Volume Element Of Material Research Articles

A rolling contact fatigue life prediction model for bearing steel considering its gradient structure due to cyclic hardening

This paper presents a rolling contact fatigue life prediction model for bearing steel. In the initial stage of rolling contact, the gradient structure appears in the subsurface region of bearing steel and the hardness shows a gradient distribution along the contact depth direction due to the roller compaction effect. This indicates that the fatigue resistance of bearing steel varies in the subsurface region. Besides, each subsurface material volume element is subject to different stress cycles. Based on Weibull theory, the survival possibility of subsurface volume elements is formulated. Then the rolling contact fatigue life is evaluated considering the stress state and anti-fatigue performance of each elementary volume of the subsurface material. According to the phenomenon and assumption, the model proposed for gradient material was applied in the rolling contact fatigue life prediction of the bearing steel GCr15 and validated with the fatigue experiment data in the open literature. Furthermore, the accuracy of the proposed model results was compared with the traditional empirical rolling contact fatigue life prediction models.

Simulated annealing framework for generating representative volume elements of materials with complex ligamentous microstructures

At the microscale, various materials from biological tissues to nanoporous metals are formed by networks of ligaments. Here we propose a highly efficient simulated annealing (SA) framework for generating synthetic representative volume elements (RVE) of such materials. It can produce RVE where the microstructural characteristics both on the network level (e.g., node valency and ligament length) and on the level of individual ligaments (e.g., curvature) can be predefined by the user via probability distributions. As an application example of our framework, we generate a large variety of RVEs, analyze their mechanical properties by the finite element method, and establish through this approach links between microstructural descriptors and macromechanical properties of materials with ligamentous microstructures.

Multiscale modeling of ferroelectrics with stochastic grain size distribution

Macroscopic properties of ferroelectrics are controlled by processes on the microscale, in particular the switching of crystal unit cells and the movement of domain walls, respectively. Besides these microscopic levels, the grains of a polycrystalline material constitute the mesoscopic scale. Interactions of grains with statistically distributed orientations, as a consequence of mechanical and electrostatic mismatch, give rise to for example, residual stress which in turn affects domain switching. A multiscale modeling thus has to incorporate at least three interacting scales. In this context, the condensed method has recently been elaborated as an efficient tool with low computational cost and effort of implementation. It is extended toward statistical distributions of grain sizes in a representative material volume element and amended with regard to the modeling of domain evolution. Each of the few parameters of the constitutive approach has a unique physical meaning and is adapted to available experimental values of macroscopic quantities of barium titanate taken from various sources.

A numerical investigation of the statistical size effect in non-crimp fabric laminates under homogeneous compressive loads

The compressive strength of fiber reinforced composites is typically limited by a shear localization phenomenon known as microbuckling and is very sensitive to local imperfections of fiber alignment. Local misalignments act as randomly distributed flaws and introduce a dependence of the compressive strength on the size of material volume element under consideration. For homogeneously loaded material elements, weakest-link theory in combination with a Weibull power law is a frequently employed statistical model for microbuckling strength. This implies a dependence of strength on the size of volume under consideration. The present contribution investigates the strength–size relation for a non-crimp fabric via a numerical approach. Characteristics of the misalignment flaws used in simulations are derived from a comprehensive data set collected via large-scale measurements of roving dislocations on dry fiber material. Predictions resulting from the weakest-link Weibull theory are compared against strength–size statistics gathered by numerical analysis. In this manner, the size effects in single plies and laminates are quantified. The main findings are that weakest-link Weibull theory is well suited to predict size related strength loss in individual plies. However, it is also found that when plies are bonded to form laminates, misalignments in individual plies are mitigated in a way that is inconsistent with the weakest-link assumption. In all situations considered here, the strength loss expected from weakest-link Weibull theory was outweighed by a strength increase due to the mitigation effect when the volume was increased by adding extra layers to a laminate.

Modeling of deformation and fracture of porous media taking into account their morphological composition

This paper investigates the mechanical behavior and fracture of porous materials with an aluminum matrix. The purpose of the work was to create numerical models of failure of representative volume elements of such materials and to reveal the dependences of the nature of the failure processes on their structural morphology. Representative volume elements of these materials are random non-uniform structures of closed-cell and open-cell types. To create three-dimensional geometric models of the closed-cell structures, methods of sequential synthesis the possibility of their mutual intersection were used. For creation of models of interpenetrating structures of the open-cell type, methods based on the analytical determination of surfaces separating the two phases are used. In this paper, three approaches to fracture mechanics of representative volume elements of porous materials were studied and implemented. The first approach is an implementation of the elastic model and damage accumulation based on elastic properties degradation in accordance with the criterion of maximum stresses with reduction of the stiffness matrix coefficients in individual elements. The second approach is an implementation of the same model, but with removal of the failed elements. The third approach is based on the Johnson-Cook elastic plastic behavior and fracture model. Numerical modeling of the representative volumes was carried out with finite element analysis using each of the above approaches. The influence of the internal structure of the representative volumes of the porous materials on the processes of deformation and failure was studied on the example of several structures of open-cell and closed-cell types. The influence of stress concentrators on the distribution of stresses in representative volumes and character of their subsequent failure has been studied.

Design of thermal insulation based on Open-Cell carbon materials for spacecraft

High-temperature open-cell carbon materials with a porosity between 60 and 95% are the object of study. The aim of this work is to find ways to reduce the weight of spacecraft thermal insulation due to the rational choice of porosity level and characteristic pore size of high-temperature carbon materials. This aim is achieved by building geometric models of representative volume elements of open-cell carbon materials based on Voronoi diagrams, modeling radiation-conductive heat transfer in representative volume elements of materials, theoretical calculation of thermophysical properties, and performing thermal design of the solar probe heat shield using materials of the optimized structure.

A damage model predicting moderate temperature and size effects on concrete in compression

Experimental isotherm compressive tests show that concrete behaviour is dependent on temperature. The aim of such tests is to reproduce how concrete will behave under environmental changes within a moderate range of temperature. In this paper, a novel constitutive elastic damage behaviour law is proposed based on a free energy with an apparent damage depending on temperature. The proposed constitutive behaviour leads to classical theory of thermo-elasticity at small strains. Fixed elastic mechanical characteristics and fixed evolution law of damage independent of temperature and the material volume element size are considered. This approach is applied to compressive tests. The model predicts compressive strength and secant modulus of elasticity decrease as temperature increases. A power scaling law is assumed for specific entropy as function of the specimen size which leads to a volume size effect on the stress-strain compressive behaviour. The proposed model reproduces theoretical and experimental results from literature for tempertaures ranging between 20oC and 70oC. The effect of the difference in the coefficient of thermal expansion between the mortar and coarse aggregates is also considered which gives a better agreement with FIB recommendations. It is shown that this effect is of a second order in the considered moderate range of temperature.

Porous Metal Bonds Increase the Resource Efficiency for Profile Grinding

Profile grinding is irreplaceable for the machining of various brittle and hard workpieces, e.g. cutting tools for milling and drilling, seal components made of ceramics and bearing components. Grinding is rather inefficient regarding the energy demand for the machining of one volume element of material compared to other manufacturing processes. However, the process forces can be reduced without influencing the tool wear by using grinding wheels with a porous metal bond and grains that tend to splinter. This allows higher material removal rates without increasing the process forces, ultimately reducing the energy consumption per workpiece manufactured. Additionally thermal and mechanical loads on the workpiece are reduced leading to increased life cycles of grinded products. The application of these grinding wheels is currently on hold for profile grinding since the dressing process is not in control. Therefore, this paper investigates the dressing operation for grinding wheels with a porous metal bond in order to reduce the energy consumption in profile grinding of brittle and hard materials.

Micro–macro mechanical relations in Palmetto wood by numerical homogenisation

In this paper we investigate micro–macro mechanical relations in Palmetto wood, a natural composite, by means of a computational homogenisation-based multi-scale modeling approach. The homogenisation procedure is solved sequentially at multiple length scales by following a bottom-up approach. The proposed strategy consists of homogenising the three length-scales by means of representative volume elements of material. The resulting effective macroscopic properties allow us to compute the mechanical response of individual macro-fibres and bulk wood specimens by means of a standard finite element procedure. The numerical predictions are compared successfully with experimental measurements. Following the validation, a parametric study is performed to investigate the micro–macro relations in Palmetto wood. The influence of several micro-structural features present at different length scales are investigated, such as the cellulose content and its crystallinity, the microfibril angle, the cell-wall thickness of micro-fibres, and the porosity of macro-fibres and their surrounding matrix, among others. The effects of such microscopic parameters are investigated on the Young’s modulus and density of Palmetto bulk wood. These results show that the present strategy can pave the way for further studies on complex microstructure–property relationships existing in different wood species, and on the possible development of more advanced bio-inspired composite materials.

A computational multi-scale approach for the stochastic mechanical response of foam-filled honeycomb cores

This paper investigates the uncertainty in the mechanical response of foam-filled honeycomb cores by means of a computational multi-scale approach. A finite element procedure is adopted within a purely kinematical multi-scale constitutive modelling framework to determine the response of a periodic arrangement of aluminium honeycomb core filled with PVC foam. By considering uncertainty in the geometric properties of the microstructure, a significant computational cost is added to the solution of a large set of microscopic equilibrium problems. In order to tackle this high cost, we combine two strategies. Firstly, we make use of symmetry conditions present in a representative volume element of material. Secondly, we build a statistical approximation to the output of the computer model, known as a Gaussian process emulator. Following this double approach, we are able to reduce the cost of performing uncertainty analysis of the mechanical response. In particular, we are able to estimate the 5th, 50th, and 95th percentile of the mechanical response without resorting to more computationally expensive methods such as Monte Carlo simulation. We validate our results by applying a statistical adequacy test to the emulator.

Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: I. Strength, static crack growth, lifetime and scaling

Engineering structures must be designed for an extremely low failure probability such as 10 −6, which is beyond the means of direct verification by histogram testing. This is not a problem for brittle or ductile materials because the type of probability distribution of structural strength is fixed and known, making it possible to predict the tail probabilities from the mean and variance. It is a problem, though, for quasibrittle materials for which the type of strength distribution transitions from Gaussian to Weibullian as the structure size increases. These are heterogeneous materials with brittle constituents, characterized by material inhomogeneities that are not negligible compared to the structure size. Examples include concrete, fiber composites, coarse-grained or toughened ceramics, rocks, sea ice, rigid foams and bone, as well as many materials used in nano- and microscale devices. This study presents a unified theory of strength and lifetime for such materials, based on activation energy controlled random jumps of the nano-crack front, and on the nano-macro multiscale transition of tail probabilities. Part I of this study deals with the case of monotonic and sustained (or creep) loading, and Part II with fatigue (or cyclic) loading. On the scale of the representative volume element of material, the probability distribution of strength has a Gaussian core onto which a remote Weibull tail is grafted at failure probability of the order of 10 −3. With increasing structure size, the Weibull tail penetrates into the Gaussian core. The probability distribution of static (creep) lifetime is related to the strength distribution by the power law for the static crack growth rate, for which a physical justification is given. The present theory yields a simple relation between the exponent of this law and the Weibull moduli for strength and lifetime. The benefit is that the lifetime distribution can be predicted from short-time tests of the mean size effect on strength and tests of the power law for the crack growth rate. The theory is shown to match closely numerous test data on strength and static lifetime of ceramics and concrete, and explains why their histograms deviate systematically from the straight line in Weibull scale. Although the present unified theory is built on several previous advances, new contributions are here made to address: (i) a crack in a disordered nano-structure (such as that of hydrated Portland cement), (ii) tail probability of a fiber bundle (or parallel coupling) model with softening elements, (iii) convergence of this model to the Gaussian distribution, (iv) the stress-life curve under constant load, and (v) a detailed random walk analysis of crack front jumps in an atomic lattice. The nonlocal behavior is captured in the present theory through the finiteness of the number of links in the weakest-link model, which explains why the mean size effect coincides with that of the previously formulated nonlocal Weibull theory. Brittle structures correspond to the large-size limit of the present theory. An important practical conclusion is that the safety factors for strength and tolerable minimum lifetime for large quasibrittle structures (e.g., concrete structures and composite airframes or ship hulls, as well as various micro-devices) should be calculated as a function of structure size and geometry.

How to Compute the Atomic Stress Objectively?

Atomistic simulation has been a powerful study tool in mechanics research, but how to objectively compute the atomic stress equivalent to Cauchy stress is still controversial, especially on the velocity-related part in the virial stress definition. In this paper, by strictly following the classical definition of the Cauchy stress for continuum medium, the fundamental Lagrangian atomic stress is proposed and can be used to obtain the correct Cauchy stress under any circumstances. Furthermore, the Lagrangian virial stress is proposed, which is still in virial form but does not include velocities to avoid controversial velocity treatments. It is also found that the widely used classical virial stress is actually the Eulerian virial stress, which includes the velocities of atoms, and is valid only when the impulse-momentum theorem is applicable to estimate the internal forces. However this requirement for the Eulerian atomic stress can not always be met in practical cases, such as the material volume element in rotation and the examples presented in this paper, but the proposed Lagrangian atomic stress can avoid these velocity-related nonobjectivities.

Strength distribution of dental restorative ceramics: Finite weakest link model with zero threshold

Ensuring a small enough failure probability is important for the design and selection of restorative dental ceramics. For this purpose, the two-parameter Weibull distribution, which is based on the weakest link model with infinitely many links, is usually adopted to model the strength distribution of dental ceramics. This distribution has been thoroughly validated for perfectly brittle materials. However, dental ceramics are generally quasibrittle because the inhomogeneity size is not negligible compared to the size of the ceramic part. For such materials, the experimental histograms of many quasibrittle materials have been shown to exhibit strong deviations from the two-parameter Weibull distribution. As a remedy, the three-parameter Weibull distribution, which has a nonzero threshold, has been proposed. However, the improvement of the fits of histograms of quasibrittle materials has been only partial. Instead of making the threshold non-zero, the correct remedy is to consider the weakest link model to have a finite number of links, each of them representing one finite-size representative volume element of material. This model has recently been justified on the basis of the probability of random jumps of atomic lattice cracks over the activation energy barriers on the free energy potential of the lattice. It is shown that, in similarity to other quasibrittle materials, this new model allows excellent fits of the experimental strength histograms of various types of dental ceramics.

A Nanoindentation Study of Photo-Induced Changes in Polymers Containing Azobenzene

Irradiation of materials by light of suitable photon energy can cause structural alterations of either permanent or temporary character that are reflected in their mechanical properties. Nanoindentation is particularly well suited to characterise small representative volume elements of materials with gradual or discrete changes of their mechanical properties. It is, therefore, an ideal technique for investigation of thin films, nanocomposites and other specimens with inhomogeneities in the nanometer to micrometer length scale that are either inherently existing in the material or externally stimulated. In this article, nanoindentation was applied to study the mechanical properties of pDR1 m surface relief gratings containing azobenzene. Hardness and elastic modulus increase after and during irradiation of the pDR1 m film correlated with an increase in the polymer density after light exposure. Permanent structural changes in the pDR1 m film are related to crosslinking and photo-isomerisation. In-situ nanoindentation under controlled periodic light illumination exhibits a week photomechanical switching and results finally in a permanent stiffer polymer network.

Numerical modelling of damage susceptibility of an inhomogeneous representative material volume element of polymer composites

A microstructure sample size capturing the inhomogeneity of the reinforcing phase of a composite material was described in a preceding paper. Here using a finite element method, the transverse damage behaviour of a number of microstructural samples has been investigated. The samples ranged in size from 300 μm × 300 μm to 180 μm × 180 μm, and the finite element model was based on fibre centroid and fibre diameter data from a cross-ply laminate. Comparing normalized stiffness curves for the samples suggested that a sample size of 300 μm × 300 μm, containing approximately 400 fibres, is necessary to capture the effects of microstructural inhomogeneity on the damage susceptibility of the analysed microstructure. Selecting this size as suitable for a representative volume element (RVE) corresponds to a 0.10 significance level in the Kolmogorov goodness- of-fit statistical test described in an earlier paper.

Mechanics-based statistics of failure risk of quasibrittle structures and size effect on safety factors.

In mechanical design as well as protection from various natural hazards, one must ensure an extremely low failure probability such as 10(-6). How to achieve that goal is adequately understood only for the limiting cases of brittle or ductile structures. Here we present a theory to do that for the transitional class of quasibrittle structures, having brittle constituents and characterized by nonnegligible size of material inhomogeneities. We show that the probability distribution of strength of the representative volume element of material is governed by the Maxwell-Boltzmann distribution of atomic energies and the stress dependence of activation energy barriers; that it is statistically modeled by a hierarchy of series and parallel couplings; and that it consists of a broad Gaussian core having a grafted far-left power-law tail with zero threshold and amplitude depending on temperature and load duration. With increasing structure size, the Gaussian core shrinks and Weibull tail expands according to the weakest-link model for a finite chain of representative volume elements. The model captures experimentally observed deviations of the strength distribution from Weibull distribution and of the mean strength scaling law from a power law. These deviations can be exploited for verification and calibration. The proposed theory will increase the safety of concrete structures, composite parts of aircraft or ships, microelectronic components, microelectromechanical systems, prosthetic devices, etc. It also will improve protection against hazards such as landslides, avalanches, ice breaks, and rock or soil failures.

A Three-Dimensional Unit Cell Model With Application Toward Particulate Composites

A formulation of a fully three-dimensional unit cell model is presented for uniform general deformation at a point in a composite material. The unit cell model is constructed as a finite element discretization of the unit cube. General displacement periodicity boundary conditions are prescribed such that the cell may be considered as a representative volume element of material. As a particular application of the model, the problem of determining the least anisotropic periodic model of a particulate composite is considered, and comparisons are made with bounds for elastic two-phase composites possessing cubic symmetry.

A proposal for the onset of chip segmentation in machining

Abstract The results of machining tests on certain steels are described and the proposal is made that a particular type of serrated chip is the result of inhomogeneous plastic deformation which leads to ductile fracture. Chip segmentation is deemed to occur when the steel accumulates a certain amount of damage in the machining operation. The steels can be classified by a critical accumulated damage (CAD) factor, which can be calculated on the basis of upsetting tests and a ductile fracture model. The lower the CAD value, the easier it is for the steel to exhibit chip segmentation. In order to evaluate the damage incurred by volume elements of material in the machining process it is necessary to have a knowledge of the stress and strain within the deformation zone. To this end a slip line field solution is proposed which allows, in an approximate manner, for the influence of strain, strain rate and temperature in the machining process. The slip line field is used to demonstrate that damage can accumulate and that it can accumulate at a faster rate as the cutting speed is increased—hence the transition from continuous to segmented chips as the cutting velocity is increased.

Damage Coupled With Heat Conduction in Uniaxially Reinforced Composites

This paper presents a continuum damage model for a unidirectionally reinforced composite based upon fundamental concepts of continuum mechanics and irreversible thermodynamics. Damage is incorporated by two symmetric, second-rank, tensor-valued, internal state variables which represent the total areas of “active” and “passive” cracks contained within a representative material volume element. Constitutive relations are derived for both the mechanical response and heat flux in the presence of damage. It is shown that damage growth contributes to dissipation in the coupled heat conduction process. A specific fracture mechanics solution is employed to relate “microlevel” crack growth processes to “macrolevel” damage growth expressions. This approach lends itself to a probabilistic formulation of the continuum damage model.

Failure Analysis of Cast Irons Under General Three-Dimensional Stress States

Observations of microscale behavior of cast irons indicate that considerable plastic flow occurs between microflaws prior to rupture. Therefore, the plastic limit load for small representative volume elements of material treated as structures should be a close approximation to the load at rupture. A representative volume element of cast iron is analyzed using limit analysis techniques to obtain bounds on the limit, or failure, surface under three dimensional average stress. Comparisons with other cast iron failure criteria and experimental data are made.