Showing papers in "Journal of The Mechanics and Physics of Solids in 1967"

A generalized dynamical theory of thermoelasticity

Abstract: In this work a generalized dynamical theory of thermoelasticity is formulated using a form of the heat transport equation which includes the time needed for acceleration of the heat flow. The theory takes into account the coupling effect between temperature and strain rate, but the resulting coupled equations are both hyperbolic. Thus, the paradox of an infinite velocity of propagation, inherent in the existing coupled theory of thermoelasticity, is eliminated. A solution is obtained using the generalized theory which compares favourably with a known solution obtained using the conventional coupled theory.

Plane strain deformation near a crack tip in a power-law hardening material

Abstract: C rack-tip strain singularities are investigated with the aid of an energy line integral exhibiting path independence for all contours surrounding a crack tip in a two-dimensional deformation field of an elastic material (or elastic/plastic material treated by a deformation theory). It is argued that the product of stress and strain exhibits a singularity varying inversely with distance from the tip in all materials. Corresponding near crack tip stress and strain fields are obtained for the plane straining of an incompressible elastic/plastic material hardening according to a power law. A noteworthy feature of the solution is the rapid rise of triaxial stress concentration above the flow stress with increasing values of the hardening exponent. Results are presented graphically for a range of hardening exponents, and the interpretation of the solution is aided by a discussion of analogous results in the better understood anti-plane strain case.

On the description of anisotropic workhardening

Abstract: A model of workhardening is proposed which generalizes the known rules of isotropic and kinematic workhardening by introducing the concept of a ‘field of workhardening moduli.’ This field is defined by a configuration of surfaces of constant workhardening moduli in the stress space. For any loading history the instantaneous configuration can be determined by calculating the translation and expansion or contraction of all surfaces; the material behaviour can thus be determined for complex loading paths, in particular for cyclic loadings. Several examples for a plane stress state are presented.

The essential structure of constitutive laws for metal composites and polycrystals

Abstract: T his is a rigorous general study of the macro-mechanics of heterogeneous elastoplastic systems. The problem is to uncover those structural features of overall constitutive laws which are unaffected by the heterogeneity, of whatever kind. The outcome is a theoretical framework describing the principal phenomena at pointed vertices on overall yield surfaces. This makes possible a reassessment of the classical rules for yield and flow, and also of recent observational data.

A theory for the deceleration of long rods after impact

Abstract: A modified hydrodynamic theory which takes some account of strength effects is used to predict the deceleration of a long rod after striking a target. The results are then compared with experimental data from X-ray observations.

Elastic moduli of perfectly disordered composite materials

Abstract: Using the concept of correlation functions the overall elastic moduli of perfectly disordered composite materials are derived from the local elastic moduli in a rigorous manner. Explicit results which compare well with experiment are obtained for the shear modulus of aggregates of cubic crystallites. The results are close to those derived by the self-consistent method.

A comparison of the fracture criteria of griffith and barenblatt

Abstract: The tendancy of a crack to extend under applied loads is governed by the cohesive forces acting near the crack tip. The crack-extension criteria of Griffith and Barenblatt take account of the cohesive forces in rather different ways, but are both of the same form. Therefore, if they are to agree, the surface energy T appearing in the Griffith criterion must be related in a definite way to Barenblatt's modulus of cohesion K. It is shown in this paper, from a detailed consideration of the cohesive forces, that the required relationship holds, in an asymptotic sense, if the forces act only over a short range; this is true in practice. Extension of the analysis to uniformly moving cracks then shows that K is a function of velocity, even for a perfectly elastic-brittle body. This has not been noted previously. Finally, the relative advantages of the two formulations are compared.

Tensile properties of fibre-reinforced metals: Fracture mechanics

Abstract: Experiments have been designed to study how a material reinforced with aligned fibres fails at the root of a notch. Ductile and brittle tungsten wires and silica fibres have been introduced into a copper matrix made either by casting or by electrodeposition. A completely notch-insensitive composite can be produced provided splitting is tolerated parallel to the fibres. In these experiments splitting appears to occur in shear. If splitting does not occur then it is shown experimentally for thin sheets that fracture is governed by the established rules of fracture mechanics. The contribution of the matrix to the work of fracture is assessed. An important result is that the work of fracture varies linearly with the fibre diameter and in a copper matrix at room temperature is less than 10 7 ergs. cm −2 for a fibre diameter of 20 μ.

Eigenmodal deformations in elastic/plastic continua

Abstract: The Possibility of incremental deformation under all-round dead loading, independently of creep or thermal effects, is explored from a general standpoint. The analysis applies to any material whose constitutive law in differential form is a homogeneous relation between stress-rate and strain-rate, either linear or nonlinear. Circumstances in which such eigenstates are encountered on quasi-stable and non-bifurcating strain paths are described. The critical conditions and the character of eigenmodal deformations is examined in some detail, with illustrations.

Coupled modes of elastic buckling

Abstract: I t is shown that when the critical loads corresponding to the modes of buckling of a two-degrees-of-freedom system are equal, or nearly so, then elastic post-buckling involves coupling between the modes. The forms taken by these coupled modes are discussed for the general case and for special cases of symmetry; the forms are illustrated by reference to a simple structural model with two degrees of freedom.

Crack propagation in viscoelastic media

Abstract: The problem is solved of a uniformly moving crack in an infinite body, in anti-plane strain, driven by loads which travel with it. The body is taken to be a ‘standard linear solid.’ A force-velocity relation is then derived by applying a Barenblatt type of fracture criterion. This shows that the dissipation tends to stabilize the motion, if the limiting speed of the crack, which cannot be obtained from a linear analysis, is sufficiently high.

Streamlined wire drawing dies of minimum length

Abstract: P revious work has established that ideal frictionless dies can be profiled to give perfect efficiency and strain uniformity in the reduction of ideal perfectly-plastic solids. The design principle for such dies, called die streamlining, requires that the principal stress directions be everywhere tangent to streamlines. This is not sufficient, however, to give a unique profile for each reduction. In the present work streamlined wire drawing profiles are determined which are believed to be the shortest possible for each reduction. These profiles have a generally concave shape with zero entrance angle, but finite exit angle. They are about half as long as the earlier sigmoidal streamlined dies which had zero exit as well as zero entrance angles. The shortest streamlined dies are believed to be advantageous for actual metal reduction since they tend to minimize the actual frictional force, thus approaching most nearly the theoretical frictionless model. The similarity of metal-forming die design and supersonic fluid nozzle design is briefly described.

Plastic instability in rate-dependent materials

Abstract: T he Paper presents a theoretical analysis of the time variation of strain gradients in a tensile specimen of rate-dependent material, the analysis being based on the assumption that the strain rate in the material is a function of the local values of stress and strain. The theory is used to determine the criterion for the growth of strain gradients, and it is shown that for a given material there exists a region of the stress-strain plane defining conditions under which these gradients increase indefinitely with time. The theory is applied to materials with specific types of rate-dependence, and the results are related to experimental data obtained at constant rates of strain.

The plastic constraint of V-notched tension bars

Abstract: Close bounds are obtained to the plastic constraint factor for any symmetrically V-notched tension bar of rigid/plastic material under conditions of plane strain. The estimated factors are well represented by 1 + In {(ω + 1)/2} or 1 + 12 π-α, whichever is the lesser, where 2α is the notch angle and ω is the ratio of bar width to root width. When these expressions are equal the geometry is critical in the sense that for smaller ω or α the plastic flow is not localized around the root section.

A study of the fractures produced in glass blocks by impact

Abstract: This paper discusses the fractures produced by the impact of steel balls on glass plates for a wide range of impact velocities. Four aspects of the problem have been considered. First, the appearance and geometrical form of the fractures were examined as a function of the impact velocity, and it was found that the Hertzian cone fracture grew and penetrated further and further into the glass plates as the velocity of impact was increased. At higher velocities two or more Hertzian cracks were formed and radial fractures also appeared; these eventually fractured the whole specimen. Also, at the highest velocities a powdering of the glass surface was observed immediately under the centre of the impact. This was presumably due to formation of a multitude of tiny cracks, which were then subjected to large compressive forces, which crushed the glass fragments. The second study was concerned with measuring the coefficient of restitution as a function of velocity, when fracture took place. The most remarkable result here was that the formation of small fractures produced no noticeable effect on the rebound height and thus appeared to dissipate very little energy. At higher velocities of impact much more energy was dissipated and the scatter observed in the values of the rebound heights was found to be very large. The third investigation was to record the surface waves produced on glass blocks by impacts of steel balls and observe the effect that fracture has on the stress pulse which propagated out from the impact. These experiments showed very clearly the presence of fractures, when they occurred, as sharp jumps in the output of the strain gauges used to detect the stress waves. A number of interesting results emerged from these experiments; thus when fracture occurred the pulse shape differed when observed in different radial directions. As the velocity of impact increased, the fracture occurred earlier and earlier in the impact process, and the amplitude and number of jumps also increased. For the impacts of 0.5 in. dia. balls the pulse shape did not change appreciably with distance of travel in the region examined, and at radial distances greater than about 1.5 in., the amplitude decreased approximately inversely as the square root of the radial distance of travel. For shorter radial distances the amplitude fell off more rapidly than this. The fourth study was an approximate mathematical treatment of the wave propagation produced by such Hertzian impacts on an elastic half-space. A number of simplifying assumptions had to be made in order to keep the mathematical and numerical analysis tractable. Thus the area of contact was taken to be fixed throughout the impact and the stress distribution in the contact area was assumed to be uniform. The analysis, however, gave surprisingly close agreement with the experimental pulse shapes.

A series-method for constructing plastic slipline fields

Abstract: A power-series method is developed for the numerical computer construction of any rigid/plastic plane-strain slipline field defined by two suitably smooth orthogonal slipline segments. Extensions to other slipline problems are examined and as an example the stress-free surface starting from an initial known slipline is determined. The radii of curvature, cartesian coordinates, tractions and moments acting on line-segments for the field defined by two equal circular arcs are tabulated at 10° intervals of net angle turned through up to 90°.

The ring cracking of glass by spherical indenters

Abstract: T he W eibull statistical treatment of brittle strength is used to compute the mean fracture load and the standard deviation of fracture load for the ring cracking of glass by spherical indenters. An extension of the Weibull treatment allows the mean crack location also to be calculated. Predictions of mean load, standard deviation of load and mean crack location, all as a function of indenter radius, are made from material properties determined in bending tests. Good agreement is found with experiment over a wide range of indenter sizes.

Discontinuous yielding of commercially-pure aluminium

Abstract: T he discontinuous repeated yielding of commercially-pure aluminium under slowly applied dead weight loading is examined as a dynamic instability problem. The destabilizing factor is shown to be the negative slope of the flow stress—strain rate relation which, for aluminium, is a consequence of rapid strain ageing. The essentials of the phenomenon are illustrated by a dry friction model which incorporates a counterpart of strain hardening. More generalized models show the equivalent of progressive yielding over the specimen length and delayed yielding under incremental loading. A stepped stress—strain curve is derived from the experimental flow stress-strain rate relation which agrees well with the observations in dead weight loading tests. Additional experiments on an Instron testing machine were performed with variable machine flexibility and strain gauge recording. These test results support the explanation of the origin of the stepped stress-strain curves.

Boussinesq problems for an anisotropic half-space

Abstract: A fourier transform method is used to solve some mixed boundary value problems of Boussinesq type for a generally anisotropic half-space. When the prescribed normal displacement is any polynomial function, the analytic form of the associated pressure distribution is determined. The parameters contained in it satisfy a set of linear algebraic equations, whose coefficients involve contour integrals which can be evaluated numerically. The problem of indentation by a smooth flat punch is examined in detail.

The influence of strain rate and strain ageing on the flow stress of commercially-pure aluminium

Abstract: T ensile tests on annealed, commercially-pure, aluminium specimens were performed in a hard machine at different temperatures for a number of constant strain rates. The results show that a range of strain, strain rate, and temperature exists for which the flow stress decreases with strain rate. This material property, when coupled with suitable mechanical conditions, is believed to be the cause of discontinuous repeated yielding. Various experiments, especially relaxation and reloading tests, show that the decrease of flow stress with strain rate can be directly attributed to rapid strain ageing due to impurity diffusion. Other experiments indicate the same effect for different loading histories. An analytical representation of the strain rate and strain ageing of the material is developed which predicts results in very good agreement with the experiments.

Ideal forming operations for perfectly plastic solids

Abstract: I t is shown that steady-state forming operations of a general kind can be designed so that in the working zone the streamlines are trajectories of principal stress, either exactly or to a close approximation. Initially transverse planes are then perpetually converted as surfaces orthogonal to the streamlines, and so finish as transverse planes. Advantages of this type of deformation in regard to a low energy consumption are discussed.

On stress intensifications in specimens of charpy geometry prior to general yield

Abstract: Stress intensifications prior to general yield are estimated for notched specimens of Charpy geometry deformed in four-point bending. The results show a less rapid increase of local stress with applied stress than that predicted by other analyses, and the consequences of this conclusion for some aspects of cleavage fracture initiation at a notch are briefly discussed.

A theoretical and experimental investigation of the flaw distribution on glass surfaces

Abstract: T he apparent increase in the tensile strength of glass plates observed when measurements are made by pressing small spherical indenters on the surfaces of the plates is discussed in terms of the statistics of the surface flaw distribution. An analytic flaw distribution function is derived from the experimentally-determined relation between radius of indenter and apparent tensile strength, and this function is used to forecast the scatter of observations about their mean values. It is found that for indenters of large radius a purely statistical explanation of the observation applies, but for indenters with small radii of curvature, other mechanisms come into play. Some measurements have also been carried out to compare the values of the tensile strength found in quasi-static tests with those obtained in impact experiments.

On the nature of the stress at the tip of a perfectly brittle crack

Abstract: A particular distribution of interatomic cohesive forces across the tip of a cleavage crack in a perfectly brittle solid is proposed and the corresponding interatomic force-distance curve is derived by the methods of the continuum theory of elasticity. The detailed shape of this curve is unrealistic but by comparing its bulk properties with those of real materials it is concluded that large cohesive stresses exist over regions of a few atomic radii only.

On the vectorial superposition of Hencky-Prandtl nets

Abstract: In many problems of plane plastic flow the construction of a net of sliplincs, or their traces in the hodograph, is an essential part of the solution. It is shown how any two nets can be combined vectorially to produce a third; especially, every net can be generated from centred fans. As well as reducing computation, the method promotes the solution of boundary-value problems by exposing significant aspecis of the field geometry. The potentialities are illustrated by several examples.

Difference method for wave analysis of the split hopkinson pressure bar with a viscoelastic specimen

Abstract: This paper presents an original method, using finite difference approximations, for describing stress-wave propagation in a split Hopkinson pressure bar with a viscoelastic specimen. The difference scheme for the elastic bar is briefly reviewed, and the schemes for the viscoelastic bar and interface boundaries are derived in detail. An experiment with a split Hopkinson pressure bar is described, and the experimental data obtained are subjected to conventional analysis and to wave analysis by the difference method. A method of obtaining the complex moduli of the viscoelastic specimen by parameter conversion is also demonstrated.

On the theory of plastic wave propagation in a bar

Abstract: A study is presented of the problem of plastic wave propagation in a semi-infinite bar subjected to an axially-applied impact stress. Malvern's theory is applied and it is shown on theoretical grounds that we should expect the strain plateau to appear always and that, for a given static stress-strain curve and magnitude of applied stress, the time needed for the appearance of a given length of plateau depends on a characteristic time τ char of the material. It is shown that the Karman-Taylor theory is the special case for which τ char = 0. Particular emphasis is placed on the influence of strain hardening on the instantaneous strain distribution along the bar. A bilinear static stress-strain curve is considered and the calculations are made by using Malvem's theory, the method of characteristics, and an IBM 7040–7094 digital computer. It is shown that a decrease in strain hardening increases the time needed for a given length of plateau to appear. It is therefore possible, when solving this problem numerically, to miss the plateau if calculations are not carried out for a sufficient duration of time.

Towards a general statistical theory of imperfection-sensitivity in elastic post-buckling

Abstract: T he Paper represents a first step towards the development of a complete general statistical theory of imperfection-sensitivity in elastic post-buckling. A general conservative structural system described by n generalized coordinates, a loading parameter and an imperfection parameter is considered, and is assumed to exhibit either an asymmetric or an unstable-symmetric point of bifurcation. A general statistical distribution is introduced for the magnitude of the imperfection parameter, and asymptotic relationships are derived for the distribution of failure loads in the two points of bifurcation. The distinguishing features are observed, and the relative insensitivity of the results to the local form of the imperfection distribution is noted.

A theory of random fatigue

Abstract: A theory of the fatigue process is given which enables the survival probability under general broad-band random loading to be calculated. In this theory the growth of fatigue damage is regarded as taking place continuously in time, rather than discretely, per cycle. The total damage increase in a time interval is obtained by integrating a damage-rate function over the interval for the particular stress—time curve which has operated. Forming averages over different stress-time functions for the random problem can then be carried out. For a normally-distributed random stress and a particular (realistic) damage-rate function we have calculated the mean damage, its standard deviation, and the survival function. We have also established that this approach gives good results for fixed-level tests.

Experiments on rebound of steel balls from blocks of polymer

Abstract: E xperiments are performed in which steel balls are impacted onto blocks of polymer. The time of contact and height of rebound are recorded as functions of ball size, velocity of impact and temperature of the specimen. These results are compared with results of similar tests as well as those of different methods of dynamic testing.