biphasic model at the expense of numerical stability. Moreover, the value of βmin is also accepted as a good estimate for αmin (cf. (4.44)2) to stabilise the monolithic scheme if uniform equal-order elements are used, although α is independent of the FE mesh size, cf. Huang et al. [94].
4.4 Treatment of Unbounded Domains 83
(3) In the framework of absorbing boundary condition (ABC) schemes, the method of perfectly matched layers (PML) appears as an important strategy to simulate the response of unbounded half-space dynamics, see, e. g., Basu [8], Basu & Chopra [9] and Oskooi et al. [141] for details. This method is originally developed for electromagnetic wave problems and then extended to elastic wave propagation in infinite domains. The PML is an unphysical wave absorbing layer, which is placed adjacent to the truncated near-field boundaries that are supposed to extend to infinity. The purpose of the PML is basically to prevent wave reflections back to the near domain. This scheme is applied mostly when solving wave equations in the frequency domain as the solution is based on frequency-dependent stretching functions.
In the current contribution, the simulation of wave propagation into infinity is realised in the time domain. Here, the near field is discretised with the FEM, whereas the spatial discretisation of the far field is accomplished using the mapped IEM in the quasi-static form as given in the work by Marques & Owen [127]. This insures the representation of the far-field stiffness and its quasi-static response instead of implementing rigid bound-aries surrounding the near field, cf. Wunderlich et al. [182]. This mapped IEM has al-ready been successfully applied by Schrefler & Simoni to simulate the isothermal and the non-isothermal consolidation of unbounded biphasic porous media, see [156, 158]. In particular, they have performed a coupled analysis under quasi-static conditions, where infinite elements with different decay functions are applied to the solid displacement, the pore pressure and the temperature fields. Moreover, they have calibrated the numerical results by comparison with respective analytical reference solutions.
However, in dynamical applications, some additional considerations should be taken into account. Here, when body waves approach the interface between the FE and the IE domains, they partially reflect back to the near field as the quasi-static IE cannot capture the dynamic wave pattern in the far field12. To overcome this, the waves are absorbed at the FE-IE interface using the viscous damping boundary (VDB) scheme, which basically belongs to the ABC class, cf. Figure 4.6, right.
000000000 000000000 000000000 111111111 111111111 111111111 FEM
IEM
VDB load
near field
far field
structure soil half space
0 1
0 1 00 11
0
1 0011
0 1 0
1 01
0 1
0 1
ΓFI
∞
∞ IE FE
θ
r1
r2
Figure 4.6: Viscous damping boundary method (VDB): FEM-IEM coupling with VDB at the interface (left), and rheological model with applied damping forces (right)
12The shape functions (decay functions) of the quasi-static infinite elements towards the infinity, as discussed in Section 4.3.2, are chosen for an easy-to-implement treatment as simple functions of time (polynomials) and not dependent on the frequency of the loading (e. g. exponential) as for dynamic infinite elements (cf. Khaliliet al.[102]) .
4.4.1 Viscous Damping Boundary Method (VDB)
The idea of the VDB is based on the work by Lysmer & Kuhlemeyer [120], in which velocity- and parameter-dependent damping forces are introduced to get rid of artificial wave reflections. In this, the verification of the proposed VDB scheme has been carried out by studying the reflection and refraction of elastic waves at the interface between two domains, where the arriving elastic energy should be absorbed. For more information and different applications, see, e. g., the works by Haeggblad & Nordgren [79], Underwood &
Geers [168], Wunderlich et al. [182] and Akiyoshi et al. [2].
The implementation of the VDB scheme in this work is confined to the case of the mate-rially incompressible biphasic model, where the considered set of equations is the uvp (2) formulation, cf. (3.45). Following this, the weak form of the overall momentum balance is split for the treatment of the far field into two parts: a quasi-static part discretised in space with the mapped IE method and a dynamic part replaced by damping forces integrated over the FE-IE interface ΓF I. In particular, one obtains
Z
Ω
gradδuS·(TSE −pI) dv− Z
Ω
δuS·ρbdv
| {z }
quasi-static (→IE)
+ Z
ΓF I
δuS·(aiρ ci)vSda
| {z }
dynamic (→VDB)
= 0.
(4.86)
In general, three apparent modes of bulk waves, which are two compressional and one shear wave, can be observed in biphasic solid-fluid aggregates, cf. Footnote 11 in Page 80.
However, as the treatment of unbounded domains in the current contribution is oriented to geotechnical problems (such as fully saturated silty sand under seismic excitation), only two body waves need to be considered. Here, the low-frequency excitation and the low permeability make the relative motion between the solid matrix and the viscous pore fluid under dynamic loading very slow. Thus, it is accepted that far from the permeable boundary the pore fluid is almost trapped in the solid matrix, and therefore, the fluid can approximately be treated as an incompressible material together with the solid phase. In other words, only biphasic poroealstic media with intrinsically incompressible solid and fluid constituents in the low frequency regime are addressed, which gives rise to only two types of bulk waves that have to be damped out. In particular, these are the longitudinal pressure wave and the transverse shear wave transmitted through the elastic structure of the solid skeleton.
Following this, in order to develop boundary conditions that ensure the absorption of the elastic energy arriving at a certain boundary, Lysmer & Kuhlemeyer [120] developed damping expressions for the boundary conditions. For two-dimensional (2-d) problems, the damping relations read
σ = (a ρ cp)vS1 : p-waves, τ = (b ρ cs)vS2 : s-waves.
(4.87) These equations are formulated for incident primary (p) and secondary (s) waves that act at an angle θ from the x1-axis, cf. Figure 4.6, right. In (4.86) and (4.87), ρ = ρS+ρF
4.4 Treatment of Unbounded Domains 85
is the density of the overall aggregate, vS1 and vS2 represent the nodal solid velocities in x1- and x2-direction, cp and cs are the velocities of the p- and s-waves given in equations (3.77) and (3.79), and a, b are dimensionless parameters (cf. [79]) given as
a = 8
15π(5 + 5c+ 2c2) , b = 8
15π(3 + 2c) with c = s
µS
λS+ 2µS . (4.88) The implementation of the method is fairly simple since one adds nothing more than dashpots with damping constants (a ρ cp) and (b ρ cs) to the degrees of freedom (DOF) of the FE-IE interface elements (Figure 4.6, right). The effectiveness of the VDB depends strongly on the wave incident angleθ. Indeed, it is shown in the work by Lysmer & Kuh-lemeyer [120] that a nearly perfect absorption of the incident waves can only be achieved for θ < 60, whereas some reflections occur for bigger angles. In the weak formulation, the damping terms are written in an integral form over the boundary ΓF I, which for 2-d problems read
r =
r1
r2
= Z
ΓF I
δuS·
a ρ cp 0 0 b ρ cs
vSda . (4.89)
Due to the dependency on the primary unknown vS, the arising damping terms in (4.89) enter the weak formulation of the problem in a form of nonlinear boundary integrals.
Thus, an unconditional stability of the numerical solution requires that these terms are treated implicitly by integrating over the boundary at the current time step in the sense of a weakly-imposed Neumann boundary, cf. Ehlers & Acart¨urk [59]. Here, relation (4.54) is employed for the numerical generation of the respective mass, stiffness and Jacobi matrices.
Chapter 5:
Liquefaction of Saturated Granular Materials
This chapter is devoted to the investigation of liquefaction phenomena, which usually appear in saturated granular materials after dynamic excitation. This includes definitions and descriptions of liquefaction mechanisms, factors affecting saturated soil behaviour and an in-situ example of seismic-induced liquefaction. Moreover, the elasto-viscoplastic constitutive model, as introduced in Section 3.2, is used to capture the basic features of liquefaction events, such as the pore-fluid pressure accumulation and softening of the granular structure.