A Hamiltonian-based methodology is presented to study the fracture behaviors of the thermo-viscoelastic materials based on the Laplace transform. The governing equations and associated boundary conditions are rebuilt in a Hamiltonian form by using the symplectic mathematics in the frequency domain (s-domain). The fundamental unknown vector composed of both displacements and stresses variables is expanded in terms of the symplectic eigensolutions. The corresponding unknown coefficients of the symplectic series are determined from the outer boundary conditions. Thus, the main unknowns are obtained and transformed into the time domain (t-domain). The fracture parameters including stress intensity factors (SIFs) and J-integrals are derived simultaneously. Numerical examples as well as convergence studies are given and are found to be in good agreement with the ANSYS results. A parametric study of thermo-viscoelastic parameters is included also.
Nomenclatureu i component of displacements σ ij , ε ij component of stresses and strains s ij , e ij component of deviatoric stresses and deviatoric strains α T coefficient of thermal expansion η coefficient of viscosity G shear modulus K bulk modulus υ Poisson's ratio (x 1 , x 2 ) Cartesian coordinates (r, θ ) Polar coordinates H Hamiltonian function q, p mutually dual vectors H Hamiltonian operator matrix μ eigenvalue of Hamiltonian matrix ψ (α) n , ψ (β) n eigensolution of α and β group K I , K II Mode I and II stress intensity factors J
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