A newly designed triaxial testing machine was used to investigate the ultimate strength of sandstone and crystalline schists under multiaxial stresses. Each specimen used was 5.5 cubic cm in size, and the capacity of the testing machine was 200 tons vertically and 100 tons in each horizontal axis. The experiments were carried out under stress-controlled condition (8tons/min). Rubber sheets with silicon grease were always used to minimize friction between the specimen and piston plates; thus the coefficient of friction could be reduced to 0.002. The unconfined compressive strength of sandstone was found to be σc=1190kg/cm2 with variation of about ±3.9%. For the sake of non-dimensional representation of multiaxial test data, all the results are expressed as the value divided by σc. Introducing the principal stress parameter λ=2σ2-(σ1+σ3)/σ1-σ3(-1≤λ≤1), experimental results of the axial compression test (λ=-1) with high confining stress are in accordance with Mohr-Coulomb's failure criterion, deriving the strength parameters φ0=33°40' and c0/σc=0.46. The influence of the intermediate principal stress can be expressed by the following extended Mohr-Coulomb's criterion: σ1-σ3/σ1+σ3+2c0cotφ0=sinφ0/1-α*/2√1-λ2=sinφλ, provided we choose the parameter α*=0.07. Under low confining stress, however, the experimental results deviate from the above criterion; we cannot regard them as shear failure. The final representation of the test data is the failure curve on the octahedral stress plane as shown in Fig. 8, indicating a slight influence of the intermediate principal stress on the strength of sandstone. With respect to the multiaxial compression test using schists, attention will be called to the main object which consists in investigating the change in the ultimate strength due to the inclination angle of jointed plane and the variation in these characteristics under high confining pressures. As is clear from uni-axial compression tests on cylindrical specimens, the influence of the inclination of jointed plane on strength is maximum for α≈30° (α: angle between maximum principal stress and jointed plane), the decrease in strength being 75-90% of that for α=90°(see Fig. 10). Failure pattern is generally divided into the cleavage failure along the jointed plane, the shear fracture along it and the shear independent of the jointed plane. From the result of triaxial compression tests on the cubic specimens it is known that the decrease in anisotropy of rock strength cannot be expected to be so large; for α=30°-specimens there occures 50% decrease in strength even under the confining pressure as high as 200kg/cm2 (see Fig. 11). From a series of triaxial compression test under the condition of αoct=const, the failure plane of actual rocks with laminated cracks in the principal stress space is determined as shown in Fig. 13.
It is generally considered recently that the mechanical model which describes the rheological behaviour of matter consists of three elements, i.e. spring, dash-pot and slider combined. But there are some complex mechanical behaviours of rocks that the combination of these three elements can not account for such as a cresent shaped hysteresis loop, a stress-strain diagram which opens upwards, the difference between Young's moduli for tensile and compressive stress, etc. In order to account for these characteristics of the mechanical behaviour of rocks, the author presents a concept of new element of the mechanical model which is named as the “stopper”. This new element behaves as a rigid body over or under the designated stress level. Some effects of the properties of this new element are discussed in the present paper, and it is revealed that (1) the “stopper” presents a non-linearity of the mechanical behaviour like the slider, (2) the “stopper” does not present any residual strains, therefore has nothing to do with the hysteresis loss. Finally, using the “stopper” and other conventional elements of mechanical model, the mechanical behaviour of some rocks are described.
It has been suggested that the rheological equation of anisotropic viscoelastic material should be expressed in the form of differential equation, and its differential operators can be determined by consideration of Biot's equation derived on the basis of irreversible thermodynamics. The differential equation is given in two different forms, one of which is applicable to the analysis of relaxation phenomenon, and the other is to the creep analysis. There exist some relations among the differential operators in the two different forms of the equation. The correspondence principle proposed by Lee can be extended to the anisotropic case by introducing into it the suggested rheological equation. The extended correspondence principle has been adopted to making the analysis of the creep strain of orthotropic viscoelastic material under the plane strain condition, and its results may be applied to determining the viscoelastic constants of the material through experimental data. Discussion has also been made on the analysis of a circular tunnel subjected to uniformly distributed internal pressure in an orthotropic viscoelastic medium.
A new triaxial compression technique has made it possible to make studies on laws of fracture and yielding of rocks under general triaxial stress states, in which all the three principal stresses are different. By this new method, the effects of the stress states to fracture and yielding of rocks were experimentally studied. Fracture and flow properties of rocks are markedly affected not only by the least compression σ3, but also by the intermediate compression σ2. The stress states, which produce fracture and yielding, are determined by the following formulas: τoct=f1(σ1+σ3) for fracture, τoct=f2(σ1+σ2+σ3) for yielding, where f1 and f2 are functions of monotonic increase. The failure criteria, corresponding to von Mises' criteria generalized, are physically interpreted as follows: fracture or yielding will occur when the distorsional strain energy reaches a critical value which increases monotonically with the effective mean pressure: (σ1+σ3)/2 for fracture and (σ1+σ2+σ3)/3 for yielding.
The rock-type materials in general show a definite irreversible deformation above a certain stress state. It is a well known fact that under large confining pressure or high temperature they behave as plastic and viscoplastic materials under loading. This type of behavior is called ductile deformation, and may be explained by the dislocation theory. On the other hand, the rock-type materials show another kind of irreversible deformation under relatively low confining pressure or temperature. This may be called brittle irreversible deformation mainly caused by grain boundary separations and crack development in grains. Generally, the ductile deformation is accompanied by no volumetric change, but the volumetric expansion is common in the brittle deformation. It is not easy to construct such constitutive equations of the material as well explain the whole behavior of the material. In this paper, the constitutive equations which describe these two irreversible deformations are derived from the 2nd law of thermodynamics. As initiative step to evaluating the constitutive equations, the direct shear test was conducted by measuring the shear and the normal displacements ux and wz, respectively. By plotting log ux and log wz with respect to the average applied shear stress τ for various normal pressure po, we find four characteristic points. By these characteristic points in the τ vs. po diagram, the various zones of the stress states are defined, elastic, viscoplastic, stable brittle, unstable brittle and failure states. The constitutive equations for ductile deformation may be applicable to the viscoplastic state, while that for brittle one to stable brittle state. It may also be said in conclusion that the ductile yield function does not necessarily depend on the mean stress as the brittle yield function does. Finally, we find that the relation between the strain rate and the applied stress of the viscoplastic component of the material behavior is expressed by the power function φ=(τ0/K0-1)n and with n=2.6 obtained for halite.
In the previous tests performed of cubic specimens of cement mortar, the authors showed that the material would turn anisotropic with increase of the deviatoric stress component, and that the texture of the material underwent changes with increase of its hydrostatic stress component. From these results it is supposed that cement mortar which seems to be macroscopically homogeneous and isotropic will exhibit anisotropy not by the intrinsic property of the material, but by the transition of stress in state, and that the internal change of the texture of the material depends upon the degree of porosity contained in the material. The triaxial compression tests by experiments I and II were therefore performed of cement mortar and marble specimens with considerable difference in porosity in them respectively under various states of stress, in order to investigate the relationship between the textural change and the characteristics of deformation due to the transition of stress from isotropic state to anisotropic state. From the results of these tests the authors found the following facts. (1) The internal texture of the material undergoes changes and resistance against deformation decreases with increase of hydrostatic stress under different levels of stress on mortar and marble. The characteristics of deformation tend also to be anisotropic as deviatoric stress increase. (2) The characteristics of deformation are definitely determined by the state of stress regardless of its path if the given stress of the material is below its yield stress.
The present paper is concerned with studies on initiation and propagation of brittle fracture in rock-like materials under compression. In the first place, theoretical conditions for initiation of fracture are derived from a single open or closed crack inclined arbitrarily to the major principal axis. As the generalization of the Griffith theory the criterion for initiation of fracture from a non-flat elliptical crack is also briefly mentioned. In the second place, the results of experimental studies are reported on the initiation and propagation of fracture in uniaxial and biaxial compression from a single slit contained in a polymethylmethacrylate plate, in flyash cement paste and in neat cement paste, and also on the initiation and propagation of fracture from a single inclusion made with a sheet of vinylchloride or steel in flyash cement paste. In the third place, the initiation and propagation of fracture in uniaxial compression in flyash cement paste with multiple slits and/or inclusions are described. The results indicate that the generalized Griffith criterion is not adequate to describe the initiation of fracture from a single open slit, although it explains that from a single soft inclusion. The modified Griffith criterion may also be applicable to the latter. Cracks were observed to develop even from hard inclusions. In the model containing multiple flaws, crack is initiated on the boundary of the most vulnerably sized and orientated flaw as if it were a single flaw. At the stage of propagation of this crack, a new crack takes place at the point of the less vulnerable flaw. As the load increases, a large number of cracks are initiated in the similar way, and they so formed propagate over a short distance and align themselves with the major principal axis. In the course of the complex process of fracture propagation, unstabilized zones are locally formed as the result of the intermeshed network of cracks. Collapses or macroscopic fractures occur as an integrated instability over the entire specimen. It may be said in conclusion that initiation and propagation of fracture in real rock materials under compression are characterized by the phenomena that are similar to what has been described above.
In this paper the authors report the results of the test of andesite respecting its fatigue failure under pulsating compressive stress. The discussion has been made on cases where fatigue lives have been observed as of wide fluctuation, on the assumption that fatigue failure takes place in stochastic process. Using a graphical method in which the logarithms of probability of survival are plotted against fatigue lives, the authors show that the probability of survival is given by p=ξjexp(-λjN)+ξ2exp(-λ2N) where p: the probability of survival, ξi: the probability that failure of the test piece occurs by the fracture process with the rate constant λi, and ξ1+ξ2=1, N: the number of loading cycles. This result means that fatigue failure in the rock takes place in either of the two disjunctive Poisson's processes of the first order. Of these two processes the one with the smaller rate constant would correspond to the intergrannular fracture of the rock structure, while the other with the greater rate constant would correspond to the intergrannular fracture of the rock structure. Based on these test results, the rate constants λi are evaluated for various mean stresses and stress amplitudes. The effects of the mean stress and stress amplitude on the rate constants are discussed, and expressed as λi=Ai·σmmi·σani where σm: the mean stress, σa: the stress amplitude. Using the method of least squares, the unknown parameters in this equation are evaluated.
The investigation of rock deformation under high pressure and temperature has been made by means of the uniaxial test apparatus at various strain rates ranging from about 10-2sec-1 to 10-6sec-1. The ximum value of confining and axial pressure was 8kb and 20kb respectively. The rocks (Dunite) for the present experiments were collected from Horoman. Hokkaido. The stress-strain curves of Horoman dunite were obtained at temperatures of 900°C, 1000°C and 1100°C under a confining pressure of 8kb and strain rates from 1.2×10-3sec-1 to 6.0×10-7sec-1. Under these conditions the deformation pattern of the dunite showed ductile flow. The microscopic observation showed that the deformation mechanism of dunite was mainly dislocation due to sliding. At the temperatures from 900°C to 1100°C, the kink bands with gliding direction  were observed in olivine. The relation between the strain rate and the differential stress was studied by applying Eyring's equation to the present experimental ranges ε=Dexp(-E/kT)sinh(ατ/kT) (1) where α is the shear activation volume, ε the strain rate, τ the differential stress, E the activation energy, D the constant, k the Boltzman constant, and T the absolute temperature. The full line in Fig. 10 shows equation (1). From the linear relation as shown in Fig. 10 we determined the values of E and α:α=90cm3/mol, E=77kcal/mol and 99kcal/mol at the temperatures of 900°C and 1100°C, respectively. Using these values and the geologic strain rate 3×10-14sec-1, we estimated the values of viscosity coefficient η and sustaining stress τ for the physical conditions of the upper mantle. η thus obtained is in the order of 1014∼1017 poises and τ becomes less than 1b. This suggests the possibility of the mantle convection which may have been the cause of the continental drift.
This paper is the third report following up the first and the second, the first having been published in “Jour. Soc. Materi. Sci. Japan, ” Vol. 14, p. 507, and the second in the same journal, Vol. 17, p. 925. Our experiments were started on Aug. 7, 1957 in a basement laboratory of the Geological and Mineralogical Institute, Kyoto University. Owing to the reconstruction of the Institute buildings, the test-pieces had to be removed (very carefully) on Oct. 14, 1967 to a distantly located laboratory in the Kyoto University compounds, where the experiments have been carried on likewise for nearly 3 years up to date (Aug., 1970). The deflection curve of the test-piece is represented by y=T(t)·X(x), where T(t) is a function of time t that has a reciprocal dimension of Young's modulus E and T(O)=1/E. From T(t) we can derive S(t) which is the sag of the central point. Plotting the values of T(t) against t, it is found that the moving of the test-pieces produces no abrupt change in the general trends of T(t) for both the center-loaded and unloaded beams (the two test-pieces). In the general trend the values of T(t) of both the beams increase very slowly with the time for the whole period of 13 years. This impels us to postulate that granite undergoes viscous flow or plastic flow that has a very small yield stress. It has been already metioned in our second report that the mean curves of T(t) show clear dependence on humidity for the first 5 to 10 years. The range of annual changes in the humidity in the old laboratory was 35% to 85%, but less range of 75% to 95% has been shown in the new laboratory. Accordingly T(t) obtained there does not show so clear dependence on humidity as T(t) did in the old laboratory. In the later part of this paper is given the report of the examination of the correlations between T(t) (and S(t)) and humidity for the first 5 to 10 years. It is found that the correlation coefficients in S(t) for both the beams are nearly equal. In Fig. 6 are shown the graghs of T(t) corrected for humidity. The general trend of the T(t) is almost the same when correction has been made regarding humidity as that before correction, indicating that the correction for change in humidity does not sensibly affect the values of viscosity. In the previous reports it is reported that, from the beginning, minute folds have been formed on the upper surfaces of both the test-pieces. It is confirmed in this paper that the mean amplitude of the fold has been very slowly increasing, as shown by the variation of ε (Fig. 4), for each test-piece during the whole period of 13 years.
In this paper, mechanical behaviors of the rock during the long period of million years are considered on the scale of the earth's crust of Southwest Japan. As the result of neotectonical studies by Huzita (1969), the structural elements of the Quaternary crustal movements named“Rokko Movements”in Southwest Japan are shown schematically in Fig. 3. The anticlinal uplifts and synclinal depressions due to foundation folding (Makiyama, 1956) have developed associated with thrust faults, which run parallel to the folding axes, and are perpendicular to the direction of the Japan Island Arc. Another fault system is represented by conjugate sets of strike-slip faults of NW-SE and NE-SW trends, of which the former is left-lateral and the latter right-lateral. Such fold and fault systems suggest that Southwest Japan has been under the horizontal compressive stress state of E-W trend. On the other hand, the recent studies of Hagiwara (1967) on the Bouguer anomalies in Japan give us valuable informations on the bottom surface of the crust, which may have been deformed isostatically corresponding to the foundation folding of the crustal surface in the Inner Zone of Southwest Japan. On the basis of the geological data in Rokko and Osaka areas, let us assume 1) that“Rokko Movements”began one million years ago and the foundation folding has developed at uniform rate as the first approximation, and 2) that the maximun amount of depressions and uplifts formed during the period of one million years reaches 600m in each. Provided that the thickness of the crust is 40km, the second assumption gives rise to vertical expansions of the crust reaching 7% on the average, from which horizontal contraction en of the crust is derived to be 10%. From the first assumption, we get the strain rate en=3×10-15/sec. Kumagai and Itô have been continuing the experiments extending over 10 years on the flow of granite under small stresses, and they have reached the conclusion that granite undergoes viscous flow or plastic flow with a very small yield stress. In order to explain the above mentioned crustal contraction en=10% based on this conclusion, it seems to be proper to consider that the crust makes viscous flow or plastic flow with very small yield stress. Assuming the viscous flow of the crust, we get the simple relation pn=3ηen, where pn is compressive stress horizontally acting on the crust and η is its viscosity. Using this relation, we can get the viscosity of the crust without consideration of the motion of the mantle. The compressive stress pn may be estimated by using the data of stress releases calculated by the analyses of earthquake faults and those of absolute rock-stress measurements performed in mines. Thus we have been enabled to get the viscosity of the crust to be in the order of 1022 poises.
To contribute to promoting rational design of mine pillars, the stresses in and near the pillars have been analyzed by the finite element method under the assumed condition that one of the principal stresses is horizontally directed along the axes of the pillars while the others may take any direction and be of any magnitude. Model experiments have also been carried out to examine the relationship between the stresses in the pillars and the state of fracture in and near the same pillar if it occurs. From these results of the investigations the following conclusions have been drawn: (1) Sometimes tensile fracture may appear as a single crack either at two corners of an opening or at the central part of its roof and floor according to the original stress state. It seems that the tensile fracture has little connection with the collapse of the pillars. (2) As the earth pressure acting on the pillar increases, compressive fracture may occur first or shortly after the initiation of tensile cracks at the end of the pillar, and this fracture grows larger in scale with increase in the earth pressure. (3) The stress related to the collapse of the pillar may be the high compressive stress occurring at the end of the pillar. (4) For evaluation of the maximum compressive stress, the formula is presented in this paper.
It is believed that rock breakage by blasting is normally completed both by the dynamic breakage under intense stress waves, and by the quasi-static breakage under the pressure of gas explosion. First in this study the stress condition caused by the time dependent pressure in an elastic medium was computed from time to time as it worked on the spherical cavity, and thereby were inferred what were the stresses in the medium that were affected by the variation of the rise time of the pressure in the cavity. It is concluded that the values of the dynamic stresses in the medium caused by the stress wave decrease with increase of the rise time of the pressure, and that the stress condition in the medium is controlled by the quasi-static stress caused by the pressure in the cavity. It is considered therefore that the breakage of rocks caused by the explosion of the explosives having low velocity of defragration is completed chiefly by the quasi-static pressure of the gas explosion. Next the stress condition caused by the quasi-static pressure was analyzed by using the finite element method as it worked on the circular boreholes, assuming the cases in which the boreholes ranged linearly with constant spacing of 10a, where a was the radius of the hole. At first pressure was applied to every other borehole, and the stress condition around the hole where no pressure was applied was carefully observed, and then the pressure was applied to every hole, and probable change in the stress distribution was studied upon the assumed growth of four radial cracks that would be originated from the inner surfaces of the holes, as two cracks grew along the line which connected the centre of each hole, and two other cracks grew along the line perpendicular to the above. These cracks are supposed to have been produced from the inner surface of each hole when the value of the pressure exceeds the tensile strength of the material. It is clarified as the result that the cracks along the line which connects the centre of each hole will grow further while those which grow perpendicular to the above are repressed. By comparing the results of these stress analyses with the strain measured in the material and the pattern of the crack produced actually by the pressure of the gas explosion, it is concluded finally that the quasi-static gas pressure contributes very much to the completion of smooth wall in the smooth blasting operation.