Secant modulus reduction curve for Hardening soil model with small strain stiffness (HSS model)

Question

Based on PLAXIS material introduction, the tangent and secant modulus reductions vs shear strain satisfy: 
 
 To capture the strain hardening at plastic zone, a lower bound was introduced to HSS model, as with Figure 7.3 below. 
 
 In terms of tangent shear modulus reduction, there is a significant cut-off, at which Gur = Gt as per Figure 7.3 above. 
 I would like to know: 
 
 1) Could this cut-off be employed to secant modulus reduction? 
 2) What&rsquo;s the equation for the &ldquo;new secant modulus reduction curve&rdquo; when we take this cut-off into consideration (for example, the results that I obtained from PLAXIS in the Data Example diagram ? 
 Unfortunately, I was not able to find the equation for this &ldquo;new secant modulus reduction curve&rdquo; from PLAXIS material manual. Please let me know. Thank you very much!

Stefanos Papavasileiou · Answer

Dear Wang, 
 In Finite Element theory, the constitutive model plays a key role in 2 operations: 
 (1) definition of the element stiffness matrix (which will be later assembled into the global stiffness matrix) and (2) computation of stress change corresponding to a given strain change. 
 The first operation consists only of writing the elastic (or elasto-plastic) tangent constitutive matrix for the current stress state. The second operation involves the integration of the constitutive model at Gauss point level. 
 For an elasto-plastic constitutive model (such as HS Small), a numerical algorithm is required to perform accurately the integration of the constitutive equations. In such conditions, tangent elastic (and also plastic) quantities are used. Therefore, it is only required to establish a lower limit cut-off for the tangent shear modulus. 
 Among other reasons, the presentation of the secant shear modulus reduction curve predicted by the constitutive model intends to assist the calibration process, since results of conventional laboratory tests (e.g. resonant column test) are often presented in terms of secant shear modulus as a function of the shear strain level (as well as damping ratio evolution with shear strain level). 
 In addition, linear-equivalent methods (which use closed-form solutions in the frequency domain) typically use secant quantities and, therefore, there is a large accumulation of experience in terms of secant shear modulus curves for different types of materials. 
 For further details on the cyclic non-linear elastic component of the HS small, please refer to the following article: https://communities.bentley.com/products/geotech-analysis/w/plaxis-soilvision-wiki/46056/hysteretic-damping-in-a-small-strain-stiffness-model .