Multiscale modelling of materials,MMM - Introduction and Explanatory Notes; Refs.,Images, on a Hot Interdisciplinary field

"There is much to be gained by creating theory appropriate for materials science, in particular that which is experimentally verifiable and which does not compromise with the complexity of technology." states Cambridge Uni's, Prof. Harry Bhadeshia in his paper, 'Mathematical models in materials science,' Materials Science and Technology 2008 VOL 24 NO 2;(1a and cf. related posts below) (freely available to members of the Institute of Materials,Minerals and Mining, IOM3.)

"Modelling as a subject has become as much a part of materials science as experimental characterisation for two reasons. First, to compete, industry must achieve solutions using minimal resources. Second, scientists are excited by the quantitative expression of multivariate problems." continues Harry Bhadeshia(1a).

During much of my own mature professional life, it was the first factor which motivated me, the second factor, more personal and profound, led me to decide on science as a career and is also the main motivation to write this blog post.

"Modelling has now become a routine part of materials science. It is appropriate therefore to assess some of the successes and failures of the method and to understand how and if modelling differs from ordinary quantitative science. The subject is now sufficiently mature to bear some constructive self-criticism and to temper exaggerated claims," says Bhadeshia(1a) who does just that in his paper. (1a cf. my related post below)

Here, I am much less ambitious, but hope my contribution will encourage readers to point out insufficiencies in my current understanding of these highly specialised and interdisciplinary fields, and with readers and specialists assistance, drive these "materials matter" to beneficial socio-economic benefit as wide as the subject matters themselves."

The information that triggered my post.

The R & D pages of the December 09 issue of Materials World (1), figured an interview and news report entitled Model behaviour of ceramic and intermetallic alloys, MW Dec.09 describing work and research aims in multi-scale physical modelling of advanced materials in particular intermetallic alloys and ceramic coatings which act as thermal barriers (TBC's-Fig 1) on single crystal turbine blades.

The work brings together an international team from academia and industry whose aims , according to team leader, Prof N. Fleck of Cam. Univ., are to bridge the knowledge gaps between the materials behaviour and current models and methodology from the atomistic level through micro-, meso- to the macroscopic level in such advanced materials systems.)

The scale and scope of these multiple challenges is illustrated by a 2D-length versus time, graphical representation of main pointers on the state of the art and the problems still to be solved. The graph, available only via the paper edition of the IOM3 members house journal Materials World (MW) but shown here as a photograph, is extremely dense to say the least.(Clicked-upon the graph Fig. 2, to magnify)

Lengths are shown on the Y-axis. They range from 10^-12 m or 1 picometre (1pm, a sub atomic size characteristic of electrons and radiations ) bottom left to 1 m top left. Time is shown on the X-axis. They range from 10^-14s bottom left (10 femtoseconds) to 10^8s (just over 1 year). Of course any graph of length vs time leads naturally to velocity. The series of parallel dotted lines represent different features occurring at the same velocity-iso-velocity lines. Different theoretical approaches and materials science features phenomena and technologies may be represented with reference to their size and iso-velocities.


The graph maps the team's areas of current capabilities (circular areas delimited by dotted lines, near the areas, delimited or hidden by the squares A,B,C,D). Proposed extensions are circular areas delimited by full lines. In a bottom-up approach (bottom left to top right) they correspond roughly with the following methods and models developed by the team; 1st principles ab-initio quantum and more especially molecular dynamics (MD), Single defect calibration, Extended discrete dislocation-dislocation dynamics(DD). Dislocation homogenisation (maths not heat-treatment), Enhanced non-local plasticity theories, strain gradients theories and macro finite element simulations. These models are responses to the many physical and chemical, often competitive phenomena at play in complex alloy and ceramic systems. Four materials phenomena and technologies have been singled out for proposed model extensions: A. Defect-Grain boundary interaction, B. Interfacial toughness and Hall-Petch (HP) effects [strength increase as grain size decreases and it's break down at small grain size, less than 30 nm.

To the material scientist and engineer they speak for themselves. Nevertheless let me try to elaborate. At the most fundamental level is shown the basic crystal unit the lattice comprised of regular arrangements of atoms, the white dots. At a higher level, dislocations, discontinuities from the crystal lattice arrangements move through the material (DD).The unit dislocation is a line defect, a misaligned row of atoms which both holds and carries a discrete quantity of deformation (represented in magnitude and direction by its vector, the Burger vector (b). Dislocation behaviour is central to material strength. Higher again at the subgrain level all sorts of phenomena are found; precipitates or inclusions and precipitated phases of differing chemical composition and/or crystallographic-lattice structure depending on the type of material (s) in question. Next are the polycrystal-grain structures. The difference in crystalline grain shading indicates differences in individual or single grain crystal orientation, delimited by their grain boundaries. Compared to dislocations, grain boundaries are major atomic misalignment (several atoms in width, 1 nm in nanostructures ) They arise from the differences in orientation between neighbouring crystal-grains. cf Fig 4, 5? Dislocations, interfaces, grains, forms orientations

Finally engineered products, here, a punched cup, to represent a continuum-"homogeneous continuous medium"-the engineered product (Figs. 2, 3) cf. also, next paragraph). More helpful highly visual presentations on the materials science and engineering issues just mentioned, from Prof. D. Raabe, Max Planck Inst.(3) in english. Above.









CONTINUUM -UPPER RIGHT Figs 2, 3. above.

The traditional analytical and computational framework in mechanics has been the continuum. Materials are assumed to be comprised of an infinitely indivisible continuous medium, imbued with a constitutive behaviour that remains unchanged regardless of how small the structure of interest may be. By careful fitting of the mathematical form of the constitutive laws to experimental observations, the behaviour of real materials is introduced into the continuum framework, and structures can then be analyzed through the solution of boundary value problems.

As already indicated in the title, most materials phenomena are manifestations of processes that are operative over a vast range of length and time scales. A complete understanding of the behaviour of materials thereby requires theoretical and computational tools that span the atomic-scale detail of first-principles methods and the more coarse-grained description provided by continuum equations. Continuum equations, typically in the form of deterministic or stochastic partial differential equations, are at the pinnacle of the coarse-graining hierarchy. The underlying atomic structure of matter is neglected altogether and is replaced with a continuous and differentiable mass density. Analogous replacements are made for other physical quantities such as energy and momentum. Differential equations are then formulated either from basic physical principles, such as the conservation of energy or momentum, or by invoking approximations within a particular regime. There are many benefits of a continuum representation of materials phenomena. Foremost among these is the ability to examine macroscopic regions in space over extended periods of time. This is facilitated by extensive libraries of numerical methods for integrating deterministic and stochastic differential equations. The best known of these is the finite element method (Vvedensky 4) in the family of Multi-grid methods).

"Fundamental materials problems at the nanoscale involve interactions between microstructure, for example grain boundaries, and defects, such as dislocations and cracks. ‘Conventional continuum descriptions fail to predict the dependence of strength upon microstructural size, and the associated evolution of microstructure with deformation,’ says Fleck.( 1)

[Indeed classical continuum modeling methods such as FEM, have no intrinsic length scale. Length scale effects must be introduced through the material models in the finite element calculation) cf. Meyers (9), Fleck et al. (14) for example.]


Gaps in the modelling space map therefore reflect gaps in materials understanding, such as the strengthening mechanisms of grain boundaries leading to the Hall-Petch and inverse Hall-Petch effects (methods of strengthening materials by changing their average grain size)" (1) This core metallurgical "anomaly" the Inverse Hall-Petch Effect (IHPE) intrigued me. Also, I saw it as a way to compliment my own metallurgical knowledge while venturing into the new field of "metallurgical nanomaterials." Some important features of my trip into IHPE, figure below. §Remarkable size effects.


ATOMS-UP.

Both Vvedensky (4), and Ghoniem, Busso, Kioussis, Hanchen Huang (5) provide overviews describing in some detail the different underlying methods used to improve classical continuum such as quasi-continuum methods (Tadmor Philips, Ortiz, (1992-96) ref.6, : ab-initio methods leading, larger scale, larger numbers of atoms in molecular dynamics modelling etc. Of several illustrations I have chosen on from D. Raabe (3) cf. Fig.6. The micrographic illustrations, also help further the notion of homogenisation, smoothing the steps on the path to deeper knowledge and understanding across the spectrum: atomic (quantum mechanical) and molecular levels through micro-and meso-structures to a quasi-continuum, full description of real materials in service, to their optimisation and to improved predicative capacities for new materials design.


HIERARCHY and HOMOGENISATION.
The absence of an intrinsic length scale in the finite element method is valuable computationally since it allows the simulations of crystalline systems having dimensions several orders of magnitude appart [104] with approximately the same amount of computational work (9).

A length scale is introduced into a finite element model of a polycrystal material by
1. Using a material model having a length scale.
2. Modifying the behavior of the grain boundary, thereby introducing a length scale as the ratio of the grain-boundary volume to its surface area.
3. Modeling the grain boundary with a finite thickness, which directly imposes a length scale. (9)

But to span such scales as in Fig 2, ref 1, it has become primordial, for the consortium team of 7 led by Fleck, to identify a set of macroscopic equations from the relevant internal variables which describe in a robust form the constitutive laws which govern these systems and their behaviour.

To achieve this, these techniques must be capable of discriminating between the complex, competitive materials phenomena and their phenomenological expression or theories. To mention only one of the many examples of competitive phenomena in materials under stress, consider an all too well known macroscopic failure, fracture: cracks can either propagation or not (blunting) both mechanism are found in materials. At atomic and molecular scales Ionic or covalent solids are more prone to crack propagation (brittle, crack before blunting)as are BCC metals-body centre cubic lattice) than others (ductile, blunt before cracking) such as FCC metals face centred cubic lattice) In steels things are more complicated they cannot be catalogued as either brittle or ductile, indeed steel demonstrates a transition from brittle to ductile as temperature increases.

‘The [almost self-imposed] main objective of the project is to develop a hierarchy of methods involving information transfer from one level to the next,’ says Fleck. ‘For example, thermal barrier coatings in gas turbines comprise thin ceramic coatings bonded to an underlying superalloy. These new modelling techniques will help understand the effect of interfaces [within ceramic or metallic crystals, or between disparate phases] upon mechanical properties such as creep strength.’ (1)

In order to do this, the team established a series of models focused on molecular dynamics, discrete dislocation and diffusion calculations, to predict strength, creep resistance and fracture toughness.

They used homogenisation techniques to identify a set of macroscopic equations for the relevant internal variables and to develop robust macroscopic constitutive laws that discriminate between the competing phenomenological theories.

Hierarchy.
Two approaches to multi-scale modelling are Fig 7 opposite:
I. hierarchic whereby properties calculated at one scale may be passed to the next higher scale in a serial and hierarchical way. Both Figs 1 and 2 are representations of such hierarchic approaches. This is the approach underlined by Prof Fleck in order to meet the as the challenges raise by tackling advance ceramic and TBCs.
II. In contrast the other approach is known as concurrent. Concurrent methods build
around the idea of describing the physics of different regions of a material with different models and linking them via a set of boundary conditions. The archetype of concurrent methods divides the space into atomistic regions coupled with a continuum modelled via the FE-Finite element method. (3) So called coarse graining is achieved by coupling seamlessly a molecular dynamics (MD) a coarse grained molecular dynamic (CGMD) region to a FE mesh. (Rudd et al 7). Rudd and Broughton (7) provide a critical review of many of the current methods currently used.

Earlier Tadmor,Phillips and M. Ortiz,originators of the quasicontinuum approach 1992-96, (6) in their paper Hierarchical modelling in the mechanics of materials (Ref 8) describe MMM as of the most exciting threads of research in the mechanics of materials, namely, the attempt to develop models of material response that are constructed on the basis of elementary mechanisms rather than curve-fitting. The strategy proposed to face this challenge is to set up a hierarchy of modeling efforts in which models at one scale are used to inform those at another. They give a concrete example of this approach as it applies to the development of a quantitative theory of single crystal plasticity.

In their excellent overview of such approaches Tadmor, Phillips and Ortiz (8) who examine hierarchical approaches to modeling problems in multi-scale mechanics of solid materials with special reference to the way in which information can be fed from one scale to the next in models that simulate single crystal plasticity from underlying dislocation dynamics and atomic processes. In the case of crystal plasticity. For example one approach in the attempt to inform models of single crystal plasticity on the basis of mechanistic understanding is to seek an understanding of processes such as hardening directly on the basis of the dynamics of interacting dislocations.

Fleck et al. give an further example of homogenization in their study of a non-linear two phase composite, whereby they consider an elastoplastic composite subjected to loading which varies spatially in a sufficiently slow manner with respect to the microstructure that its homogenised response is (approximated to be)without strain gradient effects on the macro-scale.

SOME REMARKABLE MATERIALS PHENOMENA AND SIZE EFFECTS HIGH-LIGHTED BY THE MINITURIZATION DRIVE.

Food for though: Dispersoid Hardening in nanostructured and bulk metal matrix composites, a case of competitive forces on added internal particle fracure Kelly and Fine vs Orowan mechanism, Fig.8.opposite.

Fleck underlines the team's main area of interest lies in the range 10 nm to 10μm.

Like many of his colleagues in MMM, Fleck re-iterates the importance of size effects, as an increasing concern in engineering materials due to the development of fine-structure metallic alloys and ceramic coatings. (MEMS,Thin film multi-layered structural composite materials etc). It is almost self-evident, that in nano-scale and micro-scale applications, the ratio of interface to volume becomes very large compared with bulk applications, and hence interfacial effects are expected to become increasingly dominant(Vvedensky (ref 4).

The Inverse Hall-Petch (H-P) Effects (IHPE).

The Hall–Petch relationship predicts that the yield stress increases with the inverse of the square root of the grain size.
That the H-P relationship break down in the limit as the grain size tends to zero, yield strength tends to infinity, was of course realised by the metallurgists, Hall(1951) and Petch(1953) at the time of their establishing the empirical law named after them and which has served metallurgists and materials scientists well for many decades. However it is due to the relentless drive to smaller size applications (Moore's Law effect) that experimental evidence brought this effect (IHHE) to the fore and is an intensive area of current research and modelling in order to elucidate all the facets likely to be encountered in practice, The premium for success is optimum strength and general mechanical behaviour at very small grain sizes,(less than 30nm) thin films thickness, and multiple layer composite design to name a few.

Many recent works can help us up-date on understanding of micro- and nanostructured materials processing in general and of apprehending the "homogenising" role of H-P and deviations such as IHPE. In particular M.A.Meyers et al (ref.9) reviewed comprehensively the mechanical properties of nanocrystalline materials, with emphasis on their constitutive response and on the fundamental physical mechanisms, including the deviation from the Hall–Petch (H-P) slope and possible negative slope [Inverse H-P effect (IHPE)], first reported in 1989. They conclude that: "The Hall–Petch curve for the nanocrystalline range clearly shows a deviation from the regular trend in the micro-crystalline range; there is a significant decrease in the slope for small grain sizes. But, there is no clear evidence on the nature of the curves at grain sizes below about 10–15 nm. Though researchers have debated the existence of the negative Hall Petch effect, there is insufficient information to validate the existence of this effect. The most probable behaviour is that the yield strength flattens below a critical grain size. The real trend is still to be determined along with the knowledge of whether it varies for different materials."(9) Aware of the influence of the various special manufacturing process for nanostructured materials, Meyers et al (9) summarize the main one at the outset of their review with the spurious effects engendered in attempts to analyse data for IHPE.

Carlton and Ferreira (10) (2007) have classed previous models into 4 types: namely dislocation-based models, diffusion-based models, grain-boundary-shearing models, and two-phase-based models, then critically review them against 8 published TEM measurements all but one (1993) the other 7 carried out between (2000-2005). They offer an explanation for the inverse Hall–Petch effect based on the statistical absorption of dislocations by grain boundaries, showing that the
yield strength is dependent on strain rate and temperature and deviates from the Hall–Petch relationship below a critical grain size. Their approach gives the following modified Hall–Petch relation:



where, Pdis, is the probability of a dislocation being absorbed by the grain boundary.
The Carlton-Ferreira (C-F) Eqn.. reverts to the classical Hall–Petch equation when the boundary is rigid and dislocation probability, Pdis = 0.

The essential effect of grain-boundary dislocation absorption is to reduce the number of dislocations, n,in the pile-up, which decreases the stress at the leading dislocation.

If all moving dislocations are absorbed by the existing grain boundaries (Pdis = 1).
then C-F Eqn.. (11) is reduced to sigma y = sigma 0, i.e., grain boundaries do not play any role in strengthening the material.

As the term Pdis is negative in the C-F Eqn.. reducing the Hall–Petch coefficient, k, is simply an inversion of the Hall–Petch relation.

The larger the grain size, the larger must be the number of atoms absorbed by the grain boundary, if the dislocation is also to be absorbed. Thus, for larger grain sizes, the probability for dislocation absorption by the grain boundary is lowered. The predictions provided by the model can be tested in the laboratory, keeping in mind that the strain rate, the temperature, grain-boundary distribution and grain size distribution are all important parameters to be considered.

It is interesting to compare a C-F IHPE graph Fig 9, LHS below with one from one from Kim and Estrin(2005), Fig 10. RHS below.


Kim and Erin (RHS) modelled a polycrystalline material as a mixture of two phases: grain interior material whose plastic deformation is governed by dislocation and diffusion mechanisms and grain boundary phase whose plastic flow is controlled by a boundary diffusion mechanism. (11) The main target of this study was the effect of grain size on the flow stress and its strain rate sensitivity as well as on the strain hardening. Conventional Hall–Petch behaviour in coarse grained materials at high strain rates governed by the dislocation glide mechanism was shown to be replaced with inverse Hall–Petch behavior in ultrafine grained materials at low strain rates, when both phases deform predominantly by diffusion controlled mechanisms.

Lastly a precious paper from George Weng giving both insights into homogenisation, many highly relevent Inverse H-P effects:

George Weng in his review, highlights a micromechanics-based homogenization scheme that has wide applicability for calculations of the overall rate-independent plasticity, time-dependent creep, strain-rate sensitivity, effect of porosity, and void growth for nanocrystalline materials. Based on the morphology disclosed in molecular dynamic simulation, he too establishes a composite model to represent the grain interior and the grain-boundary zone (GB zone). The nonlinear rate-independent plasticity is formulated in terms of the secant moduli of the constituent phases, whereas the rate-dependent viscoplasticity is formulated in terms of their secant viscosity. In both cases the heterogeneous stress and strain fields of the constituent phases are analytically determined. Through two related field fluctuation approaches, the effective stresses of the grain interior and the GB zone are derived through the variation of the overall secant moduli and the overall secant viscosity with respect to the constituent property. The overall behavior then can be calculated from the effective secant moduli or effective secant viscosity. He demonstrate how this approach provides the overall stress-strain relation as the grain size decreases from the coarse grain to the nanometer range, and how the slope of the Hall-Petch plot continues to decrease and eventually turns into negative below certain critical grain size. This critical grain size also gives rice to the maximum yield strength, and is an important factor in material design. He also shows how the creep resistance increases with decreasing grain size and then decline, how the strain-rate sensitivity of the nanocrystalline materials is affected by grain size, and how porosity and grain size compete with each other under a constant strain rate loading. We conclude by the study of void growth during
viscoplastic deformation of nanocrystalline materials.
quoted from the abstract. (12)
STRAIN GRADIENT (SG.)THEORY .

-TIME-LINE:A useful time-line and review of gradient-dependent theory with references from its origins recent developments can be found in Abu Al-Ruba et al 2007,(ref.13) " One class of gradient theories have been advocated in the last decade that assume higher-order gradients of the displacement field (e.g. [Fleck et al., 1994], [Fleck and Hutchinson, 1993], [Fleck and Hutchinson, 1997], [Fleck and Hutchinson, 2001], [Nix and Gao, 1998], [Gao et al., 1999], [Gao and Huang, 2001], [Hwang et al., 2002], [Gurtin, 2000] and [Gurtin, 2003]). This group of theories is in fact a particular case of the generalized continua, such as micromorphic continua (Eringen, 1968), or continua with microstructure (Mindlin, 1964), which were all inspired by the pioneering work of the Cosserat brothers (Cosserat and Cosserat, 1909)." cf. also the timeline in Fleck et al.(15).

-BACKGROUND TO CURRENT RENEWED SG. INTEREST highlighted by the work of Fleck and Hutcheson amoung others.

Although not predicted by classical models, an increase in flow stress is seen during deformation when the observed phenomena is on the order of a micron and inhomogeneties are present. For example, Fleck et.al. [1] showed that when loaded in torsion, a wire displays greater strength for smaller radii. Others authors have observed this type of effect in other systems, including bending [2], indentation hardness [3], and particle hardened alloys [4]. The increase in hardness under these conditions is attributed to the additional dislocations needed for compatibility. These dislocations are commonly referred to as geometrically necessary dislocations [5]. Reference Details in Annexe: Strain gradient approach below.

Theories to extend plasticity to the micron scale have been in existence for over a decade, complemented by a growing body of experimental data.

The Latest Critical Assessement.The latest critical assessement of these methods (2009) has been carried out by one of the originators of the Fleck-Hutchinson, FH model, Prof. JW.Hutchinson together with his colleague (sadly) the late Prof. AG. Evans renowned materials scientists and engineering professors and researchers.

Here, materials and mechanics aspects of two prominent strain gradient theories of plasticity, due to Nix and Gao (NG model) and to Fleck and Hutchinson, are assessed within the context of simple bending. Differences between the theories are highlighted. The theories predict different trends relative to the size dependence of initial yielding and rate of hardening. The dislocation mechanics underpinning the two theories is addressed. Distinctions between lower-order theories and higher-order theories are also drawn, emphasizing the flexibility of higher-order theories to solve problems for a wide range of boundary conditions, especially those where, locally, the dislocations are blocked (pile up) and the plastic strain is zero.

There are several important conclusion made. Let me mention only two : [full paper(16)]
1. The difference between lower-order (conventional NG) and higher-order (FH) theory was emphasized and illustrated by the example of surface passivated foils in bending. The distinction is that the low-order theory is restricted in terms of the boundary conditions that can be imposed, excluding the solution of various important problems. The higher-order theory does not have this restriction, because it introduces a new quantity (with dimensions stress x length) governing the plastic work within the strain gradient: analogous to the dissipation within dislocation pile ups. Accordingly, higher-order theory can solve problems involving locations where the plastic strain is zero owing to the blockage of dislocations.

2. The uncertainties that remain from this assessment suggest continued critical evaluation of the basic formulations with input guided by fundamentals of dislocation mechanics. Further progress requires better experimental data for a range of materials using bending, torsion and shear as benchmarks.

NB. To check the validity of their models, the team lead by Fleck carried out the following physical testing experiments: Room temperature (RT) and high temperature (HT) torsion (twisting-shear) and bending results are reported to confirmed that the discrete dislocation models have been improved and that high temperature deformation in turbine blades (aero- engines, power generation...) and similar materials can be more accurately predicted.


Last but not least :
MICROSCOPY-METROLOGY.(microscope and typical resolution on space map., Fig 11 opposite)
If the macro-level is visible to the naked eye, the same cannot be said of the lower size levels. Indeed little or no modern MMM would have been possible without the spectacular progress in microscopy and metrology. Meso, and micro levels can be fairly readily studied with quality optical methods and more fundamental levels by Electron Microscopy. For example, transmission electon microscopics (TEM) images of dislocation on thin metal foil were first reported in the 1950's (Hirsch et al., Bollmann) Current high-energy transmission and scanning transmission electron microscopics TEM and STEM (Fig. 12 below) have resolutions at the nano level. STM-Scanning Tunnelling Microsope reaches(0.1nm). They can now visulise individual atoms. Dislocations, their motion and interactions with sub-grain features and at grain boundaries, so important in the understanding of the strength of materials are today conveniently seen in TEM and measurement made of sizes and velocities. Quantitative measurements of internal stress distributions (Mughrabi 1975) are now current whereby atom probe and electron microscopy are performed at the atomic level accompanied by strain measurement and chemical analysis on the same level.

Tadmor et al's historical introduction to modern multi-scale problems is well chosen. In their paper "Hierarchical modelling in the mechanics of materials (REF 8)" . They give the example of century old known size effects, based on the British Metallurgist W.C. Roberts-Austen reported series of experiments studying the effect of impurities on the tensile strength of gold. His results indicated a strong and regular dependence of the tensile strength on the size (atomic volume) of the impurity. FIG. 13. LHS








Before Richard Feynman's (1918–1988) famous retorical question and his answer:
Q."If in some cataclysm all scientific knowledge were to be destroyed and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words?

A. I believe it is the atomic hypothesis that all things are made of atoms - little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see there is an enormous amount of information about the world, if just a little imagination and thinking are applied."

Hume-Rothery(1899-1968)had taken the atomistic view and layed down a number much used empirical rules concerning size effects in metallic alloying The Hume-Rothery Rules

The team led by Freck, also report improved understanding at the atomic scale (10^-9, 1 nm or 0.1Å , Ångströms) of creep and fracture in silicon single crystals (Si chips) at the heart of electronic integrated circuits. At this level focus was given to ab-initio MD calculations. Mechanisms of atomic rearrangement at the crack tip explain observed instabilities during crack propagation.

RELATED POSTS:

It's not HSLA-Bainite"Nanostructured Steels"-Green Light by Irvine-based Materials Science Co-MMFX Tech Corp - Corrosion and Toughness Themes

Mathematical Modeling in Materials Science Review by H.Bhadeshia,Univ. of Cambridge UK

References and further reading.

1a. Bhadeshia H. Mathematical models in materials science, Materials Science and Technology 2008 VOL 24 NO 2

1. Model behaviour of ceramic and intermetallic alloys, MW Dec.09

2.Ortiz, Multiscale Models of Materials: Linking Microstructure and Macroscopic Behaviour SIAM Conference on Computational Science and Engineering Orlando, Florida Feb. 12, 2005.

3.Helpful highly visual presentations on the materials science and engineering from Prof D. Raabe, Max Planck Inst. in english.

4. Multiscale modelling of nanostructures, Dimitri D Vvedensky, J. Phys.: Condens. Matter 16 (2004) R1537–R1576.

5. Multiscale modelling of nanomechanics and micromechanics: an overview N.M.Ghoniemy, E. P. Busso, N. Kioussis, H. Huang, Phil. Mag, Nov– Dec 2003, Vol. 83, Nos. 31–34, 3475–3528.

6.Tadmor,Phillips and M. Ortiz, originators of the quasicontiuum approach 1992-96, and

7. Coarse-grained molecular dynamics: Nonlinear finite elements and finite temperature Robert E. Rudd, Jeremy Q. Broughton Arxiv 22 Aug.2005.

7bis. Concurrent multiscale model of an atomic crystal coupled with elastic continua
P. A. Deymier and J. O. Vasseur, PHYSICAL REVIEW B 66, 134106 ,2002.


8. Tadmor, Phillips and Ortiz, Hierarchical modeling in the mechanics of materials,
International Journal of Solids and Structures 37 (2000) 379±389

9.Mechanical properties of nanocrystalline materials, Meyers,Mishra,and Benson, Progress in Materials Science 51 (2006) 427–556.

10. Carlton and Ferreira, What is behind the inverse Hall–Petch effect in nanocrystalline materials? 2007 Acta Materialia, doi:10.1016/j.actamat.2007.02.021.

11. Kim and Estrin, Phase mixture modeling of the strain rate dependent mechanical behaviour of nanostructured materials, Acta Materialia 53 (2005) 765–772.

12. George J. Weng, A HOMOGENIZATION SCHEME FOR THE PLASTIC
PROPERTIES OF NANOCRYSTALLINE MATERIALS, Rev.Adv.Mater.Sci. 19(2009)41-62.

13. Abu Al-Ruba, Voyiadjisa, and Bammann, A thermodynamic based higher-order gradient theory for size dependent plasticity, International Journal of Solids and Structures 44 (2007) 2888–2923.

14. Fleck et al. Strain Gradient Plasticity Theory:Theory and Experiment. acta metall.materV42,N°2 pp475-487, 1994.

15. Fleck and Willis, Bounds and estimates for the e%ect of strain gradients upon the e%ective plastic properties of an isotropic two-phase composite, Journal of the Mechanics and Physics of Solids, 52 (2004) 1855 – 1888.

16. A.G. Evans, J.W. Hutchinson A critical assessment of theories of strain gradient plasticity, Acta Materialia 57 (2009) 1675–1688.

Annexe Strain Gradient Approach:REFERENCES1. N. A. Fleck, G. M. Muller, M. F. Ashby, and J. W. Hutchinson, Acta Metall. Mater. 42, 475
(1993).
2. J. S. Stölken and A. G. Evans, Acta. Mater. 46, 5109 (1998).
3. N. A. Stelmashenko, M. G. Walls, L. M. Brown, and Y. V. Milman, in Mechanical Properties
and Deformation Behavior of Materials Having Ultra-Fine Microstructures, edited by M.
Nastasi, D. M. Parkin, and H. Gleiter. (NATO ASI Series E 233, 1993) pp. 605-610.
4. D. J. Lloyd, Int. Mater. Rev. 39, 1 (1994).
5. M. F. Ashby, Philos. Mag. 21, 399 (1970).