Discontinuous Bifurcation of a Soft-Impact System

In this paper, we investigate discontinuous bifurcations of a soft-impact system, which is nonsmooth at the switching boundary consisting of two parts. We find that there are no periodic orbits located only in the impact zone, and when grazing bifurcation on one part of the switching boundary occurs, the tangency point changes may occur for different bifurcation parameters, which is also verified by numerical simulation. In particular, we discover degenerate inner and external corner bifurcations, which can produce chaotic behavior, for example, period-doubling cascades and a degenerate inner corner bifurcation that induce chaotic responses. In this way, our investigation reveals the presence of narrow bands of chaotic motion induced by the afore mentioned dynamical phenomena.

Discontinuous phenomena in bioreactor and membrane reactor systems

In this work, we analyze the existence of discontinuous bifurcation and stability issues in discontinuous flow of bioreactor and membrane reactor models with or without recycle. The reaction is assumed to be governed by certain types of discontinuities in Monod growth kinetics curve leading to discontinuous dynamical system. The criteria for the existence and stability of steady-states of these models are established. More generally, our analysis highlights the presence of several types of bifurcation depending upon the effect of the dilution factor (residence time), biomass concentration and solid-liquid-gas separator efficiency. As well, we present bifurcation conditions defining the dynamics near steady-state branches on the border, providing a possible framework for existing of saddle-node, nonsmooth fold, persistence and grazing-sliding scenarios. It is shown that the critical values of residence time dependence upon recycle ratio, decay rate and existence of discontinuity surface. Further, the performance of the reactor at largest residence times will be discussed. In addition, numerical simulations to illustrate and confirm the results will be carried out.

Contact Impact Forces at Discontinuous 2-DOF Vibroimpact

Dynamic behaviour of contact impact forces in strongly nonlinear discontinuous vibroimpact system is studying. Contact impact force is one of the most significant vibroimpact system characteristics. We investigate the 2-DOF vibroimpact system by numerical parameter continuation method in conjunction with shooting and Newton-Raphson methods. We simulate the impact by nonlinear contact interactive force according to Hertz’s contact law. This paper is the continuation of the previous works [1,2]. We have determined the instability zones and bifurcations points for loading curves [1] and frequency-amplitude response [2] under variation of excitation amplitude and frequency. In this paper we investigate the behaviour of contact forces at bifurcation points particularly at discontinuous bifurcation points where set-valued Floquet multipliers cross the unit circle by jump that is their moduli becoming more than unit by jump. It is phenomenon unique for nonsmooth systems with discontinuous right-hand side. We observe also the contact forces increase at nT -periodical multiple impacts regimes. We also learn the change of contact forces behaviour when the impact between system bodies became the soft one due the change of system parameters.

Recent Advances in Simulating Failure Evolution with the Material Point Method

To predict a complete process of failure evolution, discontinuous bifurcation analysis has been performed to link elastoplasticity and damage models with decohesion models. To simulate multi-phase interactions involving failure evolution, the Material Point Method (MPM) has been developed to discretize localized large deformations and the transition from continuous to discontinuous failure modes. In a recent study for the Sandia National Laboratories (SNL) challenge, the decohesion modeling is improved by making the failure mode adjustable and by replacing the critical normal and tangential decohesion strengths with the tensile and shear peak strengths, in order to predict the cracking path in a complex configuration with the least computational cost,. It is found that there is a transition between different failure modes along the cracking path, which depends on the stress distribution around the path due to the nonlocal nature of failure evolution. Representative examples will be used to demonstrate the recent advances in simulating failure evolution with the MPM.

Geometry Method for Localization Analysis in Gradient-Dependent J2 Plasticity

In this work the geometrical method for the assessment of discontinuous bifurcation conditions is extended to encompass gradient-dependent plasticity. To this end, the gradient-dependent localization condition is cast in the form of an elliptical envelope condition in the coordinates of Mohr. The results in this work demonstrate the capability of thermodynamically consistent gradient-dependent elastoplastic model formulations to suppress the localized failure modes of the classical plasticity that take place when the hardening/softening modulus H¯ equals the critical value for localization H¯c, provided the characteristic length l remains positive.

Constitutive Modeling and Discontinuous Bifurcation Assessment in Unsaturated Soils

In this work an elastoplastic constitutive theory for unsaturated soils is presented. The proposed material model is formulated in the general framework of the theory of porous media and of the flow theory of plasticity. The model is based on an extension of the well-known MRS Lade model whereby the suction and the effective stress tensor are introduced as additional independent and dependent stress components, respectively. Consequently the cap and cone yield conditions of the MRS Lade model both in hardening and softening as well as the internal evolution laws in these regimes are redefined to include the dependency on the suction. The paper illustrates the predictive capability of the extended MRS Lade model for partially saturated soils. Finally, the condition for discontinuous bifurcation in elastoplastic partially saturated porous media as well as the localized failure predictions of the proposed material formulation for different suctions and stress states are also analyzed and discussed.