Equations For Finite Deformation Research Articles

NEW DEVELOPMENTS IN AN ANATOMICAL FRAMEWORK FOR MODELING CARDIAC ISCHEMIA

A new, anatomically accurate, mathematical model of the right and left porcine ventricular myocardium is described based on measurements of the geometry and fibrous-sheet structure. Passive and active properties of the myocardium are calculated using an orthotropic constitutive law based on the fibrous-sheet structure and a biophysical cellular based model of cardiac contraction. Using Galerkin finite element techniques, the equations of finite deformation are solved to determine deformation and regional wall stress through the heart cycle. The mechanics model is coupled via myocardial wall stress, to a one-dimensional coronary blood flow model embedded in the myocardium. Bidomain electrical activation of the myocardium is also modeled, with ionic current based electrophysiological equations and reaction–diffusion equations based on orthotropic conductivity tensors referred to the fibrous-sheet material axes. Metabolic models are used to couple energy supply to contraction and excitation in the heart, and at the body surface, a framework for quantifying the effect of ischemic heart disease is developed.

Altered T wave dynamics in a contracting cardiac model.

The implications of mechanical deformation on calculated body surface potentials are investigated using a coupled biophysically based model. A cellular model of cardiac excitation-contraction is embedded in an anatomically accurate two-dimensional transverse cross-section of the cardiac ventricles and human torso. Waves of activation and contraction are induced by the application of physiologically realistic boundary conditions and solving the bidomain and finite deformation equations. Body surface potentials are calculated from these activation profiles by solving Laplace's equation in the passive surrounding tissues. The effect of cardiac deformation on electrical activity, induced by contraction, is demonstrated in both single-cell and tissue models. Action potential duration is reduced by 7 msec when the single cell model is subjected to a 10% contraction ramp applied over 400 msec. In the coupled electromechanical tissue model, the T wave of the ECG is shown to occur 18 msec earlier compared to an uncoupled excitation model. To assess the relative effects of myocardial deformation on the ECG, the activation sequence and tissue deformation are separated. The coupled and uncoupled activation sequences are mapped onto the undeforming and deforming meshes, respectively. ECGs are calculated for both mappings. Adding mechanical contraction to a mathematical model of the heart has been shown to shift the T wave on the ECG to the left. Although deformation of the myocardium resulting from contraction reduces the T wave amplitude, cell stretch producing altered cell membrane kinetics is the major component of this temporal shift.

The structure of constitutive equations for finite deformation of elastic–plastic materials involving strain‐induced anisotropy with applications

AbstractConstitutive relations based on finite‐deformation‐valid analysis require the formulation of coupling effects that were not needed in earlier developments which included small deformation restrictions. However, the feasibility to introduce such coupling effects mathematically requires that the new terms introduced must be in accord with physically based concepts of the phenomena involved. An example came to light in which objectivity of a constitutive relation, considered to be generalized for finite deformation applications involving strain‐induced plastic anisotropy, had been achieved by use of the Jaumann derivative in place of the material derivative for the growth of a tensor variable which expressed the anisotropic characteristics. The spin terms thus added to the infinitesimal displacement analogue were not in accord with a physically based assessment of the underlying micro‐mechanical phenomena and in fact led to the prediction of anomalous stress response. This paper comprises comments on the subsequent development of the theory, including a tensor equation for the evolution of the anisotropy. By solving a sample test problem with different initial conditions both the numerical implementation of the constitutive relation and the response according to the proposed growth law of the anisotropy are described and discussed.

A note on finite deflections of circular ring plates

We solve the problem of the symmetrically deforming edge loaded polar-orthotropic circular ring plate for the case of large deformations of a radially rigid plate, and compare the results with the corresponding solutions in accordance with the small finite deformation equations of von Karman.

Shock Thickness in Viscoplastic Solids

In this paper a system of constitutive relations for plane-strain finite deformation in strain-rate sensitive elastic-plastic materials is developed. A method is presented for computing the main features of steady-state one-dimensional plastic-strain waves by elementary numerical techniques in materials of this type. This method is applicable to a wide variety of material constitutive relations and provides an alternative to computing complete wave profiles for investigators interested primarily in the effects of certain numerical parameters on the principal features of waves. The method is illustrated by use of particular constitutive relations but is applicable to a much wider class of relations. Maximum normal stress, pressure, and shear stress are computed, using this method, as a function of wave speed. For two different plastic strain-rate relations the maximum plastic strain rate and total strain are computed as a function of wave speed. Making use of these results, estimates are provided for the wave-front thickness in terms of wave speed and the parameters of the particular viscoplastic constitutive equations used. These results suggest ways in which plastic wave experiments can be used to motivate the construction of constitutive equations for finite deformation in strain-rate sensitive plastic solids.