A strengthening model of particle-matrix interaction based on an axisymmetric strain gradient plasticity analysis

Abstract

Precipitation of fine particles into the base material of a metal is a potent strengthening mechanism. This is numerically analyzed within a continuum framework based on a higher order strain gradient plasticity theory and by use of an axi-symmetric unit cell model. The unit cell contains a spherical particle which is resilient to inelastic deformation and embedded in a homogeneous matrix material. An interface with special characteristics, that separates the particle from the matrix, plays a key role for the overall strengthening. Based on a systematic parametric study a closed form relation is deduced and proposed for the increase in the overall yield stress. This relation is limited to materials containing elastic particles with spacing smaller than the material length scale and volume fractions less than 10 %. It these conditions are met, the plastic strain field in the material becomes essentially constant on the scale of particle spacing. The character of the solution suggests that this result is general despite the simplicity of the unit cell employed in the parametric study. Predictions from the closed form relation is compared with published experimental results, and good agreement is observed in some metal matrix composites and alloys. The influence of a mismatch in elastic modulus between particles and matrix is elucidated, and effects on post yield strain hardening are discussed.

Analytic prediction of yield stress and strain hardening in a strain gradient plasticity material reinforced by small elastic particles

“…In the present study a purely energetic formulation will be employed and ḊI is taken to be zero. Regarding the specific form of ψ, Asgharzadeh and Faleskog (2021b) and Faleskog and Gudmundson (2021) find that it is possible to obtain precipitation strengthening in line with existing experimental results using a ψ-function that depends linearly on the current value of effective plastic strain ε p e = 2 3 ε ij ε ij at the interface S I . This suggests that the moment traction can be obtained as…”

“…A convenient and appropriate definition of the proportionality factor is ψ = σ 0 α, when prediction of yield strength is of primary interest, cf. Asgharzadeh and Faleskog (2021b), Faleskog and Gudmundson (2021). Here, α is a non-dimensional parameter in the range [0, 1] that determines the plastic constraint at an interface: a value of zero puts no constraint, whereas a value of one puts full constraint on the plastic strains.…”

“…where the integration over V has been split up into sub-volumes V p and V m . Supported by the numerical studies by Asgharzadeh and Faleskog (2021b) and Asgharzadeh and Faleskog (2021a), the assumption that plastic deformation develops in the whole matrix simultaneously as soon as the condition for initiation of plastic strain is met is successfully employed in Faleskog and Gudmundson (2021). Furthermore, these authors note that the plastic strain field evolving in the matrix is essentially homogeneous provided that a/ and f are sufficiently small.…”

“…Several thermodynamically consistent interface models within the framework of higher order isotropic SGP theories have been proposed over the years, e.g. Fleck and Willis (2009b); Fredriksson and Gudmundson (2005); Asgharzadeh and Faleskog (2021b); ; Voyiadjis and Deliktas (2009).…”

“…Investigations using isotropic SGP theories to study above mentioned metals include Chen and Wang (2002); Liu et al (2003); Zhang et al (2007); Yueguang (2001); Qu et al (2005); Dai et al (1999); Chen and Wang (2002); Zhu et al (1997); Xue et al (2002). In a recent work by Asgharzadeh and Faleskog (2021b), numerical simulations were performed on a metal matrix reinforced with impenetrable particles using the higher order SGP theory proposed by Gudmundson (2004). A contribution to the composite yield stress proportional to the total surface area of the particles was identified and a model was proposed.…”