Showing papers in "Journal of The Mechanics and Physics of Solids in 1995"

Thermomechanical aspects of NiTi

Abstract: The uniaxial behavior of a nearly equiatomic NiTi alloy is studied experimentally. Experiments are conducted in a temperature and deformation regime in which the alloy exhibits the shape memory effect and pseudoelasticity. These characteristics are due to the displacive nature of transformation between the two major phases of the material, austenite and martensite, and to the fact that in this alloy martensite accommodates deformation by twinning. A series of uniaxial experiments is conducted on NiTi wire at temperatures in the range of approximately −20 to 100 °C where the fundamental material response changes drastically. In addition, the loading rate and the choice of ambient medium were found to have a significant influence on the recorded stress-strain responses due to a complex interaction between the inherent mechanical properties of the material and the prevailing heat transfer conditions of the experiment. It is demonstrated that local measurements of strain and temperature can help clarify events that take place at different stages of a typical loading-unloading history. These local measurements are used to track the movement of the interfaces between phases during stress-induced phase transformations at different loading rates.

The effect of spatial distribution on the effective behavior of composite materials and cracked media

Abstract: Estimates of the Hashin-Shtrikman type are developed for the overall moduli of composites consisting of a matrix containing one or more populations of inclusions, when the spatial correlations of inclusion locations take particular “ellipsoidal” forms. Inclusion shapes can be selected independently of the shapes adopted for the spatial correlations. The formulae that result are completely explicit and easy to use. They are likely to be useful, in particular, for composites that have undergone a prior macroscopically uniform large deformation. To the extent that the statistics that are assumed may not be realized exactly, the formulae provide approximations. Since, however, they are derived as variational approximations for composites with some explicit statistics that are realizable, they are free from some of the drawbacks of competitor approximations such as that of Mori and Tanaka (1973 Acta Metall. 21, 571–574), which can generate tensors of effective moduli which fail to satisfy a necessary symmetry requirement. The new formulae are also the only ones known that take explicit account, at least approximately, of inclusion shape and spatial distribution independently.

Ductile crack growth-I. A numerical study using computational cells with microstructurally-based length scales

Abstract: Many metals which fail by a void growth mechanism display a macroscopically planar fracture process zone of one or two void spacings in thickness characterized by intense plastic flow in the ligaments between the voids; outside this region, the voids exhibit little or no growth. To model this process a material layer containing a pre-existing population of similar sized voids is assumed. The thickness of the layer, D, can be identified with the mean spacing between the voids. This layer is represented by an aggregate of computational cells of linear dimension D. Each cell contains a single void of some initial volume. The Gurson constitutive relation for dilatant plasticity describes the hole growth in a cell resulting in material softening and, ultimately, loss of stress carrying capacity. The collection of cells softened by hole growth constitutes the fracture process zone of length l1. Two fracture mechanism regimes can be identified corresponding to l1 ≈ D and l1 ⪢ D. The connection between these mechanisms and fracture resistance is discussed. Finite element calculations have been carried out to determine crack growth resistance curves for plane strain, mode I crack growth under small scale yielding. A row of voided cells is placed on the symmetry plane ahead of the initial crack. These cell elements are embedded within a conventional elastic-plastic continuum. Under increasing load, the voids in the cells grow and coalesce to form a new crack surface thereby advancing the crack. Resistance curves are calculated for crack growth exceeding many multiples of D. The parameters affecting fracture resistance are discussed emphasizing the roles of microstructural parameters and continuum properties of the material. The effect of crack tip constraint on fracture resistance is examined under small scale yielding by way of the T-stress. As a final application, resistance curves for a deep and a shallow crack bend bar are computed. These are compared with experimental data.

Mechanics of the hydrogendashdislocationdashimpurity interactions-I. Increasing shear modulus

Abstract: The effect of hydrogen on dislocationdashdislocation and dislocationdashimpurity atom interactions is studied under conditions where hydrogen is in equilibrium with local stresses and in systems where hydrogen increases the shear modulus. In the case of two edge dislocations (plane strain) the effect of hydrogen is modeled by a continuous distribution of dilatation lines whose strength depends on the local hydrogen concentration. The hydrogen distribution in the atmospheres is adjusted to minimize the energy of the system as the dislocations approach each other. The iterative finite element analysis used to calculate the hydrogen distribution accounts for the stress relaxation associated with the hydrogen induced volume and the elastic moduli changes due to hydrogen. The interactions between the dislocations are calculated accounting for all the stress fields due to dislocations and hydrogen atmospheres. Modeling of the hydrogen effects on the edge dislocationdashinterstitial solute atom interaction and on the screw dislocationdashinterstitial solute atom interaction is discussed using a finite element analysis and the atom interaction energies are calculated in the presence of hydrogen. For the case where hydrogen increases the shear modulus, a significant hydrogendashrelated decrease of the edge dislocationdashinterstitial solute atom interaction energy was observed when the edge dislocationdashsolute distance is approximately less than two Burgers vectors. Depending on the orientation of the tetragonal axis of the interstitial solute distortion field, hydrogen may strengthen or weaken the interaction between the screw dislocationdashinterstitial solute.

Origin of crack tip instabilities

Abstract: This paper demonstrates that rapid fracture of ideal brittle lattices naturally involves phenomena long seen in experiment, but which have been hard to understand from a continuum point of view. These idealized models do not mimic realistic microstructure, but can be solved exactly and understood completely. First it is shown that constant velocity crack solutions do not exist at all for a range of velocities starting at zero and ranging up to about one quarter of the shear wave speed. Next it is shown that above this speed cracks are by and large linearly stable, but that at sufficiently high velocity they become unstable with respect to a nonlinear microcracking instability. The way this instability works itself out is related to the scenario known as intermittency, and the basic time scale which governs it is the inverse of the amount of dissipation in the model. Finally, we compare the theoretical framework with some new experiments in Plexiglas, and show that all qualitative features of the theory are mirrored in our experimental results.

A computational approach to ductile crack growth under large scale yielding conditions

Abstract: Mode I crack initiation and growth under plane strain conditions in tough metals is computed using an elastic-plastic continuum model which accounts for void growth and coalescence ahead of the crack tip. The material parameters are the Young’s modulus, yield stress and strain hardening exponent of the metal, along with the parameters characterizing the spacing and volume fraction of voids in material elements lying in the plane of the crack. For a given set of these parameters and a specific specimen, or component, subject to a specific loading, relationships among load, load-line displacement and crack advance can be computed with no restrictions on the extent of plastic deformation. Similarly, there is no limit on crack advance, except that it must take place on the symmetry plane ahead of the initial crack. Suitably defined measures of crack tip loading intensity, such as those based on the J-integral, can also be computed, thereby directly generating crack growth resistance curves. In this paper, the model is applied to five specimen geometries which are known to give rise to significantly different crack tip constraints and crack growth resistance behaviors. Computed results are compared with sets of experimental data for two tough steels for four of the specimen types. Details of the load, displacement and crack growth histories are accurately reproduced, even when extensive crack growth takes place under conditions of fully plastic yielding.

Self-healing slip pulse on a frictional surface

Abstract: Guided by seismic observations of short-duration radiated pulses in earthquake ruptures, Heaton (1990) has postulated a mechanism for the frictional sliding of two identical elastic solids that consists in the subsonic propagation of a self-healing slip velocity pulse of finite duration along the interface. The same type of pulse may be conjectured for inhomogeneous slip along sufficiently large, and compliant, technological surfaces. We analyze such pulses, first as steady traveling waves which move at constant speed, and without alteration of shape, on the interface between joined elastic half-spaces, and later as transient disturbances along such an interface, arising as slip rupture propagates spontaneously from an over-stressed nucleation site. The study is conducted in the framework of antiplane elastodynamics; normal stress is uniform and alteration of it is not considered. We show that not all constitutive models allow for steady traveling wave pulses: the static friction threshold subsequent to the relocking of the fault must increase with time. That is, such solutions do not exist for pure velocity-dependent constitutive models, in which the stress-resisting slip on the ruptured surface is a continuously decreasing function of the instantaneous sliding rate (but not of its previous history or of other measures of the evolving state of the surface). Further, even for constitutive models that include both the rate- and state-dependence of friction, such as the laboratory-based constitutive models for friction as developed by Dieterich (1979, 1981) and Ruina (1983), steady pulse solutions do not exist for versions, like one discussed by Ruina (1983), which do not allow (rapid) restrengthening in truly stationary contact. For a particular class of rate- and state-dependent laws which includes such restrengthening, we establish parameter ranges for which steady pulse solutions exist, and use a numerical method stabilized by a Tikhonov-style regularization to construct the solutions. The numerical method used for the transient analysis adopts Fourier series representations for the spatial dependence of stress and slip along the interface, with the (time-dependent) coefficients in those Fourier series being related to one another in a way which obtains from exact solution to the equations of elastodynamics. This allows an efficient numerical method, based on use of the Fast Fourier Transform in each time step, with the frictional constitutive law enforced at the FFT sample points along the interface. Solutions based on a law that includes restrengthening in stationary contact show that spontaneous rupture propagation will occur either in the self-healing slip pulse mode (but not generally as a steady pulse) or in the classical enlarging-crack mode, depending on the values of parameters which enter the constitutive law. This analysis suggests that the strictly steady, traveling wave pulse solutions may either be unstable or have a limited basin of attraction.

An algorithm for computing the effective linear elastic properties of heterogeneous materials: Three-dimensional results for composites with equal phase poisson ratios

Abstract: An algorithm based on finite elements applied to digital images is described for computing the linear elastic properties of heterogeneous materials. As an example of the algorithm, and for their own intrinsic interest, the effective Poisson's ratios of two-phase random isotropic composites are investigated numerically and via effective medium theory, in two and three dimensions. For the specific case where both phases have the same Poisson's ratio (ν1 = ν 2), it is found that there exists a critical value ν ∗ , such that when ν 1 = ν 2 > ν ∗ the composite Poisson's ratio ν always decreases and is bounded below by ν ∗ when the two phases are mixed. If ν 1 = ν 2 ∗ , the value of ν always increases and is bounded above by ν ∗ when the two phases are mixed. In d dimensions, the value of ν ∗ is predicted to be 1 (2d − 1) using effective medium theory and scaling arguments. Numerical results are presented in two and three dimensions that support this picture, which is believed to be largely independent of microstructural details.

On the cold compaction of powders

Abstract: Constitutive models are developed for stage I cold compaction of powders under general loading. Densification is assumed to occur by plastic deformation at the isolated contacts between particles. The shape of the yield surface is found to be sensitive to the cohesive strength between particles and to be less sensitive to the degree of inter-particle friction. An internal state variable model is used to describe the evolution of anisotropy under general loading. The theory assumes that the distribution of contacts between particles can be approximated by a second order tensor B; a prescription is given for updating B as deformation proceeds. The predicted compaction behaviour for a state of uniaxial strain is in good agreement with experimental observations reported in the literature.

An analysis of fully plastic Brinell indentation

Abstract: Indentation of a hard sphere into inelastic solids, Brinell indentation, is examined theoretically and numerically by aid of classical plastic flow theory. With the main interest focused on fully plastic behaviour at indentation the mechanical analysis is carried out for power-law hardening rigid-plastic materials where self-similarity features play a dominant role. It is explained in detail how the problem of a moving contact boundary may be reduced to a stationary one by an appropriate transformation of field variables. Within this setting classical empirical findings by Meyer (1908) and O'Neill (1944) are established on a rigorous theoretical ground. In particular, it is shown to advantage also for nonlinear materials how intermediate solutions for a flat die may by cumulative superposition generate solutions for a class of curved indenters. In the case of perfect plasticity it turns out in the present context that indentation hardness is independent of die profiles. For hardening solids when the material behaviour is history dependent, reduction to a stationary geometry is achieved also by expressing the accumulated strain by cumulative superposition. The intermediate flat die problem is then solved for a variety of hardening exponents by a finite element procedure designed to account for material incompressibility. With finite element computations as a basis desired solutions are obtained by straightforward numerical superposition procedures. Detailed results are then given for bulk quantities such as the mean contact pressure as well as relevant field variables. The influence of hardening characteristics on sinking-in and piling-up of indented surfaces and contact pressure distributions are discussed in the light of earlier findings based on deformation theory of plasticity and available discriminating experiments. Correlation is particularly sought with the celebrated universal hardness parameters proposed by Tabor (1951) and the existence of representative strain measures. Attention is also given to the elastic-plastic transition region of Brinell indentation in search for loading levels sufficiently high that the results tend to an asymptotic fully plastic state. A standard finite element technique employing contact elements for a moving boundary is used to analyse with tolerable accuracy the influence of elasticity and more elaborate hardening behaviour. Some relevant features are shown for a sequence of solutions from elastic Hertzian to fully plastic behaviour.

A spectral method for three-dimensional elastodynamic fracture problems

Abstract: We present a numerical formulation for three-dimensional elastodynamic problems of fracture on planar cracks and faults. Stress and displacement components are given a spectral representation as finite Fourier series in space coordinates parallel to the fracture plane. The formulation is based on an exact representation, involving a convolution integral for each Fourier mode, of the elastodynamic relation existing between the time-dependent Fourier coefficients for the tractions acting on the fracture plane and for the resulting displacement discontinuities. A wide range of constitutive models can be used to relate the local value of the strength on the fracture plane with the displacement and velocity history. Efficiency of the code is achieved by using an explicit time integration scheme and by computing the conversion between the spatial and spectral distributions through a FFT algorithm. The method is particularly suited to implementation on massively parallel computers; a CM-5 was used in this work. The stability and precision of the formulation are discussed for tensile (mode 1) situations in a detailed modal analysis, and numerical results are compared with existing three-dimensional elastodynamic solutions. The adequacy of the method to investigate various three-dimensional dynamic fracture problems involving non-propagating and propagating tensile cracks is illustrated, including crack growth along a plane of heterogeneous fracture toughness.

A three dimensional analytical solution of the longitudinal wave propagation in an infinite linear viscoelastic cylindrical bar. Application to experimental techniques

Abstract: This paper presents an original three dimensional (3D) analytical general solution of the longitudinal wave propagation in an infinite linear viscoelastic cylindrical bar and its applications to some experimental methods of material behaviour testing to improve their accuracy. One application is to take into account the wave dispersion effects in the split Hopkinson pressure bar (SHPB) setup composed of viscoelastic bars. Another is to eliminate the geometrical effects in an impulse test in which the linear viscoelastic material properties can be deduced from the change in the wave shape due to the propagation between two points of measurement in a specimen bar.

Orientation dependence of the pseudoelastic behavior of single crystals of CuAlNi in tension

Abstract: Uniaxial tension experiments were performed on single crystals of Cu-13.95 wt% Al-3.93 wt% Ni. Three specimens were prepared with tension axes in directions that were chosen based on Schmid law calculations using the 96 possible Austenite-Martensite (AM) interface orientations in this alloy. Specimen number one was chosen to have a tensile axis of [2.43,1,0] which results in a very near minimum value for its predicted tension transformation stress. Specimen number two was oriented 15 degrees from [111] direction and has a [1,1,1.73] tensile axis direction. The third specimen has the [111] direction as its tensile axis, which is the direction of maximum tensile transformation stress. A strong relationship is found between the mechanical behavior of the specimens in tension and their observed microstructure. Specimen one exhibits an extremely flat stress plateau during transformation and almost no hysteresis. The microstructure observed in this specimen consists of two nearly perpendicular AM interfaces that interact to form an X-structure that results in a purely uniaxial deformation. This microstructure is completely reversible and seems to present no restriction on the motion of either interface. Specimen two was observed to have only a single AM interface after transformation. This interface appears to preclude the formation of any other interfaces. Specimen three required five times the normal stress of that needed to transform specimen one. This specimen also exhibited a large amount of hysteresis. The microstructure observed consisted of two A M interface systems that meet to form wedges. Because the interfaces must end at the wedge apex, the formation of the wedges resulted in a kinematic coupling between the two AM interface systems. The amount of coupling between the interfaces in the microstructure correlates to the amount of hysteresis observed.

Ductile crack growth—II. Void nucleation and geometry effects on macroscopic fracture behavior

Abstract: Many metals that fail by void growth and coalescence display a macroscopically planar fracture process zone of one or two void spacings in thickness; outside this region, the voids exhibit little or no growth. A finite element model of this mode of failure was described in Part I of this paper [J. Mech. Phys. Solids, 43, 233–259 (1995)]. A row of void-containing cell elements is placed on the symmetry plane ahead of the initial crack. The cells incorporate the softening characteristics of hole growth and dependence on stress triaxiality. These cells are embedded within a conventional elastic-plastic continuum. Under increasing strain, the voids grow and coalesce to form new crack surfaces, thereby advancing the crack. Parametric studies reveal that the important microstructural parameters in the model are D and f0, characterizing the spacing and the initial volume fraction of voids on the fracture plane. Using this model, Xia et al. [J. Mech. Phys. Solids, 43, 389–413 (1995)] have successfully predicted details of the load, displacement and crack growth histories—collectively referred to as the macroscopic fracture behavior—of four specimen geometries, which give rise to significantly different crack tip constraints under fully plastic conditions. Here we study the quantitative effects of void nucleation by a stress and strain criterion on the macroscopic fracture behavior. This behavior is compared with predictions using a similar volume of voids present from the very beginning. Geometry effects on macroscopic fracture behavior under contained and fully yielded conditions are discussed for the three-point-bend specimen and the center-crack-panel loaded in tension. Here the objective is to show the connection between the crack growth resistance and the fracture environment, namely, the constraint ahead of the crack tip and the tensile stress on the fracture plane.

Dynamic strain aging and plastic instabilities

Abstract: A constitutive model proposed by McCormick [(1988) Theory of flow localization due to dynamic strain ageing. Acta. Metall. 36, 3061–3067] based on dislocation-solute interaction and describing dynamic strain aging behavior, is analyzed for the simple loading case of uniaxial tension. The model is rate dependent and includes a time-varying state variable, representing the local concentration of the impurity atoms at dislocations. Stability of the system and its post-instability behavior are considered. The methods used include analytical and numerical stability and bifurcation analysis with a numerical continuation technique. Yield point behavior and serrated yielding are found to result for well defined intervals of temperature and strain rate. Serrated yielding emerges as a branch of periodic solutions of the relaxation oscillation type, similar to frictional stick-slip. The distinction between the temporal and spatial (loss of homogeneity of strain) instability is emphasized. It is found that a critical machine stiffness exists above which a purely temporal instability cannot occur. The results are compared to the available experimental data.

Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The vigdergauz microstructure

Abstract: For modeling coherent phase transformations, and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. The existence of a particularly simple microstructure with extremal elastic behavior, in the context of two-phase composites made from isotropic components in two space dimensions, has previously been shown. This “Vigdergauz microstructure” consists of a periodic array of appropriately shaped inclusions. We provide an alternative discussion of this microstructure and its properties. Our treatment includes an explicit formula for the shape of the inclusion, and an analysis of various limits. We also discuss the significance of this microstructure (i) for minimizing the maximum stress in a composite, and (ii) as a large volume fraction analog of Michell trusses in the theory of structural optimization.

An exact closed form solution for fragmentation of Weibull fibers in a single filament composite with applications to fiber-reinforced ceramics

Abstract: Exact equations are derived governing the evolution of fiber fragments in a Weibull fiber loaded according to the “single filament composite test”. These equations differ from those formulated by others who have made a priori assumptions on the shape of the fragment distribution that are shown to be incorrect. An explicit closed form solution of the governing equations is derived for arbitrary Weibull modulus ϱ and for random initial breaks with exponentially distributed spacings of a given normalised rate α along the fiber. In particular, the special case of unique fiber strength (ϱ = ∞), which is adapted from an exact solution of Widom [J. Chem. Phys. 44, 3888–3894 (1966)], is a limiting case of our solution. Furthermore, the solution for the case of ϱ = 0 can be expressed in terms of elementary functions. The limiting distribution function for normalised fragment length is also obtained in closed form for all ϱ ⩾ 0. The convergence of this distribution function to that for the case of unique fiber strength, where the normalised fragment lengths x lie between 1 2 and 1, is very slow being O(ϱ −1 2 ) . For finite ϱ the lower tail asymptotics of the limiting distribution function are proportional to (2x)2ϱ + 1, x ⩾ 0, so that a Weibull plot there has a slope of 2ϱ + 1. In fact, in the limit as ϱ → 0, the normalised fragment length follows an exponential distribution, which turns out to be a good approximation for 0 ⩽ ϱ s ∗ for all ϱ > 0, contrary to assertions made by others. In particular, s ∗ is estimated to within 2% by the formula s ∗ = [6(1 − √ 1 − 2 3ϱ )] 1 (1 + ϱ) for ϱ ⩾ 1, and we also give excellent approximations for general ϱ > 0.

Microbuckle initiation in fibre composites : A finite element study

Abstract: A finite strain continuum theory is presented for unidirectional fibre reinforced composites under in-plane loading. The constitutive response is expressed in terms of couple stress theory, and is deduced from a unit cell of a linear elastic Timoshenko beam embedded in a non-linear elastic-plastic matrix. The continuum theory is implemented within a finite element framework and is used to analyse compressive failure of polymer matrix composites by fibre microbuckling. It is assumed that microbuckling initiates from an imperfection in the form of a finite elliptical region of fibre waviness. The calculations show that the compressive strength decreases with increasing imperfection spatial size from the elastic bifurcation value of Rosen (1965, Fibre Composite Materials, pp. 37–75, American Society Metals Seminar) to the imperfection-sensitive infinite band strength given by Fleck et al. [1995, J. Appl. Mech. 62, 329–337.].

Shear dominated transonic interfacial crack growth in a bimaterial-i. experimental observations

Abstract: In this work we describe a series of impact experiments performed on PMMA/4340 steel edge cracked bimaterial plates Specimens were impacted at 20 m s−1 in a one point bend configuration using a high speed gas gun Dynamic interfacial crack propagation was observed using the optical method of Coherent Gradient Sensing and high speed photography Very high crack tip accelerations (108 m s−2) and very high crack tip speeds (up to 15cRPMMA) were measured and are reported Quantitative measurements show that in experiments in which the crack tip speed entered the intersonic range for PMMA, the stress field surrounding the crack tip was shear dominated The observation of high shear around the crack tip can also be explained using wave propagation arguments It is found that the reason for attainment of intersonic (with respect to PMMA) crack tip speeds is directly related to the large amounts of energy necessary to initiate the crack tip under shear dominated conditions A comparison with the theoretical results of Part II in this study is also made There seems to be an unfavorable region of stable crack propagation velocities in the intersonic regime This region is cspmma < v < √2csPMMA In all experiments performed, the propagating crack accelerated quickly out of this region In the few interferograms that do actually correspond to crack propagation in this unfavorable velocity range, crack face contact was observed This observation is also in agreement with the findings of Part II of this investigation

Dynamic weight functions for a moving crack. I. Mode I loading

Abstract: Dynamic weight functions are constructed for general time-dependent shear loading of a plane semi-infinite crack propagating with constant speed in an infinite isotropic elastic body. The use of Fourier transforms reduces the problem to the analysis of a matrix Wiener-Hopf equation. The solution of the Wiener-Hopf problem is presented. An expression is derived for the first-order perturbation to the stress intensity factors induced by a small time-dependent deviation from straightness of the crack front. The asymptotic procedure requires consideration of two terms of the asymptotic expansions of the displacement and stress tensor components in a neighbourhood of the crack front.

The influence of mobility of dissolved hydrogen on the elastic response of a metal

Abstract: The general principles of the mechanics of materials are used to describe the effect of interstitial mobile hydrogen on the linear elastic behavior of metals and alloys. The linear field equations reveal that during transient hydrogen diffusion the Laplacian of the hydrostatic stress is related to the Laplacian of the hydrogen concentration in the lattice, and it is not zero, as has often been assumed in calculations involving stress-driven diffusion of hydrogen under plane strain conditions. When the hydrogen reaches equilibrium with the local stress and diffusion terminates, the linear elastic constitutive response of the solid accounting for the hydrogen effect can be described by the standard Hooke's law of infinitesimal elasticity in which the stiffness moduli are termed moduli at fixed solute chemical potential and are calculated in terms of the moduli at fixed solute composition, the nominal hydrogen concentration, and the material parameters of the system. These moduli at fixed solute chemical potential can be viewed as the counterparts of those characterizing the drained deformation at constant pressure of fluid-infiltrated porous geomaterials, or the adiabatic deformation of thermoelastic materials. Next the linear transient field equations are solved in the case of a dislocation and a line force in an infinite medium under plane strain conditions by using analytic function theory. The range of validity of the solution to the linear field equations for an isolated edge dislocation is investigated for specific materials. Lastly, the implications of the constitutive behavior of the hydrogen-metal binary system on the fracture and dislocation behavior are discussed when the hydrogen is in equilibrium with local stress.

The crack tip solution for hydraulic fracturing in a permeable solid

Abstract: This paper extends previous work on self-similar analytical solutions for a hydraulically driven fracture propagating in a solid which is in a state of plane strain In particular, we examine the effect of fluid loss to the formation modelled by the Carter leak-off mechanism Our main new results are asymptotic solutions for arbitrary rock permeability; it is shown how these may be represented by an expansion whose leading term is the near-tip solution in the high permeability limit This term gives an integrable singularity in the near-tip fluid pressure which is slightly stronger than the singularity which arises in the impermeable case; it follows that there is always a region immediately adjacent to the fluid front where the solution is dominated by fluid loss Our most important conclusion for applications is that the solution in the practical case of intermediate permeability may be constructed as a series which starts from the loss dominated limit We also provide a detailed comparison of our predictions with results from a numerical model which includes a fluid lag zone The results are found to be in good agreement in all of the various permeability regimes

Wrinkling of the oxide scale on an aluminum-containing alloy at high temperatures

Abstract: To survive in an oxygen-bearing gas, an alloy grows an oxide scale to cover itself. The scale thickens slowly at a high temperature, until the compressive stress generated by oxidation and cooling causes it to spall off. A widely known model assumes that the scale-alloy interface has unbonded flaws, which allow the scale to buckle under the compression, followed by delamination and cracking. For the scale to buckle under a typical stress, this model requires flaws with radii exceeding ten times the scale thickness. In general, such flaws are too large to be produced during oxidation. This paper proposes a different model. It is commonly observed that wrinkles appear at the high temperature, caused by the oxidation-induced compressive stress. Initially the interface remains bonded, and the alloy conforms to the shape change by creep or diffusion. If the high temperature is maintained long enough, voids will grow on the interface, and the scale will slide, fold and crack. Cooling can intervene at any point; for example, even when the scale only wrinkles without extensive voiding, large thermal stress on the wavy interface may cause the scale to spall off. The initial wrinkling is analyzed as a time-dependent bifurcation by using a variational principle. The base alloy is assumed to conform by metal diffusion along the interface, motivated by the strain energy release due to wrinkling. The results are discussed in connection with experimental observations.

Shear dominated transonic interfacial crack growth in a bimaterial I-II. Asymptotic fields and favorable velocity regimes

Abstract: Motivated by experimental observations of transonic crack tip speeds (Lambros and Rosakis, 1994c, J. Mech. Phys. Solids 43(2), 169–188), the problem of intersonic interfacial crack growth in an elastic-rigid bimaterial system is analysed. Following the analytical procedure employed in Liu et al. (1993, J. Mech. Phys. Solids 41, 1887–1954), the two-dimensional in-plane asymptotic deformation field surrounding the tip of a crack propagating intersonically along an elastic-rigid bimaterial interface, is obtained. The theoretical results show that the near-tip stress field does not exhibit oscillations, while a stress singularity weaker than 0.5 still exists and is a function of the crack tip speed. In addition, due to the intersonic nature of crack growth, a singular line emanating from the moving crack tip is present in the near-tip field. Across this line, stresses and particle velocities suffer infinite jumps. The theoretical analysis also shows that the near-tip deformation field is shear dominated. It is also shown that in the velocity range cs < v < √2cs, either crack face contact or negative normal tractions ahead of the crack tip exist. Visual evidence of such contact is reported in Part I of this study. These observations, together with additional experimental results of Part I, lead to the conclusion that crack growth is favorable in the velocity regimes 0 < v < cs and √2cs < v < c1.

Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. I: The confocal ellipse construction

Abstract: For modeling coherent phase transformations and for applications to structural optimization, it is of interest to identify microstructures with minimal energy or maximal stiffness. We present a new and appealingly simple class of extremal microstructures, which we call the confocal ellipse construction, for the case of a two-dimensional elastic composite made from two isotropic elastic materials. When the macroscopic stress and strain are isotropic, our construction reduces to the well-known “coated sphere” microstructure.

Creep models for metal matrix composites with long brittle fibers

Abstract: Creep models for metal matrix composites reinforced by long brittle fibers with weak interfaces are presented. These models extend the work of McLean (1985) to include effects of fiber breaks and the consequential stress relaxation in the broken fibers. The creep strain and the creep rupture time are calculated when global load sharing occurs. Analyses are conducted for composites with a wide range of fiber volume fractions, Young's modulus of the fibers and the matrix, interfacial sliding stress and Weibull properties for the strength of the fibers. The results derived from this study are compared with those predicted by McLean's (1985) model. The creep life is found to be sensitive to the extent of fiber stress relaxation in the broken fibers. Models, which ignore this effect overestimate the creep rupture time especially when the composite is subjected to a low or moderate level of stress.

Void growth due to creep and grain boundary diffusion at high triaxialities

Abstract: The growth of grain boundary voids at elevated temperatures by coupled creep and grain boundary diffusion is studied numerically using a cylindrical unit cell model. Emphasis is on the influence of the remote stress triaxiality, which is taken to cover the full range of axisymmetric stress states, from purely effective to purely hydrostatic states of stress. The motivation for extending previous results stems from the need for an accurate cavity growth model to analyse damage due to hydrogen attack, where the grain boundary voids are internally pressurized. Because of the wide range of stress states considered, numerical stability requires the use of two normalizations of the variational principle for the coupled void growth problem; one when the effective stress is dominant and the other when the mean stress is dominant. In the regime where deformation is primarily by creep, two distinct modes of deformation appear for each level of porosity; one for low triaxialities and one that takes over for sufficiently high triaxialities. Approximate models found in the literature for a dilute concentration of voids, or for finite concentrations, are explored to check their ability to represent the stress state dependence of the volumetric void growth rate. A novel approximate formula is derived for creep dominated growth and is shown to give good agreement with numerically computed void growth rates in the high triaxiality regime and for finite concentrations. A fairly abrupt transition between creep dominated void growth and diffusion dominated void growth is found when the stress triaxiality is very high, so that the interaction between creep and diffusion is then relatively unimportant. Finally, formulae are presented which give an approximate, yet fairly accurate, expression for the void volume growth rate due to coupled diffusional and creep growth over the full range of axisymmetric stress states.

Effects of local constraint along three-dimensional crack fronts—a numerical and experimental investigation

Abstract: A two-parameter characterization of the crack front state in a variety of geometries (mode I) is explored by means of three-dimensional finite element analysis, including finite strain effects. In particular the approximate J-Q theory is scrutinized and it is found that Q appears to be a good measure of the deviation in stress triaxiality ahead of a crack tip as compared to the highly constrained plane strain SSY-solution, also in cases where the crack front is relatively curved. The implications for cleavage fracture in the upper transition region are elucidated by appraising the results from an extensive experimental program, where both tension and bending type of plane specimens as well as surface cracked plate specimens had been tested. It appears that the J-Q concept together with some cleavage failure criteria, e.g. the RKR-model, can be applied locally along three-dimensional crack fronts in a structure in order to assess cleavage fracture. To the extent that one dominating cleavage fracture spot could be located at a three-dimensional crack border, this was in general found at the position which had undergone the most critical J-Q sequence, in the light of the RKR-criteria. Microstructural features of both ductile and cleavage fracture are elucidated by a fractographical survey performed.

On the thermomechanical deformation behavior of duplex-type materials

Abstract: Two-phase duplex-type materials possess microstructures containing roughly the same amounts of the constituent phases whose grains form interwoven networks. Duplex stainless steels are typical representatives of this material group. In these steels the constituent phases austenite and ferrite have different coefficients of thermal expansion. On pure thermal loading or thermomechanical loading the yield strength of the phases can be exceeded. Specimens of a forged duplex steel with a uniaxially anisotropic micro-structure deform irreversibly even under pure thermal cycling conditions with a monotonic accumulation of strain. The results of a systematic finite element based micromechanical analysis of the thermomechanical deformation behavior of duplex steels are presented and discussed. The analysis is based on a quantitative characterization of both the real and model microstructures. Additionally, an extended constitutive material law for the thermomechanical loading of the duplex steel is proposed. For dual-phase materials this description incorporates an additional thermomechanical strain increment as a very important contribution to the total strain increment. Both the micromechanical model and the analytical model are used to analyse the experimental findings from dilatometer tests. The micromechanical approach allows the evolution of the irreversible strains in the two phases generated in a thermal cycle to be modeled. It is shown that the matrix-phase is always more deformed than the inclusion-phase, irrespective of which of the two phases (austenite or ferrite) forms the matrix. This prediction is confirmed by electron microscopic observations of a thermally cycled duplex steel. Based on these results a mechanism driving the ratchet effect is proposed.

Nonlocal continuum effects on bifurcation in the plane strain tension-compression test

Abstract: The paper examines nonlocal effects on bifurcation phenomena. A gradient plasticity model is used where a characteristic length is introduced in the yield criterion. Hill's well known framework of bifurcation theory is shown to hold in the presence of normality and a sufficient condition for uniqueness is given. Further, the regularizing effects of nonlocality are underlined. It is also shown that the underlying local continuum, obtained when the length scale goes to zero, always provides a lower bound for bifurcation stresses for the nonlocal continuum. Detailed analysis of bifurcation phenomena in the plane strain tension-compression test is carried out and compared to the results of Hill and Hutchinson for the local continuum. The results are qualitatively the same in the long wavelength domain while they differ markedly in the short wavelength domain. In this last case and in the elliptic regime, bifurcation modes disappear in tension while the corresponding stresses are significantly increased in the compressive regime.