Official J. Lin, D. Dielectric and piezoelectric properties of K0.

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Zhang, S. Dielectric and piezoelectric properties of Ba0. Fu, P. Piezoelectric, ferroelectric and dielectric properties of La2O3-doped Bi0. Uetsuji, Y. First-principles study on crystal structure and piezoelectricity of perovskite-type silicon oxides. Solid Mech. Superconducting and electro-optical thin films prepared by pulsed laser deposition technique. Absolute hardness: companion parameter to absolute electronegativity. Key words: Generalized plasticity, double structure, swelling clays, sealing material Abstract.

The paper presents a double structure constitutive model based on a generalized plasticity formalism. The behaviour of macrostructure, microstructure and their interactions are described. A coupled hydromechanical formulation is then presented that assumes no hydraulic equilibrium between structural levels. Constitutive law and formulation are applied to the simulation of the behaviour during hydration of a heterogeneous mixture of bentonite powder and bentonite pellets.

A satisfactory reproduction of observed behaviour is achieved.

## Computational Plasticity

The behaviour of swelling clays is better understood if the effect of the pore size structure on their hydromechanical behaviour is taken into account. In compacted and therefore unsaturated swelling clays, the pore size structure is set up during compaction but it may change significantly in response to various actions such as loading and hydration. Although the distribution of pore sizes is of course continuous, useful insights can be obtained by considered only two structural levels: microstructural and macrostructural as well as their interactions.

This dual material aspect is reinforced when the material is composed of a mixture of powder and highly compacted pellets.

## Plasticity (physics) - Wikipedia

This mixture is an attractive sealing material in radioactive waste disposal schemes because, even when only modest compaction efforts are applied, a sufficiently high density value is achieved after hydration has taken place. However, the heterogeneity of the material gives rise to a complex hydromechanical behaviour that must be well understood if a sufficient degree of confidence in the performance of the seal is to be achieved. The generalized plasticity model adopted for the description of the behaviour of the double structure material is presented first followed by the formulation of the hydromechanical problem.

Constitutive law and formulation are then applied to description of a number of swelling pressure tests of a mixture of bentonite powder and highly compacted bentonite pellets being studies as a potential sealing material for a dep geological repository for high level nuclear waste.

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Those additional mechanisms can often be attributed to the interaction between the macrostructure and the microstructure. This kind of irreversible behaviour generally appears at any value of applied suction and it is difficult to determine the initiation of the yielding. Those facts encourage the use of the generalized plasticity theory to describe these plastic mechanisms.

In a generalized plasticity model the yield function is not defined or it is not defined in an explicit way [1]. The advantages in using the generalized plasticity theory to model the plastic mechanisms ascribed to the interaction between structures are presented in detail in [2]. The model is defined in terms of the three stress invariants p, J, and suction s.

To formulate the double structure model is necessary to define laws for: i the macrostructural level, ii the microstructural level, and iii the interaction between both structural levels. The BBM considers two stress variables to model the unsaturated behaviour: the net stress computed as the excess of the total stresses over the gas pressure, and the matric suction s , computed as the difference between gas pressure and liquid pressure pg-pl.

To complete the definition of the yield surface as set out in 1 , it is possible, in principle, to adopt any suitable Lodes angle function, g. The trace of the yield function on the isotropic p-s plane is called LC Loading-Collapse yield curve because it represents the locus of activation of irreversible deformations due to loading increments or to hydration collapse.

The strains arising from microstructural phenomena are considered nonlinear. The microstructural strains are proportional to the microstructural effective stress p through a microstructural bulk modulus according to:. The parameter is included only to account for the possibility that the microstructure may become unsaturated. In very active expansive clays this assumption can be supported by the high affinity of the active clay minerals by water, which maintain the interlayer space and micro-pores saturated even at relatively high suction.

Under this condition mean effective stress controls the mechanical behaviour at microstructural level. The concept of a Neutral Line NL is introduced corresponding to constant p and no microstructural deformation Fig. The NL divides the p-s plane into two parts, defining two main generalized stress paths, which are identified as: MC microstructural contraction when there is an increase in p and MS microstructural swelling in the opposite case. A hypothesis of the model is that the plastic deformations of the macrostructure vpM induced by microstructural effects are proportional to the microstructural strains vm according to interaction functions f [2].

The total plastic macrostructural strains i. When this ratio is low, it implies a dense packing of the material and it is expected that, under this condition dense macrostructure , the microstructural swelling MS path affects strongly the global arrangements of clay aggregates, inducing large macrostructural plastic strains.

In this case the microstructure effects induce a more open macrostructure, which implies a macrostructural softening. Under this path the clay tends towards a more dense state, which implies a hardening of the macrostructure.

In this way the effect of microstructural processes on the global arrangements of aggregates is taken into account. To fully describe the soil behaviour, the definition of specific elasto-plastic laws for each domain is required according to the microstructural stress path followed MC or MS. Generalized plasticity theory can deal with such conditions, allowing the consideration of two directions of different behaviour and the formulation of proper elasto-plastic laws for each region.

Thus, a complete description of a generalized model includes the definition of the: i loading and unloading direction, ii plastic flow direction, and iii a plastic modulus. One vector indicates the MC direction and the other the MS direction. Given a generalized stress state and stress increment, the criterion to identify the microstructural stress path is illustrated in Figure 1. In classical plasticity theory, it is assumed that the material behaves either as an elastic or an elasto-plastic solid.

In generalized plasticity theory, the state of the material is determined directly from the control variables: generalized stresses, strains and a finite number of internal variables. A process of loading is defined as elastic if the set of internal variables remains unchanged. In the case of a purely nonlinear elastic loading, the stress increment is related to the increment of strains and suction by the following relationship:.

When a loading process is inelastic, the material behaviour is described by the elastoplastic mechanisms that are activated during the loading process. A multidissipative approach [5] has been adopted to derive the general elasto-plastic relations that can be expressed as follows:. The expressions for the vectors and matrices in 6 and 7 together with the details of numerical implementation are presented in [6].

In the following, subscript M will stand for the macrostructure and subscript m for the microstructure.

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Accordingly, macroporosity and microporosity are denoted as M and m respectively. Macroporosity and microporosity are defined as the volume of macropores and micropores, respectively, divided by the total volume of the soil. Thus, total porosity equals M m. The degree of saturation of the macroporosity, SwM, is the volume of macropores occupied by water over the volume of the macropores; an equivalent definition holds for the microporosity degree of saturation, Swm. An important feature of the formulation is that hydraulic equilibrium between the two continua is not assumed, i.

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For simplicity, a linear relationship is assumed e. It is assumed that only matric and gravitational potential contribute to the total potential of the macrostructure but an additional osmotic component may also contribute to the microstructural potential [8]. Here, potential is defined in pressure units. As the water exchange is local in space, the gravitational potential will be the same for the two media.