Discrete mechanics and the Finite Element Method

Summary We explore applications of the Finite Element Method (FEM) to both Veselov and Lee discrete mechanics in this paper. Based on the FEM, disretizations of continuous Lagrangians are developed and corresponding integrators are obtained. Error estimates for variational integrators are also given. This work is supported by the National Natural Science Foundation of China (grant Nos. 90103004, 10171096) and the National Key Project for Basic Research of China (G1998030601).     

Trans-scale mechanics： looking for the missing links between continuum and micro/nanoscopic reality

Problems involving coupled multiple space and time scales offer a real challenge for conventional frame-works of either particle or continuum mechanics. In this paper, four cases studies （shear band formation in bulk metallic glasses, spallation resulting from stress wave, interaction between a probe tip and sample, the simulation of nanoindentation with molecular statistical thermodynamics） are provided to illustrate the three levels of trans-scale problems （problems due to various physical mechanisms at macro-level, problems due to micro-structural evolution at macro/micro-level, problems due to the coupling of atoms/ molecules and a finite size body at micro/nano-level） and their formulations. Accordingly, non-equilibrium statistical mechanics, coupled trans-scale equations and simultaneous solutions, and trans-scale algorithms based on atomic/molecular interaction are suggested as the three possible modes of trans-scale mechanics.     

A discrete element model for simulating saturated granular soil

A numerical model is developed to simulate saturated granular soil, based on the discrete element method. Soil particles are represented by Lagrangian discrete elements, and pore fluid, by appropriate discrete elements which represent alternately Lagrangian mass of water and Eulerian volume of space. Macro-scale behavior of the model is verified by simulating undrained biaxial compression tests. Micro-scale behavior is compared to previous literature through pore pressure pattern visualization during shear tests. It is demonstrated that dynamic pore pressure patterns are generated by superposed stress waves. These pore-pressure patterns travel much faster than average drainage rate of the pore fluid and may initiate soil fabric change, ultimately leading to liquefaction in loose sands. Thus, this work demonstrates a tool to roughly link dynamic stress wave patterns to initiation of liquefaction phenomena.     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.     

Intermediate asymptotics,scaling laws and renormalization group in continuum mechanics

Scaling laws and self-similar solutions are very popular concepts in modern continuum mechanics. In the present paper these concepts are analyzed both from the viewpoint of intermediate asymptotics, known in classical mathematical physics and fluid mechanics, and from the viewpoint of the renormalization group technique, known in modern theoretical physics. The definition of the renormalization group is proposed, related to the intermediate asymptotics with incomplete similarity. The general presentation is illustrated by examples of essentially non-linear problems where all analytical properties of the solutions and their asymptotics are rigorously proved, as well by an example from turbulence, where the rigorous problem statement is missing. General lecture delivered at the 11th Italian National Congress of Theoretical and Applied Mechanics (AIMETA), Trento, Sept./Oct. 1992.     

Investigation of the damage variable basic issues in continuum damage and healing mechanics

Consistent mathematics and mechanics are used here to properly interpret the damage variable within the confines of the concept of reduced area due to damage. In this work basic issues are investigated for the damage variable in conjunction with continuum damage and healing mechanics. First, the issue of the additive decomposition of the damage variable into damage due to voids and damage due to cracks in continuum damage mechanics is discussed. The accurate decomposition is shown to be non-additive and involves a term due to the interaction of cracks and voids. It is shown also that the additive decomposition can only be used for the special case of small damage. Furthermore, a new decomposition is derived for the evolution of the damage variable. The second issue to be discussed is the new concept of independent and dependent damage processes. For this purpose, exact expressions for the two types of damage processes are presented. The third issue addressed is the concept of healing processes occurring in series and in parallel. In this regard, systematically and consistently, the equations of healing processes occurring either consecutively or simultaneously are discussed. This is followed by introducing the new concept of small healing in damaged materials. Simplified equations that apply when healing effects are small are shown. Finally, some interesting and special damage processes using a systematic and original formulation are presented.     

Linking discrete particle simulation to continuum process modelling for granular matter: Theory and application

Two approaches are widely used to describe particle systems: the continuum approach at macroscopic scale and the discrete approach at particle scale. Each has its own advantages and disadvantages in the modelling of particle systems. It is of paramount significance to develop a theory to overcome the disadvantages of the two approaches. Averaging method to link the discrete to continuum approach is a potential technique to develop such a theory. This paper introduces an averaging method, including the theory and its application to the particle flow in a hopper and the particle-fluid flow in an ironmaking blast furnace.     

Application of Discrete Element Method for Continuum Dynamic Problems

A new method based on the principle of minimum potential energy is presented, aiming to overcome some weakness of the present discrete element method (DEM). Our primary research is to put forward the DEM with a tight theory base and a fit technique for treating continuum dynamic problems. By using this method, we can not only extend the existing seven-disc model, but also establish a new nine-disc model in a general way. Moreover, the equivalences of two kinds of models have been verified. In addition, three numerical examples of stress wave propagation problems are given in order to validate accuracy and efficiency of the present DEM models and their algorithms. Finally, the dynamic stress concentration problem of a square plate with a circular hole is analyzed.This work was supported by Nation Natural Science Foundation of China (nos. 10232040 and 10572002).     

Discrete and continuum modelling of excavator bucket filling

Two-dimensional discrete and continuum modelling of excavator bucket filling is presented. The discrete element method (DEM) is used for the discrete modelling and the material-point method (MPM) for continuum modelling. MPM is a so-called particle method or meshless finite element method. Standard finite element methods have difficulty in modelling the entire bucket filling process due to large displacements and distortions of the mesh. The use of a meshless method overcomes this problem. DEM and MPM simulations (plane strain) of bucket filling are compared to two-dimensional experimental results. Cohesionless corn grains were used as material and the simulated force acting on the bucket and flow patterns were compared with experimental results. The corn macro (continuum) and micro (DEM) properties were obtained from shear and oedometer tests. As part of the MPM simulations, both the classic (nonpolar) and the Cosserat (polar) continuums were used. Results show that the nonpolar continuum is the most accurate in predicting the bucket force while the polar and DEM methods predict lower forces. The DEM model does not accurately predict the material flow during filling, while the polar and nonpolar methods are more accurate. Different flow zones develop during filling and it is shown that DEM, the polar and the nonpolar methods can accurately predict the position and orientation of these different flow zones.     

Combining X-ray microtomography with computer simulation for analysis of granular and porous materials

The use of X-ray microtomographic (XMT) methods in analysing particulate systems has expanded rapidly in recent years with the availability of affordable desk-top apparatus. This review presents a summary of the major applications in which computer simulations are explicitly coupled with XMT in the area of granular and porous materials. We envisage two main ways of establishing the coupling between both techniques, based on the transference or exchange of information by using physical or geometrical paramet...