Microplane modeling of the elasto-viscoplastic constitution

In this paper, the elasto-viscoplastic Constitutive model is applied within the Microplane framework. The use of strain-dependent models allows measuring the effect of loading speed on the soil. Additionally, rate-based behavior models in simulation modeling avoid the uniqueness of the ruling equation. The proposed model can plot the stress-strain history on plates with different angles inside the soil. Therefore, valuable information can be obtained about the failure plane. Using the Microplane framework enables this hybrid behavior model to predict local strain.


INTRODUCTION
Microplane refers to planes within materials that have different angles. This model is used to examine the structure of fine materials. Taylor proposed the original idea for this model and was called Slip Panels (Pham et al, 2011). Taylor's idea was formulated by Roters (2019). Many researchers have used this model to study metals, soils, and rocks.
Before 1984, microscopic Constitutive models were developed on a consistent fixed basis. But proving satisfactory coping conditions was a difficult issue for this Constitutive model. This ambiguity was resolved by Bazant and Gambarova (1984). In this method, instead of the stress tensor image, the strain tensor image is applied to the image planes. As a result, the compatibility condition is fulfilled and the equilibrium conditions are created using the Principle of Virtual Work.
The elasto-viscoplastic Constitutive model identified in this paper was presented by Teichtmeister et al., (2017). This Constitutive model was developed using concepts of critical state and viscoplasticity. The benefits of this Constitutive model include primary and secondary pressure modeling, predicting the effect of stress rate on Untrained shear tests, and creep modeling. In this constitutive model, the elastic strain is considered to be non-rate dependent. An advantage of this assumption is the ease of use of this constitutive model. Besides, the dependence of plastic strain on time seems more rational (Perzyna, 1966;Wu et al., 2018;Diehl et al., 2017).
Microplane modeling of the elasto-viscoplastic behavior, allows us to use the unique features of both models simultaneously. On the one hand, the ratedependent behavior of the soil is simulated by the elastoviscoplastic model, and on the other, the behavior of the fine or microstructure is described by the Microplane framework.

Formulation of elasto-viscoplastic Constitutive model
Teichtmeister The viscoplastic strain is calculated from the following equation: Where the parameter  is calculated from the compatibility conditions. Three surfaces are used in this framework: 1- The yield surface that determines the state of the sample at the reference time ( t ). The definition of such a surface is to obtain the state of the soil sample at the required time ( t ).
2-yield surface under loading ( 0  f ): A surface that contains the stress state.
The surface that determines the path and rate of strain. ( fˆ) can be greater or smaller than ( f ). These three yield surfaces have the same formula but their sizes are different. The location of the surface with the horizontal axis is the parameter that determines the size of each surface ( Figure 1). This parameter is equal to ( L p ) for the loading yield surface and the reference yield surface is ( 0 p ) and for the potential surface is ( 0 p ).
The value of ( L p ) changes as a result of loading, but This increase causes a viscoplastic strain in the model.
The ( 0 p ) value is calculated from the following equation:  The relationship between 0 p and 0 p is calculated from the following formula: is the vector perpendicular to the potential surface in the stress path.

Microplane formulation
The Virtual work performed by stress on the sphere volume of a per-unit radius can be obtained from the stress multiplier integral in the strain development.
In the above relation ij δε is the strain tensor and ij σ  is the stress tensor. The strain vector ) n dε ( can be divided into any plane by the perpendicular to the strain ( N dε ) and the tangent to the plane ( T dε ).
i n is the vector components perpendicular to the plane and ij δ is the function of the delta kronker. Using the principle of virtual work, we can find the following relation for virtual work in terms of stress and strain components that are depicted on the desired plane: Where A is the hemisphere surface with a unit radius, N σ  is the perpendicular stress vector and T σ  is the tangential stress vector. If there is a Constitutive relationship between stress and strain, the following relationships can be rewritten for stress with the strain depicted on the plane:    Using the definitions of the perpendicular and tangent components of the stress on the plane, the following relation is obtained:

Applying elasto-viscoplastic Constitutive model in Microplane framework
Using the Pastor-Zienkiewicz Constitutive Model, the following relation can be presented for plastic strain on plates: The total stress development can be calculated by placing the equation 25 in equation 20.  As shown in Figures 2 and 3, the modeling results are in good agreement with the experimental results.
As the strain rate increases, the constitution of the consolidated normal soil will be similar to that of the preconsolidated soil. In addition, by increasing the strain rate (1% / min), the pore water pressure initially increases and decreases at the end of the test, which indicates constitution similar to pre-consolidated clay. However, for the low strain rate, the pore water pressure increases steadily and then remain constant.    Figures 6 and 7 illustrate the changes in normal stress relative to normal strain. The normal stress on planes 5,6,7 and 8 shows a maximum value as the strain rate increases. Using Figures 8 and 9, it can be concluded that the maximum shear stress on planes 5,6,7 and 8 is more likely due to the local strain on these planes.

CONCLUSION
A constitutive model was presented to describe the mechanical behavior of the clay. This constitutive model was created by applying an elasto-viscoplastic constitutive model in the Microplane framework. The proposed model is capable of depicting behavior on planes within the material. we can detect failure mechanisms with this feature. The angles of fracture planes are determined with using this model. In addition, the constitutive dependence of material to time can also be modeled well. Briefly, the presented model has the advantages of both constitutive models.