A. Zervos

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain-softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C-1 continuity element. The required higher-order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill-posedness caused by strain-softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post-peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear-band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values.

Modelling of localisation and scale effect in thick-walled cylinders with gradient elastoplasticity

We model the progressive localisation of deformation which causes failure around thick-walled cylinders under external radial pressure. The study is based on a second-gradient elastoplastic model developed to regularise the ill-posedness caused by material strain-softening behaviour. The stress increment is related to both the strain increment and its Laplacian. The gradient terms introduce an internal length scale to the material allowing robust modelling of its post-peak behaviour. The numerical implementation is based on a C-1 finite element displacement formulation. Mesh insensitivity in terms of load-displacement and failure mechanism is demonstrated. The internal length in the constitutive equations enables modelling of the scale effect in thick-walled cylinders, according to which the load required to induce failure appears to be much larger for small holes than for large holes.

Numerical investigation of granular interfaces kinematics

The numerical method of Contact Dynamics is used in order to simulate the Ring Simple Shear Experiment of a granular medium model (Schneebeli material). The numerical results on the discrete medium are then post-processed in order to construct the displacement fields of an equivalent continuum, and the formation of a shear interface layer is observed. The existence and magnitude of individual grain rotations imply that the equivalent continuum could be a Cosserat continuum in this case, at least for the material inside the interface layer. Porosity profiles are presented, showing the progressive loosening of the granular medium, while its dilation is also measured.

What causes large submarine landslides on low gradient (<2°) continental slopes with slow (∼0.15 m/kyr) sediment accumulation?

Submarine landslides can cause damaging tsunamis, the height of which scales up with the volume of the displaced mass. The largest underwater landslides are far bigger than any landslides on land, and these submarine megaslides tend to occur on open continental slopes with remarkably low gradients of less than 2°. For geohazard assessments it is essential to understand what preconditions and triggers slope failure on such low gradients. Previous work has suggested that generation of high excess pore pressure due to rapid sediment deposition plays a key role in such failures. However, submarine slope failure also occurs where sedimentation rates are low (<0.15 m/kyr), such as off northwest Africa. We use a fully coupled stress and fluid flow finite element model to test whether such low sedimentation rates can generate sufficient excess pore pressures to cause failure of a 2° slope. The sensitivity of overpressure generation and slope stability is assessed with respect to different sedimentation rates and patterns, sediment consolidation properties, and stratigraphic layer configurations. The simulations show that, in general, it is difficult to generate significant excess pore pressure if sediment accumulation is slow and the only pressure source. However, we identify a sediment compression behavior that can lead to submarine landslides in locations worldwide. Our results imply that compressibility is an important factor for the stability of low gradient continental slopes.

Elastoplastic finite element analysis of inclined wellbores

An effective finite element model which can be used for elastoplastic analysis of inclined wellbores is presented. The same discretised model can be used for any wellbore inclination and azimuth by varying only the applied initial stress field or boundary conditions. The model is applied for calculating the optimum drilling mud-pressure assuming an elastoplastic rock behaviour and different failure criteria. The rock material parameters were derived from triaxial compression tests. The results show that the difference in mud- pressure predictions between elasticity and plasticity depends strongly on the employed failure criteria. The difference between elasticity and plasticity becomes more pronounced with a criterion which allows the material near the borehole wall to reach a critical plastic strain determined from the calibration tests. In such a case differences from 45% to 80% were encountered. Calibration factors given by the ratio of plasticity predictions over the elasticity predictions have been derived and can be used with analytical models for quick field wellbore stability computations.

How Do ∼2° Slopes Fail in Areas of Slow Sedimentation? A Sensitivity Study on the Influence of Accumulation Rate and Permeability on Submarine Slope Stability

Overpressure generation due to rapid sediment deposition can result in low effective stresses within the sediment column. It has been proposed that these large overpressures are the main preconditioning factor for causing large-scale submarine slope failure on passive continental margins, such as those in the Gulf of Mexico and offshore Norway. The rate of overpressure generation depends on the sedimentation rate, sediment compressibility and permeability. The Gulf of Mexico and the Norwegian continental slope have experienced comparatively high sediment input, but large-scale slope failure also occurs in locations with very low sedimentation rates such as the Northwest African continental margin. Here we show results from 2D numerical modelling of a 2° continental slope subjected to deposition rates of 0.15 m/ka. These results do not indicate any evidence for significant overpressure or slope instability. We conclude that factors other than overpressure must be fundamental for initiating slope failure, at least in locations with low sedimentation rates.

Three-dimensional stress analysis of a wellbore with perforations and a fracture

This study presents results of large scale 3-D elastic analysis of a wellbore with perforations. Both vertical and horizontal wellbores with perforations at different orientations are considered. The extra stresses imposed by a propped fracture at the unfractured perforations are also evaluated. A propped fracture results in increase of the compressive stress around the perforations which is higher in the perforations closer to the fracture and increases with propped width. The increase of compressive stress is much higher at the top/lower faces of perforations which is already less than the existing compressive stress at the lateral faces of the perforations. The results are also discussed in relation to the problem of hydraulic fracture initiation

Shear Localisation in Thick-Walled Cylinders Under Internal Pressure Based on Gradient Elastoplasticity

We studied failure of thick-walled cylinders under external confinement and internal pressurisation. The material is assumed to be pressure-sensitive with dilatant and strain-softening response. The analysis was carried out using Gradient Elastoplasticity, a higher order theory developed to regularise the ill-posed problem caused by material strain-softening. In this theory the stress increment is related to both the strain increment and its Laplacian. The gradient terms in the constitutive equations introduce an extra parameter of internal length related to material micro-structure, allowing robust modelling of the post-peak material behaviour. The governing equations were solved numerically with the displacement finite element formulation, using C1-continuity elements. Numerical results show that at a critical loading threshold the initial axisymmetry of deformation breaks spontaneously and an instability of finite wavenumber develops. With increasing pressurisation, a curved shear-band of finite thickness forms and propagates progressively towards the outer boundary. For high confining pressures this mode of shear failure is more critical than the trivial tensile failure mode. Practical applications can be found in wellbore stability and hydraulic fracturing in petroleum engineering, and in pile driving design and the interpretation of pressuremeter and penetrometer tests in geotechnical engineering.

On natural boundary conditions in linear 2nd-grade elasticity

This work aims at drawing the attention of mechanicians interested in the development of extended continuum theories on the unresolved issue of the physical interpretation of the additional boundary conditions introduced by 2nd-grade models. We discuss this issue in the context of the linearized theory of elasticity as an appropriate platform for discussion. Apart from lineal densities of edge-forces, 2nd-grade models allow for the prescription of force-like quantities energy-conjugated to the gradient of the velocity field on the boundary. Previous works proposed reductionistic interpretations, treating 2nd-grade models as particular cases of continua with affine microstructure; from the latter one can deduce field equations reminiscent of 2nd-grade models, either in the “low-frequency, medium wavelength” limit or by constraining the microstructural degrees of freedom to the gradient of the velocity field. The interpretation we propose here is based on the concept of ortho-fiber, and has the merit of achieving a simple physical interpretation of the boundary conditions without recourse to extensive algebra or the need to invoke microstructure.

Continua with microstructure: second-gradient theory: Theory, examples and computational issues

ABSTRACT Second-gradient theories represent a frequently used subset of theories of continua with microstructure. This paper presents an extended overview of second-gradient theories, starting from a simple one-dimensional example, proceeding with a thorough description of gradient elasticity and additionally briefly describing some other theories of this kind. A series of characteristic examples is presented to demonstrate the main aspects and applications of second-gradient theories. Finally, the complications in the finite-element implementation of second-gradient theories are presented, along with a review of the finite elements that have been developed for this purpose.