George Exadaktylos

Gradient elasticity with surface energy: Mode-I crack problem

Abstract The solution of the mode-I crack problem is given by using an anisotropic strain-gradient elasticity theory with surface energy, extending previous results by Vardoulakis and co-workers, as well as by Aifantis and co-workers. The solution of the problem is derived by applying the Fourier transform technique and the theory of dual integral and Fredholm integral equations. Asymptotic analysis of the solution close to the tip gives a cusping crack with zero slope of the crack displacement at the crack tip. Cusping of the crack tips is caused by the action of “cohesive” double forces behind and very close to the tips, that tend to bring the two opposite crack lips in close contact. Consideration of Griffiths energy balance approach leads to the formulation of a fracture criterion that predicts a linear dependence of the specific fracture surface energy on increment of crack propagation for such crack length increments that are comparable with the characteristic size of materials microstructure. This important theoretical result agrees with experimental measurements of the fracture energy dissipation rate during fracturing of polycrystalline, polyphase materials such as rocks and ceramics. The potential of the theory to interpret the size effect, i.e. the dependence of fracture toughness of the material on the size of the crack, is also presented. Also, the theory predicts an inverse first power relation between the tensile strength and the size of the pre-existing crack which is in accordance with experimental evidence. Furthermore, it is shown that the effect of the volumetric strain-gradient term is to shield the applied loads leading to crack stiffening, hence the theory captures the commonly observed phenomenon of high-effective fracture energies of rocks and ceramics; the effect of the surface strain-energy term is to amplify the applied loads leading to crack compliance and essentially captures the development of the “process zone” or microcracking zone around the main crack in a brittle material. Thus, the present anisotropic gradient elasticity theory with surface energy provides an effective tool for understanding phenomenologically main crack-microdefect interaction phenomena in brittle materials.

Microstructure in linear elasticity and scale effects: a reconsideration of basic rock mechanics and rock fracture mechanics

Abstract An account on the role of higher order strain gradients in the mechanical behavior of elastic-perfectly brittle materials, such as rocks, is given that is based on a special grade-2 elasticity theory with surface energy as this was originated by Casal and Mindlin and further elaborated by the authors. The fundamental idea behind the theory is that the effect of the granular and polycrystalline nature of geomaterials (i.e. their microstructural features) on their macroscopic response may be modeled through the concept of volume cohesion forces, as well as surface forces rather than through intractable statistical mechanics concepts of the Boltzmann type. It is shown that the important phenomena of the localization of deformation in macroscopically homogeneous rocks under uniform tractions and of dependence of rock behavior on the specimens dimensions, commonly known as size or scale effect, can be interpreted by using this ‘non-local’, higher order theory. These effects are demonstrated for the cases of the unidirectional tension test, and for the small circular hole under uniform internal pressure commonly known as the inflation test. The latter configuration can be taken as a first order approximation of the indentation test that is frequently used for the laboratory or in situ characterization of geomaterials. In addition, it is shown that the solution of the three basic crack deformation modes leads to cusping of the crack tips that is caused by the action of ‘cohesive’ double forces behind and very close to the tips, that tend to bring the two opposite crack lips in close contact, and further, it is demonstrated that the fracture toughness depends on the size of the crack, and thus it is not a fundamental property of the material. This latter outcome agrees with experimental results which indicate that materials with smaller cracks are more resistant to fracture than those with larger cracks.

A closed-form elastic solution for stresses and displacements around tunnels

Abstract A closed-form plane strain solution is presented for stresses and displacements around tunnels based on the complex potential functions and conformal mapping representation. The tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed geomaterial. Further, the tunnel is considered to be deep enough such that the stress distribution before the excavation is homogeneous. Needless to say that tunnels of semi-circular or “D” cross-section, double-arch cross-section, or tunnels with arched roof and parabolic floor, have a great number of applications in soil/rock underground engineering practice. For the specific type of semi-circular tunnel the distribution of stresses and displacements around the tunnel periphery predicted by the analytical model are compared with those of the FLAC 2D numerical model, as well as, with Kirschs “circular” solution. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of the tunnel.

A semi-analytical elastic stress–displacement solution for notched circular openings in rocks

Abstract A semi-analytical plane elasticity solution of the circular hole with diametrically opposite notches in a homogeneous and isotropic geomaterial is presented. This solution is based on: (i) the evaluation of the conformal mapping function of a hole of prescribed shape by an appropriate numerical scheme and (ii) the closed-form solutions of the Kolosov–Muskhelishvili complex potentials. For the particular case of circular notches––which resemble to the circular cavity breakout in rocks––it is demonstrated that numerical results pertaining to boundary stresses and displacements predicted by the finite differences model FLAC 2D , as well as previous analytical results referring to the stress-concentration-factor, are in agreement with analytical results. It is also illustrated that the solution may be easily applied to non-rounded diametrically opposite notch geometries, such as “dog-eared” breakouts by properly selecting the respective conformal mapping function via the methodology presented herein. By employing a stress-mean-value brittle failure criterion that takes into account the stress-gradient effect in the vicinity of the curved surfaces in rock as well as the present semi-analytical solution, it is found that a notched hole, e.g. borehole or tunnel breakout, may exhibit stable propagation. The practical significance of the proposed solution lies in the fact that it can be used as a quick-solver for back-analysis of borehole breakout images obtained in situ via a televiewer for the estimation of the orientation and magnitude of in situ stresses and of strain–stress measurements in laboratory tests.

Applications of an explicit solution for the transversely isotropic circular disc compressed diametrically

Abstract This paper focuses on the Brazilian test configuration of anisotropic rocks. The proposed methodology follows that of Pinto (Proceedings of the Fourth ISRM Cong, Montreux, 1979); however, it is more accurate since we adopt Amadeis method of analysis (Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 33 (1996) 293) that accounts for the influence of anisotropy on the stress distribution. In a first step, explicit expressions for stresses and strains for the anisotropic circular disc compressed diametrically are presented by employing Lekhnitskiis formalism. It is shown that the proposed analytical solution can be effectively used as a “back-analysis” tool for the characterization of rock elasticity and strength properties. This is illustrated both in the case of published experimental results on schist and gneiss rock types, and finally in the case of specially designed Brazilian tests on Dionysos marble.

Influence of nonlinearity and double elasticity on flexure of rock beams — II. Characterization of Dionysos marble

Abstract A technical bending theory of beams accounting for nonlinearity due to damage and bimodularity of brittle rocks was proposed in Part I. In order to check the validity of the above theory, a series of three-point bending (3PB) tests has been carried out using Dionysos marble beams that have been sampled from the same extracted block. Although the modeling of the 3PB test is considerably more complicated than that of the four-point bending test, the experimental procedure in the former test is simpler than in the later test and most importantly, the location of the fracture is better controlled in the 3PB test. Herein, it is demonstrated that the test results have very good repeatability and they support the above technical theory. The bending tests also indicate that Dionysos marble is characterized by different elastic modulus in compression ( E c ) and in tension ( E t ) at small loads, such that the relation m = E c / E t ≅0.8 holds true. This relationship of elastic moduli for this type of marble is also supported independently by uniaxial compression and direct tension tests on test specimens cored from the same marble block. A plausible physical explanation for this type of marble anisotropy has yet to be made. This observed difference cannot be explained by considering the rock simply as a material with cracks. It may be attributed to pure micromechanical reasons such as the complex microstructure of this type of rock, characterized by a complex previous loading history (metamorphism). Until such an explanation is available, the apparent behavior can be used in analyzing the stress–strain behavior of rocks. Further, the 3PB experiments indicate that fracture of marble starts always at the bottom fiber of the middle cross-section of the beam and the failure extension strain is the same with that occurring in the direct tension test. This last result is due to the fact that the central section of marble beam is almost under extensional strain, which in turn is caused by the combination of the concentrated load and Poisson’s effects. The damage parameter that enters the direct tension stress–strain law was obtained independently from longitudinal strain measurements at the outermost compression and extension fibers, as well as, from bending curvature and deflection measurements. This value of the damage factor is in accordance with the damage measured from the direct tension tests. It is also demonstrated that a linear Timoshenko-type theory containing an intrinsic length scale is able to approximate the nonlinear deflection behavior of Dionysos marble beams. Finally, based on a suggestion by Ludwig Prandtl, the stress–strain relationships in unconfined compression and direct tension, as well as Poisson’s ratio, of Dionysos marble were derived from bending tests.

Surface instability in gradient elasticity with surface energy

Abstract Biots theory of plane strain surface instability of an isotropic elastic body under initial stress in finite strain is extended to include higher order strain-gradients. Higher order straingradients are properly introduced in the definition of the strain energy density, leading to an anisotropic gradient elasticity theory with surface energy. Accordingly, the present theory includes two material lengths characterizing the volume strain energy and the surface energy of the elastic body. The consideration of these two material lengths leads to the occurrence of a boundary layer. This in turn, gives rise to interesting phenomena related to the stability of the half-space, i.e. extra surface instability modes, thin skin effects and significant weakening of the half-space. It is also shown that the appearance of surface instability is associated with the vanishing velocity of propagation of Rayleigh waves. Furthermore, results derived in the context of the present theory on the dependence of the critical buckling stress of the layer on the thickness, suggest that it can be used effectively for the homogenization of elastic bodies containing periodic arrays of collinear Griffith cracks.

On the constraints and relations of elastic constants of transversely isotropic geomaterials

In the present work two basic aspects of the theory of anisotropic elasticity are studied, namely the estimation of the lower and upper bounds in the variation of elastic moduli of transversely isotropic rocks and the possibility of the existence of phenomenological (empirical) relations among the elastic moduli of orthotropic or transversely isotropic rocks that may be derived experimentally. If such relationships exist then the inversion of laboratory and field measurements pertaining to the characterization of the deformability of orthotropic or transversely isotropic rocks in engineering applications is greatly simplified. These two aspects are investigated here for the plane stress and plane strain configurations by recourse to a special formulation of anisotropic elasticity theory and experimental evidence. Finally, several examples of application of the proposed formulation are given and it is illustrated as to how the hydrostatic and deviatoric concepts of the isotropic elasticity and plasticity theories are effectively generalized for the study of failure of anisotropic geomaterials.

Influence of nonlinearity and double elasticity on flexure of rock beams — I. Technical theory

Abstract As a rule, solids display nonlinearity during loading in the relation between strains and stresses. Deviations from Hooke’s linear constitutive law were also registered in the range of initial, small loads both in uniaxial compression and tension of crystalline rocks. Nonlinearity of strain in rocks is manifested primarily in the stress dependency of tangent or secant elasticity modulus and Poisson’s ratio and is caused by closure, initiation, propagation and linkup of pre-existing and new microcracks, frictional sliding along cracks, growth of dislocations, etc. Many experimenters and standardization procedures assume that the dependence of the strain on the applied stress is linear and for practical calculations only two elasticity constants are used: the tangent or secant elasticity modulus at 50% of the failure load in compression and Poisson’s ratio at the same stress level. Apart from nonlinearity many rock types and concretes have quite different stress–strain relations in tension and compression. Yet direct tensile testing is seldom performed because of its many inherent difficulties. Such unrecognized double elasticity and nonlinearity of rocks can invalidate a stress analysis, and in addition, produce a meaningless overestimate (or underestimate) of tensile strength based upon the modulus of rupture derived from a bending test. In Part I of the present study, it is shown that both double elasticity and nonlinearity have a profound effect on flexural strength of rocks as predicted by application of fundamental continuum damage mechanics relations and an appropriate technical theory. The proposed theory is validated in Part II of this work, in which an appropriate back-analysis procedure is suggested for the characterization of the mechanical properties of Dionysos marble in the uniaxial tension and compression regime from properly designed three-point bending tests.

Rayleigh wave propagation in intact and damaged geomaterials

Abstract An analytical model of an elastically deforming geomaterial with microstructure and damage is assumed to be a material where the second spatial gradients of strain are included in the constitutive equations. Based on this assumption, a linear second gradient (or grade-2) elasticity theory is employed, to investigate the propagation of surface waves in either intact or weathered—–although homogeneous and isotropic at the macroscale—materials with microstructure such as soils, rocks and rock-like materials. First, it is illustrated that in contrast to classical (grade-1) elasticity theory, the proposed higher-order elasticity theory yields dispersive Rayleigh waves, as it is also predicted by the atomic theory of lattices (discrete particle theory), as well as by viscoleasticity theory. Most importantly, it is demonstrated that the theory: (a) is in agreement with in situ non-destructive measurements pertaining to velocity dispersion of Rayleigh waves in monumental stones, and (b) it may be used for back analysis of the test data for the quantitative characterization of degree of surface cohesion or damage of Pendelikon marble of the Parthenon monument of Athens.