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Kinematic and Constitutive Modeling of Elastic and Thermoelastic Continua with Finite Deformation using Multiplicative Decomposition of Deformation Gradient

Darijani, Hossein | 2011

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 41310 (08)
  4. University: Sharif University of Technology
  5. Department: mechanical engineering
  6. Advisor(s): Naghdabadi, Reza; Kargarnovin, Mohammad Hassan
  7. Abstract:
  8. In this thesis, a deformation measure is introduced which leads to a class of strain measures in the Lagrangian and Eulerian descriptions. In order to develop a constitutive equation, a second-order constitutive relation based on these strain measures is considered for modeling the mechanical behavior of solids at finite deformation. For this purpose and performance evaluation of the proposed strains, a Hookean-type constitutive equation is considered and the uniaxial loading as well as simple shear and pure shear tests are examined and the results are compared with the test data. Also, in order to characterize the mechanical behavior of elastic continua, constitutive equations through a strain energy density function are presented. For this purpose, the strain energy density is expressed as sum of the mathematical functions in terms of the principal stretches. These functions are determined by applying the governing postulates on the form of the strain energy density. In addition, the strain energy density function of the Saint Venant–Kirchhoff type is modified and expressed in terms of the proposed strain measures for behavior modeling of elastic continua with finite deformation. The proposed strain energy density functions cover modeling both of compressible and incompressible materials. Moreover, the materials parameters of these models are calculated based on the correlation between the values of the strain energy density (rather than the stresses) cast from the test data and the theory. In order to investigate the appropriateness of the proposed models, several experimental data for incompressible and compressible isotropic materials under homogeneous deformations are examined in which the predictions of the proposed models show a good agreement with experimental data. Furthermore, some fundamental kinematic and kinetic aspects of finite deformation thermoelasticity theory within the framework of the multiplicative decomposition of the deformation gradient are considered. At first, the kinematics of the thermoelastic continua in the purely thermal process of nonisothermal deformation is investigated for finite deformation thermoelasticity. For this purpose, a linear relation between the thermal expansion tensor and the rate of the thermal deformation tensor is presented. Also, based on the multiplicative decomposition of the deformation gradient, the mechanical and thermal strain measures are defined in the power and exponential forms. The decomposition of the total strain into the mechanical and thermal strains is investigated for implementation in the governing equations of finite deformation thermoelasticity. Using this decomposition, an explicit analytical solution for the thermoelastic stress distribution in the thick-walled cylindrical vessels with finite deformations under the thermal and mechanical loadings is achieved. Finally, in order to solve a problem in the context of continuum thermodynamics using energy viewpoint, the stress power, the rate of the heat transferred to the system and the rate of the specific internal energy are presented in different descriptions for thermoelastic continua. In order to model the mechanical behavior of thermoelastic continua in the stress-producing process of nonisothermal deformation, an isothermal effective stress-strain equation based on the proposed strains is considered. Using this constitutive equation and assuming a linear dependence of the specific heat on temperature, the state functions including the internal energy, free energy, entropy and stress tensor are derived in terms of four possible pairs of independent state variables. In addition, the mechanical and thermal material parameters are determined using the mechanical tests done at constant temperature and free thermal expansion test data, respectively
  9. Keywords:
  10. Multiplicative Decomposition ; Constitutive Modeling ; Strain Energy Density ; Strain Energy Function ; Compressible/Incompressible Algorithm ; Materials Parameters ; Strain Measures ; Thermoelastic Continua ; Conjugacy ; Elastic Continua

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