Modified Elastic Model
It is clear that for soils the linear behavior is acceptable only for relatively low magnitudes of applied loads. This becomes evident upon unloading that usually shows a rather small number of elastic deformation compare to the overall deformation. The modified linear model attempts at least to some extent to take this into account by considering different modulus for loading and unloading as plotted in the Figure.
A drop in the material stiffness along a given loading path attributed to the plastic yielding is reflected through a modulus of elasticity E, which can be imagined as a secant modulus associated with a certain stress level.
An elastic response is assumed upon unloading. To increase the clarity of model formulation the elastic modulus for the unloading branch is replaced by the unloading-reloading modulus Eur that governs the response of a soil upon unloading and subsequent reloading up to the level of stress found in the material point prior to the unloading.
With reference to the following Figure these moduli are given by:
where: | E | - | Modulus of elasticity (secant) [MPa] |
Εur | - | Unloading/reloading modulus [MPa] |
(a) Real stress-strain diagram of soil, (b) Simplified stress-strain diagram for MLM
During the primary loading the response of a soil is therefore governed by the modulus of elasticity E while upon unloading it follows the path set by the unloading-reloading modulus Eur. An approximate value of this modulus is (2-5)*modulus of elasticity E. In every case, both parameters should be obtained from reliable experimental measurements.