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.     

On deformation of beryllium under high‐velocity oblique collision

The highvelocity oblique collision of samples of beryllium (beryllium and stainless steel) was studied experimentally. The disturbance amplitudes of beryllium, magnesium, aluminum, copper, and steel were compared. It is established that for the same Mach numbers, the disturbance amplitude for beryllium is maximal. The low plasticity and high brittleness of beryllium determine the nature of formation of a welded joint. Fusion and mixing of the metals occur in a very narrow zone, which practically cannot be seen in microsections. Under oblique collision of beryllium and steel, a solid solution layer of elevated hardness is attached to the interface.     

Oscillations of the plastic wave front under high‐rate loading

Two models of elastoplastic wave propagation in metals under uniaxial deformation are considered. The first model treats plastic deformation as being due to dislocation motion during heterogeneous formation of dislocations. The second model assumes that plastic deformation occurs by motion of dipoles of partial disclinations. It is shown that in both cases, certain conditions can give rise to damped oscillations of the plastic wave front, which were detected in shock loading experiments with flat specimens made of 28Kh3SNMFA steel.     

Extracting the stress–strain properties of non-linear viscoelastic materials using the bubble inflation test

In this study, an inverse method based on the Levenberg–Marquardt algorithm was evaluated in a numerical experiment to determine the large strain viscoelastic properties from the bubble inflation test. The properties were determined by iteratively matching the calculated bubble pressure–piston displacement data from finite element simulations to a single set of bubble pressure–piston displacement data. The strain-dependent behaviour was characterised by a two-parameter Mooney–Rivlin hyperelastic model, while the time-dependent behaviour was characterised by a three-parameter power law equation. Different initial guesses were used to evaluate the inverse method, and transformation functions were applied to constrain the intermediate guesses to be within bounds. It was found that estimates of the viscoelastic properties could be obtained reasonably using only one set of bubble pressure–piston displacement data. Estimates of the properties were likely affected by the limited time duration of the test, as the behaviour at shorter and particularly larger time scales was less accurately predicted.     

Elastic–plastic behavior of steel sheets under in-plane cyclic tension–compression at large strain

Elastic–plastic behavior of two types of steel sheets for press-forming (an aluminum-killed mild steel and a dual-phase high strength steel of 590 MPa ultimate tensile strength) under in-plane cyclic tension–compression at large strain (up to 25% strain for mild steel and 13% for high strength steel) have been investigated. From the experiments, it was found that the cyclic hardening is strongly influenced by cyclic strain range and mean strain. Transient softening and workhardening stagnation due to the Bauschinger effect, as well as the decrease in Young's moduli with increasing prestrain, were also observed during stress reversals. Some important points in constitutive modeling for such large-strain cyclic elasto-plasticity are discussed by comparing the stress–strain responses calculated by typical constitutive models of mixed isotropic–kinematic hardening with the corresponding experimental observations.     

Investigation on the effect of density ratio on the convergence behavior of partitioned method for fluid–structure interaction simulation

In order to investigate the effect of density ratio of fluid and solid on the convergence behavior of partitioned FSI algorithm, three strong-coupling partitioned algorithms (fixed-point method with a constant under-relaxation parameter, Aitken’s method and Quasi-Newton inverse least squares (QN-ILS) method) have been considered in the context of finite element method. We have employed the incompressible Navier–Stokes equations for a Newtonian fluid domain and the total Lagrangian formulation for a non-linear motion of solid domain. Linear-elastic (hyper-elastic) model has been employed for solid material with small (large) deformation. A pulsatile inlet-flow interacting with a 2D circular channel of linear-elastic material and a pressure wave propagation in a 3D flexible vessel have been simulated. Both linear-elastic and hyper-elastic (Mooney–Rivlin) models have been adopted for the 3D flexible vessel. From the present numerical experiments, we have found that QN-ILS outperforms the others leading to a robust convergence regardless of the density ratio for both linear-elastic and hyper-elastic models. On the other hand, the performances of the fixed-point method with a constant under-relaxation parameter and the Aitken’s method depend strongly on the density ratio, relaxation parameter selected for coupling iteration, and degree of deformation. Although the QN-ILS of this work is still slower than a monolithic method for serial computation, it has an advantage of easier parallelization due to the modularity of the partitioned FSI algorithm.     

Approximate estimates of the high‐temperature creep of structural elements

The creep and longterm strength of structural elements under temperatures much higher than operational temperatures are studied. The actual nonuniform stress–strain state is reduced to a fictitious uniform state by averaging the specific dissipated power over the volume of the body. This allows one to estimate the intensity of creep and longterm strength of structural elements in terms of volumeaveraged energy parameters by using methods of ideal plasticity and considering statically possible stress fields and kinematically possible velocity and strain rate fields.     

Discretization of the plane problem for a cracked body with nonlinear stress–strain diagram under tension

The plane problem for a cracked body with a piecewise-linear stress–strain diagram under tension is reduced by the Fourier transformation to a system of nonlinear algebraic equations. The system is numerically solved for plane strain and stress states of a perfect elastoplastic material to study plastic zones, stress and strain distributions, and displacements of crack faces     

Comparative analysis of the two‐constant generalizations of hooke's law for isotropic elastic materials at finite strains

For ten models of the isothermal behavior of materials, the solutions of boundaryvalue problems are studied for five types of the experimentally reproducible uniform stress–strain state with unchanged directions of the principal axes. It is found that, for three models, the governing equations are similar to the relations of Hooke's law and valid within the same range of the ratio between the shear and bulk moduli. In these models, the specific strain energy can be represented as a sum of the energies due to changes in volume and shape. The ranges where the other three known models exhibit incorrect behavior are determined.     

A micro–macro method for predicting the shear strength of brittle rock under compressive loading

Microcracks have great significance for shear strength of brittle rock in compression. A major challenge of this area is to establish the correlation of microcracks and macroscopic shear strength. A new micro–macro method is presented to predict the shear strength of brittle rock in compression. This method incorporates the microcrack model suggested by Ashby, Mohr–Coulomb failure criterion and a crack-strain relation. This crack–strain relation is presented to link the crack growth and axial strain by combining the micro and macro definitions from rock damage. The shear strength and stress–strain relationship of Jinping marble are theoretically investigated in detail. The rationality of this suggested method is verified by using the experimental results founded on Jinping marble. Effects of the initial microcrack size, friction coefficient and confining pressure on internal friction angle, cohesion, and shear strength are also discussed.     

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.     

On the “viscosity overshoot” during the uniaxial extension of a low density polyethylene

An experimental investigation of the viscosity overshoot phenomenon observed during uniaxial extension of a low density polyethylene is presented. For this purpose, traditional integral viscosity measurements on a Münstedt-type extensional rheometer are combined with local measurements based on the in-situ visualization of the sample under extension. For elongational experiments at constant strain rates within a wide range of Weissenberg numbers (Wi), three distinct deformation regimes are identified. Corresponding to low values of Wi (regime I), the tensile stress displays a broad maximum, but such maximum is observed with various polymeric materials deformed at low rates and it should not be confused with the “viscosity overshoot” phenomenon. Corresponding to intermediate values of Wi (regime II), a local maximum of the integral extensional viscosity is systematically observed. Moreover, within this regime, a strong discrepancy between integral measurements and the space average of the local elongational viscosity is observed which indicates large deviations from an ideal uniaxial deformation process. Images of samples within this regime reinforce this finding by showing that, corresponding to the maximum of the integral viscosity, secondary necks develop along the sample. The emergence of a maximum of the integral elongational viscosity is, thus, related to the distinct inhomogeneity of deformation states and most probably not to the rheological properties of the material. In the fast stretching limit (high Wi, regime III), the overall geometric uniformity of the sample is well preserved, no secondary necks are observed and both the integral and the space averaged transient elongational viscosity show no maximum. A detailed but yet incomplete comparison of the experimental findings with results from the literature is presented and several open questions are stated.     

A method for calculating the stress–strain state in the general boundary-value problem of metal forming—part 1

To simulate metal-forming processes, one has to calculate the stress–strain state of the metal, i.e. to solve the relevant boundary-value problems. Progress in the theory of plasticity in that respect is well known, for example, via the slip-line method, the finite element method, etc.) , yet many unsolved problems remain. It is well known that the slip-line method is scanty. In our opinion the finite element method has an essential drawback. (No one is against the idea of the discretization of the body being deformed and the approximation of the fields of mechanical variables.) The results of calculation of the stress state by the FEM do not satisfy Newtonian mechanics equations (these equations are said to be softened, i.e, satisfied approximately) and stress fields can be considered poor for solution of the subsequent fracture problem. We believe that it is preferable to construct an approximate solution by the FEM and soften the constitutive relations (not Newtonian mechanics equations) , especially as, in any event, they describe the rheology of actual deformable materials only approximately. We seem to have succeeded in finding the solution technique.Here we present some new results for solving rather general boundary-value problems which can be characterized by the following: the anisotropy of the materials handled; the heredity of their properties and compressibility; finite deformations; non-isothermal flow; rapid flow, with inertial forces; a non-stationary state; movable boundaries; alternating and non-classical boundary conditions, etc.Solution by the method proposed can be made in two stages: (1) integration in space with fixed time, with an accuracy in respect of some parameters; (2) integration in time of certain ordinary differential equations for these parameters.In the first stage the method is based on the principle of virtual velocities and stresses. It is proved that a solution does exist and that it is the only possible one. The approximate solution softens (approximately satisfies) the constitutive relations, all the rest of the equations of mechanics being satisfied precisely. The method is illustrated by some test examples.     

Analysis of the localization of deformation and the complete stress–strain relation for mesoscopic heterogeneous brittle rock under dynamic uniaxial tensile loading

Stress redistribution induced by excavation results in the tensile zone in parts of the surrounding rock mass. It is significant to analyze the localization of deformation and damage, and to study the complete stress–strain relation for mesoscopic heterogeneous rock under dynamic uniaxial tensile loading. On the basis of micromechanics, the complete stress–strain relation including linear elasticity, nonlinear hardening, rapid stress drop and strain softening is obtained. The behaviors of rapid stress drop and strain softening are due to localization of deformation and damage. The constitutive model, which analyze localization of deformation and damage, is distinct from the conventional model. Theoretical predictions have shown to consistent with the experimental results.     

Influence of the cavitation on the stress–strain fields of compressible Blatz–Ko materials at finite deformation

The aim of this article is the analysis of fracture growth in media characterized by random distribution of micro-failure mechanisms per unit volume. The deformation behavior of the material was investigated in terms of a spherical unit cell model, containing an initially spherical cell of porous. The effective elastic bulk modulus as a function of micro-failures concentration was computed and using the Griffith critirium and certain boundary conditions the rate at which the void area varies was determined too. Along the analysis a special form of the strain energy function for compressible Blatz–Ko material was used. The applied traction on the unit cell of the material was determined as a function of the porosity of the material, as well as the strain field within the solid. At low values of the porosity, as the applied external traction was increased instabilities were observed in the void growth.     

Modelling large deformation behaviour under loading–unloading of semicrystalline polymers: Application to a high density polyethylene

In this work, the large deformation behaviour under monotonic loading and unloading of a high density polyethylene (HDPE) is studied. To analyze the nonlinear time-dependent response of the material, mechanical tests were conducted at room temperature under constant true strain rates and stress relaxation conditions. A physically-based inelastic model written under finite strain formulation is proposed to describe the mechanical behaviour of HDPE. In the model, the inelastic mechanisms involve two parallel elements: a visco-hyperelastic network resistance acting in parallel with a viscoelastic–viscoplastic intermolecular resistance where the amorphous and crystalline phases are explicitly taken into consideration. The semicrystalline polymer is considered as a two-phase composite. The influence of the crystallinity on the loading and unloading behaviour is investigated. Numerical results are compared to experimental data. It is shown that the model is able to accurately reproduce the experimental observations corresponding to monotonic loading, unloading and stress relaxation behaviours at different strain levels. Finally, the model capabilities to capture cyclic loading–unloading behaviour up to large strains are discussed. To demonstrate the improved modelling capabilities, simulations are also performed using the original model of Boyce et al. [Boyce, M.C., Socrate, S., Llana, P.G., 2000. Constitutive model for the finite deformation stress–strain behavior of poly(ethylene terephthalate) above the glass transition. Polymer 41, 2183–2201] modified by Ahzi et al. [Ahzi, S., Makradi, A., Gregory, R.V., Edie, D.D., 2003. Modeling of deformation behavior and strain-induced crystallization in poly(ethylene terephthalate) above the glass transition temperature. Mechanics of Materials 35, 1139–1148].     

Exact solutions and mathematical properties of boundary‐value problems for dynamic‐diffusion boundary layers

The paper studies boundaryvalue problems for dynamicdiffusion boundary layers occurring near a vertical wall at high Schmidt numbers and for dynamic boundary layers whose inner edge is adjacent to the dynamicdiffusion layers. Exact solutions for boundary layers at small and large times are derived. The wellposedness of the boundaryvalue problem for a steady dynamicdiffusion layer is studied.     

Hybrid liquid metal–water cooling system for heat dissipation of high power density microdevices

The recent decades have witnessed a remarkable advancement of very large scale integrated circuits (VLSI) and electronic equipments in micro-electronic industry. Meanwhile, the ever increasing power density of microdevices leads to the tough issue that thermal management becomes rather hard to solve. Conventional water cooling is widely used, but the convective coefficient is not high enough. Liquid metal owns much higher convective coefficient and has been identified as an effective coolant recently, but the high cost greatly precludes its large scale utilization. In this paper, a hybrid liquid metal–water cooling system which combines the advantages of both water and liquid metal cooling was proposed and demonstrated. By utilizing a liquid metal “heat spreader” in front of the water cooling module, this system not only owns more excellent cooling capability than that based on water alone, but also has much lower initial cost compared with absolute liquid metal cooling system. A series of experiments under different operation conditions have been performed to evaluate the cooling performance of this hybrid system. The compared results with absolute water cooling and liquid metal cooling system showed that the cooling capability of the new system is competitive with absolute liquid metal cooling, but the initial cost could be much lower. The theoretical thermal resistance model and economic feasibility also have been analyzed and discussed, which shows that the hybrid liquid metal–water cooling system is quite feasible and useful.     

Derivatives of the three-dimensional Green’s functions for anisotropic materials

A concise formulation is presented for the derivatives of Green’s functions of three-dimensional generally anisotropic elastic materials. Direct calculation for derivatives of the Green’s function on the Cartesian coordinate system is a common practice, which, however, usually leads to a complicated course. In this paper the Green’s function derived by Ting and Lee [Ting, T.C.T., Lee, V.G., 1997. The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids. The Quarterly Journal of Mechanics and Applied Mathematics 50 (3) 407–426] is extended to obtain the derivatives. Using a spherical coordinate system, the Green’s function can be shown as the composition of two independent functions, one depends only on the radial distance of the field point to the origin and the other is in spherical angles. The method of derivation is based on the total differential scheme and then takes its partial differentiation accordingly. With the application of Cauchy residue theorem, the contour integral can be evaluated in terms of the Stroh eigenvalues of a sextic equation. For the degenerate case, evaluation of residues at multiple poles is also given. Applications of the present result are made to examine the Green’s functions and stress components for isotropic and transversely isotropic materials. The results are in exact agreement with existing solutions.     

Special virtual fields for the direct determination of material parameters with the virtual fields method. 2––Application to in-plane properties

This paper deals with the direct identification of mechanical parameters that govern the in-plane constitutive law of orthotropic materials. Those parameters are extracted from heterogeneous strain fields that occur in a short beam specimen tested in a Iosipescu fixture. The procedure used is the virtual fields method with special virtual fields. The case of linear elasticity is first addressed. It is shown that the parameters are directly extracted with this method: no iterative calculations are required. The stability is also discussed in different cases. A non-linear shear response is then considered. The parameter that governs this non-linearity is also directly identified with the special virtual fields.