ex-ample, the uniaxial strength of freshwater ice strength in compression is 2-10 times larger than in tension. It is also suggested that hydrostatic pressure would result in a phase change (Kim, 2013).
During the impact between the iceberg and structure, large confinement and compres-sion appears within the contact area of ice. The existence of the high-pressure zone shown in the Figure2.8(b) makes ice features next to the edge easily spall and extrusion. There is also the phase change from the ice to water in these small areas. Thus, the resistance to deviatoric stresses is reduced in high hydrostatic pressure. To model this phenomenon, a shear-cap yield surface, such as "Tsai-Wu" yield surface, is normally used to simulate ice-impact behaviours.
Temperature Dependence
A comprehensive study on the temperature profile is performed by Loset. It is found that there exists a temperature gradient from the surface to the core region for the iceberg in sea because of the low thermal conductivity of ice. The submerged part of iceberg is sub-jected to a steep temperature gradient and the core temperature is reached approximately 3 meters from the surface (Loset, 1993). When the iceberg is modelled in the simula-tion tool, this gradient needs to be considered. The yield criterion should be modelled as temperature-dependent and the temperature gradient should be applied. For example, a linear interpolation of temperature gradient is used for elements between the surface and core area using the computer code LS-DYNA by Liu (Liu et al., 2011).
2.3 Analytical Ice Material Model
It is impossible to develop an universal model of the ice for every condition due to its var-ious existence in different condition. In this thesis, only the impact problem is considered.
As mentioned previous, several special properties need to be included in the modelling of iceberg ice. An elastic-perfect plastic model based on the data from triaxial experiments to describe the ice material behaviour during iceberg impacts is developed by Liu (Liu et al., 2011). In this section, the analytical model is introduced in detail with respect to the yield surface, return mapping algorithm, failure criterion, erosion technique and flow rule.
2.3.1 Yield Surface
The yield surface is formulated as an algebraic combination of the invariants of stress tensorΣij.
Due to the dependence on the hydrostatic pressure, some widely used yield-surface techniques such as Von Mises, Drucker-Pager and Mohr-Coulomb are not suitable. In the iceberg-structure impact, the ice particles in the center contact area would be confined by neighbouring particles, which means the ice is in a triaxial stress state. A suitable yield surface should be adopted according to triaxial experiment results.
An elliptical yield envelop for the iceberg is proposed by Derradji-Aouay as shown in Equation2.2.
Chapter 2. Theory Review
3 is the octahedral stress, Sij is the deviatoric stress andpis the hydrostatic pressure.
Mathematically, this yield surface is same as the "Tsai-Wu" yield surface on the con-ditionη= 0. For an isotropic material, it is usually written as:
f(p, q) =q−p
a0+a1p+a2p2 (2.3)
,where p = Σ3kk = I31 is the hydrostatic pressure,q = 32p
Sij:Sij is the Von Mises stress, anda1,a2anda3are the constant that requires fitting to triaxial experimental data.
In order to make the implementation convenient, the "Tsai-Wu" yield surface is written as:
f(p, J2) =J2−(a0+a1p+a2p2) = 0 (2.4) , whereJ2is the second invariant of deviatoric stress tensor.
This yield function is defined that there is no loading if it is a negative value. The value zero means the elastic limit and plasticity will occur when the function lies on this limit.
The positive value is inadmissible. During plasticity deformation, the yield surface should change to lay on the limit. The shapes of the yield surface inp−J2space adopted from different data sources are shown in the Figure2.9. It is clear that the difference exists due to different data sources and fitting methods. To reflect the temperature dependence, the a1,a2anda3are set as functions of the temperature in the Equation2.4.
Figure 2.9:Illustration of "Tsai-Wu" yield surface inp−J2space (Liu et al., 2011)
2.3 Analytical Ice Material Model
2.3.2 Return Mapping Algorithm
As mentioned above, when the plasticity happens, the stress should map back to the yield surface.
In the model, the Cutting Plane Algorithm (CPA) is used. First, the elastic equations are integrated with total strain increments to obtain an elastic predictor. Then, the elasti-cally predicted stresses are mapped to a suitably updated yield surface by correcting the plastic strain increments iteratively (Huang and Griffiths, 2009). The procedure is shown schematically in the Figure2.10.
Figure 2.10: Illustration of the stress return method corresponding to CPA (Huang and Griffiths, 2009).
The plastic correction phase is driven by a plastic multiplier, 4λn+1. Thus, during the plastic-predictor stage, the plastic strain remains fixed, and during the plastic-corrector stage, the total strain is fixed. Enforcing an normality rule at the beginning of the step, we have:
{4σ(k)}={σ(k+1)} − {σ(k)}=−[De]{4σp(k)} (2.5)
{4εp(k)}=4λ(k){a(k)} (2.6)
, where k is the iteration number and[De]is the constitutive matrix.
At every iteration, linearizing the yield function f around the current stress values {σ(k)}, we have:
fk+1=fk+{a(k)}T({σ(k+1)} − {σ(k)}) (2.7) By settingfk+1= 0, we have:
Chapter 2. Theory Review
4λ(k)= fk
{a(k)}T[De]{a(k)} (2.8)
2.3.3 Failure criterion
Depending on the stress state and hydrostatic pressure, different failure modes of the ice may occur.
In the iceberg impact problem, the ice is well confined and the hydrostatic pressure is an important influencing factor. It is shown taht the hydrostatic pressure and friction may trigger different failure mechanisms, namely frictional or Coulombic faults and non-frictional or plastic faults, which are shown in the Figure2.11(a) (Schulson, 2009). At a lower confinement, the friction sliding of the ice is not suppressed and the shear force results in the Coulombic failure. At a higher confinement, the friction sliding is restricted and the plastic failure occurs.
(a) Schematic sketched of two kinds of com-pressive shear fault (Coulombic faults form under lower degree of confinement and plas-tic faults form under higher degrees of triaxial confinement) (Schulson, 2009)
(b) Illustration of the U-shape strain based failure criterion curve proposed by Liu (Liu et al., 2011)
Figure 2.11:Illustration of failure criterion of iceberg ice
When it is related to iceberg impact, the Coulombic appears at the beginning of contact.
For those elements in this state, reaching a certain level of shear force would trigger its deletion from analysis using erosion technique. This is easy to satisfy because the pressure is increasing during Coulombic faults. As the interaction continues, the appearance of plastic failure makes the failure criterion hard to reach because the increasing hydrostatic pressure results in a much stiffer ice.
In Liu’s model, the usage of the "Tsai-Wu" yield surface and the CPA makes it possible to introduce more advanced failure criteria through the strain rate. Based on the hypothesis that the stiffness of ice will not change significantly in the loading process until later when the plastic limit load is approached, a U-shaped strain-based failure criterion as shown in
2.3 Analytical Ice Material Model the Figure2.11(b) is proposed (Liu et al., 2011). This failure criterion to simulate the ice-fracture mechanics is empirical and based on the effective plastic strain and hydrostatic pressure as shown below:
,whereεpeq is the equivalent plastic strain,εpij is the plastic strain tensor,εf is the failure stain,ε0is the initial failure strain, which should be determined by fitting experimental data andp2is the larger root of the yield function. The erosion is activated if theεpeq > εf or the pressure is not greater than the cut-off pressurepcut. The cut-off pressure is described as the strength difference between the tensile and compression states.
2.3.4 Erosion Technique
The erosion technique means that the elements violating the failure criterion will be deleted from the analysis.
In this implemented model, when the erosion occurs, the deviatoric stresses on the element are set to zero, but not the hydrostatic pressure. This can be viewed as a rapid softening process due to the fact that the removal of an element is equivalent to setting stress level zero (Kjerstad, 2019).
In practice, there are two problems existing related to the "erosion". One is that it does not simulate any additional loads caused by the extrusion of the crushed ice, which may play an important role during the crushing. Another is that this technique largely depends on the meshing of the ice , thus it is necessary to check the convergence property.
2.3.5 Flow Rule
As mentioned in the part of the yield surface, the associated flow rule, a nominal flow rule, is used. In this case, the plastic strain increment is normal to the yield surface. It means that the yield surface, as the plastic potential, could used to derive the plastic strains. The plastic strain increment,dεp, is determined by:
dεp=dλ∂f
∂δ =dλ4f (2.11)
,where f is the yield function anddλ is the plastic multiplier, which means the plastic strain magnitude is normal to the yield surface.
Chapter 2. Theory Review
Chapter 3
IACS Unified Requirements Review
In this chapter, as a design reference of the ships constructed of steel and intended for inde-pendent navigation in ice-infested polar waters, the IACS UR is reviewed. The mechanics of the ice load determination is introduced in detail and the same algorithm is used in the derivation of the ice mechanics model later.