Description of the deformation properties of cross-linked polymers in the framework of the linear theory of heredity using a new formalization of the instantaneous component (Q2836565)

The authors examine a dense mesh with a spatially homogeneous topological structure. The model is described by the system of equations NEWLINE\[NEWLINEu_{ik} = (1/2)\mathbf J\tau_{ik}-(1/3)B_{\infty}p\delta_{ik},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\mathbf J = J_{\infty}[w_{J, \beta} + (1-w_{J, \beta})\mathbf J_{N, \alpha}],NEWLINE\]NEWLINE NEWLINE\[NEWLINE\mathbf J_{N, \alpha} = \int_{-\infty}^{\infty}L_{J, \alpha}(\theta)(1-\exp(-t/\theta))\mathrm{d}\ln\theta,NEWLINE\]NEWLINE where \(u_{ik}\) is the tensor of specific deformation, \(\mathbf J\) the relaxation operator of the shear yield, \(\tau_{ik}\) the tensor of shear stress, \(B_{\infty}\) the equilibrium volume yield, \(p\) the pressure, \(\delta_{ik}\) the Kroneker symbol, \(J_{\infty}\) the equilibrium yield at given temperature, \(J_{\infty}\) the weighted coefficient, \(\mathbf J_{N, \alpha}\) the fractional-exponential operator, \(\mathbf J_{N, \alpha}\) the normalized relaxation time, \(\theta\) the relaxation time, and \(t\) the current time. This model reflects all states of dense mesh polymers, including the high-elasticity state, the glassy state, and the transitional state.