Elastic instabilities in dry, mesoporous minerals and their relevance to geological applications

2010 ◽  
Vol 74 (2) ◽  
pp. 341-350 ◽  
Author(s):  
E. K. H. Salje ◽  
J. Koppensteiner ◽  
W. Schranz ◽  
E. Fritsch
Keyword(s):  
Bulk Modulus ◽  
Porous Materials ◽  
Power Law ◽  
Master Curve ◽  
Elastic Moduli ◽  
External Stress ◽  
Effective Elastic Moduli ◽  
Critical Porosity ◽  
Elastic Instabilities ◽  
Linear Decay

AbstractThe collapse of minerals and mineral assemblies under external stress is modelled using a master curve where the stress failure is related to the relative, effective elastic moduli which are in turn related to the porosity of the sample. While a universal description is known not to be possible, we argue that for most porous materials such as shales, silica, cement phases, hydroxyapatite, zircon and also carbonates in corals and agglomerates we can estimate the critical porosity ϕc at which small stresses will lead to the collapse of the sample. For several samples we find ϕc ~0.5 with an almost linear decay of the bulk moduli with porosity at ϕc <0.5. The second scenario involves the persistence of elasticity for porosities until almost 1 whereby the bulk modulus decreases following a power law κ ~ (1–ϕm, m>2, between ϕ = 0.5 and ϕ = 1.

Hysteresis effects studied by numerical simulations: Cyclic loading-unloading of a realistic sand model

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. F13-F20 ◽  
Author(s):  
Xavier García ◽  
Ernesto A. Medina
Keyword(s):  
Cyclic Loading ◽  
Bulk Modulus ◽  
Power Law ◽  
Elastic Moduli ◽  
Effective Medium Theory ◽  
Hysteretic Behavior ◽  
Dynamics Simulation ◽  
Spherical Particles ◽  
Elastic Response ◽  
Power Law Exponent

When Hertz-Mindlin force laws are considered in the context of the effective-medium theory, the predictions yield a constant Poisson coefficient and bulk/shear elastic moduli that scale with pressure with a 1/3 power law exponent [Formula: see text]. This prediction contradicts early and recent experimental findings that conclude moduli grow faster with a 1/2 power law exponent ([Formula: see text]). Such a conclusion is also reached by recent second-order corrections to linear elastic theory. In this work we use a discrete-particle method to study the elastic response of a model of sand that is unconsolidated because of cyclic loading. We use a detailed molecular dynamics simulation that accounts for Hertz-type grain interactions and history-dependent shear forces. The porous sand model is constructed from spherical particles whose size distribution mimics well-sorted unconsolidated sands. The geometry of the model is obtained by simulating critical processes in sedimentary rock formations. Hysteretic behavior and relations between the sample bulk modulus, strain, and stress are obtained. The simulated sample reproduces experimental transient and stationary loading-unloading behavior. We find good correspondence of pressure and strain dependence of elastic moduli in our model with semilinear elasticity theory predictions. Simple arguments explain low coordination numbers observed on force-transmitting samples and the tendency to reduce dissipation under cyclic loading. Our approach clearly shows that a Hertz-Mindlin grain interaction is not inconsistent with the experimental [Formula: see text] behavior of the bulk modulus.

Effect of the Transition Zone on the Bulk Modulus of Concrete

1994 ◽  
Vol 370 ◽  
Author(s):  
Melanie P. Lutz ◽  
Paulo J. M. Monteiro
Keyword(s):  
Bulk Modulus ◽  
Transition Zone ◽  
Cement Paste ◽  
Power Law ◽  
Elastic Moduli ◽  
Volume Fraction ◽  
Closed Form Expression ◽  
Form Expression ◽  
Effective Bulk Modulus ◽  
Model Predictions

AbstractIn concrete, non-uniformities in the hydration process, due to the the “wall effect” produeed by the aggregate (inclusion) particles, lead to an interfacial transition zone (ITZ) that is characterized by an increase in porosity near the inclusions. This increase in porosity may in turn be expected to cause a local decrease in the elastic moduli. We have modeled the effect of the ITZ by assuming that the elastic moduli vary smoothly in the vicinity of the inclusions, according to a power law. The exponent in the power law is chosen based on the estimated thickness of the ITZ. For this model, a closed-form expression can be found for the overall effective bulk modulus. The predicted bulk modulus of the concrete depends on known properties such as the elastic moduli of the bulk cement paste and the inclusions, the volume fraction of the inclusions, as well as on the elastic moduli at the interface. By comparing the model predictions to measured data, we can obtain estimates of the elastic moduli at the interface. Application of this inverse procedure to a set of data from the literature on mortar containing sand inclusions leads to the conclusion that the modulus at the interface is about 15-50% lower than in bulk cement paste.

Modes of Wave Propagation and Dispersion Relations in Inclusion Reinforced Composite Plates

2010 ◽  
Author(s):  
Yu Cheng Liu ◽  
Jin Huang Huang
Keyword(s):  
Composite Plate ◽  
Elastic Moduli ◽  
Lamb Waves ◽  
Plate Thickness ◽  
Composite Plates ◽  
Dispersion Relations ◽  
Elastic Behavior ◽  
Effective Elastic Moduli ◽  
Aspect Ratios ◽  
Reinforced Composite

This paper mainly analyzes the wave dispersion relations and associated modal pattens in the inclusion-reinforced composite plates including the effect of inclusion shapes, inclusion contents, inclusion elastic constants, and plate thickness. The shape of inclusion is modeled as spheroid that enables the composite reinforcement geometrical configurations ranging from sphere to short and continuous fiber. Using the Mori-Tanaka mean-field theory, the effective elastic moduli which are able to elucidate the effect of inclusion’s shape, stiffness, and volume fraction on the composite’s anisotropic elastic behavior can be predicted explicitly. Then, the dispersion relations and the modal patterns of Lamb waves determined from the effective elastic moduli can be obtained by using the dynamic stiffness matrix method. Numerical simulations have been given for the various inclusion types and the resulting dispersions in various wave types on the composite plate. The types (symmetric or antisymmetric) of Lamb waves in an isotropic plate can be classified according to the wave motions about the midplane of the plate. For an orthotropic composite plate, it can also be classified as either symmetric or antisymmetric waves by analyzing the dispersion curves and inspecting the calculated modal patterns. It is also found that the inclusion contents, aspect ratios and plate thickness affect propagation velocities, higher-order mode cutoff frequencies, and modal patterns.

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