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Large deformation of elastic tubes
/. B~~mrrchun~cs. Vol. 3. pp. 593400.
LARGE
Pergamon Press. 1970.
Printed in Great Britain
DEFORMATION W. H. HOPPMANN
OF
ELASTIC
TUBES”
II and LEE WAN
College of Engineering, University of South Carolina, Columbia, S.C. 19108, U.S.A. Abstract- In this paper a study is made of the large deformation of elastic tubes. Analysis is confined to the case of statical deformation. The energy of deformation is considered to be a function of the three strain invariants only. In order to determine its form for actual materials it is expanded in its Taylor’s series. From the theoretical analysis of the inflation and extension of tubes, linear algebraic equations in the elastic constants are developed. Using these and appropriate measurements of diametral change and longitudinal stretch. obtained from experiments, elastic constants for the non-linear theory of elasticity are explicitly determined with the aid of a computer program. The highly non-linear forms of deformation of tubes computed on the basis of the theory are compared with our experimental findings for a latex rubber tubing and for some bio-physical data taken from the literature. INTRODUCTIOK
is an increasing need for systematic studies of the large elastic deformation of biological as well as other kinds of materials. Important examples of the biological type are arteries, veins, aortae, lymphatic ducts, colons, and bladders. A probably useful formulation of the problem of deformation can be made from the standpoint of continuum mechanics. In the last score of years there has been a simultaneous growth of our knowledge of the non-linear theory of elastic deformation and of our need for understanding the deformability of the various parts of flow systems in the animal body. The non-linear field theory of the continuum is rather complex and the history of the subject is sometimes difficult to follow. However, we are fortunate in that an attempt by Truesdell (I 965) to treat both the theory and its history turns out to be a veritable row- deform. Along with D. Noll, he has presented an extensive and fairly readable account in the Handbuch der Physik (I 965). An earlier work by Truesdell (1952) on this subject is also informative. The pioneer study of Murnaghan (195 1) deserves consideration but. as noted by Truesdell. it is somewhat confusing and unsatisfactorily developed.
One purpose of the present paper is to systematically apply the theory of elasticity in order to determine the elastic constants for the case of very large strains in various kinds of materials. A good reference for the pertinent theory is the textbook by Green and Zema (1968). Another purpose of the paper is to apply the theory to experimental data given by Glagov and Wolinsky ( 1963, 1964) for the very large deformation of aortae of rabbits.
THERE
*Receiced
9 Jamrap
ENERGY
FUNCTIONS
In the present study, the energy function, from which the constitutive equations are derived, is considered to be a function of the strain invariants I,, I?, and I3 only. in order that the energy vanish when the strains vanish it is written as follows: w=
W(l,-3,1,-3,1,--
1).
0)
All of the materials studied in the present paper are practically incompressible so that I3 is equal to unity and the energy may be written more simply as: w=
W(I,-3,12-3).
The energy function may be represented
1970. 593
(3) in
W. H. HOPPMANN
594
and L. WAN
If r is the radius to a point in the tube before loading and R is the radius to the same material point after loading, we may write:
a Taylor’s series as follows: WI,-3,1,--3)=
II
E (A,,(I,-3)+A?,(I,-3))” n=1 (3)
in which the constant term has been placed equal to zero. The terms A,, and A.), represent the usual partial derivatives in a Taylor’s expansion, evaluated at I, equal 3 and I, equal 3. Although it is not stated explicitly, a power series expression given by Treloar (1958) is equivalent to the series expansion in equation (3). It is given in the following double series form: w =
5
crj(fl-3)i(lp-3jy.
i=OJ=O
(4)
R(r) = rQ(r) . The incompressibility expressed as:
(3
condition
can
hrdr = RdR
(6)
where A is the axial stretch factor. Using equation (5) the differential equation (6) may be written: Q(Q+rQA
= h .
(7)
Its solution is: R = rQ = (A(? + K)}liz
Both equations (3) and (4) are power series expansions in (I, - 3) and (I, - 3). In the present paper, the form given by equation (3) will be used. The first order terms represent the form proposed by Mooney for some studies of rubber as discussed by Treloar ( 1958). In order to determine the elastic constants in (3) for some specific materials, experiments are performed on tubing which has been both inflated and simultaneously stretched. In order to be able to evaluate all of the constants for the homogeneous, isotropic, incompressible materials under analysis, it is essential to experiment with the combined loading. In the present state of development of the subject it is well to state that the material is also non-polar: that is, it is one in which body couples and couple stresses are not considered. The solution of such a problem in terms of a general energy function W is given by Green and Zerna (1968). Their results will be briefly summarized for our purposes.
be
(8)
where K is the constant of integration. Because of the symmetry in the problem, the solution of the equations of equilibrium give the following simple relation: P =-$+-($+Q2)$-L(r)+H
(9)
where H is another constant of integration and L(R) is as follows:
(10) The physical stress components tangential, and axial directions ively as follows: u
in the radial, are respect-
,,=H--L(R)
cT2=H-L(R)+(&$+($-Q2)+ and
RESUME OF THEORY
The following resume is taken from the solution of the problem of the inflated and extended tube, given by Green and Zema (I 968).
and
atR=R, atR=R,
VII = 0 (+I1=-PO
.
(12)
LARGE
DEFORMATION
in our experiments the boundary conditions are actually as follows: atR=R, andatR=R?
Cl1
=
atmospheric
crll = gage pressure.
pressure (13)
It is easy to show that the two sets of conditions give identically the same results. The function L (R) contains all of the elastic constants and by considering as many different values of pressure P as there are elastic constants, for a given stretch ratio A, we obtain n linear algebraic equations in the n unknown elastic constants in order to numerically evaluate the constants. It is quite simple to solve these equations with a computer, As previously noted, it is necessary to introduce stretch into the problem in order to evaluate all of the possible elastic coefficients. In this connection observe that the function L (R) given by equation (10) is a function of the stretch factor X. EXPERIMENTAL METHOD FOR DETERMINATION OF ELASTIC CONSTANTS OF LATEX RUBBER
In order to determine the elastic constants for latex rubber, using the theory in the present paper, a length of commercially available tubing was used. It was nominal Ain. i.d. tube of 27 in. length and l/32 in. thickness. The tube was suspended vertically with the lower end closed by a solid plug. The upper end was attached to a compressed air line which provided a predetermined pressure inside the tube. A suitable control valve was used for limiting the pressure to the amount desired. The gas pressure was precisely measured by means of a mercury column. The desired axial load was obtained as the sum of the load from the internal pressure acting on the bottom plug and weights attached externally to the plug. At a given pressure and axial load the external diameter of the section at mid-length was determined micrometrically. In addition, the change in length of the tube, over a 15-in.
OF
ELASTIC
395
TtiBES
The measurements base, was measured. provided the data to calculate the change of radius ratio p and the stretch factor A for given pressure and axial load. In order to investigate the possible variation of diameter of cross-section along the length at a given pressure, a series of measurements were made at stations appropriately located. It turns out that the maximum variation is less than 5 per cent from the average of ail variations measured, except within a few diameters of the ends. Also the stretch factor h was measured for a series of base lengths from 15 in. to the full length of the tube and it was found to vary less than 5 per cent for any applied pressure and axial load. In order to determine the change in external radius with pressure for the case of A equal unity, that is no axial stretch, a special experimental arrangement was used. A large closed loop of tubing with a 117-in. Long straight horizontal portion was filled with water rather than air and was attached to a variable pressure water column for applying the desired pressure. In this manner no appreciable axial load is developed. All measurements of diametral change were made at the midlength of the horizontal portion. All of the experiments were conducted at room temperature, which was about 7- 1°C. RESULTS OF EXPERIMENTS
WiTH
LATEX TUBING
The purpose of the present study was to determine only the elastic response to static loading so some preliminary experiments were made in order to determine at least approximately the limits of the elastic region. The failure load, that is the load causing rupture, occurs approximately at 7 equal to 0.4 and without axial stretch. The assumed limit of elastic performance is for 7~ equal to 0.24 and X equal to 1.55. Obviously, the limits of the elastic region cannot be determined with great precision using the measurement techniques of the present investigation. However, the experimental data obtained are
W. H. HOPPMANN
596
considered to represent reasonably well and they Fig. I. These data were elastic constants assuming material. The theoretical
the latex rubber are presented in used to compute a Mooney type results are also
Iland L. WAN
shown in Fig. 2. Since the Mooney curves are essentially simple parabolas, with no change in sign of curvature, they obviously cannot represent the experimental curves which do have a noticeable change in sign of
12
Theoretical - Mooney material ----
Experimental for latex rubber
6 4 2
-12 -14 -16 -18
0 Fig.
2
4
6
14 16 8 10 12 PRESSURE RATIO, in x102
'18
20
2i
I. For latex rubber, the relation of inflation ratio p to pressure ratio w for various values of A.
24
LARGE
0
2
4
6
DEFORMATION
8
lo
PRESSURE
OF ELASTIC
12 RATIO,
14 x
16
TUBES
18
591
20
22
24
x lo2
Fig. 1. For latex rubber. the relation of inflation ratio p to pressure ratio r for A equal unity.
curvature. That is, in the neighborhood of 7~ equal to zero for the various stretch factors A, the experimental curves have a different sign for curvature than the Mooney type curves whereas at the higher strain levels the curvatures are the same. Using a higher order theory of elasticity than the Mooney theory it can be seen that the experimental data can be better represented in the neighborhood of r equal to zero. For this purpose the finite series representing the energy function was taken to include terms containing second powers of the invariants. COMPARISON OF THEORETICAL RESULTS WITH EXPERIMENTAL DATA OBTAINED WITH AORTAE OF RABBITS
An interesting application of the theory used in the present paper was to determine the
elastic constants for the aorta of a rabbit. The experiments were conducted by Glagov and Wolinsky (1963, 1964, 1967). Inflation experiments were performed on the aortae much the same as those described in the present paper. Air was used to obtain the desired pressures which were sufficiently high to produce very large strains. Also, data for experiments with aortae are given by Pieper and Paul (1969). These latter experiments were confined to low values of pressure. In order to cover the complete strain range for the experiments of Wolinsky and Glagov (1963), calculations were made by the present authors using powers of the invariants up to and including the fifth. The comparison of the experimental data and the theoretical curves are shown in Fig. 3.
W. H. HOPPMANN
598
II and L. WAN
Theoretical -----
Sth order
elasticity
Experimental for aorta of rabbit
wall thickness
20
40
60
80 100 PRESSURE ( mm
120
140
160
180
200
Hg )
Fig. 3. Relation of rabbit radius and wall thickness to distending pressure.
DISCLJSSIOZ(AND CONCLUSIONS
We consider that the results of the present research are sufficiently informative so that we may come at least to some tentative conclusions. The elastic constants have been sufficiently well determined to be used at least in preliminary studies of flow of liquids through distensible tubes as analogues to the recirculatory cardiovascular system. It is clear from the present results that considerable thought must be given to the possible effects of large strains and also to the effects of axial deformation in developing the theory of fluid flow in elastic tubes as proposed, for example, by Campbell and Yang (1969). The question of possible anisotropy in materials of which tubes are constructed must be investigated, but the material for
which the energy W is a function only of the strain invariants is isotropic. It is considered that systematic use of the method proposed in the present paper may help to avoid some of the confusion in biomechanical literature as evidenced in a recent paper by Tichner and Sacks (1967). In their paper the authors, probably inspired by the theory of plasticity, introduced the incremental theory of strain. Also, there seems to be some confusion about the nature of anisotropy as represented in the energy function W. No justification seems to be given for their complex proposal for analyzing the deformation of the aorta. In the development of hemodynamics many ad hoc type of assumptions concerning the compliance of arteries, aortae, and other blood vessels have been used. Both in the in vitro as
LARGE
DEFORMATlON
well as in the in cico studies of biological flows more detailed information concerning the constitutive equations of the materials should be welcomed. It seems improper to use overly simple assumptions concerning the elasticity of the tubes in order to simplify the flow theory. It would appear much more satisfactory to determine the most nearly exact constitutive equations for the tube material and then if necessary, simplify in order to make approximate analyses of biological flows. Campbell and Yang (1969) in studying a flow problem, allowing for deformability of tubes, used a rather simple relation between pressure and change of area of cross-section of tubing, as experimentally determined by Campbell ( 1968), without considering the effect of axial stretch. It is true that their wave analysis for the flow in the tube is difficult enough, even with a linear relation assumed for the elasticity of the tube. however it would seem more suitable in these cases to properly determine the elastic constants for the material. It is considered that the inflation with extension experiments should be routinely made for any particular tubing used in in c&o experiments. Statical experiments using this method will provide some preliminary knowledge at least of the elasticity properties. A further extension to studies of viscoelastic properties will require a development of the method to include dynamic loading. It would certainly be wortwhile to study the response of the tube to pulsating pressures and oscillatory axial loads. Increasing the pressure to upper limits would also give some indication of plastic effects. It is hoped that our results may encourage further research which will determine the most effective form of constitutive equations for studying the problem of flow of liquids in elastic tubes. which is so important in current physiology and biomechanics. Also, it may be of value to associate the experimentally determined regimes of strain with normal
B.M.-VoL
3. ho. 6-F
OF
ELASTIC
TUBES
599
conditions of the and with traumatic organism. Before ending the present paper it may be useful to cite a few references in the recent literature which pertain to the subject. An example of a flow study without consideration of the distensibility of walls is discussed in a recent paper by Lew and Fung (1970). Studies of flow with the usual assumption of linear elasticity for the tubes are exemplified by Miekisz ( I96 I, 1961). In a very interesting recent book edited by Schwan (1969) a fairly extensive survey of the elastic tube concept for the linear case is given in an article on Hemodynamics by Abraham Noordergraft. His list of references is very complete, extending from 1546 to the present. Finally, mention should be made of a survey article by Fung ( 1968). Of special pertinence to the material of the present paper is his general discussion of stress-strain laws for soft tissue. REFERENCES Campbell. J. L. III t 1968) Method of determining some distensible tube properties. Terms. ,-tS.W& J. bm. Eqng 90D. 133. Campbell. J. L. III and Yang. T. (1969) Pulsatile how behavior in elastic systems containing wave reflection sites. Tmrrs. ,4SME 1. bru. Engng 91D. 9% 101. Fung. Y. C. B. (1968) Biomechanics-iti scope. history, and some problems of continuum mechanics in physiology. rlppl. nrech. Rec. 21. I-10. Galgov. S. and Wolinsky. H. ( 1963) Aorric wall as a ‘two-phase‘ material. ~Vnmrr. Lond. 199.606. Green, A. E. and Zerna. W. (1968) T1reorrriccrl Elasricir.v. 2nd Edn. p. 87. Oxford University Press. Oxford. Green. A. E. and Adkins. J. E. (1960) Large EIosric Deformurions. p. 16. Oxford University Press. Oxford. Lew, H. S. and Fung. Y. C. CL9701 Entry How into blood vessels at arbitrary Reynolds number. J. Biomechnics 3. 23-38. Miekisz. S. ( I96 I ) On the dispersion of the propagation velocity of a disturbance in a liouid tlowine through elastic walled vessels. Actor ph~s. pbl. 20.827-830. hliekisz, S. ( 1961) Non-linear theory of viscous flow in elastic tubes. Phxs. .Lled. Biol. 6. 103-109. Mumaghan. F. D. (1950 Finite Dejbrmcrrion of‘ cm Elnsric Solid. p.63. Wiley. New York. Pieper, H. P. and Paul. L. T. ( 1969) Response of aortic smooth muscle studied in intact dogs. .im. J. Plrysiol. 217. 15-t-160. Schwan. Herman P. ( 1969) Biolopicnl Engineering. pp. 556. McGraw-Hill. New York. Tickner. E. G. and Sacks. A. H. t 1967) .A theory for the static elastic behavior of blood vessels. Biorheo1oy.v 4. 141-168.
W. H. HOPPMANN
600
Treloar, L. R. G. (1958) The Physics ofRubber Efasricity. 2nd. Edn. p. 155. Oxford Press. Oxford. Truesdell. C. and NOR. D. (1965) The Non-Lineur Field Theories of Mech&ics‘ (Die N&Linemen Feld Theorien), Encyclopedia of Physics (Handbuck der Physik) (Edited by S. Fliigge), p. 50 and p. I I8 et seq. Truesdell. C. ( 1952)J. rur. Mech. Analysis 1, 179.
Wolinsky, H. and Glagov. S. (1964) Structural basis for the mechanical properties of the aortic media. Circulation Res. 14.400-413. Wolinsky, H. and Glagov, S. (1967) Lamellar unit of aortic structure and function in mammals. Circulation Res. 20,99- I I 1. NOMENCLATURE Reference: Textbook by Green and Zema (1968) W = W(I,,I,,I,) = energy function I, = g”G, = 3 + y: = I st variant 12= G”g,ls = 3 + 4y:+ 2 (y:y: - y;y$) = 2nd variant 1s = ij$+ 2y:= s = 3rd invariant y. = j(GU-g,) Yl =
= strain
PYkJ
P,~= metrical coefficients in unstrained bodv
Y”
GiJ= metrical coefficients in strained body 6: = Kronecker delta G = det Gil
II and L. WAN g=detgil +i = tensorial stress uU = (GJG”) ~~ = physical A = axial stretch ratio r = radius to a material before loading R = corresponding radius with loading r, = radius to outer surface R, = radius to outer surface RI-r
P = L=rt
point in tube cross-section to the same material point before loading after loading
radius in strained state
*S-E P pressure ratio PO PI,= atmospheric pressure P = gage pressure r”~~g’J+$B;J+pG’J
2dW &=xz 2dW -’ = 12 d12 p=21,$
components of stress
3
B” = f,gu-gg”gbG,.
LARGE
Pergamon Press. 1970.
Printed in Great Britain
DEFORMATION W. H. HOPPMANN
OF
ELASTIC
TUBES”
II and LEE WAN
College of Engineering, University of South Carolina, Columbia, S.C. 19108, U.S.A. Abstract- In this paper a study is made of the large deformation of elastic tubes. Analysis is confined to the case of statical deformation. The energy of deformation is considered to be a function of the three strain invariants only. In order to determine its form for actual materials it is expanded in its Taylor’s series. From the theoretical analysis of the inflation and extension of tubes, linear algebraic equations in the elastic constants are developed. Using these and appropriate measurements of diametral change and longitudinal stretch. obtained from experiments, elastic constants for the non-linear theory of elasticity are explicitly determined with the aid of a computer program. The highly non-linear forms of deformation of tubes computed on the basis of the theory are compared with our experimental findings for a latex rubber tubing and for some bio-physical data taken from the literature. INTRODUCTIOK
is an increasing need for systematic studies of the large elastic deformation of biological as well as other kinds of materials. Important examples of the biological type are arteries, veins, aortae, lymphatic ducts, colons, and bladders. A probably useful formulation of the problem of deformation can be made from the standpoint of continuum mechanics. In the last score of years there has been a simultaneous growth of our knowledge of the non-linear theory of elastic deformation and of our need for understanding the deformability of the various parts of flow systems in the animal body. The non-linear field theory of the continuum is rather complex and the history of the subject is sometimes difficult to follow. However, we are fortunate in that an attempt by Truesdell (I 965) to treat both the theory and its history turns out to be a veritable row- deform. Along with D. Noll, he has presented an extensive and fairly readable account in the Handbuch der Physik (I 965). An earlier work by Truesdell (1952) on this subject is also informative. The pioneer study of Murnaghan (195 1) deserves consideration but. as noted by Truesdell. it is somewhat confusing and unsatisfactorily developed.
One purpose of the present paper is to systematically apply the theory of elasticity in order to determine the elastic constants for the case of very large strains in various kinds of materials. A good reference for the pertinent theory is the textbook by Green and Zema (1968). Another purpose of the paper is to apply the theory to experimental data given by Glagov and Wolinsky ( 1963, 1964) for the very large deformation of aortae of rabbits.
THERE
*Receiced
9 Jamrap
ENERGY
FUNCTIONS
In the present study, the energy function, from which the constitutive equations are derived, is considered to be a function of the strain invariants I,, I?, and I3 only. in order that the energy vanish when the strains vanish it is written as follows: w=
W(l,-3,1,-3,1,--
1).
0)
All of the materials studied in the present paper are practically incompressible so that I3 is equal to unity and the energy may be written more simply as: w=
W(I,-3,12-3).
The energy function may be represented
1970. 593
(3) in
W. H. HOPPMANN
594
and L. WAN
If r is the radius to a point in the tube before loading and R is the radius to the same material point after loading, we may write:
a Taylor’s series as follows: WI,-3,1,--3)=
II
E (A,,(I,-3)+A?,(I,-3))” n=1 (3)
in which the constant term has been placed equal to zero. The terms A,, and A.), represent the usual partial derivatives in a Taylor’s expansion, evaluated at I, equal 3 and I, equal 3. Although it is not stated explicitly, a power series expression given by Treloar (1958) is equivalent to the series expansion in equation (3). It is given in the following double series form: w =
5
crj(fl-3)i(lp-3jy.
i=OJ=O
(4)
R(r) = rQ(r) . The incompressibility expressed as:
(3
condition
can
hrdr = RdR
(6)
where A is the axial stretch factor. Using equation (5) the differential equation (6) may be written: Q(Q+rQA
= h .
(7)
Its solution is: R = rQ = (A(? + K)}liz
Both equations (3) and (4) are power series expansions in (I, - 3) and (I, - 3). In the present paper, the form given by equation (3) will be used. The first order terms represent the form proposed by Mooney for some studies of rubber as discussed by Treloar ( 1958). In order to determine the elastic constants in (3) for some specific materials, experiments are performed on tubing which has been both inflated and simultaneously stretched. In order to be able to evaluate all of the constants for the homogeneous, isotropic, incompressible materials under analysis, it is essential to experiment with the combined loading. In the present state of development of the subject it is well to state that the material is also non-polar: that is, it is one in which body couples and couple stresses are not considered. The solution of such a problem in terms of a general energy function W is given by Green and Zerna (1968). Their results will be briefly summarized for our purposes.
be
(8)
where K is the constant of integration. Because of the symmetry in the problem, the solution of the equations of equilibrium give the following simple relation: P =-$+-($+Q2)$-L(r)+H
(9)
where H is another constant of integration and L(R) is as follows:
(10) The physical stress components tangential, and axial directions ively as follows: u
in the radial, are respect-
,,=H--L(R)
cT2=H-L(R)+(&$+($-Q2)+ and
RESUME OF THEORY
The following resume is taken from the solution of the problem of the inflated and extended tube, given by Green and Zema (I 968).
and
atR=R, atR=R,
VII = 0 (+I1=-PO
.
(12)
LARGE
DEFORMATION
in our experiments the boundary conditions are actually as follows: atR=R, andatR=R?
Cl1
=
atmospheric
crll = gage pressure.
pressure (13)
It is easy to show that the two sets of conditions give identically the same results. The function L (R) contains all of the elastic constants and by considering as many different values of pressure P as there are elastic constants, for a given stretch ratio A, we obtain n linear algebraic equations in the n unknown elastic constants in order to numerically evaluate the constants. It is quite simple to solve these equations with a computer, As previously noted, it is necessary to introduce stretch into the problem in order to evaluate all of the possible elastic coefficients. In this connection observe that the function L (R) given by equation (10) is a function of the stretch factor X. EXPERIMENTAL METHOD FOR DETERMINATION OF ELASTIC CONSTANTS OF LATEX RUBBER
In order to determine the elastic constants for latex rubber, using the theory in the present paper, a length of commercially available tubing was used. It was nominal Ain. i.d. tube of 27 in. length and l/32 in. thickness. The tube was suspended vertically with the lower end closed by a solid plug. The upper end was attached to a compressed air line which provided a predetermined pressure inside the tube. A suitable control valve was used for limiting the pressure to the amount desired. The gas pressure was precisely measured by means of a mercury column. The desired axial load was obtained as the sum of the load from the internal pressure acting on the bottom plug and weights attached externally to the plug. At a given pressure and axial load the external diameter of the section at mid-length was determined micrometrically. In addition, the change in length of the tube, over a 15-in.
OF
ELASTIC
395
TtiBES
The measurements base, was measured. provided the data to calculate the change of radius ratio p and the stretch factor A for given pressure and axial load. In order to investigate the possible variation of diameter of cross-section along the length at a given pressure, a series of measurements were made at stations appropriately located. It turns out that the maximum variation is less than 5 per cent from the average of ail variations measured, except within a few diameters of the ends. Also the stretch factor h was measured for a series of base lengths from 15 in. to the full length of the tube and it was found to vary less than 5 per cent for any applied pressure and axial load. In order to determine the change in external radius with pressure for the case of A equal unity, that is no axial stretch, a special experimental arrangement was used. A large closed loop of tubing with a 117-in. Long straight horizontal portion was filled with water rather than air and was attached to a variable pressure water column for applying the desired pressure. In this manner no appreciable axial load is developed. All measurements of diametral change were made at the midlength of the horizontal portion. All of the experiments were conducted at room temperature, which was about 7- 1°C. RESULTS OF EXPERIMENTS
WiTH
LATEX TUBING
The purpose of the present study was to determine only the elastic response to static loading so some preliminary experiments were made in order to determine at least approximately the limits of the elastic region. The failure load, that is the load causing rupture, occurs approximately at 7 equal to 0.4 and without axial stretch. The assumed limit of elastic performance is for 7~ equal to 0.24 and X equal to 1.55. Obviously, the limits of the elastic region cannot be determined with great precision using the measurement techniques of the present investigation. However, the experimental data obtained are
W. H. HOPPMANN
596
considered to represent reasonably well and they Fig. I. These data were elastic constants assuming material. The theoretical
the latex rubber are presented in used to compute a Mooney type results are also
Iland L. WAN
shown in Fig. 2. Since the Mooney curves are essentially simple parabolas, with no change in sign of curvature, they obviously cannot represent the experimental curves which do have a noticeable change in sign of
12
Theoretical - Mooney material ----
Experimental for latex rubber
6 4 2
-12 -14 -16 -18
0 Fig.
2
4
6
14 16 8 10 12 PRESSURE RATIO, in x102
'18
20
2i
I. For latex rubber, the relation of inflation ratio p to pressure ratio w for various values of A.
24
LARGE
0
2
4
6
DEFORMATION
8
lo
PRESSURE
OF ELASTIC
12 RATIO,
14 x
16
TUBES
18
591
20
22
24
x lo2
Fig. 1. For latex rubber. the relation of inflation ratio p to pressure ratio r for A equal unity.
curvature. That is, in the neighborhood of 7~ equal to zero for the various stretch factors A, the experimental curves have a different sign for curvature than the Mooney type curves whereas at the higher strain levels the curvatures are the same. Using a higher order theory of elasticity than the Mooney theory it can be seen that the experimental data can be better represented in the neighborhood of r equal to zero. For this purpose the finite series representing the energy function was taken to include terms containing second powers of the invariants. COMPARISON OF THEORETICAL RESULTS WITH EXPERIMENTAL DATA OBTAINED WITH AORTAE OF RABBITS
An interesting application of the theory used in the present paper was to determine the
elastic constants for the aorta of a rabbit. The experiments were conducted by Glagov and Wolinsky (1963, 1964, 1967). Inflation experiments were performed on the aortae much the same as those described in the present paper. Air was used to obtain the desired pressures which were sufficiently high to produce very large strains. Also, data for experiments with aortae are given by Pieper and Paul (1969). These latter experiments were confined to low values of pressure. In order to cover the complete strain range for the experiments of Wolinsky and Glagov (1963), calculations were made by the present authors using powers of the invariants up to and including the fifth. The comparison of the experimental data and the theoretical curves are shown in Fig. 3.
W. H. HOPPMANN
598
II and L. WAN
Theoretical -----
Sth order
elasticity
Experimental for aorta of rabbit
wall thickness
20
40
60
80 100 PRESSURE ( mm
120
140
160
180
200
Hg )
Fig. 3. Relation of rabbit radius and wall thickness to distending pressure.
DISCLJSSIOZ(AND CONCLUSIONS
We consider that the results of the present research are sufficiently informative so that we may come at least to some tentative conclusions. The elastic constants have been sufficiently well determined to be used at least in preliminary studies of flow of liquids through distensible tubes as analogues to the recirculatory cardiovascular system. It is clear from the present results that considerable thought must be given to the possible effects of large strains and also to the effects of axial deformation in developing the theory of fluid flow in elastic tubes as proposed, for example, by Campbell and Yang (1969). The question of possible anisotropy in materials of which tubes are constructed must be investigated, but the material for
which the energy W is a function only of the strain invariants is isotropic. It is considered that systematic use of the method proposed in the present paper may help to avoid some of the confusion in biomechanical literature as evidenced in a recent paper by Tichner and Sacks (1967). In their paper the authors, probably inspired by the theory of plasticity, introduced the incremental theory of strain. Also, there seems to be some confusion about the nature of anisotropy as represented in the energy function W. No justification seems to be given for their complex proposal for analyzing the deformation of the aorta. In the development of hemodynamics many ad hoc type of assumptions concerning the compliance of arteries, aortae, and other blood vessels have been used. Both in the in vitro as
LARGE
DEFORMATlON
well as in the in cico studies of biological flows more detailed information concerning the constitutive equations of the materials should be welcomed. It seems improper to use overly simple assumptions concerning the elasticity of the tubes in order to simplify the flow theory. It would appear much more satisfactory to determine the most nearly exact constitutive equations for the tube material and then if necessary, simplify in order to make approximate analyses of biological flows. Campbell and Yang (1969) in studying a flow problem, allowing for deformability of tubes, used a rather simple relation between pressure and change of area of cross-section of tubing, as experimentally determined by Campbell ( 1968), without considering the effect of axial stretch. It is true that their wave analysis for the flow in the tube is difficult enough, even with a linear relation assumed for the elasticity of the tube. however it would seem more suitable in these cases to properly determine the elastic constants for the material. It is considered that the inflation with extension experiments should be routinely made for any particular tubing used in in c&o experiments. Statical experiments using this method will provide some preliminary knowledge at least of the elasticity properties. A further extension to studies of viscoelastic properties will require a development of the method to include dynamic loading. It would certainly be wortwhile to study the response of the tube to pulsating pressures and oscillatory axial loads. Increasing the pressure to upper limits would also give some indication of plastic effects. It is hoped that our results may encourage further research which will determine the most effective form of constitutive equations for studying the problem of flow of liquids in elastic tubes. which is so important in current physiology and biomechanics. Also, it may be of value to associate the experimentally determined regimes of strain with normal
B.M.-VoL
3. ho. 6-F
OF
ELASTIC
TUBES
599
conditions of the and with traumatic organism. Before ending the present paper it may be useful to cite a few references in the recent literature which pertain to the subject. An example of a flow study without consideration of the distensibility of walls is discussed in a recent paper by Lew and Fung (1970). Studies of flow with the usual assumption of linear elasticity for the tubes are exemplified by Miekisz ( I96 I, 1961). In a very interesting recent book edited by Schwan (1969) a fairly extensive survey of the elastic tube concept for the linear case is given in an article on Hemodynamics by Abraham Noordergraft. His list of references is very complete, extending from 1546 to the present. Finally, mention should be made of a survey article by Fung ( 1968). Of special pertinence to the material of the present paper is his general discussion of stress-strain laws for soft tissue. REFERENCES Campbell. J. L. III t 1968) Method of determining some distensible tube properties. Terms. ,-tS.W& J. bm. Eqng 90D. 133. Campbell. J. L. III and Yang. T. (1969) Pulsatile how behavior in elastic systems containing wave reflection sites. Tmrrs. ,4SME 1. bru. Engng 91D. 9% 101. Fung. Y. C. B. (1968) Biomechanics-iti scope. history, and some problems of continuum mechanics in physiology. rlppl. nrech. Rec. 21. I-10. Galgov. S. and Wolinsky. H. ( 1963) Aorric wall as a ‘two-phase‘ material. ~Vnmrr. Lond. 199.606. Green, A. E. and Zerna. W. (1968) T1reorrriccrl Elasricir.v. 2nd Edn. p. 87. Oxford University Press. Oxford. Green. A. E. and Adkins. J. E. (1960) Large EIosric Deformurions. p. 16. Oxford University Press. Oxford. Lew, H. S. and Fung. Y. C. CL9701 Entry How into blood vessels at arbitrary Reynolds number. J. Biomechnics 3. 23-38. Miekisz. S. ( I96 I ) On the dispersion of the propagation velocity of a disturbance in a liouid tlowine through elastic walled vessels. Actor ph~s. pbl. 20.827-830. hliekisz, S. ( 1961) Non-linear theory of viscous flow in elastic tubes. Phxs. .Lled. Biol. 6. 103-109. Mumaghan. F. D. (1950 Finite Dejbrmcrrion of‘ cm Elnsric Solid. p.63. Wiley. New York. Pieper, H. P. and Paul. L. T. ( 1969) Response of aortic smooth muscle studied in intact dogs. .im. J. Plrysiol. 217. 15-t-160. Schwan. Herman P. ( 1969) Biolopicnl Engineering. pp. 556. McGraw-Hill. New York. Tickner. E. G. and Sacks. A. H. t 1967) .A theory for the static elastic behavior of blood vessels. Biorheo1oy.v 4. 141-168.
W. H. HOPPMANN
600
Treloar, L. R. G. (1958) The Physics ofRubber Efasricity. 2nd. Edn. p. 155. Oxford Press. Oxford. Truesdell. C. and NOR. D. (1965) The Non-Lineur Field Theories of Mech&ics‘ (Die N&Linemen Feld Theorien), Encyclopedia of Physics (Handbuck der Physik) (Edited by S. Fliigge), p. 50 and p. I I8 et seq. Truesdell. C. ( 1952)J. rur. Mech. Analysis 1, 179.
Wolinsky, H. and Glagov. S. (1964) Structural basis for the mechanical properties of the aortic media. Circulation Res. 14.400-413. Wolinsky, H. and Glagov, S. (1967) Lamellar unit of aortic structure and function in mammals. Circulation Res. 20,99- I I 1. NOMENCLATURE Reference: Textbook by Green and Zema (1968) W = W(I,,I,,I,) = energy function I, = g”G, = 3 + y: = I st variant 12= G”g,ls = 3 + 4y:+ 2 (y:y: - y;y$) = 2nd variant 1s = ij$+ 2y:= s = 3rd invariant y. = j(GU-g,) Yl =
= strain
PYkJ
P,~= metrical coefficients in unstrained bodv
Y”
GiJ= metrical coefficients in strained body 6: = Kronecker delta G = det Gil
II and L. WAN g=detgil +i = tensorial stress uU = (GJG”) ~~ = physical A = axial stretch ratio r = radius to a material before loading R = corresponding radius with loading r, = radius to outer surface R, = radius to outer surface RI-r
P = L=rt
point in tube cross-section to the same material point before loading after loading
radius in strained state
*S-E P pressure ratio PO PI,= atmospheric pressure P = gage pressure r”~~g’J+$B;J+pG’J
2dW &=xz 2dW -’ = 12 d12 p=21,$
components of stress
3
B” = f,gu-gg”gbG,.