Determination of the Inhomogeneous Preliminary Stress–Strain State in a Piezoelectric Disk

The problem of steady radial vibrations of a thin electroelastic hollow disk in the presence of a preliminary inhomogeneous plane stress–strain state is solved. Vibrations are induced by applying a potential difference across the electrodes placed on the end surfaces of the disk. Equations of the vibrations and boundary conditions are formulated. The preliminary stress state corresponding to the solution of the Lam´e problem was investigated. The direct problem of determining the displacement function is solved numerically by the shooting method. The inverse problem of determining a pre-stress parameter from the change in the natural frequency of the disk is formulated and solved. The accuracy of determining the prestressed state for initial data specified with an error is analyzed.     

Measurement of Temperature-Dependent Young’s Modulus at a Strain Rate for a Molding Compound by Nanoindentation

The mechanical properties of a molding compound on a packaged integrated circuit (IC) were measured by spherical nanoindentation using a 50 μm radius diamond tip. The molding compound is a heterogeneous material, consisting of assorted diameters of glass beads embedded in an epoxy. Statistical analysis was conducted to determine the representative volume element (RVE) size for a nanoindentation grid. Nanoindentation was made on the RVE to determine the effective viscoelastic properties. The relaxation functions were converted to temperature-dependent Young’s modulus at a given strain rate at several elevated temperatures. The Young’s modulus values at a given strain rate from nanoindentation were found to be in a good agreement with the corresponding data obtained from tensile samples at or below 90 °C. However, the values from nanoindentation were significantly lower than the data obtained from tensile samples when the temperature was near or higher than 110 °C, which is near the glass transition. The spatial distribution of the Young’s modulus at a given strain rate was determined using nanoindentation with a Berkovich tip. The spatial variation of the Young’s modulus at a given strain rate is due to the difference in nanoindentation sites (glass beads, epoxy or the interphase region). A graphical map made from an optical micrograph agrees reasonably well with the nanoindentation results.     

Decomposition of the Deformations of a Thin Shell. Asymptotic Behavior of the Green-St Venant’s Strain Tensor

We investigate the behavior of the deformations of a thin shell, whose thickness δ tends to zero, through a decomposition technique of these deformations. The terms of the decomposition of a deformation v are estimated in terms of the L ²-norm of the distance from ∇ v to SO(3). This permits in particular to derive accurate nonlinear Korn’s inequalities for shells (or plates). Then we use this decomposition technique and estimates to give the asymptotic behavior of the Green-St Venant’s strain tensor when the “strain energy” is of order less than δ ^3/2.     

Study of the Elastoplastic Axisymmetric Stress–Strain State of Irradiated Laminated Shells with a Loading History

A technique is proposed to determine the axisymmetric state of laminated shells under thermoradiation loading. The deformation is assumed simple. The axisymmetric elastoplastic stress–strain state of a two-layered continuous plate under a distributed load and a neutron flux incident on one of the plate surfaces is studied within the framework of the Kirchhoff–Love hypothesis for a layer stack. The results obtained with and without regard for the loading history are presented. An important role of the loading history in a stress analysis is confirmed     

Independent‐Stress Method for Analysis of Nonlinear Antiplane Strain

The stress field in a cylindrical body under antiplane strains is studied using the nonlinear theory of elasticity in actual variables under assumptions of the absence of body forces and weak nonlinearity of the elastic potential. The stresses are determined by solving the nonlinear boundaryvalue problem for two independent stresses in polar coordinates of the physical and stress planes. Analytical solutions of the nonlinear problems are obtained. The effect of potential nonlinearity is studied. It is shown that the nonlinear problem can be solved using the harmonicequation solution corresponding to the linear potential.     

Obtaining a Wide-Strain-Range True Stress–Strain Curve Using the Measurement-In-Neck-Section Method

Background

Measuring true stress–strain curve over a large-strain-range is essential to understand mechanical behavior and simulate non-linear plastic deformation. The digital image correlation (DIC) technique, a non-contact full-field optical measurement technique, is a promising candidate to obtain a long-range true stress–strain curve experimentally.

Objective

This paper proposes a method for measuring true stress–strain curves over a large-strain-range during tensile testing using DIC.

Methods

The wide-strain-range true stress–strain curves of dual-phase and low carbon steels were extracted on the transverse direction in the neck region. The axial strain on the neck section was estimated by averaging the inhomogeneous deformation on the cross-section of the tensile specimen. The true stress was calculated from the engineering stress and the cross-sectional area of the neck.

Results

The validity of the proposed method was assessed by comparing the experimental load–displacement responses during tensile testing with the finite element method (FEM) simulation results. The stress and strain on the neck section estimated using the FEM and DIC, respectively, were proven to satisfy the uniaxial condition and successfully obtained.

Conclusions

The experimental results agree well with the FEM results. The proposed concept can be applied to various deformation modes for accurately measuring long-range true stress–strain curves.

     

Exploration of Saint-Venant’s Principle in Inertial High Strain Rate Testing of Materials

Current high strain rate testing procedures of materials are limited by poor instrumentation which leads to the requirement for stringent assumptions to enable data processing and constitutive model identification. This is the case for instance for the well known Split Hopkinson Pressure Bar (SHPB) apparatus which relies on strain gauge measurements away from the deforming sample. This paper is a step forward in the exploration of novel tests based on time and space resolved kinematic measurements obtained through ultra-high speed imaging. The underpinning idea is to use acceleration fields obtained from temporal differentiation of the full-field deformation maps measured through techniques like Digital Image Correlation (DIC) or the grid method. This information is then used for inverse identification with the Virtual fields Method. The feasibility of this new methodology has been verified in the recent past on a few examples. The present paper is a new contribution towards the advancement of this idea. Here, inertial impact tests are considered. They consist of firing a small steel ball impactor at rectangular free standing quasi-isotropic composite specimens. One of the main contributions of the work is to investigate the issue of through-thickness heterogeneity of the kinematic fields through both numerical simulations (3D finite element model) and actual tests. The results show that the parasitic effects arising from non-uniform through-the-thickness loading can successfully be mitigated by the use of longer specimens, making use of Saint-Venant’s principle in dynamics.     

An Asymptotic Elastic Plastic Analysis in Plane Strain Deformation near a Crack Tip

In this paper,an asymptotic elasto-plastic analysis in plane strain deformation near a crack tip is established.In this article,special emphasis is laid on the well-known Irwin’s solution of elastic material.Thus,all the field quantity necessary for the elasto-plastic analysis of fracture mechanics can be obtained in this paper.     

The Thermoviscoelastoplastic Axisymmetric Stress–Strain State of Laminated Flexible Orthotropic Shells

A technique is proposed to determine the thermoviscoelastoplastic axisymmetric stress–strain state of laminated shells made of isotropic and orthotropic materials. The paper deals with processes of shell loading such that both instantaneous elastoplastic and creep strains occur in isotropic materials and elastic and creep strains in orthotropic materials. The technique is developed within the framework of the Kirchhoff–Love hypotheses for a stack of layers with the use of the equations of the geometrically nonlinear theory of shells in a quadratic approximation. The deformation of isotropic materials is described by the equations of the theory of deformation along slightly curved trajectories, while the deformation of orthotropic materials is described by Hooke's law with additional terms allowing for creep. A numerical example is given     

Fiber Bragg Grating Strain Rosette

0Introduction OpticalFiberBraggGratings(FBGs)sensorshavefounddiverseapplicationsinvariousfields sincetheirdevelopmentinthelate1980s.FBGshaveanumberofsignificantadvantagesoverother fiberopticsensorssuchasmultiplexing,self referencing,optical,mechanicalandt…     

On Nonexistence for Stationary Solutions to the Navier–Stokes Equations with a Linear Strain

We consider stationary solutions to the three-dimensional Navier–Stokes equations for viscous incompressible flows in the presence of a linear strain. For certain class of strains we prove a Liouville type theorem under suitable decay conditions on vorticity fields.     

Very Large Poisson’s Ratio with a Bounded Transverse Strain in Anisotropic Elastic Materials

In a recent paper by Ting and Chen [18] it was shown by examples that Poisson’s ratio can have no bounds for all anisotropic elastic materials. With the exception of cubic materials, the examples presented involve a very large transverse strain. We show here that a very large Poisson’s ratio with a bounded transverse strain exists for all anisotropic elastic materials. The large Poisson’s ratio with a bounded transverse strain occurs when the axial strain is in the direction very near or at the direction along which Young’s modulus is very large. In fact the transverse strain has to be very small for the material to be stable. If the non-dimensionalized Young’s modulus is of the order δ⁻¹, where δ is very small, the axial strain, the transverse strain and Poisson’s ratio are of the order δ, δ^1/2 and δ^−1/2, respectively. Mathematics Subject Classifications (2000) 74B05, 74E10.T.C.T. Ting: Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing a plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite plane strain elastic body, which differs from that in earlier studies using the three-dimensional Green’s function. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is suppressed. The problem of a cylindrical inclusion embedded concentrically in a finite plane strain cylindrical elastic matrix of an enhanced continuum is analytically solved for the first time by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical elasticity-based Eshelby tensor for the cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are not considered. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

The Elastoplastic Axisymmetric Stress–Strain State of Flexible Laminated Shells Exposed to Radiation

A procedure is proposed to numerically study the thermoelastoplastic axisymmetric stress–strain state of laminated flexible shells exposed to radiation. The equations of thermoradiation plasticity describing simple processes are used. Results of an analysis of the elastoplastic state of a three-layer shell with regard for radiation effects are presented     

Equations of dynamics of a mixture composed of a gas and hollow selective‐permeable microspheres

Based on the laws of conservation of mass, momentum, and energy, equations of dynamics of multiphase systems, which are gas mixtures with hollow microspheres with selectively permeable shells, are obtained under the assumption of quasisteadiness of the process offilling the microspheres by the gas. Acoustic characteristics of the system composed of a uniform gas and hollow permeable microspheres are studied using a simplified (onevelocity and onetemperature) model. The frequency dependences of velocity and damping coefficient of sound are determined with regard for gas density (pressure) relaxation inside the microspheres.     

Flow of a Viscoplastic Fluid on a Rotating Disk

ＦＬＯＷＯＦＡＶＩＳＣＯＰＬＡＳＴＩＣＦＬＵＩＤＯＮＡＲＯＴＡＴＩＮＧＤＩＳＫＦａｎＣｈｕｎ（范椿）（ＩｎｓｔｉｕｉｅｏｆＭｅｃｈａｎｉｃｓ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ）（ＲｅｃｅｉｖｅｄＮｏｖ．２０，１９９２；Ｃｏｍｍｕｎｉｃａｔ...     

The Stress–Strain State of Geometrically and Physically Nonlinear Plates with One-Sided Corrosive Wear

A technique is proposed to investigate one-sided corrosive wear. The problem is solved with regard for geometric and physical nonlinearity. Two, Dolinskii's and Gutman's corrosion models are considered. The quasistatic problem is solved by the method of variational iterations, which reduce ordinary differential equations to a system of nonlinear equations with approximation o(h ²) to be solved by Newton's method. At each step, to allow for physical nonlinearity, the method of variable elastic parameters is used. Also a technique is developed to consider various boundary conditions and ⁱ(e ⁱ) diagrams. Specific numerical results are presented.     

Biological control of a predator–prey system through provision of a super predator

In this paper, we make a systematic analysis of the dynamics of a predator–prey system with type-II functional response, in which the predator growth rate is affected by the presence of a super predator. The main aim of this research is to study the consequences of the presence of a super predator on the system dynamics. The existence and stability of the different possible equilibrium points are studied, and we conclude that the maximum consumption rate of a super predator plays a key role in determining the eventual state of the ecosystem. A detailed bifurcation analysis is carried out through numerical simulations, and we observe that theoretically it is possible to control the dynamics of the system by manipulating the consumption rate of the super predator.     

Stability of Filtration of a Gas—Liquid Mixture

The stability of steady regimes of filtration of a gas—liquid mixture at pressure lower than the saturation pressure is studied for the case of a nonmonotonic dependence of the relative phase permeability of the liquid on the gas saturation. It is shown that periodic self–oscillations can appear, and their evolution leads to deterministic chaos due to the appearance and destruction of quasiperiodic motions.