ACME 2011 Proceedings of the 19 UK National Conference of the ...

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through individual rock blocks and along joint systems. Consequently, there is a need to model Darcy type seepage within the continuous rock and pipe (sheet in 3D) network flow along the joint planes. This fluid response is strongly coupled to the deformation of the rock arising from loading and progressive fracturing, which in turn is driven by the fluid pressure imposed on the rock. Computationally, these problems can be treated by employing the same mesh that is used to model fracture within individual blocks to represent seepage flow and using compatible joint elements to simulate inter-block fluid flow. In order to understand the complex interplay between fluid and fractured rock, several approaches and theories have been proposed. One strategy aims to model the flow through cracks without taking into account the flow within the material itself. This simplification is reasonable for soils or rock-like materials with low permeabilities, since the flow along the fractures is dominant. However, an accurate analysis of the in situ stress field can only be obtained if the rock material itself is treated as a porous medium. This is particularly important for rock masses that contain a high degree of small scale fractures resulting in high permeabilities so that the seepage behaviour becomes prominent. The main modelling issues are briefly discussed below. Constitutive & governing equations for porous materials The widely used “effective stress” in porous media employs the Biot number which is introduced to take into account the volumetric deformability of the particles. It is related to the bulk modulus of the skeleton and the bulk modulus of the grains. The effective stress is related to the incremental strain and rotation by means of an incremental constitutive relationship involving the rotational stress in the Green-Naghdi rate and a fourth order tensor defined by state variables and the direction of the increment. The governing equations can be written in a u-p formulation that involve (i) the total momentum equilibrium equation for the partially saturated solid-fluid mixture where the acceleration of the fluid relatively to the solid and the convective terms are neglected. This assumption is valid for medium speed and dynamics of lower frequency phenomena, (ii) an equation that ensures mass conservation of the fluid flow in the seepage field and (iii) a mass conservation equation based on a cubic law for the flow within the fracture where an individual fracture is treated as a single confined aquifer. The system defined by the above three governing equations and appropriate boundary conditions describes a well-defined problem which can then be discretized by standard finite elements in space and by a central difference approximation in time. Figure 3. Vertical stress profile in the plane of the Figure 4. Influence of the leak-off on the size of the fracture for elastic and elasto-plastic simulations. fracture. 4
Numerical example The numerical model presented here is validated against the experimental results published by van Dam et al. 2000. The experiment consists of cubic blocks 0.30m in size, which are loaded in a true triaxial machine to simulate in-situ stress states. After reaching the desired stress state, a high-pressure pump injects fluid to propagate the crack. Artificial rock samples made of cement, plaster and diatomite have been used. The fracturing fluid employed was silicon oil, which approximately behaves as a Newtonian fluid. Further details of the experiments can be found in van Dam et al. 2000. Three different cases have been considered: elastic, elastoplastic and poroelastic analyses. In the elastic and elastoplastic cases the vertical stress (σyy) profile has been compared with the ones provided in van Dam et al (2000). In the poroelatic case, different permeabilities for the rock were evaluated in order to verify the influence of the leak-off upon parameters such as length and aperture of the fracture. The permeability varies from 0 to 50000 miliDarcy (mD). The mechanical behavior of the rock is described through a Mohr-Coulomb model which employs the Newton-Raphson method in the return mapping scheme. The cohesion and friction angles are assumed to be given functions of the effective plastic strain. An interface law based on a cohesive-zone model has been used to describe the rupture process at the fracture tip. In this model, the softening curve is obtained through the fracture energy release rate. Within this framework, the fracture is opened when the tensile strength and fracture width reach a critical value. Figure 3 shows that the vertical stress of the current model (σyy) agrees well with the results of van Dam et al (2000). The aperture in the elastoplastic case is larger than the elastic case due to the inelastic deformations. Also, the net pressure necessary to propagate the fracture in the elastoplastic case must be higher than in the elastic case. This can be seen in the vertical stress profile, where an increase in (σyy) is experienced near the fracture tip, which is at the same position in both analyses. Figures 4 shows the aperture profiles in the poroelastic analyses for rocks with different permeabilities. It can be seen that in rocks with higher permeabilities, the length and width of the fracture reduces due to an increase in the leak-off from the fracture to the porous rock. Rock blasting. In this application coupling takes place through interaction between the gas pressure due to explosive detonation and the progressively fracturing rock. The most appropriate route to solution is provided by superposing a background Eulerian grid over the Lagrangian mesh used for fracture modelling (Owen et al. 2004). Within this regular Eulerian grid the gas pressure modelling is based on the mass conservation and momentum equations for gas flow employing directional porosities derived from the rock fracture simulation. The coupling takes place through an interdependence between the evolving gas pressure distribution driving the fracturing process which, in turn, provides the porosity distribution which controls the gas pressure. Computationally, solution can be effectively provided through use of a staggered solution scheme based upon time integration of the two fields with partitioned time stepping. The theory of gas flow through rock cracks involves a mass conservation equation in which the component velocities of the gas are related to the gas density and the local directional porosities of the rock mass. The gas velocities are obtained through the momentum equation as a function of the pressure gradients. Since the gas pressure arises from the detonation process, the initial time/pressure relation is obtained using the equation of state for the explosive. Numerical results are presented below for a HPE test. This is a plan view (plane strain) test case where a borehole of 165 mm diameter, containing an ANFO explosive charge, is located at the centre of a 3m x 3m square rock mass. The meshes employed for the rock fracture and gas pressure development are shown respectively in Figure 5 (a) and (b) and the crack patterns developed at 1000µs are illustrated in Figure 5 (c), where the surface cracking due to reflection of the compressive pressure pulse as a tensile wave is readily apparent. 5
Page 1: ACME 2011 Proceedings of the 19 th
Page 4 and 5: First published 2011 by Heriot-Watt
Page 7 and 8: CONTENTS Invited Lectures CHALLENGE
Page 9 and 10: A DAMAGE-MECHANICS-BASED RATE-DEPEN
Page 11: A PROJECTED QUASI-NEWTON METHOD FOR
Page 14 and 15: minimise the strain energy density.
Page 18 and 19: (a) Finite element mesh (b) Gas mes
Page 21 and 22: Proceedings of the 19 th UK Confere
Page 23 and 24: T E −1 ( ) ( ) T ∂s ∂T − du
Page 27 and 28: nondimensional tangential velocity
Page 31 and 32: Figure 2: Results for an open arter
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that R Γ Spin[N]XdΓ = 0, the func
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Proceedings of the 19 th UK Confere
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4 INCORPORATING EPRCM IN FEA The de
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No distinct yield point is observed
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stress [MPa] 1 0 -1 -2 -3 -2.5 -2 -
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ecomes available, the material mode
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where ∂fin ∂UM force 2.5 2 1.5
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Practicallly, the perturbation fiel
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3 SIMULATING CONCRETE MESO-STRUCTUR
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¯K i is an approximation of the co
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h = 5.36 matrix 4.3 z P E f νf 0.9
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From the total hoop strainε θ , t
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3 EXTERIOR POINT ESHELBY BASED CRAC
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is applied uniformly on the edge to
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(a) (b) (c) (d) Figure 1: Influence
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On each subdomain, we then have to
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( i�1) ( i�1) �Un�1 � ε
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� ~ � 1 ψ ~ � � ~ (2) �
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For the case of uni-axial tension,
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3 Numerical results Preliminary num
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The coefficients An cannot be deriv
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The spatial semi-discretisation is
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3 CRITICAL TIME STEP INCREASED WITH
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v/vo 2E-05 1.5E-05 1E-05 5E-06 0 -3
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3 RESULTS AND DISCUSSION Fig. 1a sh
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Model updating from eigenvalue and
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1. If R = ω 2 n, there is no eigen
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F = ´ Ω STv CSv dΩ A = ´ Γ U
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Fig. 3 Wall normal pressure in the
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contact contact (ii). f ≈ f ( d,
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In the analysis of the incident P-w
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elow a certain threshold. In both t
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K/d 2 Effective Permeability 0.0032
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METHOD 2 (Mellor and Herring Type)
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(a) Free Surface Profile from Janos
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2 IMMERSED FLUID SOLVER Consider th
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References [1] R. Mittal and G. Iac
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a second impact on the forward fuse
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Acknowledgement L. Papagiannis is f
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acceptable probability throughout t
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algorithm was then responsible for
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strategy of saving the computationa
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(a) (b) Figure 5. Comparison of (a)
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Thermal load step i i i i Update ,
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Displacement (mm) Conclusions 45 40
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load carrying of the structure to b
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compressive stress field between co
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Unbonded flexible pipes and risers
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Figure 2: (a) Riser configuration (
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Furthermore the total strain varies
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Displacement 14 12 10 8 6 4 2 0 Mem
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As this project was in the conceptu
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kg and additional mass to represent
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1 Step make box 2 Step make cylinde
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element method. The solid model is
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just new element classes) without t
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Fig.4. Frame model and the displace
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capacities to meet with such demand
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esponses. Severe cases are to be co
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2 Numerical results In this section
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the frequency intially increases un
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and F (R) = ⎡ ⎢ ⎣ ... UαT (R
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Merit function g 10 1 10 0 10 -1 10
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2 STRUCTURAL CHARACTERISTICS OF PTM
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Fig. 5. Simplified FE model of a PT
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general the computational results o
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References [1] Ainsworth M, Oden JT
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Error Cause Interpolation Error Ele
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(a) Before Improvement (b) After Im
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een applied to FE fluid flow proble
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Acknowledgements Figure 2: Flowchar
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1 1 ′′ 1 ′′′ 3 ′′ 2
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3 COMBINING SHAPE FUNCTIONS The exp
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electron current). For the case of
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electron concentration (1/cm 3 ) x
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2 CONSERVATION LAW FOR A MOVING CV
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5 NUMERICAL RESULTS A 1-D cellular
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FEM can be found in [3]. In this pa
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h Y 1 0 −1 −2 −3 −4 −5
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applications. The partition of unit
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1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 2
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2 NURBS BASIS FUNCTIONS NURBS are a
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4 RESULTS To illustrate the isoBEM
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290
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1 1 2 { σ ( ξ, s) } = [ D] { ε (
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4.4 Example 4 - A single edge crack
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