Determination of the microscale stress–strain curve and strain gradient effect from the micro-bend of ultra-thin beams

A simple method is established to determine the microscale uniaxial stress–strain curve from the load and deflection data for a doubly clamped beam. The method is based on the fact that, for beam deflection much larger than the beam thickness, the axial stretching dominates the deformation in the doubly clamped beam and the doubly clamped beam behaves like a simple plastic hinge. The microscale uniaxial stress–strain curve, together with the cantilever beam experiments, is used to determine the strain gradient effect in Au thin beams. The effect of finite rotation is also discussed.     

Stress–strain state of nonthin orthotropic spherical shells of variable thickness

The stress–strain state of an orthotropic spherical shell with thickness varying in two coordinate directions is analyzed. Different boundary conditions are considered, and a refined problem statement is used. A numerical analytic method based on spline-approximation and discrete orthogonalization is developed. The stress–strain state of spherical orthotropic shells with variable thickness is studied     

Modelling of pressure–strain correlation in compressible turbulent flow

Previous studies carried out in the early 1990s conjectured that the main compressible effects could be associated with the dilatational effects of velocity fluctuation. Later, it was shown that the main compressibility effect came from the reduced pressure-strain term due to reduced pressure fluctuations. Although better understanding of the compressible turbulence is generally achieved with the increased DNS and experimental research effort, there are still some discrepancies among these recent findings. Analysis of the DNS and experimental data suggests that some of the discrepancies are apparent if the compressible effect is related to the turbulent Mach number, Mt. From the comparison of two classes of compressible flow, homogenous shear flow and inhomogeneous shear flow （mixing layer）, we found that the effect of compressibility on both classes of shear flow can be characterized in three categories corresponding to three regions of turbulent Mach numbers： the low-Mr, the moderate-Mr and high-Mr regions. In these three regions the effect of compressibility on the growth rate of the turbulent mixing layer thickness is rather different. A simple approach to the reduced pressure-strain effect may not necessarily reduce the mixing-layer growth rate, and may even cause an increase in the growth rate. The present work develops a new second-moment model for the compressible turbulence through the introduction of some blending functions of Mt to account for the compressibility effects on the flow. The model has been successfully applied to the compressible mixing layers.     

Analyzing the stress–strain state of thick cylindrical shells

Two approaches to the analysis of the stress–strain state of thick cylindrical shells are elaborated. The shell is divided by concentric cross-sectional circles into several coaxial cylindrical shells. Each of these shells has its own curvature determined on its midline. The stress–strain state of the original shell is described by satisfying the interface conditions between the component shells. The distribution of unknown functions throughout the thickness is determined by finding the analytic solution of a system of differential equations in the first approach and is approximated by polynomial functions in the second approach. The stress–strain state of thick shells is analyzed. It is revealed that the effect of reduction becomes stronger with increasing curvature     

Spline-approximation solution of stress–strain problems for beveled cylindrical shells

The problem of bending of beveled circular cylindrical shells is solved by parametrizing the shell and reducing the two-dimensional boundary-value problem to a one-dimensional one by the spline-collocation method. This problem is solved by the stable discrete-orthogonalization method. The effect of the variability of the geometrical parameters on the displacement fields of circular cylinders is analyzed     

Stress–strain state of flexible ring plates of variable stiffness in a magnetic field

A nonlinear two-dimensional model of a magnetoelastic flexible current-carrying ring plate is developed.Asystem of nonlinear equations describing the stress–strain state of flexible current-carrying plates in nonstationary mechanical and electromagnetic fields is derived. The stress state of a flexible plate of variable stiffness in a magnetic field is determined     

Stress–strain curve and stored energy during uniaxial deformation of polycrystals

The subject of this paper is an attempt to obtain information about the energy stored during plastic deformation from experimentally measured stress–strain curve.Theoretical analysis of the stress–strain curve for elastic-perfectly plastic polycrystalline material has shown that only the part of stored energy can be calculated from the stress–strain curve. This part is the energy stored during non-homogeneous plastic deformation.The results of such calculation have been compared with the total stored energy determined experimentally. It has been shown that part of total stored energy related to non-homogeneous plastic deformation of investigated materials is much lower than that corresponding to homogeneous one.     

Development of temperature-compensated resistance strain gages for use to 800°C

This paper describes the development and evaluation of temperature-compensated resistance strain gages for use to 800°C. These gages included single-element gages and double-element half-bridge gages. The filament of single-element gages was fabricated from specially developed Fe–Cr–Al–V–Ti–Y alloy wire. When bonded to high-temperature Ni-based alloy GH39 after stabilization at 800°C for one hour, the apparent strain from room temperature to 800°C was less than 2000 m/m. Double-element gages were fabricated from Pt–W–Re–Ni–Cr–Y alloy wire (active grid) and Pt–Ir alloy wire (compensating grid). When bonded to different high-temperature alloy specimens and stabilized, and when ballast resistance in series with the compensating grid adjusted suitably, the gages' apparent strains from room temperature to 800°C were less than 2400 m/m.Effects of preoxidization of Fe–Cr–Al wire on the characteristics of the single-element gages are described.     

Micro–macro analysis of granular material behavior along proportional strain paths

When granular materials are subjected to proportional strain loading paths, they manifest a variety of behaviors depending on the initial void ratio of the specimen as well as the imposed dilatancy/contractancy rate. In some cases, the stress components may vanish over the duration of the test, and the specimen may progressively liquefy. To investigate this behavior, the authors have developed a kinematic approach to be deployed in two parts. First, numerical simulations are performed by means of a discrete element method. Secondly, two micromechanical models have corroborated the DEM results. The performance of these models may explain a number of microstructural mechanisms responsible for the macroscopic constitutive behavior.     

Shear response of β-SiC bulk dependent on temperature and strain rate

The shear responses of β-SiC are investigated using molecular dynamics simulation with the Tersoff interatomic potential. Results show a clear decreasing trend in critical stress,fracture strain and shear modulus as temperature increases. Above a critical temperature, β-SiC bulk just fractures after the elastic deformation. However, below the critical temperature, an interesting pattern in β-SiC bulk emerges due to the elongation of Si-C bonds before fracture. Additionally, the shear deformation of β-SiC at room temperature is found to be dependent on the strain rate. This study may shed light on the deformation mechanism dependent on temperature and strain rate.     

Stress–strain state of the alveolar crest under the primary strut of a subperiosteal implant

The stress–strain state in the alveolar bone crest is analyzed over a wide range of stiffness ratios between the bone and the primary strut of the subperiosteal implant. Recommendations on the rational design of subperiosteal implants are given     

Prediction of stress–strain and fracture behaviour of an 8-Harness satin weave ceramic matrix composite

A computationally economic finite-element-based approach has been developed to predict the stress–strain and fracture behaviour of an 8-Harness satin woven ceramic matrix composite with strain-induced damage. The finite element analysis utilises a solid element to model the behaviour of the homogenised orthotropic uni-directional tow and its matrix. The underpinning models of the tow and matrix, (Tang et al., 2009) capture the physics of the interactions between fibres and matrix; and, in this way, permit modelling that bridges the length scales of the fibres and full-scale components. The non-linear multi-axial stress–strain behaviour of the composite has been discretised by multi-linear elastic curves; and the latter has been used as input to a user defined subroutine, UMAT, in the commercial finite element package, ABAQUS. A partial unit cell model has been constructed of the 8-Harness satin weave composite of carbon fibres embedded in an amorphous carbon matrix, HITCO C/C. Predictions of the global stress–strain curve, which include the effects of fibre waviness, have been made for two failure modes: the first by deformation localisation, and the second by dynamic tow failure on fibre fracture, triggered by instantaneous pull-out deactivation. Comparisons have been made between the predictions and experimental data that exhibit two classes of fracture behaviour: brittle and quasi-ductile. The predicted results, both with and without tow waviness, compare well with the experimental data; however, the predictions for waviness are slightly better. The two extremes of experimental behaviour have been found to correspond with the two tow fracture criteria modelled.     

Finite strain––isotropic hyperelasticity

This paper presents a strain energy density for isotropic hyperelastic materials. The strain energy density is decomposed into a compressible and incompressible component. The incompressible component is the same as the generalized Mooney expression while the compressible component is shown to be a function of the volume invariant J only. The strain energy density proposed is used to investigate problems involving incompressible isotropic materials such as rubber under homogeneous strain, compressible isotropic materials under high hydrostatic pressure and volume change under uniaxial tension. Comparison with experimental data is good. The formulation is also used to derive a strain energy density expression for compressible isotropic neo-Hookean materials. The constitutive relationship for the second Piola–Kirchhoff stress tensor and its physical counterpart, involves the contravariant Almansi strain tensor. The stress stretch relationship comprises of a component associated with volume constrained distortion and a hydrostatic pressure which results in volumetric dilation. An important property of this constitutive relationship is that the hydrostatic pressure component of the stress vector which is associated with volumetric dilation will have no shear component on any surface in any configuration. This same property is not true for a neo-Hookean Green’s strain–second Piola–Kirchhoff stress tensor formulation.     

Finite strain––beam theory

An appropriate strain energy density for an isotropic hyperelastic Hookean material is proposed for finite strain from which a constitutive relationship is derived and applied to problems involving beam theory approximations. The physical Lagrangian stress normal to the surfaces of a element in the deformed state is a function of the normal component of stretch while the shear is a function of the shear component of stretch. This paper attempts to make a contribution to the controversy about who is correct, Engesser or Haringx with regard to the buckling formula for a linear elastic straight prismatic column with Timoshenko beam-type shear deformations. The derived buckling formula for a straight prismatic column including shear and axial deformations agrees with Haringx’s formula. Elastica-type equations are also derived for a three-dimensional Timoshenko beam with warping excluded. When the formulation is applied to the problem of pure torsion of a cylinder no second-order axial shortening associated with the Wagner effect is predicted which differs from conventional beam theory. When warping is included, axial shortening is predicted but the formula differs from conventional beam theory.     

On the stress–strain states of cellular materials under high loading rates

A virtual Taylor impact of cellular materials is analyzed with a wave propagation technique, i.e. the Lagrangian analysis method, of which the main advantage is that no pre-assumed constitutive relationship is required. Time histories of particle velocity, local strain, and stress profiles are calculated to present the local stress–strain history curves, from which the dynamic stress–strain states are obtained.The present results reveal that the dynamic-rigid-plastic hardening(D-R-PH) material model introduced in a previous study of our group is in good agreement with the dynamic stress–strain states under high loading rates obtained by the Lagrangian analysis method. It directly reflects the effectiveness and feasibility of the D-R-PH material model for the cellular materials under high loading rates.     

Carrier fringe method of moiré interferometry for tiny strain measurements in micro-field

In this paper, we demonstrate a new optical method for tiny strain measurements based on the principle of carrier fringes of moire interferometry. A cross-line grating with frequency of 1200 lp/mm is replicated on the specimen surface, and the strain can be deduced from the changes in carrier fringes before and after the deformation of an object. Four coherent laser beams are used to obtain the carrier fringe patterns of field U and V. Both theoretical analysis and numerical simulation indicate that the ideal accuracy of strain can be controlled within a range of ±1με. Case study of a plane extension experiment shows that the measurement accuracy of strain can be controlled within the range of ±10με. The average strain values of every row of field U and every column of field V can be obtained by using this method, and approximated strain of every pixel in the whole-field can be further acquired, and thus it is possible to measure tiny strains occurred in a micro-field. The technology in this paper can provide comprehensive information for analyzing related mechanical content in the field of MEMS.     

DEM investigation of strain behaviour and force chain evolution of gravel–sand mixtures subjected to cyclic loading

The strain characteristic and load transmission of mixed granular matter are different from those of homogeneous granular matter. Cyclic loading renders the mechanical behaviours of mixed granular matter more complex. To investigate the dynamic responses of gravel–sand mixtures, the discrete element method (DEM) was used to simulate the cyclic loading of gravel–sand mixtures with low fines contents. Macroscopically, the evolution of the axial strain and volumetric strain was investigated. Mesoscopically, the coordination number and contact force anisotropy were studied, and the evolution of strong and weak contacts was explored from two dimensions of loading time and local space. The simulation results show that increasing fines content can accelerate the development of the axial strain and volumetric strain but has little effect on the evolution of contact forces. Strong contacts tend to develop along the loading boundary, presenting the spatial difference. Weak contacts are firstly controlled by confining pressure and then controlled by axial stress, while strong contacts are mainly controlled by axial stress throughout the whole cyclic loading. Once compression failure occurs, the release of axial stress causes the reduction of strong contact proportion in all local regions. These findings are helpful to understand the dynamic responses of gravel–sand mixtures, especially in deformation behaviours and the Spatio-temporal evolution of contact forces.