A gradient-enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non-linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non-uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher-order stress tensor are derived based on an extension of the Hill-Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher-order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects.

http://www.mate.tue.nl/mate/pdfs/2091.pdf

Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme

Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy

It is shown that the phenomenon of strength dependence on size, for otherwise geometrically similar specimens, may be interpreted on the basis of gradient elasticity and gradient plasticity arguments. This is illustrated by adopting a simple strength of materials approach for considering torsion and bending of solid bars and subsequent comparison with available experimental data. Solutions of boundary value problems based on gradient elasticity and gradient plasticity including those of fracture and shear banding can also be used to interpret size effects in more complex situations. This is not discussed here, however, in order to maintain simplicity and clarity when illustrating the theoretical predictions in comparison with the experimental trends.

Strain gradient interpretation of size effects

A conventional theory of mechanism-based strain gradient plasticity

Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments

Constitutive modeling for the particle size effect on the strength of particulate-reinforced metal matrix composites is investigated. The approach is based on a gradient-dependent theory of plasticity that incorporates strain gradients into the expression of the flow stress of matrix materials, and a finite unit cell technique that is used to calculate the overall flow properties of composites. It is shown that the strain gradient term introduces a spatial length scale in the constitutive equations for composites, and the dependence of the flow stress on the particle size/spacing can be obtained. Moreover, a nondimensional analysis along with the numerical result yields an explicit relation for the strain gradient coefficient in terms of particle size, strain, and yield stress. Typical results for aluminum matrix composites with ellipsoidal particles are calculated and compare well with data measured experimentally.

Strain gradients and continuum modeling of size effect in metal matrix composites

The instability of biaxial stretching leading to the development of shear bands in viscoplastic materials with both isotropic hardening and kinematic hardening is examined The main issues are to evaluate the effect of anisotropy and inertia on the onset of instability and to examine the development and evolution of an inhomogeneous deformation structure, using both a linear analysis and a nonlinear finite element analysis It is shown that the back stress and plastic spin, modeling deformation induced anisotropy and texture development, have a significant influence on the critical state at which the first instability occurs The corotational rate of the back stress and its relation to the spin of the substructure produces apparent softening, causing instability at positive strain hardening (in the absence of back stress and corotational rates instability occurs at zero hardening) When decreasing the magnitude of the spin of the substructure by varying the plastic spin parameter the critical value of the hardening parameter decreases towards zero, illustrating again the softening effect of the spin It is also shown that at the onset of instability, the linear analysis predicts the development of a periodic structure of square cells, implying that although the deformation has become inhomogeneous itmore » is far from being localized The nonlinear finite element analysis also predicts the formation of a square cell pattern at the onset of instability This pattern however evolves into localized shear bands far into the post localization regime as shown by the nonlinear analysis 34 refs, 9 figs« less

H. T. Zhu

Papers

Strain gradients and continuum modeling of size effect in metal matrix composites

On the Effect of Anisotropy and Inertia on Shear Banding: Instability of Biaxial Stretching