PREISACH MODEL USED TO DECRIBE RESPONSE OF CLAY IN DIRECT SHEAR TEST

Summary: Mathematical modeling of engineering materials is important issue in design of structures. Model itself should satisfy two conditions: first to explain in the best way mechanical behavior of material and second to be as simple as possible for easy everyday application in engineering. In the present paper the Preisach model is applied to behaviour of clay as an elasto-plastic material under direct shear test. The clay sample was taken by exploratory drilling on the territory of Šimanovci from a depth of 9.1 m to 9.3 m. First, an experimental examination of a clay sample was performed in order to classify and determine the resistance-deformable characteristics. The clay is classified as CH-clay of high plasticity. The sample was then subjected to direct shear test. All obtained results clearly show advantages of the Preisach model for describing behavior of elastoplastic material. This analytical model allows an accurate computation of all points on the result curve from the direct shear test.


INTRODUCTION
Preisach model originally is used for defining hysteretic phenomenon in magnetism [1], the model quickly found application in other fields of physics. The first implementation of this model in continuum mechanics describes the behavior of ductile materials under cyclic loading [2,3]. Later on model was applied for elastoplastic cyclic bending of beams  [7,8]. Generally, the Preisach model is hysteretic operator used for defining cyclic behavior of ductile materials. In addition to the primary characteristic of a model to describe the cyclic behavior of materials, it has capability to accurately describe the monotonic behavior of the material. Existing models, mapping strain ε(t) as input into stress σ(t) as output, are based on bilinear working diagrams (Fig. 2). They are used for modeling cyclic and monotonic behavior of ideally elasto-plastic materials and ideally elasto-plastic materials with linear hardening. In this paper we used this model to describe behavior of clay in direct shear test.

THE PREISACH MODEL OF HYSTERESIS
According to Mayergoyz [6], the Preisach model implies the mapping of an input u(t) to the output f(t) in the integral form: where Gα,β is an elementary hysteresis operator given in Figure 1

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From formula (1) it is obtained: Denoting the output value at u= β by fα,β from the limiting triangle, it follows that: By differentiating expression (3) twice, with respect to α and β, the Preisach weight function is derived in the form: The Preisach model explained above possesses two properties: wiping out and congruency properties. Those properties and much more about Preisach model is explained in the Lubarda et al. [2] and [3].

THE PREISACH MODEL FOR CYCLIC BEHAVIOR OF DUCTILE MATERIALS
One dimensional hysteretic behavior of elasto-plastic material can be successfully described by the Preisach model. Ductile material is represented in various ways by a series or parallel connections of elastic (spring) and plastic (slip) elements, explained in Lubarda, at al. [2]. Elastic-linearly hardening material behavior, characterized by the stress-strain curve shown in Figure

A THREE-ELEMENT UNIT CONNECTED IN PARALLEL
In this case the Preisach function can be determined from the hysteresis nonlinearity shown in Figure 2a again, taking into consideration that strain is input and stress is output.
The Preisach function in this case has support along the lines −= and −=Y/E, i.e.it is given by: The expression for stress as a function of applied strain is, consequently: The first and second term on the right-hand side of (6) are elastic and plastic stress, respectively. For a system consisting of infinitely many of three-element units, connected in a parallel and with uniform yield strength distribution within the range Ymin≤Y≤Ymax, the total stress is: In (7) the integration domain A is the area of the band contained between the lines −=2Ymin/E and −=2Ymax/E in the limiting triangle, shown in Figure 3

RESULTS OF EXPERIMENTAL TESTS OF CLAY
In this paper, the application of Preisach model for describing the behavior of clay in the direct shear test will be presented. The clay sample was obtained by exploratory drilling on the territory of the municipality of Šimanovci at a depth of 9.1 m to 9.3 m. Based on the conducted experiments, it was determined that the soil is a highly plastic clay-CH clay with a yield limit wp = 56.5% and a plasticity index Ip = 37.1%. The bulk density in the natural state with pores is ρ = 2.016 g / cm 3 , while the dry bulk density is ρ = 1.631 g / cm 3 . The compressibility modulus for the stress level 100kPa-200kPa is Mv = 10MPa. A drained direct shear test was performed. This test is standardly applied in practice and gives parameters expressed by effective stresses. For three levels of normal stresses 0kPa, 100kPa and 200kPa, a dependence is established between shear stress τ and shear displacements δ between the upper and lower part of the sample Figure 4. The maximum values of shear stresses are plotted on the (σ, τ) diagram according to Figure 5 so that they define points on the stress envelope that is approximated by the linear dependence-Coulomb line τ = σtgϕ + c. Where, ϕ is the angle of internal friction, while c is the cohesion of the material. For the tested sample, the angle of internal friction is ϕ = 20˚, and the cohesion is c = 7kPa.  Depending on the level of normal stresses, the soil can show a whole range of behaviors from brittle-plastic, through plastic, to tough fracture. As Preisach model finds application in elasto-plastic materials, clay in the experiment of direct shear expresses the properties of elasto-plasticity for the highest possible stress level. For these reasons, Preisach model will be applied for a stresses level of 200 kPa. Shear strain is obtained dividing the shear displacement by the height of the sample.

ILLUSTRATIVE EXAMPLE
To In the calculation it is used G and Gh as a slip module and hardening slip module and Ymax and Ymin as upper yield and lower yield point. The curve is divided into two parts. The first part of curve is described by modules G and Gh, while the second part of curve for the calculation peak shear strength is described by modules Gh and Ga as shown in Figure 6. For the calculation of the peak shear strength, the coordinate origin was moved to the stress level of 67.4 kPa, so the part of the curve to the peak shear strength was observed as a separate curve. All parametars for calculation are given in the Table 1.

Figure 6. Parametars for caculation adopting from experimental curve
The calculation was performed in three steps. For γel= Ymin/G=0.00526, the first step covers the slip area from 0 to 5γel, the second area is from 0 to 10 γel, and the third area is from 0 to 15γel. Figure 7 showes Preisach triangles for two divided parts of curve.  The stress under limit Ymin is linear.

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The third step of calculation 0 < γ < 15 γel: The Figure 8 shows comparasion of calculated values with experimental curve.

CONCLUSION
In present paper is considered the application of Preisach model for the calculation of soil shear resistance for a known shear deformation. Preisach model was primarily used to determine the response of elasto-plastic materials under cyclic loading. As Clay, due to the action of normal stress in the direct shear test, has the properties of elasto-plastic materials, the possibility of applying Preisach model for determining the curve of shear resistance-deformation was tested. It is difficult to perform a cyclic test of direct shear due to the limitations of the apparatus, also when shearing the soil, it is difficult to register the elactic deformation. The clay sample was exposed to direct shear for three stress levels.
Preisach model was applied in the case of a normal stress of 200 kPa to peak shear strength.
The obtained values indicate that Preisach model can describe all phases of shear behavior of clay: elastic phase, hardening and the peak strength as well.