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.     

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.     

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.     

Ply cracking and stiffness degradation in cross-ply laminates under biaxial extension, bending and thermal loading

Transverse ply cracking often leads to the loss of stiffness and reduction in thermal expansion coefficients. This paper presents the thermoelastic degradation of general cross-ply laminates, containing transverse ply cracks, subjected to biaxial extension, bending and thermal loading. The stress and displacement fields are calculated by using the state space equation method [Zhang D, Ye JQ, Sheng HY. Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading. Compos Struct [in press].]. By this approach, a laminated plate may be composed of an arbitrary number of orthotropic layers, each of which may have different material properties and thickness. The method takes into account all independent material constants and guarantees continuous fields of all interlaminar stresses across interfaces between material layers. After introducing the concept of the effective thermoelastic properties of a laminate, the degradations of axial elastic moduli, Poisson’s ratios, thermal expansion coefficients and flexural moduli are predicted and compared with numerical results from other methods or available test results. It is found that the theory provides good predictions of the stiffness degradation in both symmetric and antisymmetric cross-ply laminates. The predictions of stiffness reduction in nonsymmetric cross-ply laminates can be used as benchmark test for other methods.     

On error estimates and adaptivity in elastoplastic solids: Applications to the numerical simulation of strain localization in classical and Cosserat continua

The a posteriori error estimates based on the post-processing approach are introduced for elastoplastic solids. The standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy. This is known to represent a natural metric for the class of elastoplastic problems of evolution. In addition, the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates. A posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem. As a model problem, the constitutive equations describing a generalization of standard J₂-elastoplasticity within the Cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post-instability region. The proposed a posteriori error estimates are appropriately modified to account for the Cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems. Superior behaviour of the Cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material, resulting in convergent solutions with an h-refinement and almost uniform error distribution in all considered error norms.     

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。     

Vibration and damping analysis of a three-layered composite annular plate with a viscoelastic mid-layer

The natural frequencies and modal loss factors of the three-layered annular plate with a viscoelastic core layer and two polar orthotropic laminated face layers are considered. The discrete layer annular finite element is employed to derive the equations of motion for the three-layered annular plate. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by the complex quantities. Complex eigenvalued problems are then solved, and the frequencies and modal loss factors of the composite plate are extracted. The results of the symmetric and non-symmetric composite annular plates are both presented. The effects of material properties, radius to thickness ratio, stacking sequences and thickness of face layers, and thickness of the viscoelastic core layer are discussed.     

Propagation of impact-induced stress waves in composite plates

A theoretical investigation is carried out of the response of a laminated plate to a surface impact. The plate consists of four layers of a uni-directional fiber composite in a symmetric cross-ply configuration. The composite material is modeled as a transversely isotropic, homogeneous, elastic continuum. The four layers are assumed to be of finite depth, infinite lateral extent, and perfectly bonded to each other. The surface impact has been modeled by three different conditions, namely, a delta function impulse, a square pulse, and a half-sine pulse. The variation with depth of the normal component of stress is examined as a step toward gaining insight into the through thickness response of the plate.     

Multiscale continuum modeling of a crack in elastic media with microstructures

Cosserat type continuum theories have been employed by many authors to study cracks in elastic solids with microstructures. Depending on which theory was used, different crack tip stress singularities have been obtained. In this paper, a microstructure continuum theory is used to model a layered elastic medium containing a crack parallel to the layers. The crack problem is solved by means of the Fourier transform. The resulting integrodifferential equations are discretized using the Chebyshev polynomial expansion method for numerical solutions. By using the present theory, the explicit internal length effects upon the crack opening displacement and stress field can be observed. It is found that the stress field near the crack tip is not singular according to the microstructure continuum solution although the level of the opening stress shows an increasing trend until it gets very close to the crack tip. The rising portion of the near tip opening stress is used to project the stress intensity factor which agrees fairly well with that obtained using the FEM to perform stress analyses of the cracked layered medium with the exact geometry. The numerical solutions also indicate that treating the layered medium as an equivalent homogeneous classical elastic solid is not adequate if cracks are present and accurate stress intensity factors in the original layered medium is desired.     

On wave propagation in laminated composites—II. Propagation normal to the laminates

A continuum model is developed for elastic waves propagating in a direction normal to the laminates of a laminated medium. The developments are analogous to those used in a companion paper for waves traveling along the laminates and indicate that the behavior is “nonlocal” in time as a result of the history-dependence of the current state. A zeroth-order model is deduced which consists of a system of integro-differential equations. The behavior of this system is then analysed and it is found that during an early phase, the motion is confined to a boundary layer and consists of highly damped waves. During a later phase the behavior is found to approach that of a macroscopically homogeneous medium. The behavior during both phases is described by two distinct differential systems. The early phase behavior is then determined for a laminated half-space subjected to a step normal stress.     

A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials

A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu–Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts.     

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.     

Longitudinal elastic wave propagation in laminated composites with bonds

We extended the finite element displacement method to study the propagation of longitudinal elastic waves in laminated composite with bonds. The geometric arrangement of the composite model considered in this paper is treated as a special type of a trilaminated composite in which each of its major constituents is sandwiched between two bonding layers. The dispersion characteristics of this model are presented here and compared with some exact results. The exact dispersion relation for the trilaminated composite is formally obtained by solving the field equations subjected to continuity conditions at materials' interfaces. Also included in the comparisons are the results obtained with continuum theory with microstructure. It is found that numerical results of finite element analysis, continuum theory and exact analysis corelate well especially for the lower modes of wave propagation.     

Effective moduli and failure considerations for composites with periodic fiber waviness

A finite element micromechanical model for fibrous materials introduced in a previous work [J. Compos. Mater. 38 (4) (2004) 273] is used to further study the effects of periodic and localized fiber waviness. A periodic unit cell based on hexagonal fiber packing and sinusoidal fiber waviness was assumed as a representative volume element. Equivalent to this wavy-shaped unit cell, a straight unit cell but with wavy material-orientation is introduced. This type of homogenized continuum modeling simplifies the analysis since the wavy geometry with details of constituent materials is avoided. Thus, stiffness parameters associated with individual lamina with waviness are estimated when subject to the constraining effects of neighboring isotropic or straight fiber material layers. It is shown that the shear constraint of the added layers increases the effective moduli of the wavy layer by inhibiting the fiber straightening deformation mechanism. The local stress distribution is also examined and the potential for material failure is investigated. The methodology provides a platform to study the behavior of wavy fiber composites in a systematic manner.     

Dynamic behavior of cross-ply laminated beams with piezoelectric layers

The dynamic behavior of cross-ply non-symmetric composite beams, having uniform piezoelectric layers is analysed. A first-order Timoshenko type analysis is applied to obtain the equations of motion, which include shear deformation, rotary inertia, bending-stretching coupling terms and induced axial strains caused by the piezoelectric material. Using the principle of virtual work, the coupled equations of motion and the relevant boundary conditions are obtained. For a laminated beam having uniform piezoelectric layers the induced strains appear only in the boundary conditions yielding time dependent ones. Therefore, a special procedure involving orthogonality of the coupled Timoshenko type natural vibrational modes of the beam is applied to help understanding of the dynamic behavior of the non-symmetric laminated beam and to investigate the influence of the induced strains (by the piezoelectric layers) on the dynamic behavior while keeping an ‘open-loop’ control. Typical types of laminates and piezoelectric materials are used to calculate natural frequencies and mode shapes. Numerical results for various parameters of laminated beams are presented to stress the better applicability and suitability of the present approach to the analysis of dynamic behavior of laminated composite beams with piezoelectric layers.     

Elastic characterization of laminated composites based on multiaxial tests

This paper presents a technique that continuously identifies the elastic moduli of laminated composites from multiaxial tests and its application to multiaxial identification. Unlike the conventional characterization where materials are characterized uniaxially, the technique identifies the elastic moduli by considering specimens on a continuum basis. The technique further controls the multiaxial testing machine by extracting quantities from the matrix used for identification and maximizing the quantities. Numerical examples first investigate the significance, robustness and efficiency of the proposed technique where its efficacy has been then confirmed via various quantifications. The proposed technique was finally applied to a realistic characterization problem, and its practicality has been demonstrated.