Simulation of fiber-reinforced viscoelastic structures subjected to finite strains: multiplicative approach

Abstract

The study is devoted to the geometrically nonlinear simulation of fiber-reinforced composite structures. The applicability of the multiplicative approach to the simulation of viscoelastic properties of a composite material is assessed, certain improvements are suggested. For a greater accuracy in applications involving local compressive fiber buckling, a new family of hyperelastic potentials is introduced. This family allows us to account for the variable critical compressive stress, which depends on the fibermatrix interaction. For the simulation of viscoelasticity, the well-established Sidoroff decomposition of the deformation gradient is implemented. To account for the viscosity of the matrix material, the model of Simo and Miehe (1992) is used; highly efficient iteration-free algorithms are implemented. The viscosity of the fiber is likewise described by the multiplicative decomposition of the deformation gradient, leading to a scalar differential equation; an efficient iteration-free algorithm is proposed for the implicit time stepping. The accuracy and convergence of the new iteration-free method is tested and compared to that of the standard scheme implementing the Newton iteration. To demonstrate the applicability of the approach, a pressurized multi-layer composite pipe is modelled; the so-called stretch inversion phenomenon is reproduced and explained. The stress distribution is obtained by a semi-analytical procedure; it may serve as a benchmark for FEM computations. Finally, the issue of the parameter identification is addressed.

Geometrically nonlinear modelling of pre-stressed viscoelastic fibre-reinforced composites with application to arteries

“…Such materials sustain large cyclic strains and show viscoelastic anisotropic mechanical behaviour. In recent years many material models were introduced to take numerous mechanical phenomena into account: not only anisotropic hyperelasticity [5], [44], [17], [34], [45], viscoelasticity [18], [24], [42], [28] and elasto-plasticity [36], [14], [37] were modelled, but also damage accumulation [21], [1], [16] as well as growth and remodelling of living tissues [7], [3], [23] were taken into account. From the mechanical standpoint, the main goal of these studies is to predict the stress response of the analyzed tissue on the macroscopic level.…”

“…Some modelling approaches to residual stresses inside arteries were already considered in [17], [2], [4] dealing with elastic (hyperelastic) stress response. To enable accurate and efficient simulations, a combination of constitutive assumptions and numerical schemes presented in [42] will be generalised here to cover the presence of pre-stresses. In contrast to the mentioned publications, the interaction between residual stresses and viscous effects is incorporated.…”

“…It is based on the assumption that the overall stress response of the material is a sum of stress contributions provided by matrix and fibre, both of them experiencing the same deformation. Thanks to simplified kinematics, it allows one to employ efficient numerical methods used to assess the overall behaviour of macroscopic structures [17], [11], [42]. In this paper, in order to further reduce the computational costs, the viscoelastic properties are modelled using the Maxwell body approach (so-called "spring-dashpot" model).…”

“…In general, the Maxwell approach is more efficient than the use of convolution integrals [18]: the Maxwell approach employs a limited number of internal variables whereas the convolution integral with a general relaxation function requires the entire deformation history. Moreover, for the considered versions of the Maxwell bodies, efficient iteration-free time-stepping methods are already available [38], [35], [42]. The applicability of the advocated approach to fibre-reinforced composites was already assessed in [42].…”

“…Moreover, for the considered versions of the Maxwell bodies, efficient iteration-free time-stepping methods are already available [38], [35], [42]. The applicability of the advocated approach to fibre-reinforced composites was already assessed in [42]. Since the integration methods are non-iterational, much higher computational efficiency can be achieved, which is especially important for the analysis of clinical applications [32].…”