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.     

A kinematically and thermodynamically consistent volumetric growth model based on the stress-free configuration

A continuum volumetric growth theory is presently elaborated based on the stress-free configuration. The total deformation gradient is the composition of a growth mapping followed by an elastic mapping ensuring the compatibility of the body. A measure of the growth rate fully lying in the stress-free configuration is elaborated, and the balance of momentum accounting for mass changes due to growth is written in the same configuration, highlighting measures of kinematic incompatibilities. An Eshelby stress is identified as the driving force for growth, in the sense that it appears as the true dissipative thermodynamic force conjugated to the growth rate. Examples of incompatible cylindrical growth with residual stress generation and tumor growth with mechanics coupled to diffusion of nutrients illustrate the proposed theory.     

Understanding the influence of structural hierarchy and its coupling with chemical environment on the strength of idealized tropocollagen–hydroxyapatite biomaterials

Hard biomaterials such as bone, dentin, and nacre have primarily an organic phase (e.g. tropocollagen (TC)) and a mineral phase (e.g. hydroxyapatite (HAP) or aragonite) arranged in a staggered arrangement at the nanoscopic length scale. Interfacial interactions between the organic phase and the mineral phase as well as the structural effects arising due to the staggered arrangement significantly affect the strength of such biomaterials. The effect of such factors is intricately intertwined with the chemical environment of such materials. In the present investigation, an idealized TC–HAP composite system under tensile loading is analyzed using explicit three-dimensional (3-D) molecular dynamics (MD) simulations to develop an understanding of these factors. The material system is analyzed in three different environments: (1) in the absence of water molecules (non-hydrated), (2) in the presence of water molecules (hydrated), and (3) in the presence of water molecules with calcium ions (ionized water). The analyses focus on understanding the correlations among factors such as the structural arrangement, the peak stress during deformation, Young's modulus, the peak interfacial strength, and the length scale of the localization of peak stress during deformation. Analyses show that maximizing the contact area between the TC and HAP phases results in higher interfacial strength as well as higher fracture strength. Due to the staggered arrangement, the orientation of HAP crystals has insignificant effect on the biomaterial strength. Analyses based on strength scaling as a function of structural hierarchy level reveal that while peak strength follows a multiscaling relation, the fracture strength does not. The peak strain for failure was found to be independent of the changes in levels of structural hierarchy. Overall, the analyses, being limited in size due to the computational time constraint, point out important correlations between the mechanical strength and chemically influenced structural hierarchy of biomaterials.     

A spectral streamline diffusion finite element coupling method of unsteady transport equation in the field of neutron logging

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Ludwig Prandtl and the growth of fluid mechanics in Germany

Ludwig Prandtl (1875–1953) has been called the father of modern aerodynamics. His name is associated most famously with the boundary layer concept, but also with several other topics in 20th-century fluid mechanics, particularly turbulence (Prandtl's mixing length). Among his disciples are pioneers of modern fluid mechanics like Heinrich Blasius, Theodore von Kármán, and Walter Tollmien. Furthermore, Prandtl founded the Aerodynamische Versuchsanstalt (AVA) and the Kaiser-Wilhelm-Institut für Strömungsforschung in Göttingen, nuclei for the growth of fluid mechanics in Germany. In this article I trace this development on the basis of my recent biography of Prandtl.     

Transport of mass and energy of polyatomic gases in magnetic fields computed by a Monte Carlo algorithm

The transport of mass through a rectangular channel and of energy between parallel surfaces of polyatomic gases in the Knudsen regime and in the presence of external magnetic fields is calculated by means of a Monte Carlo algorithm. A four-parameter mathematical model is proposed that takes into account the dynamical aspects of molecule-surface interactions and the influence of external magnetic fields on the angular momentum of polyatomic molecules. The Monte Carlo algorithm makes use of the Latin super-cube sampling method, correlated samplings, and the concept of importance sampling. The four parameters are determined by an optimised method based on the Monte Carlo algorithm and on experimental results for the mass flux rate of the gases N₂ and CO in the presence of external magnetic fields through a rectangular channel with surfaces coated with Au. The optimised values of the parameters are used to determine the behaviour of the mass flux rate through a rectangular channel and the heat flux between two parallel plates for the gases N₂ and CO as a function of the applied magnetic field. The calculated curves fit the experimental data well.Received: 18 February 2003, Accepted: 2 September 2003, Published online: 12 December 2003PACS: 05.60.-k, 51.60. + aCorrespondence to: G.M. Kremer     

Modeling of growth and remodeling in soft biological tissues with multiple constituents

Among the various important characteristics of biological tissues is their ability to grow and remodel. It is well-known that one of the primary triggers behind the growth and remodeling process is changes in the mechanical environment, for instance changes in stress, strain, etc. These mechanisms of mechanotransduction are the driving force behind many changes in structure and function including growth and remodeling. The purpose of this article is to formulate better constitutive equations for the stress in tissues with multiple constituents undergoing growth and remodeling. This is a very complex problem and is of tremendous importance. Here, we do the modeling from a mechanics point of view, utilizing the theory of natural configurations coupled with population dynamics to accurately model the production and removal of the different constituents that comprise the tissue. This is accomplished by deriving a generalized McKendrick equation for growth and remodeling and has the advantage of directly including the age distribution of constituents into the model. The population distribution function is then used to determine the stress in the tissue.     

机电耦联系统失稳振荡的动态分岔研究

应用机电分析动力学的理论建立了交流电机组的机电耦联振动方程组。运用微分动力学系统理论深入分析了交流电机的机电耦联失稳振荡问题。对于该系统出现的高余维分岔问题,通过中心流形定理、多参数稳定性理论和归一化方法得到了原系统的Normal Form形式,并详细讨论了系统的分岔情况以及分岔解的稳定性,并进行了详细的数值计算分析,很好地验证了理论分析结果。取得了交流电机失稳振荡更深入一步的研究成果。     

Fatigue crack growth of cable steel wires in a suspension bridge: Multiscaling and mesoscopic fracture mechanics

Intrinsically, fatigue failure problem is a typical multiscale problem because a fatigue failure process deals with the fatigue crack growth from microscale to macroscale that passes two different scales. Both the microscopic and macroscopic effects in geometry and material property would affect the fatigue behaviors of structural components. Classical continuum mechanics has inability to treat such a multiscale problem since it excludes the scale effect from the beginning by introducing the continuity and homogeneity assumptions which blot out the discontinuity and inhomogeneity of materials at the microscopic scale. The main obstacle here is the link between the microscopic and macroscopic scale. It has to divide a continuous fatigue process into two parts which are analyzed respectively by different approaches. The first is so called as the fatigue crack initiation period and the second as the fatigue crack propagation period. Now the problem can be solved by application of the mesoscopic fracture mechanics theories developed in the recent years which focus on the link between different scales such as nano-, micro- and macro-scale.On the physical background of the problem, a restraining stress zone that can describe the material damaging process from micro to macro is then introduced and a macro/micro dual scale edge crack model is thus established. The expression of the macro/micro dual scale strain energy density factor is obtained which serves as a governing quantity for the fatigue crack growth. A multiscaling formulation for the fatigue crack growth is systematically developed. This is a main contribution to the fundamental theories for fatigue problem in this work. There prevail three basic parameters μ^∗, σ^∗ and d^∗ in the proposed approach. They can take both the microscopic and macroscopic factors in geometry and material property into account. Note that μ^∗, σ^∗ and d^∗ stand respectively for the ratio of microscopic to macroscopic shear modulus, the ratio of restraining stress to applied stress and the ratio of microvoid size ahead of crack tip to the characteristic length of material microstructure.To illustrate the proposed multiscale approach, Hangzhou Jiangdong Bridge is selected to perform the numerical computations. The bridge locates at Hangzhou, the capital of Zhejiang Province of China. It is a self-anchored suspension bridge on the Qiantang River. The cables are made of 109 parallel steel wires in the diameter of 7 mm. Cable forces are calculated by finite element method in the service period with and without traffic load. Two parameters α and β are introduced to account for the additional tightening and loosening effects of cables in two different ways. The fatigue crack growth rate coefficient C₀ is determined from the fatigue experimental result. It can be concluded from numerical results that the size of initial microscopic defects is a dominant factor for the fatigue life of steel wires. In general, the tightening effect of cables would decrease the fatigue life while the loosening effect would impede the fatigue crack growth. However, the result can be reversed in some particular conditions. Moreover, the different evolution modes of three basic parameters μ^∗, σ^∗ and d^∗ actually have the different influences on the fatigue crack growth behavior of steel wires. Finally the methodology developed in this work can apply to all cracking-induced failure problems of polycrystal materials, not only fatigue, but also creep rupture and cracking under both static and dynamic load and so on.     

Macroscopic modelling of transport phenomena in porous media. 2: Applications to mass,momentum and energy transport

In this second paper, the averaging rules presented in Part 1 are employed in order to develop a general macroscopic balance equation and particular equations for mass, mass of a component, momentum and energy, all of a phase in a porous medium domain. These balance equations involve averaged fluxes. Then macroscopic equations are developed for advective, dispersive and diffusive fluxes, all in terms of averaged state variables of the system. These are combined with the macroscopic balance equations to yield field equations that serve as the core of the mathematical models that describe the transport of extensive quantities in a porous medium domain. It is shown that the methodology of averaging leads to a better understanding of the effective stress concept employed in dealing with transport phenomena in deformable porous media.     

Incremental boundary value problems in the presence of coupling of elastic and plastic deformations: A rock mechanics oriented theory

In some elastoplastic systems (typically rocklike media) the elastic properties are affected by plastic yielding. This “coupling” phenomenon is dealt with in this paper in the context of the incremental theory of plasticity with nonassociated (lacking normality) flow laws. Some extremum properties and uniqueness conditions are given for solution of the rate problem. Stability conditions are established. The results achieved are equally applicable to media with nonassociative flow rules without coupling, with or without worksoftening, and are believed to be novel in this more traditional area. The results are cast into algebraic formulations for finite element models, in view of their practical applications.     

On the mechanics of mother-of-pearl: A key feature in the material hierarchical structure

Mother-of-pearl, also known as nacre, is the iridescent material which forms the inner layer of seashells from gastropods and bivalves. It is mostly made of microscopic ceramic tablets densely packed and bonded together by a thin layer of biopolymer. The hierarchical microstructure of this biological material is the result of millions of years of evolution, and it is so well organized that its strength and toughness are far superior to the ceramic it is made of. In this work the structure of nacre is described over several length scales. The tablets were found to have wavy surfaces, which were observed and quantified using various experimental techniques. Tensile and shear tests performed on small samples revealed that nacre can withstand relatively large inelastic strains and exhibits strain hardening. In this article we argue that the inelastic mechanism responsible for this behavior is sliding of the tablets on one another accompanied by transverse expansion in the direction perpendicular to the tablet planes. Three dimensional representative volume elements, based on the identified nacre microstructure and incorporating cohesive elements with a constitutive response consistent with the interface material and nanoscale features were numerically analyzed. The simulations revealed that even in the absence of nanoscale hardening mechanism at the interfaces, the microscale waviness of the tablets could generate strain hardening, thereby spreading the inelastic deformation and suppressing damage localization leading to material instability. The formation of large regions of inelastic deformations around cracks and defects in nacre are believed to be an important contribution to its toughness. In addition, it was shown that the tablet junctions (vertical junctions between tablets) strengthen the microstructure but do not contribute to the overall material hardening. Statistical variations within the microstructure were found to be beneficial to hardening and to the overall mechanical stability of nacre. These results provide new insights into the microstructural features that make nacre tough and damage tolerant. Based on these findings, some design guidelines for composites mimicking nacre are proposed.     

Arnoldi and Crank–Nicolson methods for integration in time of the transport equation

A comparison is made between the Arnoldi reduction method and the Crank–Nicolson method for the integration in time of the advection–diffusion equation. This equation is first discretized in space by the classic finite element (FE) approach, leading to an unsymmetric first‐order differential system, which is then solved by the aforementioned methods. Arnoldi reduces the native FE equations to a much smaller set to be efficiently integrated in the Arnoldi vector space by the Crank–Nicolson scheme, with the solution recovered back by a standard Rayleigh–Ritz procedure. Crank–Nicolson implements a time marching scheme directly on the original first‐order differential system. The computational performance of both methods is investigated in two‐ and three‐dimensional sample problems with a size up 30 000. The results show that in advection‐dominated problems less then 100 Arnoldi vectors generally suffice to give results with a 10⁻³–10⁻⁴ difference relative to the direct Crank–Nicolson solution. However, while the CPU time with the Crank–Nicolson starts from zero and increases linearly with the number of time steps used in the simulation, the Arnoldi requires a large initial cost to generate the Arnoldi vectors with subsequently much less expensive dynamics for the time integration. The break‐even point is problem‐dependent at a number of time steps which may be for some problems up to one order of magnitude larger than the number of Arnoldi vectors. A serious limitation of Arnoldi is the requirement of linearity and time independence of the flow field. It is concluded that Arnoldi can be cheaper than Crank–Nicolson in very few instances, i.e. when the solution is needed for a large number of time values, say several hundreds or even 1000, depending on the problem. Copyright © 2001 John Wiley & Sons, Ltd.     

Modeling fluid and heat transport in the reactive, porous bed of downdraft (biomass) gasifier

A fluid flow and heat transfer model has been developed for the reactive, porous bed of the biomass gasifier to simulate pressure drop, temperature profile in the bed and flow rates. The conservation equations, momentum equation and energy equation are used to describe fluid and heat transport in porous gasifier bed. The model accounted for drag at wall, and the effect of radial as well as axial variation in bed porosity to predict pressure drop in bed. Heat transfer has been modeled using effective thermal conductivity approach. Model predictions are validated against the experiments, while effective thermal conductivity values are tested qualitatively using models available in literature. Parametric analysis has been carried out to investigate the effect of various parameters on bed temperature profile and pressure drop through the gasifier. The temperature profile is found to be very sensitive to gas flow rate, and heat generation in oxidation zone, while high bed temperature, gas flow rate and the reduction in feedstock particle size are found to cause a marked increase in pressure drop through the gasifier. The temperatures of the down stream zones are more sensitive to any change in heat generation in the bed as compared to upstream zone. Author recommends that the size of preheating zone may be extended up to pyrolysis zone in order to enhance preheating of input air, while thermal insulation should not be less than 15 cm.     

A review on the application of modified continuum models in modeling and simulation of nanostructures

Analysis of the mechanical behavior of nanostructures has been very challenging. Surface energy and nonlocal elasticity of materials have been incorporated into the traditional continuum analysis to create modified continuum mechanics models. This paper reviews recent advancements in the applications of such modified continuum models in nanostructures such as nanotubes, nanowires, nanobeams,graphenes, and nanoplates. A variety of models for these nanostructures under static and dynamic loadings are mentioned and reviewed. Applications of surface energy and nonlocal elasticity in analysis of piezoelectric nanomaterials are also mentioned. This paper provides a comprehensive introduction of the development of this area and inspires further applications of modified continuum models in modeling nanomaterials and nanostructures.     

Simulations of chemical transport and reaction in a suspension of cells I: an augmented forcing point method for the stationary case

A novel augmented forcing point method is presented to solve the problem of chemical transport in the fluid outside of a collection of suspended cells coupled with chemical reactions on the surfaces of the cells. In this method, the chemical concentrations and the forcing function values are determined simultaneously from an augmented system of equations. The method is more stable and accurate than predictor‐corrector‐type forcing point methods, yields the same solution as a corresponding ghost cell method with much less computational cost, and provides solutions that are pointwise second‐order accurate in space and time, even for closely‐spaced cells. Copyright © 2011 John Wiley & Sons, Ltd.