Plastic Enclave Research Articles

Room-temperature post-yield creep

Post-yield creep of smooth specimens at room temperature in nine different metals was studied. Significant creep was observed in eight of the nine metals tested. The average creep strain from all tests was equal to nearly half of the total strain after one hour at constant load. Creep strains of significant magnitude were also found to occur within plastic enclaves of otherwise elastically deformed notched members. The characteristics of localized creep in the notched members strongly parallel the creep behavior observed in unnotched members. The need to consider time-dependent deformation in experimental mechanics is discussed. Post-yield time-dependent deformation as a cause of failure, as a cause of residual-stress relaxation affecting fatigue life, and as a primary feature of inelastic deformation needing to be modeled, are also discussed.

Orowan mechanism for abrupt localized flow and fracture initiation in metals ahead of an yielding notch

Plane strain elastic-plastic stresses are determined in Mises yielding solid at the root of an yielding crack like notch. This external edge notch is infinitely deep, and has a small finite (fixed) flank angle with a small tip root blunting radius. A boundary value type approach has been followed throughout, to solve this famous Orowan-lrwin problem. Firstly, a fictitious elastic stress field is calculated, considering a misfit in the bulk volume loading; these elastic stress expressions are valid when the notch is fully loaded. Secondly, the plastic stresses are determined in the compressibility gradient, maintaining the continuity of stresses and their derivatives at the yielded-unyielded interface. Our calculations reveal that: Orowan mechanism is fairly dominant below the notch root, as well as on ± 45° planes. It is concluded that the flow-localization in the Mises solid is due to a reverse slip, caused by the sudden release of a favourable critical mismatch stress concentration. Some elastic strain energy density is seen to be getting released from the bulk volume, while unloading the misfit load. The mismatch has been created entirely due to the compressibility-incompressibility difference, as suggested by Orowan. Following Orowan, it is shown here that, before the onset of a stable crack extension, the increase in stress concentration at the notch tip root, is directly proportional to the strength of mismatch strain-localization below the notch, and inversely proportional to the plane strain plastic zone size on the crack extension plane. For a large scale yielding situation, compressive stresses and pure distortion regions are seen to occur at a far field within the plastic enclave.

Yield loci around fiber inclusions

An analysis of the plastic enclaves developed around the end points of rigid fiber inclusions embedded in an elastic plate is undertaken. The plate is subjected to a uniform uniaxial stress at any angle of inclination with the axis of the fiber. It is assumed that the material of the plate is elastic-perfectly plastic and that the deformations are small. A singular solution of the stress field in the vicinity of the end points of the fiber is used in conjunction with the Mises yield criterion to obtain the radius of the plastic zone under conditions of plane strain and generalized plane stress. The dependence of the size and shape of the plastic deformation on the orientation of the fiber, the Poisson's ratio of the plate and the state of stress (plane strain or generalized plane stress) is established.

Equilibrium and kinetic studies of the hydrogen-assisted cracking of steel

With improved apparatus, the equilibrium relationship p ∗ (K) between hydrogen gas pressure and stress-intensity parameter for AISI 4340 steel of 250-ksi yield strength has been measured at room temperature. In addition, the hysteresis effect following unloading has been investigated, and the sonic emissions during cracking with hydrogen have been studied. It is shown that the adsorption model does not fit the experimental p ∗ (K) data. Our decohesion theory, elaborated to incorporate a K-dependence of the effective crack-tip radius, ρ, fits the experimental data and permits the deduction that ρ increases linearly with K. Furthermore, the physical quantities derived from fitting the decohesion model to the p ∗ (K) data are in qualitative agreement with the premises of the decohesion theory, and theory also permits the understanding of the hysteresis due to unloading. The frequency of sonic emissions is found to vary as the first power of the crack velocity and the 6th power of K. It is inferred that the sporadic character of the crack advance is not intrinsic to the action of hydrogen. Cracking in steels proceeds by decohesioning and by plastic tearing. Hydrogen affects directly and importantly only the decohesioning mode at regions directly at the crack tip and at regions of high stress due to dislocation interactions within the plastic enclave.

鋼の水素脆化と鋼中水素の微視的挙動に関する研究(第2報)

The effect of deformation on hydrogenn diffusion has been studied with an electro-chemical flux monitoring technique sensitive to 10-13 atomsH/cm2-s during tensile test and constant load rupture test.Tensile test and constant load test have carried out, over a range of aging time and strain rates, on notched specimens of steel, hydrogen charged and quenched.When specimen was allowed to stand at room temperature under removed stress after deformation, the amount of evolved hydrogen decreased with increasing plastic strain and diffusion coefficient of hydrogen at near by notched part decreased. The result is explained by suggesting that hydrogen is trapped in defects and dislocations in plastic enclave.During tensile test, with increasing plastic strain, the amount of evolved hydrogen from plastic enclave increased than from elastic range in specimen and increased with decreasing deformed velosity, as cross head speed. As aging time increases, evolved hydrogen from near by notch increased and N.T.S. decreased, with more increasing aging time, N.T.S. recovered from embrittlement and that hydrogen decreased. The results are explained by suggesting that hydrogen trapped mainly dislocations is trasported with dislocation to plastic enclave under slower C.H.S. and with more increasing aging time removed out from dislocation. However, under quick C.H.S., trapped hydrogen is broken away from dislocation.On the constant load rupture test, the amount of evolved hydrogen from plastic enclave near by notched part and crack tip increased remarkably right before hydrogen delayed cracking.

The modified Dugdale-Barenblatt model adapted to various fracture configurations in metals

The Dugdale-Barenblatt model is extended to encompass the influence of strain hardening on the plastic enclaves developed at the tips of a crack in a plate subjected to tension at infinity. While in the DB-model the distribution of internal stresses along the plastic zone in the extension of the crack length was taken constant and equal to the yield stress, in the modified version of this model this distribution is taken variable with a minimum value the yield stress (σ0) and a maximum value (σmax depending on the loading step and the amount of strain-hardening of the material. Six different configurations of stress distribution in the plastic enclaves were considered with various values of the ratio σmax/σ0. For each stress configuration and for various loading steps the shape of the respective caustic corresponding to the singularity created at the plastic zone near the crack tip was computed by modifying appropriately Dugdale's theory to each of the six configurations in the plastic enclave. The caustic was formed by reflections of a parallel coherent light beam at the vicinity of the crack-tip.

The kinetics of hydrogen embrittlement: A quantitative diffusion model

Stress assisted diffusion of hydrogen towards the tip of a notch or crack is described by adding a stress-dependent term to Fick's first and second law of diffusion. It is postulated that a critical combination of hydrogen concentration and stress will develop at the edge of the plastic enclave, and that this will produce an incremental crack growing from the edge of the plastic zone towards the notch or crack tip. Crack propagation is treated as a succession of incremental crack bursts. Equations are developed from which the primary and secondary incubation times can be calculated. Approximate solutions are presented for the case of a crack in an infinitely wide flat plate.

Influence of material properties on dynamic fracture toughness of steels

Recent studies of plastic enclave formation at running brittle cracks were extended to account for the influence of crack tip boundary conditions on the temperature at which the enclaves start to develop. The En 2A and three other steels were used in the analysis. It was found that this temperature depends very strongly both on the magnitude and on the distribution of the stresses in the discrete crack tip zone. This suggests that the onset of enclave formation and the rate of their growth are governed by the balance of two sets of material characteristics. The first set consists of at least two parameters describing the microscopic fracture resistance which promotes enclave formation. The second set includes the macroscopic yield and flow properties which may make enclave formation more difficult in higher strength steels. These findings are related to the dynamic or crack arrest fracture toughness which is found to be derived from two different sources. One is connected with the microscopic plastic deformation of the fracturing metal in the crack tip zone and is present at all temperatures. The other is the result of enclave formation, it is present only at higher temperatures and is responsible for the energy transition. In contrast to the case of crack initiation, the dynamic fracture toughness depends not only on the microscopic fracture strength or strain but on the complete stress-displacement relationship of the weakened material which is governed by the microscopic fracture mechanism at the tip of a running crack. It is noted that the present results can be expected to be valid for all steels which fracture in the cleavage or quasi-cleavage modes.

Plastic deformation at a stably growing crack tip

The deformations around the tip of a straight, through-the-thickness crack in sheets of cellulose nitrate, celluloid, in tension have been studied, as well before as after the onset of stable crack growth. A tensile testing apparatus and a device for engraving a fine grid directly on the specimen sheet have been designed. The development of the plastic enclave also after the start of stable crack extension has been studied in connexion with investigations of the deformations in gross fields of size up to 30 × 40 mm2 performed on a digital computer (line element length 1 or 0.5 mm, virgin crack length 80 mm). Unexpected deformations very close to the crack tip were found in the near field where line elements of length 50 µm were analyzed. A modified Dugdale model where the cohesive stresses vary linearly is examined and is applied on polyvinylehloride, PVC, specimen the thickness of which is strongly reduced near the crack tip. A method for measurement of thickness changes of the order of a few µm by tracing isohypses is described. Finally, to enable comparison with the experimental results, the elastic deformation pattern for a square mesh in the plane strain and plane stress case is obtained by a digital computer.

An energy balance criterion for crack growth under fatigue loading from considerations of energy of plastic deformation

An approximate analysis using a bilinear representation of the stress-strain behaviour has been made for the energy of plastic deformation at the tip of a crack growing under sinusoidal loading of constant amplitude. The energy of plastic deformation results from hysteretic and non-hysteretic plastic deformation. It is shown that the energy due to hysteresis is independent of the rate of growth of the crack whereas energy due to non-hysteretic plastic deformation is dependent on growth rate. Work hardening due to hysteretic plastic deformation is not considered in the analysis. The energy balance criterion which is basic to fracture mechanics has been applied to the problem of crack growth under cyclic loading, considering the energy due to hysteretic plastic deformation in the plastic enclave as obtained in the analysis. The equation of energy balance results in an expression for crack growth rate, consistent with the general trends observed in experiments. Some of the merits and limitations of the energy formulation of fatigue crack growth have been discussed.

A model of brittle fracture propagation part I—continuum aspects

The micromechanism of crack propagation in steel is described and analyzed in continuum terms and related to the macroscopic fracture behavior. It is proposed that propagation of cleavage microcracks through favorably oriented grains ahead of the main crack tip is the principal weakening mode in brittle fracture. This easy cleavage process proceeds in the Griffith manner and follows a continuous, multiply connected, nearly planar path with a very irregular front which spreads both forward and laterally and leaves behind disconnected links which span the prospective fracture surface. A discrete crack zone which extends over many grains thus exists at the tip of a running brittle crack. Final separation of the links is preceeded by plastic straining within the crack zone and occurs gradually with the increasing crack opening displacement. It is suggested that in low stress fracture, straining of the links is the only deformation mode. However, it is recognized that under certain conditions plastic enclaves may adjoin the crack zone. This deformation mode is associated with high stress fracture, energy transition and eventually with crack arrest. Energy dissipation resulting from the two deformation mechanisms is related to crack velocity, applied load and temperature and the crack velocity in a given material is expressed as a function of the external conditions. Fracture initiation and crack arrest are then discussed in terms of the conditions which are necessary to maintain the propagation process. Finally, the dimensions of a small scale crack tip zone for a steady state, plane strain crack are evaluated as functions of material properties and the elastic stress intensity factor. The microstructural aspects of brittle fracture will be discussed in a separate Part 2 [1].

Formation of plastic enclaves at running brittle cracks

Onset of enclave formation and their growth at the tip of a steady state crack which propagates under plane strain conditions in a mild steel plate is investigated. It is recognized that the crack tip boundary conditions are determined by the stresses transmitted in the weakened ductile links within a discrete crack zone which forms as a result of selective microcrack propagation in the elastic material ahead of the main crack. A general condition for the beginning of nonuniform viscoplastic yielding is proposed for an arbitrary path in the stress-time space and used for investigation of initial yielding and for an approximate description of enclave growth, while the loading path is derived from a dynamic elasticity solution of the crack problem. The plastic enclaves form initially at some distance from the crack line and may adjoin the crack zone only in later stages of their development. It is suggested that enclave formation and their eventual interference with the local separation process are respectively responsible for energy transition and crack arrest in brittle fracture.

Estimation Of Local Cyclic Stress Ratio Due To Confined Notch Plasticity

Estimation Of Local Cyclic Stress Ratio Due To Confined Notch Plasticity

On the calculation of plastic energy dissipation rate during stable crack growth

An approximate analysis is presented for the calculation of the plastic energy dissipation rate during stable growth of a centrally located through crack. in a sheet subjected to gradually increasing uniaxial tension normal to the crack plane. It is shown that the plastic energy dissipation rate is a function of the slow growth parameter (δωp/δσ)·(dσ/da)+(δωp/δa), where ωp is the plastic enclave width in the plane of the crack and dσ/da is the rate of increase of the gross stress with respect to stable growth. At the point of instability this parameter becomes equal to δωp/δa. By assuming that this parameter is zero at the point of instability, a simple expression is obtained for the plastic energy dissipation rate. The analysis excludes the energy dissipation rate resulting from energy changes in an inner “fracture zone” in the immediate neighborhood of the crack tip in which it is presumed that fracture processes such as vacancy formation, crack initiation by dislocation pile ups etc., are active. The analysis is not applicable in this inner zone as deformation is not homogeneous.

Elasto‐plastic solutions of engineering problems ‘initial stress’, finite element approach

AbstractThe paper presents first a general formulation of the elasto‐plastic matrix for evaluating stress increments from those of stresses for any yield surface with an associated flow rule. A new ‘initial stress’ computational process is proposed which is shown (1) to yield more rapid convergence than alternative approaches (2) to permit large load increments without violating the yield criteria and thus simply to establish lower bound solutions. Several solutions showing stress distribution, strain development and growth of plastic enclaves are given both for the von Mises and for Coulomb (Drucker) type yield surfaces. Load reversal and thermoplastic behaviour are dealt with.

Plastic yield on inclined slip planes at the tip of a crack deformed in anti-plane strain

The characteristics of the plastic enclave at the tip of a crack in a medium subjected to stress are investigated by means of a dislocation model. A model in which dislocations can glide along a plane coplanar with the crack has previously been given by Bilby, Cottrell and Swinden(1), and it is the purpose of this work to extend these authors' results to more general situations. We first of all consider the case of a single slip plane at the crack tip which is inclined to the plane of the crack, and develop an approximation technique for solving the equations of equilibrium of the dislocations. This yields the size of the plastic zone, the density of dislocations and the plastic displacement at the crack tip as explicit functions of the crack-length, applied stress, angle between crack and slip-plane and yield stress and elastic modulus of the medium. These functions become particularly simple for either low or high stresses. A more difficult problem is that in which slip can occur on two different slip-planes at the crack-tip. The same approximation method can be used, but the results are no longer expressible in terms of elementary functions. A computer program has been used to calculate numerical solutions in some cases and in one case to evaluate higher approximations to the solution.

Plastic enclave sizes for internal cracks emanating from circular cavities within elastic plates

The size of the plastic enclave at the tip of a crack within an externally loaded elastic medium is important for the complete understanding of the stress distribution in the cracked medium. One of the first useful approximations of zone size was given by Irwin[1]. Later, Dugdale[2] significantly aided in the estimation of plastic enclave lengths by introducing his elastic-plastic strip model of the enclave region. Experimental results from tests made by him and others [3] with mild steel substantiate the model's use to reduce the analytical complexity of the non-linear plasticity effect in the enclave problem for materials with similar mechanical properties.

The effect of plasticity on the stress-distribution of thin notched plates in tension

An explicit solution of the stress-distribution in notched plates under conditions of plane-stress is given in the elasto-plastic domain of straining. The material is considered as elastic-perfectly plastic and isotropic obeying the Mises yield condition and the Prandtl-Reuss incremental stress-strain relations. Elastic and plastic components of strains are determined by using information from birefringent coatings cemented on the surface of the specimens and observed in normal circularly or linearly polarized light. These measurements are complemented with data from an electrical analogy which yield the ϵ x -strain distribution. Thin fat plates containing either symmetrical semi-circular grooves or deep notches of various flank angles are subjected to a uniform tension. Four cases of flank angles are considered lying between 0 and 120 deg. The minimum width of all plates is taken constant and equal to the half of the width of the plates. The analysis of the state of stress is restricted to the initial part of the loading path from an incipient plastic deformation up to an impending plastic flow. The variation of stress distribution with the evolution of plastic enclaves is studied and the influence of the severity of notches on the mode of spreading of plastic zones is determined.

Elastic-plastic analysis of perforated thin strips of a strain-hardening material

The paper considers the problem of determining the elastic-plastic strain and stress distribution that occurs in thin perforated strips of a strain-hardening material when the applied stress is increased monotonically from the elastic region of loading to values producing an impending plastic flow. The values of elastic and plastic components of strains were determined by using the birefringent coating method complemented with the electrical analogy method which yields the strain distribution normal to the surface of the strips. The components of stresses were evaluated by introducing the usual assumptions of generalized plane stress to a material conforming to the Mises yield condition and the incremental flow-type stress-strain relations. Two cases were considered of thin perforated strips, where the ratio δ of the diameter of the hole d to the width of the strip L was taken equal to one third and one half. The mode of evolution of plastic enclaves was studied and the influence of the ratio of the diameter of the hole to the width of the strain and stress distribution was determined.