For soils, the behaviour depend on the stress path followed. The total deformation of such materials can be decomposed into a recoverable part and an irrecoverable part. Hypoelasticity constitutes a generalized incremental law in which the behaviour can be simulated from increment to increment rather than for the entire load or stress at a time. In hypoelasticity, the increment of stress is expressed as a function of stress and increment of strain. The Hypoelastic concept can provide simulation of constitutive behaviour in a smooth manner and hence can be used for hardening or softening soils.
Hypoelastic models can be considered as modification of linear elastic models. However, it may incrementally reversible, with no coupling between volumetric and deviatoric responses and is path-independent.
5.2 Use sketches to explain the physical (geometric) meaning of all 7 parameters (only 5 independent) in a cross-anisotropic elastic soil model ().
Figure 5.1 Parameters in cross-anisotropic elastic model
- Young’s modulus in the depositional direction;
- Young’s modulus in the plane of deposition ;
- Poisson’s ratio for straining in the plane of deposition due to the stress acting in the direction of deposition;
- Poisson’s ratio for straining in the direction of deposition due to the stress acting in the plane of deposition;
- Poisson’s ratio for straining in the plane of deposition due to the stress acting in the same plane;
- Shear modulus in the plane of the direction of deposition;
- Shear modulus in the plane of deposition.
Due to symmetry requirements, only 5 parameters are independent.
Assignment 6 (Lecture 6 – Elasto-plastic behaviour):
6.1
(a) Explain and discuss (i) yield, (ii) yield criterion, (iii) potential surface, (iv) flow rule, (v) normality, (vi) consistency condition.
(i) The yield strength or yield point of a is defined in and as the at which a material begins to . Prior to the yield point the material will deform and will return to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the deformation will be permanent and non-reversible. In the uniaxial situations the yield stress indicates the onset of plastic straining. In the multi-axial situation it is not sensible to talk about a yield stress. Instead, a yield function is defined which is a scalar function of stress and state parameters.
(ii) A yield criterion, often expressed as yield surface, or yield locus, is an hypothesis concerning the limit of elasticity under any combination of stresses. There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide a justification based on established physical principles. Since stress and strain are tensor qualities they can be described on the basis of three principal directions, in the case of stress these are denoted by , and .
(iii) Potential surface is the segment of a plastic potential surface plotted in principal stress space, as shown in Figure 6.1 (a). A two dimensional case was shown in Figure 6.1 (b).
(iv) Flow rule:
– a scalar multiplier;
– plastic potential function;
{} – location of surface (a vector), not in the final equation
Figure 6.1 Plastic potential presentation
(v) Assuming the plastic potential function to be the same as the yield function as a further simplification:
The incremental plastic strain vector is then normal to the yield surface and the normality condition is said to apply.
(vi) Having defined the basic ingredients of an elasto-plastic constitutive model, a relationship between incremental stresses and incremental strains then can be obtained. When the material is plastic the stress state must satisfy the yield function. Consequently, on using the chain rule of differentiation, gives:
This equation is known as the consistency equation or consistency condition.
(b) Explain and discuss the associate flow rule and non-associate flow rule and how the two rules affect the volumetric deformation and the bearing capacity of a strip footing on sand.
Sometimes simplification can be applied by assuming the plastic potential function to be the same as the yield function (i.e. ). In this case the flow rule is said to be associated. The incremental plastic strain vector is then normal to the yield surface and the normality condition is said to apply. In the general case in which the yield and plastic potential functions differ (i.e. ), the flow rule is said to be non-associated. If the flow rule is associated, the constitutive matrix is symmetric and so is the global stiffness matrix. On the other hand, if the flow rule is non-associated both the constitutive matrix and the global stiffness matrix become non-symmetric. The inversion of non-symmetric matrices is much more costly, both of storage and computer time.
As noted, it occurs in a special class of plasticity in which the flow rule is said to be associated. Substitution of a symmetric for all elements in a finite element mesa, into the assembly process, results in a symmetric global stiffness matrix. For the general case in which the flow rule is non-associated and the yield and plastic potential functions differ, the constitutive matrix is non-symmetric. When assembled into the finite element equations this results in a non- symmetric global stiffness matrix. The inversion of such a matrix is more complex and requires more computing resources, both memory and time, than a symmetric matrix. Some commercial programs are unable to deal with non-symmetric global stiffness matrices and, consequently, restrict the typo of plastic models that can be accommodated to those which have an associated flow rule.
(c) Explain plastic strain hardening and plastic work hardening or softening.
The state parameters, , are related to the accumulated plastic strains . Consequently, if there is a linear relationship between and so that
then on substitution, along with the flow rule, the unknown scalar,, cancels and A becomes determinant. If there is not a linear relationship between and , the differential ratio on the left hand side of the above equation is a function the plastic strains and therefore a function of . When substituted, along with the flow rule given, the A's do not cancel and A becomes indeterminate. It is then not possums to evaluate the []. In practice all strain hardening/
softening models assume a linear relationship between the state parameters and the plastic strains .
In this type of plasticity the state parameters}, are related to the accumulated plastic work, ,which is dependent on the plastic strains it can be shown, following a similar argument to that parented above for strain hardening/softening plasticity, that as long as there is a linear relationship between the state parameters }, and the plastic work, , the parameter defined becomes independent of the unknown scalar, , send therefore is determinant. If the relationship between and is not linear, become a function of and it is not possible to evaluate the constitutive matrix.
6.2 Show steps to derive the elastic plastic constitutive matrix [] in (6.16).
The incremental total strains can be split into elastic and plastic , componets. The incremental stress, are related to the incremental elastic strains, by the elastic constitutive matrix:
Or alternatively
Combining gives
The incremental plastic strains are related to the plastic potential function, via the flow rule. This can be written as
Substituting gives
When the material is plastic the stress state must satisfy the yield function. Consequently, which, on using the chain rule of differentiation.
This equation is known as the consistency equation. It can be rearranged to give
Combining, we can get
Where
Substituting again
So that
6.3 The dimension of a slope is shown in Figure 6.2. Calculate the factor of safety of the following cases:
(a) Without tension crack, the properties of Soil (1) are kPa, , kN/m3; The properties of Soil (2) are kPa, , kN/m3 (no water table).
(b) With tension crack filled with water, repeat the calculation in (a).
(c) Without tension crack, the properties of Soil (1) are kPa, , kN/m3 (below water table) and kN/m3 (above water table); the properties of Soil (2) are kPa, , kN/m3 (below water table) and kN/m3 (above water table). Water table is shown.
Figure 6.2 Dimension of the slope and water table
(a)
Figure 6.3 Model without tension crack or water table
Factor of Safety: 1.498
Figure 6.4 Results without tension crack or water table
Figure 6.5 Slice 1 – Morgenstern-Price Method
(b)
Figure 6.6 Model with tension crack filled with water
Figure 6.7 Results with tension crack filled with water
The safety factor : 1.406
Figure 6.8 Slice 1 – Morgenstern-Price Method
(c)
Figure 6.9 Model without tension crack but with water table
Figure 6.10 Results without tension crack but with water table
Factor of Safety: 1.258
Figure 6.11 Slice 1 – Morgenstern-Price Method
