J Biomechanws Vol. 12, pp. 651.-655 @ Pcrgamon Press Ltd 1979. Pnnted
@321-92Wi79/0901-0651 in Great
SO2WO
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INCREMENTAL ELASTIC MODULUS FOR ORTHOTROPIC INCOMPRESSIBLE ARTERIES* ANTAL
G. HUDETZ
Experimental Research Department, Semmelweis Medical University, Budapest, Hungary Abstract Mechanics of incremental deformations has been applied to introduce an elastic modulus (H) for characterizing incremental elastic properties of thick walled, straight, cylindrically orthotropic, incompressible blood vessels pressurized at fixed length. The modulus H characterizes the radial and tangential stiffness complexly. Volume distensibility, wave velocity and characteristic impedance are directly related to H. For isotropic incompressible vessels, modulus H is equal to 4/3 of the isotropic incremental Young’s modulus. Formulae of both moduli contain an additional term representing the influence of the initial stress, This implies that the classical formula of the isotropic Young’s modulus introduced by Bergel for arteries is not applicable to incremental deformations. The incremental modulus His proposed as a suitable parameter describing arterial elasticity under in oioo circumstances.
INTRODUCTION
Blood vessels are generally subjected to large deformations in citlo and they possess nonlinear viscoelastic characteristics as well. For an adequate description of their mechanical behaviour, nonlinear or incremental continuum theories may be appropriate (Pate1 and Vaishnav, 1972). Mechanics of incremental deformations (Biot, 1965) is actually a linear continuum theory. It has been established for describing small deformations of an elastic material in its arbitrary prestressed initial state. Incremental moduli are defined as elastic constants relating small changes in stress to small incremental strains at any initial state. Thus nonlinear elastic properties of a material are characterized by the dependence of incremental moduli on the initial strain state. It was Bergel (1961) who first applied an ‘incremental’ modulus to intact artery segments. He assumed that the well known expression of the Young’s modulus of a pressurized isotropic cylindrical tube. derived by Love (1927) on the basis of the classical linear theory, is valid for incremental deformations, too. It will be shown in this work that this generalization was not correct, since incremental moduli are, in general. different from the respective moduli defined in the linear theory. due to the presence of initial stress. The proper expression of the isotropic Young’s modulus applicable for incremental deformations will be derived using a linear-incremental correspondence principle. This method is established for obtaining solutions of incremental boundary value problems if respective solutions in the linear theory are known. Later. it has been shown that large arteries are not isotropic. but cylindrically orthotropic (Pate1 and Fry, __~_ * Received 28 July 1978.
1969) and incompressible (Carew et al., 1968). Therefore, at least three independent moduli are necessary for characterising the anisotropic elastic properties of arteries under ‘physiological’ loading (i.e. intraluminal pressure and axial tension). In spite of this, as an approximation many authors have used Bergel’s isotropic modulus (Cox, 1975b, 1976; Goedhard et al., 1973; Newman et al., 1971; Saito et al., 1975, etc.), as it can be determined from a simple inflation test of a cylindrical vessel segment, and for practical purposes it is often more convenient to use a single modulus instead of three moduli. We show in this work that elasticity of a cylindrically orthotropic incompressible artery may be correctly characterized by a single incremental modulus, too, if the vessel is pressurized at fixed length. A formula useful for experimental determination of the new modulus for thick walled vessels will be derived. It will be shown that volume elastic modulus, pressure modulus, wave velocity and characteristic impedance of vessels are directly related to this modulus. THEORETICALPRELIMINARIES
Let us consider an elastic body in an arbitrary deformed state, called the reference state. The initial stress tensor in the reference state will be denoted by t:‘. Superimposing a small incremental deformation field described by the displacement vector uk, the stress tensor tz’ will be changed to t”. For vanishing body force the equilibrium equations for incremental stresses are represented by Suhubi (1975) : 5”..k - P. o,k Ilk.m = 0.
(1)
where ?’ = tk’ - td’ is the incremental stress tensor. The semi colons (;) indicate covariant partial differentiation. The stress tensors appearing in equation (1) are Cauchy stress tensors, which measure mechanical forces per unit area in the deformed body.
651
652
ANTALG. Humrrz
For vanishing initial stress, i.e. if t$’ = 0, equilibrium equations of the linear theory of elasticity are obtained : TL’, = 0.
(2)
Stresses referred to unit area in the reference configuration are specified by the first Kirchoff-Piola tensor T”, which is related to the Cauchy stress tensor by
ponents of the initial stress tensor are also present in the equilibrium equation. It is well known that the stress field is inhomogeneous and depends on the radial coordinate in an elastic tube. Since, in general, the initial stress distribution is unknown, the above equilibrium equation cannot be solved for the incremental Cauchy stresses. Equation (4) does not contain the initial stress tensor and under the same conditions reduces to
~kl= Paxk/axlm. Here xk and x” denote the coordinates of a material particle in the reference and deformed state, respectively, referred to the same coordinate system, and J is the Jacobian defined by J = det(ax”/@). If ?’
@321-92Wi79/0901-0651 in Great
SO2WO
Brilm
INCREMENTAL ELASTIC MODULUS FOR ORTHOTROPIC INCOMPRESSIBLE ARTERIES* ANTAL
G. HUDETZ
Experimental Research Department, Semmelweis Medical University, Budapest, Hungary Abstract Mechanics of incremental deformations has been applied to introduce an elastic modulus (H) for characterizing incremental elastic properties of thick walled, straight, cylindrically orthotropic, incompressible blood vessels pressurized at fixed length. The modulus H characterizes the radial and tangential stiffness complexly. Volume distensibility, wave velocity and characteristic impedance are directly related to H. For isotropic incompressible vessels, modulus H is equal to 4/3 of the isotropic incremental Young’s modulus. Formulae of both moduli contain an additional term representing the influence of the initial stress, This implies that the classical formula of the isotropic Young’s modulus introduced by Bergel for arteries is not applicable to incremental deformations. The incremental modulus His proposed as a suitable parameter describing arterial elasticity under in oioo circumstances.
INTRODUCTION
Blood vessels are generally subjected to large deformations in citlo and they possess nonlinear viscoelastic characteristics as well. For an adequate description of their mechanical behaviour, nonlinear or incremental continuum theories may be appropriate (Pate1 and Vaishnav, 1972). Mechanics of incremental deformations (Biot, 1965) is actually a linear continuum theory. It has been established for describing small deformations of an elastic material in its arbitrary prestressed initial state. Incremental moduli are defined as elastic constants relating small changes in stress to small incremental strains at any initial state. Thus nonlinear elastic properties of a material are characterized by the dependence of incremental moduli on the initial strain state. It was Bergel (1961) who first applied an ‘incremental’ modulus to intact artery segments. He assumed that the well known expression of the Young’s modulus of a pressurized isotropic cylindrical tube. derived by Love (1927) on the basis of the classical linear theory, is valid for incremental deformations, too. It will be shown in this work that this generalization was not correct, since incremental moduli are, in general. different from the respective moduli defined in the linear theory. due to the presence of initial stress. The proper expression of the isotropic Young’s modulus applicable for incremental deformations will be derived using a linear-incremental correspondence principle. This method is established for obtaining solutions of incremental boundary value problems if respective solutions in the linear theory are known. Later. it has been shown that large arteries are not isotropic. but cylindrically orthotropic (Pate1 and Fry, __~_ * Received 28 July 1978.
1969) and incompressible (Carew et al., 1968). Therefore, at least three independent moduli are necessary for characterising the anisotropic elastic properties of arteries under ‘physiological’ loading (i.e. intraluminal pressure and axial tension). In spite of this, as an approximation many authors have used Bergel’s isotropic modulus (Cox, 1975b, 1976; Goedhard et al., 1973; Newman et al., 1971; Saito et al., 1975, etc.), as it can be determined from a simple inflation test of a cylindrical vessel segment, and for practical purposes it is often more convenient to use a single modulus instead of three moduli. We show in this work that elasticity of a cylindrically orthotropic incompressible artery may be correctly characterized by a single incremental modulus, too, if the vessel is pressurized at fixed length. A formula useful for experimental determination of the new modulus for thick walled vessels will be derived. It will be shown that volume elastic modulus, pressure modulus, wave velocity and characteristic impedance of vessels are directly related to this modulus. THEORETICALPRELIMINARIES
Let us consider an elastic body in an arbitrary deformed state, called the reference state. The initial stress tensor in the reference state will be denoted by t:‘. Superimposing a small incremental deformation field described by the displacement vector uk, the stress tensor tz’ will be changed to t”. For vanishing body force the equilibrium equations for incremental stresses are represented by Suhubi (1975) : 5”..k - P. o,k Ilk.m = 0.
(1)
where ?’ = tk’ - td’ is the incremental stress tensor. The semi colons (;) indicate covariant partial differentiation. The stress tensors appearing in equation (1) are Cauchy stress tensors, which measure mechanical forces per unit area in the deformed body.
651
652
ANTALG. Humrrz
For vanishing initial stress, i.e. if t$’ = 0, equilibrium equations of the linear theory of elasticity are obtained : TL’, = 0.
(2)
Stresses referred to unit area in the reference configuration are specified by the first Kirchoff-Piola tensor T”, which is related to the Cauchy stress tensor by
ponents of the initial stress tensor are also present in the equilibrium equation. It is well known that the stress field is inhomogeneous and depends on the radial coordinate in an elastic tube. Since, in general, the initial stress distribution is unknown, the above equilibrium equation cannot be solved for the incremental Cauchy stresses. Equation (4) does not contain the initial stress tensor and under the same conditions reduces to
~kl= Paxk/axlm. Here xk and x” denote the coordinates of a material particle in the reference and deformed state, respectively, referred to the same coordinate system, and J is the Jacobian defined by J = det(ax”/@). If ?’
