- Email: [email protected]
Waves in initially stressed fluid-filled thick tubes
J. Biomechanic~. Vol. 30, No. 3. pp 173-376, 1997 i;~ 1997 Elsevier Saence Ltd. All rights reserved Prmted in Great Britain GfX?---9290:9? $I 7.00 + 00
PII: ~21-92~%)~146-7
ELSEVIER
TECHNICAL
WAVES IN INITIALLY
STRESSED
NOTE
FLUID-FILLED
THICK
TUBES
Hilmi Demiray* Marmara Research Centre, Research Institute for Basic Sciences, Department of Mathematics, P.O. Box 21, Gebze-Kocaeli, Turkey Abstract--In this paper, treating the artery as a thick wailed cylindrical shell made of an incompressible, elastic and isotropic material and the blood as an incompressible inviscid fluid, by taking the inertia of the wall into account, the propagation of harmonic waves in an initially stressed tube, filled with an inviscid fluid, is studied. Utilizing inner-pressure-inner-cross-sectional-area relation in the linear momentum equation of the fluid, together with the continuity equation, we obtained two nonlinear equations governing the axial velocity and the crosssectional area of the tube. Assuming that the dynamical motion superposed on large initial static deformation is small, a harmonic wave type of solution to incremental equations is sought and the dispersion relation is obtained as a function of transmural pressure, axial stretch, thickness ratio and the wave number. The wave speed is evaluated numerically for various materials and thickness/radius, and the results are depicted in graphical forms. The result indicates that, due to the inertial component of pressure, the wave is dispersive. The present formulation is compared with some previously pubiished works. $0 1997 Elsevier Science Ltd. All rights reserved. i(r~words: Artery; Harmonic waves; Thick tube; Initial stress.
INTRODUCTION
Wave measurement techniques are effectively used, without causing any damage to the specimen, in determining the mechanical and geometrical characteristics of the material under consideration. In the past, several theoretical and experimental studies have been conducted to determine the pulse wave velocity in distensible tubes containing inviscid or viscous fluids (Moodie et nl., 1986; Reuderink er al., 1989). The wave speed changes with material and geometrical characteristics of the artery, as well as the frequency of the wave. Moreover, considering that for a healthy human being the mean transmural pressure is about 100 mm Hg, and in in uivo conditions the axial stretch is about 1.5, one may assume that, during blood flow. the blood vesselsare subjected to large initial static deformations (cf. Demiray, 1976). In order to see how the wave speed changes with large initial deformation, as well as material and geometrical characteristics of the arteries, one should know the explicit expression of the wave speed as a function of these parameters. The majority of theoretical studies either employed small deformation theories (Womersley, 1957) or treated the arteries as thin-walled membranes (Rachev, 1979; Demiray, 1992). Thin shell theories are applicable only when the ratio of thickness of the shell to the mean radius is less than l/20. However, for most arteries this ratio varies between t/4 and l/6. This consideration clearly indicates that membrane or thin shell theories cannot be employed in the analysis of arterial mechanics. Moreover, in assessing the transmurai pressure and cross-sectional inner area relation, the wall inertial effects have been neglected by previous researchers. However, for thick wailed cylindrical tubes the contribution of wall inertia to pressure variation may become important and it changes the character of the wave. Furthermore, the arteries are subjected to large initial deformation. Therefore, in assessing the speed of propagation these physiological, mechanical and geometrical characteristics should be taken into account. This work aims at studying the wave propagation in an initially stressed cylindrical elastic thick tube filled with an incompressible inviscid Huid by using the large deformation theories and the dynamical
pressure area relation. Although blood is known to be an incompressible non-Newtonian fluid, for simplicity in mathematical analysis, as a first approximation we shall treat it as an inviscid fluid (Rudinger, 1970). Moreover, as pointed out by Gow and Taylor (1968) the arterial wall material is known to be incompressible, viscoelastic and anisotropic but we shall treat it here as an incompressible and isotropic elastic material. Considering the physiological conditions that the arteries experience, the propagation of small amplitude harmonic waves superimposed on, the initial large static field is studied. Some special cases, such as a thin tube case, vanishing initial deformation and wall inertial effects are also discussed, and compared with previous reports. Finally, the explicit expression of wave speeds for some typical elastic materials is given and their numerical values are obtained as a function of axial stretch ratio and transmural pressure. The results indicate that, as opposed to previous work, the present approach predicts dispersive waves, due to inertial effects of the wall. BASIC
EQUATIONS
AND
THEORETICAL
PRELIMINARIES
As a result of the assumption that the fluid is inviscid the variation of field quantities in the radial direction of the artery may be neglected. However, the radial changes are included by taking the variation of the cross-sectional area into account. The equation governing the conservation of mass may be given by
where Ai denotes the internal cross-sectional area of the tube, t is the time, z denotes the axial distance, and w is the Auid velocity in the axial direction. The equation of balance of linear momentum in the axial direction may be given by SW SW 1 l?P, 7$+1vz+--=o,
p&
where p is the fluid density, Pi the pressure difference between the inside and outside of the tube. Equations (1) and (2) give only two relations to determine the unknowns Ai, w and Pi. For the complete determination of these field quantities. an additional relationship between these parameters is
Received in final form 16 September 1996. *Also at: Istanbul Technical University, Faculty of Sciences and Letters. Department of Engineering Sciences, Maslak 80626 Istanbul, Turkey. 273
Technical Note
274
needed. For an isotropic and incompressible elastic material this relationship was studied before by Knowles (1960) in connection with his work on large radial oscillations of a cylindrical tube and is given by
and defining the phase velocity r = .Q’< the dispersion relation igt reduces to
(3)
where Q, denotes the phase velocity in the absence 01‘the wall’s inertial effects. Here 1’0 also corresponds to the wave speed obtained by the method of characteristics, which is the same as that given by Simon rt al. (1972). This speed may also be considered as the long wave limit of the general case. We should point out that. in general? the wave is dispersive which results from the inertial effect of the tube wali. So far we have kept the generality of the problem and treated the artery as a chick-walled cylindrical shell. As a special case of the general derivation it might be instructive to study the thin-~valled tube case.
where C is the strainenergy density function, ii; is the axial stretch ratio. Bi is the ratio of deformed inner cross-sectional area to the undeformed inner cross-sectional area A? and other quantities are defined by B, = K(Bi
Rb - R; ;’ =: -, &Rf
f j+>
Here Ri and R, are the inner and outer radii of the tube before deformation. Introducing equation (3) into equation (2) and using the Leibnitz rule for differentiation, we have
Dispersion
relation
,for t~~j?i-~~,fi/~~~ ttthes
The dispersion relation for thin-walled tubes may be obtained from this general formulation by assuming that the thickness h of the wall is quite small as compared to its inner radius R,. that is i: = h/R, is small compared to unity. Expanding the phase velocity, given in equation (1 l), into a power series of E and keeping only the linear terms in E we obtain
(3 where ,Uis the shear modulus of the tube material and S(B,) is defined bY 1 dX(Bi) K dWJ ,&(BJ=-----. (6) i.=Bi - 1 dB, I,& - 1 dB, Also, dividing both sides of equation (11) by A: and using the definition of B, one has
As might be seen from equation (5), due to the contribution of wall inertia term to the inner pressure, the character of this equation is quite different from those employed by previous researchers (cf. Moodie er al., 1986). If the mass density of the wall is set equal to zero in equation (5) we obtain the relation used by Moodie et al. (1986). HARMONIC
WAVE
If the initial deformation vanishes. i.e. I; = 1. BY = 1, then, in terms of the real physical quantities the phase velocity i:O takes the following form
where cHK is the Moens-Korteweg speed for pulse waves in arteries and E is the classical Young’s modulus (p = E/3). Equation (13) is exactly the same as the one obtained by Bergel(l961) for thin-walled tubes. After examining some special cases and comparing the present results with previous reports we may return to investigate the more general case. Thus far we have not used any specific form for the strain energy function. In what follows we shall choose some strain energy functions existing in the current literature. (if For the Mooney-Rivlin elastic material the strain energy function is given by z =; [b(l, - 3) + (1 - h)(l, - 3)-j,
PROPAGATION
Considering that for a healthy human being the systolic pressure is around 120 mmHg, diastolic pressure is about 80 mmHg and the axial stretch i, is around 1.5, the arteries may be assumed to be subjected to a large inner static pressure PO and axial stretch 2,. In the course of blood flow, a small pressure increment is added to this static field and, due to the tethering effect, the axial motion of the tube may be neglected. Hence, under these assumptions, the field equations can be linearized. Denoting the value of Bi after finite static deformation by BP and the increment of it by &. from equations (5) and (7) we have
(8) Seeking a harmonic wave type of solution to equation (8) we obtain the following dispersion relation
where w is the angular frequency and k is the wave number. Introducing the following non-dimensionalized quantities
(141
where p is the shear modulus, b a material constant and I, and l2 are the principal invariants of the Finger deformation tensor. (ii) The strain energy function proposed by Demiray (1972) for soft biological tissues is given by ?1.= f
[exp[r*(l, - 3)] - 1 i,
(13
where N is a material constant to be determined from experimental measurements. (iii) Finally, the strain energy density function proposed by Demiray (1976b) for soft biological tissues as I: = $ (exp[ll(rz
- 3)3 - 1j.
(16)
where J is another material constant to be determined from experimental measurements. Introducing these strain energy density functions into the definition of S(B$, given in equation (6), we may obtain an explicit expression for it, for each case, as a function of initial deformation, mechanical and geometrical characteristics. But in order to save the space we shali not list them here. Having determined the explicit expression of SfB$‘), we can calculate the wave speed as a function of deformation and geometrical characteristics, which will be explored in the following section.
Technical NUMERICAL
RESULTS
AND
275
DISCUSSION
In this section we shall study the variation of wave speed with initial deformation and the thickness parameter. To do this we need the numerical values of material constants b, c( and /% For a typical rubbery material the numerical value of material constants b may be taken as b = 0.9 (Fung, 1972). For the remaining strain energy functions proposed for soft biological tissues, we compared these analytical models with the experimental results by Simon et al. (1972) on canine abdominal artery with geometrical characteristics Ri = 0.31 cm, R, = 0.38 cm and the values of material constants are found to be r* = 1.948 and /I = 0.82. Utilizing these values of material constants in the wave speed Do corresponding to longwave limit, or neglecting the wall inertial effects, for each material the speed is evaluated numerically and the results are depicted in Figs l-3. Figure 1 shows that, for MooneyyRivlin material (engineering material) the wave speed decreases with circumferential and axial stretch ratios while it increases with thickness ratio. The variations of wave speeds of the materials corresponding to exponential models proposed for soft biological tissues are, respectively, given in Figs 2 and 3. For both cases the wave speed increases with circumferential and axial stretch ratios and the thickness parameter. The main reason for such a different behavior of engineering and biological materials is that, due to inner structure, the tangent modulus, or incremental Young modulus, of an engineering material decreases with deformation whereas for soft biological tissues it gets stiffer with increasing deformation. This result is also consistent with the findings of Atabek and Lew (1966). As a final remark, we note that for the models of soft biological tissues, the variations of wave
---x,=1.3 -h,=1.5
Note
himmy-Rivlm
o.;Io
010
Fig. 3. The variations of phase velocity of longwave limit for soft biological tissues (model II) with stretches and thickness ratio.
speeds follow similar patterns and the first model gives larger propagation speed. Before concluding this work the following remarks can be made. As pointed out in the main text, the present formulation may be considered as the generalization of the previous work, in the sense that the present derivation takes the initial stress, the effect of thickness and the wall inertia (d’Alembert force) into account. As a result of this and as opposed to the previous work, the present work gives dispersive wave and the speed of it is smaller than that of previous studies. This work can be extended to include the anisotropic and viscoelastic effects. The result reported here may be used in modern biomechanics, like other works in the literature. in estimating the values of elastic coefficients and the changes in some of the geometrical characteristics of the arterial wall. For instance, the increase of wave speed from time to time may be the result of increasing elastic stiffness and/or the thickness parameter or the development of high blood pressure, each of which may be the cause of a disease related to circulatory system. Acknowledyements-This work is supported by Turkish Academy of Sciences and TUBITAK, Mechanics and Applied Mathematics Research Unit.
REFERENCES
Fig. 1. The variations of phase velocity of longwave limit for Mooney-Rivlin material with stretches and thickness ratio.
6.00 :
---x,=1.3 -&=1.5 /
4.00
Li&$pModel I
2=0.2
w
4 /
/ ./ * ,f,f
/ // /
2.00
/ / ,' ,' /
/
0.00 /I 1.00
2.00
A;
3.00
Fig. 2. The variations of phase velocity of longwave limit for biological tissues (model I) with stretches and thickness ratio.
soft
Anliker. M., Moritz, W. E. and Ogden, E. (1968) Transmission characteristics of axial waves in blood vessels. J. Bionwchanics 1, 235-246. Atabek, H. B. and Lew. H. S. (1966) Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophys. J. 7,480-503. Bergel, D. H. (1961) The static elastic properties of arterial wall. /. Physiol. 156, 44-457. Demiray, H. (1972) On the elasticity of soft biological tissues. J. Biomechanics 5, 309-3 11. Demiray, H. (1976a) Some basic problems in biophysics, Bull. Math. Biol. 38, 701-711. Demiray, H. (1976b) Stresses in ventricular wall. J. Appl. Mech. Trarzs. ASME 98, 194-197. Demiray, H. (1992) Wave propagation through a viscous Auid contained in a prestressed thin elastic tube. Int. J. Enyny Sci. 30, 1607-1620. Fung, Y. C. (1972) Stress-strain history relation of soft tissues in simple elongation. In Biomechanics: Its Foundation and Objectives (Edited by Fung, Y. C., Peronne, N. and Anliker, M.). Prentice-Hall, Englewood Cliffs, NJ. Gow, B. S. and Taylor, M. G. (1968) Measurement of viscoelastic properties of arteries in the living dog. Circ. Rex 23, 11 l--122. Knowles, J. K. (1960) Large amplitude oscillations of a tube of incompressible elastic material. Q. Appl. Math. 18, 71-77.
770
Technical
Moodre. ‘I‘. B.. Barclay, D. W. and Greenwald. S. E. (1986) Impulse propagation in liquid tilled distensible tubes: theory and experiment for intermediate to long wavelengths. .4ctir Mechunicc~ 59, 47758. Rachev. A. I. (1979) Effect of transmural pressure and muscular activity on pulse waves in arteries. J. Biornech. Eng. ASME 102, 119-123. Reuderink, P. J., Hoogstraten. H. W.. Sipkema, P.. Hillen, B. and Westerhof. N. (1989) Linear and nonlinear one-dimensional models of pulse wave transmission at high Womersley numbers. J. Biornechtrnic~s 22, 8 19m 827.
Note Rudinger, G. (1970) Shock waves in mathematical models r~f the aotta .I. Appl. Mrch. 37, 34-m37. Simon B. R.. Kobayashi. A. S.. Stradness. D. E. and Wiederhielm. c‘. A. (1977) Re-evaluation of arterial constitutive laws. C~IY. Raps. 30, 491m-500. Womersley. J. R. (1957) An elastic tube theory of pulse transmissron and oscillatory flow in mammalian arteries. WADC Tech. Rep. TR 56-614.
PII: ~21-92~%)~146-7
ELSEVIER
TECHNICAL
WAVES IN INITIALLY
STRESSED
NOTE
FLUID-FILLED
THICK
TUBES
Hilmi Demiray* Marmara Research Centre, Research Institute for Basic Sciences, Department of Mathematics, P.O. Box 21, Gebze-Kocaeli, Turkey Abstract--In this paper, treating the artery as a thick wailed cylindrical shell made of an incompressible, elastic and isotropic material and the blood as an incompressible inviscid fluid, by taking the inertia of the wall into account, the propagation of harmonic waves in an initially stressed tube, filled with an inviscid fluid, is studied. Utilizing inner-pressure-inner-cross-sectional-area relation in the linear momentum equation of the fluid, together with the continuity equation, we obtained two nonlinear equations governing the axial velocity and the crosssectional area of the tube. Assuming that the dynamical motion superposed on large initial static deformation is small, a harmonic wave type of solution to incremental equations is sought and the dispersion relation is obtained as a function of transmural pressure, axial stretch, thickness ratio and the wave number. The wave speed is evaluated numerically for various materials and thickness/radius, and the results are depicted in graphical forms. The result indicates that, due to the inertial component of pressure, the wave is dispersive. The present formulation is compared with some previously pubiished works. $0 1997 Elsevier Science Ltd. All rights reserved. i(r~words: Artery; Harmonic waves; Thick tube; Initial stress.
INTRODUCTION
Wave measurement techniques are effectively used, without causing any damage to the specimen, in determining the mechanical and geometrical characteristics of the material under consideration. In the past, several theoretical and experimental studies have been conducted to determine the pulse wave velocity in distensible tubes containing inviscid or viscous fluids (Moodie et nl., 1986; Reuderink er al., 1989). The wave speed changes with material and geometrical characteristics of the artery, as well as the frequency of the wave. Moreover, considering that for a healthy human being the mean transmural pressure is about 100 mm Hg, and in in uivo conditions the axial stretch is about 1.5, one may assume that, during blood flow. the blood vesselsare subjected to large initial static deformations (cf. Demiray, 1976). In order to see how the wave speed changes with large initial deformation, as well as material and geometrical characteristics of the arteries, one should know the explicit expression of the wave speed as a function of these parameters. The majority of theoretical studies either employed small deformation theories (Womersley, 1957) or treated the arteries as thin-walled membranes (Rachev, 1979; Demiray, 1992). Thin shell theories are applicable only when the ratio of thickness of the shell to the mean radius is less than l/20. However, for most arteries this ratio varies between t/4 and l/6. This consideration clearly indicates that membrane or thin shell theories cannot be employed in the analysis of arterial mechanics. Moreover, in assessing the transmurai pressure and cross-sectional inner area relation, the wall inertial effects have been neglected by previous researchers. However, for thick wailed cylindrical tubes the contribution of wall inertia to pressure variation may become important and it changes the character of the wave. Furthermore, the arteries are subjected to large initial deformation. Therefore, in assessing the speed of propagation these physiological, mechanical and geometrical characteristics should be taken into account. This work aims at studying the wave propagation in an initially stressed cylindrical elastic thick tube filled with an incompressible inviscid Huid by using the large deformation theories and the dynamical
pressure area relation. Although blood is known to be an incompressible non-Newtonian fluid, for simplicity in mathematical analysis, as a first approximation we shall treat it as an inviscid fluid (Rudinger, 1970). Moreover, as pointed out by Gow and Taylor (1968) the arterial wall material is known to be incompressible, viscoelastic and anisotropic but we shall treat it here as an incompressible and isotropic elastic material. Considering the physiological conditions that the arteries experience, the propagation of small amplitude harmonic waves superimposed on, the initial large static field is studied. Some special cases, such as a thin tube case, vanishing initial deformation and wall inertial effects are also discussed, and compared with previous reports. Finally, the explicit expression of wave speeds for some typical elastic materials is given and their numerical values are obtained as a function of axial stretch ratio and transmural pressure. The results indicate that, as opposed to previous work, the present approach predicts dispersive waves, due to inertial effects of the wall. BASIC
EQUATIONS
AND
THEORETICAL
PRELIMINARIES
As a result of the assumption that the fluid is inviscid the variation of field quantities in the radial direction of the artery may be neglected. However, the radial changes are included by taking the variation of the cross-sectional area into account. The equation governing the conservation of mass may be given by
where Ai denotes the internal cross-sectional area of the tube, t is the time, z denotes the axial distance, and w is the Auid velocity in the axial direction. The equation of balance of linear momentum in the axial direction may be given by SW SW 1 l?P, 7$+1vz+--=o,
p&
where p is the fluid density, Pi the pressure difference between the inside and outside of the tube. Equations (1) and (2) give only two relations to determine the unknowns Ai, w and Pi. For the complete determination of these field quantities. an additional relationship between these parameters is
Received in final form 16 September 1996. *Also at: Istanbul Technical University, Faculty of Sciences and Letters. Department of Engineering Sciences, Maslak 80626 Istanbul, Turkey. 273
Technical Note
274
needed. For an isotropic and incompressible elastic material this relationship was studied before by Knowles (1960) in connection with his work on large radial oscillations of a cylindrical tube and is given by
and defining the phase velocity r = .Q’< the dispersion relation igt reduces to
(3)
where Q, denotes the phase velocity in the absence 01‘the wall’s inertial effects. Here 1’0 also corresponds to the wave speed obtained by the method of characteristics, which is the same as that given by Simon rt al. (1972). This speed may also be considered as the long wave limit of the general case. We should point out that. in general? the wave is dispersive which results from the inertial effect of the tube wali. So far we have kept the generality of the problem and treated the artery as a chick-walled cylindrical shell. As a special case of the general derivation it might be instructive to study the thin-~valled tube case.
where C is the strainenergy density function, ii; is the axial stretch ratio. Bi is the ratio of deformed inner cross-sectional area to the undeformed inner cross-sectional area A? and other quantities are defined by B, = K(Bi
Rb - R; ;’ =: -, &Rf
f j+>
Here Ri and R, are the inner and outer radii of the tube before deformation. Introducing equation (3) into equation (2) and using the Leibnitz rule for differentiation, we have
Dispersion
relation
,for t~~j?i-~~,fi/~~~ ttthes
The dispersion relation for thin-walled tubes may be obtained from this general formulation by assuming that the thickness h of the wall is quite small as compared to its inner radius R,. that is i: = h/R, is small compared to unity. Expanding the phase velocity, given in equation (1 l), into a power series of E and keeping only the linear terms in E we obtain
(3 where ,Uis the shear modulus of the tube material and S(B,) is defined bY 1 dX(Bi) K dWJ ,&(BJ=-----. (6) i.=Bi - 1 dB, I,& - 1 dB, Also, dividing both sides of equation (11) by A: and using the definition of B, one has
As might be seen from equation (5), due to the contribution of wall inertia term to the inner pressure, the character of this equation is quite different from those employed by previous researchers (cf. Moodie er al., 1986). If the mass density of the wall is set equal to zero in equation (5) we obtain the relation used by Moodie et al. (1986). HARMONIC
WAVE
If the initial deformation vanishes. i.e. I; = 1. BY = 1, then, in terms of the real physical quantities the phase velocity i:O takes the following form
where cHK is the Moens-Korteweg speed for pulse waves in arteries and E is the classical Young’s modulus (p = E/3). Equation (13) is exactly the same as the one obtained by Bergel(l961) for thin-walled tubes. After examining some special cases and comparing the present results with previous reports we may return to investigate the more general case. Thus far we have not used any specific form for the strain energy function. In what follows we shall choose some strain energy functions existing in the current literature. (if For the Mooney-Rivlin elastic material the strain energy function is given by z =; [b(l, - 3) + (1 - h)(l, - 3)-j,
PROPAGATION
Considering that for a healthy human being the systolic pressure is around 120 mmHg, diastolic pressure is about 80 mmHg and the axial stretch i, is around 1.5, the arteries may be assumed to be subjected to a large inner static pressure PO and axial stretch 2,. In the course of blood flow, a small pressure increment is added to this static field and, due to the tethering effect, the axial motion of the tube may be neglected. Hence, under these assumptions, the field equations can be linearized. Denoting the value of Bi after finite static deformation by BP and the increment of it by &. from equations (5) and (7) we have
(8) Seeking a harmonic wave type of solution to equation (8) we obtain the following dispersion relation
where w is the angular frequency and k is the wave number. Introducing the following non-dimensionalized quantities
(141
where p is the shear modulus, b a material constant and I, and l2 are the principal invariants of the Finger deformation tensor. (ii) The strain energy function proposed by Demiray (1972) for soft biological tissues is given by ?1.= f
[exp[r*(l, - 3)] - 1 i,
(13
where N is a material constant to be determined from experimental measurements. (iii) Finally, the strain energy density function proposed by Demiray (1976b) for soft biological tissues as I: = $ (exp[ll(rz
- 3)3 - 1j.
(16)
where J is another material constant to be determined from experimental measurements. Introducing these strain energy density functions into the definition of S(B$, given in equation (6), we may obtain an explicit expression for it, for each case, as a function of initial deformation, mechanical and geometrical characteristics. But in order to save the space we shali not list them here. Having determined the explicit expression of SfB$‘), we can calculate the wave speed as a function of deformation and geometrical characteristics, which will be explored in the following section.
Technical NUMERICAL
RESULTS
AND
275
DISCUSSION
In this section we shall study the variation of wave speed with initial deformation and the thickness parameter. To do this we need the numerical values of material constants b, c( and /% For a typical rubbery material the numerical value of material constants b may be taken as b = 0.9 (Fung, 1972). For the remaining strain energy functions proposed for soft biological tissues, we compared these analytical models with the experimental results by Simon et al. (1972) on canine abdominal artery with geometrical characteristics Ri = 0.31 cm, R, = 0.38 cm and the values of material constants are found to be r* = 1.948 and /I = 0.82. Utilizing these values of material constants in the wave speed Do corresponding to longwave limit, or neglecting the wall inertial effects, for each material the speed is evaluated numerically and the results are depicted in Figs l-3. Figure 1 shows that, for MooneyyRivlin material (engineering material) the wave speed decreases with circumferential and axial stretch ratios while it increases with thickness ratio. The variations of wave speeds of the materials corresponding to exponential models proposed for soft biological tissues are, respectively, given in Figs 2 and 3. For both cases the wave speed increases with circumferential and axial stretch ratios and the thickness parameter. The main reason for such a different behavior of engineering and biological materials is that, due to inner structure, the tangent modulus, or incremental Young modulus, of an engineering material decreases with deformation whereas for soft biological tissues it gets stiffer with increasing deformation. This result is also consistent with the findings of Atabek and Lew (1966). As a final remark, we note that for the models of soft biological tissues, the variations of wave
---x,=1.3 -h,=1.5
Note
himmy-Rivlm
o.;Io
010
Fig. 3. The variations of phase velocity of longwave limit for soft biological tissues (model II) with stretches and thickness ratio.
speeds follow similar patterns and the first model gives larger propagation speed. Before concluding this work the following remarks can be made. As pointed out in the main text, the present formulation may be considered as the generalization of the previous work, in the sense that the present derivation takes the initial stress, the effect of thickness and the wall inertia (d’Alembert force) into account. As a result of this and as opposed to the previous work, the present work gives dispersive wave and the speed of it is smaller than that of previous studies. This work can be extended to include the anisotropic and viscoelastic effects. The result reported here may be used in modern biomechanics, like other works in the literature. in estimating the values of elastic coefficients and the changes in some of the geometrical characteristics of the arterial wall. For instance, the increase of wave speed from time to time may be the result of increasing elastic stiffness and/or the thickness parameter or the development of high blood pressure, each of which may be the cause of a disease related to circulatory system. Acknowledyements-This work is supported by Turkish Academy of Sciences and TUBITAK, Mechanics and Applied Mathematics Research Unit.
REFERENCES
Fig. 1. The variations of phase velocity of longwave limit for Mooney-Rivlin material with stretches and thickness ratio.
6.00 :
---x,=1.3 -&=1.5 /
4.00
Li&$pModel I
2=0.2
w
4 /
/ ./ * ,f,f
/ // /
2.00
/ / ,' ,' /
/
0.00 /I 1.00
2.00
A;
3.00
Fig. 2. The variations of phase velocity of longwave limit for biological tissues (model I) with stretches and thickness ratio.
soft
Anliker. M., Moritz, W. E. and Ogden, E. (1968) Transmission characteristics of axial waves in blood vessels. J. Bionwchanics 1, 235-246. Atabek, H. B. and Lew. H. S. (1966) Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophys. J. 7,480-503. Bergel, D. H. (1961) The static elastic properties of arterial wall. /. Physiol. 156, 44-457. Demiray, H. (1972) On the elasticity of soft biological tissues. J. Biomechanics 5, 309-3 11. Demiray, H. (1976a) Some basic problems in biophysics, Bull. Math. Biol. 38, 701-711. Demiray, H. (1976b) Stresses in ventricular wall. J. Appl. Mech. Trarzs. ASME 98, 194-197. Demiray, H. (1992) Wave propagation through a viscous Auid contained in a prestressed thin elastic tube. Int. J. Enyny Sci. 30, 1607-1620. Fung, Y. C. (1972) Stress-strain history relation of soft tissues in simple elongation. In Biomechanics: Its Foundation and Objectives (Edited by Fung, Y. C., Peronne, N. and Anliker, M.). Prentice-Hall, Englewood Cliffs, NJ. Gow, B. S. and Taylor, M. G. (1968) Measurement of viscoelastic properties of arteries in the living dog. Circ. Rex 23, 11 l--122. Knowles, J. K. (1960) Large amplitude oscillations of a tube of incompressible elastic material. Q. Appl. Math. 18, 71-77.
770
Technical
Moodre. ‘I‘. B.. Barclay, D. W. and Greenwald. S. E. (1986) Impulse propagation in liquid tilled distensible tubes: theory and experiment for intermediate to long wavelengths. .4ctir Mechunicc~ 59, 47758. Rachev. A. I. (1979) Effect of transmural pressure and muscular activity on pulse waves in arteries. J. Biornech. Eng. ASME 102, 119-123. Reuderink, P. J., Hoogstraten. H. W.. Sipkema, P.. Hillen, B. and Westerhof. N. (1989) Linear and nonlinear one-dimensional models of pulse wave transmission at high Womersley numbers. J. Biornechtrnic~s 22, 8 19m 827.
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