Aug 04, 2009 the effect of the free surface on the elastic field of an antiplane crack can be obtained from the solutions for a cracked plate under antiplane shear. For general design, the elastic section modulus is used, applying up to the yield point for most metals and other common materials. Green pointed out that the existence of an elastic strain energy required that of the 36 elastic constants relating the 6 stress. T1 estimation of the elastic properties of fractured rock masses. Elastic modulus is sometimes called youngs modulus after thomas young who published the concept back in 1807. If the material is elastic, the object will return to its initial shape and size when these forces are removed. Also, it is equal to the ratio between the stress and the associated strain in that direction. We develop an effective elasticwave scattering theory to accurately calcula.
We study the modulus of a medium containing a varying density of nonintersecting and intersecting antiplane cracks. The new theory leads to a set of differential equations for the effective elastic moduli which are easily solved. The effective elastic moduli of a solid with spherical inclusions and paralleldistributed pennyshaped cracks are also studied. The geometric model of a cracked body is a spatially periodic medium whose unit cell contains a number of arbitrarily placed aligned circular cracks. The elastic modulus, percolation, and disaggregation. The fields are constructed as expansions in the parameter \\overline na3 \ wich is assumed small n is the crack density, and a is the crack radius. Determination of effective elastic properties of microcracked rocks. Abstract extended solutions are derived, on the basis of the micromechanical methods, for the effective elastic moduli of porous media containing stiff pores and both open and closed cracks. Effective elastic moduli of cracked solid and application.
Effective moduli, nonlinear deformation and strength of a. Equation 15 gives the variation of the effective elastic compliances for any kind of distribution of cracks filled with fluids. The first one, the differential scheme, accounts at best for fluid content, crack interactions and connectivity. Calculations on the basis of the selfconsistent method are made for the elastic moduli of bodies containing randomly. An elastic modulus e can be determined for any solid material and represents a constant ratio of stress and strain a stiffness a material is elastic if it is able to return to its original shape or size immediately after being stretched or squeezed. Quantify the linear elastic stressstrain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. The elastic moduli of a solid permeated with an isotropic distribution of flat cracks have been calculated from the energy of a single crack by use of a selfconsistent approximation. Approximate evaluation for effective elastic moduli of. Overall properties of a cracked solid volume 88 issue 2 j. Many early studies focus on a general solid with a matrix containing inclusions, from which the problem of effective elastic moduli of cracked solids can be.
A stiffer material will have a higher elastic modulus. Accurate modeling of elastic properties of cracked rocks in the earths shallow crust has long been an important topic in the field of geophysics. Elastic modulus in the global x, y, and z directions. Apr 28, 2016 since strain is unitless, all the elastic moduli have units of pressure pascals, pa, and is usually on the order of tens of gpa billions of pascals for rocks. Properties of a cracked elastic solid griffith loci for cracks and external cracks the effective elastic moduli define, in stressstrain space, lines having various slopes, corresponding to equilibrium elastic deformations for a solid containing various densities of cracks and external cracks. The geometric model of a cracked body is a spatially periodic medium whose unit cell contains a number of. A variational formulation has been recently proposed by the authors xu x.
In this framework, the multiscale representations for the shale rocks are presented by introducing the microcrackweakened equivalent solid with hierarchical microstructures, whose transversely isotropic properties are obtained by performing multilevel. The stress state and effective elastic moduli of an isotropic solid containing equally oriented pennyshaped cracks are evaluated accurately. Using this formulation, explicit expressions have been obtained for the upper bounds of the elastic moduli in the case of penny. The former value is in agreement with that obtained for the extended normal hydrocarbons and also in fairly good agreement with the. Wave speeds and attenuation of elastic waves in material. Cuddalorepatta, gidong sim, han li, daniel pantuso, joost j. Residual stressdriven test technique for freestanding. Determination of effective elastic properties of microcracked. A simplified multiscale damage model for the transversely.
N2 the discontinuities in rock masses in the form of joints, fissures and interface separations are crucial to the design of various excavations. Calculations on the basis of the selfconsistent method are made for the elastic moduli of bodies containing randomly distributed flat cracks, with or without fluid. Numerical modeling of wave propagation in a cracked solid. There are two major theoretical approaches in the literature addressing the problem of effective elastic moduli of cracked rocks. Cracked solid, international journal of solids and struc tures, vol. Stress intensity factor and effective stiffness of a solid. A simplified multiscale damage model is proposed for the transversely isotropic shale rocks under tensile loading. Static and dynamic elastic moduli of calcare massiccio mudstonelimestone, the typical seismogenic rock in the italian apennines, are measured using a standard uniaxial static compression test, a dual cantilever forced oscillation test and ultrasonic measurement of elastic wave velocities. Dynamic bulk and shear moduli due to grainscale local. It distills a vast quantity of background theory and laboratory results into a series of concise chapters that provide practical solutions to problems in geophysical data interpretation. Explicit expressions, correct to lowest order in the ratio of the crack size to a wavelength, are derived here for the overall elastic parameters and the overall wave speeds and attenuation of elastic waves in cracked materials where the mean crack is circular, and the cracks are either aligned or randomly orientated. It is clearly shown that the tension and compression elastic asymmetry can result in acoustic nonlinearity.
Module 3 constitutive equations learning objectives understand basic stressstrain response of engineering materials. The modulus of nonintersecting, strongly interacting, 2d antiplane cracks obeys a meanfield theory for which the mean field on a crack inserted in a random ensemble is the applied stress. It is established that the effects of inclusions and microcracks on overall moduli are approximately decoupled for stiff inclusions, which are in. Effective elastic moduli of cracked solid and application to. Effective moduli, nonlinear deformation and strength of a cracked elastic solid. A missing ingredient of the previous theory of oconnell and budiansky is the correct accounting of crack interaction energy. The discrepancy may due to the difference of the tested materials and the experimental condition for different measurements. The effect of microcracks on the elastic moduli of brittle materials.
A sphereequivalency approach of elastic wave scattering was used to model the elastic moduli of an isotropic solid containing aligned cracks. Analytical formulas of the overall bulk and shear moduli are obtained as functions of the elastic moduli of the solid skeleton, porosity and the densities. These results are obtained by calculating the elastic parameters at 12 equally spaced pressures identified by the dots in figure figure1. Utilizing the principles of linear elastic fracture mechanics lefm, the effective elastic moduli, the stability, and the strength of a solid containing a random distribution of interacting cracks is calculated. It is defined as the ratio of tensile stress to tensile strain. An elastic modulus also known as modulus of elasticity is a quantity that measures an object or substances resistance to being deformed elastically i. General concepts are outlined for arbitrary cracks and explicit derivations together with numerical results are given for.
Jan 23, 2019 the three elastic moduli of nacl and mgo under hydrostatic conditions in relevant pressure ranges are shown in figure figure1, 1, while the associated pressure derivatives are shown in figure figure2. The rock physics handbook addresses the relationships between geophysical observations and the underlying physical properties of rocks. For a solid containing a crack distribution with mirror symmetry, the effective elastic constants. Green pointed out that the existence of an elastic strain energy required that of the 36 elastic constants. The nonuniqueness of the elastic moduli, and, especially, poissons ratio in. The modulus of elasticity formula is simply stress divided by strain. In their model, solid grains and soft pores are assumed to make up a modified solid phase, while stiff pores are assumed to occupy the main pore space in which fluid pressure is. Pdf effective elastic moduli in solids with high crack density. Section modulus equations and calculators common shapes. Study of effective elastic moduli of cracked solid. Elastic modulus applied stress displacement field elastic modulo dynamical. When microcracks exist at the interface, the tensile and compressive effective moduli of the cracked interface are considered to be different.
Calculations on the basis of the selfconsistent method are made for the elastic moduli of bodies containing randomly distributed flat cracks, with or without fluid in. Elastic moduli gpa clay content volume compressional modulus shear modulus b. Explicit bounds on elastic moduli of solids containing. The controlling parameters are the crack density, the crack aspect ratio, the fluid compressibility and the solid grain matrix elastic properties youngs modulus and poisson ratio. Elastic moduli of a cracked solid international journal of. The elastic modulus of an object is defined as the slope of its stressstrain curve in the elastic deformation region. International journal of solids and structures issue date. A sphereequivalency approach for calculating the elastic. The calculated values for polyethylene and polytetrafluoroethylene are 3. Dynamic bulk and shear moduli due to grainscale local fluid. Dispersion and anisotropy of elastic waves in cracked rocks. Abstract a selfconsistent theory for the determination of elastic moduli of cracked solids is presented, and worked out for an isotropic distribution of cracks.
Furthermore, they exhibit similar trend with these published data. There are various elastic moduli, such as youngs modulus, the shear modulus, and the bulk modulus, all of which are measures of the inherent elastic properties of a material as a resistance to deformation under an applied load. Hoenig,international journal of solids and structures 15 1979 7154. Estimation of the elastic properties of fractured rock masses. Stress is applied to force per unit area, and strain is proportional change in length. A selfconsistent mechanics method for solids containing inclusions. Seismic shear wave anisotropy in cracked rocks and an.
Shear modulus, \\mu\ or \g\, a measure of shear elasticity. The influence of ellipsoidal inclusions and elliptic cracks on the overall effective moduli of a twophase composite and of a cracked body, respectively, is investigated by means of moritanakas theory for three types of inclusion and four types of crack arrangements. The elastic properties of cracked rocks depend on a number of facts. International journal of solids and structures 12, 8197. Elastic moduli of a cracked solid international journal. Abstract computations of effective elastic moduli of porouscracked rocks have been performed by using three distinct theories.
Residual stressdriven test technique for freestanding ultrathin films. Anisotropic effective moduli of microcrack damaged media. Youngs modulus e describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis. Extended analytical solutions for effective elastic moduli. Elastic behavior and residual strain volume 34 issue 20 gayatri k. Pdf we investigate the weakening of elastic materials through randomly distributed circles and cracks numerically and compare the results. Aug 12, 2008 in this paper, effective moduli of cracked solid material were investigated. Subsurface rocks often contain cracksfractures with various orientations.
For a linear elastic material, the elastic modulus in a certain direction is defined as the stress value in that direction that causes a unit strain in the same direction. Aug 01, 2012 a variational formulation has been recently proposed by the authors xu x. We find bounds on the effective elastic moduli of cracked materials in terms of the effective conductivity of such media. The modulus of elasticity, also known as youngs modulus, is a material property and a measure of its stiffness under compression or tension. Shearwave anisotropy in cracked rocks and its application. Dispersion and anisotropy of elastic waves in cracked. Elastic anisotropy due to aligned cracks in porous rock1. The three elastic moduli of nacl and mgo under hydrostatic conditions in relevant pressure ranges are shown in figure figure1, 1, while the associated pressure derivatives are shown in figure figure2.
In this paper, effective moduli of cracked solid material were investigated. Transducer and bond phase shifts in ultrasonics and their effects on measured pressure derivatives of elastic moduli, g. Study of effective elastic moduli of cracked solid pressure. Solid objects will deform when adequate forces are applied to them. An analytical approach is discussed for a cracked solid containing randomly oriented inclusions by using elastic potential and a standard tensorial basis.
From the fields, a stressstrain relationship is calculated for the body, and effective elastic moduli are derived. The displacement of the walls of a crack of size 2 l c cutting a plate fig. Calculations on the basis of the selfconsistent method are made for the elastic moduli of bodies containing randomly distributed flat cracks, with or without fluid in their interiors. The effective elastic moduli of microcracked composite. Elastic moduli of heterogeneous solids with ellipsoidal. The various moduli apply to different kinds of deformation. The elastic moduli and corresponding attenuation are overall comparable with the values mentioned in the literature, especially for youngs modulus. A method of calculating elastic moduli of simple helical polymers on the basis of the urey. Extended analytical solutions for effective elastic moduli of. Differences between static and dynamic elastic moduli of a. Cracks, crack density, effective elastic moduli, fatigue damage. This paper derives a novel analytical solution for acoustic nonlinearity evaluation of the cracked interface.
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