Fracture mechanics of multiple load path structure

The strain energy approach to linear elastic fracture mechanics is combined with a ‘virtual work’ technique for general structural analysis in order to provide a means of predicting the crack propagation behaviour in a multiple load path structure. Progressive failure in one member may lead to significant changes in the relative stiffness of the load paths, and consequent re-distribution of loads within the structure. A simple example is presented, to illustrate the general approach.     

Geometric continuum mechanics

Geometric Continuum Mechanics ( GCM) is a new formulation of Continuum Mechanics ( CM) based on the requirement of Geometric Naturality ( GN). According to GN, in introducing basic notions, governing principles and constitutive relations, the sole geometric entities of space-time to be involved are the metric field and the motion along the trajectory. The additional requirement that the theory should be applicable to bodies of any dimensionality, leads to the formulation of the Geometric Paradigm ( GP) stating that push-pull transformations are the natural comparison tools for material fields. This basic rule implies that rates of material tensors are Lie-derivatives and not derivatives by parallel transport. The impact of the GP on the present state of affairs in CM is decisive in resolving questions still debated in literature and in clarifying theoretical and computational issues. As a consequence, the notion of Material Frame Indifference ( MFI) is corrected to the new Constitutive Frame Invariance ( CFI) and reasons are adduced for the rejection of chain decompositions of finite elasto-plastic strains. Geometrically consistent notions of Rate Elasticity ( RE) and Rate Elasto-Visco-Plasticity ( REVP) are formulated and consistent relevant computational methods are designed.     

Nonextensive statistical mechanics: A brief introduction

Boltzmann-Gibbs statistical mechanics is based on the entropy . It enables a successful thermal approach to ubiquitous systems, such as those involving short-range interactions, markovian processes, and, generally speaking, those systems whose dynamical occupancy of phase space tends to be ergodic. For systems whose microscopic dynamics is more complex, it is natural to expect that the dynamical occupancy of phase space will have a less trivial structure, for example a (multi)fractal or hierarchical geometry. The question naturally arises whether it is possible to study such systems with concepts and methods similar to those of standard statistical mechanics. The answer appears to be yes for ubiquitous systems, but the concept of entropy needs to be adequately generalized. Some classes of such systems can be satisfactorily approached with the entropy (with , and ). This theory is sometimes referred in the literature as nonextensive statistical mechanics. We provide here a brief introduction to the formalism, its dynamical foundations, and some illustrative applications. In addition to these, we illustrate with a few examples the concept of stability (or experimental robustness) introduced by B. Lesche in 1982 and recently revisited by S. Abe.Received: 27 May 2003, Accepted: 27 August 2003, Published online: 11 February 2004PACS: 05.10.-a, 05.20.Gc, 05.20.-y, 05.45.-a, 05.70.Ln, 05.90. + m, 05.60.CdCorrespondence to: C. Tsallis     

Shock capturing viscosities for the general fluid mechanics algorithm

The performance of different shock capturing viscosities has been examined using our general fluid mechanics algorithm. Four different schemes have been tested, both for viscous and inviscid compressible flow problems. Results show that the methods based on the second gradient of pressure give better performance in all situations. For instance, the method constructed from the nodal pressure values and consistent and lumped mass matrices is an excellent choice for inviscid problems. The method based on L₂ projection is better than any other method in viscous flow computations. The residual based anisotropic method gives excellent performance in the supersonic range and gives better results in the hypersonic regime if a small amount of residual smoothing is used. © 1998 John Wiley & Sons, Ltd.     

Closed trans-scale statistical microdamage mechanics

Damage and failure due to distributed microcracks or microvoids are on the challenging frontiers of solid mechanics. This appeals strongly to tools not yet fully developed in continuum damage mechanics, in particular to irreversible statistical thermodynamics and a unified macroscopic equations of mechanics and kinetic equations of microstructural transformations. This review provides the state of the art in statistical microdamage mechanics.

Conservation laws in classical mechanics for quasi-coordinates

Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.     

Solution of one problem of fracture mechanics

This paper considers a model for the opening-mode fracture separation process based on the introduction of an interaction layer. This layer is defined as the region of localization of the fracture process. The stress-strain state of the layer material is uniform in the cross section of the layer. A study is made of the deformation of a double-cantilever beam weakened by a notch whose width is equal to the thickness of the interaction layer. The problem is solved in a linearly geometrical approximation. The thickness of the interaction layer is estimated, and a method for solving the formulated problem is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 121–127, July–August, 2007.     

Fracture mechanics of laminated glass subjected to blast loading

A failure criterion based on energy balance approach is introduced for the laminated glass panel subjected to blast loading. Based on this failure criterion, a damage factor is developed to assess the failure of the laminated glass panel. If the damage factor is less than one, the plate is safe otherwise unsafe. Trigonometric function is employed to express the transverse deflection and the Airy’s stress function in von Karman’s large deflection equations of a thin plate. The nonlinear ordinary differential equation of motion obtained using the Galerkin method is solved using Runge–Kutta method. The predicted results indicate that the breakages of the laminated glass may be caused by the negative phase of the blast load if the positive phase blast load is not violent enough to cause failure. Also, the size of glass shards the laminated glass plies breaks in to is predicted using the surface energy based failure model.     

Joseph Boussinesq's legacy in fluid mechanics

Joseph Boussinesq was the most prolific of all French contributors to nineteenth-century fluid mechanics. His scientific production included a novel theory of solitary waves, the KdV equation for finite deformations of the water surface in an open channel, a systematic study of open channel and pipe flow based on the concept of effective viscosity, pioneering derivations of boundary layers and entrance effects, new exact solutions of the Navier–Stokes equation under geometrically simple boundary conditions, and the ‘Boussinesq approximation’ for heat convection in a moving fluid under gravity. Although his extraordinary skills were quickly recognized and rewarded, other experts in the field were often unaware even of his most important results and they ended up rediscovering some of them. Boussinesq's unusual background and the resulting peculiarities of his style explain this problematic diffusion. They also account for the richness of his legacy.     

Mesofracture mechanics: a necessary link

Classical fracture mechanics is based on the premise that small scale features could be averaged to give a larger scale property such that the assumption of material homogeneity would hold. Involvement of the material microstructure, however, necessitates different characteristic lengths for describing different geometric features. Macroscopic parameters could not be freely exchanged with those at the microscopic scale level. Such a practice could cause misinterpretation of test data. Ambiguities arising from the lack of a more precise range of limitations for the definitions of physical parameters are discussed in connection with material length scales. Physical events overlooked between the macroscopic and microscopic scale could be the link that is needed to bridge the gap. The classical models for the creation of free surface for a liquid and solid are oversimplified. They consider only the translational motion of individual atoms. Movements of groups or clusters of molecules deserve attention. Multiscale cracking behavior also requires the distinction of material damage involving at least two different scales in a single simulation. In this connection, special attention should be given to the use of asymptotic solution in contrast to the full field solution when applying fracture criteria. The former may leave out detail features that would have otherwise been included by the latter. Illustrations are provided for predicting the crack initiation sites of piezoceramics. No definite conclusions can be drawn from the atomistic simulation models such as those used in molecular dynamics until the non-equilibrium boundary conditions can be better understood. The specification of strain rates and temperatures should be synchronized as the specimen size is reduced to microns. Many of the results obtained at the atomic scale should be first identified with those at the mesoscale before they are assumed to be connected with macroscopic observations. Hopefully, “mesofracture mechanics” could serve as the link to bring macrofracture mechanics closer to microfracture mechanics.     

Kinematic variables bridging discrete and continuum granular mechanics

It is known that there is wide, and at present, unbridgeable, gap between discrete and continuum granular mechanics. In this contribution, first, microscopic kinematic variables neglected in classical continuum granular mechanics are investigated based on the kinematics of discs in contact. Then, a kinematic variable called the averaged pure rotation rate (APR) is proposed for an assembly of circular discs of different sizes, which is then used to produce another two kinematic tensors with one equal to the deformation rate tensor and the other unifying the spin tensor and the APR. As an example, the kinematic variables are incorporated into the unified double-slip plasticity model. Finally, these theoretical analyses are verified using a two-dimensional discrete element method. The study shows that these kinematic variables can be used to bridge discrete and continuum granular mechanics.     

High frequency homogenization for structural mechanics

We consider a net created from elastic strings as a model structure to investigate the propagation of waves through semi-discrete media. We are particularly interested in the development of continuum models, valid at high frequencies, when the wavelength and each cell of the net are of similar order. Net structures are chosen as these form a general two-dimensional example, encapsulating the essential physics involved in the two-dimensional excitation of a lattice structure whilst retaining the simplicity of dealing with elastic strings.Homogenization techniques are developed here for wavelengths commensurate with the cellular scale. Unlike previous theories, these techniques are not limited to low frequency or static regimes, and lead to effective continuum equations valid on a macroscale with the details of the cellular structure encapsulated only through integrated quantities. The asymptotic procedure is based upon a two-scale approach and the physical observation that there are frequencies that give standing waves, periodic with the period or double-period of the cell. A specific example of a net created by a lattice of elastic strings is constructed, the theory is general and not reliant upon the net being infinite, none the less the infinite net is a useful special case for which Bloch theory can be applied. This special case is explored in detail allowing for verification of the theory, and highlights the importance of degenerate cases; the specific example of a square net is treated in detail. An additional illustration of the versatility of the method is the response to point forcing which provides a stringent test of the homogenized equations; an exact Green's function for the net is deduced and compared to the asymptotics.     

Surrogate-based modeling and dimension reduction techniques for multi-scale mechanics problems

Successful modeling and/or design of engineering systems often requires one to address the impact of multiple "design variables" on the prescribed outcome.There are often multiple,competing objectives based on which we assess the outcome of optimization.Since accurate,high fidelity models are typically time consuming and computationally expensive,comprehensive evaluations can be conducted only if an efficient framework is available.Furthermore,informed decisions of the model/hardware’s overall performance rely on an adequate understanding of the global,not local,sensitivity of the individual design variables on the objectives.The surrogate-based approach,which involves approximating the objectives as continuous functions of design variables from limited data,offers a rational framework to reduce the number of important input variables,i.e.,the dimension of a design or modeling space.In this paper,we review the fundamental issues that arise in surrogate-based analysis and optimization,highlighting concepts,methods,techniques,as well as modeling implications for mechanics problems.To aid the discussions of the issues involved,we summarize recent efforts in investigating cryogenic cavitating flows,active flow control based on dielectric barrier discharge concepts,and lithium(Li)-ion batteries.It is also stressed that many multi-scale mechanics problems can naturally benefit from the surrogate approach for "scale bridging."     

Combination of fracture and damage mechanics for numerical failure analysis

A new approach for the analysis of crack propagation in brittle materials is proposed, which is based on a combination of fracture mechanics and continuum damage mechanics within the context of the finite element method. The approach combines the accuracy of singular crack-tip elements from fracture mechanics theories with the flexibility of crack representation by softening zones in damage mechanics formulations. A super element is constructed in which the typical elements are joined together. The crack propagation is decided on either of two fracture criteria; one criterion is based on the energy release rate or the J-integral, the other on the largest principal stress in the crack-tip region. Contrary to many damage mechanics methods, the combined fracture⧹damage approach is not sensitive to variations in the finite element division. Applications to situations of mixed-mode crack propagation in both two- and three-dimensional problems reveal that the calculated crack paths are independent of the element size and the element orientation and are accurate within one element from the theoretical (curvilinear) crack paths.     

以力学竞赛促进基础力学教改的创新与实践

以组织与辅导力学竞赛为契机,促进基础力学教学改革. 着眼于提高学生的力学素养与创新 能力,围绕如何创新基础力学教学新模式,从激发学生学习的热情与动力、改革教学实验、 提升力学课程建设等方面介绍了基于大学生力学竞赛培养力学人才与促进高校基础力学教学 改革的基本做法、实践效果与创新成果.     

Elastic-plastic mechanics of steady crack growth under anti-plane shear

For a crack with steady growth under anti-plane shear, analysis shows a primary plastic zone included in an angle of ±19.7° ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks, each included in an angle of 0.37°. Numerical solutions give the shape of the plastic zones which determine the active and residual plastic strains, and give the crack tip displacement, which is approximately 0.07 of that for monotonic loading without growth. The length of the primary plastic zone is almost the same as that without growth, but the thickness is about 3/5 as great. Coupled with ductile fracture criteria, the present results predict initially stable crack growth, whereas analyses based on the simplification of yielding on just one plane predict unstable fracture immediately following initiation.     

The mechanics of soil cutting by a rotating wire

The cutting of soil by a rotating wire analogous to the tip of a rotary tiller blade while cutting a two-dimensional soil slice over a range of ‘fetch-ratios’ (bite length/depth-ratios) in a quasi-static condition is presented. A theoretical models based on Mohr-Coloumb soil mechanics has been proposed to predict forces on the wire (tip). The model is dependent upon observed passive general shear failure of the soil slice towards the curved free surface of a previous cut and the lateral local shear failure towards the undeformed soil. The predicted forces in a frictional soil and in a pure cohesive medium (artificial clay) agreed well with experimental results.