Higher Order Thermo-mechanical Gradient Plasticity Model: Nonproportional Loading with Energetic and Dissipative Components

  • George Z. Voyiadjis
  • Yooseob Song
Reference work entry


In this chapter, two cases of thermodynamic-based higher order gradient plasticity theories are presented and applied to the stretch-surface passivation problem for investigating the material behavior under the nonproportional loading condition. This chapter incorporates the thermal and mechanical responses of microsystems. It addresses phenomena such as size and boundary effects and in particular microscale heat transfer in fast-transient processes. The stored energy of cold work is considered in the development of the recoverable counterpart of the free energy. The main distinction between the two cases is the presence of the dissipative higher order microstress quantities \( {\mathcal{S}}_{ijk}^{\mathrm{dis}} \). Fleck et al. (Soc. A-Math. Phys. 470:2170, 2014, ASME 82:7, 2015) noted that \( {\mathcal{S}}_{ijk}^{\mathrm{dis}} \) always gives rise to the stress jump phenomenon, which causes a controversial dispute in the field of strain gradient plasticity theory with respect to whether it is physically acceptable or not, under the nonproportional loading condition. The finite element solution for the stretch-surface passivation problem is also presented by using the commercial finite element package ABAQUS/standard (User’s Manual (Version 6.12). Dassault Systemes Simulia Corp., Providence, 2012) via the user-subroutine UEL. The model is validated by comparing with three sets of small-scale experiments. The numerical simulation part, which is largely composed of four subparts, is followed. In the first part, the occurrence of the stress jump phenomenon under the stretch-surface passivation condition is introduced in conjunction with the aforementioned three experiments. The second part is carried out in order to clearly show the results to be contrary to each other from the two classes of strain gradient plasticity models. An extensive parametric study is presented in the third part in terms of the effects of the various material parameters on the stress-strain response for the two cases of strain gradient plasticity models, respectively. The evolution of the free energy and dissipation potentials are also investigated at elevated temperatures. In the last part, the two-dimensional simulation is given to examine the gradient and grain boundary effect, the mesh sensitivity of the two-dimensional model, and the stress jump phenomenon. Finally, some significant conclusions are presented.


Higher order gradient plasticity Energetic Dissipative Stress jump Non-proportional loading 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringLouisiana State UniversityBaton RougeUSA

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