Non-symmetric Stress Research Articles

Finite element analysis of localization and micro–macro structure relation in granular materials. Part I: Formulation

The formation of strain localization influences the stability and stiffness of the soil mass or geosystem. The thickness of shear bands provides insight into overall strength and stiffness inside the granular body, and the shear band angle gives information about the failure surface in a given soil or soil mass. Thus, it is important to be able to predict when a shear band forms and how this zone of intense deformation is located and oriented within the granular medium. A rational finite element analysis for capturing the formation and development of shear bands has been formulated by using a micropolar continuum in finite element analysis. Hardening parameters including nonsymmetric stress and length scale are implemented for a realistic stress–strain analysis. Implementation and simulations of the model will be discussed in the Part II of this research. The micro–macro structure relation for finite element analysis based on this formulation paper will be simulated in the corresponding paper.

Peristaltic Flow of a Magneto‐Micropolar Fluid: Effect of Induced Magnetic Field

We carry out the effect of the induced magnetic field on peristaltic transport of an incompressible conducting micropolar fluid in a symmetric channel. The flow analysis has been developed for low Reynolds number and long wavelength approximation. Exact solutions have been established for the axial velocity, microrotation component, stream function, magnetic‐force function, axial‐induced magnetic field, and current distribution across the channel. Expressions for the shear stresses are also obtained. The effects of pertinent parameters on the pressure rise per wavelength are investigated by means of numerical integrations, also we study the effect of these parameters on the axial pressure gradient, axial‐induced magnetic field, as well as current distribution across the channel and the nonsymmetric shear stresses. The phenomena of trapping and magnetic‐force lines are further discussed.

Computations of size effects in granular bodies within micro-polar hypoplasticity during plane strain compression

The numerical investigations of size effects in granular bodies during a plane strain compression test are performed. To describe a mechanical behaviour of a cohesionless granular material during a monotonous deformation path in a plane strain compression test, a micro-polar hypoplastic constitutive model was used. It includes particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. In the paper, deterministic and statistical size effects in geometrically similar granular specimens are analysed. The deterministic calculations were carried out with a uniform distribution of the initial void ratio. To investigate a statistical size effect, in order to reduce the number of realizations without loosing the accuracy of the calculations, a Latin hypercube method was applied to generate Gaussian truncated random fields in a granular specimen. The results show that the statistical size effect is significantly stronger than the deterministic one. The shear resistance decreases and the rate of softening increases with increasing specimen size. The effect of the boundary roughness on shear localization is pronounced.

FE-investigations of a deterministic and statistical size effect in granular bodies within a micro-polar hypoplasticity

The paper deals with numerical investigations of a deterministic and statistical size effect in granular bodies during shearing of an infinite layer under plane strain conditions and free dilatancy. For a simulation of the mechanical behavior of a cohesionless granular material during a monotonous deformation path, a micro-polar hypoplastic constitutive was used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. To describe a deterministic size effect, the calculations were carried out with an uniform distribution of the initial void ratio for four different heights of the granular layer: 5, 50, 500 and 2,000 mm. To investigate a statistical size effect, the distribution of the initial void ratio in infinite granular layers was assumed to be spatially correlated. As only primary stochastic calculations were performed, single examples of different random fields of the initial void ratio were generated. For this purpose a conditional rejection method was used.

Effect of Cross Section Geometry on the Response of an SMA Column

This numerical study investigates the effect of cross section geometry on the bending of a beam and also buckling of a column made of shape memory alloy (SMA). It incorporates superelastic SMA's non-linear stress—strain relations, including the non-symmetric behavior in tension and compression. Primarily, an arbitrary rectangular cross section is selected for analysis of the beam and column, loaded beyond the linear stress—strain relations. Then similar analysis is carried out for two different equivalent cross sections, namely, square of same area and square of same inertia, for comparing the effect of cross section geometry on the response of the beam and column. Using the equations of static equilibrium, the variations of effective modulus and bending moment for different equivalent cross sections are presented. The critical stress of the column is calculated by assuming an appropriate deflection curve for both ends hinged condition. It is found that the effective modulus depends only on the stress—strain curve. The bending moment, on the other hand, is a function of the stress—strain curve and cross section geometry. For a given load, the square of same area develops the minimum bending stress, while the square of same inertia develops the maximum bending stresses. Because of the non-symmetric stress—strain curve, the compressive stress is found to be higher than the tensile stress. It is also found that the buckling load for a column is distinctly high if the non-symmetric stress—strain curve is used in the analysis. Moreover, for a very high critical stress, the square of same area has the highest slenderness ratio, while the rectangle has the lowest slenderness ratio.

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.

On anisotropic flow rules in multiplicative elastoplasticity at finite strains

Hill’s anisotropic formulation of the flow rule is extended so as to fit in the realm of multiplicative finite strain plasticity. The anisotropic Hill-type yield criterion is formulated in terms of purely material quantities. The list of arguments of the flow function includes a non-symmetric material Eshelby-like stress tensor, as well as structural tensors that describe the anisotropy at hand. The formulation is exemplified on orthotropy, where three structural tensors are employed. The fact that the stress tensor is not symmetric necessitates a special treatment of the flow function, where representation theorems of tensor valued function with non-symmetric arguments are invoked. The consequences of such a definition on the resulting inelastic rate are discussed in full. It is shown that the corresponding resulting rate, as defined at the actual configuration, is not symmetric any more. Accordingly, the rate naturally includes a plastic material spin. Moreover, we deal with the theoretically interesting question of how to define spin-free rates. It is also demonstrated that the flow function must depend not only on the stress tensor and on adequate structural tensors, but also on the deformation itself in form of the right Cauchy–Green tensor C. However, this surprising dependency, which must obey a specific form, can be justified as physically meaningful. Various numerical examples of large plastic deformations of structural components are presented, that underpin the capabilities of the formulation.

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C e . The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.

FE Analysis of Contractant Shear Zones in Loose Granular Materials

The paper focuses on the formation of contractant shear zones in initially loose cohesionless granular materials subject usually to continuous densification. For a simulation of the mechanical behaviour of a granular material during monotonous deformation paths, a micro-polar hypoplastic constitutive model was used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The FE investigations of shear localization were carried out with initially very loose quartz sand during four different rate boundary value problems: shearing of an infinite layer between two very rough walls, plane strain compression under constant lateral pressure, biaxial compression with rigid and deformable boundaries and passive earth pressure on a horizontally translating retaining wall. The calculations were carried out with a simple random distribution of the initial void ratio under conditions of large deformations and curvatures.

A numerical study of the effect of penalty parameters for membrane elements with independent rotation fields and penalized equilibrium

We offer a numerical investigation into the effect of penalty parameters in elements with drilling degrees of freedom and penalized equilibrium, for which the stress tensor σ is not a priori assumed to be symmetric. The parameters under investigation are γ , which relates the in-plane translations to the rotations, and α , which results due to enforcement of stress equilibrium in assumed stress implementations of elements with drill rotations. Rather than simply reporting on the quantitative influence of the penalty parameter γ on measures like displacement, rotation and stress, we directly assess the skewness of the nonsymmetric stress tensor σ . Results are presented for both isotropic and orthotropic constitutive relationships.

Effect of fluctuation of current void ratio on the shear zone formation in granular bodies within micro-polar hypoplasticity

The paper focuses on the effect of the distribution of the initial void ratio and fluctuations of the current local void ratio on the evolution of the internal shear localization within a granular body under plane strain conditions during a compression test. For a simulation of the mechanical behavior of a cohesionless granular material during a monotonous deformation path, a micro-polar hypoplastic constitutive was used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length of micro-structure. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. For the calibration of the constitutive constants, the data of a medium quartz sand are used. The effect of various stochastic distributions of the initial void ratio (both spatially correlated and non-correlated ones) on shear localization in the granular specimen was investigated. In addition, the calculations were carried out with different amplitudes and frequencies of fluctuations for the change of the current void ratio.

Modeling of a cyclic plane strain compression-extension test in granular bodies within a polar hypoplasticity

In a recent paper, the effect of cyclic shearing on forced shear localization in an infinite granular strip between rough boundaries was numerically investigated. The present paper focuses on the evolution of spontaneous developed shear localization within an granular body under plane strain conditions, constant lateral pressure and cyclic vertical compression-extension. For a simulation of the mechanical behavior of a cohesionless granular material, a micro-polar hypoplastic constitutive is used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. For the calibration of the constitutive constants, the data of a medium quartz sand are used. The attention of numerical simulations is laid on the influence of the number of cycles, the magnitude of the vertical deformation amplitude and the initial density on the evolution of shear zones in an initially prismatic granular specimen.

Thermomechanical lattice instability and phase field theory of martensitic phase transformations, twinning and dislocations at large strains

Using the second law of thermodynamics, a criterion for the instability of a crystal lattice with respect to a change in order parameters is derived for finite strains and lattice rotations. An explicit relation for lattice rotation is derived. A Gibbs (Landau) potential describing martensitic phase transformations, twinning, and dislocation nucleation is derived for a prescribed nonsymmetric nominal stress tensor and small elastic strain, but finite transformation strain and rotations. The equilibrium and transformation conditions are obtained. Martensitic phase transformations in NiAl, BN and C are analyzed and the importance of finite-strain corrections is demonstrated.

Cell model simulation of void growth in nodular cast iron under cyclic loading

The failure of cast iron under high plastic cyclic strains is controlled by the mechanisms of formation, growth and coalescence of voids. A cell model approach is used to simulate nodular cast iron as a periodic array of loosely bonded spherical inclusions in the matrix material. The models are analyzed by the finite element method under cyclic loading while keeping the stress triaxiality constant. Different types of matrix hardening are used: isotropic, kinematic and combined hardening. The graphite inclusions are simulated by a rigid body. Deformation and void growth are studied in dependence on stress triaxiality and strain range. In most cases after a few cycles a non-symmetric stationary mesoscopic cyclic stress–strain curve is established. The deformation response and the development of the void volume fraction are strongly affected by the value of triaxiality. The void volume is incrementally increasing with each load cycle in a ratcheting manner. The void growth rate depends on the chosen hardening type and is smallest for kinematic hardening. The comparison with simulations in absence of graphite inclusions revealed that void evolution is favored by the inclusions.

Linear micropolar elastic crack-tip fields under mixed mode loading conditions

Micropolar elasticity laws provide a possibility to describe constitutive properties of materials for which internal length scales may become important. They are characterized by the presence of couple stresses and nonsymmetric Cauchy stress tensor. Beyond the classical displacement field, the kinematical variables are augmented by a so-called microrotation field and its gradient, the latter introducing an internal length scale in the theory. For an isotropic, linear micropolar elastic material, the near-tip asymptotic field solutions for mode I and mode II cracks are derived. It is shown that these solutions behave similar to those according to the so-called couple stress theory, which has been investigated by Huang et al. (1997a), or similar to those derived for cellular materials by Chen et al. (1998). In particular, the singular fields have an order of singularity r−1/2 and are governed by some amplitude factors, having the meaning of stress intensity factors as in the classical linear elastic theory. The effect of material parameters on the stress intensity factors is studied by applying the finite element method to calculate the values of the stress intensity factors for an edge-cracked specimen of finite width.

Linear and Nonlinear Analysis of a Smart Beam Using General Electrothermoelastic Formulation

Coupled electrothermoelastic equations applicable to the analysis of smart structures have been derived from first principles. Using the equations and applying a layer-by-layer finite element model, the induced potential and mechanical deformations in the piezo and nonpiezo core material have been obtained for various cases of actuation and sensing of a smart beam under external mechanical load and actuation potential. The present study clearly brings out the essential difference between sensing and actuation. It is also brought out that the interaction between polarization and electric field in the piezo continuum leads to nonlinear distributed body force and nonsymmetric stress tensor. These nonlinear effects are found to have significant influence on the deformation of a smart beam under actuation. Shape control studies of multipatch smart beams have also been investigated.

Evolution of shear bands in sand

Localisation of deformation in narrow shear bands is a fundamental phenomenon of granular material behaviour. When modelling granular materials, shear localisation has therefore to be considered. The model must include an adequate constitutive law with an intrinsic length determining the thickness of the localised shear zones, and a representation of microscopic inhomogeneities triggering shear localisation. In the framework of a Cosserat continuum the behaviour of non-cemented granular material can be described by a hypoplastic law including the evolution of non-symmetric stresses and couple stresses. Void ratio fluctuations are introduced by a physically justified probability density function. The assumption is that fluctuations are minimal in the densest state, and maximal in the critical state. It is shown that shear bands can evolve spontaneously in the interior of a granular body even under homogeneous stress or displacement boundary conditions. Experimental verifications are given. Numerical results are compared with laboratory tests performed with a Cornforth and a Hamblytype biaxial apparatus. Shear localisation in the experiment is visualised by means of an optical method called particle image velocimetry. Further numerical results are compared with experimental results taken from publications.

Guest Editorial

Guest Editorial

A mixture theory for geophysical fluids

Abstract. A continuum theory is developed for a geophysical fluid consisting of two species. Balance laws are given for the individual components of the mixture, modeled as micropolar viscous fluids. The continua allow independent rotational degrees of freedom, so that the fluids can exhibit couple stresses and a non-symmetric stress tensor. The second law of thermodynamics is used to develop constitutive equations. Linear constitutive equations are constituted for a heat conducting mixture, each species possessing separate viscosities. Field equations are obtained and boundary and initial conditions are stated. This theory is relevant to an atmospheric mixture consisting of any two species from rain, snow and/or sand. Also, this is a continuum theory for oceanic mixtures, such as water and silt, or water and oil spills, etc.