On a possibility of reconstruction of Cosserat moduli in particulate materials using long waves

The paper proposes a method of reconstruction of the Cosserat elastic moduli using the measurements of velocities of the p-wave and the high-frequency twist wave as well as the low-frequency asymptotics of a shear wave dispersion relationship. It is shown that in the case of a general isotropic Cosserat continuum, the information obtained from these wave measurements is insufficient for the complete moduli reconstruction. The reconstruction is shown to be possible in the case of a 3D isotropic Cosserat continuum governed by at most four independent parameters. Such a continuum is suggested for a particulate material consisting of spherical particles connected by normal, shear and rotational links. Another case when the full reconstruction is possible consists of 2D orthotropic Cosserat continuum modelling particulate material with square packing of cylindrical particles and 2D isotropic Cosserat continuum modelling with hexagonal packing of cylindrical particles. In the 2D materials, the measurements of p-wave velocity and the shear wave dispersion relationship are sufficient for complete reconstruction of all moduli. A phase shift method and reconstruction algorithms are presented.     

Inclusion of microscopic rotation of powder particles during compaction in finite element method using Cosserat continuum theory

The effect of microscopic rotation of powder particles in compaction is included in the rigid-plastic finite element method on the basis of the Cosserat continuum theory. In the Cosserat continuum theory, couple stress induced from the microscopic rotation is introduced, and the equilibrium equations of moment for the couple stress are solved simultaneously with those of force. A yield criterion for the Cosserat porous continuum is proposed by taking the effect of the couple stress into consideration, and constitutive equations for the rigid-plastic porous material are derived from the yield criterion on the basis of the associated flow rule. The equilibrium equations of force and moment for the Cosserat continuum are formulated by the use of the Galerkin method. The effect of microscopic rotation of powder particles in plane-strain closed-die compaction is examined. In addition, the calculated result is compared with that for the conventional continuum without the microscopic rotation. © 1998 John Wiley & Sons, Ltd.     

Size effects in the mechanical behavior of cellular materials

Effective mechanical properties of cellular materials depend strongly on the specimen size to the cell size ratio. Experimental studies performed on aluminium foams show that under uniaxial compression, the stiffness of these materials falls below the corresponding bulk value, when the ratio of the specimen size to the cell size is small. Conversely, in the case of simple shear and indentation, the overall stiffness rises above the bulk value. Classical continuum theory, lacking a length scale, cannot explain this size dependent mechanical behaviour. One way to account for these size effects is to explicitly model the discrete cellular morphology. We performed shear, compression and bending tests using discrete models, for hexagonal (regular and irregular) microstructures. Even though discrete models give a very good agreement with the experiments, they are computationally expensive for complex microstructures, especially in three dimensions. To overcome this, one can use a generalized continuum theory, such as Cosserat continuum theory, which incorporates a material length scale. We fit the Cosserat elastic constants of the models by comparing the discrete calculations with the analytical Cosserat continuum solutions in terms of macroscopic properties. We critically address the limitations of the Cosserat continuum theory.     

饱和多孔介质中动力渗流耦合分析的Biot-Cosserat连续体模型与应变局部化有限元模拟

提出了适用于饱和多孔介质中应变局部化分析及动力渗流耦合分析的Biot-Cosserat连续体模型。基于饱和多孔介质动力渗流耦合分析的Biot理论,将固体骨架看作Cosserat连续体,并考虑旋转惯性,建立了饱和多孔介质动力渗流耦合分析的Biot-Cosserat连续体模型。基于Galerkin加权余量法,对所发展的模型推导了以固体骨架广义位移(包含旋转)及孔隙水压力为基本未知量的有限元公式。利用所发展的数值模型,对包含压力相关弹塑性固体骨架材料的饱和多孔介质进行了动力渗流耦合分析与应变局部化有限元模拟,结果表明,所发展的两相饱和多孔介质动力渗流耦合分析的Biot-Cosserat连续体模型能保持饱和两相介质应变局部化问题的适定性及模拟饱和多孔介质中由应变软化引起的应变局部化现象的有效性。     

基于参变量变分原理的三维Cosserat体模型弹塑性分析与应变局部化模拟

基于参变量变分原理,该文发展了三维Cosserat连续体模型弹塑性有限元分析的二次规划算法。由于Cosserat连续体模型的本构方程中存在材料内尺度参数,该模型可以解决经典连续介质理论在分析应变软化问题时病态的有限元网格依赖性问题。数值结果表明所发展的三维Cosserat连续介质弹塑性有限元模型可以有效的模拟应变局部化现象并且该算法具有很好的数值稳定性,同时获得的数值结果具有良好的非网格依赖性。     

Dynamic finite element formulation for Cosserat elastic plates

A displacement and rotation‐based dynamic finite element formulation for Cosserat plates is discussed in detail in this paper. Special attention is given to the validation of the element through adequate benchmarks and patch tests. The choice of the interpolation functions and the order of integration of the stiffness and the mass matrices are extensively argued. The possibility of local system deficiencies is explored, and a shear locking investigation specifically elaborated for Cosserat plates is carried out. It is shown how the present formulation has interesting computational properties as compared to a classical continuum‐based formulation and how it can provide suitable results despite the use of reduced integration. An example of application of the finite element is given, in which the natural frequencies of a masonry panel modelled by means of discrete elements are computed and compared with the finite elements solution. Copyright © 2014 John Wiley & Sons, Ltd.     

Size effects and micromechanics of a porous solid

The rigidity of rods of a polymeric foam in bending and torsion is measured as a function of diameter. The dependence of rigidity upon specimen size is found to be inconsistent with a classically viscoelastic continuum model. The Cosserat continuum, which admits additional degrees of freedom associated with rotation of the microstructure, describes the foam more accurately than the classical continuum. Evidence is presented that additional degrees of freedom associated with the deformation of the microstructure, must be incorporated in a complete continuum model of foamed materials.     

A highly efficient membrane finite element with drilling degrees of freedom

In this paper, the development of a new quadrilateral membrane finite element with drilling degrees of freedom is discussed. A variational principle employing an independent rotation field around the normal of a plane continuum element is derived. This potential is based on the Cosserat continuum theory where skew symmetric stress and strain tensors are introduced in connection with the rotation of a point. From this higher continuum theory a formulation that incorporates rotational degrees of freedom is extracted, while the stress tensor is symmetric in a weak form. The resulting potential is found to be similar to that obtained by the procedure of Hughes and Brezzi. However, Hughes and Brezzi derived their potential in terms of pure mathematical investigations of Reissner’s potential, while the present procedure is based on physical considerations. This framework can be enhanced in terms of assumed stress and strain interpolations, if the numerical model is based on a modified Hu-Washizu functional with symmetric and asymmetric terms. The resulting variational statement enables the development of a new finite element that is very efficient since all parts of the stiffness matrix can be obtained analytically even in terms of arbitrary element distortions. Without the addition of any internal degrees of freedom the element shows excellent performance in bending dominated problems for rectangular element configurations.     

A hyperbolic augmented elasto-plastic model for pressure-dependent yield

A three-dimensional elasto-plastic model for the deformation and flow of granular materials which generalises the plastic potential model and contains an additional term analogous to that appearing in the double-shearing model is presented. It is shown that for planar flows the resulting system of first-order partial differential equations is hyperbolic. This is in distinct contrast to both the non-associated plastic potential and double-shearing models, which fail to be hyperbolic. The ill-posedness of the Cauchy problem for the planar double-shearing model is due to the presence of the rotation rate of the principal axes of stress while that of the non-associated plastic potential model is due to distinct quasi-static spatial stress and velocity characteristics. The present model attains well-posedness by replacing the planar rotation rate of the principal stress axes by the vector intrinsic spin of a Cosserat continuum and using it to ensure identical spatial stress and velocity characteristics. Flows in which the intrinsic spin vector is constant in both space and time correspond to flows in an ordinary continuum. The model governing such flows is embedded into a Cosserat model in such a way that the characteristic structure is preserved.     

Numerical solution of axisymmetric nonlinear elastic problems including shells using the theory of a Cosserat point

Starting with a recently developed three-dimensional eight node brick Cosserat element for nonlinear elastic materials, a simplified Cosserat element is developed for torsionless axisymmetric motions. The equations are developed within the context of the theory of a Cosserat point and the resulting theory is hyperelastic and is valid for dynamics of nonlinear elastic materials. The axisymmetric Cosserat element has four nodes with a total of eight degrees of freedom. As in the more general element, the constitutive equations are algebraic expressions determined by derivatives of a strain energy function and no integration is needed throughout the element region. Examples of large deformations of a nearly incompressible circular cylindrical tube and large deflections of a compressible clamped circular plate are considered to test the accuracy of the element.     

Static equations of the Cosserat continuum derived from intra-granular stresses

I present a derivation of the static equations of a granular mechanical interpretation of Cosserat continuum based on a continuum formulated in the intra-granular fields. I assume granular materials with three-dimensional, non-spherical, and deformable grains, and interactions given by traction acting on finite contact areas. Surface traction is decomposed into a mean and a fluctuating part. These account for forces and contact moments. This decomposition leads to a split of the Cauchy stress tensor into two tensors, one of them corresponding to the stress tensor of the Cosserat continuum. Macroscopic variables are obtained by averaging over representative volume. The macroscopic Cauchy stress tensor is shown to be symmetric even in non-equilibrium. The stress tensor of the Cosserat continuum becomes asymmetric when the sum of the contact moments acting on the boundary of the representative volume is different from zero.     

Waves in a rotating elastic solid

Wave propagation in a uniformly rotating elastic solid is discussed based on displacement equations in a moving frame. The time-harmonic Green’s dyadic for a point body force is obtained in closed form. It is reconfirmed that two quasi dilatational and shear waves are coupled to each other, and the deformation decomposition into the dilatation and rotation is not possible for the rotating solid. Further, it is also confirmed that the velocity of the Rayleigh surface wave depends not only on the rotational velocity but also on its direction and that the Rayleigh wave vanishes when the rotational velocity approaches the Rayleigh wave velocity of the immovable solid.     

A frictional Cosserat model for the flow of granular materials through a vertical channel

Summary A rigid-plastic Cosserat model has been used to study dense, fully developed flow of granular materials through a vertical channel. Frictional models based on the classical continuum do not predict the occurrence of shear layers, in contrast to experimental observations. This feature has been attributed to the absence of a material length scale in their constitutive equations. The present model incorporates such a material length scale by treating the granular material as a Cosserat continuum. Thus, localized couple stresses exist, and the stress tensor is asymmetric. The velocity profiles predicted by the model are in close agreement with available experimental data. The predicted dependence of the shear layer thickness on the width of the channel is in reasonable agreement with data. In the limit of small (ratio of the particle diameter to the half-width of the channel), the model predicts that the shear layer thickness scaled by the particle diameter grows as ^-1/3.     

The influence of non-coaxiality on shear banding in viscous-plastic materials

Many of the most important and commonly occurring, emergent geological structures such as shear bands, fault zones and folds may be understood as a consequence of changes in the type of the governing model equations. Such changes, or bifurcations, depend strongly on the details of the constitutive relationships including the presence of strain softening, thermal or chemical effects, and whether the flow rule is associated or non-associated, coaxial or non-coaxial. Here we focus on the influence of non-coaxiality of the flow rule on the development and evolution of shear bands. The term non-coaxial refers to the non-coincidence of the principal axes of the stress and of the plastic strain rate tensor. Non coaxial plasticity models were originally proposed by Josselin de Jong and A. J. M. Spencer. The geometric structure of most non-coaxial models is defined by the assumption that the plastic deformation is carried by one (single slip) or two (double slip) slip systems that are inclined at ±(p/4+n/2), 0 ￡ n ￡ f{\pm (\pi /4+\nu /2), 0\leq \nu \leq \phi } to the less compressive principal stress axis; f{\phi } is the internal angle of friction of the material. Both cases –single and double slip– are considered here. A particular feature of the double slip model is the appearance of an additional variable requiring an additional constitutive relationship. Geometrically, the additional variable may be interpreted as a spin. Spencer assumed that, in 2D, the additional spin is equal to the rate of rotation of the principal axes of the stress tensor. Here we show that Spencer’s assumption is very similar (up to a factor of proportionality) to the assumption that the internal slip systems are material. We also propose an alternative closure, based on experimental observations, indicating that the ratio between the average particle spin and the spin of a spatial element of the granular assembly is constant, in simple shear. Finally, we propose a closure for the double slip model within the framework of a Cosserat Continuum theory. In a Cosserat continuum, a material point possesses the degrees of freedom of an infinitesimal rigid body: three translations and three rotations. In this case the indeterminacy of the double slip model is removed by the angular momentum balance equation associated with the rotational degrees of freedom. We propose constitutive relationships for the Cosserat model and illustrate the behavior of the model by means of an analytical solution of the simple shear problem.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Micro–macro homogenization of granular materials based on the average-field theory of Cosserat continuum

This paper performs further study on the micro–macro homogenization approach of granular materials (Li et al., 2010) based on the advancement of Hill’s lemma for Cosserat continuum (Liu, 2013). Firstly, the average couple stress tensor, expressed as the volume integration of quantities over the representative volume element (RVE) in the average-field theory of Cosserat continuum, is further deduced and expressed in terms of discrete quantities on the discrete particle assembly RVE of granular materials. The expression is also discussed and compared with other typical definitions of the effective couple stress tensor for granular materials in the literature. Then, rate forms of micromechanically based constitutive models consistent with different types of RVE boundary conditions are derived and discussed. Since the presented micro–macro homogenization approach is used, not only the micro–macro energy equivalence is satisfied, but also the microstructure and its evolution can be taken into account in the constitutive formulation with no need of specifying macroscopic phenomenological constitutive model.