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Physical modelling of the arterial wall
Physical modelling Dear
of the arterial wall
Sir,
I read with interest the paper by G.L. Papageorgiou and N.B. Jones (Physical Modelling of the arterial wall. Part 1. Testing of tubes of various materials. .J. Biomed. Eng. 1987; 9: 153-6). I agree with the authors’ statement, ‘from the results on static elasticity of tubes of various materials, it appears that the latex tubes are those most suitable for arterial simulation since their incremental modulus of elasticity is close to that of arteries’. Arterial walls do possess viscoelastic properties’,“. Variations which are a function of time, temperature and the frequency components of the loading conditions are important parameters which act simultaneously in the arteries. It seems probable that elaborate thermophysical and chemical considerations would also play an important role. It cannot be too highly stressed that the next step in this
Three-dimensional Dear
investigation should take into account firstly the effects of pulsating flow in latex tubes and then to model the effect of the surrounding media (muscles) on the behaviour of the arteries; it would then be possible to compare the behaviour oflatex tubes with that of real arteries. P.A.A.
Laura
Institute of Applied Mechanics, Puerto Bclgrano Naval Base, 8 111 Argentina.
1. Brrgcl DH, The z~iscoelastic properties OJ [he arterial wall. PhD ‘I’hcsis, University of London, 1960. 2. Lraroyd BM, ‘l’aylor MG. Alterations with age in the viscorlastic properties of human arterial walls. Cir Res 1966; XVIII : 278-92. (Krfercncrs
cited
by the authors.)
curved beam stress analysis of the human femur
Sir,
First let us express our congratulations to the authors of ‘Three-dimensional curved beam stress analysis of the human femur’ (D.D. Raftopoulos and W. Quassem, J Biomed Eng 1987; 9: 356-66) for attempting a three-dimensional strength-of-materials treatment of such an important problem. As is clearly stated by the authors of this paper, ‘since the longitudinal axis of the femur is curved and the femoral cross-section is a composite anisotropic material, it is of great interest to investigate stresses on the femur. This can be done by viewing the femur as a three-dimensional composite curved beam’. However, the authors also state that ‘composite beam theory assumes that a cross-sectional plane remains plane before and after bending, and that stresses and strains are proportional for both cortical and spongy bone interfaces’. We feel that a rational justification for these two points is needed, especially concerning the assumption that a plane cross-section remains plane. It seems unlikely that in the case of a three-dimensional composite curved structural element subjected to axial loadings, shear forces, biaxial bending and twisting moments, a plane cross-section will remain in the same plane.
c:: 1988 Buttrrworth bi Co (Publishers) Ol41-5425/88/040373~1 $03.00
Ltd
It is well known that even in the case of free torsion of prismatic members the cross-sections do not remain plane (St Venant’s theory of the mathematical theory of elasticity) when dealing with an arbitrary cross-section. The only condition for which the cross-section remains plane is when it is circular. The validity of expressions ( 1 I ) and ( 12)) which predict shearing stresses, is certainly open to question. It would also be helpful to have details of the derivation of equations (4) through ( IO). It seems that a reasonable, although expensive, approach to this three-dimensional, non-homogeneous and non-isotropic mechanical system would be to use a finite element formulation, which would take all the complicating factors into account. E.A. Department Universidad
Romanelli
of Engineering, National de1 Sur, 8000 Bahia Blanca and
P.A.A.
Laura
Institute of Applied Mechanics, Purrto Brlgrano Naval Base, 81 1 I Argentina
,J. Biomcd.
Eng. 1988, Vol. lO,_July
373
of the arterial wall
Sir,
I read with interest the paper by G.L. Papageorgiou and N.B. Jones (Physical Modelling of the arterial wall. Part 1. Testing of tubes of various materials. .J. Biomed. Eng. 1987; 9: 153-6). I agree with the authors’ statement, ‘from the results on static elasticity of tubes of various materials, it appears that the latex tubes are those most suitable for arterial simulation since their incremental modulus of elasticity is close to that of arteries’. Arterial walls do possess viscoelastic properties’,“. Variations which are a function of time, temperature and the frequency components of the loading conditions are important parameters which act simultaneously in the arteries. It seems probable that elaborate thermophysical and chemical considerations would also play an important role. It cannot be too highly stressed that the next step in this
Three-dimensional Dear
investigation should take into account firstly the effects of pulsating flow in latex tubes and then to model the effect of the surrounding media (muscles) on the behaviour of the arteries; it would then be possible to compare the behaviour oflatex tubes with that of real arteries. P.A.A.
Laura
Institute of Applied Mechanics, Puerto Bclgrano Naval Base, 8 111 Argentina.
1. Brrgcl DH, The z~iscoelastic properties OJ [he arterial wall. PhD ‘I’hcsis, University of London, 1960. 2. Lraroyd BM, ‘l’aylor MG. Alterations with age in the viscorlastic properties of human arterial walls. Cir Res 1966; XVIII : 278-92. (Krfercncrs
cited
by the authors.)
curved beam stress analysis of the human femur
Sir,
First let us express our congratulations to the authors of ‘Three-dimensional curved beam stress analysis of the human femur’ (D.D. Raftopoulos and W. Quassem, J Biomed Eng 1987; 9: 356-66) for attempting a three-dimensional strength-of-materials treatment of such an important problem. As is clearly stated by the authors of this paper, ‘since the longitudinal axis of the femur is curved and the femoral cross-section is a composite anisotropic material, it is of great interest to investigate stresses on the femur. This can be done by viewing the femur as a three-dimensional composite curved beam’. However, the authors also state that ‘composite beam theory assumes that a cross-sectional plane remains plane before and after bending, and that stresses and strains are proportional for both cortical and spongy bone interfaces’. We feel that a rational justification for these two points is needed, especially concerning the assumption that a plane cross-section remains plane. It seems unlikely that in the case of a three-dimensional composite curved structural element subjected to axial loadings, shear forces, biaxial bending and twisting moments, a plane cross-section will remain in the same plane.
c:: 1988 Buttrrworth bi Co (Publishers) Ol41-5425/88/040373~1 $03.00
Ltd
It is well known that even in the case of free torsion of prismatic members the cross-sections do not remain plane (St Venant’s theory of the mathematical theory of elasticity) when dealing with an arbitrary cross-section. The only condition for which the cross-section remains plane is when it is circular. The validity of expressions ( 1 I ) and ( 12)) which predict shearing stresses, is certainly open to question. It would also be helpful to have details of the derivation of equations (4) through ( IO). It seems that a reasonable, although expensive, approach to this three-dimensional, non-homogeneous and non-isotropic mechanical system would be to use a finite element formulation, which would take all the complicating factors into account. E.A. Department Universidad
Romanelli
of Engineering, National de1 Sur, 8000 Bahia Blanca and
P.A.A.
Laura
Institute of Applied Mechanics, Purrto Brlgrano Naval Base, 81 1 I Argentina
,J. Biomcd.
Eng. 1988, Vol. lO,_July
373