Showing papers on "Stress field published in 1971"

Fracture mechanics of interfacial cracks

Abstract: This paper contains a two-dimensional analysis of the stress field around a crack on the plane interface between two bonded dissimilar anisotropic elastic half-spaces. This analysis is then combined with the usual local form of the Griffith virtual work argument to give an explicit fracture criterion which involves a suitably defined ‘stress concentration vector’ and the specific surface energy of the bonded surfaces. This criterion has a simple structure and reduces to the conventional form of Irwin when the two half-spaces are isotropic and identical. The analysis is then extended to cracks moving uniformly and a local fracture criterion with the same structure as the static criterion is derived by an energy balance argument. The criterion is specialized to isotropic half-spaces for illustration, when it predicts that the speed of a crack on an interface between such media will be limited by a speed Vc which is slightly greater than the smaller of the two Rayleigh wave speeds. A by-product of the analysis is an expression for the displacement field of an arbitrary interfacial dislocation, either stationary or moving uniformly.

The Hertzian fracture test

Abstract: The crack growth in the non-uniform stress field due to the contact loading of a spherical ball on an elastic half-space is described quantitatively for the case of an isotropic brittle solid. This theoretical understanding provides a basis for the Hertzian test which may be used to measure three important surface properties of strong solids and consequently their strength. These are fracture toughness, surface crack size densities and residual stress. All examples of applications of the test are described in detail for glass but the application to a wider variety of strong materials is implied and gives the test a wider significance.

A review of the three-dimensional stress problem for a cracked plate

Abstract: An analytical study has been carried out to examine the influence of plate thickness on the stress distribution around the crack. The qualitative feature of the three-dimensional solution is first determined by an asymptotic expansion of the stresses and displacements in terms of the cylindrical polar coordinates r, θ, z for small values of r which is referenced from the border of a semi-infinite crack. It is found that the stresses σrr, σθθ, σ zz , and σ rθ are singular of the order r $$r{\text{ }}^{{\text{ - }}\tfrac{1}{2}} $$ , but the transverse shear stresses σ rz and σθz , are bounded for plates under stretching and bending. The intensity of crack-border stress field becomes a function of the thickness coordinate z. Knowing that the problem prohibits any exact analytical solutions of a quantitative nature, the three-dimensional equations of elasticity will be approximated by appealing to minimum principles in the calculus of variations. Guided by the results obtained from the asymptotic expansions, each one of the six stress components is assumed to be the product of two functions, one being assigned to describe the stress distribution in the plane of the plate and the other across the thickness. The z-distribution of the stresses may either be pre-assigned arbitrarily or determined from the plane strain condition ahead of the crack. On the basis of the principle of minimum complementary energy, a system of three simultaneous differential equation in two variables is obtained and solved for the problem of an infinite plate containing a through crack by means of integral representations. Determined in closed elementary form are the detailed structure of the three-dimensional crack-edge stress field. The stress-intensity factor, which varies in the thickness direction, is shown to be a function of the ratio of plate thickness to crack length and is found to increase rapidly in magnitude as the plate thickness is perturbed slightly from zero. The present analysis suggests a method by which the effect of a finite plate thickness can be incorporated into an examination of the fracture toughness of cracked sheet specimens.

Photoelastic determination of stress-intensity factors

Abstract: The increasing number of analytical and numerical solutions for the crack-tip stress-intensity factor has greatly widened the scope of application of linear elastic fracture-mechanics technology. Experimental verification of a particular solution by elastic stress analysis is often a necessary supplement to provide the criteria for proper application to actual design problems. In this paper, it is shown that the photoelastic technique can be used to obtain rather good estimates of the stress-intensity factor for various specimen geometries and loading conditions. Treated are the following cases: wedge-opening load specimen, several notched rotating-disk configurations, and a notched pressure vessel. A sharp crack is simulated by a relatively narrow notch terminating in a root radius of 0.010 in or less. Stress distributions along the section of symmetry ahead of the notch tip are obtained using three-dimensional frozen-stress photoelasticity. The results are used to determine the stress-intensity factor, cK I , by three methods. Two of these are based on Irwin's expressions for the elastic stress field at the tip cf a crack, and the other is a result of Neuber's hyperbolic-notch analysis. Agreement, with available analytical solutions is good.

The stamp-load bearing strength of rock an experimental and theoretical investigation

Abstract: The Stamp-Load Bearing Strength of Rock An Experimental and Theoretical Investigation Though the stamp-load bearing strength of rock is one of the more relevant rock properties in many mining problems it has been neglected in comparison with the efforts devoted to the strength of rock under uniform compression. As stamp-load bearing experiments offer a simple and effective means for studying the strength of rock under triaxial conditions the bearing strength of the rock has been investigated in detail. Using fundamental equations derived by Love and Sneddon the axisymmetric stress field induced by the indentation of a flat-ended circular stamp into a plane rock surface has been studied. In addition to the normal representation of a stress field by means of stress trajectories and lines of equal principal stresses, potential fracture zones have been calculated for a typical Witwatersrand quartzite. The theoretical analysis of the stampload bearing problem resulted in the acquisition of a deeper knowledge of the complexity of the induced stress field and the fracture mechanism. It has also been shown that there exist serious restrictions of the elastic treatment of this problem. In order to get a more detailed understanding of the actual failure mechanism and to establish the effects of the non-linear behaviour of rock near to failure some rock specimens were instrumented with strain gauges and cantilevers to measure surface strains and deflection outside the contact area of the stamp. From these experiments the conclusion can be drawn that lateral expansion of fractural rock is an integral part of the failure of rock under stamp loads. Based on these measurements a qualitative model of the failure process has been developed. The experimental work on the stamp-load bearing strength of rock has resulted in the following findings: 1. The load-displacement curve of the stamp-loading experiments is more sensitive to changes in rock properties than that obtained from compression tests. 2. The stamp-load bearing strength of rocks has been found to be affected by the size of the stamp. The size-dependent behaviour of rock is more obvious on brittle than on ductile rocks. It can be described by an experimental law of the formσ st =Q a a , withα being between 0 and −0.5 for ductile and brittle rocks respectively. At large stamp diameters the bearing strength tends towards a constant value of approximately four times the uniaxial compressive strength of rock. 3. A direct relation between the exponentα and the inelastic part of the lateral expansion of rock at failure has been found. This enables the determination of the lateral expansion of rock from stamp-loading experiments using different stamp sizes.

Chain scission and plastic deformation in the strained crystalline polymer

Abstract: Depending on temperature and load rate an unoriented polymer solid under sufficiently large applied stress either deforms plastically (high temperature, low rate of loading) or undergoes brittle fracture (low temperature, high rate of loading). The fracture proceeds through the areas of minimum strength, i.e., along the boundaries between adjacent spherulites and between parallel lamellae. Very few chains, i.e., interlamella tie molecules and molecules bridging the crack if a crystal lamella is broken, are ruptured during this process. Hence the number of radicals per cm2 of new surface is small, about 1013/cm2. The plastic deformation gradually transforms the sample into the extremely well oriented fiber structure of much higher elastic modulus and strength but of smaller strain to break. The basic element of the fiber structure is the long and narrow microfibril formed by micronecking of the crystal lamella. The microfibril consists of fully oriented folded chain blocks connected by a great many tie molecules obtained by partial chain unfolding in the micronecks. Their number per amorphous layer increases with the draw ratio and so does the tensile strength of the microfibril. Under load the fiber structure breaks at very small strain. The fracture proceeds through the weakest elements of the structure, i.e., through the boundary between adjacent microfibrils and the amorphous layers between subsequent blocks of the microfibrils. With axial stress the fracture must cut at least one full cross section of the fiber and hence rupture about 1014 intrafibrillar tie molecules per cm2. ESR experiments show long before the final break a much higher number of radicals throughout the whole strained sample upto 1017/cm2. The number of broken chains depends on strain and not on stress. One imagines that at every applied strain one breaks the most strained tie molecules which as a consequence of stress concentration have to carry most of the load. The stress concentration caused by the special morphology of the microfibrils and tie molecules reduces the tensile strength of the polymer far below the theoreticall limit calculated for uniform stress field even if one considers that the number of chains passing through the fracture plane is only a fraction of the maximum number calculated with all macromolecules extended and perfectly aligned.

Stress Gradients in Laminated Composite Cylinders

Abstract: In a recent paper [1], (see also [2]) it was shown that extremely severe stress gradients can exist in the wall of a unidirectional, helical-wound cylinder under the common loadings applied in the laboratory, i.e., axial loading, internal pressurization, and torsion. In that study it was found that the stress gradients were drastically reduced in an orthotropic, symmetric laminated cylinder, such as an angle-ply. In fact, for the case of torsion, the shear stress distribution almost coincides with that which occurs in an isotropic cylinder. While an analogous smoothing of the stress field occurs under other loadings, the axial normal stress under axial loading and the hoop stress under internal pressure are somewhat different from the respective isotropic results. For example, refer to Figure 12 in [1]. In order to complete the treatment of the geometric design of tubular characterization specimens, the remaining stress components, e.g., the hoop and shear stresses induced in the axial loading experiment, should be considered. It is therefore the purpose of this note to present some results in this regard for laminated cylinders.

Elastic strain recovery in Proterozoic rocks near Elliot Lake, Ontario

Abstract: Elastic strain recovery in boreholes was measured in underground mines near Elliot Lake, Ontario, Canada. Maximum elastic strain recovery is horizontal and parallel to the local postorogenic joint sets. Close to the mine workings this relationship is disturbed by the stress field induced by mining. The magnitude of the stress tensor increases with depth. The in situ stresses are interpreted as remanent tectonic stresses that were imprinted onto the rock during the Hudsonian orogeny (1700 m.y. ago). Unloading and reorientation of the stresses was achieved by long-lived arching along an easterly trending axis. A parallelism of maximum elastic strain recovery (maximum compressive stress) and axes of regional arching in eastern North America is inferred from available data.

An examination of the effects of some idealized models of fracture on accelerating cracks

Abstract: A new representation , based on the solution of a diffraction problem given by F.G. Friedlander, is presented for the stress field produced by a semi-infinite accelerating crack in a longitudinal shear field. This is employed to give an explicit expression for the field at any point generated by an incident plane stress wave and also to obtain asymptotic expressions for the components of strain and velocity near the crack tip when the crack is loaded in any prescribed manner. The relationship of the Griffith-Irwin energy balance to the Dugdale-Barenblatt model of small-scale yielding is next examined, and it is concluded that the Griffith-Irwin energy balance is acceptable for all but the most rapid loading conditions, provided the yielding is small-scale. Finally, some effects of large-scale yielding are investigated by adopting the Dugdale model, with a plastic zone of finite size, the motion of a crack subjected to a step load being investigated in detail. An important simplifying approximation, which is adequate in this case and likely also to be so in most practical contexts, is noted, which very drastically reduces the complexity of the mathematical analysis. It is concluded that a crack with a finite plastic zone accelerates less rapidly than one for which the yielding is small-scale, although the accelerations predicted are still unrealistically high. It is expected that this defect will be remedied by the future inclusion of rate-dependence into the model.

On variational principles for a non-newtonian fluid

Abstract: For the boundary value problem for creeping flow of a fluid, called "a generalized Newtonian fluid", which is defined by the constitutive relations τij= 2η (IIe, IIIe)eij, the uniqueness of its solution are proved, and two equivalent variational problems are formulated. This fluid contains plastic fluids. The flow field, therefore, may consist of flow regions and stationary regions, or may include surfaces of discontinuous velocity. In stationary regions the stress field is not unique. Present work is a partial extention and development of studies by Hill and Prager.

The effect of diffusivity gradients on diffusion to dislocations

Abstract: The kinetic theory developed by Cottrell and Bilby (1949) and by Ham (1959) to describe the stress-assisted migration of point defects to dislocations is extended to take into account the gradient in diffusivity that exists in the stress field of dislocations. This diffusivity gradient arises from the interaction between the activation volume of diffusion of the point defects and the stress field of the dislocations. The partial differential equation defining the rate of loss of point defects due to diffusion, drift and diffusivity gradient is presented. The results of numerical integrations of this equation for the case of the dislocation acting as a perfect sink are also reported. It is shown that while the diffusivity gradient influences the rate of defect migration to the dislocation it has no effect on the form of the kinetics. In this respect the results are in substantial agreement with those obtained by Ham in showing t2/3 kinetics only at early times. An important qualitative conclusion to emerge...

Interfacial drag and the growth of martensite

Abstract: A new model for the growth of massive-martensite in Fe−Ni−C alloys is presented and predictions from it are compared with the results of hot-stage metallographic experiments on five Fe-10 Ni−C alloys. The comparison involves six steps: 1) Development of a specific model of the interface for a particular crystallography. The single array of parallel dislocations calculated for this interface is compatible in every way with the requirements of the crystallographic theories; 2) calculation of the elastic interaction energy between the strain fields of dissolved carbon atoms in the martensite lattice and the stress field of the dislocation array in the interface; 3) development of diffusion models for the drag force created by this interaction on a moving interface during the formation of the martensite phase; 4) establishment of a balance of forces at the moving interface; 5) prediction of growth rates of the product; and 6) comparison of these predictions with the experimental data. The rate of growth of the product and its strong dependence on carbon concentration can both be explained if it is assumed that the rate controlling mechanism for isothermal growth is the drag caused by the movement of Cotrell atmospheres with the migrating interface. The Zener-Hillert model for growth control by the diffusion of carbon away from the tip of a growing martensite plate is shown to be incorrect in principle because it ignores the geometrical prerequisites of the transformation. In the present model growth is controlled by the sidewise movement of the planar transformation interface, and length growth is a geometrical consequence of this motion. This view is supported by the observation that the activation enthalpies for length and width growth in these alloys are almost identical. Possible improvements to the model are discussed, and their predicted effects on the results are indicated.

Note on the growth of a penny-shaped crack in a general uniform applied stress field

Abstract: Energetic arguments are used to discuss the growth of a penny-shaped crack situated within an infinite solid which is subject to tensile and shear stresses that are respectively normal and parallel to the crack plane. The most favourable growth mode is that for which the circular periphery becomes an ellipse, such that there is no growth perpendicular to the direction of application of the. shear stress; the appropriate growth condition is derived and compared with that obtained by assuming the circular crack to expand uniformly.

Theory of Dislocation Cells. IV. Hybrid Cells

Abstract: Extending the investigations of Papers I–III, infinitely tall ``hybrid cells'', i.e., dislocation cells derivable from mixtures of different orders of multipoles, are considered. A symbolism is developed aiding in the analysis of such cells. It is shown that at different distances the stress fields of hybrid cells are dominated, in turn, by the multipoles composing them. For example, a cell that may be regarded as an assembly of dipoles, tripoles, and hexapoles exhibits a stress field that has dipole nature at the farthest distances, has tripole nature in an intermediate range, and has hexapole nature at the closest distance just beyond the true short‐range stress field of the cell walls. The latter is the same as that of the corresponding infinitely extended boundaries, as was shown in Paper III.

Elastic analysis of an axisymmetric stress field perturbed by a spheroidal inhomogeneity

Abstract: An infinite elastic medium contains an elastic spheroidal inclusion. Both materials are transversely isotropic. Assuming that the stress field in the absence of any inhomogeneity is prescribed, it is desired to calculate the modification caused by the inclusion. This paper presents a general solution to this elasticity problem with the restriction that the prescribed stress field is axisymmetric. The analysis is based upon some new identities in Legendre functions, which are derived in this paper. The solution is in the form of combinations of Legendre functions. An example of a spheroidal cavity in a tension field is given.

Computer-simulated stress-optic patterns

Abstract: A new method for producing theoretical fringe patterns using a small real-time computer interfaced to analog equipment is presented The method applies equally well to isochromatic, isopachic or interferometric stress patterns of any stress field for which the analytical solution is known