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

Rashid K. Abu Al-Rub, George Z. Voyiadjis, Douglas J. Bammann

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85 Scopus citations


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.

Original languageBritish English
Pages (from-to)2888-2923
Number of pages36
JournalInternational Journal of Solids and Structures
Issue number9
StatePublished - 1 May 2007


  • Gradient plasticity
  • Non-local
  • Size effect
  • Thin films


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