Relaxation: Stress relaxation describes how materials relieve stress under constant strain. Its nonlinearity is described by both stress relaxation and creep.
Strain rate effects: When creep happen under the undrained normal-consolidation condition and drained over-consolidation condition, the creep strength decreases along with the
decreasing of the strain rate.
Time effect: When creep happens under the undrained normal-consolidation condition and drained over-consolidation condition, the creep strength Su decreases with the extending of loaded time.
(c) Explain the concept of “equivalent time” for creep.
“Equivalent time” for creep is the time of creep down from the reference time line and has the same creep rate as that at the same stress-strain state for any other loading history.
8.2
(a) Show steps to derive Yin & Graham’s 1-D elastic visco-plastic (1-D EVP) model in Eqn. (8) in a paper by Yin and Graham (1994).
(b) Discuss how to determine all model parameters.
The 1-D EVP model parameters cab be determined as follows.
Or a approximate method can be used.
8.3
(a) Derive the differential equation of Maxwell rheological model.
The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected consecutively, as shown in the diagram. In this configuration, under an applied axial stress, the total stress, σTotal and the total strain, εTotal can be defined as follows:
σTotal = σD = σS
εTotal = εD + εS
where the subscript D indicates the stress/strain in the damper and the subscript S indicates the stress/strain in the spring. Taking the derivative of strain with respect to time, we obtain:
where E is the elastic modulus and η is the material coefficient of viscosity. This model describes the damper as a and models the spring with .
In a Maxwell material, σ, ε and their rates of change with respect to time t are governed by equations of the form:
or, in dot notation:
(b) Calculate the creep strain for any given time and under a given stress σ using the Maxwell rheological model (assuming initial strain and stress of and).
(c) Discuss the limitations of Maxwell rheological model.
It can only deal with the linear situation, however in most cases it is non-linear and creep is non-linear in accordance with the reality, say, compression of clays in oedometer condition.
(d) Discuss the similarity and difference between Maxwell rheological model and Yin and Graham’s 1-D EVP model.
Similarity: We can both calculate the total strain which consist elastic strain and visco-plastic strain using these two models.
Differences: In Maxwell’s Rheoloical Model, the strain rate is linear which mean we can we only use this model in which condition the property of strain rate perform linear. Whereas the Yin and Graham’s 1-D EVP model can calculate the strain in accordance with the reality as they assume the strain rate is Non-linear, no matter the elastic one or the visco-plastic one expressed in the equation. The 1-D EVP model can take more factors into consider like the time effect which is not demonstrated in the Maxwell’s Rheoloical Model.