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Papers by David Durban

Research paper thumbnail of Advances in the Mechanics of Plates and Shells

Springer eBooks, 2002

The aim of this series is to provide lucid accounts written by authoritative researchers giving v... more The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages.

Research paper thumbnail of The Influence of Crack-Face Normal and Shear Stress Loading on Hydraulic Fracture-Tip Singular Plastic Fields

Rock Mechanics and Rock Engineering, Mar 8, 2018

We investigated the singular plastic fields at the crack tip of a fracture that is loaded with no... more We investigated the singular plastic fields at the crack tip of a fracture that is loaded with normal and shear loads due to a viscous flow in a hydraulic fracturing. The level of the expected shear load in comparison with the normal load is examined. The lubrication flow and plastic deformation were decoupled assuming that the relation between applied shear load and normal load follows a linear friction-type relation. This assumption allows to investigate extreme bounds of the solution. Both the applied normal and shear loads are assumed to exhibit singular behavior near the tip which is consistent at the fracture surfaces with the plastic singular stress fields that are investigated. The fractured material is assumed to obey a non-associative Drucker-Prager solid with power law hardening response. The singular values and the corresponding fields were determined over a range of material parameters. For both von Mises material and associative Drucker-Prager material, we found that the level of singularity is given by 1/n where n is the power coefficient of the hardening relation. This level of singularity is stronger than the HRR value, 1/(n + 1), which has been determined for traction free crack surfaces. We found that the shear loading does not influence the level of singularity but it changes the shape of the developed plastic zones with the emergence of a boundary layer near the fracture surface. Deviation from material associativity produces consistent small increases in the level of singularity. The near-tip stress, strain and displacement profiles are illustrated for a few representative cases.

Research paper thumbnail of Longitudinal shock waves in solids: the piston shock analogue

Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, Apr 8, 2014

Propagation of longitudinal deformation in hardening elastoplastic solids is investigated by the ... more Propagation of longitudinal deformation in hardening elastoplastic solids is investigated by the way of analogy with the shock tube pattern in fluid mechanics. Conditions for the appearance of a shock discontinuity are formulated and steady-state continuous and discontinuous solutions are derived. Field characteristics are then investigated for a representative family of hardening elastoplastic Mises solids accounting for finite strains. A critical limit value of the imposed velocity for the emergence of a shock wave is found and sensitivity to material parameters is assessed. Evaluation of dissipation effects is conducted and field response is compared with other uniaxial stress fields. In agreement with available experimental results, it is established that the field may consist of both an elastic precursor and a plastic shock separated by a continuous elastoplastic range. Or, alternatively, when the imposed velocity is higher, the plastic shock overtakes those regions allowing for a variety of resulting fields.

Research paper thumbnail of Thermo-elastic/plastic behaviour of a strain-hardening thick-walled sphere

International Journal of Solids and Structures, 1983

ABSTRACT

Research paper thumbnail of Asymptotic Solution for Extrusion of Sintered Powder Metals

Journal of Applied Mechanics, Jun 1, 1991

Research paper thumbnail of Bifurcation of Elastoplastic Pressure-Sensitive Hollow Cylinders

Journal of Applied Mechanics, Mar 1, 1999

A linear bifurcation analysis is presented for pressure sensitive elastoplastic hollow cylinders ... more A linear bifurcation analysis is presented for pressure sensitive elastoplastic hollow cylinders under radial surface loads. Material response is modeled by flow and deformation theories of the Drucker-Prager solid accounting for arbitrary hardening. Sample calculations are given for cylinders that deform in axially symmetric patterns under uniform radial pressure applied at the boundaries. No bifurcation points were found with flow theory in the realistic range of stress though the primary equilibrium path is nearly identical for both theories. For thick-walled cylinders the dominant bifurcation mode predicted by deformation theory appears to be a circumferential surface instability. Deformation theory results for bifurcations are apparently not sensitive to deviations from associativity.

Research paper thumbnail of Radial flour of sintered powder metals

International Journal of Mechanical Sciences, 1989

Research paper thumbnail of Thermo-elastic/plastic behaviour of a strain-hardening thick-walled sphere

International Journal of Solids and Structures, 1983

ABSTRACT

Research paper thumbnail of On two stress concentration problems in plane-stress anisotropic plasticity

International Journal of Solids and Structures, 1987

... 0. For an elastic linearhardening solid we have the stressstrain characteristic Ep = N(i iy) ... more ... 0. For an elastic linearhardening solid we have the stressstrain characteristic Ep = N(i iy) (14) where N=i11 "=1' (15) TI Ci Ci and ET is the tangent modulus. Combining eqn (13) with eqn (14) we find that eqns (9) and (10) become ,=(!+ Ns2) (v + Ns Is + 5Sy (16) fiZ.+NNEy. (17) ? ...

Research paper thumbnail of Incremental Behaviour of an Elasto-Plastic Continuum

Abstract : A three dimensional incremental theory which describes the behaviour of an elasto-plas... more Abstract : A three dimensional incremental theory which describes the behaviour of an elasto-plastic continuum is proposed. At any instant of the deformation process a constitutive relation between infinitesimal quantities is constructed. By passing to the limit an exact rate theory is obtained. The deformation can be of any magnitude. For infinitesimal deformation the theory degenerates to the classical elasto-plastic theory. Tensors are considered as invariant objects rather than as a collection of components. By using full tensorial notation a simple invariant formation is obtained. The work contains a complete derivation of a rate theory; kinematics, equilibrium, boundary conditions, virtual principles and constitutive equations.

Research paper thumbnail of Self Similar Dynamic Expansion of a Spherical Cavity in Elastoplastic Media

Research paper thumbnail of Plane-strain incremental response and sensitivity of stretched plates

European Journal of Mechanics - A/Solids

A family of rate boundary value problems for initially stretched plates is investigated, under th... more A family of rate boundary value problems for initially stretched plates is investigated, under the plane strain constraint. The governing strain-rate compatibility equation is solved, in terms of Fourier series, for normal and shear traction rates applied to the longitudinal faces. Specific examples are provided for the Blatz-Ko material, including a mapping of bifurcation stretches in the elliptic range. The perturbed fields induced by the incremental loads are highly sensitive to initial stretch. Near bifurcation stretches in tension and in compression, there is large amplification of traction rates, strain rates and velocities whenever the applied load is in resonance with the corresponding bifurcation mode. The plate appears to be increasingly sensitive to small perturbations as bifurcation is approached. Numerical illustrations of that sensitivity are supported by asymptotic expansions.

Research paper thumbnail of Plastic Bifurcation in the Triaxial Confining Pressure Test

Journal of Applied Mechanics, 2000

Bifurcations of a circular cylinder are studied, within the context of the triaxial confining pre... more Bifurcations of a circular cylinder are studied, within the context of the triaxial confining pressure test, for pressure sensitive solids. Material response is modeled by large strain versions of flow and deformation theories of plasticity in conjunction with the Drucker-Prager solid. An axially symmetric deformation pattern is assumed prior to bifurcation and only diffuse modes within the elliptic regime are considered. The governing equations are solved analytically in terms of Bessel functions and a search procedure is employed to trace bifurcation loads. Deformation theory predicts critical stresses which are consistently below flow theory results, and provides practical upper bounds on experimentally observed values of peak stresses. [S0021-8936(00)01403-3]

Research paper thumbnail of Elastoplastic Buckling of Annular Plates in Pure Shear

Journal of Applied Mechanics, 1989

A linear buckling analysis is presented for annular elastoplastic plates under shear loads. The s... more A linear buckling analysis is presented for annular elastoplastic plates under shear loads. The standard plate buckling equations are used in conjunction with the small strain J2 flow and deformation theories of plasticity. The main numerical finding is that deformation theory predicts critical loads which are considerably below the predictions obtained with the flow theory. Furthermore, comparison with experimental data for different metals shows a good agreement with the deformation theory results over a wide range of geometries. The limiting buckling problem of a long narrow panel under shear stresses is treated separately. This problem admits an exact solution and it is shown that the critical loads for the panel are approached asymtotically by the annular plate results. Contact is made with earlier studies on the buckling of elastic-orthotropic and elastoplastic shear panels.

Research paper thumbnail of Interface Stress Singularities at a Notch Tip in Antiplane Shear

Journal of Applied Mechanics, 1987

Research paper thumbnail of Singular Plastic Fields in Steady Penetration of a Rigid Cone

Journal of Applied Mechanics, 1992

The essential features of the active plastic zone at the tip of a penetrating rigid cone are inve... more 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.

Research paper thumbnail of On the Problem of a Spherical Cavity in an Infinite Elasto-Plastic Medium

Journal of Applied Mechanics, 1976

The nonlinear problem of a spherical cavity surrounded by an infinite elasto-plastic medium, and ... more The nonlinear problem of a spherical cavity surrounded by an infinite elasto-plastic medium, and subjected to uniform radial loads, is considered. The material is assumed to be incrementally elasto-plastic. No restriction on the magnitude of the deformation and stress is imposed. For the cases of internal or external pressure, the governing nonlinear differential equations are solved in terms of closed integrals. Numerical results obtained for some metals are also shown.

Research paper thumbnail of Stress and Deformation Fields Around a Cylindrical Cavity Embedded in a Pressure-Sensitive Elastoplastic Medium

Journal of Applied Mechanics, 1998

The problem of an internally pressurized cylindrical cavity under remote nonequibiaxial compressi... more The problem of an internally pressurized cylindrical cavity under remote nonequibiaxial compression is examined within the framework of small strain theory. The cavity is embedded in a medium with a pressure-sensitive elastoplastic, strain-hardening and nonassociative response. The stress and deformation fields around the cavity are derived using a Drucker-Prager type deformation theory under the assumption of plane strain. The symmetry conditions allow the solution to be expanded as a Fourier cosine series in the circumferential direction. The Fourier coefficients are functions of the radial coordinate and are governed by a coupled system of ordinary differential equations. Numerical examples illustrate the evolution of the elastoplastic interface, together with the variation in both the stress concentration factor and the displacements at the cavity surface, due to increasing remote load.

Research paper thumbnail of A Method for Approximate Stability Analysis and Its Application to Circular Cylindrical Shells Under Circumferentially Varying Edge Loads

Journal of Applied Mechanics, 1973

A procedure is presented which relates the solution of the stability problem for an elastic struc... more A procedure is presented which relates the solution of the stability problem for an elastic structure subjected to a complex prebuckling stress distribution, to that of a solved stability problem for the same structure but with a “simple” stress distribution. The procedure reduces to obtaining the characteristic roots of a symmetric matrix. It is then applied to the buckling problem of a circular cylindrical shell subjected to a circumferentially varying compressive stress field and boundary conditions of the SS1-2 types. The main result is that for the boundary conditions and number of harmonic terms in the edge load taken, the buckling parameter approaches, as h/R → 0, that of a cylinder uniformly subjected to the maximum circumferential stress.

Research paper thumbnail of Asymptotic Solution for Extrusion of Sintered Powder Metals

Journal of Applied Mechanics, 1991

Research paper thumbnail of Advances in the Mechanics of Plates and Shells

Springer eBooks, 2002

The aim of this series is to provide lucid accounts written by authoritative researchers giving v... more The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages.

Research paper thumbnail of The Influence of Crack-Face Normal and Shear Stress Loading on Hydraulic Fracture-Tip Singular Plastic Fields

Rock Mechanics and Rock Engineering, Mar 8, 2018

We investigated the singular plastic fields at the crack tip of a fracture that is loaded with no... more We investigated the singular plastic fields at the crack tip of a fracture that is loaded with normal and shear loads due to a viscous flow in a hydraulic fracturing. The level of the expected shear load in comparison with the normal load is examined. The lubrication flow and plastic deformation were decoupled assuming that the relation between applied shear load and normal load follows a linear friction-type relation. This assumption allows to investigate extreme bounds of the solution. Both the applied normal and shear loads are assumed to exhibit singular behavior near the tip which is consistent at the fracture surfaces with the plastic singular stress fields that are investigated. The fractured material is assumed to obey a non-associative Drucker-Prager solid with power law hardening response. The singular values and the corresponding fields were determined over a range of material parameters. For both von Mises material and associative Drucker-Prager material, we found that the level of singularity is given by 1/n where n is the power coefficient of the hardening relation. This level of singularity is stronger than the HRR value, 1/(n + 1), which has been determined for traction free crack surfaces. We found that the shear loading does not influence the level of singularity but it changes the shape of the developed plastic zones with the emergence of a boundary layer near the fracture surface. Deviation from material associativity produces consistent small increases in the level of singularity. The near-tip stress, strain and displacement profiles are illustrated for a few representative cases.

Research paper thumbnail of Longitudinal shock waves in solids: the piston shock analogue

Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, Apr 8, 2014

Propagation of longitudinal deformation in hardening elastoplastic solids is investigated by the ... more Propagation of longitudinal deformation in hardening elastoplastic solids is investigated by the way of analogy with the shock tube pattern in fluid mechanics. Conditions for the appearance of a shock discontinuity are formulated and steady-state continuous and discontinuous solutions are derived. Field characteristics are then investigated for a representative family of hardening elastoplastic Mises solids accounting for finite strains. A critical limit value of the imposed velocity for the emergence of a shock wave is found and sensitivity to material parameters is assessed. Evaluation of dissipation effects is conducted and field response is compared with other uniaxial stress fields. In agreement with available experimental results, it is established that the field may consist of both an elastic precursor and a plastic shock separated by a continuous elastoplastic range. Or, alternatively, when the imposed velocity is higher, the plastic shock overtakes those regions allowing for a variety of resulting fields.

Research paper thumbnail of Thermo-elastic/plastic behaviour of a strain-hardening thick-walled sphere

International Journal of Solids and Structures, 1983

ABSTRACT

Research paper thumbnail of Asymptotic Solution for Extrusion of Sintered Powder Metals

Journal of Applied Mechanics, Jun 1, 1991

Research paper thumbnail of Bifurcation of Elastoplastic Pressure-Sensitive Hollow Cylinders

Journal of Applied Mechanics, Mar 1, 1999

A linear bifurcation analysis is presented for pressure sensitive elastoplastic hollow cylinders ... more A linear bifurcation analysis is presented for pressure sensitive elastoplastic hollow cylinders under radial surface loads. Material response is modeled by flow and deformation theories of the Drucker-Prager solid accounting for arbitrary hardening. Sample calculations are given for cylinders that deform in axially symmetric patterns under uniform radial pressure applied at the boundaries. No bifurcation points were found with flow theory in the realistic range of stress though the primary equilibrium path is nearly identical for both theories. For thick-walled cylinders the dominant bifurcation mode predicted by deformation theory appears to be a circumferential surface instability. Deformation theory results for bifurcations are apparently not sensitive to deviations from associativity.

Research paper thumbnail of Radial flour of sintered powder metals

International Journal of Mechanical Sciences, 1989

Research paper thumbnail of Thermo-elastic/plastic behaviour of a strain-hardening thick-walled sphere

International Journal of Solids and Structures, 1983

ABSTRACT

Research paper thumbnail of On two stress concentration problems in plane-stress anisotropic plasticity

International Journal of Solids and Structures, 1987

... 0. For an elastic linearhardening solid we have the stressstrain characteristic Ep = N(i iy) ... more ... 0. For an elastic linearhardening solid we have the stressstrain characteristic Ep = N(i iy) (14) where N=i11 "=1' (15) TI Ci Ci and ET is the tangent modulus. Combining eqn (13) with eqn (14) we find that eqns (9) and (10) become ,=(!+ Ns2) (v + Ns Is + 5Sy (16) fiZ.+NNEy. (17) ? ...

Research paper thumbnail of Incremental Behaviour of an Elasto-Plastic Continuum

Abstract : A three dimensional incremental theory which describes the behaviour of an elasto-plas... more Abstract : A three dimensional incremental theory which describes the behaviour of an elasto-plastic continuum is proposed. At any instant of the deformation process a constitutive relation between infinitesimal quantities is constructed. By passing to the limit an exact rate theory is obtained. The deformation can be of any magnitude. For infinitesimal deformation the theory degenerates to the classical elasto-plastic theory. Tensors are considered as invariant objects rather than as a collection of components. By using full tensorial notation a simple invariant formation is obtained. The work contains a complete derivation of a rate theory; kinematics, equilibrium, boundary conditions, virtual principles and constitutive equations.

Research paper thumbnail of Self Similar Dynamic Expansion of a Spherical Cavity in Elastoplastic Media

Research paper thumbnail of Plane-strain incremental response and sensitivity of stretched plates

European Journal of Mechanics - A/Solids

A family of rate boundary value problems for initially stretched plates is investigated, under th... more A family of rate boundary value problems for initially stretched plates is investigated, under the plane strain constraint. The governing strain-rate compatibility equation is solved, in terms of Fourier series, for normal and shear traction rates applied to the longitudinal faces. Specific examples are provided for the Blatz-Ko material, including a mapping of bifurcation stretches in the elliptic range. The perturbed fields induced by the incremental loads are highly sensitive to initial stretch. Near bifurcation stretches in tension and in compression, there is large amplification of traction rates, strain rates and velocities whenever the applied load is in resonance with the corresponding bifurcation mode. The plate appears to be increasingly sensitive to small perturbations as bifurcation is approached. Numerical illustrations of that sensitivity are supported by asymptotic expansions.

Research paper thumbnail of Plastic Bifurcation in the Triaxial Confining Pressure Test

Journal of Applied Mechanics, 2000

Bifurcations of a circular cylinder are studied, within the context of the triaxial confining pre... more Bifurcations of a circular cylinder are studied, within the context of the triaxial confining pressure test, for pressure sensitive solids. Material response is modeled by large strain versions of flow and deformation theories of plasticity in conjunction with the Drucker-Prager solid. An axially symmetric deformation pattern is assumed prior to bifurcation and only diffuse modes within the elliptic regime are considered. The governing equations are solved analytically in terms of Bessel functions and a search procedure is employed to trace bifurcation loads. Deformation theory predicts critical stresses which are consistently below flow theory results, and provides practical upper bounds on experimentally observed values of peak stresses. [S0021-8936(00)01403-3]

Research paper thumbnail of Elastoplastic Buckling of Annular Plates in Pure Shear

Journal of Applied Mechanics, 1989

A linear buckling analysis is presented for annular elastoplastic plates under shear loads. The s... more A linear buckling analysis is presented for annular elastoplastic plates under shear loads. The standard plate buckling equations are used in conjunction with the small strain J2 flow and deformation theories of plasticity. The main numerical finding is that deformation theory predicts critical loads which are considerably below the predictions obtained with the flow theory. Furthermore, comparison with experimental data for different metals shows a good agreement with the deformation theory results over a wide range of geometries. The limiting buckling problem of a long narrow panel under shear stresses is treated separately. This problem admits an exact solution and it is shown that the critical loads for the panel are approached asymtotically by the annular plate results. Contact is made with earlier studies on the buckling of elastic-orthotropic and elastoplastic shear panels.

Research paper thumbnail of Interface Stress Singularities at a Notch Tip in Antiplane Shear

Journal of Applied Mechanics, 1987

Research paper thumbnail of Singular Plastic Fields in Steady Penetration of a Rigid Cone

Journal of Applied Mechanics, 1992

The essential features of the active plastic zone at the tip of a penetrating rigid cone are inve... more 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.

Research paper thumbnail of On the Problem of a Spherical Cavity in an Infinite Elasto-Plastic Medium

Journal of Applied Mechanics, 1976

The nonlinear problem of a spherical cavity surrounded by an infinite elasto-plastic medium, and ... more The nonlinear problem of a spherical cavity surrounded by an infinite elasto-plastic medium, and subjected to uniform radial loads, is considered. The material is assumed to be incrementally elasto-plastic. No restriction on the magnitude of the deformation and stress is imposed. For the cases of internal or external pressure, the governing nonlinear differential equations are solved in terms of closed integrals. Numerical results obtained for some metals are also shown.

Research paper thumbnail of Stress and Deformation Fields Around a Cylindrical Cavity Embedded in a Pressure-Sensitive Elastoplastic Medium

Journal of Applied Mechanics, 1998

The problem of an internally pressurized cylindrical cavity under remote nonequibiaxial compressi... more The problem of an internally pressurized cylindrical cavity under remote nonequibiaxial compression is examined within the framework of small strain theory. The cavity is embedded in a medium with a pressure-sensitive elastoplastic, strain-hardening and nonassociative response. The stress and deformation fields around the cavity are derived using a Drucker-Prager type deformation theory under the assumption of plane strain. The symmetry conditions allow the solution to be expanded as a Fourier cosine series in the circumferential direction. The Fourier coefficients are functions of the radial coordinate and are governed by a coupled system of ordinary differential equations. Numerical examples illustrate the evolution of the elastoplastic interface, together with the variation in both the stress concentration factor and the displacements at the cavity surface, due to increasing remote load.

Research paper thumbnail of A Method for Approximate Stability Analysis and Its Application to Circular Cylindrical Shells Under Circumferentially Varying Edge Loads

Journal of Applied Mechanics, 1973

A procedure is presented which relates the solution of the stability problem for an elastic struc... more A procedure is presented which relates the solution of the stability problem for an elastic structure subjected to a complex prebuckling stress distribution, to that of a solved stability problem for the same structure but with a “simple” stress distribution. The procedure reduces to obtaining the characteristic roots of a symmetric matrix. It is then applied to the buckling problem of a circular cylindrical shell subjected to a circumferentially varying compressive stress field and boundary conditions of the SS1-2 types. The main result is that for the boundary conditions and number of harmonic terms in the edge load taken, the buckling parameter approaches, as h/R → 0, that of a cylinder uniformly subjected to the maximum circumferential stress.

Research paper thumbnail of Asymptotic Solution for Extrusion of Sintered Powder Metals

Journal of Applied Mechanics, 1991