A thermo-mechanical continuum theory with internal length for cohesionless granular materials

This article continues Part I. Here the non-equilibrium responses of the constitutive variables t (Cauchy stress tensor), q (heat flux vector), h (equilibrated stress vector), Γ (flux term associated with the internal length ℓ), Π (production term associated with ℓ) and f (equilibrated intrinsic body force) as well as the Helmholtz free energy Ψ are postulated by use of a quasi-linear theory for three of four models deduced in Part I. In so doing, together with the equilibrium responses gained in Part I, a complete set of constitutive equations for the constitutive quantities for each model is obtained. The implemented models are applied to investigate typical isothermal steady granular shearing flows with incompressible grains, namely, simple plane shear flow, inclined gravity-driven flow and vertical channel-flow. The emphasis is on the models in which ℓ is considered a material constant (Model I) and an independent dynamic field quantity (Model III). Numerical results show that Model III is more appropriate than Model I since in the former model the effect of the motion of an individual grain can better be taken into account. Such a result is in particular significant for avalanches, since it verifies the existence of a thin layer immediately above the base of an avalanche, in which the grains are colliding strongly with one another, and provides a quantitative means to measure such a thin layer.     

Anisotropic damage mechanics for viscoelastic ice

We present a formulation of continuum damage in glacier ice that incorporates the induced anisotropy of the damage effects but restricts these formally to orthotropy. Damage is modeled by a symmetric second rank tensor that structurally plays the role of an internal variable. It may be interpreted as a texture measure that quantifies the effective specific areas over which internal stresses can be transmitted. The evolution equation for the damage tensor is motivated in the reference configuration and pushed forward to the present configuration. A spatially objective constitutive form of the evolution equation for the damage tensor is obtained. The rheology of the damaged ice presumes no volume conservation. Its constitutive relations are derived from the free enthalpy and a dissipation potential, and extends the classical isotropic power law by elastic and damage tensor dependent terms. All constitutive relations are in conformity with the second law of thermodynamics.PACS 83.60.Df, 62.20.Mk     

A thermo-mechanical continuum theory with internal length for cohesionless granular materials

A thermodynamically consistent continuum theory for single-phase, single-constituent cohesionless granular materials is presented. The theory is motivated by dimensional inconsistencies of the original Goodman-Cowin theory [1–3]; it is constructed by removing these inconsistencies through the introduction of an internal length ℓ. Four constitutive models are proposed and discussed in which ℓ is (i) a material constant (Model I), (ii) an independent constitutive variable (Model II), (iii) an independent dynamic field quantity (Model III) and (iv) an independent kinematic field quantity (Model IV). Expressions of the constitutive variables emerging in the systems of the balance equations in these four models in thermodynamic equilibrium are deduced by use of a thermodynamic analysis based on the Müller-Liu entropy principle. Comments on the validity of these four models are given and discussed; the results presented in the current study show a more general formulation for the constitutive quantities and can be used as a basis for further continuum-based theoretical investigations on the behaviour of flowing granular materials. Numerical results regarding simple plane shear flows will be discussed and compared in Part II of this work.     

Induced anisotropy in large ice shields: theory and its homogenization

For polycrystalline ice, an isothermal flow law is derived from microscopic considerations concerning constitutive equations and kinematic assumptions. On the basis of an elasto-plastic decomposition of the deformation gradient on the grain level and by assuming a continuous distribution of different orientated grains in the vicinity of each material point the classical macroscopic field quantities are obtained by calculating the weighted mean values of the associated microscopic quantities. The weighting function is represented by a so called Orientation Distribution Function (ODF). For the general two dimensional (plane and rotationally symmetric) flow regime analytical representations of the ODF are derived under the assumption of a uniform stress distribution over all polycrystals (Sachs-Condition) and a plane or rotationally symmetric orientation distribution. Additionally, the influence of the macroscopic constitutive relations on the microscopic level is restricted to isotropic parts only. Simple examples are used to demonstrate the ability of the ODF to perform the evolving texture. The microscopic constitutive relation for the dissipation potential is assumed to be an objective function of the stress deviator and is expressed as a polynomial law up to the power , as proposed by Lliboutry (1993). A second order structure tensor which depends on the ODF is introduced to consider induced anisotropy. The resulting macro fluidities (inverse viscosities) are then calculated from the analytical representation of the ODF for the case of uniaxial loading underlying linear and nonlinear material behaviour. Received March 30, 1998     

Quantitative Sputter Depth Profiling of Silicon- and Aluminium Oxynitride Films

 Silicon- and aluminium oxynitride films have gained attention because of their interesting properties in various fields of technology. The specific properties strongly depend on the concentration of oxygen and nitrogen in the films. For the quantitative analysis of homogeneous silicon- and aluminium oxynitride films, EPMA has been proven a very reliable and precise method of analysis. In order to characterise films with graded composition or interface effects between the film and the substrate it is necessary to use sputter depth profiling techniques such as SIMS, hf-SNMS, AES, or hf-GD-OES. Unfortunately, stoichiometric silicon- and aluminium oxynitride films are insulating and therefore charge compensation has to be applied. For the quantification it was necessary to prepare calibration samples which have been analysed by different bulk analytical techniques such as NRA, RBS and EPMA. With these calibration samples, sensitivity factors have been determined and the functional dependence of the sensitivity factors on the composition has been derived. The advantages and disadvantages of the different sputtering techniques and the applicability of the obtained sensitivity functions for the quantitative depth profiling of silicon- and aluminium oxynitride films are discussed.     

A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi's effective stress principle

Summary The principle of virtual power is used to derive the equilibrium field equations of a porous solid saturated with a fluid, including second density-gradient effects; the intention is the elucidation and extension of the effective stress principle of Terzaghi and Fillunger. In the context of a first density-gradient theory for a saturated solid we interpret the porewater pressure as a Lagrange multiplier in the expression for the deformation energy, assuring that the saturation constraint is verified. We prove that this saturation pressure is distributed among the constituents according to their respective volume fraction (Delesse law) only if they are both true density-preserving. We generalize the Delesse law to the case of compressible constituents. If a material-dependent effective stress contribution is to arise, it is, in general, nonvanishing simultaneously in both the solid and fluid constituents. Moreover, saturation pressure, effective stresses and compressibility constitutive equations determine the exchange volume forces. In a theoretical formulation without non-isotropic strain measures, second density-gradient effects must be incorporated, not only to accommodate for the equilibrium-solid-shear stress and the interaction among neighboring solid-matrix pores, but also to describe internal capillarity effects. The earlier are accounted for by a dependence of the thermodynamic energy upon the density-gradient of the solid, while the latter derives from a mixed density-gradient dependence. Examples illustrate the necessity of these higher gradient effects for properly posed boundary value problems describing the mechanical behaviour of the disturbed rock zone surrounding salt caverns. In particular, we show that solid second-gradient strains allow for the definition of the concept of static permeability, which is distinct from the dynamic Darcy permeability. Received 1 February 1999; accepted for publication 9 March 1999     

Robust automated three-dimensional segmentation of secondary ion mass spectrometry image sets

Three-dimensional (3-D) element distributions generated by scanning secondary ion mass spectrometry (SIMS) are usually noisy and blurred and contain objects which do not usually have sharp edges or may have noise induced boundaries. Additionally, there are local intensity differences due to sensitivity differences of the channelplate. As a result, traditional segmentation techniques become difficult and yield rather poor results. We present a novel methodology which combines a restoration process (using a combination of channelplate sensitivity compensation with a 3-D de-noising technique based on the wavelet transform) with a fuzzy logic 3-D gray level segmentation which can be used to successfully segment 3-D SIMS image sets. The restoration algorithm removes the artifacts produced by the channelplate inhomogeneities as well as noise aberrations from the image sets and the gray level thresholding algorithm segments their features. The algorithm is designed for minimal user interaction to achieve a high automation level. The methodology is discussed and experimental results using real 3-D images are presented.