A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.     

A criterion of classification for non-linear hypo-elastic materials in simple shear

Summary Hypo-elastic materials can be classified according to their static response to deformations of simple shear. Since the principal components of stress must satisfy a non-linear ordinary differential system, the derivatives of the stresses at the origin characterize either the hard or the soft behaviour of the material; moreover the sign of a bilinear form on the strain rates defines the so-called stability of the material.

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Sommario I materiali ipo-elastici possono essere classificati secondo la loro risposta statica a deformazioni di scorrimento puro. Poiché le componenti principali della tensione devono soddisfare un sistema di equazioni differenziali ordinarie non lineari, le derivate delle tensioni all'origine caratterizzano sia il comportamento incrudente sia quello soffice del materiale; inoltre il segno di una forma bilineare sulle velocità di deformazione definisce la cosiddetta stabilità del materiale.

     

The relationship between simple shear and pure shear for a simple fluid without coupling effects

Similarities between simple shear and pure shear or planar extension are exploited to derive equations relating stress in pure shear at constant extension rate to the stress in simple shear at constant shear rate. For the class of materials considered it follows that there are only two independent material functions required to describe simple shear. The relationships derived may also be used to estimate the ratio of first to second normal-stress differences in simple shear using experimental results from pure shear experiments.     

Tresca-type plastic materials in the theory of hypoelasticity II. Optical constitutive equations and birefringence in simple shear

Behavior of a Tresca type plastic dielectric is investigated theoretically from a continuum mechanical point of view. The optical constitutive equations are defined as special cases of a hypo-elastic dielectric of grade two. The singularity condition of the constitutive equations satisfies the Tresca yield criterion. The index deviator tensor is proportional to the stress deviator tensor and, then, the birefringence and the extinction angle are expressed by the stress deviator. Their numerical variations with the angle of shear in simple shear deformation are shown.     

A simple shear cell for the direct visualization of step-stress deformation in soft materials

We introduce a custom-built stress-controlled shear cell coupled to a confocal microscope for direct visualization of constant-stress shear deformation in soft materials. The torque generator is a cylindrical Taylor–Couette system with a Newtonian fluid between a rotating inner bob and a free-to-move outer cup. A spindle/cone assembly is coaxially coupled to the cup and transfers the torque exerted by the fluid to the sample of interest in a cone-and-plate geometry. We demonstrate the performance of our device in both steady-state and transient experiments with different viscoelastic materials. Our apparatus can conduct unidirectional constant-stress experiments as accurately as most commercial rheometers, with the capability to directly visualize the flow field using tracer particles. Further, our step-stress experiments on viscoelastic materials are devoid of creep ringing, which is an advantageous aspect of our torque generation mechanism. We believe that the device presented here could serve as a powerful and cost-effective tool to investigate the microstructural determinants of nonlinear rheology in complex fluids.     

A constitutive model for shape memory alloys accounting for thermomechanical coupling

This paper presents a generalized Zaki-Moumni (ZM) model for shape memory alloys (SMAs) [cf. Zaki, W., Moumni, Z., 2007a. A three-dimensional model of the thermomechanical behavior of shape memory alloys. J. Mech. Phys. Solids 55, 2455-2490 accounting for thermomechanical coupling. To this end, the expression of the Helmholtz free energy is modified in order to derive the heat equation in accordance with the principles of thermodynamics. An algorithm is proposed to implement the coupled ZM model into a finite element code, which is then used to solve a thermomechanical boundary value problem involving a superelastic SMA structure. The model is validated against experimental data available in the literature. Strain rate dependence of the mechanical pseudoelastic response is taken into account with good qualitative as well as quantitative accuracy in the case of moderate strain rates and for mechanical results in the case of high strain rates. However, only qualitative agreement is achieved for thermal results at high strain rates. It is shown that this discrepancy is mainly due to localization effects which are note taken into account in our model. Analyzing the influence of the heat sources on the material response shows that the mechanical hysteresis is mainly due to intrinsic dissipation, whereas the thermal response is governed by latent heat. In addition, the variation of the area of the hysteresis loop with respect to the strain rate is discussed. It is found that this variation is not monotonic and reaches a maximum value for a certain value of strain rate.     

A simple theory of plasticity

An internal-variable model of rate-independent plastic behavior, based on loading-unloading irreversibility, is proposed. The model is compatible with thermodynamics and assumes no yield or loading function, stability postulate or specific nature of the internal variables. It is shown that current theories of plasticity are restricted forms of the proposed theory.     

On the analysis of simple shear problem using the micro-polar hypoelasticity Cosserat theory

In this paper, an analysis of kinematics of the isotropic elastic Cosserat continuum is presented in infinitesimal and finite deformations. Emphasis is given on the applicability of corotational stress rates for hypoelasticity in micro-polar continua. A non-linear finite element analysis is performed with an explicit formulation of tangent stiffness matrices in the case of Truesdell stress and couple stress rates. A comprehensive path-dependent procedure is employed based on the arc-length method to calculate the stability points and handle the snap-back problem. Finally, the accuracy and efficiency of method are illustrated by numerical examples.     

On the thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening by means of incremental energy minimization

The thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening is analyzed within the present paper. This coupling is of utmost importance in many applications, e.g., in those showing low cycle fatigue (LCF) under large strain amplitudes. Since the by now classical thermomechanical coupling originally proposed by Taylor and Quinney cannot be used directly in case of kinematic hardening, the change in heat as a result of plastic deformation is computed by applying the first law of thermodynamics. Based on this balance law, together with a finite strain plasticity model, a novel variationally consistent method is elaborated. Within this method and following Stainier and Ortiz (2010), all unknown variables are jointly and conveniently computed by minimizing an incrementally defined potential. In sharp contrast to previously published works, the evolution equations are a priori enforced by employing a suitable parameterization of the flow rule and the evolution equations. The advantages of this parameterization are, at least, twofold. First, it leads eventually to an unconstrained stationarity problem which can be directly applied to any yield function being positively homogeneous of degree one, i.e., the approach shows a broad range of application. Secondly, the parameterization provides enough flexibility even for a broad range of non-associative models such as kinematic hardening of Armstrong–Frederick-type. Different to Stainier and Ortiz (2010), the continuous variational problem is approximated by a standard, fully-implicit time integration. The applicability of the resulting numerical implementation is finally demonstrated by analyzing the thermodynamically coupled response for a loading cycle.     

Instabilities in shear and simple shear deformations of gold crystals

We use the tight-binding potential and molecular mechanics simulations to study local and global instabilities in shear and simple shear deformations of three initially defect-free finite cubes of gold single crystal containing 3480, 7813, and 58,825 atoms. Displacements on all bounding surfaces are prescribed while studying simple shear deformations, but displacements on only two opposite surfaces are assigned during simulations of shear deformations with the remaining four surfaces kept free of external forces. The criteria used to delineate local instabilities in the system include the following: (i) a component of the second-order spatial gradients of the displacement field having large values relative to its average value in the body, (ii) the minimum eigenvalue of the Hessian of the energy of an atom becoming non-positive, and (iii) structural changes represented by a high value of the common neighborhood parameter. It is found that these criteria are met essentially simultaneously at the same atomic position. Effects of free surfaces are evidenced by different deformation patterns for the same specimen deformed in shear and simple shear. The shear strength of a specimen deformed in simple shear is more than three times that of the same specimen deformed in shear. It is found that for each cubic specimen deformed in simple shear the evolution with the shear strain of the average shear stress, prior to the onset of instabilities, is almost identical to that in an equivalent hyperelastic material with strain energy density derived from the tight-binding potential and the assumption that it obeys the Cauchy-Born rule. Even though the material response of the hyperelastic body predicted from the strain energy density is stable over the range of the shear strain simulated in this work, the molecular mechanics simulations predict local and global instabilities in the three specimens.     

A thermomechanical theory for a porous anisotropic elastic solid with inclusions

We construct a mixture theory which describes a porous elastic anisotropic solid with inclusions. Thermal effects are taken into account. The theory is in accord with classical thermodynamics. Fully nonlinear isotropic and anisotropic materials are considered, and field equations are also given for a nontrivial special case which, though nonlinear, is controlled by a few material functions. When properly specialized, the theory reduces to the P- model, a model widely used to describe porous solids.This work is dedicated to Jerald L. Ericksen on the occasion of his 60^th birthday