A continuum dislocation-based model of wedge microindentation of single crystals

Abstract Recent Electron Backscatter Diffraction (EBSD) experiments have revealed the emergence of heterogeneous dislocation microstructures forming under a wedge indenter in fcc crystals, where micro-meter dislocation patterns challenge the predictions of traditional models of plasticity. In order to explain the formation of these features and develop a relationship between the force-displacement curve and the dislocation substructure, we present here a model of wedge indentation based on the continuum theory of dislocations. The model accounts for large deformation kinematics through the multiplicative split of the deformation gradient tensor, where the incompatible plastic component of deformation results from the flux of dislocations on different and interacting slips systems. Constitutive equations for dislocation fluxes are determined from a dissipative variational principle. As a result, each dislocation density satisfies an initial-boundary value problem with convective-diffusive character, which is coupled to the macroscopic stress and displacement fields governing the deformation process. Solution to the self-consistent continuum formulation is found using the finite element method. Computer simulations mimic the experimental conditions of wedge micro-indentation experiments into Ni single-crystals used by Kysar et al. (2010a). A comparison of overall dislocation density distribution and macroscopic mechanical response shows good overall agreement with the experimental results in terms of the detailed features of dislocation patterns and lattice rotations as well as the macroscopic force-displacement response.

[1]  S. J. Tupper,et al.  The theory of wedge indentation of ductile materials , 1947, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[2]  J. Kysar,et al.  High strain gradient plasticity associated with wedge indentation into face-centered cubic single crystals: Geometrically necessary dislocation densities , 2007 .

[3]  M. Hütter,et al.  Microscopically derived free energy of dislocations , 2015 .

[4]  A. Minor,et al.  Indentation across size scales and disciplines: Recent developments in experimentation and modeling , 2007 .

[5]  Bharat Bhushan,et al.  Fracture Mechanisms of Thin Amorphous Carbon Films in Nanoindentation , 1997 .

[6]  G. Pharr,et al.  The Indentation Size Effect: A Critical Examination of Experimental Observations and Mechanistic Interpretations , 2010 .

[7]  J. Lothe.  Theory of Dislocation Mobility in Pure Slip , 1962 .

[8]  Brian R. Lawn,et al.  A Critical Evaluation of Indentation Techniques for Measuring Fracture Toughness: I , 1981 .

[9]  En-Jui Lee.  Elastic-Plastic Deformation at Finite Strains , 1969 .

[10]  C. Schuh.  Nanoindentation studies of materials , 2006 .

[11]  F. Roters,et al.  Three dimensional investigation of the texture and microstructure below a nanoindent in a Cu single crystal using 3D EBSD and crystal plasticity finite element simulations , 2006 .

[12]  Frank Reginald Nunes Nabarro,et al.  Theory of crystal dislocations , 1967 .

[13]  A. C. Fischer-Cripps,et al.  Critical review of analysis and interpretation of nanoindentation test data , 2006 .

[14]  David L. Olmsted,et al.  Atomistic simulations of dislocation mobility in Al, Ni and Al/Mg alloys , 2005 .

[15]  W. Oliver,et al.  Hardness measurement at penetration depths as small as 20 nm , 1983 .

[16]  D. Tabor.  Hardness of Metals , 1937, Nature.

[17]  S. Qu,et al.  Indentation of a hard film on a soft substrate: Strain gradient hardening effects , 2007 .

[18]  R. Hannink,et al.  Slip system determination in cubic carbides by hardness anisotropy , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[19]  Anter El-Azab,et al.  Computational modelling of mesoscale dislocation patterning and plastic deformation of single crystals , 2015 .

[20]  B. Bhushan,et al.  A Review of Nanoindentation Continuous Stiffness Measurement Technique and Its Applications , 2002 .

[21]  G. Pharr,et al.  An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments , 1992 .

[22]  J. Kysar,et al.  Geometrically necessary dislocation density measurements associated with different angles of indentations , 2014 .

[23]  Subra Suresh,et al.  A new method for estimating residual stresses by instrumented sharp indentation , 1998 .

[24]  Jee-Gong Chang,et al.  Molecular dynamics analysis of temperature effects on nanoindentation measurement , 2003 .

[25]  M. Hütter,et al.  Free energy of dislocations in a multi-slip geometry , 2016 .

[26]  Yang-Tse Cheng,et al.  Scaling, dimensional analysis, and indentation measurements , 2004 .

[27]  R. Kriz,et al.  Size effects in indentation response of thin films at the nanoscale: A molecular dynamics study , 2008 .

[28]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[29]  G. Györgyi,et al.  Variational approach in dislocation theory , 2009, 0904.1105.

[30]  Amit Acharya,et al.  A model of crystal plasticity based on the theory of continuously distributed dislocations , 2001 .

[31]  Bob Svendsen,et al.  Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations , 2002 .

[32]  J C Hamilton,et al.  Dislocation emission around nanoindentations on a (001) fcc metal surface studied by scanning tunneling microscopy and atomistic simulations. , 2002, Physical review letters.

[33]  Christoph Wehrli,et al.  The Derivation of Constitutive Relations from the Free Energy and the Dissipation Function , 1987 .

[34]  Marc Fivel,et al.  Three-dimensional modeling of indent-induced plastic zone at a mesoscale , 1998 .

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

[36]  Dierk Raabe,et al.  Dislocation density distribution around an wedge indent in single- crystalline nickel: Comparing non-local crystal plasticity finite element predictions with experiments , 2014 .

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

[38]  T. Bieler,et al.  Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications , 2010 .

[39]  Michael Zaiser,et al.  Continuum dislocation dynamics: Towards a physical theory of crystal plasticity , 2014 .

[40]  A. Needleman,et al.  Discrete dislocation plasticity analysis of the wedge indentation of films , 2006, Journal of the Mechanics and Physics of Solids.

[41]  C. F. Niordson,et al.  Length-scale effect due to periodic variation of geometrically necessary dislocation densities , 2013 .

[42]  F. Roters,et al.  On the origin of deformation-induced rotation patterns below nanoindents , 2008 .

[43]  R. Dewit.  The direction of the force on a dislocation and the sign of the Burgers vector , 1965 .

[44]  J. Kysar,et al.  Resolving geometrically necessary dislocation density onto individual dislocation types using EBSD-based continuum dislocation microscopy , 2016 .

[45]  R. Pippan,et al.  Stacking fault energy and indentation size effect: Do they interact? , 2008 .

[46]  A. Sully,et al.  An X-ray investigation of pure iron-nickel alloys. Part 4: the variation of lattice-parameter with composition , 1937 .

[47]  A. Hartmaier,et al.  Continuum simulation of the evolution of dislocation densities during nanoindentation , 2012 .

[48]  M. Hütter,et al.  Collective behaviour of dislocations in a finite medium , 2014 .

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

[50]  Paul Steinmann,et al.  Views on multiplicative elastoplasticity and the continuum theory of dislocations , 1996 .

[51]  K. Le,et al.  Dislocation structure during microindentation , 2015 .

[52]  J. Zimmerman,et al.  Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation , 2003 .

[53]  M. Gurtin,et al.  On the characterization of geometrically necessary dislocations in finite plasticity , 2001 .

[54]  Giacomo Po,et al.  Recent Progress in Discrete Dislocation Dynamics and Its Applications to Micro Plasticity , 2014 .

[55]  Yanfei Gao,et al.  Lattice rotation caused by wedge indentation of a single crystal: Dislocation dynamics compared to crystal plasticity simulations , 2014 .

[56]  R. Wiť.  The Continuum Theory of Stationary Dislocations , 1960 .

[57]  E. Giessen,et al.  A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations , 2004 .

[58]  S. Bargmann,et al.  On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation , 2010 .

[59]  S. Mesarovic,et al.  Thermodynamic coarsening of dislocation mechanics and the size-dependent continuum crystal plasticity , 2010 .

[60]  David L. McDowell,et al.  Coarse-Grained Atomistic Simulations of Dislocations in Al Ni and Cu Crystals. , 2012 .

[61]  O. Kraft,et al.  High cycle fatigue of thin silver films investigated by dynamic microbeam deflection , 1999 .

[62]  P. Gumbsch,et al.  Dynamic aspects of dislocation motion: atomistic simulations , 2005 .

[63]  W. T. Huh,et al.  Experimental lower bounds on geometrically necessary dislocation density , 2010 .

[64]  W. Lee,et al.  Simulation of micro-indentation hardness of FCC single crystals by mechanism-based strain gradient crystal plasticity , 2010 .

[65]  Y. Gan,et al.  NUMERICAL AND EXPERIMENTAL STUDIES OF DEEP INDENTATION ON SINGLE CRYSTALS , 2008 .

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

[67]  Zhenyu Xue,et al.  The Influence of Indenter Tip Radius on the Micro-Indentation Hardness , 2002 .

[68]  Vasily V. Bulatov,et al.  On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals , 2004 .

[69]  S. Forest,et al.  Crystal plasticity analysis of cylindrical indentation on a Ni-base single crystal superalloy , 2013 .

[70]  Cristian Teodosiu,et al.  Elastic Models of Crystal Defects , 1982 .

[71]  D. Raabe,et al.  Investigation of the indentation size effect through the measurement of the geometrically necessary dislocations beneath small indents of different depths using EBSD tomography , 2009 .

[72]  M. Fivel,et al.  3D simulation of a nanoindentation test at a mesoscopic scale , 1997 .

[73]  J. Kysar,et al.  Cylindrical void in a rigid-ideally plastic single crystal. Part I: Anisotropic slip line theory solution for face-centered cubic crystals , 2005 .

[74]  Ting Zhu,et al.  Atomistic mechanisms governing elastic limit and incipient plasticity in crystals , 2002, Nature.

[75]  E. Kröner,et al.  Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen , 1959 .

[76]  Guy T. Houlsby,et al.  Application of thermomechanical principles to the modelling of geotechnical materials , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[77]  J. Kysar,et al.  Wedge indentation into elastic–plastic single crystals. 2: Simulations for face-centered cubic crystals , 2012 .