David Durban

Squeeze-flow of a Herschel–Bulkley fluid

Abstract Squeeze-flow experiments of a Herschel–Bulkley material between two rigid plates, investigated both experimentally and computationally by Adams et al. (J. Non-Newtonian Fluid Mech. 71 (1997) 41) are compared against an approximate analysis for generalised Newtonian fluids presented by Sherwood and Durban (J. Non-Newtonian Fluid Mech. 62 (1996) 35), for the case in which the interface between the material and the plates is lubricated. The analysis presented here assumes a rigid-viscoplastic material, rather than the elastic-viscoplastic material of Adams et al., but the viscoplastic model for flow is identical. However, the shear stress boundary condition at the plates differs from the Coulomb friction law of Adams et al.: the shear stress is here assumed to be a constant fraction of the effective stress (and consequently turns out to be independent of position). A simple expression for the total force required to push the plates together is obtained for the case when friction at the plates is small. Agreement between this expression and the experimentally-measured force is good at high strain, though the elastic deformation observed prior to yield is not captured by the rigid-viscoplastic analysis.

Squeeze flow of a power-law viscoplastic solid

Abstract A cylinder of height h is squeezed between two parallel circular plates of radius R >>h. The cylinder is assumed to behave as a generalised Newtonian material in which the stress and strain rate are coaxial: the particular cases of a rigid-plastic solid and power-law fluid are considered in detail. It is assumed that the frictional stress at the walls is a fixed fraction m of the yield stress in shear, k, in the case of the plastic material, and a fixed fraction of the effective Mises stress in the case of the power-law fluid. This boundary condition, often used in plasticity analysis, leads in both cases to a constant shear stress at the walls, rather than a no-slip boundary condition. Hoop stresses are included in an approximate analysis in which stresses and velocities are expanded as series in inverse powers of the radial coordinate r: these expansions break down near the axis r = 0 of the cylinder. The force required to compress the rigid-plastic cylinder is F= 2 3 mkϵR 3 h − + 1 2 3 kϵR 2 [(1−m 2 ) 1 2 + m − sin − m]+ O(kRh) independent of the speed of compression. The analysis can be extended to other solids and fluids characterised by a coaxial constitutive relation: by way of example, results are presented for the Bingham fluid.

Elastoplastic buckling of rectangular plates in biaxial compression/tension

Abstract This paper examines the elastoplastic buckling of a rectangular plate, with various boundary conditions, under uniform compression combined with uniform tension (or compression) in the perpendicular direction. The analysis is based on the standard linear buckling equations and material behaviour is modelled by the small strain J2 flow and deformation theories of plasticity. A detailed parametric study has been made for Al 7075 T6 over a range of plate geometries (a/b=0.25–4,a/h≈20–100) and with three sets of boundary conditions (four simply supported boundaries and the symmetric combinations of clamped/simply supported sides). For sufficiently thin plates we recover with both theories the classical elastic results. However, for thicker plates there is a remarkable difference in the buckling loads predicted by these two theories. Apart from the expected observation that deformation theory gives lower critical stresses than those obtained from the flow theory, we report on the existence of an optimal loading path for the deformation theory model. Buckling loads attained along the optimal path—specified by particular compression/tension ratios—are the highest possible over the entire space of loading histories. By contrast, no similar optimum has been found with the flow theory. This discrepancy in the buckling behaviour, obtained from the two competing plastic theories, provides a possibly new illustration of the plastic buckling paradox.

Experimentally Validated Microstructural 3D Constitutive Model of Coronary Arterial Media

Accurate modeling of arterial response to physiological or pathological loads may shed light on the processes leading to initiation and progression of a number of vascular diseases and may serve as a tool for prediction and diagnosis. In this study, a microstructure based hyperelastic constitutive model is developed for passive media of porcine coronary arteries. The most general model contains 12 independent parameters representing the three-dimensional inner fibrous structure of the media and includes the effects of residual stresses and osmotic swelling. Parameter estimation and model validation were based on mechanical data of porcine left anterior descending (LAD) media under radial inflation, axial extension, and twist tests. The results show that a reduced four parameter model is sufficient to reliably predict the passive mechanical properties. These parameters represent the stiffness and the helical orientation of each lamellae fiber and the stiffness of the interlamellar struts interconnecting these lamellae. Other structural features, such as orientational distribution of helical fibers and anisotropy of the interlamellar network, as well as possible transmural distribution of structural features, were found to have little effect on the global media mechanical response. It is shown that the model provides good predictions of the LAD media twist response based on parameters estimated from only biaxial tests of inflation and extension. In addition, good predictive capabilities are demonstrated for the model behavior at high axial stretch ratio based on data of law stretches.

Spherical Cavity Expansion in a Drucker-Prager Solid

A finite strain analysis is presented for the pressurized spherical cavity embedded in a Drucker-Prager medium. Material behavior is modeled by a nonassociated deformation theory which accounts for arbitrary strain-hardening. The governing equations of spherically symmetric response are reduced to a single differential equation with the effective stress as the independent variable. Some related topics are discussed including the elastic-perfectly plastic solid, the thin-walled shell, and the Mohr-Coulomb material. Spontaneous growth (cavitation limit) of an internally pressurized cavity is treated as a self-similar process and a few numerical examples are presented. These illustrate, for different hardening characteristics, the pressure sensitivity of material response and that deviations from normality always reduce the caviation pressure.

Dynamic spherical cavity expansion in an elastoplastic compressible Mises solid

The elastoplastic field induced by a self-similar dynamic expansion of a pressurised spherical cavity is investigated for the compressible Mises solid. The governing system consists of two ordinary differential equations for two stress components where radial velocity and density are known functions of these stresses. Numerical illustrations of radial profiles of field variables are presented for several metals. We introduce a new solution based on expansion in powers of the nondimensionalized cavity expansion velocity, for both elastic/perfectly plastic response and strain-hardening behavior. A Bernoulli-type solution for the dynamic cavitation pressure is obtained from the second-order expansion along with a more accurate third-order solution. These solutions are mathematically closed and do not need any best fit procedure to numerical data, like previous solutions widely used in the literature. The simple solution for elastic/perfectly plastic materials reveals the effects of elastic-compressibility and yield stress on dynamic response. Also, an elegant procedure is suggested to include strain-hardening in the simple elastic/perfectly plastic solution. Numerical examples are presented to demonstrate the validity of the approximate solutions. Applying the present cavitation model to penetration problems reveals good agreement between analytical predictions and penetration depth tests.

ELASTOPLASTIC ANALYSIS OF CYLINDRICAL CAVITY PROBLEMS IN GEOMATERIALS

A large-strain elastoplastic analysis is presented for a cylindrical cavity embedded in an infinite medium under uniform radial pressure. The investigation employs invariant, non-associated deformation-type theories for Mohr-Coulomb (M-C) and Drucker-Prager (D-P) solids, accounting for arbitrary hardening, with the equivalent stress as the independent variable. The M-C model results in a single first-order differential equation, whereas for the D-P solid an algebraic constraint supplements the governing differential equation. Material parameters and response characteristics were determined by calibrating the models with data from triaxial compression tests on Castlegate sandstone and on Jurassic shale. A comparison is presented between predictions obtained from the two models and experimental data from hollow cylinder tests under external loading. A sensitivity of the results to material parameters, like friction and dilation angles, is provided for the case of a cavity subjected to internal pressure in terms of limit pressure predictions. In all cases it has been found that the results of the D-P inner cone model are in close agreement with those obtained from the M-C model.

Singular Plastic Fields in Steady Penetration of a Rigid Cone

The essential features of the active plastic zone at the tip of a penetrating rigid cone are investigated for a rigid/perfectly plastic solid. An exact solution is suggested for the plastic zone. A rigid zone exists ahead of the cone and is separated from the plastic zone by a conical surface of discontinuity. It is assumed that the material yields instantaneously by going through a “shear shock” across the rigid/plastic interface. The orientation of the interface is determined by an ad hoc requirement for minimum shear strain jump at the shear shock. Results are presented for different cone angles and friction factors. The stresses within the plastic zone admit a logarithmic singularity whose level increases with cone angle and wall friction.

A General Solution for the Pressurized Elastoplastic Tube

The problem of a thick-walled cylindrical tube subjected to internal pressure is investigated within the framework of continuum plasticity. Material behavior is modeled by a finite strain elastoplastic flow theory based on the Tresca yield function. The deformation pattern is restricted by the plane-strain condition but arbitrary hardening and elastic compressibility are accounted for. A general solution is given in terms of quadratures. The analysis also includes treatment of a second plastic phase, characterized by corner relations, that may develop at the inner boundary. It is shown that the interface between the two plastic regions moves initially outwards and then, beyond a certain strain level, it moves back inwards. Some useful and simple results are given for thin-walled tubes of hardening materials and for thick-walled elastic/perfectly plastic tubes.

Elastoplastic buckling of axially compressed circular cylindrical shells

Buckling of axially compressed circular cylindrical shells is examined within the framework of small strain elastoplasticity. A linear bifurcation model is used along with the J2 flow and deformation theories. All four possibilities of simple supports (SS1–SS4) and clamped edges (CL1–CL4) are considered. A general solution is given with special emphasis on the axially symmetric modes. Numerical results show that the weakening effect of the relaxed simple supports (SS1–SS2) is considerably reduced in the plastic range. A detailed comparison with available experimental data points in favour of the deformation theory.