Force Moment Tensor Research Articles

Relevance of system size to the steady-state properties of tapped granular systems.

We investigate the steady-state packing fraction ϕ and force moment tensor Σ of quasi-two-dimensional granular columns subjected to tapping. Systems of different height h and width L are considered. We find that ϕ and Σ, which describe the macroscopic state of the system, are insensitive to L for L>50d (with d the grain diameter). However, results for granular columns of different heights cannot be conciliated. This suggests that comparison between results of different laboratories on this type of experiments can be done only for systems of same height. We show that a parameter ɛ=1+(Aω)2/(2gh), with A and ω the amplitude and frequency of the tap and g the acceleration of gravity, can be defined to characterize the tap intensity. This parameter is based on the effective flight of the granular bed, which takes into account the h dependency. When ϕ is plotted as a function of ɛ, the data collapses for systems of different h. However, this parameter alone is unable to determine the steady state to be reached since different Σ can be observed for a given ɛ if different column heights are considered.

Ensemble theory for slightly deformable granular matter.

Given a granular system of slightly deformable particles, it is possible to obtain different static and jammed packings subjected to the same macroscopic constraints. These microstates can be compared in a mathematical space defined by the components of the force-moment tensor (i.e. the product of the equivalent stress by the volume of the Voronoi cell). In order to explain the statistical distributions observed there, an athermal ensemble theory can be used. This work proposes a formalism (based on developments of the original theory of Edwards and collaborators) that considers both the internal and the external constraints of the problem. The former give the density of states of the points of this space, and the latter give their statistical weight. The internal constraints are those caused by the intrinsic features of the system (e.g. size distribution, friction, cohesion). They, together with the force-balance condition, determine which the possible local states of equilibrium of a particle are. Under the principle of equal a priori probabilities, and when no other constraints are imposed, it can be assumed that particles are equally likely to be found in any one of these local states of equilibrium. Then a flat sampling over all these local states turns into a non-uniform distribution in the force-moment space that can be represented with density of states functions. Although these functions can be measured, some of their features are explored in this paper. The external constraints are those macroscopic quantities that define the ensemble and are fixed by the protocol. The force-moment, the volume, the elastic potential energy and the stress are some examples of quantities that can be expressed as functions of the force-moment. The associated ensembles are included in the formalism presented here.

Fluctuations in shear-jammed states: A statistical ensemble approach

Granular matter exists out of thermal equilibrium, i.e. it is athermal. While conventional equilibrium statistical mechanics is not useful for characterizing granular materials, the idea of constructing a statistical ensemble analogous to its equilibrium counterpart to describe static granular matter was proposed by Edwards and Oakshott more than two decades ago. Recent years have seen several implementations of this idea. One of these is the stress ensemble, which is based on properties of the force moment tensor, and applies to frictional and frictionless grains. We demonstrate the full utility of this statistical framework in shear-jammed (SJ) experimental states, a special class of granular solids created by pure shear, which is a strictly non-equilibrium protocol for creating solids. We demonstrate that the stress ensemble provides an excellent quantitative description of fluctuations in experimental SJ states. We show that the stress fluctuations are controlled by a single tensorial quantity: the angoricity of the system, which is a direct analog of the thermodynamic temperature. SJ states exhibit significant correlations in local stresses and are thus inherently different from density-driven, isotropically jammed (IJ) states.

Contact network topology in tapped granular media

We analyze the contact network of simulated two-dimensional granular packings in different states of mechanical equilibrium obtained by tapping. We show that topological descriptors of the contact network allow one to distinguish steady states of the same mean density obtained with different tap intensities. These equal-density states were recently proven to be distinguishable through the mean force moment tensor. In contrast, geometrical descriptors, such as radial distribution functions, bond order parameters, and Voronoi cell distributions, can hardly discriminate among these states. We find that small-order loops of contacts-the polygons of the network-are especially sensitive probes for the contact structure.

Master curves for the stress tensor invariants in stationary states of static granular beds. Implications for the thermodynamic phase space

We prepare static granular beds under gravity in different stationary states by tapping the system with pulsed excitations of controlled amplitude and duration. The macroscopic state---defined by the ensemble of static configurations explored by the system tap after tap---for a given tap intensity and duration is studied in terms of volume, $V$, and force moment tensor, $\Sigma$. In a previous paper [Pugnaloni et al., Phys. Rev. E 82, 050301(R) (2010)], we reported evidence supporting that such macroscopic states cannot be fully described by using only $V$ or $\Sigma$, apart from the number of particles $N$. In this work, we present an analysis of the fluctuations of these variables that indicates that $V$ and $\Sigma$ may be sufficient to define the macroscopic states. Moreover, we show that only one of the invariants of $\Sigma$ is necessary, since each component of $\Sigma$ falls onto a master curve when plotted as a function of $\rm{Tr}(\Sigma)$. This implies that these granular assemblies have a common shape for the stress tensor, even though it does not correspond to the hydrostatic type. Although most results are obtained by molecular dynamics simulations, we present supporting experimental results.Received: 11 May 2011, Accepted: 15 July 2011; Edited by: J-C. Géminard; Reviewed by: B. Tighe, Instituut-Lorentz, Universiteit Leiden, Netherlands; DOI:http://dx.doi.org/10.4279/PIP.030004Cite as: L. A. Pugnaloni, J. Damas, I. Zuriguel, D. Maza, Papers in Physics 3, 030004 (2011)

Towards a relevant set of state variables to describe static granular packings

We analyze, experimentally and numerically, the steady states, obtained by tapping, of a two-dimensional granular layer. Contrary to the usual assumption, we show that the reversible (steady state branch) of the density-acceleration curve is nonmonotonous. Accordingly, steady states with the same mean volume can be reached by tapping the system with very different intensities. Simulations of dissipative frictional disks show that equal volume steady states have different values of the force moment tensor. Additionally, we find that steady states of equal stress can be obtained by changing the duration of the taps; however, these states present distinct mean volumes. These results confirm previous speculations that the volume and the force moment tensor are both needed to describe univocally equilibrium states in static granular assemblies.

Quantitative acoustic emission source characterization of fatigue cracks in a thin-plate of 7075-T6 aluminum: Journal Applied Physics, Vol. 64, No. 11, pp. 6264–6275 (Dec. 1988)

Acoustic emission (AE) signals generated by microcrack formation in front of a propagating fatigue crack in a thin‐plate compact tension specimen of 7075‐T6 aluminum alloy were analyzed to investigate the dynamics of microcrack formation. The dominant wave motion was in‐plane so that the dynamic elastic wave field could be analyzed by generalized plane stress theory. The acoustic signals were detected by an array of four miniature piezoelectric transducers which were mounted on the edge of a specimen ahead of the advancing crack. A second‐rank force moment tensor was used to model the microcrack formation and a time‐domain double‐iterative signal deconvolution method was used to process the detected acoustic signals to remove the effects of the structure and to recover the temporal and spatial characteristics of the AE source. It was found that a force moment tensor model can be used to represent the growth of the microcrack.

Quantitative acoustic emission source characterization of fatigue cracks in a thin-plate of 7075-T6 aluminum

Acoustic emission (AE) signals generated by microcrack formation in front of a propagating fatigue crack in a thin-plate compact tension specimen of 7075-T6 aluminum alloy were analyzed to investigate the dynamics of microcrack formation. The dominant wave motion was in-plane so that the dynamic elastic wave field could be analyzed by generalized plane stress theory. The acoustic signals were detected by an array of four miniature piezoelectric transducers which were mounted on the edge of a specimen ahead of the advancing crack. A second-rank force moment tensor was used to model the microcrack formation and a time-domain double-iterative signal deconvolution method was used to process the detected acoustic signals to remove the effects of the structure and to recover the temporal and spatial characteristics of the AE source. It was found that a force moment tensor model can be used to represent the growth of the microcrack.

Scattering characteristics of elastic waves by an elastic heterogeneity

Elastic wave scattering by a general elastic heterogeneity having slightly different density and elastic constants from the surrounding medium is formulated using the equivalent source method and Born approximation. In the low‐frequency range (Rayleigh scattering) the scattered field by an arbitrary heterogeneity having an arbitrary variation of density and elastic constants can be equated to a radiation field from a point source composed of a unidirectional force proportional to the density contrast between the heterogeneity and the medium, and a force moment tensor proportional to the contrasts of elastic constant. It is also shown that the scattered field can be decomposed into an “impedance‐type” field, which has a main lobe in the backscattering direction and no scattering in the exact forward direction, and a “velocity type” scattered field, which has a main lobe in the forward scattering direction and no scattering in the exact backward direction. For Mie scattering we show that the scattered far field is a product of two factors: (1) elastic Rayleigh scattering of a unit volume, and (2) a scalar wave scattering factor for the parameter variation function of the heterogeneity which we call “volume factor.” For the latter we derive the analytic expressions for a uniform sphere and for a Gaussian heterogeneity. We show the relations between volume factors and the 3-D Fourier transform (or 1-D Fourier transform in the case of spherical symmetry) of the parameter variations of the heterogeneity. The scattering spatial pattern varies depending upon various combinations of density and elastic‐constant perturbations. Some examples of scattering pattern are given to show the general characteristics of the elastic wave scattering.

The determination of source properties by the linear inversion of seismograms

abstract A method has been developed allowing the direct inversion of seismograms to obtain an estimate of the seismic source, as characterized by its equivalent force moment tensor. With reasonable assumptions the problem can be shown to be linear in both the frequency and time domains and least-squares methods are used to estimate the source mechanism. In the synthetic tests that have been run, several possible uses of the method have been investigated. With a satisfactory station distribution the source mechanism can be recovered using complete seismograms from three component stations or several P arrivals on vertical components. The method is useful in distinguishing multiple events including explosions. The errors of the fits are determined in the procedure and yield information such as the bandwidth of useful data in the frequency domain and error estimates for fault-plane solutions in the time domain fits.

Interpreting the seismic glut moments of total degree two or less

The stress glut Γ(x, t) (Backus & Mulcahy 1976a) is a symmetric second-order tensor field which completely describes the mechanical properties of an indigenous seismic source. The equivalent force γ(x,t)=-ΔΓ(x,t) is all that can be determined by observing the motion. The glut moment tensor Λ(m,n)(ζ, r) is the polynomial moment of Λ=ΔtΓ relative to position ζ and time r with spatial degree m and temporal degree n. It affects the seismic motion only through the force moment tensor γ(m+1,n)(ζ,τ), which is obtained by symmetrizing Λ(m,n)f(ζ,τ) on its last m+1 indices. Here we show how to obtain the force moment tensors of low total degree m+ n+ 1 from the long-period motion produced by the source. We show that in faulting with rupture propagation, all the Λ(m,n)(ζ, τ) with m + n = p will usually produce motion of comparable intensity, and that they are all parts of a single tensor Λ(ρ, r) of order p+ 2 over R4, four-dimensional space time. We extend the definitions of centroid and variance tensor to tensor fields and show that the centroid of Λ in R4 gives the position and time of the source centroid. The variance tensor estimates the time duration of anelastic behaviour at the source, the rupture velocity, and the size, shape and orientation of the source region. The hypocentral coordinates, the time duration, and the rupture velocity are always calculable from the glut moments with m+n < 2, m < 1, and these can be measured from the long-period motion. The geometry of the source region requires knowledge of Γ(2,0), only part of which can be calculated from γ(3,0) for a general indigenous seismic source. We illustrate the general theory with four different types of seismic sources, and we find that when all six tensors γ(m+1,n)(ζ, τ) with m + n<2 have been measured, there are three ways to resolve the fault-plane ambiguity, one using long-period information, and the other two using measurements of the final static deformation produced by the source.