Cylindrical void in a rigid-ideally plastic single crystal III: Hexagonal close-packed crystal

The analytical solution is derived for the plane strain stress field around a cylindrical void in a hexagonal close-packed single crystal with three in-plane slip systems oriented at the angle π/3 with respect to one another. The critical resolved shear stress on each slip system is assumed to be equal. The crystal is loaded by both internal pressure and a far-field equibiaxial compressive stress. The deformation field takes the form of angular sectors, called slip sectors, within which only one slip system is active; the boundaries between different sectors are radial lines. The stress fields are derived by enforcing equilibrium and a rigid, ideally plastic constitutive relationship, in the spirit of anisotropic slip line theory. The results show that each slip sector is divided into smaller regions denoted as stress sectors and the stress state valid within each stress sector is derived. It is shown that stresses are unique and are continuous within stress sectors and across stress sector boundaries, but the gradient of stresses is not continuous across the boundaries between stress sectors. The solution shows self-similarity in that the stresses over the entire domain can be determined from the stresses within a small region adjacent to the void by invoking certain scaling and symmetry properties. In addition, the stress state exhibits periodicity along logarithmic spirals which emanate from the void. The results predict that the mean value of in-plane pressure required to activate plastic deformation around a void in a single crystal can be higher than that necessary for a void in an isotropic material and is sensitive to the orientation of the slip systems relative to the void.     

Cylindrical void in a rigid-ideally plastic single crystal II: Experiments and simulations

Experimental results and finite element simulations of plastic deformation around a cylindrical void in single crystals are presented to compare with the analytical solutions in a companion paper: Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals [Kysar, J.W., Gan, Y.X., Mendez-Arzuza, G., 2005. Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals, International Journal of Plasticity, 21, 1481–1520]. In the first part of the present paper, the theoretical predictions of the stress and deformation field around a cylindrical void in face-centered cubic (FCC) single crystals are briefly reviewed. Secondly, electron backscatter diffraction results are presented to show the lattice rotation discontinuities at boundaries between regions of single slip around the void as predicted in the companion paper. In the third part of the paper, the finite element method has been employed to simulate the anisotropic plastic deformation behavior of FCC single crystals which contain cylindrical voids under plane strain condition. The results of the simulation are in good agreement with the prediction by the anisotropic slip line theory.     

Wedge indentation into elastic-plastic single crystals. 2: Simulations for face-centered cubic crystals

The asymptotic stress and deformation fields associated with the contact point singularity of a nearly-flat wedge indenter impinging on a specially-oriented single face-centered cubic crystal are derived analytically in a companion paper. In order to investigate the extent of the asymptotic fields, the indentation process is simulated numerically using single crystal plasticity. The quasistatically translating asymptotic fields consist of four angular elastic sectors separated by plastically deforming sector boundaries, as predicted in the companion paper. The asymptotic stress distributions are in accord with the analytical predictions. In addition, simulations are performed for a wedge indenter with a 90° included angle in order to investigate the consequences of finite deformation and finite lattice rotation. Several salient features of the deformation field for the nearly-flat indenter persist in the deformation field for the 90° wedge indenter. The existence of the salient features is validated by comparison to experimental measurements of the lower bound on geometrically necessary dislocation (GND) densities.     

A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier

A phenomenological, flow theory version of gradient plasticity for isotropic and anisotropic solids is constructed along the lines of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain-gradient plasticity. J. Mech. Phys. Solids 52, 1379-1406]. Both energetic and dissipative stresses are considered in order to develop a kinematic hardening theory, which in the absence of gradient terms reduces to conventional J₂ flow theory with kinematic hardening. The dissipative stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition with associated plastic flow. The theory includes interfacial terms: elastic energy is stored and plastic work is dissipated at internal interfaces, and a yield surface is postulated for the work-conjugate stress quantities at the interface. Uniqueness and extremum principles are constructed for the solution of boundary value problems, for both the rate-dependent and the rate-independent cases. In the absence of strain gradient and interface effects, the minimum principles reduce to the classical extremum principles for a kinematically hardening elasto-plastic solid. A rigid-hardening version of the theory is also stated and the resulting theory gives rise to an extension to the classical limit load theorems. This has particular appeal as previous trial fields for limit load analysis can be used to generate immediately size-dependent bounds on limit loads.     

Theoretical latent hardening of crystals in double slip—I. F.C.C. crystals with a common slip plane

The hardening of all slip systems in f.c.c. crystals, deforming in finite double-slip on two systems with a common slip plane, is determined according to the “simple theory” of rotation-dependent plastic anisotropy. Both tensile and compressive axial-loading are considered. Of particular interest are predictions of crystal response after the loading axis has rotated into a higher symmetry position (6-fold in tension and 4-fold in compression). In contrast with classical Taylor hardening, the simple theory predicts that the axis will “overshoot” the higher symmetry position. A postulate of minimum plastic work plays a significant role in the theoretical analyses of multiple-slip positions.     

Non-local theory solution for a Mode I crack in piezoelectric materials

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials subjected to a uniform tension loading. The permittivity of the air in the crack is considered. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the materials constants, the electric boundary conditions and the lattice parameter on the stress and the electric displacement fields near the crack tips. It can be obtained that the effects of the electric boundary conditions on the electric displacement fields are large. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allowing us to use the maximum stress as a fracture criterion.     

Time-domain Green’s functions for unsaturated soils. Part I: Two-dimensional solution

In this article, primarily a brief discussion about the formulation of unsaturated soils including the equilibrium, air and moisture transfer equations is presented. Then the closed form two-dimensional Green’s functions of the governing differential equations for an unsaturated deformable porous medium with linear elastic behavior for a symmetric polar domain in both Laplace transform and time domains have been introduced, for the first time. Using the linear form of the governing differential equations and considering the effects of non-linearity of the governing equations, the Green’s functions have been derived exactly and verified in both Laplace transform and time domains.     

AN ANALYTICAL SOLUTION OF G. I. TAYLOR’S THEORY OF PLASTIC DEFORMATION IN IMPACT OF CYLINDRICAL PROJECTILES AND ITS IMPROVEMENT

The theory of plastic deformation in the impact of cylindrical projectiles on rigid targets was first introduced by G. I. Taylor(1948)^[1]. The importance of this theory lies in the fact that the dynamic yield strength of the materials can be determined from the measurement of the plastic deformation of flat-ended cylindrical projectiles. From the experimental results^[2] we find that the dynamic yield strength is independent of impact velocity, and that it is higher than the static yield strength in general, and several times higher than the static yield strength in certain cases. This gives an important foundation for the study of elastoplastic impact problems in general. However, it is well known that the complexity of differential equations in Taylor’s theory compelled us to use the troublesome numerical solution. In this paper, the analytical solution of all the equations in Taylor’s theory is given in parametrical form and the results are discussed in detail.In the latter part of this paper, the method of calculation of impulse of impact is improved by considering the processes of radial’ movement of materials. The analytical solution of the improved theory shows that it gives better agreement with the experimental results than that of original Taylor’s theory.     

Strain gradient elasticity theory for antiplane shear cracks. Part I: Oscillatory displacements

Solutions of crack problems for a strain gradient elastic theory whose crack opening displacements are monotonic rather than oscillatory in profile are examined in this (portion of a two-part) paper. A new analytical solution having a simple form is also obtained through the introduction of parabolic cylindrical coordinates and expansion of the imposed displacement along the crack surfaces in terms of orthogonal polynomials. Uniqueness of solution is also discussed, as well as finite-difference approximations of the governing equations for use in numerical solutions of boundary value problems.     

A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: Formulation

In this Part I, of a two-part paper, we present a detailed continuum-mechanical development of a thermo-mechanically coupled elasto-viscoplasticity theory to model the strain rate and temperature dependent large-deformation response of amorphous polymeric materials. Such a theory, when further specialized (Part II) should be useful for modeling and simulation of the thermo-mechanical response of components and structures made from such materials, as well as for modeling a variety of polymer processing operations.     

Free surface on a simple fluid between rotating eccentric cylinders. Part I: Analytical solution

The steady motion of a simple fluid between vertical cylinders which rotate about non-concentric axes is solved by means of domain perturbations. The theory is developed as a perturbation of the rest state in powers of the angular frequency ω of the inner cylinder, and the solution is carried out to O (ω²). The stress is expanded in a series of Rivlin-Ericksen tensors. At the second order only one material parameter, the climbing constant, enters the analysis. A procedure is developed for predicting the shape of the free surface on the fluid. Secondary motions generated by the eccentricity are shown to appear at the second order.     

Carbon dioxide flow boiling in a single microchannel - Part I: Pressure drops

Carbon dioxide two-phase flow pressure drops have been investigated in a single horizontal stainless-steel micro-tube having a 0.529 mm inner diameter. Experiments were carried out in adiabatic conditions for four saturation temperatures of −10; −5; 0; 5 °C and mass fluxes ranging from 200 to 1400 kg/m² s, for inlet qualities up to unity. Measurements have been compared to the predictions of well-known methods. The Müller-Steinhagen and Heck correlation and the Friedel correlation gave the best fit as well as the homogeneous model when the liquid viscosity is used to represent the apparent two-phase viscosity. Further, an analysis based on the homogeneous model has not shown any clear appearance of the laminar or the transition regimes in any given range of Reynolds number. The apparent viscosity of the two-phase mixture was found larger than the liquid viscosity at low vapour qualities, namely at the lowest temperatures. Hence, a new expression to determine the equivalent viscosity was suggested as a function of the reduced pressure. Lastly, the Chisholm parameter from the Lockhart-Martinelli correlation was found lower than expected and also mainly dependent on the saturation temperature.     

The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part I.: Full-field solution and asymptotics

A lamellar (zero-thickness) rigid inclusion, so-called ‘stiffener’, is considered embedded in a uniformly prestressed (or prestrained), incompressible and orthotropic elastic sheet, subject to a homogeneous far-field deformation increment. This problem is solved under the assumption of plane strain deformation, with prestress principal directions and orthotropy axes aligned with the stiffener. A full-field solution is obtained solving the Riemann-Hilbert problem for symmetric incremental loading at infinity (while for shear deformation the stiffener leaves the ambient field unperturbed). In addition to the full-field solution, the asymptotic Mode I near-tip representation involving the corresponding incremental stress intensity factor are derived and these results are complemented with the Mode II asymptotic solution. For null prestress, the full-field stress state is shown to match correctly with photoelastic experiments performed by us (on two-part epoxy resin samples containing an aluminum lamina). Our experiments also confirm the fracture patterns for a brittle material containing a stiffener, which do not obey a hoop-stress criterion and result completely different from those found for cracks. Issues related to shear band formation and evaluation of energy release rate for a stiffener growth (or reduction) are deferred to Part II of this article.     

A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations

This study develops a small-deformation theory of strain-gradient plasticity for isotropic materials in the absence of plastic rotation. The theory is based on a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows:

•

the microstresses to depend on , the gradient of the plastic strain-rate, and

•

the free energy ψ to depend on the Burgers tensor .

The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for the plastic strain. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the gradient microstress is partially energetic, and this, in turn, leads to a back stress and (hence) to Bauschinger-effects in the flow rule. It is further shown that dependencies of the microstresses on lead to strengthening and weakening effects in the flow rule.Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and, as an aid to numerical solutions, a weak (virtual power) formulation of the nonlocal flow rule is derived.     

Buoyancy-driven instability in a vertical cylinder: Binary fluids with Soret effect. Part I: General theory and stationary stability results

Buoyancy-driven instability of a monocomponent or binary fluid which is completely contained in a vertical circular cylinder is investigated, including the influence of the Soret effect for the binary mixture. The Boussinesq approximation is used, and the resulting linear stability problem is solved using Galerkin's technique. The analysis considers various types of fluid mixtures, ranging from gases to liquid metals, in cylinders with a variety of radius-to-height ratios. The flow structure is found to depend strongly on both the cylinder aspect ratio and the magnitude of the Soret effect. Comparisons are made with experiments and other theories, and the predicted stability limits are shown to agree closely with observations.     

Boundary film shear elastic modulus effect in hydrodynamic contact. Part I: theoretical analysis and typical solution

Boundary film shear elastic modulus effect is analyzed in a hydrodynamic contact. The contact is one-dimensional composed of two parallel plane surfaces, which are, respectively, rough rigid with rectangular micro projections in profile periodically distributed on the surface and ideally smooth rigid. The whole contact is consisted of cavitated area and hydrodynamic area. The hydrodynamic area consists of many micro Raleigh bearings which are discontinuously and periodically distributed in the contact. Analysis is thus carried out for a micro Raleigh bearing in this contact. The hydrodynamic contact in this micro Raleigh bearing consists of boundary film area and fluid film area which, respectively, occur in the outlet and inlet zones. In boundary film area, the film slips at the upper contact surface due to the limited shear stress capacity of the film–contact interface, while the film does not slip at the lower contact surface due to the shear stress capacity large enough at the film–contact interface. In boundary film area, the viscosity, density and shear elastic modulus of the film are varied across the film thickness due to the film–contact interactions, and their effective values are used in modeling, which depend on the film thickness. The analytical approach proposed by Zhang (J Mol Liq 128:60–64, 2006) and Zhang et al. (Int J Fluid Mech Res 30:542–557, 2003) is used for boundary film area. In fluid film area, the film does not slip at either of the contact surfaces, and the shear elastic modulus of the film is neglected. Conventional hydrodynamic analysis is used for fluid film area. The present paper presents the theoretical analysis and a typical solution. It is found that for the simulated case the boundary film shear elastic modulus effects on the mass flow through the contact, the overall film thickness of the contact and the carried load of the contact are negligible but the boundary film shear elastic modulus effect on the local film thickness of the contact may be significant when the boundary film thickness is on the 1 nm scale and the contact surfaces are elastic. In Part II will be presented detailed results showing boundary film shear elastic modulus effects in different operating conditions.

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The edge dislocation in a three-quarter plane. Part I: Influence functions

The problem of an edge dislocation, having arbitrary Burgers vector, and lying on one of the projection lines of a three-quarter plane is studied. The complete internal state of stress is found, with particular attention focused on the correct asymptotic behaviour. The solution is found in a form which makes it particularly suitable as the kernel of an integral equation formulation for an interface crack.