received 13 January 1974; in revised form 5 August 1974)
Abstract-In discussion of red cells in suction there is some doubt about the importance of slip at the pipette mouth. The extreme assumption that there is no slip is considered here. Two theoretical models, Hookean and Mooney materials, are used to investigate the inflation of a spherical cap. The results of this analysis are then compared with the experimental data already available from suction experiments in an attempt to determine a lower bound for elastic moduli. The values derived are at the lower end of the generally accepted range, being close to those of Hochmuth and Mohandas. The no-slip assumption is then critically reviewed in the light of these predictions. INTRODUCTION
With the introduction of artificial organs, mechanical haemolysis has become a significant clinical problem and this has led to increasing interest in the rheological properties of red cells over a large range of strain-rates. One of the types of experiment frequently performed to investigate low strain-rate behaviour is suction, utilising a technique which was first developed by Mitchison and Swann [1] for work on sea urchins' eggs and was later adapted by Rand and Burton [2] for use on red cells. In essence, a micropipette is placed in contact with the cell surface, slight suction is applied to attach the cell, and then the pressure drop is increased and the displacement of all or part of the cell into the pipette is observed. As red cells are so small it is difficult to obtain accurate quantitative information on their suction, particularly as diffraction problems occur in using visible light to measure lengths of, say, twice the wavelength of such light. This has led to the use both of other types of cell and of synthetic models in search of insight into the events of suction. The following have been studied in this context: Sea urchins' eggs, which are spherical, of dia. 100 ~m and have thickness 1·5 ~m [1], Rubber balls of various thicknesses and dia. [1], Liquid drops [2], Semipermeable spherical nylon microcapsules containing erythrocyte haemolysate, [3]. Since the behaviour of these other materials in suction is markedly different from that of red cells we shall, in fact, pay limited attention to them here. The intention in this paper is to concentrate particularly on the conditions at the pipette mouth. There appears to be no final concensus on whether the cell slips, slips partially, or does not slip at all into the pipette. Below, it will be shown that one of the possible cases, that of no slip, is at least consistent with the experimental evidence. If there were no slip, all the displacement of the membrane into the pipette at a given pressure would result from stretching of the membrane originally inside the pipette. Analysis of this situation would give greater values for deformation at given pressure than if some slip did occur. Thus it would produce a lower bound for the elastic modulus, whichever theoretical model might be adopted for analysis. Well known theoretical
*Address for correspondence: Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, CB3 9EW, U.K. 39
ELSPETH RICHARDSON
40
models will be employed here in an attempt to establish such a lower bound. The author does not wish to assert that the evidence suggests conclusively that no slip actually occurs-indeed criticism of the assumption will subsequently be made-but merely that it provides a complement to the work of Skalak's group [4] using the perfect slip hypothesis. The actual situation might be expected to lie between the two. DISCUSSION OF AV AILABLE EXPERIMENTAL RESULTS
In attempting to analyse this experimental situation there are many assumptions to be made before examining the predictions of various rheological models for the cell membrane. (A definition sketch of the quantities mentioned is given in Fig. 1).
Pressure-displacement curves We notice from the published results [1,3] that the rubber and nylon models all produced curves which are very nearly straight lines or are concave to the displacement axis. The viscoelastic nature of the sea urchin membrane was exhibited by the change in slope for various rates of loading, each loading curve again being nearly a straight line [1]. The curve shown for red cells (Fig. 2) was produced using one of the largest pipettes, while results from smaller pipettes were qualitatively similar. We note particularly that the slope increases with increasing pressure, perhaps indicative of an extension-limiting mechanism. This is in marked contrast to the other materials and provides one of the major reasons for considering these other substances to be inadequate models for red cell behaviour. Weed and Lacelle[5] used "the negative pressure required to draw into the micropipette a portion of cell membrane having a diameter equal to that of the pipette itself" as a measure of membrane deformability. Pooled data from more than 20 cells in a 3 /.Lm pipette gave the value 4 mm H 2 0 (390 dyn/cm 2 ) for this quantity.
Critical pressure The concept of "critical pressure" was defined by Rand and Burton [2] from their work with liquid drops. The interface between isobutyl alcohol in a large hanging drop and the water in the pipette was observed to be very unstable and to vibrate until it moved right into the pipette extremely suddenly at a critical pressure. Similar, though less well-defined, effects were seen with other materials and smaller drops. They recorded that for red cells at low suction pressures,
_. ____ ...1~
Rp
---0
Fig. I. Definition sketch for suction experiments. x-tongue length from pipette end; xo-the value of x at zero inflating pressure; Pi-pressures; R-cell radius (if spherical); Rp-pipette radius.
Analysis of suction experiments on red blood cells
41
12
oN J:
E
E
•
8
t~critical
0..1 N
0..
./
•
pressure
./ o
2
Tongue length x,
/-Lm
Fig. 2. Pressure-displacement curve for a red cell. From Rand and Burton [2].
P2-PI, the membrane position was unstable and it vibrated, With increasing suction it suddenly became stable and remained stationary, this point being used to define a "critical pressure". At this stage the pressure-displacement curve gradient increased sharply and this occurred when tongue-length was a little greater than pipette diameter. Further increase of suction pressure brought little further tongue movement. Measurements of critical pressure were made on large drops, small drops, normal biconcave cells, crenated shrunken spherical cells, spheroidal swollen cells, and spherical but not haemolysed cells [2],
Hysteresis Marked hysteresis was observed in the case of nylon microcapsules and sea urchins' eggs, whilst hysteresis only occurred in red ceIIs when the pressure was increased beyond the critical point. The hysteresis obvious in many of the pressure-displacement curves may be a genuine viscoelastic effect or, as Mitchison and Swann [1] pointed out, the product of different slippage at the pipette mouth during loading and unloading. Since there is only a very limited amount of hysteresis and we are here concerned with very low strain-rate, we shall concentrate on initial elastic response to changing suction pressures and treat elastic models only. Slip
Sea urchin's eggs are sufficiently large for conditions at the pipette mouth to be directly observed and, indeed, results were only taken from those eggs where slip was clearly seen. Red ceIIs, on the other hand, are so small that direct evidence is not available, The very limited increase in extension at high suction pressures suggests that there might be very little slip in the later stages of suction. The direct evidence from this experiment, i.e. the lack of significant hysteresis and the existence of a limiting extension, is consistent with the assumption of no slip but far from conclusive. It therefore seems justifiable to investigate the consequences of this assumption in
42
ELSPETH RICHARDSON
analysis of the problem, even though the resulting predictions may be criticised in the light of evidence from other situations.
Other qualitative observations on red cells The effect of changing osmotic pressure was marked. For isotonic and slightly hypotonic media the portion of the cell outside the pipette did not sphere and Rand and Burton [2] concluded that only the portion of the membrane within the tube was stretched. There also seemed to be little difference between the rim and dimple of such cells and so we shall ignore the possible effects of an internal structure of the kind suggested by Shrivastav and Burton [61. However, crenated cells, in hypertonic media, were more readily deformed, i.e. required lower pressure for the same displacement into the same pipette, than biconcave celIs. They moved further into the pipette at critical pressure while having a generally similar pressure-displacement curve. These crenated cells were also seen to move far more freely into the pipette showing no hysteresis or evidence of adhesion or friction with the glass tube wall. Sphered cells showed no apparent movement of the tongue without haemolysis, even with a pressure of 100 mm Hg (1·3 x lOs dyn/cm 2 ). For a variety of qualitative reasons fully discussed in Richardson [7] we consider flexible membranes only. Internal pressure and surface tension make a significant difference to the behaviour of membranes under inflation as may be seen from Mitchison and Swann's resu!t[I] that a model membrane already partially inflated gives a different pressure-displacement curve from one initially unstretched. There is little direct evidence, as distinct from that derived from theoretical and experimental models for suction and sphering problems, of the magnitude of internal pressure in an isotonic medium. We shall therefore investigate the consequences of varying initial internal pressure or surface tension to throw some light on this question. THEORETICAL MODELS
Because of the small size there is no published detail of the shape of the portion of red cell within the pipette, Only for sphered celIs, eggs, and models, is it certain that the initial shape is a spherical cap. For a portion of red ceIl, providing it is small, then either a flat sheet or a spherical cap of appropriate curvature appear sensible, and, being axisymmetric, tractable approximations. Each of the theoretical models used makes predictions of the shape of the membrane at given pressures and this might eventualIy help discriminate between such models. The geometry of the situation is such that a model must be chosen with regard to tractability and its predictions compared with experiment, rather than using the data directly to determine material stress-strain laws or the strain energy function. The simple models considered here are: Surface tension theory, popular with many authors; A Hookean material; A Mooney material.
Surface tension theory Surface tension theory was used by Rand and Burton [2] themselves to interpret their suction experiments in the following way: The Law of Laplace is used, i.e. jj.p = T(1IR, + I/R 2 ) where jj.p = pressure drop, R j = principal radii of curvature, T = membrane tension. It is assumed that both portions, i.e. inside and outside the pipette, are portions of spheres, i.e. R, = R 2 , and that T is a constant of the material. This predicts a pressure-displacement curve of the shape shown in Fig. 3 where the P scale is arbitrary and the shape otherwise depends on
43
Analysis of suction experiments on red blood cells
/e
I f0.-1 o..N
/i
/1
.,.,..---e
I
e
e
0
I
I I
/
I
I
I
xo/Rp
I
Fig. 1 Pressure-displacement curve predicted by surface tension theory in the case Rp/K xo/Rp = 0·2oR.
~
0·5 and
Rp IR·. The point at which xl Rp = 1, the hemispherical case, is defined as the critical pressure-the point at which liquid drops shoot up the tube and R, = R" where R, is the radius of curvature of the portion of cell in the tube. Then Po - P, = 2T/Rp - 2T/R == 2T/Rp - Pd , where Pd is the internal pressure. The surface tension theory fits the liquid drops used by Rand and Burton admirably as one would expect. Difficulties arise, however, when attempting to apply it to cells. With respect to the pressure-displacement curve it should be noted that this theory predicts a curve concave to the displacement axis, while the results of Mitchison and Swann are more nearly a straight line and those of Rand and Burton concave to the pressure axis. The model fits more nearly the microcapSUle experiments [3], the divergence at larger displacements suggesting increasing tension with deformation, i.e. an elastic situation. We also note that the predicted curve becomes parallel to the displacement axis at the point xl Rp = 1, whereas the two sets of cell experiments do not show this. The identification of the point of critical pressure for cells is also in doubt. As Fung and Tong [8] commented, the displacement at which the cell appeared suddenly to stiffen need not correspond to the displacement x = Rp at which liquid drops disappear down the pipette. Indeed Rand and Burton reported that the critical pressure did not always occur at the same x IRp and that it occurred more nearly when x = 2Rp, i.e. tongue length = pipette diameter. This suggests reservations about applying the theory to red cells. Further difficulties arise when the outer portion of the cell is not spherical in knowing how to deal with the pressure drop there. We note too that for any finite size cell or bubble there is automatically a finite value of T IR and thus the necessity of internal pressure initially, which is not the case with an elastic model.
44
ELSPETH RICHARDSON
A Hookean material A formulation is used which Mushtari and Galimov [9] produced to include displacements of greater order of magnitude than the more usual classical theories permit, i.e. it is assumed that, while elongations and shears are small compared with unity, displacements and changes of curvature may be of finite and even of considerable magnitude. Since we are here concerned with a flexible material the elasticity relations remain conventional as are the definitions of strain. This being so we shall merely quote results but try to review critically the underlying approximations in the light of this particular application. We shall also overstep the limit of small elongations in our calculation but with a corresponding decrease in confidence in the results. The particular case we consider is the inflation of a shallow flexible spherical cap by uniform internal pressure where the displacement of the pole is comparable with the base radius, i.e. with w(O) ~ a. (See definition sketch, Fig. 4.) For a shallow shell we require a ~ Rc and make the approximation (drlda) = 1, where r is the distance from the axis of revolution to a point on the middle surface and a the distance from the pole to the same point measured along a meridian. It must be appreciated that as we proceed from smaller values of pressure to ones producing displacements w(O) ~ a we can have decreasing confidence in the results obtained. In this theory we identify the metrics of the deformed and undeformed shells, which is adequate for small deformations but appears increasingly dubious as the deformation and displacements increase. The equations we need here are the integrated equation of compatibility
one integrated equation of equilibrium dw I 2 = 0, T lr 2k 2 - T lrTr-zqr and the further equilibrium equation
p=o
I
p=o
I
~ p=o
I
I
I
I
/
/
/
R,
~/ It
(0)
/
f-P=q
0
I
I
I
I I
lei
V
I
I
/
/ /R
/
' (b)
Fig. 4. Definition sketch for inflation of a spherical cap of Hookean material. (a) Undeformed. a = Rp = pipette radius. Rc = initial radius of curvature of the spherical cap. E = alRc = sin
ANALYSIS OF SUCTION EXPERIMENTS ON RED BLOOD CELLS ELSPETH RICHARDSON*
Bioengineering Unit, University of Strathclyde, Glasgow, Scotland (Received 13 January 1974; in revised form 5 August 1974)
Abstract-In discussion of red cells in suction there is some doubt about the importance of slip at the pipette mouth. The extreme assumption that there is no slip is considered here. Two theoretical models, Hookean and Mooney materials, are used to investigate the inflation of a spherical cap. The results of this analysis are then compared with the experimental data already available from suction experiments in an attempt to determine a lower bound for elastic moduli. The values derived are at the lower end of the generally accepted range, being close to those of Hochmuth and Mohandas. The no-slip assumption is then critically reviewed in the light of these predictions. INTRODUCTION
With the introduction of artificial organs, mechanical haemolysis has become a significant clinical problem and this has led to increasing interest in the rheological properties of red cells over a large range of strain-rates. One of the types of experiment frequently performed to investigate low strain-rate behaviour is suction, utilising a technique which was first developed by Mitchison and Swann [1] for work on sea urchins' eggs and was later adapted by Rand and Burton [2] for use on red cells. In essence, a micropipette is placed in contact with the cell surface, slight suction is applied to attach the cell, and then the pressure drop is increased and the displacement of all or part of the cell into the pipette is observed. As red cells are so small it is difficult to obtain accurate quantitative information on their suction, particularly as diffraction problems occur in using visible light to measure lengths of, say, twice the wavelength of such light. This has led to the use both of other types of cell and of synthetic models in search of insight into the events of suction. The following have been studied in this context: Sea urchins' eggs, which are spherical, of dia. 100 ~m and have thickness 1·5 ~m [1], Rubber balls of various thicknesses and dia. [1], Liquid drops [2], Semipermeable spherical nylon microcapsules containing erythrocyte haemolysate, [3]. Since the behaviour of these other materials in suction is markedly different from that of red cells we shall, in fact, pay limited attention to them here. The intention in this paper is to concentrate particularly on the conditions at the pipette mouth. There appears to be no final concensus on whether the cell slips, slips partially, or does not slip at all into the pipette. Below, it will be shown that one of the possible cases, that of no slip, is at least consistent with the experimental evidence. If there were no slip, all the displacement of the membrane into the pipette at a given pressure would result from stretching of the membrane originally inside the pipette. Analysis of this situation would give greater values for deformation at given pressure than if some slip did occur. Thus it would produce a lower bound for the elastic modulus, whichever theoretical model might be adopted for analysis. Well known theoretical
*Address for correspondence: Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, CB3 9EW, U.K. 39
ELSPETH RICHARDSON
40
models will be employed here in an attempt to establish such a lower bound. The author does not wish to assert that the evidence suggests conclusively that no slip actually occurs-indeed criticism of the assumption will subsequently be made-but merely that it provides a complement to the work of Skalak's group [4] using the perfect slip hypothesis. The actual situation might be expected to lie between the two. DISCUSSION OF AV AILABLE EXPERIMENTAL RESULTS
In attempting to analyse this experimental situation there are many assumptions to be made before examining the predictions of various rheological models for the cell membrane. (A definition sketch of the quantities mentioned is given in Fig. 1).
Pressure-displacement curves We notice from the published results [1,3] that the rubber and nylon models all produced curves which are very nearly straight lines or are concave to the displacement axis. The viscoelastic nature of the sea urchin membrane was exhibited by the change in slope for various rates of loading, each loading curve again being nearly a straight line [1]. The curve shown for red cells (Fig. 2) was produced using one of the largest pipettes, while results from smaller pipettes were qualitatively similar. We note particularly that the slope increases with increasing pressure, perhaps indicative of an extension-limiting mechanism. This is in marked contrast to the other materials and provides one of the major reasons for considering these other substances to be inadequate models for red cell behaviour. Weed and Lacelle[5] used "the negative pressure required to draw into the micropipette a portion of cell membrane having a diameter equal to that of the pipette itself" as a measure of membrane deformability. Pooled data from more than 20 cells in a 3 /.Lm pipette gave the value 4 mm H 2 0 (390 dyn/cm 2 ) for this quantity.
Critical pressure The concept of "critical pressure" was defined by Rand and Burton [2] from their work with liquid drops. The interface between isobutyl alcohol in a large hanging drop and the water in the pipette was observed to be very unstable and to vibrate until it moved right into the pipette extremely suddenly at a critical pressure. Similar, though less well-defined, effects were seen with other materials and smaller drops. They recorded that for red cells at low suction pressures,
_. ____ ...1~
Rp
---0
Fig. I. Definition sketch for suction experiments. x-tongue length from pipette end; xo-the value of x at zero inflating pressure; Pi-pressures; R-cell radius (if spherical); Rp-pipette radius.
Analysis of suction experiments on red blood cells
41
12
oN J:
E
E
•
8
t~critical
0..1 N
0..
./
•
pressure
./ o
2
Tongue length x,
/-Lm
Fig. 2. Pressure-displacement curve for a red cell. From Rand and Burton [2].
P2-PI, the membrane position was unstable and it vibrated, With increasing suction it suddenly became stable and remained stationary, this point being used to define a "critical pressure". At this stage the pressure-displacement curve gradient increased sharply and this occurred when tongue-length was a little greater than pipette diameter. Further increase of suction pressure brought little further tongue movement. Measurements of critical pressure were made on large drops, small drops, normal biconcave cells, crenated shrunken spherical cells, spheroidal swollen cells, and spherical but not haemolysed cells [2],
Hysteresis Marked hysteresis was observed in the case of nylon microcapsules and sea urchins' eggs, whilst hysteresis only occurred in red ceIIs when the pressure was increased beyond the critical point. The hysteresis obvious in many of the pressure-displacement curves may be a genuine viscoelastic effect or, as Mitchison and Swann [1] pointed out, the product of different slippage at the pipette mouth during loading and unloading. Since there is only a very limited amount of hysteresis and we are here concerned with very low strain-rate, we shall concentrate on initial elastic response to changing suction pressures and treat elastic models only. Slip
Sea urchin's eggs are sufficiently large for conditions at the pipette mouth to be directly observed and, indeed, results were only taken from those eggs where slip was clearly seen. Red ceIIs, on the other hand, are so small that direct evidence is not available, The very limited increase in extension at high suction pressures suggests that there might be very little slip in the later stages of suction. The direct evidence from this experiment, i.e. the lack of significant hysteresis and the existence of a limiting extension, is consistent with the assumption of no slip but far from conclusive. It therefore seems justifiable to investigate the consequences of this assumption in
42
ELSPETH RICHARDSON
analysis of the problem, even though the resulting predictions may be criticised in the light of evidence from other situations.
Other qualitative observations on red cells The effect of changing osmotic pressure was marked. For isotonic and slightly hypotonic media the portion of the cell outside the pipette did not sphere and Rand and Burton [2] concluded that only the portion of the membrane within the tube was stretched. There also seemed to be little difference between the rim and dimple of such cells and so we shall ignore the possible effects of an internal structure of the kind suggested by Shrivastav and Burton [61. However, crenated cells, in hypertonic media, were more readily deformed, i.e. required lower pressure for the same displacement into the same pipette, than biconcave celIs. They moved further into the pipette at critical pressure while having a generally similar pressure-displacement curve. These crenated cells were also seen to move far more freely into the pipette showing no hysteresis or evidence of adhesion or friction with the glass tube wall. Sphered cells showed no apparent movement of the tongue without haemolysis, even with a pressure of 100 mm Hg (1·3 x lOs dyn/cm 2 ). For a variety of qualitative reasons fully discussed in Richardson [7] we consider flexible membranes only. Internal pressure and surface tension make a significant difference to the behaviour of membranes under inflation as may be seen from Mitchison and Swann's resu!t[I] that a model membrane already partially inflated gives a different pressure-displacement curve from one initially unstretched. There is little direct evidence, as distinct from that derived from theoretical and experimental models for suction and sphering problems, of the magnitude of internal pressure in an isotonic medium. We shall therefore investigate the consequences of varying initial internal pressure or surface tension to throw some light on this question. THEORETICAL MODELS
Because of the small size there is no published detail of the shape of the portion of red cell within the pipette, Only for sphered celIs, eggs, and models, is it certain that the initial shape is a spherical cap. For a portion of red ceIl, providing it is small, then either a flat sheet or a spherical cap of appropriate curvature appear sensible, and, being axisymmetric, tractable approximations. Each of the theoretical models used makes predictions of the shape of the membrane at given pressures and this might eventualIy help discriminate between such models. The geometry of the situation is such that a model must be chosen with regard to tractability and its predictions compared with experiment, rather than using the data directly to determine material stress-strain laws or the strain energy function. The simple models considered here are: Surface tension theory, popular with many authors; A Hookean material; A Mooney material.
Surface tension theory Surface tension theory was used by Rand and Burton [2] themselves to interpret their suction experiments in the following way: The Law of Laplace is used, i.e. jj.p = T(1IR, + I/R 2 ) where jj.p = pressure drop, R j = principal radii of curvature, T = membrane tension. It is assumed that both portions, i.e. inside and outside the pipette, are portions of spheres, i.e. R, = R 2 , and that T is a constant of the material. This predicts a pressure-displacement curve of the shape shown in Fig. 3 where the P scale is arbitrary and the shape otherwise depends on
43
Analysis of suction experiments on red blood cells
/e
I f0.-1 o..N
/i
/1
.,.,..---e
I
e
e
0
I
I I
/
I
I
I
xo/Rp
I
Fig. 1 Pressure-displacement curve predicted by surface tension theory in the case Rp/K xo/Rp = 0·2oR.
~
0·5 and
Rp IR·. The point at which xl Rp = 1, the hemispherical case, is defined as the critical pressure-the point at which liquid drops shoot up the tube and R, = R" where R, is the radius of curvature of the portion of cell in the tube. Then Po - P, = 2T/Rp - 2T/R == 2T/Rp - Pd , where Pd is the internal pressure. The surface tension theory fits the liquid drops used by Rand and Burton admirably as one would expect. Difficulties arise, however, when attempting to apply it to cells. With respect to the pressure-displacement curve it should be noted that this theory predicts a curve concave to the displacement axis, while the results of Mitchison and Swann are more nearly a straight line and those of Rand and Burton concave to the pressure axis. The model fits more nearly the microcapSUle experiments [3], the divergence at larger displacements suggesting increasing tension with deformation, i.e. an elastic situation. We also note that the predicted curve becomes parallel to the displacement axis at the point xl Rp = 1, whereas the two sets of cell experiments do not show this. The identification of the point of critical pressure for cells is also in doubt. As Fung and Tong [8] commented, the displacement at which the cell appeared suddenly to stiffen need not correspond to the displacement x = Rp at which liquid drops disappear down the pipette. Indeed Rand and Burton reported that the critical pressure did not always occur at the same x IRp and that it occurred more nearly when x = 2Rp, i.e. tongue length = pipette diameter. This suggests reservations about applying the theory to red cells. Further difficulties arise when the outer portion of the cell is not spherical in knowing how to deal with the pressure drop there. We note too that for any finite size cell or bubble there is automatically a finite value of T IR and thus the necessity of internal pressure initially, which is not the case with an elastic model.
44
ELSPETH RICHARDSON
A Hookean material A formulation is used which Mushtari and Galimov [9] produced to include displacements of greater order of magnitude than the more usual classical theories permit, i.e. it is assumed that, while elongations and shears are small compared with unity, displacements and changes of curvature may be of finite and even of considerable magnitude. Since we are here concerned with a flexible material the elasticity relations remain conventional as are the definitions of strain. This being so we shall merely quote results but try to review critically the underlying approximations in the light of this particular application. We shall also overstep the limit of small elongations in our calculation but with a corresponding decrease in confidence in the results. The particular case we consider is the inflation of a shallow flexible spherical cap by uniform internal pressure where the displacement of the pole is comparable with the base radius, i.e. with w(O) ~ a. (See definition sketch, Fig. 4.) For a shallow shell we require a ~ Rc and make the approximation (drlda) = 1, where r is the distance from the axis of revolution to a point on the middle surface and a the distance from the pole to the same point measured along a meridian. It must be appreciated that as we proceed from smaller values of pressure to ones producing displacements w(O) ~ a we can have decreasing confidence in the results obtained. In this theory we identify the metrics of the deformed and undeformed shells, which is adequate for small deformations but appears increasingly dubious as the deformation and displacements increase. The equations we need here are the integrated equation of compatibility
one integrated equation of equilibrium dw I 2 = 0, T lr 2k 2 - T lrTr-zqr and the further equilibrium equation
p=o
I
p=o
I
~ p=o
I
I
I
I
/
/
/
R,
~/ It
(0)
/
f-P=q
0
I
I
I
I I
lei
V
I
I
/
/ /R
/
' (b)
Fig. 4. Definition sketch for inflation of a spherical cap of Hookean material. (a) Undeformed. a = Rp = pipette radius. Rc = initial radius of curvature of the spherical cap. E = alRc = sin
