Elastic modulus structural relaxation model

B. Relation of Entropy Theory to Elastic Modulus Model of Structural Relaxation... 

However, amorphous water-soluble materials, such as food materials, deform viscoelastically. The deformation and relaxation behavior of such materials can be described by means of various viscoelastic models. Depending on the nature of the stress/strain applied, either the storage and loss modulus or the elasticity and the viscosity are included as material parameters in these models. These rheological material parameters depend on the temperature and the water content as well as on the applied strain rate. The viscoelastic deformation enlarges the contact area and decreases the distance between the particles (see Fig. 7.3). If the stress decreases once again, the achieved deformation is partially reversed (structural relaxation). 

As it is known [2, 12], within the frameworks of cluster model the elasticity modulus E value is defined by stiffness of amorphous polymers structure both components local order domains (clusters) and loosely packed matrix. In Fig. 13.3, the dependences E(v J are adduced, obtained for tensile tests three types with constant strain rate, with strain discontinuous change and on stress relaxation. As one can see, the dependences E(yJ are approximated by three parallel straight lines, cutting on the axis E loosely packed matrix elasticity modulus E different values. The greatest value E is obtained in tensile tests with constant strain rate, the least one - at strain discontinuous change and in tests on stress relaxation E = 0 [1]. 

Our goal was to measure the viscoelastic properties of the human brain under practical conditions. Therefore, we used the tactile resonant sensor with the stress-strain function that simulated manual palpation. In this study, the stiffness was 2.837 0.709 (N), Young s elastic modulus was E = 5.08 1.31, and the shear modulus was G = 1.94 0.49 for a depth of 3.0 mm. Poisson s ratio (u) was calculated as 0.31-0.62 using the equation E = 2G (1h- u). These values were approximately equal to those previously reported for the viscoelasticity properties of the brain in vivo [1-7]. The results of indentation fitted the Maxwell model as expressed by the equation G = Ge - Gi exp (-t/x), where Ge is the instantaneous modulus in shear, Gi is the relaxation in the shear modulus, t is time, and x is the relaxation time. Thus, G = 1.94-1- 3.3 exp(-t/0.5) under the assumption that Ge = 1.94, Gi = 3.3, t = h/1.5, and x = 0.5. The results obtained in this study by an indentation method, reflected those of a previous model [9-12]. However, this measurement method evaluated brain viscoelasticity via multiple structural layers including the skin, subcutaneous tissues, muscle fascia, and dura. Moreover, some assumptions had to be made to approximate the expression for elasticity. 

For concentrated emulsions and foams, Princen [182, 183] proposed a stress-strain theory based on a two-dimensional cell model. Consider a steady state shearing of such a system. Initially, at small values of strain, the stress increases linearly as in elastic body. As the strain increases, the stress reaches a yield point, and then at higher deformation it catastrophically drops to negative values. The reason for the latter behavior is the creation of unstable cell structure that relaxes by recoil. For real emulsions the shear modulus and yield stress are expected to follow the expressions ... 

The modulus recovery experiments allowed measuring the terminal relaxation time of reptation motion of bulk and surface immobilized chains, supporting the hypothesis that theie is no interphase per se when nano-scale is considered. In order to bridge the gap between the continuum interphase on the microscale and the discrete molecular structure of the matrix consisting of freely reptating chains in the bulk and retarded reptatiug chains in contact with the inclusions, higher order elasticity combined with a suitable molecular dynamics model could be utilized [151-155]. 

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