A thermodynamic based higher-order gradient theory for size dependent plasticity

A physically motivated and thermodynamically consistent formulation of small strain higher-order gradient plasticity theory is presented. Based on dislocation mechanics interpretations, gradients of variables associated with kinematic and isotropic hardenings are introduced. This framework is a two non-local parameter framework that takes into consideration large variations in the plastic strain tensor and large variations in the plasticity history variable; the equivalent (effective) plastic strain. The presence of plastic strain gradients is motivated by the evolution of dislocation density tensor that results from non-vanishing net Burgers vector and, hence, incorporating additional kinematic hardening (anisotropy) effects through lattice incompatibility. The presence of gradients in the effective (scalar) plastic strain is motivated by the accumulation of geometrically necessary dislocations and, hence, incorporating additional isotropic hardening effects (i.e. strengthening). It is demonstrated that the non-local yield condition, flow rule, and non-zero microscopic boundary conditions can be derived directly from the principle of virtual power. It is also shown that the local Clausius–Duhem inequality does not hold for gradient-dependent material and, therefore, a non-local form should be adopted. The non-local Clausius–Duhem inequality has an additional term that results from microstructural long-range energy interchanges between the material points within the body. A detailed discussion on the physics and the application of proper microscopic boundary conditions, either on free surfaces, clamped surfaces, or intermediate constrained surfaces, is presented. It is shown that there is a close connection between interface/surface energy of an interface or free surface and the microscopic boundary conditions in terms of microtraction stresses. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given. Applications of the proposed theory for size effects in thin films are presented.

[1]  D. Stone,et al.  Indentation size effect in polycrystalline F.C.C. metals , 2002 .

[2]  René de Borst,et al.  Gradient-dependent plasticity: formulation and algorithmic aspects , 1992 .

[3]  Castrenze Polizzotto,et al.  A thermodynamics-based formulation of gradient-dependent plasticity , 1998 .

[4]  Huajian Gao,et al.  A finite deformation theory of strain gradient plasticity , 2002 .

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

[6]  P. M. Naghdi,et al.  SOME REMARKS ON ELASTIC-PLASTIC DEFORMATION AT FINITE STRAIN , 1971 .

[7]  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 .

[8]  D. Clarke,et al.  Size dependent hardness of silver single crystals , 1995 .

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

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

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

[12]  R. D. Mindlin.  Micro-structure in linear elasticity , 1964 .

[13]  Amit Acharya,et al.  Lattice incompatibility and a gradient theory of crystal plasticity , 2000 .

[14]  A. Eringen,et al.  On nonlocal elasticity , 1972 .

[15]  Morton E. Gurtin,et al.  Surface stress in solids , 1978 .

[16]  M. M. Chaudhri,et al.  The effect of the indenter load on the nanohardness of ductile metals : an experimental study on polycrystalline work-hardened and annealed oxygen-free copper , 1999 .

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

[18]  Pranav Shrotriya,et al.  On the measurement of the plasticity length scale parameter in LIGA nickel foils , 2003 .

[19]  K. Gall,et al.  Atomistic simulation of the structure and elastic properties of gold nanowires , 2004 .

[20]  Amit Acharya,et al.  Grain-size effect in viscoplastic polycrystals at moderate strains , 2000 .

[21]  M. Frémond,et al.  Damage, gradient of damage and principle of virtual power , 1996 .

[22]  Hussein M. Zbib,et al.  On the localization and postlocalization behavior of plastic deformation. I: On the initiation of shear bands , 1988 .

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

[24]  Jerzy Pamin,et al.  Some novel developments in finite element procedures for gradient-dependent plasticity , 1996 .

[25]  John L. Bassani,et al.  Incompatibility and a simple gradient theory of plasticity , 2001 .

[26]  H. Zbib,et al.  Damage and size effect during superplastic deformation , 2002 .

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

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

[29]  N.,et al.  A PHENOMENOLOGICAL THEORY FOR STRAIN GRADIENT EFFECTS IN PLASTICITY , 2002 .

[30]  P. Steinmann,et al.  Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity , 2001 .

[31]  A. Cemal Eringen,et al.  Theory of Micropolar Elasticity , 1999 .

[32]  William D. Nix,et al.  The Role of Indentation Depth on the Measured Hardness of Materials , 1993 .

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

[34]  Rashid K. Abu Al-Rub,et al.  A Finite Strain Plastic-damage Model for High Velocity Impact using Combined Viscosity and Gradient Localization Limiters: Part I - Theoretical Formulation , 2006 .

[35]  Abu Al-Rub,et al.  Material length scales in gradient-dependent plasticity/damage and size effects: theory and computation , 2004 .

[36]  G. Voyiadjis,et al.  Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments , 2004 .

[37]  Paul Steinmann,et al.  On the continuum formulation of higher gradient plasticity for single and polycrystals , 2000 .

[38]  M. Gurtin.  On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients , 2003 .

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

[40]  D. Wilkinson,et al.  Plastic flow and fracture of a particulate metal matrix composite , 1996 .

[41]  Huajian Gao,et al.  Discrete dislocation dynamics simulations of surface induced size effects in plasticity , 2006 .

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

[43]  A. Needlemana,et al.  A comparison of nonlocal continuum and discrete dislocation plasticity predictions , 2002 .

[44]  H. Zbib,et al.  A superdislocation model for the strengthening of metal matrix composites and the initiation and propagation of shear bands , 1994 .

[45]  Jianmin Qu,et al.  Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films , 2005 .

[46]  Pradeep Sharma,et al.  Size-Dependent Eshelby’s Tensor for Embedded Nano-Inclusions Incorporating Surface/Interface Energies , 2004 .

[47]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[48]  J. Pamin,et al.  Gradient-dependent plasticity in numerical simulation of localization phenomena , 1994 .

[49]  H. Zbib,et al.  Flow strength and size effect of an Al-Si-Mg composite model system under multiaxial loadings , 1995 .

[50]  George Z. Voyiadjis,et al.  Multiscale Analysis of Multiple Damage Mechanisms Coupled with Inelastic Behavior of Composite Materials , 2001 .

[51]  M. E. Gurtin,et al.  A general theory of curved deformable interfaces in solids at equilibrium , 1998 .

[52]  D. Borst,et al.  Fundamental issues in finite element analyses of localization of deformation , 1993 .

[53]  Z. Bažant,et al.  Nonlocal Continuum Damage, Localization Instability and Convergence , 1988 .

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

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

[56]  Anthony N. Palazotto,et al.  Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for dynamic localization problems using gradient theory , 2004 .

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

[58]  George Z. Voyiadjis,et al.  A direct finite element implementation of the gradient‐dependent theory , 2005 .

[59]  K. C. Valanis,et al.  A gradient theory of internal variables , 1996 .

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

[61]  T. Belytschko,et al.  Localization limiters in transient problems , 1988 .

[62]  David R. Clarke,et al.  The influence of particle size and particle fracture on the elastic/plastic deformation of metal matrix composites , 1996 .

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

[64]  Morton E. Gurtin,et al.  A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations , 2002 .

[65]  C. Polizzotto.  Unified thermodynamic framework for nonlocal/gradient continuum theories , 2003 .

[66]  J. Vlassak,et al.  Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments , 1998 .

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

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

[69]  Morton E. Gurtin,et al.  A comparison of nonlocal continuum and discrete dislocation plasticity predictions , 2003 .

[70]  George Z. Voyiadjis,et al.  A Finite Strain Plastic-damage Model for High Velocity Impacts using Combined Viscosity and Gradient Localization Limiters: Part II - Numerical Aspects and Simulations , 2006 .

[71]  Peter Gudmundson,et al.  Size-dependent yield strength of thin films , 2005 .

[72]  Norman A. Fleck,et al.  The prediction of a size effect in microindentation , 1998 .

[73]  Morton E. Gurtin,et al.  On the plasticity of single crystals: free energy, microforces, plastic-strain gradients , 2000 .

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

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

[76]  D. Lloyd.  Particle reinforced aluminium and magnesium matrix composites , 1994 .

[77]  John W. Hutchinson,et al.  The mechanics of size-dependent indentation , 1998 .

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

[79]  M. Ashby,et al.  Micro-hardness of annealed and work-hardened copper polycrystals , 1996 .

[80]  Jian Chen,et al.  Identification of the intrinsic material length in gradient plasticity theory from micro-indentation tests , 2001 .

[81]  P. Dawson,et al.  Using Spatial Gradients to Model Localization Phenomena , 1999 .

[82]  Norman A. Fleck,et al.  Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity , 2001 .

[83]  Hussein M. Zbib,et al.  On the localization and postlocalization behavior of plastic deformation. II: On the evolution and thickness of shear bands , 1988 .

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

[85]  Z. Bažant,et al.  Nonlocal damage theory , 1987 .

[86]  H. Zbib,et al.  A thermodynamical theory of gradient elastoplasticity with dislocation density tensor. I: Fundamentals , 1999 .

[87]  Anthony N. Palazotto,et al.  Non-local coupling of viscoplasticity and anisotropic viscodamage for impact problems using the gradient theory , 2003 .

[88]  G. Voyiadjis,et al.  Determination of the Material Intrinsic Length Scale of Gradient Plasticity Theory , 2004 .

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

[90]  Lucia Nicola,et al.  Discrete dislocation analysis of size effects in thin films , 2003 .

[91]  Huajian Gao,et al.  Taylor-based nonlocal theory of plasticity , 2001 .