Initiation and growth of wrinkling due to nonuniform tension in sheet metal forming

An experimental investigation was conducted on the initiation and growth of wrinkling due to nonuniform tension using the Yoshida buckling test. The initiation of wrinkling was detected by strain gages mounted on both surfaces of the samples in the loading and transverse directions. The bifurcation of aluminum auto body sheets appeared to be smooth and much less abrupt than that observed in a steel sheet. A special fixture was designed to, perhaps for the first time, continuously measure the in situ growth of the buckle heights so that the rates of buckle growth were monitored as functions of strain and stress in the loading direction. In contrast to what is commonly believed, it was found that the buckle height is not predominantly determined by the material yield strength, and lower averager value does not increase the rate of buckle growth. Crystallographic texture components and pole figures of the test materials were also measured, and the relationship of plastic anisotropy with wrinkling behavior was investigated by experiments with specimens aligned in the rolling direction, the transverse direction and 45-deg to the rolling direction of the sheet materials.     

Effect of stress triaxiality on porosity evolution in notched bars: Quantitative agreement between a recent dilatational model and X-ray tomography data

In this paper, it is shown that a micromechanically motivated macroscopic model can predict with accuracy the role of the stress state on void evolution in engineering materials. Specifically, a recent criterion that accounts for the influence of all stress invariants on the dilatational response of porous metals is used to predict porosity evolution and strength reduction in aluminum alloy AA 6016-T4. A very good quantitative agreement between the simulation results and X-ray tomography damage measurements in specimens of different notch acuities is obtained. In contrast to existing models, the void volume fraction evolution correlates very well with the X-ray data for all stress triaxialities.     

A constitutive model dealing with damage due to cavity growth and the Mullins effect in rubber-like materials under triaxial loading

In this work, we attempted to describe the evolution of damage in rubber-like materials due to the Mullins effect and the cavity growth process. To this end we introduced two distinct internal variables into the constitutive laws; the first one essentially describes the Mullins damage and the second describes the cavity growth. The Mullins effect was considered as a continuous type of damage that can be modelled within the continuum damage theory. The cavity growth, being discontinuous at the microscopic scale, was also modelled by a continuous variable after a homogenization procedure. These analyses allow the establishment of a compressible constitutive law describing the strain-softening phenomena for rubber-like materials. In order to identify the material parameters and to verify the efficiency of the model, we carried out experimental studies involving uniaxial, biaxial, and hydrostatic tensions under monotonic and cyclic loading. Comparison between the model-predicted results and the experimental data shows that the present model can efficiently describe both the Mullins damage and the porosity evolution of rubber-like materials under triaxial monotonic or cyclic loading with a satisfactory accuracy. The proposed concept is simple and easy to apply to engineering calculations.     

Dynamic void nucleation and growth in solids: A self-consistent statistical theory

We present a framework for a self-consistent theory of spall fracture in ductile materials, based on the dynamics of void nucleation and growth. The constitutive model for the material is divided into elastic and “plastic” parts, where the elastic part represents the volumetric response of a porous elastic material, and the “plastic” part is generated by a collection of representative volume elements (RVEs) of incompressible material. Each RVE is a thick-walled spherical shell, whose average porosity is the same as that of the surrounding porous continuum, thus simulating void interaction through the resulting lowered resistance to further void growth. All voids nucleate and grow according to the appropriate dynamics for a thick-walled sphere made of incompressible material. The macroscopic spherical stress in the material drives the response in all volume elements, which have a distribution of critical stresses for void nucleation, and the statistically weighted sum of the void volumes of all RVEs generates the global porosity. Thus, macroscopic pressure, porosity, and a distribution of growing microscopic voids are fully coupled dynamically. An example is given for a rate-independent, perfectly plastic material. The dynamics of void growth gives rise to a rate effect in the macroscopic material even though the parent material is rate independent.     

On the localization and failure in aluminum shells due to crushing induced bending and tension

The paper is concerned with the accurate numerical simulation of localized deformation that can develop into necking and failure, induced by combined bending and tension in shell structures. The study is motivated by the need to establish the onset and evolution of such failures in imploding underwater structures. Such localized zones of deformation are shown to develop under controlled conditions in experiments on Al-6061-T6 cylindrical shells crushed laterally by rigid punches. The crushing induces gradually developing local depressions in the shells, at radially constrained locations. As the crushing progresses, the depressions with a width of the order of the shell wall thickness, deepen, increase their span, become neck-like and develop inclined failures. In the experimental set-up used, the crushing was terminated when the first of four such depressions that develop ruptured. The shell was sliced along the principal plane of crushing and the most deformed cross sections of the necks were measured. The crushing experiments were simulated numerically using solid FE models. The material was modeled as a finitely deforming elastic–plastic solid that hardens isotropically using the von Mises, the non-quadratic isotropic Hosford and anisotropic Yld04–3D yield functions suitably calibrated. While the overall structural response was reproduced well by all models, differences were observed in the evolution of localization in the depressions. For the von Mises yield function, the localized deformation was significantly milder than in the experiments. The isotropic Hosford yield function produced necks that were closer to the experimental ones, while Yld04–3D produced results that were very close to the measurements. Clearly, and in concert with other applications, the adoption of a non-quadratic yield function is necessary for reproduction of localized and other challenging deformation histories in Al alloys. The addition of anisotropy in such models improves further the predictions.     

Recoverable creep deformation and transient local stress concentration due to heterogeneous grain-boundary diffusion and sliding in polycrystalline solids

Numerical simulations are used to investigate the influence of heterogeneity in grain-boundary diffusivity and sliding resistance on the creep response of a polycrystal. We model a polycrystal as a two-dimensional assembly of elastic grains, separated by sharp grain boundaries. The crystal deforms plastically by stress driven mass transport along the grain boundaries, together with grain-boundary sliding. Heterogeneity is idealized by assigning each grain boundary one of two possible values of diffusivity and sliding viscosity. We compute steady state and transient creep rates as functions of the diffusivity mismatch and relative fractions of grain boundaries with fast and slow diffusion. In addition, our results show that under transient conditions, flux divergences develop at the intersection between grain boundaries with fast and slow diffusivity, which generate high local stress concentrations. The stress concentrations develop at a rate determined by the fast diffusion coefficient, and subsequently relax at a rate determined by the slow diffusion coefficient. The influence of the mismatch in diffusion coefficient, loading conditions, and material properties on the magnitude of this stress concentration is investigated in detail using a simple model problem with a planar grain boundary. The strain energy associated with these stress concentrations also makes a small fraction of the plastic strain due to diffusion and sliding recoverable on unloading. We discuss the implications of these results for conventional polycrystalline solids at high temperatures and for nanostructured materials where grain-boundary diffusion becomes one of the primary inelastic deformation mechanisms even at room temperature.     

A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics

Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.     

Size effects in material strength due to crack growth and material non-linearity

Failure by strength and fracture collapse tend to compete with one another when the specimen sizes are varied. Material testing dealing with the determination of tensile strength and hardening is usually carried out with small specimens while the evaluation of fracture mechanics parameters such as critical stress-intensity factor or strain energy density factor requires specimens that are larger in size. The formation of cracks in small specimens does not appreciably affect failure by strength collapse. On the other hand, the fracture process is not disturbed by the development of plastic hinges in the unbroken ligament of the larger specimens.     

Free surface flow under gravity and surface tension due to an applied pressure distribution: I Bond number greater than one-third

We consider steady free surface two-dimensional flow due to a localized applied pressure distribution under the effects of both gravity and surface tension in water of constant depth, and in the presence of a uniform stream. The fluid is assumed to be inviscid and incompressible, and the flow is irrotational. The behavior of the forced nonlinear waves is characterized by three parameters: the Froude number, F, the Bond number, τ > 1/3, and the magnitude and sign of the pressure forcing parameter ɛ. The fully nonlinear wave problem is solved numerically by using a boundary integral method. For small amplitude waves and F < 1 but not too close to 1, linear theory gives a good prediction for the numerical solution of the nonlinear problem in the case of bifurcation from the uniform flow. As F approaches 1, the nonlinear terms need to be taken account of. In this case the forced Korteweg-de Vries equation is found to be an appropriate model to describe bifurcations from an unforced solitary wave. In general, it is found that for given values of F < 1 and τ > 1/3, there exists both elevation and depression waves. In some cases, a limiting configuration in the form of a trapped bubble occurs in the depression wave solutions.     

Experimental and numerical study on laminar natural convection in a cavity heated from bottom due to an inclined fin

Natural convection heat transfer in an inclined fin attached square enclosure is studied both experimentally and numerically. Bottom wall of enclosure has higher temperature than that of top wall while vertical walls are adiabatic. Inclined fin has also adiabatic boundary conditions. Numerical solutions have been done by writing a computer code in Fortran platform and results are compared with Fluent commercial code and experimental method. Governing parameters are Rayleigh numbers (8.10⁵ ≤ Ra ≤ 4 × 10⁶) and inclination angle (30° ≤ and ≤ 120°). The temperature measurements are done by using thermocouples distributed uniformly at the wall of the enclosure. Remarkably good agreement is obtained between the predicted results and experimental data. A correlation is also developed including all effective parameters on heat transfer and fluid flow. It was observed that heat transfer can be controlled by attaching an inclined fin onto wall.     

A void nucleation and growth based damage framework to model failure initiation ahead of a sharp notch in irradiated bcc materials

A continuum damage framework is developed and coupled with an existing crystal plasticity framework, to model failure initiation in irradiated bcc polycrystalline materials at intermediate temperatures. Constitutive equations for vacancy generation due to inelastic deformation, void nucleation due to vacancy condensation, and diffusion-assisted void growth are developed. The framework is used to simulate failure initiation at dislocation channel interfaces and grain boundaries ahead of a sharp notch. Evolution of the microstructure is considered in terms of the evolution of inelastic deformation, vacancy concentration, and void number density and radius. Evolution of the damage, i.e., volume fraction of the voids, is studied as a function of applied deformation. Effects of strain rate and temperature on failure initiation are also studied. The framework is used to compute the fracture toughness of irradiated specimens for various loading histories and notch geometries. Crack growth resistance of the irradiated specimens are computed and compared to that of virgin specimens. Results are compared to available experimental data.     

Relations between twin and slip in parent lattice due to kinematic compatibility at interfaces

The relationships between a slip system in the parent lattice and its transform by twinning shear are considered in regards to tangential continuity conditions on the plastic distortion rate at twin/parent interface. These conditions are required at coherent interfaces like twin boundaries, which can be assigned zero surface-dislocation content. For two adjacent crystals undergoing single slip, relations between plastic slip rates, slip directions and glide planes are accordingly deduced. The fulfillment of these conditions is investigated in hexagonal lattices at the onset of twinning in a single slip deforming parent crystal. It is found that combinations of slip system and twin variant verifying the tangential continuity of the plastic distortion rate always exist. In all cases, the Burgers vector belongs to the interface. Certain twin modes are only admissible when slip occurs along an 〈a〉 direction of the hexagonal lattice, and some others only with a 〈c + a〉 slip. These predictions are in agreement with EBSD orientation measurements in commercially pure Ti sheets after plane strain compression.     

Stress relaxation in broken fibers in unidirectional composites: modeling and application to creep rupture analysis

This paper describes a model of stress relaxation in broken fibers in unidirectional metal matrix composites reinforced with long brittle fibers. A cylindrical cell with a broken fiber embedded in a power law creeping matrix is employed, and the broken fiber is assumed to have a bilinear distribution of axial stress. Then, on the basis of energy balance in the cell under constant overall strain, a relaxation equation of interfacial shear stress acting on stress recovery segments is derived in a simple form. The relaxation equation is approximated rationally and integrated to obtain an analytical solution, which is shown to agree fairly well with the numerical analysis of Du and McMeeking. (Du, Z.-Z., McMeeking, R.M., 1995. Creep models for metal matrix composites with long brittle fibres. J. Mech. Phys. Solids 43, 701–726.) Moreover, the relaxation equation is combined with Curtin's model (Curtin, W.A., 1991. Theory of mechanical properties of ceramic-matrix composites. J. Am. Ceram. Soc. 74, 2837–2845.), so that rupture time in long term creep is evaluated analytically and explicitly on the assumption of global load sharing. It is shown that the resulting relation represents well the dependence of creep rupture time on applied stress observed experimentally on a unidirectional metal matrix composite.     

Instationary conjugate optical-thermal fields in thin films due to pulsed laser heating: A comparison between back and front treatment

Manufacturing of multilayer thin films deposited on a glass substrate can be accomplished by means of pulsed laser sources. Thermal and optical nonlinearities are induced during transient heating, since the response of weakly absorbing thin films depends on the temperature. The heat source can either directly impinge the film surface,␣front treatment, or the glass substrate, back treatment. In this paper a one-dimensional conjugate non-linear thermal-optical time dependent problem is numerically analysed. The investigation is accomplished in order to compare the two processes. For a Nd-YAG laser with wavelength of 1064 nm, a a-Si/TCO and Al/a-Si/TCO multilayers are investigated. Results are presented in form of spatial and temporal temperature profiles as well as absorbed heat transfer rates inside the layers. The accomplished analysis clearly shows that the back treatment is more efficient in terms of manufacturing process, oriented to the production of photovoltaic cells. Received on 4 November 1997     

Interaction between Low and High-Frequency Modes in a Nonlinear System: Gas-Filled Cylinder Covered by a Movable Piston

A simple mechanical system containing a low-frequency vibration mode andset of high-frequency acoustic modes is considered. The frequencyresponse is calculated. Nonlinear behaviour and interaction betweenmodes is described by system of functional equations. Two types ofnonlinearities are taken into account. The first one is caused by thefinite displacement of a movable boundary, and the second one is thevolume nonlinearity of gas. New mathematical models based on nonlinearequations are suggested. Some examples of nonlinear phenomena arediscussed on the base of derived solutions.     

Interaction between diffusion of palm biodiesel and large strain in rubber: Effect on stress-softening during cyclic loading

In addition to fluctuating multiaxial mechanical loading, many engineering rubber components are exposed to hostile environments such as oil rich environment. In this case, the mechanical response of rubbers is affected by the interaction existed between mechanical loading and diffusion of liquid into the material. The present work attempts to investigate the above interaction and the resulting mechanical response under cyclic loading conditions in nitrile butadiene rubber (NBR) and chloroprene rubber (CR). More precisely, our focus is on the well-known stress-softening (Mullins effect) phenomenon classically observed in rubbers under cyclic loading conditions.     

A micro-mechanical model of knitted fabric and its application to the analysis of buckling under tension in wale direction： buckling analysis

With the aid of the micro-mechanical model of knitted fabric proposed in Part 1 we analyze the buckling of a knitted fabric sheet when it is subjected to a tension along the wale direction. The large deformation of the fabric sheet in the critical configuration is considered and, to avoid possible deviation due to the approximation of the theory of thin plate, the three-dimensional theory of instability is used. The fabric sheet is considered as a three-dimensional body and all boundary conditions are satisfied. It is shown that the buckling of the fabric sheet is possible, two buckling modes and the corresponding buckling conditions are obtained, but only the flexural mode is physically possible as observed in experiments.The project supported by the National Natural Science Foundation of China (10272079)