Uni-axial compaction of a granular material (Q1319066)

The model describing the reversible compression and shear by hypoelastic laws, and the irreversible compaction and shear by evolutionary laws, is discussed. At first, it is reviewed the investigations on the compaction of granular materials under load and under repeated loading including authors' studies. The previous analysis of specific constitutive functions in the evolution laws introduced eighteen different models considering three alternatives, namely, the deviatoric strain, the deviatoric stress, and the deviatoric compaction strain. However, at some simplifying assertions only two of them permit valid initial response and are described by four material functions and two constants. The considered analysis adopts the simpler model using the total deviatoric strain as state measure. Further, it is considered the compaction of a granular material in uni- axial strain which is described by two simultaneous differential equations for the axial stress and compaction with the axial strain as independent variable, together with algebraic relations for the pressure and lateral stress. Following reformulation of the initial equations shows that the pressure derivative depends only on two of the material functions and one constant. Latter are defined by an optimization procedure from the complete load-unload pressure data of a test on dry sand, but the remaining two functions and the second material constant are defined from correlation of the principal stress difference data. To minimize the sums of squares received for each of the defined polynomials, the Polak-Ribiere conjugate gradient search algorithm is used. Then three sets of four model functions which satisfy the initial conditions have been explored. Finally, these correlations are applied to the construction of the unloading curves. Observed results for three sets have an approximate common feature, independent of the prior maximum stress, which is not evident from the governing differential equations. It is shown that there is a common unloading wave speed for all material elements and the motion is governed by a simple homogeneous wave equation. It is noted that the correlations are with data derivatives and not with points, so that accuracy of the model construction must be tested by comparison of the complete loading and unloading curves determined by solutions of the model differential equations with the experimental stress-strain data.