APPLICATION OF THE FINITE-DIFFERENCE METHOD IN LAGRANGE VARIABLES TO CALCULATION OF WAVE FIELDS IN COMPLEXLY STRUCTURED MEDIA

Questions and comments are provided for (1). I find Yulii Shikhmurzaev's opinion paper on the contact line problem interesting to read. It is helpful as it has allowed me to better identify the points where I disagree. First, he defines microscopic and macroscopic levels of continuum description (that may be applied to the contact line problem) as mutually exclusive. The definition is done in a way that excludes the mesoscopic models for the contact line problem used in the literature. Then he introduces his own model that in my view is also a mesoscale model. In contrast, I think that already his definition of levels of continuum description is not a good representation of reality, and I will below offer a slightly different one. If one accepts my classification, the widely employed mesoscale models become admitted again and Yulii Shikhmurzaev's model becomes one of them, and competes on a common ground. My first major concern is his classification of continuum descriptions. I do not un- derstand the way Yulii Shikhmurzaev draws the line between models in Continuum Mechanics and Statistical Physics. If statistical physics deals with "field variables describing statistical distributions of molecular properties" (1) then continuum me- chanics deals with field variables describing mean values of molecular properties (i.e., the lowest order moments of statistical distributions). Continuum mechanics is a part of statistical physics 1 . To obtain a valid description based on mean values (as in

We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable u having a viscous damping with relaxation time varepsilon ^alpha and an internal variable z with relaxation time varepsilon we obtain different limits for the three cases alpha in (0,1), alpha =1 and alpha >1. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.

The starting point of every theory of continuum materials is the choice of the set of independent variables defining the configuration of a material. The aim is that of determining the configuration of the material when it is subjected to given external actions. Both general principles and constitutive equations are exploited. I t is assumed not only that the constitutive equations of the material can, at least in principle, be experimentally determined, but also that the constitutive quantities arc sufficient to characterize the material for the purposes of the theory. Often, no relation between the constitutive quantities is introduced at the outset: they are, therefore, to be considered independent of each other. Consider the case of homogeneous isotropic elastic materials. When thermal effects are taken into account, the set of independent variables is composed of the deformation gradient F, the absolute temperature 0 and the temperature gradient g = grad 0. The constitutive quantities are the stress tensor T, the specific internal energy E, the specific entropy V and the heating-flux vector q. The relevant constitutive equations

In Part 1 of this paper, a machining force model was developed based on an enhanced version of Zheng’s Continuum Mechanics model that incorporates microstructural effects. Machining experiments identified Secondary Dendrite Arm Spacing (SDAS) as a significant microstructure feature of 319 aluminum in terms of machinability. A new material constitutive relationship that incorporates SDAS microstructure effects on the flow stress was proposed. In this part of the paper, disk turning tests are performed to simulate the orthogonal cutting process. The cutting forces obtained from some of these tests are used in concert with an inverse form of the continuum mechanics machining model to estimate the parameters in the material constitutive equation. The enhanced continuum mechanics orthogonal cutting model is then applied to predict cutting forces when machining Al319. Comparison of the model predicted and experimentally acquired cutting forces is demonstrated to show good agreement.

Mechanical models and numerical simulations in nanomechanics: A review across the scales

Smart materials are active and multifunctional materials, which play an important part for sensor and actuator applications. These materials have the potential to transform passive structures into adaptive systems. However, a prerequisite for the design and the optimization of these materials is, that reliable models exist, which incorporate the interaction between the different combinations of thermal, electrical, magnetic, optical and mechanical effects. Polymeric electroelastic materials, so-called (EAP), own the characteristic to deform if an electric field is applied. EAP's possesses the benefit that they share the characteristic of polymers, these are lightweight, inexpensive, fracture tolerant, elastic, and the chemical and physical structure is well understood. However, the description electroactive polymer is a generic term for many kinds of different microscopic mechanisms and polymeric materials. Based on the laws of electromagnetism and elasticity, a visco-electroelastic model is developed and implemented into the finite element method (FEM). The presented three-dimensional solid element has eight nodes and trilinear interpolation functions for the displacement and the electric potential. The continuum mechanics model contains finite deformations, the time dependency and the nearly incompressible behavior of the material. To describe the possible, large time dependent deformations, a finite viscoelastic model with a split of the deformation gradient is used. Thereby the time dependent characteristic of polymeric materials is incorporated through the free energy function. The electromechanical interactions are considered by the electrostatic forces and inside the energy function.

Using geologic structures to constrain constitutive laws not accessible in the laboratory

The basis operator method intended for constructing systems of difference approximations to differential operators in vector and tensor analysis is extended to orthogonal systems on a surface. A class of completely conservative differential-difference schemes for continuum mechanics in Lagrangian variables is constructed. Basis operators are constructed using the finite volume equation, consistency conditions for discrete operators of the first derivative, and consistent projection operators for grid functions. A system of differential-difference continuum mechanics equations on a surface is obtained, which implies all conservation laws typical of the continuum case, including additional ones. A stability estimate is derived for discrete equations of an incompressible viscous fluid.

Transient conduction simulation of a nano-scale hotspot using finite volume lattice Boltzmann method

In this paper, we present a new explicit second-order accurate structure-preserving finite volume scheme for the first-order hyperbolic reformulation of the Navier–Stokes–Korteweg equations. The model combines the unified Godunov-Peshkov-Romenski model of continuum mechanics with a recently proposed hyperbolic reformulation of the Euler–Korteweg system. The considered PDE system includes an evolution equation for a gradient field that is by construction endowed with a curl-free constraint. The new numerical scheme presented here relies on the use of vertex-based staggered grids and is proven to preserve the curl constraint exactly at the discrete level, up to machine precision. Besides a theoretical proof, we also show evidence of this property via a set of numerical tests, including a stationary droplet, non-condensing bubbles as well as non-stationary Ostwald ripening test cases with several bubbles. We present quantitative and qualitative comparisons of the numerical solution, both, when the new structure-preserving discretization is applied and when it is not. In particular for under-resolved simulations on coarse grids we show that some numerical solutions tend to blow up when the curl-free constraint is not respected.

The motion with uniform deformation is a subclass of motions inherent in many models of continuum mechanics. We concentrate on the l-plane model of atmosphere averaged over the height in various modifications. We show that there exist several steady patterns corresponding to equilibria points. They are prototypes of atmospheric movements such as vortices, saddle points, and shears. These equilibria can be stable at certain values of the parameters. We give a complete stability analysis for all cases.

In this paper, a novel parameter determination technique is developed for material models in continuum mechanics aimed at describing metamaterials. Owing to their peculiar mechanical properties and behaviors, such as extreme elasticity or high strength‐to‐weight ratio, metamaterials are of interest to be simulated by reduced‐order modeling by means of the generalized mechanics. Such models incorporate constitutive parameters to be determined; we develop an automatized optimization process specifically for obtaining metamaterials parameters. The process aims at minimizing a mechanically meaningful error function measuring the deviation of the continuum from a detailed description by using the Trust Region Reflective optimization method. The parameter identification procedure is tested for an exemplary extension experiment of a metamaterial, proving to be robust and reliable.

In Part 1 of this paper, a continuum mechanics model of the orthogonal cutting process was developed based on finite deformation theory. In this part of the paper, constitutive equations for O1 and L6 tool steels are developed using the results from split Hopkinson pressure bar tests. Statistically designed orthogonal cutting experiments are conducted to secure process results across a range of cutting conditions. The continuum mechanics model established in Part 1 of this paper is used to simulate all the cutting tests. All the model outputs are calculated and compared with the corresponding cutting experiment results. Good agreement is observed between the model predictions and the experimental results. The continuum mechanics model is successfully used to predict the cutting force, shear angle, and temperature.

Friction is everywhere and important in our daily life as well as in industry. In general, dry friction acts as a resistance to the relative motion and dissipates energy of a system; however, under certain conditions, it can cause self-excited vibration of a system, which is known as friction-induced vibration. Friction-induced vibration can potentially cause problems like wear, fatigue failure, and noise, among which brake squeal is a typical engineering problem. As a comprehensive understanding of friction-induced vibration has not been achieved, friction-induced vibration is still a challenging research topic. The aim of this research is to carry out a theoretical study of the dynamic behaviour of nonlinear/non-smooth friction-induced vibration of phenomenological mechanical models. Discrete and continuous mechanical models with dry friction that involve nonlinear contact stiffness, stick-slip motion, or separation and reattachment at the contact interface are proposed, and numerical simulations of the transient dynamic behaviour of the non-smooth frictional systems are implemented. Their complex dynamic behaviour and the influences by various system parameters are predicted. In addition, a reduction strategy for the complicated frictional systems is presented and validated via theoretical and experimental results, which is a preliminary step in analysing complicated systems (real structures) with nonlinear/non-smooth friction behaviour in future research. The main objectives of the research work reported in this thesis are: 1. To carry out the transient dynamic analysis of non-smooth friction-induced vibration. A varying time-step numerical algorithm, which combines Runge-Kutta method that is specifically for the second-order differential equation of motion and the bisection method, is proposed (Chapter 3). This algorithm ensures the accuracy of the results of the non-smooth vibration in which the motion states keep switching among distinct motion states of separation and reattachment, or stick and slip, which is testified by providing the same results of a classic non-smooth stick-slip vibration of a single-degree-of-freedom model. 2. In consideration of improving previous research on theoretical mechanical models with the assumptions that the contact stiffness is linear or separation is ignored, a nonlinear 2-degree-of-freedom slider-on-moving-belt model developed from Hoffmann’s model and the theoretical formulations are proposed (Chapter 4), in which a cubic contact spring is included; loss of contact (separation) at the slider-belt interface is allowed and importantly reattachment of the slider to the belt after separation is also considered. The stability and dynamic behaviour of the system are investigated. Complex eigenvalue analysis (CEA) indicates that the roles of the preload and the nonlinear stiffness on the stability of the nonlinear system are not monotonous. Transient dynamic analysis (TDA) shows that separation and reattachment could happen. Ignoring separation between bodies in sliding frictional contact in vibration is unsafe as this may underestimate the vibration amplitude, and predicts incorrect effects of the key parameters on the vibration, thus considering separation is very important. Moreover, frequency domain results show the necessity of implementing both CEA and TDA in the study of nonlinear friction-induced vibration and the importance of considering separation from the frequency domain point of view. Finally, non-smooth Coulomb’s law of friction is introduced in the nonlinear 2-degree-of-freedom (2-DoF) slider-belt model. Numerical results show that diverse dynamic behaviour of this 2-DoF system with nonlinearity/non-smoothness can be generated when both of the stick-slip and mode-coupling instability are involved. 3. Separation, reattachment and impact are considered in the study of friction-induced vibration of a system having an elastic disc, excited by the in-plane stick–slip vibration of a moving mass-damper-spring slider attached to a rigid wall that is dragged around on the disc surface at a constant rotating speed (Chapter 5). Theoretical formulations and the numerical procedure for the current non-smooth system are developed. Numerical results show that separation and reattachment could occur in a low speed range well below the critical disc speed in the context of a constant rotating load. Poincare maps of the system of the two distinct cases (considering separation and ignoring separation) are plotted which exhibit the diversity of nonlinear dynamic behaviour of the system and the importance of considering separation. Furthermore, the roles of the key system parameters on the vibration are investigated. Time-frequency analysis reveals the time-varying properties of this system and the contributions of separation and in-plane stick-slip vibration to the system frequencies. One major finding is that ignoring separation, as is usually done, often leads to very different dynamic behaviour and possibly misleading results. 4. Based on the idea of mode synthesis method, a reduction strategy for complicated frictional systems is put forward, in which the natural contact interfaces and the tangential friction force are involved, and its applications and experimental validation are presented (Chapter 6). Firstly, its application to a theoretical multi-degree-of-freedom model with linear contact verifies the accuracy and feasibility of the strategy. The influence of the system parameter, and the mode number that is used in the reduced model on the stability of the reduced model are investigated. The reduced model is capable of preserving the key features (bifurcation of the eigenvalue and unstable frequencies) of the original model. Furthermore, a specific reduction strategy, for a real pad-on-disc test rig and its corresponding finite element model which involve direct contact of the interface, is proposed. The results of the reduced model with a small number of modes of the substructures correlate fairly well with theoretical results of the full model and the test results in terms of predicting mode-coupling instability and unstable frequencies, which validates this promising method for future work on friction-induced vibration of complicated frictional systems or real structures.

Helical extension springs are used as flexible joints in surgical robots to achieve large bending angles. As the spring wire undergoes complex forces, continuum mechanics is essential for characterizing the spring deformation for precise joint bending control. In this work, a mechanical model of a tendon‐actuated bending spring joint is developed based on Cosserat rod theory. A reduced model is further derived to achieve dimension reduction. The contact between adjacent spring coils complicates the mechanical modeling. To address this issue, implicit variables of contact positions and forces are introduced to express geometric constraints at the contact points. The resultant model consists of Cosserat differential equations and algebraic constraint equations, and is solved by the shooting method. The Levenberg–Marquardt algorithm and numerical integration are used to find optimal solutions numerically. Simulation results show the feasibility of the reduced model. Experiments are conducted to calibrate spring parameters and validate the mechanical model. Results show that the model can predict nonlinear relationships of tendon displacements, actuation forces, and bending angles of the spring joint.