Monitoring the change in horizontal stress with multi-wave time-lapse seismic response based on nonlinear elasticity theory

Monitoring the change in horizontal stress from the geophysical data is a tough challenge, and it has a crucial impact on broad practical scenarios which involve reservoir exploration and development, carbon dioxide (CO_(2)) injection and storage, shallow surface prospecting and deep-earth structure description. The change in in-situ stress induced by hydrocarbon production and localized tectonic movements causes the changes in rock mechanic properties (e.g. wave velocities, density and anisotropy) and further causes the changes in seismic amplitudes, phases and travel times. In this study, the nonlinear elasticity theory that regards the rock skeleton (solid phase) and pore fluid as an effective whole is used to characterize the effect of horizontal principal stress on rock overall elastic properties and the stress-dependent anisotropy parameters are therefore formulated. Then the approximate P-wave, SV-wave and SH-wave angle-dependent reflection coefficient equations for the horizontal-stress-induced anisotropic media are proposed. It is shown that, on the different reflectors, the stress-induced relative changes in reflectivities (i.e., relative difference) of elastic parameters (i.e., P- and S-wave velocities and density) are much less than the changes in contrasts of anisotropy parameters. Therefore, the effects of stress change on the reflectivities of three elastic parameters are reasonably neglected to further propose an AVO inversion approach incorporating P-, SH- and SV-wave information to estimate the change in horizontal principal stress from the corresponding time-lapse seismic data. Compared with the existing methods, our method eliminates the need for man-made rock-physical or fitting parameters, providing more stable predictive power. 1D test illustrates that the estimated result from time-lapse P-wave reflection data shows the most reasonable agreement with the real model, while the estimated result from SH-wave reflection data shows the largest bias. 2D test illustrates the feasibility of the proposed inversion method for estimating the change in horizontal stress from P-wave time-lapse seismic data.

Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient,and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model(FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators.

THE GENERAL SOLUTION OF SECOND ORDER EFFECTS IN ELASTIC HALF-SPACE ACTED UPON BY A NON-UNIFORM SHEAR LOAD

This paper is a continuation of [1]. A closed form solution to the second order elasticity problem, when an isotropic compressible elastic half-space undergoes a deformation owing to a non-uniformly distributed shear load, is presented. The method of integral transform is employed to determine the solutions.

THE ILLUSTRATION CALCULATIONS OF SECOND ORDER EFFECTS IN ELASTIC HALF-SPACE ACTED UPON BY A NON-UNIFORM SHEAR LOAD

This paper is a continuation of [1]. An example is discussed in derail to illustrate the second order effects. Numerical calculations for the second order elastic material for the z-direction displacement and the stress t(rz) are carried out. It is found that the second order effect is to reduce z-direction displacement and to decrease t(rz)inside the circle but to increase its value outside the circle.

Dynamic Stress and Pore Pressure Around Circular Cavity in Saturated Porous Elastic Half-space Under Harmonic Plane Dilatational Waves

Dynamic stress concentration and pore pressure concentration around an infinitely long cylindrical cavity of circular cross-section subjected to harmonic plane dilatational waves in fluid-saturated porous elastic half-space were obtained by a complex function method based on potential function and multi-polar coordinate. The steady state Biot’s dynamic field equations of porous elastic solid with a viscous liquid were uncoupled into Helmholtz equations via given potential functions. A circular cavity with large radius is used to replace the straight boundary of the saturated porous elastic half-space. The stresses and pore pressures were obtained by using complex functions in multi-polar coordinates with certain boundary conditions of the solid matrix and the fluid matrix. The approximate solutions were compared to existing numerical solutions. Then the variations of the coefficients of dynamic stress concentration and the pore pressures concentration on boundaries of the cavity were discussed with different parameter conditions. The results of the given numerical example indicate that the method used is useful and efficient to the scattering and dynamic stress concentration of plane dilatational waves in saturated porous elastic half-space.

Completeness of the system of eigenvectors of off-diagonal operator matrices and its applications in elasticity theory

This paper deals with off-diagonal operator matrices and their applications in elasticity theory. Two kinds of completeness of the system of eigenvectors are proven, in terms of those of the compositions of two block operators in the off-diagonal operator matrices. Using these results, the double eigenfunction expansion method for solving upper triangular matrix differential systems is proposed. Moreover, we apply the method to the two-dimensional elasticity problem and the problem of bending of rectangular thin plates on elastic foundation.

Bending of small-scale Timoshenko beams based on the integral/differential nonlocal-micropolar elasticity theory: a finite element approach

A novel size-dependent model is developed herein to study the bending behavior of beam-type micro/nano-structures considering combined effects of nonlocality and micro-rotational degrees of freedom. To accomplish this aim, the micropolar theory is combined with the nonlocal elasticity. To consider the nonlocality, both integral (original) and differential formulations of Eringen’s nonlocal theory are considered. The beams are considered to be Timoshenko-type, and the governing equations are derived in the variational form through Hamilton’s principle. The relations are written in an appropriate matrix-vector representation that can be readily utilized in numerical approaches. A finite element (FE) approach is also proposed for the solution procedure. Parametric studies are conducted to show the simultaneous nonlocal and micropolar effects on the bending response of small-scale beams under different boundary conditions.

ENERGY PRINCIPLES IN THEORY OF ELASTIC MATERIALS WITH VOIDS

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author, various energy principles in theory of elastic materials with voids can be established systematically, In this paper, an important integral relation is given, which can be considered essentially as the generalized pr- inciple of virtual work. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem of work in theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field generalized variational principles, and the principle of minimum potential and complementary energies. Furthermore, with this appro ach, the intrinsic relationship among various principles can be explained clearly.

ON SOME BASIC PRINCIPLES IN DYNAMIC THEORY OF ELASTIC MATERIALS WITH VOIDS

According to the basic idea of dual-complementarity,in a simple and unified way proposed by the author,some basic principles in dynamic theory of elastic materials with voids can be established sys- tematically.In this paper, an important integral relation in terms of convolutions is given,which can be con- sidered as the generalized principle of virtual work in mechanics.Based on this relation,it is possible not on- ly to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of elastic materials with voids,but also to derive systematically the complementary functionals for the eight-field,six-field, four-field and two-field simplified Gurtin-type variational principles.Furthermore,with this approach,the in- trinsic relationship among various principles can be explained clearly.

Dynamic behavior of rectangular crack in three-dimensional orthotropic elastic medium by means of non-local theory

The dynamic behavior of a rectangular crack in a three-dimensional （3D） orthotropic elastic medium is investigated under a harmonic stress wave based on the non-local theory. The two-dimensional （2D） Fourier transform is applied, and the mixed- boundary value problems are converted into three pairs of dual integral equations with the unknown variables being the displacement jumps across the crack surfaces. The effects of the geometric shape of the rectangular crack, the circular frequency of the incident waves, and the lattice parameter of the orthotropic elastic medium on the dynamic stress field near the crack edges are analyzed. The present solution exhibits no stress singularity at the rectangular crack edges, and the dynamic stress field near the rectangular crack edges is finite.

Bolt support mechanism based on elastic theory

Mechanical model of anchorage surrounding rock considering tray effect was established based on elastic theory,in order to study the mechanism of bolt supporting.Elastic solutions of normal force at point in the interior of a semi-infnite solid were obtained by means of classical displacement function method in elasticity.The factors which influence stress of bolted surrounding rock,such as the length of bolt and tray effect,were analyzed.The absolute value of stress along bolt axes decreased rapidly with an increase in radical distance and the maximum appeared near ends of bolt.With increasing radical distance,the value of radical stress changed from positive to negative roughly and then increased to zero,with maximum at the middle of bolt.The evolution of hoop stress as radical distance increasing was similar with stress along bolt axes.With an increase in depth,the radical effect ranges of all normal stress components were reduced.These suggest that the effect from tray on stress along bolt axes of bolted surrounding rock could be neglected,except near surface of surrounding rock.

Theory of nine elastic constants of biaxial nematics

In this paper, a rotational invariant of interaction energy between two biaxial-shaped molecules is assumed and in the mean field approximation, nine elastic constants for simple distortion patterns in biaxial nematics are derived in terms of the thermal average （Dmn^（l）） （Dm＇n＇^（l＇））, where Dmn^（l） is the Wigner rotation matrix. In the lowest order terms, the elastic constants depend on coefficients Γ,Γ＇, λ, order parameters Q0 = Q0（D00^（2）） ＋Q2（D02^（2）＋D0-2^（2）） and Q2 = Q0（D20^（2）） ＋ Q2（D22^（2）＋D2-2^（2））. Here Γ and Γ＇ depend on the function form of molecular interaction energy vj′j″j （τ12） and probability function fk′k″k （τ12）, where r12 is the distance between two molecules, and λ is proportional to temperature. Q0 and Q2 are parameters related to multiple moments of molecules. Comparing these results with those obtained from Landau-de Gennes theory, we have obtained relationships between coefficients, order parameters used in both theories. In the special case of uniaxial nematics, both results are reduced to a degenerate case where K11=K33.

Density-functional theory study of the effect of pressure on the elastic properties of CaB_6

Under high pressure, the long believed single-phase material CaB6 was latterly discovered to have a new phase tI56. Based on the density-functional theory, the pressure effects on the structural and elastic properties of CaB6 are obtained. The calculated bulk, shear, and Young’s moduli of the recently synthesized high pressure phase tI56-CaB6 are larger than those of the low pressure phase. Moreover, the high pressure phase of CaB6 has ductile behaviors, and its ductility increases with the increase of pressure. On the contrary, the calculated results indicate that the low pressure phase of CaB6 is brittle. The calculated Debye temperature indicates that the thermal conductivity of CaB6 is not very good. Furthermore, based on the Christoffel equation, the slowness surface of the acoustic waves is obtained.

THEORY OF ELASTICITY OF AN ANISOTROPIC BODY FOR THE BENDING OF BEAMS

A theory of elasticity for the bending of orthogonal anisotropic beams was developed in this paper by analogy with the special case, which can be obtained by applying the theory of elasticity for bending of transversely isotropic plates to the problems of two dimensions. The authors also presented a method to solve the problems of bending of orthogonal anisotropic beams and a new theory of the deep-beam whose ratio of depth to length is larger. It is pointed out that Reissner's theory which takes into account the effect of transverse shear deformation is not suitable for the components of stress in our case.

Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory

In this paper, the free vibration of magneto- electro-elastic （MEE） nanoplates is investigated based on the nonlocal theory and Kirchhoff plate theory. The MEE nanoplate is assumed as all edges simply supported rectan gular plate subjected to the biaxial force, external electric potential, external magnetic potential, and temperature rise. By using the Hamilton＇s principle, the governing equations and boundary conditions are derived and then solved analytically to obtain the natural frequencies of MEE nanoplates. A parametric study is presented to examine the effect of the nonlocal parameter, thermo-magneto-electro-mechanical loadings and aspect ratio on the vibration characteristics of MEE nanoplates. It is found that the natural frequency is quite sensitive to the mechanical loading, electric loading and magnetic loading, while it is insensitive to the thermal loading.

Book Review “Fabrikant,V.I.Contact and Crack Problems in Linear Theory of Elasticity,Bentham Science Publishers,Sharjah,UAE(2010)”

As suggested by the title, this extensive book is concerned with crack and contact prob- lems in linear elasticity. However, in general, it is intended for a wide audience ranging from engineers to mathematical physicists. Indeed, numerous problems of both academic and tech- nological interest in electro-magnetics, acoustics, solid and fluid dynamics, etc. are actually related to each other and governed by the same mixed boundary value problems from a unified mathematical standpoint

A NEW APPROXIMATE SOLUTION TO THE MULTIPLE SCATTERING THEORY FOR THE ELASTIC PROPERTIES OF HETEROGENEOUS MATERIALS

The multiple scattering theory has been a powerful tool in determining the effective properties of heterogeneous materials. In this paper , a simple relationship between the scattering theory and the micromechanics theory based on the Eshelby principle has been suggested. According to the relationship, a new and simple approximate solution to the exact multiple scattering theory has been given in terms of Eshelby' s S-tensor. The solution easily shows those known results for isotropic composites with spherical inclusions and for tracnsversely isotropic composites, and first gives non-setf-consistent (average t-matrix) and symmetric self-consistent (effective medium or coherent potential) approximate results for isotropic composites with spheroidal inclusions.

EQUIVALENCE OF REFINED THEORY AND DECOMPOSED THEOREM OF AN ELASTIC PLATE

A connection between Cheng's refined theory and Gregory's decomposed theorem is analyzed. The equivalence of the refined theory and the decomposed theorem is given. Using operator matrix determinant of partial differential equation, Cheng gained one equation, and he substituted the sum of the general integrals of three differential equations for the solution of the equation. But he did not prove the rationality of substitute. There, a whole proof for the refined theory from Papkovich?_Neuber solution was given. At first expressions were obtained for all the displacements and stress components in term of the mid_plane displacement and its derivatives. Using Lur'e method and the theorem of appendix, the refined theory was given. At last, using basic mathematic method, the equivalence between Cheng's refined theory and Gregory's decomposed theorem was proved, i.e., Cheng's bi_harmonic equation, shear equation and transcendental equation are equivalent to Gregory's interior state, shear state and Papkovich_Fadle state, respectively.

Influence of Wires Properties on Warp-Knitted Loop Geometry in Plane Based on Theory of Elastic Rod

Based on classic theory of elastic rod,the warp-knitted loop geometry in plane is independent of yarn properties,while there is a certain gap between the geometrical model and the actual fabrics.According to this problem,further analysis of loop geometry is done based on the theory of elastic rod with theoretical calculation and experiments.The theoretical analysis found that the distance between the contacted points at the loop root affected the loop geometry,and the distance was affected by the ratio of bending rigidity and the friction between yarns.The experiments,forming simple loop by taking the yarn as an elastic rod,found that the bending rigidity affected the loop geometry.Then the relationships between warp-knitted loop geometry in plane of metallic fabrics and wires properties were studied.The results show that metallic fabrics are more suitable for the theory of elastic rod;the friction and bending rigidity of wire yarns affect the loop geometry in plane.Also,the elongation of yarn affects the loop geometry in the actual warp-knitted fabric.

THE SCATTERING OF HARMONIC ELASTIC ANTI-PLANE SHEAR WAVES BY A FINITE CRACK IN INFINITELY LONG STRIP USING THE NON-LOCAL THEORY

In this paper, the scattering of harmonic anti-plane shear wavesby a finite crack in infinitely long strip is studied using thenon-local theory. The Fourier transform is applied and a mixedboundary value problem is formulated. Then a set of dual integralequations is solved using the Schmidt method instead of the first orthe second integral equation method. A one-dimensional non-localkernel is used instead of a two-di- mensional one for the anti-planedynamic problem to obtain the stress occurring at the crack tips.Contrary to the classical elasticity solution, it is found that nostress singularity is present at the crack tip. The non-local dynamicelastic solutions yield a finite hoop stress at the crack tip, thusallowing for a fracture criterion based on the maximum dynamic stresshypothesis. The finite hoop stress at the crack tip depends on thecrack length, the width of the strip and the lattice parameters.