From Rate-Dependent to Rate-Independent Brittle Crack Propagation

On the base of many experimental results, e.g., Ravi-Chandar and Knauss (Int. J. Fract. 26:65–80, 1984), Sharon et al. (Phys. Rev. Lett. 76(12):2117–2120, 1996), Hauch and Marder (Int. J. Fract. 90:133–151, 1998), the object of our analysis is a rate-dependent model for the propagation of a crack in brittle materials. Restricting ourselves to the quasi-static framework, our goal is a mathematical study of the evolution equation in the geometries of the ‘Single Edge Notch Tension’ and of the ‘Compact Tension’. Besides existence and uniqueness, emphasis is placed on the regularity of the evolution making reference also to the ‘velocity gap’. The transition to the rate-independent model of Griffith is obtained by time rescaling, proving convergence of the rescaled evolutions and of their energies. Further, the discontinuities of the rate-independent evolution are characterized in terms of unstable points of the free energy. Results are illustrated by a couple of numerical examples in the above mentioned geometries.     

From Molecular Models¶to Continuum Mechanics

We present here a limiting process allowing us to write some continuum mechanics models as a natural asymptotic of molecular models. The approach is based on the hypothesis that the macroscopic displacement is equal to the microscopic one. We carry out the corresponding calculations in the case of two-body energies, including higher order terms, and also in the case of Thomas-Fermi type models.     

Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals

The Lie group structure of crystals which have uniform continuous distributions of dislocations allows one to construct associated discrete structures—these are discrete subgroups of the corresponding Lie group, just as the perfect lattices of crystallography are discrete subgroups of $\mathbb{R}^{3}$ , with addition as group operation. We consider whether or not the symmetries of these discrete subgroups extend to symmetries of (particular) ambient Lie groups. It turns out that those symmetries which correspond to automorphisms of the discrete structures do extend to (continuous) symmetries of the ambient Lie group (just as the symmetries of a perfect lattice may be embedded in ‘homogeneous elastic’ deformations). Other types of symmetry must be regarded as ‘inelastic’. We show, following Kamber and Tondeur, that the corresponding continuous automorphisms preserve the Cartan torsion, and we characterize the discrete automorphisms by a commutativity condition, (6.14), that relates (via the matrix exponential) to the dislocation density tensor. This shows that periodicity properties of corresponding energy densities are determined by the dislocation density.     

Atomic to Continuum Passage for Nanotubes: A Discrete Saint-Venant Principle and Error Estimates

We consider general infinite nanotubes of atoms in ${\mathbb{R}^3}$ where each atom interacts with all the others through a two-body potential. At the equilibrium, the positions of the atoms satisfy a Euler–Lagrange equation. When there are no exterior forces and for a suitable geometry, a particular family of nanotubes is the set of perfect nanotubes at the equilibrium. When exterior forces are applied on the nanotube, we compare the nanotube to nanotubes of the previous family. In part I of the paper, this quantitative comparison is formulated in our first main result as a discrete Saint-Venant principle. As a corollary, we also give a Liouville classification result. Our Saint-Venant principle can be derived for a large class of potentials (including the Lennard-Jones potential), when the perfect nanotubes at the equilibrium are stable. The approach is designed to be applicable to nanotubes that can have general shapes like, for instance, carbon nanotubes or DNA, under the oversimplified assumption that all the atoms are identical. In part II of the paper, we derive from our Saint-Venant principle a macroscopic mechanical model for general nanotubes. We prove that every solution is well approximated by the solution of a continuum model involving stretching and twisting, but no bending. We establish error estimates between the discrete and the continuous solution. More precisely we give two error estimates: one at the microscopic level and one at the macroscopic level.     

Rotation Number and Exponential Dichotomy for Linear Hamiltonian Systems: From Theoretical to Numerical Results

The paper considers the rotation number for a family of linear nonautonomous Hamiltonian systems and its relation with the exponential dichotomy concept. We propose numerical techniques to compute the rotation number and we employ them to infer when a given system enjoys or not an exponential dichotomy. Comparisons with QR-based techniques for exponential dichotomy will give new insights on the structure of the spectrum for the one-dimensional quasi-periodic Schrödinger operator. Experiments on the two dimensional Schrödinger equation will be presented as well.     

Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of the equations of motion, viewing motion as a curve in an infinite-dimensional Banach manifold of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies, where the residual stresses result from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two-dimensional curved annulus embedded in a sphere.     

Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping

Summary We call piezoelectromechanical (PEM) truss beam a truss modular beam coupled with a transmission electrical line when the coupling is obtained by piezoelectric actuators which act as bars in the module and as capacitances in the electrical line. The truss module length is assumed negligible with respect to the considered wave lengths. The transmission electrical line is assumed continuously distributed along the truss beam. Applying the method of virtual power as expounded in [2] we formulate a continuum model for PEM truss beams and we prove that there exists a critical value for the transmission electrical impedance in the neighborhood of which the electromechanical modal coupling is maximum and the possible electrical dissipation of mechanical energy is relevant. Accepted for publication 1 June 1997     

Correction to: Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

Journal of Elasticity -     

Continuum slip viscoplasticity with the Haasen constitutive model: Application to CdTe single crystal inelasticity

Small strain constitutive equations are developed for the thermomechanical behavior of semiconductor single crystals, including dislocation density as an evolving parameter. The model of Haasen, Alexander and coworkers is modified (extended) to include evolution of coefficients in the definition of internal stress. These account for an evolving dislocation substructure. The resulting model is applied in a continuum slip framework to allow multiple slip orientations. Slip system interaction rules are adapted to include slip system interaction for multiple slip conditions and to suppress secondary slip and dislocation density generation for single slip orientations. The approach is discussed relative to other models for viscoplasticity of single crystals and is examined in the context of thermodynamics with internal state variables. The framework is used to correlate experimental data from compression tests of single crystals of the compound semiconductor CdTe from room temperature to near the melting point. Sensitivity of the model to uncertainties such as initial dislocation density is explored.     

Nonlinear Thermoelastic Materials with Viscosity, and Subject to Internal Constraints: A Classical Continuum Thermodynamics Approach

A general approach to continuum thermodynamics that was advocated by R.S. Rivlin is carried out for thermoelastic materials which can also depend on strain rate. An entropy function is constructed (rather than assumed to exist). A method for treating thermomechanical internal constraints for such materials is also presented. In this method, the properties of a constrained material are inherited from those of a related equivalence class of unconstrained materials.     

Whither tube theory: From believing to measuring

This short contribution examines the difficulties that have not yet been fully overcome in the many developments made from the simplest (and original) tube model for entangled polymers.     

Effective Diffusion Coefficient: From Homogenization to Experiment

In this paper, an example of the application of the homogenization approach (asymptotic expansion technique) to predict the effective diffusion coefficient for an equivalent continuum, together with the experimental verification of the theoretical results is presented. The experimental setup was constructed for the measurements of diffusion in a model periodic porous medium made of Plexiglas. The computer program using the FEM was elaborated to solve the local boundary value problem for a period and to calculate the effective diffusion coefficient. The comparison between the theory and the experiment indicates good agreement between the numerical and experimental values of the effective diffusion coefficient. Interpretation of the test data from the point of view of the homogenization theory is also incorporated.     

Ultralight porous metals: From fundamentals to applications

Over the past few years a number of low cost metallic foams have been produced and used as the core of sandwich panels and net shaped parts. The main aim is to develop lightweight structures which are stiff, strong, able to absorb large amount of energy and cheap for application in the transport and construction industries. For example, the firewall between the engine and passenger compartment of an automobile must have adequate mechanical strength, good energy and sound absorbing properties, and adequate fire retardance. Metal foams provide all of these features, and are under serious consideration for this applications by a number of automobile manufacturers (e.g., BMW and Audi). Additional specialized applications for foam-cored sandwich panels range from heat sinks for electronic devices to crash barriers for automobiles, from the construction panels in lifts on aircraft carriers to the luggage containers of aircraft, from sound proofing walls along railway tracks and highways to acoustic absorbers in lean premixed combustion chambers. But there is a problem. Before metallic foams can find a widespread application, their basic properties must be measured, and ideally modeled as a function of microstructural details, in order to be included in a design. This work aims at reviewing the recent progress and presenting some new results on fundamental research regarding the micromechanical origins of the mechanical, thermal, and acoustic properties of metallic foams. The project supported by the U.S. Office of Naval Research (ONR/ONRIFO grant number N000140110271), the U.K. Engineering and Physical Sciences Research Council (EPSRC grant number EJA/U83), and the Chinese State Key Lab. Foundation of Xi'an Jiaotong University     

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通过对日晷、机械钟、石英钟和原子钟等计时工具的考察,阐述了计时工具从基于自然现象,到基于机械结构,再到基于高频振荡的演进过程。指出中国天文钟上的擒纵机构为机械钟的发明提供了技术准备,天文钟是基于自然现象的时钟和基于机械结构的时钟之间的过渡。     

New inroads in an old subject: Plasticity, from around the atomic to the macroscopic scale

Nonsingular, stressed, dislocation (wall) profiles are shown to be 1-d equilibria of a non-equilibrium theory of Field Dislocation Mechanics (FDM). It is also shown that such equilibrium profiles corresponding to a given level of load cannot generally serve as a travelling wave profile of the governing equation for other values of nearby constant load; however, one case of soft loading with a special form of the dislocation velocity law is demonstrated to have no ‘Peierls barrier’ in this sense. The analysis is facilitated by the formulation of a 1-d, scalar, time-dependent, Hamilton-Jacobi equation as an exact special case of the full 3-d FDM theory accounting for non-convex elastic energy, small, Nye-tensor-dependent core energy, and possibly an energy contribution based on incompatible slip. Relevant nonlinear stability questions, including that of nucleation, are formulated in a non-equilibrium setting. Elementary averaging ideas show a singular perturbation structure in the evolution of the (unsymmetric) macroscopic plastic distortion, thus pointing to the possibility of predicting generally rate-insensitive slow response constrained to a tensorial ‘yield’ surface, while allowing fast excursions off it, even though only simple kinetic assumptions are employed in the microscopic FDM theory. The emergent small viscosity on averaging that serves as the small parameter for the perturbation structure is a robust, almost-geometric consequence of large gradients of slip in the dislocation core and the persistent presence of a large number of dislocations in the averaging volume. In the simplest approximation, the macroscopic yield criterion displays anisotropy based on the microscopic dislocation line and Burgers vector distribution, a dependence on the Laplacian of the incompatible slip tensor and a nonlocal term related to a Stokes-Helmholtz-curl projection of an ‘internal stress’ derived from the incompatible slip energy.