Material length scale of strain gradient plasticity: A physical interpretation

Abstract The physical basis and the magnitude of the material length scale in theories of strain gradient plasticity are crucial for accounting for size effects in the plastic behavior of metals at small scales. However, the underlying physics of the length scale is ambiguous. The length scales in strain gradient plasticity theories in which the plastic work density can be expressed as a function of the gradient-enhanced plastic strain are here derived from known physical quantities via critical thickness theory. A connection between the length scale and the fundamental physical quantities is elucidated. The combination of the strain and strain-gradient terms within the deformation theory of strain gradient plasticity is addressed. It is shown that, compared with the harmonic sum of the strain and strain-gradient terms in Fleck-Hutchinson theory, the linear combination gives a more reasonable value of length scale, several micrometers, which is close to that in the gradient theory of Aifantis. In contrast, the value of length scale in Nix-Gao theory is much larger, in the millimeter range. The length scales determined by critical thickness theory are in good agreement with those obtained by fitting to experimental data of wire torsion.

[1]  Kai Peng,et al.  Accounting for the recoverable plasticity and size effect in the cyclic torsion of thin metallic wires using strain gradient plasticity , 2015 .

[2]  Rhj Ron Peerlings,et al.  An implicit tensorial gradient plasticity model - formulation and comparison with a scalar gradient model , 2011 .

[3]  Morton E. Gurtin,et al.  A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on the accumulation of geometrically necessary dislocations , 2008 .

[4]  J. W. Matthews,et al.  Defects in epitaxial multilayers: I. Misfit dislocations* , 1974 .

[5]  Hans Muhlhaus,et al.  A variational principle for gradient plasticity , 1991 .

[6]  B. Ehrler,et al.  Elastic limit and strain hardening of thin wires in torsion. , 2009, Physical review letters.

[7]  Norman A. Fleck,et al.  A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier , 2009 .

[8]  D. Dunstan,et al.  Strain and strain relaxation in semiconductors , 1997 .

[9]  David J. Dunstan,et al.  Grain size and sample size interact to determine strength in a soft metal , 2008 .

[10]  Lallit Anand,et al.  A one-dimensional theory of strain-gradient plasticity : Formulation, analysis, numerical results , 2005 .

[11]  H. Zbib,et al.  On the gradient-dependent theory of plasticity and shear banding , 1992 .

[12]  David L. McDowell,et al.  Modeling Dislocations and Disclinations With Finite Micropolar Elastoplasticity , 2006 .

[13]  Viggo Tvergaard,et al.  An alternative treatment of phenomenological higher-order strain-gradient plasticity theory , 2010 .

[14]  U. F. Kocks.  Laws for Work-Hardening and Low-Temperature Creep , 1976 .

[15]  D. Dunstan,et al.  Size effects in yield and plasticity under uniaxial and non-uniform loading: experiment and theory , 2011 .

[16]  Norman A. Fleck,et al.  A reformulation of strain gradient plasticity , 2001 .

[17]  L. Bragg.  A Theory of the Strength of Metals* , 1942, Nature.

[18]  Micro-plasticity and recent insights from intermittent and small-scale plasticity , 2017, 1704.07297.

[19]  Lallit Anand,et al.  A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part II: Finite deformations , 2005 .

[20]  G. Voyiadjis,et al.  Bridging of length scales through gradient theory and diffusion equations of dislocations , 2004 .

[21]  S. S. Brenner,et al.  Tensile Strength of Whiskers , 1956 .

[22]  D. Dunstan.  Critical Thickness Theory Applied to Micromechanical Testing , 2012 .

[23]  George Z. Voyiadjis,et al.  Thermo-mechanical strain gradient plasticity with energetic and dissipative length scales , 2012 .

[24]  Michael Kaliske,et al.  An implicit gradient formulation for microplane Drucker-Prager plasticity , 2016 .

[25]  L. Bardella.  A comparison between crystal and isotropic strain gradient plasticity theories with accent on the role of the plastic spin , 2009 .

[26]  N. Fleck,et al.  Guidelines for Constructing Strain Gradient Plasticity Theories , 2015 .

[27]  Douglas J. Bammann,et al.  A model of crystal plasticity containing a natural length scale , 2001 .

[28]  G. Pharr,et al.  The correlation of the indentation size effect measured with indenters of various shapes , 2002 .

[29]  Huajian Gao,et al.  Indentation size effects in crystalline materials: A law for strain gradient plasticity , 1998 .

[30]  John W. Hutchinson,et al.  Generalizing J2 flow theory: Fundamental issues in strain gradient plasticity , 2012, Acta Mechanica Sinica.

[31]  K. Le,et al.  Nonlinear continuum dislocation theory revisited , 2014 .

[32]  V. Berdichevsky.  Continuum theory of dislocations revisited , 2006 .

[33]  Rhj Ron Peerlings,et al.  The plastic rotation effect in an isotropic gradient plasticity model for applications at the meso scale , 2016 .

[34]  Vlado A. Lubarda,et al.  On the recoverable and dissipative parts of higher order stresses in strain gradient plasticity , 2016 .

[35]  Zengsheng Ma,et al.  On the intrinsic hardness of a metallic film/substrate system: Indentation size and substrate effects , 2012 .

[36]  B. Zhang,et al.  Anomalous plasticity in the cyclic torsion of micron scale metallic wires. , 2013, Physical review letters.

[37]  Lorenzo Bardella,et al.  A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations , 2006 .

[38]  W. Curtin,et al.  Stress-gradient plasticity , 2011, Proceedings of the National Academy of Sciences.

[39]  J. David,et al.  Plastic relaxation of metamorphic single layer and multilayer InGaAs/GaAs structures , 1994 .

[40]  Lorenzo Bardella,et al.  Modelling the torsion of thin metal wires by distortion gradient plasticity , 2015 .

[41]  Huajian Gao,et al.  Mechanism-based strain gradient plasticity— I. Theory , 1999 .

[42]  E. Kröner,et al.  Kontinuumstheorie der Versetzungen und Eigenspannungen , 1958 .

[43]  K. Aifantis,et al.  INTERPRETING THE INTERNAL LENGTH SCALE IN STRAIN GRADIENT PLASTICITY , 2015 .

[44]  V. Berdichevsky.  Energy of dislocation networks , 2016 .

[45]  A. Acharya,et al.  A model for rate-dependent flow of metal polycrystals based on the slip plane lattice incompatibility , 2001 .

[46]  J. Weertman,et al.  Anomalous work hardening, non-redundant screw dislocations in a circular bar deformed in torsion, and non-redundant edge dislocations in a bent foil , 2002 .

[47]  David J. Dunstan,et al.  Toward a further understanding of size effects in the torsion of thin metal wires: An experimental and theoretical assessment , 2013 .

[48]  Shaohua Chen,et al.  The coupling effect of size and damage in micro-scale metallic materials , 2017 .

[49]  K. Le.  Three-dimensional continuum dislocation theory , 2015, 1506.03570.

[50]  K. Le,et al.  On torsion of a single crystal rod , 2011 .

[51]  P. Moreau,et al.  Measurement of the size effect in the yield strength of nickel foils , 2005 .

[52]  M. Ashby.  The deformation of plastically non-homogeneous materials , 1970 .

[53]  A. Bushby,et al.  Theory of deformation in small volumes of material , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[54]  G. Pharr,et al.  A simple stochastic model for yielding in specimens with limited number of dislocations , 2013 .

[55]  Huajian Gao,et al.  Mechanism-based strain gradient plasticity—II. Analysis , 2000 .

[56]  Eugene A. Fitzgerald,et al.  Dislocations in strained-layer epitaxy : theory, experiment, and applications , 1991 .

[57]  D. Parks,et al.  Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density , 1999 .

[58]  Maria Chiricotto,et al.  Dissipative Scale Effects in Strain-Gradient Plasticity: The Case of Simple Shear , 2015, SIAM J. Appl. Math..

[59]  E. Aifantis.  Strain gradient interpretation of size effects , 1999 .

[60]  David J. Dunstan,et al.  Size effect in the initiation of plasticity for ceramics in nanoindentation , 2008 .

[61]  D. Dunstan.  Mathematical model for strain relaxation in multilayer metamorphic epitaxial structures , 1996 .

[62]  J. Nye.  Some geometrical relations in dislocated crystals , 1953 .

[63]  K. Le,et al.  On bending of single crystal beam with continuously distributed dislocations , 2013 .

[64]  C. F. Niordson,et al.  BASIC STRAIN GRADIENT PLASTICITY THEORIES WITH APPLICATION TO CONSTRAINED FILM DEFORMATION , 2011 .

[65]  T. Siegmund,et al.  A dislocation density based strain gradient model , 2006 .

[66]  F. C. Frank,et al.  One-dimensional dislocations. I. Static theory , 1949, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[67]  E. Aifantis.  On the Microstructural Origin of Certain Inelastic Models , 1984 .

[68]  Morton E. Gurtin,et al.  A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations , 2005 .

[69]  M. Kuroda,et al.  Strain hardening in bent copper foils , 2011 .

[70]  Xutao Tang,et al.  Size effects in the torsion of microscale copper wires: Experiment and analysis , 2012 .

[71]  George Z. Voyiadjis,et al.  A physically based gradient plasticity theory , 2006 .

[72]  Anthony G. Evans,et al.  A critical assessment of theories of strain gradient plasticity , 2009 .

[73]  Norman A. Fleck,et al.  A phenomenological theory for strain gradient effects in plasticity , 1993 .

[74]  Jian Lei,et al.  Individual strain gradient effect on torsional strength of electropolished microscale copper wires , 2017 .

[75]  Jerry Tersoff,et al.  Dislocations and strain relief in compositionally graded layers , 1993 .

[76]  Julia R. Greer,et al.  Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients , 2005 .

[77]  M. Ashby,et al.  Strain gradient plasticity: Theory and experiment , 1994 .

[78]  George Z. Voyiadjis,et al.  Gradient plasticity theory with a variable length scale parameter , 2005 .

[79]  M. Saif,et al.  Strain gradient effect in nanoscale thin films , 2003 .

[80]  Peter Gudmundson,et al.  A unified treatment of strain gradient plasticity , 2004 .

[81]  István Groma,et al.  Dynamics of coarse grained dislocation densities from an effective free energy , 2007 .

[82]  R. Lardner.  Dislocation dynamics and the theory of the plasticity of single crystals , 1969 .

[83]  D. Dunstan.  Validation of a phenomenological strain-gradient plasticity theory , 2016 .

[84]  Norman A. Fleck,et al.  Strain gradient plasticity under non-proportional loading , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[85]  R. Beanland.  Multiplication of misfit dislocations in epitaxial layers , 1992 .

[86]  G. Voyiadjis,et al.  Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening , 2010 .

[87]  Lallit Anand,et al.  Thermodynamics applied to gradient theories involving the accumulated plastic strain : The theories of Aifantis and Fleck and Hutchinson and their generalization , 2009 .

[88]  S. Wulfinghoff.  A generalized cohesive zone model and a grain boundary yield criterion for gradient plasticity derived from surface- and interface-related arguments , 2017 .

[89]  N. Fleck,et al.  Strain gradient plasticity , 1997 .

[90]  Elias C. Aifantis,et al.  The physics of plastic deformation , 1987 .

[91]  Anthony G. Evans,et al.  A microbend test method for measuring the plasticity length scale , 1998 .

[92]  Morton E. Gurtin,et al.  Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector , 2005 .

[93]  C. Motz,et al.  Observation of the critical thickness phenomenon in dislocation dynamics simulation of microbeam bending , 2012 .

[94]  Morton E. Gurtin,et al.  A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin , 2004 .

[95]  L. Bardella.  Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin , 2010 .

[96]  Dierk Raabe,et al.  A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations , 2006 .

[97]  E. Aifantis.  On the role of gradients in the localization of deformation and fracture , 1992 .

[98]  Y. Milman,et al.  Microindentations on W and Mo oriented single crystals: An STM study , 1993 .

[99]  Morton E. Gurtin,et al.  Gradient single-crystal plasticity within a Mises–Hill framework based on a new formulation of self- and latent-hardening , 2014 .

[100]  William D. Nix,et al.  Mechanical properties of thin films , 1989 .

[101]  J. W. Matthews.  Accommodation of misfit across the interface between single-crystal films of various face-centred cubic metals , 1966 .

[102]  David J. Dunstan,et al.  Geometrical theory of critical thickness and relaxation in strained‐layer growth , 1991 .

[103]  Michael Zaiser,et al.  Local density approximation for the energy functional of three-dimensional dislocation systems , 2015, 1508.03652.