Characteristic Length Of Micro-structure Research Articles

Mechanics and mechanisms of fatigue in a WC–Ni hardmetal and a comparative study with respect to WC–Co hardmetals

There is a major interest in replacing cobalt binder in hardmetals (cemented carbides) aiming for materials with similar or even improved properties at a lower price. Nickel is one of the materials most commonly used as a binder alternative to cobalt in these metal-ceramic composites. However, knowledge on mechanical properties and particularly on fatigue behavior of Ni-base cemented carbides is relatively scarce. In this study, the fatigue mechanics and mechanisms of a fine grained WC–Ni grade is assessed. In doing so, fatigue crack growth (FCG) behavior and fatigue limit are determined, and the attained results are compared to corresponding fracture toughness and flexural strength. An analysis of the results within a fatigue mechanics framework permits to validate FCG threshold as the effective fracture toughness under cyclic loading. Experimentally determined data are then used to analyze the fatigue susceptibility of the studied material. It is found that the fatigue sensitivity of the WC–Ni hardmetal investigated is close to that previously reported for Co-base cemented carbides with alike binder mean free path. Additionally, fracture modes under stable and unstable crack growth conditions are inspected. It is evidenced that stable crack growth under cyclic loading within the nickel binder exhibit faceted, crystallographic features. This microscopic failure mode is rationalized on the basis of the comparable sizes of the cyclic plastic zone ahead of the crack tip and the characteristic microstructure length scale where fatigue degradation phenomena take place in hardmetals, i.e. the binder mean free path.

Microstructural length scales in directionally solidified Sn-40 at.%Mn peritectic alloy containing intermetallic compounds

Sn-40 at.%Mn peritectic alloys in which both primary and peritectic phase are intermetallic compounds were directionally solidified at different growth rates (1 μm/s∼100 μm/s). Dendritic growth of primary phase has been observed even at low growth rate, and dependence of microstructural characteristic length scales on growth rate has been investigated. Liquidus slope is the dominating factor in determining non-faceted/faceted solid/liquid morphology. Experimental results show that: λ1 = 229.64v−0.20, λ2 = 33.337v−0.34, λ3 = 39.90v−0.378, R = 16.90v−0.384. Besides, both λ1/λ2 and λ1/λ3 vary greatly with increasing growth rates while λ2/R ranges from 2 to 2.3 with increasing growth rate. Both the peritectic reaction and solute distribution of intermetallic compounds during solidification influence these length scales. A modified solute distribution coefficient which is appropriate for intermetallic compounds makes calculation agree reasonably well with experimental results. The value of λ1 is influenced more by solute distribution of intermetallic compounds, while growth of λ2, λ3 and R are more significantly influenced by the peritectic reaction.

Intermittent plastic flow of single crystals: central problems in plasticity: a review

The relationships between microstructure and plastic flow properties of copper single crystals deformed at low temperatures are reviewed based on the analysis of experimental results of mechanical, structural and electrical resistivity characterisation. Electrical resistivity measurements suggest that the flow stress correlates with the average density of defects accumulated in the microstructure. The characteristic microstructural length scale that determines the flow stress corresponds closely to the size of dislocation free channels and/or cells produced during plastic flow. The correspondence between micro- and macroscopic approaches to the plastic flow has been analysed based on the behaviour of dislocation mean free path and mean slip distance and their relationship to the microstructure. Production and annihilation of nanoscale debris and point defects and their role in plastic flow and questions pertinent to the influence of microstructural instabilities on the flow stress, workhardening, storage and annihilation of lattice defects have been discussed.

A dislocation-based, strain–gradient–plasticity strengthening model for deformation processed metal–metal composites

Deformation processed metal–metal composites (DMMCs) are high-strength, high-electrical conductivity composites developed by severe plastic deformation of two ductile metal phases. The extraordinarily high strength of DMMCs is underestimated using the rule of mixture (or volumetric weighted average) of conventionally work-hardened metals. In this article, a dislocation-density-based, strain–gradient–plasticity model is proposed to relate the strain-gradient effect with the geometrically necessary dislocations emanating from the interface to better predict the strength of DMMCs. The model prediction was compared with the experimental findings of Cu–Nb, Cu–Ta, and Al–Ti DMMC systems to verify the applicability of the new model. The results show that this model predicts the strength of DMMCs better than the rule-of-mixture model. The strain-gradient effect, responsible for the exceptionally high strength of heavily cold worked DMMCs, is dominant at large deformation strain since its characteristic microstructure length is comparable with the intrinsic material length.

Modelling reinforced concrete beams under mixed shear-tension failure with different continuous FE approaches

The paper presents quasi-static numerical simulations of the behaviour of short reinforced concrete beams without shear reinforcement under mixed shear-tension failure using the FEM and four various constitutive continuum models for concrete. First, an isotropic elasto-plastic model with a Drucker-Prager criterion defined in compression and with a Rankine criterion defined in tension was used. Next, an anisotropic smeared crack and isotropic damage model were applied. Finally, an elasto-plastic-damage model was used. To ensure mesh-independent FE results, to describe strain localization in concrete and to capture a deterministic size effect, all models were enhanced in a softening regime by a characteristic length of micro-structure by means of a non-local theory. Bond-slip between concrete and reinforcement was considered. The numerical results were directly compared with the corresponding laboratory tests performed by Walraven and Lehwalter (1994). The advantages and disadvantages of enhanced models to model the reinforced concrete behaviour were outlined.

Characterization of microstructural length scales in directionally solidified Sn–36%Ni peritectic alloy

Sn–36%Ni peritectic alloys were directionally solidified at different growth rates under a constant temperature gradient (20 K/mm), the dependences of microstructural characteristic length scales on the growth rate were investigated. Experimental results are presented, including primary and higher order dendrite arm spacings λ1, λ2, λ3 and dendrite tip radius R of primary Ni3Sn2 phase. Comparisons between the theoretical predictions and the experimental results show that, for the primary dendrites, λ1=335.882v−0.21, which is in agreement with the Kurz–Fisher model; for the secondary dendrites, λ2=44.957v−0.277, which is consistent with the Bouchard–Kirkaldy model; for the tertiary dendrites, λ3=40.512v−0.274; for the dendrite tip radius, R=22.7v−0.36. The experimental results also show that the λ1/λ2 changes greatly with increasing growth rate while the λ1/λ3 has no significant change, indicating that tertiary dendrite arms have a more similar growth characteristics to primary dendrites compared with secondary dendrites. The λ1/R ranges from 2 to 2.3 with the increase of growth rate.

Computational modelling of concrete behaviour under static and dynamic conditions

Abstract The paper presents results of FE simulations of the concrete behaviour under quasi-static and dynamic loading. For quasi-static cyclic analyses, an enhanced coupled elasto-plastic-damage constitutive model has been used. To take the effect of the loading velocity into account, viscous and inertial terms have been also included. To ensure the mesh-independence and to properly reproduce strain localization in the entire range of strain rates, a constitutive formulation has been enhanced by a characteristic length of micro-structure by means of a non-local theory. Numerical results have been compared with some corresponding laboratory tests

A generalized uncertainty propagation criterion from benchmark studies of microstructured material systems

Recognizing that modern materials contain multiple phases with inherent random microstructure and in situ constituent material properties that are oft uncharacterizable with exactness, this research uses benchmark computational studies to unveil scenarios where uncertainties significantly affect macroscopic behavior. The benchmark studies, which serve as numerical experiments capturing the main features of a wide class of problems in solid mechanics, suggest a generalized uncertainty propagation criterion Δ whose assessment may be used to understand whether uncertainties may non-negligibly propagate to apparent system properties. The new Δ-criterion combines four features of a microstructured material system: the microstructure size (micro), material property correlation length (micro), structure size (macro), and global length scale of loading (macro). We compare the Δ-criterion with the familiar one from classical homogenization that relates a statistical volume element size to the size of the microstructure it contains. This work is motivated by the knowledge paucity of uncertainty’s place in multiscale analysis and the great computational expense, either in runtime or implementation, of uncertainty propagation methods. The uncertainties considered are material property uncertainty of the separate phases comprising each problem’s microstructure and the statistical description of the microstructure itself. The problems involve quasi-static elasto-plasticity, wave propagation in viscoelastic composite, and pattern transition in periodic voided elastomer. Besides the criterion, observed physical behavior for the benchmark problems is also presented in the context of input uncertainty. A close examination of the boundary condition effect on uncertainty propagation suggests that localization increases the characteristic microstructure length, which we represent with a simple deformation-dependent microstructure evolution function and that increases uncertainty in apparent system properties. This work provides a systematic framework for modelers to understand when they may reliably waive the consideration of uncertainty for their application or when their application mandates it.

Numerical Mesoscopic Analysis of Fracture in Fine-Grained Concrete

Abstract This paper presents numerical two-dimensional results for fine-grained concrete under quasi-static three-point bending at meso-scale. Concrete was modelled as a random heterogeneous three-phase material. The simulations for notched concrete beams were carried out with the standard finite element method using an isotropic damage constitutive model enhanced by a characteristic length of micro-structure by means of a non-local theory. The effect of the volume fraction, shape, size, statistical distribution and stiffness of aggregate was analysed. Moreover, the effect of the bond thickness, notch size and characteristic length of micro-structure on the material behaviour was numerically investigated. The FE results were compared with own laboratory test results and other meso-scale calculations for three-phase concrete elements.

Microviscosity, microdiffusivity, and normal stresses in colloidal dispersions

In active, nonlinear microrheology, a Brownian “probe” particle is driven through a complex fluid and its motion tracked in order to infer the mechanical properties of the embedding material. In the absence of external forcing, the probe and background particles form an equilibrium microstructure that fluctuates thermally. Probe motion through the medium distorts the microstructure; the character of this deformation, and hence its influence on probe motion, depends on the strength with which the probe is forced, Fext, compared to thermal forces, kT/b, defining a Péclet number, Pe=Fext/(kT/b), where kT is the thermal energy and b is the characteristic microstructural length scale. Recent studies showed that the mean probe speed can be interpreted as the effective material viscosity, whereas fluctuations in probe velocity give rise to an anisotropic force-induced diffusive spread of its trajectory. The viscosity and diffusivity can thus be obtained by two simple quantities—mean and mean-square displacement of the probe. The notion that diffusive flux is driven by stress gradients leads to the idea that the stress can be related directly to the microdiffusivity, and thus the anisotropy of the diffusion tensor reflects the presence of normal stress differences in nonlinear microrheology. In this study, a connection is made between diffusion and stress gradients, and a relation between the particle-phase stress and the diffusivity and viscosity is derived for a probe particle moving through a colloidal dispersion. This relation is shown to agree with two standard micromechanical definitions of the stress, suggesting that the normal stresses and normal stress differences can be measured in nonlinear microrheological experiments if both the mean and mean-square motion of the probe are monitored. Owing to the axisymmetry of the motion about a spherical probe, the second normal stress difference is zero, while the first normal stress difference is linear in Pe for Pe≫1 and vanishes as Pe4 for Pe≪1. The expression obtained for stress-induced migration can be viewed as a generalized nonequilibrium Stokes–Einstein relation. A final connection is made between the stress and an “effective temperature” of the medium, prompting the interpretation of the particle stress as the energy density, and the expression for osmotic pressure as a “nonequilibrium equation of state.”

Fictitious notch rounding concept applied to V-notches with end holes under mode 3 loading

The present technical note is aimed to provide a closed form expression for the microstructural support factor and for the fictitious notch radius in plates weakened by V-notches with root end-holes. Taking advantage of some recent closed form expressions for the stress distributions due to V-notches with end holes the fictitious notch rounding approach is applied here to mode 3 loading. The factor s for the V-notch with end holes is found to be strongly influenced by the opening angle and the new values are compared with the previous solution available in the literature and dealing with blunt V-notches. To validate the new expressions a comparison is carried out between the theoretical stress concentration factor (SCF) obtained from a rounded V-notch with a fictitiously enlarged end hole (of radius ρ f ) and the effective stress concentration factor obtained by integrating the relevant stress over the microstructural characteristic length (MCL), ρ *, in a pointed V-notch. A sound agreement is found from the comparison. The range of validity of the present equations are limited to linear elasticity or in those cases where the plastic zone is very small with respect to the MCL of the material.

FE analysis of reinforced concrete corbels with enhanced continuum models

The paper presents quasi-static FE-simulations of the behaviour of short reinforced concrete corbels with and without shear reinforcement. Concrete was modelled with the aid of three various constitutive continuum models: an isotropic elasto-plastic, an isotropic damage and an anisotropic smeared crack approach. To ensure the mesh-independent numerical results and to describe strain localization, a characteristic length of micro-structure was introduced by means of a non-local theory. To simulate the behaviour of reinforcement, an associated elasto-plastic constitutive model with isotropic hardening was assumed. The effects of a characteristic length, bond between concrete and reinforcement, tensile and compressive fracture energy and shear reinforcement on strain localization were carefully investigated. The numerical results were compared with corresponding laboratory tests.

Percolation of diffusionally evolved two-phase systems

Percolation thresholds and critical exponents for universal scaling laws are computed for microstructures that derive from phase-transformation processes in two dimensions. The computed percolation threshold for nucleation and growth processes, p(c)≈0.6612, is similar to those obtained by random placement of disks and greater than that of spinodal decomposition, p(c)≈0.4987. Three critical exponents for scaling behavior were computed and do not differ significantly from universal values. The time evolution of a characteristic microstructural length was also computed: For spinodal decomposition, this length grows according to a power law after a short incubation period; for nucleation and growth, there are several transitions in the nature of the growth law. We speculate that the transitions in nucleation and growth derive from competing effects of coalescence at short times and then subsequent coarsening. Short-range order is present, but different, for both classes of microstructural evolution.

Layered and Functionally Graded Nanocomposite Thin Films with Unique Mechanical Properties

AbstractA novel electrochemical deposition method for manufacturing functionally graded, oxide-dispersion strengthened metal matrix nanocomposites will be presented. Using a rotating disk electrode and depositing from an electrolyte containing a suspension of oxide nanoparticles, metal-ceramic nanocomposites have been produced. This method leads to precise control over the volume fraction of the oxide in the nanocomposite and allows for the manufacturing of compositionally uniform, periodically layered, or functionally graded structures. In the higher order structures the composition variation can be finely tuned with nanometer resolution, and the characteristic microstructural length scale (e.g., individual layer thickness) can range from microns up to millimeters. Using indentation methods, the nanocomposites are shown to display enhanced and tunable mechanical properties.

Crack growth behaviour of a nickel-based powder metallurgy superalloy under elevated temperature

Crack growth behaviour of a nickel-based powder metallurgy superalloy has been studied under elevated temperatures. Test results show that there is a turning point for RT, 550 °C and 650 °C crack growth curves. Above this point, the three curves merge together and influence of elevated temperature disappears. Material characteristic microstructure length (grain size) is considered to be responsible for this phenomenon. However, this turning point is suppressed at 750 °C possibly by grain boundary weakening. Dwell tests show that effect of 90 s holding time on the crack growth rate is limited at 650 °C but much significant at 750 °C due to creep damage.

A Simple Engineering Estimate of the Fatigue Notch Factor of Arbitrary Stress Concentrators

AbstractIn fatigue life estimates, the fatigue stress endured at the hot spot is usually higher than what is predicted by the elastic stress concentration factor. Recently, Neuber's and Novozhilov's approaches of introducing an effective stress by averaging the actual stress field over a characteristic microstructural length have experienced a reappraisal. The present contribution aims at complementing these recent proposals by a mathematically simple engineering assessment of arbitrarily shaped notches, resulting in a closed‐form expression for the fatigue notch factor. The limits of validity of the solution are discussed.

Fictitious Notch Rounding approach of pointed V-notch under in-plane shear

The Fictitious Notch Rounding approach (FNR) is applied here for the first time to V-shaped notches under in-plane shear loading. The fictitious radius is evaluated for different opening angles as a function of the Microstructural Characteristic Length (MCL), the actual radius and the failure hypothesis. A multiaxiality factor is introduced and found to be very sensitive to the opening angle. Under mode II loading, the problem is more complex than under mode I and mode III, mainly because the maximum elastic stress is outside the notch bisector line. The main problem is the choice of the expected crack initiation angle, which defines the direction where the relevant stress has to be integrated. This integration carried out over the MCL gives the effective stress value for the pointed V-notch. To this end, two different criteria are used, the Maximum Tangential Stress (MTS) criterion and the Minimum Strain Energy Density criterion (MSED). A large number of finite element analyses have been carried out to determine the multiaxiality factor. This factor has been evaluated by comparing the theoretical stress concentration factor (SCF) obtained from fictitiously rounded notches to the effective stress concentration factor obtained by integrating the relevant stress over the Microstructural Characteristic Length.

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 C 0 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.

Calculations of fracture process zones on meso-scale in notched concrete beams subjected to three-point bending

The paper describes investigations on fracture process zones (FPZ) at meso-scale in notched concrete beams subjected to quasi-static three-point bending. The simulations were carried out with the FEM using isotropic damage constitutive model enhanced by a characteristic length of micro-structure by means of a non-local theory. Concrete was modelled as a random heterogeneous three-phase material. The effect of the beam size, aggregate distribution, aggregate density, aggregate shape, aggregate size and characteristic length on the width and shape of FPZ and load-displacement curve was numerically investigated. The numerical results were compared with own test results using Digital Image Correlation method (Skarzynski et al., 2009a), the tests by Le Bellěgo et al., (2003) and the size effect law by Bažant (2004).

Elastic wave dispersion in microstructured membranes

The effect of microstructural properties on the wave dispersion in linear elastic membranes is addressed in this paper. A periodic spring-mass lattice at the lower level of observation is continualized and a gradient-enriched membrane model is obtained to account for the characteristic microstructural length scale of the material. In the first part of this study, analytical investigations show that the proposed model is able to correctly capture the physical phenomena of wave dispersion in microstructured membrane which is overlooked by classical continuum theories. In the second part, a finite-element discretization of microstructured membrane is formulated by introducing the pertinent inertia and stiffness terms. Importantly, the proposed modifications do not increase the size of the problem compared wiith classical elasticity. Numerical simulations confirm that the vibrational properties are affected by the microstructural characteristics of the material, particularly in the high-frequency regime.