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Large-deformation analysis of aortic valve leaflets during diastole
Engmeerinp
Frocrure
Mechanics
Vol.
2.
No. 5. PP. 773-785.
1985
0013-7944185
Pnnted I” the U.S.A.
$3.00 + .OO
0 1985 Pergamon Press Ltd.
LARGE-DEFORMATION VALVE LEAFLETS Department
ANALYSIS OF AORTIC DURING DIASTOLE
MOHAMED S. HAMID, HAN1 N. SABBAH and PAUL D. STEIN of Medicine (Division of Cardiovascular Medicine) and Department of Surgery, Henry Ford Hospital, Detroit, MI 48202, U.S.A.
Abstract-The stress distribution on the aortic valve leaflet during diastole under increasing loading conditions was calculated using nonlinear large-deformation finite elements. The geometry of the leaflet was derived based upon the actual configuration of the closed aortic valve leaflets obtained from a cast of the root of the aorta. Variability of the leaflet thickness was incorporated in the finite-element model. The leaflet material was assumed to be isotropic and three values of Young’s modulus (E) were used. The first two E values that were used were assumed constant at 300 and 5000 kPa, which corresponded to pretransition and posttransition values, respectively, in the stress-strain curve. The third E that was used assumed a stressdependent value according to a trilinearized approximation of the stress-strain curve. The stress distribution on the leaflet at diastolic aortic pressures of 9.33 and 16.00 kPa (70 and 120 mm Hg) are presented. The large-deformation analysis predicted lower maximal principal stresses than linear theory.
INTRODUCTION AN IMPORTANT advantage
of bioprosthetic valves over mechanical prosthetic valves is decreased thrombogenicity, and the elimination of the need for anticoagulation. The longevity of these valves, however, is now recognized to be limited[l]. The durability of bioprosthetic valves is thought by some to depend in part upon the optimization of design parameters[2]. Such optimization may be achieved by better understanding of the magnitude and distribution of stresses developed in natural valve leaflets under physiological loading conditions. The determination of the stress distribution within the valve leaflets may suggest sites of potential susceptibility to tissue failure, and therefore may contribute to the identification of potential sites of degeneration. The distribution of stresses within the aortic valve leaflets has been determined by some investigators using finite-element analysis[3] and it has also been determined by closed form solutions[4]. The majority of these investigations, however, has been based upon linear theory even though the aortic valve leaflets may undergo large deformation which can introduce nonlinearities. Nonlinear finite-element stress analysis of bioprosthetic valve leaflets was performed by Christie and Medland using incompressible material[5]. Their formulation and solution of the highly nonlinear membrane problems using simplex finite elements was based upon the work of Oden[6]. A nonlinear finite-element analysis of natural aortic valve leaflets using compressible material was reported by Hamid and Ghista[7]. The purpose of this study was to determine the distribution of stresses on aortic valve leaflets during diastole using nonlinear finite-element analysis. Solutions were obtained using an aortic valve leaflet of nonuniform thickness and variable elastic moduli.
METHODS Assumptions 1. The leaflet material was considered
2.
3. 4. 5.
to be compressible and isotropic with a Poisson’s ratio of 0.3. The initial relaxed leaflet geometry was approximated by an elliptic paraboloid (Fig. 1). The salient parameters, i.e. height [not including coapting surface (Fig. l)], major and minor axes were approximated from measurements taken from a mold of the root of the aorta of a normal human being, even though the mold was prepared at 100 mm Hg[8]. All three aortic valve leaflets were assumed to be equal in size and symmetrical. The aortic valve annulus deformation during loading was neglected. The coaptation surfaces of the leaflets were not incorporated in the model. 773
774
M. S. HAMID et al.
I
AORTA
VENTRICLE
Fig. 1. Top: Two-dimensional drawing of the closed aortic valve showing the height (h) used to derive the relaxed leaflet shape. The coaptafion surface was excluded from the model. Bottom: Three-dimensional view of the modeled leaflet.
6. All leaflet boundaries including the coapting edge were rigidly fixed. 7. Membrane shell elements were used in the analysis, therefore bending effects were neglected compared to in-plane membrane forces. Leajlet
thickness
Topographic variability of thickness of the aortic valve leaflet was incorporated into the model based upon data reported by Clark and Finke[9]. Figure 2 shows a map of the topographic distribution of leaflet thickness used in our model. This distribution is similar to the idealized configuration used by others[3]. The idealized leaflet thickness distribution used in our study differs somewhat from that of Clark and Finke[9] and Cataloglu et a1.[3]. Elastic
properties
The finite-element solution was obtained for three separate conditions of Young’s modulus (E). The values of E were calculated from the stress-strain curves of aortic leaflet tissue obtained by others[lO]. The first two analyses were based on constant values of E of 300 and 5000 kPa which corresponded to the pretransition and posttransition values in the stress-strain curve. The third analysis assumed that the Young’s modulus was stress dependent and a trilinear approximation was used as shown in Fig. 3.
Stress analysis of aortic valve leaflets
Fig. 2. Topographic
distribution
600
of leaflet thickness (mm) used in the finite-element
modeling.
1
500
400
300
200
100
0 ELONGATION (percent) Fig. 3. Stress-strain curve of aortic valve leaflet. based upon data used by others[lO]. A trilinearized approximation was made of the curve to evaluate E.
Finite
element
modeling
and solution
The finite-element discretization of the aortic leaflet model is shown in Fig. 4. Triangular elements were used. The mesh consisted of 276 elements with 157 nodes which required the solution of 471 equations. The derivation of the element membrane stiffness matrix was based upon the hypothesis of large deformation, large rotation and small strain. The governing equations in incremental form were obtained as
(1) where [ZCjis the incremental tangent stiffness matrix, {q} is the incremental displacement for the load increment {P}. The element formulation is shown in the Appendix and is in accordance with the procedure given by Bathe[ 111. Equation (1) was solved by using a wavefront band algorithm in which the solution progressed as soon as the equation of a node was formed[l2]. Solution of the nonlinear equation was obtained by increasing the load from 1 to 120 mm Hg. An initial equibiaxial stress of 0.001 kPa was assumed in all the elements in order to smooth out the surface[5]. The first load increment was 0.13 kPa (1 mm Hg). The load was then incremented by 0.67 kPa (5 mm Hg) up to 4.00 kPa (30 mm Hg), and after that a load increment of 1.33 kPa (10 mm Hg) was used
776
Fig. 4. Finite-element
discretization of aortic valve leaflet. 192 refers to element number. AD is the center line of the leaflet.
up to 16.00 kPa (120 mm Hg). With each increment of load, the geometry was updated to account for changes of load direction. The incremental solution procedure used in this study facilitated the incorporation of the variable Young’s modulus according to the stress level in the element evaluated at the previous load increment. The analysis was performed using a 16K word memory minicomputer (Hewlett-Packard. model 21MX) on line with a magnetic tape drive and a pen plotter. This interactive arrangement facilitated the plotting of the mesh, stresses and deformed shapes at each load increment. RESULTS Stress distribution for
E = 300 kPa For the pretransition value of E = 300 kPa. the deformed shapes of the aortic leaflet at a pressure of 9.33 and 16.00 kPa (70 and 120 mm Hg) are shown in Fig. 5. A marked difference
9.3 kPa (70mm
16.0 kPa ( 120mm Hg)
I-Id
SHAPE
SHAPE
MAX. PRINCIPAL
STRESS
013
-
15 x10’
Pa
01s
-
16 x104
Pa
MAX. PRINCIPAL
STRESS
OlO-2ox1o’Pa
Fig. 5. Deformed shape and map of maximal principal stresses for pressures of 9.33 kPa and 16.00 kPa using a constant Young’s modulus of 300 kPa.
777
Stress analysis of aortic valve leaflets
in shape was observed at 16.00 kPa in comparison to 9.33 kPa. The displacement (shape change) of the leaflet along the center line is shown in Fig. 6 for a pressure of 16 kPa. At 9.33 kPa the maximal principal stresses were of the order of 130-150 kPa and increased to 190-200 kPa as the load was increased to 16.00 kPa (Fig. 5). The maximal principal stresses were located near the aortic attachment of the leaflets within the zone of the thinnest portion of the leaflet (Figs. 2 and 5). The maximal principal stresses were in general aligned along the circumferential direction (Fig. 7). However, near the center of the leaflet the principal stresses were almost equibiaxial. This gave rise to low principal shear stresses near the center as seen in Fig. 7. Compressive stresses were only observed along the coapting edge of the leaflet and were in the range of 0.7-25 kPa at a pressure of 16.00 kPa (Fig. 7). The total reaction forces exerted by the leaflet on the surrounding support (aorta) are shown in Fig. 8. Stress distribution for E = 5ooOkPa
For the posttransition value of E = 5000 kPa, the deformed shapes of the aortic valve leaflet at a pressure of 9.33 and 16.00 kPa were not appreciably different from each other (Fig. 9). The deformed shapes were also not much different than the leaflet shape present prior to any load imposition. A distinct difference, therefore, was present in the shape of the deformed leaflet at the selected pressures when using an E value of 300 kPa in compression to an E value of 5000 kPa. The displacement of the leaflet along the center-line for a pressure of 16 kPa is shown in Fig. 6. The stress distribution at 9.33 kPa and 16.00 kPa are shown in Fig. 9. At 9.33 kPa the maximal principal stresses were of the order of 180-200 kPa and increased to 310-350 kPa as pressure was increased to 16.00 kPa. The maximal stresses were again located near the aortic attachment of the leaflets within the zone of the thinnest portion of the leaflet as shown in Fig. 9. The maximal principal stresses were also aligned along the circumferential direction (Fig. 7). Compressive stresses in the range of 8-105 kPa were present near the coapting edge of the leaflet at a pressure of 16.00 kPa (Fig. 7). A contour plot of the principal shear stresses is also shown in Fig. 7. The maximal principal shear stresses were twice as high as those observed for E = 300 kPa. The maximal total reaction force exerted by the leaflet on the surrounding support was 0.19 N at a pressure of 16.00 kPa (Fig. 8).
1. E= 300 kPa 2. E= Variable 3 E=5000 kPa 16.017 -
10.67
-
5.33
-
267
-
.Coapting
R
Fig. 6. Deformation
DIRECTION
point
( mm 1
of the center line of the leaflet (AD in Fig. 4) for various values of E. The R and Z directions are as in Fig. 1.
778
M. S. HAMID et al.
i0
SHEAR STRESS (kPa)
STRESS DIRECTION
Fig. 7. Left: Principal shear stress contours. Right: Direction of principal stresses (dotted lines refer to compressive stresses). Top: Values predicted for E = 300 kPa. Middle: Values for 5000 kPa. Bottom: Values for stress-dependent E. All diagrams are based on a pressure of 16.00 kPa.
779
Stress analysis of aortic valve leaflets 40 -
35 T
0f
1. Variable E 2. E=300 kPa 3. E=5000 kPa
30-
x 25
c
0
10
30
20
ANGLE
40
50
60
@
Fig. 8. Predicted reaction forces as a function of the angle I$ which changes from 0 to 60” along the leaflet interface with the aorta as shown in the insert diagram.
93 kPa
16.0 kPa
(7Omm HQ)
(12Omm lip)
SHAPE
0
15 - 17 rtcr’ Pa
0
18 - 20 x10* P*
MAX. PRncIAL
SHAPE
STRESS
_ m
27 - SO x10’ Pa
0
31 - 35 xto4 Pa
MAX. WAL
STRESS
Fig. 9. Deformed shape of leaflet and map of maximal principal stresses using a constant E of 5000 kPa.
780 Stress
M. S. HAMID et al. distribution
using
a trilinearized
approximation
of E
The deformed leaflet shapes predicted when using a stress dependent E for pressues of 9.33 and 16.00 kPa are shown in Fig. 10. The deformed shapes were similar to those seen for E = 300 kPa. The displacement of the leaflet along the centerline for a pressure of 16 kPa is shown in Fig. 6. The maximal principal stresses at 9.33 and 16.00 kPa were also of the same order of magnitude as those obtained with E = 300 kPa (Fig. IO), and were located in the same area of the leaflet_. The principal shear stresses were of the same order of magnitude as seen with E = 300 kPa, however, the distribution was different (Fig. 7). The total reaction forces are shown in Fig. 8. Comparison
between
the linear
and nonlinear
solution
A linear analysis of the stress distribution of the aortic valve leaflet was also performed at 16.00 kPa (Fig. 11). The stress distribution in the aortic valve leaflet predicted by the nonlinear
16.0 kPa
9.3 kPa (70mm
( 120mm Hg)
Hg)
SHAPE
SHAPE
MAX. PRINCIPAL
n
10 -
0
15 - 18
0
20 - 25
STRESS
14 x10’ x10’
Pm MAX. PRINCIPAL
STRESS
Pa 10’
Pa
Fig. 10. Deformed shape and map of maximal principal stresses using a stress-dependent E that was derived on the basis of the trilinearized approximation of the stress-strain curve.
c)30-
35
xl04Pa
@&+36-45x104Pa 0
46 - 50 x104 Pa
Fig. 11. Map of maximal principal stresses at a pressure of 16.0 kPa using small-deformation (linear) theory.
781
Stress analysis of aortic valve leaflets
solution was different than that predicted by the linear solution. In general, the linear solution predicted higher stresses which were located at a different site on the leaflet (Fig. 11). A typical relationship between stress and pressure for one element (element 192, identified in Fig. 4) for both the linear and nonlinear (large deformation) solution is shown in Fig. 12. The circumferential strain variations for element 192 for E = 300 kPa, 5000 kPa and for a stress dependent E are shown in Fig. 13.
_ 10.66
ELEMENT
2.67
STRESS
+
192
(kPa)
Fig. 12. Comparison of maximal principal stresses in a single element (element 192 shown in Fig. 4) calculated on the basis of linear theory as well as nonlinear theorv. using_ various values of E.
16.00
r
13.33
-
2 00’ no r;r
A 2 X
10.67
-
k
8.00
-
$ K a.
Element
0
5
10
CIRCUMFERENTIAL
Fig. 13. Circumferential
15
+
20 STRAIN
192
25
30
( % )
strain as a function of pressure for element 192 depicting the variation with E.
782
M. S. HAMID et al.
DISCUSSION The magnitude of the maximal principal stress predicted for a low modulus of elasticity of 300 kPa was approximately 60% of the value predicted when a higher modulus of 5000 kPa was used. This was due to large deformations which occurred during the initial loading of the leaflet. The lower maximal principal stress observed when a stress dependent modulus was used can also be attributed to such initial large deformations. The nonlinear solution applied in this study also predicted a lower maximal principal stress than the linear solution, This is primarily due to apparent increased stiffness of the leaflet as a result of the large deformations. Although a value of E of 5000 kPa was used to predict the stress distribution on the leaflet for the linear solution, the magnitude of the modulus does not affect these results. From elementary shell theory, it is well known that membrane behavior is independent of material properties as long as the deformations remain small[ IO]. The results of both the nonlinear and linear solutions showed that the maximal principal stresses were aligned along the circumferential direction as observed by others[ 131. The site of the maximal principal stresses predicted by the application of the nonlinear theory was also different than that predicted by the linear solution. Even though high stresses were located in both solutions at the thinnest portion of the leaflet, the maximal stresses were concentrated near the center of this region in the linear prediction; whereas they were located nearer to the aortic attachment of the leaflet when the nonlinear solution was used. This may suggest sites of potential failure which may not be predicted on the basis of the commonly used linear model. Our model predicted compressive stresses near the coapting edge of the aortic valve leaflet, irrespective of the value of E that was used. The presence of compressive stresses at this site may be attributed to the assumed fixed boundary conditions along the coaptation edge. However, these compressive stresses were small in comparison to the maximal tensile stresses that occurred at other sites. In our model the leaflet material was assumed to be compressible and isotropic. The assumption was made in spite of the general belief that soft biological tissues are incompressible and anisotropic. Our intent was to assess the effects of geometric and material nonlinearities on the magnitude and distribution of stresses. Linear finite-element stress analysis of natural aortic leaflets using compressible and isotropic material properties has been reported[3, 10, 141. The initial relaxed geometry of the aortic leaflet was approximated by an elliptic paraboloid. This approximation was derived from the geometry of a mold of the root of the aorta pressurized at 13.33 kPa and excluding the coapting surfaces. Even though this particular geometry may not define the exact shape of the relaxed aortic leaflet, the latter remains an unknown parameter[5]. Nevertheless, others have used the elliptic paraboloid geometry to model the aortic valve leaflet[lO]. Other shapes including spherical[3, 71, elliptical[4] and paraboloid[3] were also used to approximate the geometry of the natural aortic valve leaflet. In this study the leaflets of the aortic valve were assumed to be equal in size and symmetrical. This assumption was made in spite of the fact that the three leaflets of the natural aortic valve are usually different in size and seldom exactly symmetrical. Nevertheless, to a good level of approximation, it may be assumed that the lack of symmetry does not affect the mechanics of statically loaded valves[5]. Even though leaflet symmetry was assumed, the entire leaflet was considered for analysis. This was done to maintain continuity in future studies where the introduction of local variations would alter the symmetric conditions. Changes of the aortic root diameter were neglected in our model in order to simplify the solution. Such changes may be of the order of l-l.5 mm between systole and diastolell51. We cannot be certain of the influence of such small changes of the aortic root diameter on the overall assessment of the stress distribution within the leaflet. To date, this factor has not been included in the analysis of stress distribution of aortic valve leaflets. In our model the coapting surfaces of the leaflet were not incorporated, and therefore the coapting edge was assumed to be fixed. Even though this is a considerable departure from the natural condition, it apparently had minor influences upon the magnitude and distribution of the maximal principal stresses. Christie and Medland[S], using an incompressible isotropic model in which the coapting surfaces of the leaflet were included and were allowed to move
Stress analysis of aortic valve leaflets
783
only in the vertical plane, obtained similar magnitude and distribution of stresses as was predicted by our model. At a load of 16.00 kPa their model predicted maximal principal stresses of the order of 200 kPa. For identical loading we observed maximal principal stresses in the range of 150-250 kPa for E = 300 kPa and for the stress dependent modulus. However, for the higher modulus of 500 kPa, our model predicted maximal principal stresses in the range of 300-350 kPa. Our aortic leaflet model differs to a large extent from that reported by Hamid and Ghista[7]. In their model the aortic leaflet geometry was approximated by a sphere, constant leaflet thickness was assumed and only the initial stress was taken into account in the formulation of the tangent stiffness matrix. Due to such marked differences, direct comparison between our results and their work is not valid. In conclusion, nonlinear large-deformation stress analysis of the aortic valve leaflets predicted lower maximal principal stresses than predicted by linear analysis. The use of a stressdependent E predicted maximal principal stresses comparable to those predicted with a pretransition modulus. It is established that the state of stresses in the aortic valve leaflet are not only dependent upon the geometry of the leaflet but also upon the material properties of the leaflets. REFERENCES [II D. J. Magilligan, Jr., J. W. Lewis. Jr.. F. M. Jara. M. W. Lee. M. Alam, J. M. Riddle and P. D. Stein, Spontaneous degeneration of porcine bioprosthetic valves, Ann. Thorac. Surg. 30, 259-266 (1980). 121 D. N. Ghista and H. Reul. Optimal prosthetic aortic leaflet valve: design parametric and longevity analyses: development of the avcothane-Sl leaflet valve based on the optimum design analysis, J. Biomcch. 10, 313-324 (1977). 131 A. Cataloglu, R. E. Clark and P. L. Gould, Stress analysis of aortic valve leaflets with smoothed geometrical data. J. Biomech. 10, 153-158 (1977). 141 Y. F. Missirlis and C. D. Armeniades, Stress analysis of the aortic valve during diastole: important parameters. J. Biotnech. 9, 477-480 (1976). 151 G. W. Christie and I. C. Medland. A nonlinear finite element stress analysis of bioprosthetic heart valves. In Finite Elements in Biomechanics (Edited by R. H. Gallagher er al.) pp. 153-179. John Wiley, New York (1982). Conrinua. McGraw-Hill, New York (1972). 161 J. T. Oden. Finite Elemenrs qf Nonlinear 171 M. S. Hamid and D. N. Ghista. Finite element analysis of human cardiac structures. In Finite Elemenr Methods in Engineering (Edited by V. A. Pulmano and A. P. Kabaila), pp. 337-348. Clarendon, Australia (1976). 181 H. N. Sabbah and P. D. Stein. Effect of aortic stenosis on coronary flow dynamics: studies in an in \irro pulse duplicating system, J. Biomech. Eng. 104, 221-225 (1982). 191 R. E. Clark and E. H. Finke. Scannina and lirrht microscouv of human aortic leaflets in stressed and relaxed states. J. Thorac. Cardiovasc. Swg. 67; 792-804 (1974). ’. 1101 P. L. Gould, A. Cataloglu. G. Dhatt, A. Chattopadhyay and R. E. Clark. Stress analysis of the human aortic valve, Computers and Strrtcrwes, 3, 377-384 (1973). in Engineering Analysis. Chap. 6. Prentice Hall, New Jersey (1982). 1111 K. J. Bathe, Finite Element Procedures 1121 0. C. Zienkiewicz. The Finite Element Merbod, 3rd Edition, McGraw-Hill. London (1977). 1131 W. M. Swanson and R. E. Clark, Dimensions and geometric relationships of the human aortic valve as a function of pressure. Circ. Res. 35, 871-882 (1974). 1141 M. S. Hamid. H. N. Sabbah and P. D. Stein. Comparison of finite element stress analysis of aortic valve leaflet using either membrane elements or solid elements. Comput. Srruct. (In press). [ISI R. J. Brewer. J. D. Deck. B. Capati and S. P. Nolan. The dynamic aortic root: its role in aortic valve function. J. Thorac. Cardiowsc. Surg. 72, 413-417 (1976). (Received
4 September
1984)
APPENDIX Incremental
procedure
In nonlinear analysis, the equilibrium of the body considered must be established in the current configuration during deformation. In general, the solution can be obtained by employing incremental formulation which reduces the nonlinear problem into a sequence of linearized problems. In the Lagrangian incremental approach, the equilibrium of the body at time t + Ar can be expressed by using the principle of virtual work. Using tensor notations, the virtual work principle requires that
I
r+Ar~,J 8,+,, ei, riA’dr, = r+arR
(1)
where T,j is the Cauchy Stress tensor and e,, is the infinitesimal strain tensor. Because the configuration of the body changes continuously, the computation of stresses must also take into account the rigid body rotation of the material. This is achieved by using appropriate stress and strain measures and constitutive relations. A commonly used stress measure is the 2nd Piola-Kirchhoff stress tensor (Sij) which can be expressed in terms of the Cauchy Stress tensor and deformation gradient. The strain tensor used with the 2nd Piola-
784
M. S. HAMID ef al.
Kirchhoff stress tensor is the Green-Lagrangian
strain tensor defined as
ei, =
(uij +
f
uj,i +
(2)
uk,iuk,,)
where rci,j = du&j. The principle of virtual displacements is expressed in terms of 2nd Piola-Kirchhoff strains. The basic equations are written as
stresses and Green-Lagrange
where liA’R is the external virtual work. Equation (3) refers to the configuration at time t in updated Lagrangian formulation. The strain tensor (cij) can be written in matrix form as
where {e},refers to the linear term and {e},,refers to the nonlinear term. The stress ‘+‘:Si, is decomposed into incremental form as
and the stress increment J,, can be written in terms of strain increment (Q) as rS,j
=
(6)
CijklEhl
where is the material constitutive tensor. In matrix notation, eqn (5) is written as c,jU
““:{s}
= ‘{cr) + [D] {E)
By substituting eqn (7) into eqn (3) and approximating
I
Wr
‘+%j
(7)
= ,e;, the equation in incremental form is written as
(‘lo) + [Dl {e}) dtl = %A~jr I&r.
(8)
The left hand side of eqn (8) represents the variation of strain energy U and 6{Au} on the right hand side is the incremental virtual displacement. The second variation of strain energy is expressed as 6?U = [SZ{e}’(‘{u} + [III {e}) + 8{E}7[Dl &{t}l. The incremental {APL, i.e.
procedure is linearized by equating the incremental load {&I],,, to the incremental
(9) internal load
Taking A6V = 6’U, the linearized incremental equations are written as {ApI = [Kl,j~~~]
(II)
where {Au) is the incremental displacement, [IC:lIis the tangent stiffness coefficients evaluated prior to the application of incremental loading and {Qj is the incremental load vector. Element
stiffness
The matrix [Klr is obtained for a triangular element as follows: Assuming the variation of the displacement components uj(i = 1, 2.3) as linear within an element, the displacement matrix is written as u = [A] io)
(II)
where [A] is function of spatial coordinates and {a) are constants which can be expressed in terms of the nodal displacements. The strain {a} = {e}, + {e},,are expressed in terms of the nodal incremental displacements {I()’ as +I, = WI/ 110’.
(13a)
13,z = &In W.
(13b)
The variation S{e} and S’jc} are written as
8{e}= and
SIC}, + t5{E)n= [B],&{u}’ + WI: W”
(14a)
785
Stress analysis of aortic valve leaflets For a triangular membrane element, undergoing large deformation, given as
large rotation and small strain, the strains are
(15)
where u and 11are inplane incremental displacements and H’is the out-of-plane incremental Now the variations for 8{e}, and S*{e}~{o}are obtained as
displacement.
(16a)
and s*{r}Z {o} = std’
7]slb [ul [Bl”. qu)P.
(16b)
where [u] is the stress tensor 7ii expressed in matrix form. Using eqns (16) and (13). the tangent stiffness matrix is written as
IN: = J ([El?. lUl~Bl>% + [BlT[DWl, + [Bl:[DI[fUf + [N~TIDl[Bl, + [Bl,~TIDIIB1~)dv. The deformation depended loading condition is nonsymmetric, and is therefore computationally nonsymmetric contribution can be neglected. In mations[I I]. However. the formulation is based strain and the material is assumed to be isotropic. constitutive tensors and the generalized Hooke’s
(17)
also contributes to the tangent stiffness coefficients. This component inefficient to handle. However, for small incremental pressure, this general, evaluation of the constitutive tensor Cijk/ involves transforupon the hypothesis of large deformation, large rotation but small Therefore the transformations do not change the components of the law is employed.
Solution procedrtre Assembly of individual element stiffness coefficients
LNIW
results in a set of global equations
=
{API.
(18)
The matrix [K] is symmetric and linear and depends upon the equilibrium position at the previous incremental step. After each incremental solution, the equilibrium is checked and in case the equilibrium does not satisfy, few iterations are performed before the next incremental load is applied.
Frocrure
Mechanics
Vol.
2.
No. 5. PP. 773-785.
1985
0013-7944185
Pnnted I” the U.S.A.
$3.00 + .OO
0 1985 Pergamon Press Ltd.
LARGE-DEFORMATION VALVE LEAFLETS Department
ANALYSIS OF AORTIC DURING DIASTOLE
MOHAMED S. HAMID, HAN1 N. SABBAH and PAUL D. STEIN of Medicine (Division of Cardiovascular Medicine) and Department of Surgery, Henry Ford Hospital, Detroit, MI 48202, U.S.A.
Abstract-The stress distribution on the aortic valve leaflet during diastole under increasing loading conditions was calculated using nonlinear large-deformation finite elements. The geometry of the leaflet was derived based upon the actual configuration of the closed aortic valve leaflets obtained from a cast of the root of the aorta. Variability of the leaflet thickness was incorporated in the finite-element model. The leaflet material was assumed to be isotropic and three values of Young’s modulus (E) were used. The first two E values that were used were assumed constant at 300 and 5000 kPa, which corresponded to pretransition and posttransition values, respectively, in the stress-strain curve. The third E that was used assumed a stressdependent value according to a trilinearized approximation of the stress-strain curve. The stress distribution on the leaflet at diastolic aortic pressures of 9.33 and 16.00 kPa (70 and 120 mm Hg) are presented. The large-deformation analysis predicted lower maximal principal stresses than linear theory.
INTRODUCTION AN IMPORTANT advantage
of bioprosthetic valves over mechanical prosthetic valves is decreased thrombogenicity, and the elimination of the need for anticoagulation. The longevity of these valves, however, is now recognized to be limited[l]. The durability of bioprosthetic valves is thought by some to depend in part upon the optimization of design parameters[2]. Such optimization may be achieved by better understanding of the magnitude and distribution of stresses developed in natural valve leaflets under physiological loading conditions. The determination of the stress distribution within the valve leaflets may suggest sites of potential susceptibility to tissue failure, and therefore may contribute to the identification of potential sites of degeneration. The distribution of stresses within the aortic valve leaflets has been determined by some investigators using finite-element analysis[3] and it has also been determined by closed form solutions[4]. The majority of these investigations, however, has been based upon linear theory even though the aortic valve leaflets may undergo large deformation which can introduce nonlinearities. Nonlinear finite-element stress analysis of bioprosthetic valve leaflets was performed by Christie and Medland using incompressible material[5]. Their formulation and solution of the highly nonlinear membrane problems using simplex finite elements was based upon the work of Oden[6]. A nonlinear finite-element analysis of natural aortic valve leaflets using compressible material was reported by Hamid and Ghista[7]. The purpose of this study was to determine the distribution of stresses on aortic valve leaflets during diastole using nonlinear finite-element analysis. Solutions were obtained using an aortic valve leaflet of nonuniform thickness and variable elastic moduli.
METHODS Assumptions 1. The leaflet material was considered
2.
3. 4. 5.
to be compressible and isotropic with a Poisson’s ratio of 0.3. The initial relaxed leaflet geometry was approximated by an elliptic paraboloid (Fig. 1). The salient parameters, i.e. height [not including coapting surface (Fig. l)], major and minor axes were approximated from measurements taken from a mold of the root of the aorta of a normal human being, even though the mold was prepared at 100 mm Hg[8]. All three aortic valve leaflets were assumed to be equal in size and symmetrical. The aortic valve annulus deformation during loading was neglected. The coaptation surfaces of the leaflets were not incorporated in the model. 773
774
M. S. HAMID et al.
I
AORTA
VENTRICLE
Fig. 1. Top: Two-dimensional drawing of the closed aortic valve showing the height (h) used to derive the relaxed leaflet shape. The coaptafion surface was excluded from the model. Bottom: Three-dimensional view of the modeled leaflet.
6. All leaflet boundaries including the coapting edge were rigidly fixed. 7. Membrane shell elements were used in the analysis, therefore bending effects were neglected compared to in-plane membrane forces. Leajlet
thickness
Topographic variability of thickness of the aortic valve leaflet was incorporated into the model based upon data reported by Clark and Finke[9]. Figure 2 shows a map of the topographic distribution of leaflet thickness used in our model. This distribution is similar to the idealized configuration used by others[3]. The idealized leaflet thickness distribution used in our study differs somewhat from that of Clark and Finke[9] and Cataloglu et a1.[3]. Elastic
properties
The finite-element solution was obtained for three separate conditions of Young’s modulus (E). The values of E were calculated from the stress-strain curves of aortic leaflet tissue obtained by others[lO]. The first two analyses were based on constant values of E of 300 and 5000 kPa which corresponded to the pretransition and posttransition values in the stress-strain curve. The third analysis assumed that the Young’s modulus was stress dependent and a trilinear approximation was used as shown in Fig. 3.
Stress analysis of aortic valve leaflets
Fig. 2. Topographic
distribution
600
of leaflet thickness (mm) used in the finite-element
modeling.
1
500
400
300
200
100
0 ELONGATION (percent) Fig. 3. Stress-strain curve of aortic valve leaflet. based upon data used by others[lO]. A trilinearized approximation was made of the curve to evaluate E.
Finite
element
modeling
and solution
The finite-element discretization of the aortic leaflet model is shown in Fig. 4. Triangular elements were used. The mesh consisted of 276 elements with 157 nodes which required the solution of 471 equations. The derivation of the element membrane stiffness matrix was based upon the hypothesis of large deformation, large rotation and small strain. The governing equations in incremental form were obtained as
(1) where [ZCjis the incremental tangent stiffness matrix, {q} is the incremental displacement for the load increment {P}. The element formulation is shown in the Appendix and is in accordance with the procedure given by Bathe[ 111. Equation (1) was solved by using a wavefront band algorithm in which the solution progressed as soon as the equation of a node was formed[l2]. Solution of the nonlinear equation was obtained by increasing the load from 1 to 120 mm Hg. An initial equibiaxial stress of 0.001 kPa was assumed in all the elements in order to smooth out the surface[5]. The first load increment was 0.13 kPa (1 mm Hg). The load was then incremented by 0.67 kPa (5 mm Hg) up to 4.00 kPa (30 mm Hg), and after that a load increment of 1.33 kPa (10 mm Hg) was used
776
Fig. 4. Finite-element
discretization of aortic valve leaflet. 192 refers to element number. AD is the center line of the leaflet.
up to 16.00 kPa (120 mm Hg). With each increment of load, the geometry was updated to account for changes of load direction. The incremental solution procedure used in this study facilitated the incorporation of the variable Young’s modulus according to the stress level in the element evaluated at the previous load increment. The analysis was performed using a 16K word memory minicomputer (Hewlett-Packard. model 21MX) on line with a magnetic tape drive and a pen plotter. This interactive arrangement facilitated the plotting of the mesh, stresses and deformed shapes at each load increment. RESULTS Stress distribution for
E = 300 kPa For the pretransition value of E = 300 kPa. the deformed shapes of the aortic leaflet at a pressure of 9.33 and 16.00 kPa (70 and 120 mm Hg) are shown in Fig. 5. A marked difference
9.3 kPa (70mm
16.0 kPa ( 120mm Hg)
I-Id
SHAPE
SHAPE
MAX. PRINCIPAL
STRESS
013
-
15 x10’
Pa
01s
-
16 x104
Pa
MAX. PRINCIPAL
STRESS
OlO-2ox1o’Pa
Fig. 5. Deformed shape and map of maximal principal stresses for pressures of 9.33 kPa and 16.00 kPa using a constant Young’s modulus of 300 kPa.
777
Stress analysis of aortic valve leaflets
in shape was observed at 16.00 kPa in comparison to 9.33 kPa. The displacement (shape change) of the leaflet along the center line is shown in Fig. 6 for a pressure of 16 kPa. At 9.33 kPa the maximal principal stresses were of the order of 130-150 kPa and increased to 190-200 kPa as the load was increased to 16.00 kPa (Fig. 5). The maximal principal stresses were located near the aortic attachment of the leaflets within the zone of the thinnest portion of the leaflet (Figs. 2 and 5). The maximal principal stresses were in general aligned along the circumferential direction (Fig. 7). However, near the center of the leaflet the principal stresses were almost equibiaxial. This gave rise to low principal shear stresses near the center as seen in Fig. 7. Compressive stresses were only observed along the coapting edge of the leaflet and were in the range of 0.7-25 kPa at a pressure of 16.00 kPa (Fig. 7). The total reaction forces exerted by the leaflet on the surrounding support (aorta) are shown in Fig. 8. Stress distribution for E = 5ooOkPa
For the posttransition value of E = 5000 kPa, the deformed shapes of the aortic valve leaflet at a pressure of 9.33 and 16.00 kPa were not appreciably different from each other (Fig. 9). The deformed shapes were also not much different than the leaflet shape present prior to any load imposition. A distinct difference, therefore, was present in the shape of the deformed leaflet at the selected pressures when using an E value of 300 kPa in compression to an E value of 5000 kPa. The displacement of the leaflet along the center-line for a pressure of 16 kPa is shown in Fig. 6. The stress distribution at 9.33 kPa and 16.00 kPa are shown in Fig. 9. At 9.33 kPa the maximal principal stresses were of the order of 180-200 kPa and increased to 310-350 kPa as pressure was increased to 16.00 kPa. The maximal stresses were again located near the aortic attachment of the leaflets within the zone of the thinnest portion of the leaflet as shown in Fig. 9. The maximal principal stresses were also aligned along the circumferential direction (Fig. 7). Compressive stresses in the range of 8-105 kPa were present near the coapting edge of the leaflet at a pressure of 16.00 kPa (Fig. 7). A contour plot of the principal shear stresses is also shown in Fig. 7. The maximal principal shear stresses were twice as high as those observed for E = 300 kPa. The maximal total reaction force exerted by the leaflet on the surrounding support was 0.19 N at a pressure of 16.00 kPa (Fig. 8).
1. E= 300 kPa 2. E= Variable 3 E=5000 kPa 16.017 -
10.67
-
5.33
-
267
-
.Coapting
R
Fig. 6. Deformation
DIRECTION
point
( mm 1
of the center line of the leaflet (AD in Fig. 4) for various values of E. The R and Z directions are as in Fig. 1.
778
M. S. HAMID et al.
i0
SHEAR STRESS (kPa)
STRESS DIRECTION
Fig. 7. Left: Principal shear stress contours. Right: Direction of principal stresses (dotted lines refer to compressive stresses). Top: Values predicted for E = 300 kPa. Middle: Values for 5000 kPa. Bottom: Values for stress-dependent E. All diagrams are based on a pressure of 16.00 kPa.
779
Stress analysis of aortic valve leaflets 40 -
35 T
0f
1. Variable E 2. E=300 kPa 3. E=5000 kPa
30-
x 25
c
0
10
30
20
ANGLE
40
50
60
@
Fig. 8. Predicted reaction forces as a function of the angle I$ which changes from 0 to 60” along the leaflet interface with the aorta as shown in the insert diagram.
93 kPa
16.0 kPa
(7Omm HQ)
(12Omm lip)
SHAPE
0
15 - 17 rtcr’ Pa
0
18 - 20 x10* P*
MAX. PRncIAL
SHAPE
STRESS
_ m
27 - SO x10’ Pa
0
31 - 35 xto4 Pa
MAX. WAL
STRESS
Fig. 9. Deformed shape of leaflet and map of maximal principal stresses using a constant E of 5000 kPa.
780 Stress
M. S. HAMID et al. distribution
using
a trilinearized
approximation
of E
The deformed leaflet shapes predicted when using a stress dependent E for pressues of 9.33 and 16.00 kPa are shown in Fig. 10. The deformed shapes were similar to those seen for E = 300 kPa. The displacement of the leaflet along the centerline for a pressure of 16 kPa is shown in Fig. 6. The maximal principal stresses at 9.33 and 16.00 kPa were also of the same order of magnitude as those obtained with E = 300 kPa (Fig. IO), and were located in the same area of the leaflet_. The principal shear stresses were of the same order of magnitude as seen with E = 300 kPa, however, the distribution was different (Fig. 7). The total reaction forces are shown in Fig. 8. Comparison
between
the linear
and nonlinear
solution
A linear analysis of the stress distribution of the aortic valve leaflet was also performed at 16.00 kPa (Fig. 11). The stress distribution in the aortic valve leaflet predicted by the nonlinear
16.0 kPa
9.3 kPa (70mm
( 120mm Hg)
Hg)
SHAPE
SHAPE
MAX. PRINCIPAL
n
10 -
0
15 - 18
0
20 - 25
STRESS
14 x10’ x10’
Pm MAX. PRINCIPAL
STRESS
Pa 10’
Pa
Fig. 10. Deformed shape and map of maximal principal stresses using a stress-dependent E that was derived on the basis of the trilinearized approximation of the stress-strain curve.
c)30-
35
xl04Pa
@&+36-45x104Pa 0
46 - 50 x104 Pa
Fig. 11. Map of maximal principal stresses at a pressure of 16.0 kPa using small-deformation (linear) theory.
781
Stress analysis of aortic valve leaflets
solution was different than that predicted by the linear solution. In general, the linear solution predicted higher stresses which were located at a different site on the leaflet (Fig. 11). A typical relationship between stress and pressure for one element (element 192, identified in Fig. 4) for both the linear and nonlinear (large deformation) solution is shown in Fig. 12. The circumferential strain variations for element 192 for E = 300 kPa, 5000 kPa and for a stress dependent E are shown in Fig. 13.
_ 10.66
ELEMENT
2.67
STRESS
+
192
(kPa)
Fig. 12. Comparison of maximal principal stresses in a single element (element 192 shown in Fig. 4) calculated on the basis of linear theory as well as nonlinear theorv. using_ various values of E.
16.00
r
13.33
-
2 00’ no r;r
A 2 X
10.67
-
k
8.00
-
$ K a.
Element
0
5
10
CIRCUMFERENTIAL
Fig. 13. Circumferential
15
+
20 STRAIN
192
25
30
( % )
strain as a function of pressure for element 192 depicting the variation with E.
782
M. S. HAMID et al.
DISCUSSION The magnitude of the maximal principal stress predicted for a low modulus of elasticity of 300 kPa was approximately 60% of the value predicted when a higher modulus of 5000 kPa was used. This was due to large deformations which occurred during the initial loading of the leaflet. The lower maximal principal stress observed when a stress dependent modulus was used can also be attributed to such initial large deformations. The nonlinear solution applied in this study also predicted a lower maximal principal stress than the linear solution, This is primarily due to apparent increased stiffness of the leaflet as a result of the large deformations. Although a value of E of 5000 kPa was used to predict the stress distribution on the leaflet for the linear solution, the magnitude of the modulus does not affect these results. From elementary shell theory, it is well known that membrane behavior is independent of material properties as long as the deformations remain small[ IO]. The results of both the nonlinear and linear solutions showed that the maximal principal stresses were aligned along the circumferential direction as observed by others[ 131. The site of the maximal principal stresses predicted by the application of the nonlinear theory was also different than that predicted by the linear solution. Even though high stresses were located in both solutions at the thinnest portion of the leaflet, the maximal stresses were concentrated near the center of this region in the linear prediction; whereas they were located nearer to the aortic attachment of the leaflet when the nonlinear solution was used. This may suggest sites of potential failure which may not be predicted on the basis of the commonly used linear model. Our model predicted compressive stresses near the coapting edge of the aortic valve leaflet, irrespective of the value of E that was used. The presence of compressive stresses at this site may be attributed to the assumed fixed boundary conditions along the coaptation edge. However, these compressive stresses were small in comparison to the maximal tensile stresses that occurred at other sites. In our model the leaflet material was assumed to be compressible and isotropic. The assumption was made in spite of the general belief that soft biological tissues are incompressible and anisotropic. Our intent was to assess the effects of geometric and material nonlinearities on the magnitude and distribution of stresses. Linear finite-element stress analysis of natural aortic leaflets using compressible and isotropic material properties has been reported[3, 10, 141. The initial relaxed geometry of the aortic leaflet was approximated by an elliptic paraboloid. This approximation was derived from the geometry of a mold of the root of the aorta pressurized at 13.33 kPa and excluding the coapting surfaces. Even though this particular geometry may not define the exact shape of the relaxed aortic leaflet, the latter remains an unknown parameter[5]. Nevertheless, others have used the elliptic paraboloid geometry to model the aortic valve leaflet[lO]. Other shapes including spherical[3, 71, elliptical[4] and paraboloid[3] were also used to approximate the geometry of the natural aortic valve leaflet. In this study the leaflets of the aortic valve were assumed to be equal in size and symmetrical. This assumption was made in spite of the fact that the three leaflets of the natural aortic valve are usually different in size and seldom exactly symmetrical. Nevertheless, to a good level of approximation, it may be assumed that the lack of symmetry does not affect the mechanics of statically loaded valves[5]. Even though leaflet symmetry was assumed, the entire leaflet was considered for analysis. This was done to maintain continuity in future studies where the introduction of local variations would alter the symmetric conditions. Changes of the aortic root diameter were neglected in our model in order to simplify the solution. Such changes may be of the order of l-l.5 mm between systole and diastolell51. We cannot be certain of the influence of such small changes of the aortic root diameter on the overall assessment of the stress distribution within the leaflet. To date, this factor has not been included in the analysis of stress distribution of aortic valve leaflets. In our model the coapting surfaces of the leaflet were not incorporated, and therefore the coapting edge was assumed to be fixed. Even though this is a considerable departure from the natural condition, it apparently had minor influences upon the magnitude and distribution of the maximal principal stresses. Christie and Medland[S], using an incompressible isotropic model in which the coapting surfaces of the leaflet were included and were allowed to move
Stress analysis of aortic valve leaflets
783
only in the vertical plane, obtained similar magnitude and distribution of stresses as was predicted by our model. At a load of 16.00 kPa their model predicted maximal principal stresses of the order of 200 kPa. For identical loading we observed maximal principal stresses in the range of 150-250 kPa for E = 300 kPa and for the stress dependent modulus. However, for the higher modulus of 500 kPa, our model predicted maximal principal stresses in the range of 300-350 kPa. Our aortic leaflet model differs to a large extent from that reported by Hamid and Ghista[7]. In their model the aortic leaflet geometry was approximated by a sphere, constant leaflet thickness was assumed and only the initial stress was taken into account in the formulation of the tangent stiffness matrix. Due to such marked differences, direct comparison between our results and their work is not valid. In conclusion, nonlinear large-deformation stress analysis of the aortic valve leaflets predicted lower maximal principal stresses than predicted by linear analysis. The use of a stressdependent E predicted maximal principal stresses comparable to those predicted with a pretransition modulus. It is established that the state of stresses in the aortic valve leaflet are not only dependent upon the geometry of the leaflet but also upon the material properties of the leaflets. REFERENCES [II D. J. Magilligan, Jr., J. W. Lewis. Jr.. F. M. Jara. M. W. Lee. M. Alam, J. M. Riddle and P. D. Stein, Spontaneous degeneration of porcine bioprosthetic valves, Ann. Thorac. Surg. 30, 259-266 (1980). 121 D. N. Ghista and H. Reul. Optimal prosthetic aortic leaflet valve: design parametric and longevity analyses: development of the avcothane-Sl leaflet valve based on the optimum design analysis, J. Biomcch. 10, 313-324 (1977). 131 A. Cataloglu, R. E. Clark and P. L. Gould, Stress analysis of aortic valve leaflets with smoothed geometrical data. J. Biomech. 10, 153-158 (1977). 141 Y. F. Missirlis and C. D. Armeniades, Stress analysis of the aortic valve during diastole: important parameters. J. Biotnech. 9, 477-480 (1976). 151 G. W. Christie and I. C. Medland. A nonlinear finite element stress analysis of bioprosthetic heart valves. In Finite Elements in Biomechanics (Edited by R. H. Gallagher er al.) pp. 153-179. John Wiley, New York (1982). Conrinua. McGraw-Hill, New York (1972). 161 J. T. Oden. Finite Elemenrs qf Nonlinear 171 M. S. Hamid and D. N. Ghista. Finite element analysis of human cardiac structures. In Finite Elemenr Methods in Engineering (Edited by V. A. Pulmano and A. P. Kabaila), pp. 337-348. Clarendon, Australia (1976). 181 H. N. Sabbah and P. D. Stein. Effect of aortic stenosis on coronary flow dynamics: studies in an in \irro pulse duplicating system, J. Biomech. Eng. 104, 221-225 (1982). 191 R. E. Clark and E. H. Finke. Scannina and lirrht microscouv of human aortic leaflets in stressed and relaxed states. J. Thorac. Cardiovasc. Swg. 67; 792-804 (1974). ’. 1101 P. L. Gould, A. Cataloglu. G. Dhatt, A. Chattopadhyay and R. E. Clark. Stress analysis of the human aortic valve, Computers and Strrtcrwes, 3, 377-384 (1973). in Engineering Analysis. Chap. 6. Prentice Hall, New Jersey (1982). 1111 K. J. Bathe, Finite Element Procedures 1121 0. C. Zienkiewicz. The Finite Element Merbod, 3rd Edition, McGraw-Hill. London (1977). 1131 W. M. Swanson and R. E. Clark, Dimensions and geometric relationships of the human aortic valve as a function of pressure. Circ. Res. 35, 871-882 (1974). 1141 M. S. Hamid. H. N. Sabbah and P. D. Stein. Comparison of finite element stress analysis of aortic valve leaflet using either membrane elements or solid elements. Comput. Srruct. (In press). [ISI R. J. Brewer. J. D. Deck. B. Capati and S. P. Nolan. The dynamic aortic root: its role in aortic valve function. J. Thorac. Cardiowsc. Surg. 72, 413-417 (1976). (Received
4 September
1984)
APPENDIX Incremental
procedure
In nonlinear analysis, the equilibrium of the body considered must be established in the current configuration during deformation. In general, the solution can be obtained by employing incremental formulation which reduces the nonlinear problem into a sequence of linearized problems. In the Lagrangian incremental approach, the equilibrium of the body at time t + Ar can be expressed by using the principle of virtual work. Using tensor notations, the virtual work principle requires that
I
r+Ar~,J 8,+,, ei, riA’dr, = r+arR
(1)
where T,j is the Cauchy Stress tensor and e,, is the infinitesimal strain tensor. Because the configuration of the body changes continuously, the computation of stresses must also take into account the rigid body rotation of the material. This is achieved by using appropriate stress and strain measures and constitutive relations. A commonly used stress measure is the 2nd Piola-Kirchhoff stress tensor (Sij) which can be expressed in terms of the Cauchy Stress tensor and deformation gradient. The strain tensor used with the 2nd Piola-
784
M. S. HAMID ef al.
Kirchhoff stress tensor is the Green-Lagrangian
strain tensor defined as
ei, =
(uij +
f
uj,i +
(2)
uk,iuk,,)
where rci,j = du&j. The principle of virtual displacements is expressed in terms of 2nd Piola-Kirchhoff strains. The basic equations are written as
stresses and Green-Lagrange
where liA’R is the external virtual work. Equation (3) refers to the configuration at time t in updated Lagrangian formulation. The strain tensor (cij) can be written in matrix form as
where {e},refers to the linear term and {e},,refers to the nonlinear term. The stress ‘+‘:Si, is decomposed into incremental form as
and the stress increment J,, can be written in terms of strain increment (Q) as rS,j
=
(6)
CijklEhl
where is the material constitutive tensor. In matrix notation, eqn (5) is written as c,jU
““:{s}
= ‘{cr) + [D] {E)
By substituting eqn (7) into eqn (3) and approximating
I
Wr
‘+%j
(7)
= ,e;, the equation in incremental form is written as
(‘lo) + [Dl {e}) dtl = %A~jr I&r.
(8)
The left hand side of eqn (8) represents the variation of strain energy U and 6{Au} on the right hand side is the incremental virtual displacement. The second variation of strain energy is expressed as 6?U = [SZ{e}’(‘{u} + [III {e}) + 8{E}7[Dl &{t}l. The incremental {APL, i.e.
procedure is linearized by equating the incremental load {&I],,, to the incremental
(9) internal load
Taking A6V = 6’U, the linearized incremental equations are written as {ApI = [Kl,j~~~]
(II)
where {Au) is the incremental displacement, [IC:lIis the tangent stiffness coefficients evaluated prior to the application of incremental loading and {Qj is the incremental load vector. Element
stiffness
The matrix [Klr is obtained for a triangular element as follows: Assuming the variation of the displacement components uj(i = 1, 2.3) as linear within an element, the displacement matrix is written as u = [A] io)
(II)
where [A] is function of spatial coordinates and {a) are constants which can be expressed in terms of the nodal displacements. The strain {a} = {e}, + {e},,are expressed in terms of the nodal incremental displacements {I()’ as +I, = WI/ 110’.
(13a)
13,z = &In W.
(13b)
The variation S{e} and S’jc} are written as
8{e}= and
SIC}, + t5{E)n= [B],&{u}’ + WI: W”
(14a)
785
Stress analysis of aortic valve leaflets For a triangular membrane element, undergoing large deformation, given as
large rotation and small strain, the strains are
(15)
where u and 11are inplane incremental displacements and H’is the out-of-plane incremental Now the variations for 8{e}, and S*{e}~{o}are obtained as
displacement.
(16a)
and s*{r}Z {o} = std’
7]slb [ul [Bl”. qu)P.
(16b)
where [u] is the stress tensor 7ii expressed in matrix form. Using eqns (16) and (13). the tangent stiffness matrix is written as
IN: = J ([El?. lUl~Bl>% + [BlT[DWl, + [Bl:[DI[fUf + [N~TIDl[Bl, + [Bl,~TIDIIB1~)dv. The deformation depended loading condition is nonsymmetric, and is therefore computationally nonsymmetric contribution can be neglected. In mations[I I]. However. the formulation is based strain and the material is assumed to be isotropic. constitutive tensors and the generalized Hooke’s
(17)
also contributes to the tangent stiffness coefficients. This component inefficient to handle. However, for small incremental pressure, this general, evaluation of the constitutive tensor Cijk/ involves transforupon the hypothesis of large deformation, large rotation but small Therefore the transformations do not change the components of the law is employed.
Solution procedrtre Assembly of individual element stiffness coefficients
LNIW
results in a set of global equations
=
{API.
(18)
The matrix [K] is symmetric and linear and depends upon the equilibrium position at the previous incremental step. After each incremental solution, the equilibrium is checked and in case the equilibrium does not satisfy, few iterations are performed before the next incremental load is applied.