A line plastic-zone model for steady mode III crack growth in an elastic-plastic material

Steady-state quasi-static growth of a crack in the anti-plane shear mode through an elastic-plastic material is analyzed. The material is non-hardening and small-scale yielding conditions are assumed. The essential feature of the model is that the active plastic-zone is assumed to be a pair of discrete lines emanating from the crack tip out of the crack plane on which a suitable yield condition is satisfied. An exact solution is obtained for the plastic strain left in the wake of this active line plastic-zone. The extent of the plastic zone from the tip is determined to be 0.071 $\text{(}\text{k}\text{τ}{}_0\text{)}{}^2$ where k and τ₀ are the remote elastic stress intensity factor and the shear flow stress, respectively, and it is found that 36% of the elastic energy flowing into the crack-tip region during growth is dissipated through plastic work and 64% is trapped as residual elastic energy in the plastic-zone wake.     

On the characteristic equations of elasto-plastic flow

Three dimensional characteristic surfaces (slip surfaces) of elasto-plastic Navier's equations and the criteria for their existence are discussed, and the solutions are also applied to two dimensional cases. By making use of isotropic yield function, the following results are proved. If, and only if the plastic/elastic moduli ratio is zero and Det$\text{(φ}{}_{\mathrm{ij}}\text{)=0,(φ}{}_{\mathrm{ij}}\text{=}\text{?φ}\text{?σ}{}_{\mathrm{ij}}$, φ: yield function,σ_ij: stress tensor), characteristic surfaces exist. There are two and only two characteristic surface elements at each point, and they are identical with the surfaces of maximum shearing stress.     

Homogenization limit for electrical conduction in biological tissues in the radio-frequency rangeLimite d'homogénéisation pour la conduction électrique dans les tissus biologiques dans le domaine des radiofréquences

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a finely mixed periodic medium. We show that the homogenization limit u₀ of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation $\text{?}\text{div}\text{(σ}{}_0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+A}{}^0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+∫}{}_0{}^{\mathrm{t}}\text{A}{}^1\text{(t?τ)?}{}_{\mathrm{x}}\text{u}{}_0\text{(x,τ)}\text{d}\text{τ?}\text{F}\text{(x,t))=0}$ where σ₀>0 and the matrices A⁰, A¹ depend on geometric and material properties, while the vector function $\text{F}$ keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type. To cite this article: M. Amar et al., C. R. Mecanique 331 (2003).     

The role of shear stresses in brittle fracture

Using the three-dimensional model for brittle fracture developed earlier by S.A.F. Murrell and P.J. Digby (1970,1972) shear stress concentrations are calculated for brittle bodies and the relative roles of tensile and shear stresses in the fracture process are considered. It is found that the maximum shear stress and the maximum tensile stress occur at different places on a crack, and that there is a wide range of stress states for which they do not occur on the same crack. Furthermore, if the theoretical cleavage strength is σ_max and the theoretical shear strength is τ_max, then cleavage precedes inelastic shear and brittle fracture is possible, for suitable stress systems, when $\text{σ}{}_{\max }\text{<}\text{2τ}{}_{\mathrm{<mtext>max</mtext>}}\text{(1 ? ν)}$, where ν is the Poisson's ratio of the solid matrix. This appears to be in accordance with empirical observations.     

Viscous dissipation effects on thermophoretically augmented aerosol particle transport across laminar boundary layers

The sensitivity of aerosol particle motion to local temperature gradients has motivated this investigation of viscous dissipation effects on mass transport rates across nonisthermal, low mass-loading ‘dusty gas’ laminar boundary layers (lbl). From numerical lbl transfer calculations, including ‘ash’ particle thermophoresis and variable thermophysical properties, it has been found that for a specified wall temperature, $\text{T}{}_{\mathrm{<mtext>w</mtext>}}$, and mainstream static temperature, $\text{T}{}_{\mathrm{<mtext>e</mtext>}}$ cous dissipation within the boundary layer increases total particle deposition rates, its relative importance being dependent on $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{/T}{}_{\mathrm{<mtext>e</mtext>}}$. For combustion turbine blades which operate at near-unity Mach number, neglect of viscous dissipation is found to cause about a 25% underestimate of the fouling rate at $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{/T}{}_{\mathrm{<mtext>e</mtext>}}\text{= 0.8}$ for particle diameters between $\text{0.6 × 10}{}^{\mathrm{?2}}\text{μ}\text{m}$ and 0.3 μm. Alternatively, for conditions of fixed adiabatic wall temperature, $\text{T}{}_{\mathrm{<mtext>aw</mtext>}}$, or fixed stagnation (reservoir) temperature, $\text{T}{}_0$, dusty gas acceleration to appreciable Mach numbers is associated with reduced particle arrival rates due, in part, to the associated reduction in mainstream gas temperature. Recently developed mass transfer rate correlations are extended and found to be successful when tested against the present numerical calculations.     

Why the stress trajectories in the Dean-Hutchinson plastic sector of the growing mode III crack are an unfocused fan

The plastic zone of the growing mode III crack in an elastic perfectly plastic solid consists of two sectors in contact with each other. The sector closer to the crack plane, first studied analytically by Chitaley and McClintock (CM), consists of a fan of straight maximum shear stress trajectories that are focused on the crack tip. The other sector, first analyzed numerically by Dean and Hutchinson (DH), is a ‘radial’ fan of straight lines that are not focused at the crack tip or at another common point. In this paper it is shown with use of the dislocation density field that the need that the stress magnitude in the plastic wake be below the yield stress requires the existence of an unfocused fan in the DH sector. It appears unlikely that this result can be obtained without explicit use of dislocations.     

Effect of particulate thermophoresis in reducing the fouling rate advantages of effusion-cooling

To predict small particle diffusional mass transfer (deposition), including particle thermophoresis, transpiration cooling and variable properties, the coupled ordinary differential equations governing self-similar laminar boundary layers are solved numerically. Under typical combustion turbine conditions, although diffusional deposition rates can be dramatically reduced by transpiration cooling (eg by some 5-decades for mainstream submicron particles corresponding to a Schmidt number of about 10² (or d_p ≈ 0.7 × 10^?2μm) and a wall transpiration-cooled to $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{T}{}_{\mathrm{<mtext>e</mtext>}}\text{= 0.8}$) actual deposition rate reductions will be smaller than previously expected (by about 1 decade for particles with Sc ≈ 10²), owing to thermophoretic particle drift ‘caused’ by the colder wall. Such micro-droplets, small enough to behave like ‘heavy molecules’ in combustion systems, are often important because they can cause adherence of the much larger, supermicron, ash particles which inertially impact on the same surface     

Finite deformation analysis of COD,J-integral and crack tip intense strain region in plane strain large-scale yielding

A finite element analysis was performed to simulate crack tip blunting and the development of the intense strain region in a small compact tension specimen (0.4 T CT) of SA533B-1 under plane strain large-scale yielding, with the condition of large-geometry change around the crack tip taken into consideration. The region where the equivalent plastic strain $\text{}\text{?}$g3_p is greater than 0.15 was defined as the intense strain region, which corresponded to the recrystallized-etched zone delineated experimentally around the blunting crack tip. The development of the intense strain region was discussed as a function of the J-integral and the crack opening displacement. A linear relationship was obtained between the plastic work W_p dissipated within the intense strain region and (J/σ_y)² or b², where b is the crack opening displacement, defined as the separation of the two points at which the boundary of the intense strain region surrounding the crack tip intersects with the free surfaces of the crack.