On quasi-non-destructive strength and toughness testing of elastic–plastic materials

Non-destructive evaluation of mechanical material properties, like strength and fracture toughness, is impossible for principal reasons. However, there are possibilities of quasi-non-destructive estimation methods, which can be quite useful in practice. Instrumented indentation tests are often suitable to get information about the elastic–plastic behaviour, where the indentation depth is measured as a function of indentation force. By approximate analytical methods, key parameters like ultimate tensile strength, work-hardening exponent or even yield stress can be derived from these measurements. A mobile indenter is presented here and its use in ambulant testing is described. To obtain the uniaxial stress–strain curve more directly and more exactly, the same instrument can be used for a miniature compression test, where a small pin is machined out from the surface of the material. Furthermore, to get information about the toughness of materials, a carving instrument has been developed, which allows the energy required to introduce a defined furrow to be measured and correlated with toughness parameters.     

Matched asymptotic expansions in nonlinear fracture mechanics—II. Longitudinal shear of an elastic work-hardening plastic specimen

Earlier analysis given by T.M. Edmunds and J.R. Willis (1976) is extended to deal with cracks in elastic work-hardening plastic specimens subjected to longitudinal shear loads. Solutions are expressed in terms of a set of parameters that are determined from linear elastic solutions alone. It is proved, for any specimen geometry and any loading symmetric about the plane of the crack, that a ‘plastic-zone correction’, obtained by solving a linear elastic problem for a crack which is a length r_y longer than the actual crack, provides a two-term asymptotic expansion for the J-integral, if r_y is defined suitably in terms of the linear elastic stress concentration factor and the initial slope of the work-hardening curve. The general method is applied in detail to a strip of finite width containing an edge crack, for which the effect of the work-hardening on the maximum extent of the plastic zone and on the J-integral is summarized graphically.     

Brittle fracture: Variation of fracture toughness with constraint and crack curving under mode I conditions

The effect of constraint on brittle fracture of solids under predominantly elastic deformation and mode I loading conditions is studied. Using different cracked specimen geometry, the variation of constraint is achieved in this work. Fracture tests of polymethyl methacrylate were performed using single edge notch, compact tension and double cantilever beam specimens to cover a bread range of constraint. The test data demonstrate that the apparent fracture toughness of the material varies with the specimen geometry or the constraint level. Theory is developed using the critical stress (strain) as the fracture criterion to show that this variation can be interpreted using the critical stress intensity factorK _Cand a second parameterT orA ₃,whereT andA ₃are the amplitudes of the second and the third term in the Williams series solution, respectively. The implication of this constraint effect to the ASTM fracture toughness value, crack tip opening displacement fracture criterion and energy release rateG _Cis discussed. Using the same critical stress (strain) as the fracture criterion, the theory further predicts crack curving or instability under mode I loading conditions. Experimental data are presented and compared with the theory.     

Exponential stress and strain singularity at tip of mode I growing crack in power hardening plastic material

The stress-strain distribution near the tip of a Mode I growing crack in a power hardening plastic material is reconsidered. Two types of asymptotic equations are derived and solved numerically. It is shown that when the crack tip is approached, the stress is singular of the order r^−δ, while the strain is singular of the order r^−nδ, where r is the distance measured from the crack tip. The parameter δ is a constant; it depends on the hardening exponent n being greater than one.     

Influence of reinforcement and elastic foundation on the vibrations of shallow shells with rectangular planform

A method for determining the natural frequencies and modes of ribbed shallow shells with rectangular planform on an elastic foundation is developed. The method takes into account the discrete arrangement of the ribs. A shallow spherical shell with square planform is considered as example to analyze the effect of the number of ribs and the Winkler and Pasternak coefficients of subgrade reaction on the natural frequencies and modes. It is recommended to take into account the discrete arrangement of ribs when they are few     

Compressive response of open-cell foams. Part I: Morphology and elastic properties

This study is concerned with the understanding and modeling of the compressive response of open cell foams. The response starts with a nearly linear elastic regime which terminates into a limit load followed by an extensive load plateau. The plateau, which is responsible for the excellent energy absorption capacity of foams, is followed by a second stiff branch. Results from polyester urethane open cell foams with relative densities of about 0.025 are used to illustrate this behavior using experiments coupled with several levels of modeling. The experiments include characterization of the microstructure and the properties of the base material and measurement of the compressive response of the foams of various cell sizes.A sequence of models for predicting the complete response of such foam is developed. The foam is idealized to be periodic using the space-filling Kelvin cell assigned the major geometric characteristics found in the foams tested. The cells are elongated in the rise direction, the ligaments are assumed to be straight, to have Plateau border cross-sections and nonuniform cross-sectional area distribution. The ligaments are modeled as shear-deformable extensional beams and the base material is assumed to be linearly elastic. Prediction of the initial elastic moduli are addressed in Part I. Closed form expressions for the material constants are presented as well as results using a FE model of the characteristic cell. Comparison between measurements and predictions is very favorable. The paper finishes with results from a limited parametric study of the elastic moduli. The results demonstrate that inclusion of the geometric complexities mentioned above is essential for successful prediction of the moduli of such foams. The nonlinear parts of the response including the foam crushing behavior are addressed in Part II.     

Pair vs many-body potentials: Influence on elastic and plastic behavior in nanoindentation of fcc metals

Molecular-dynamics simulation can give atomistic information on the processes occurring in nanoindentation experiments. In particular, the nucleation of dislocation loops, their growth, interaction and motion can be studied. We investigate how realistic the interatomic potentials underlying the simulations have to be in order to describe these complex processes. Specifically we investigate nanoindentation into a Cu single crystal. We compare simulations based on a realistic many-body interaction potential of the embedded-atom-method type with two simple pair potentials, a Lennard-Jones and a Morse potential. We find that qualitatively many aspects of nanoindentation are fairly well reproduced by the simple pair potentials: elastic regime, critical stress and indentation depth for yielding, dependence on the crystal orientation, and even the level of the hardness. The quantitative deficits of the pair potential predictions can be traced back: (i) to the fact that the pair potentials are unable in principle to model the elastic anisotropy of cubic crystals and (ii) as the major drawback of pair potentials we identify the gross underestimation of the stable stacking fault energy. As a consequence these potentials predict the formation of too large dislocation loops, the too rapid expansion of partials, too little cross slip and in consequence a severe overestimation of work hardening.     

On the microstructure of open-cell foams and its effect on elastic properties

Synthetic open-cell foams have a complex microstructure consisting of an interconnected network of cells resulting from the foaming process. The cells are irregular polyhedra with anywhere from 9 to 17 faces in nearly monodisperse foams. The material is concentrated in the nearly straight ligaments and in the nodes where they intersect. The mechanical properties of such foams are governed by their microstructure and by the properties of the base material. In this study micro-computed X-ray tomography is used to develop 3D images of the morphology of polyester urethane and Duocel aluminum foams with different average cell sizes. The images are used to establish statistically the cell size and ligament length distributions, material distributions along the ligaments, the geometry of the nodes and cell anisotropy. The measurements are then used to build finite element foam models of increasing complexity that are used to estimate the elastic moduli. In the most idealized model the microstructure is represented as a regular Kelvin cell. The most realistic models are based on Surface Evolver simulations of random soap froth with N³ cells in spatially periodic domains. In all models the cells are elongated in one direction, the ligaments are straight but have a nonuniform cross sectional area distribution and are modeled as shear deformable beams. With this input both the Kelvin cell models and the larger random foam models are shown to predict the elastic moduli with good accuracy but the random foams are 5–10% stiffer.     

On the effect of interactions of inhomogeneities on the overall elastic and conductive properties

A simple method of estimating the effect of inhomogeneity interactions on the overall properties (elastic and conductive) is developed. It is formulated in terms of property contribution tensors that give the contribution of an inhomogeneity to the overall properties. The method can be viewed as further development of the approach of Rodin and Hwang (1991) and Rodin (1993) that generalized the method of analysis of crack interactions (Kachanov, 1987) to inhomogeneities. We also extend the method to the conductive properties. Considering the effect of interactions on the property contribution tensors on the example of pores we find that this effect is generally moderate, at most (even when pores touch one another) – in contrast with the effect on local fields. On example of two spheres, we compare the interaction effects on the elastic and the conductive properties, and discuss the impact of interactions on the cross-property connections.     

Numerical study of shear band instability and effect of cavitation on the response of a specimen under undrained biaxial loading

This paper presents an extension of the local second gradient model to multiphasic materials (solids particles, air, water) and including the cavitation phenomenon. This new development was made in order to model the response of saturated dilatant materials under deviatoric stress and undrained conditions and possibly, in future, the behavior of unsaturated soils. Some experiments have showed the significance of cavitation for the hydromechanical response of materials. However, to date and as far as we are aware, no attempt was made to implement the cavitation as a phase change mechanism with a control of pore pressure. The first part of the results section explores the effects of permeability, dilation angle and loading rate on the stability of shear bands during a localization event. The reasons underlying the band instability are discussed in detail, which helps defining the conditions required to maintain stability and investigating the effects of cavitation without parasite effect of materials parameters or loading rate. The model showed that, if a uniform response is obtained, cavitation triggers localization. However, in case of a localized solution, cavitation follows the formation of the shear band, with the two events being quite distinct.     

Macro-mechanical modelling of blast wave mitigation in foams. Part I: review of available experiments and models

Multiphase flows, which involve compressible or incompressible fluids with linear or nonlinear dynamics, are found in all areas of technology at all length scales and flow regimes. In this contribution, we discuss application of aqueous-foam barriers against blast wave impact. The first experiments demonstrating this behaviour were conducted in the early 1980s in free-field tests. Based on structural requirements, various foams with different blast energy contents were tested with the aim of characterizing the time history of the blast pressure reduction. A number of consistent methodologies for calculating this pressure reduction in foam are based on the effective gas flow model. For estimating the uncertainties of these methodologies, we briefly demonstrate their comparison with existing experimental data. Thereafter, we present various modifications of modelling approaches and their comparison with new results of blast wave experiments.     

Photoelastic technique to investigate the effect of temperature and strain rate on plastic flow in mild steel

Dynamic photoelastic-coating technique was used to observe successive developments of plastic flow in tension at a temperature ranging from ?157°C to 20°C. A type of plastic flow occurred which was determined by a combination of temperature and strain rate. A correlation was found to exist between photoelastic observations and the equation of thermal activation.     

Impact of an elastic rod on a deformable barrier: analytical and numerical investigations on models of a valve and a rod-shaped stamping tool

The longitudinal motion of an elastic rod is studied for the case that the rod is suddenly elastically fixed at one end and is hit by a mass at its other end. This configuration represents real settings e.g. as a valve impacts an elastic valve-seat or as a stamping device used in forging is hit by a large mass. The solution of the problem is formulated in the Laplace transformation space. The inverse transformation into the time domain is performed by engaging the so-called Laguerre polynomial technique. This method allows to calculate exact solutions for finite times from a finite number of series elements. Rigorous mathematical proofs not established up to now are given with respect to the convergence of the series encountered and the validity of exchanging the order of inversion of the Laplace transformation and summation of the established series. For comparison also a numerical solution of the problem is presented. An analysis of the energy transfer between rod, impacting mass and elastic barrier elucidates the marked influence of the deformability of the elastic barrier on the stress state in the rod.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: A very low work hardening aluminum alloy (Al6061-T6511)

In the present study, the initial and subsequent yield surfaces in Al 6061-T6511, based on 10 με deviation from linearity definition of yield, are presented. The subsequent yield surfaces are determined during tension, free end torsion, and combined tension–torsion proportional loading paths after reaching different levels of strains. The yield surfaces are also obtained after linear, bi-linear and non-linear unloading paths after finite plastic deformation. The initial yield surface is very close to the von-Mises yield surface and the subsequent yield surfaces undergo translation and distortion. In the case of this low work hardening material, the size of the yield surfaces is smaller and negative cross-effect is observed with finite plastic deformation. The subsequent yield have a usual “nose” in the loading direction and flattened shape in the reverse loading direction; the observed nose is more dominant in the case of tension and combined tension–torsion loading than in torsional loading. The size of the yield surfaces after unloading is smaller than the initial yield surface but larger than subsequent yield surfaces obtained during prior loading, show much smaller cross-effect, and the shape of these yield surfaces depends strongly on the loading and unloading paths. Elastic constants (Young’s and shear moduli) are also measured within each subsequent yield surfaces. Evolution of these constants with finite deformation is also presented. The decrease of the two moduli is found to be much smaller than reported earlier in tension by Cleveland and Ghosh [Cleveland, R.M., Ghosh, A.K., 2002. Inelastic effects on springback in metals. Int. J. Plast. 18, 769–785]. Part-II and III [(Khan et al., 2009a) and (Khan et al., 2009b)] of the papers will include experimental results on annealed 1100 Al (a very high work hardening material) and on both Al alloys (Al6061-T6511 and annealed 1100 Al) in tension- tension stress space, respectively. The results for both cases are quite different than the ones that are presented in this paper.     

Improvement on stability and convergence of A. D. I. schemes

IntroductionAlternatingdirectionimplicit(A.D.I.)schemeswhichwasdiscoveredin1950',hasbecomeoneofthemostimportantmethodsintheapproximationofthesolutionsofparabolicpartialdifferentialequationsinmulti-dimensionalspace.Someofresultsaboutstabilityandconvergencearetooweakandincomplete,we'lltrytoimprovetheminthispaper.Considerinitial-boundaryvalueproblemintwospacevariablesLetΩhbeauniformrectangularmeshofO.h>0isthespacestepinxandydireehon*ProjectsupPOI'tedbytheNahonalNaturalScienceFoundahonofChi…     

Two-phase flow in a vertical pipe and the phenomenon of choking: Homogeneous diffusion model—I. Homogeneous flow models

The paper examines the topological structure of all possible solutions which can exist in flows through adiabatic constant-area ducts for which the homogeneous diffusion model has been assumed. The conservation equations are one-dimensional with the single space variable $\text{z.}$ but gravity effects are included. The conservation equations are coupled with three equations of state: a pure substance, a perfect gas with constant specific heats, and a homogeneous two-phase system in thermodynamic equilibrium. The preferred state variables are pressure $\text{P.}$ enthalpy $\text{h.}$ and mass flux $\text{G}{}^2$.The three conservation equations are first-order but nonlinear. They induce a family of solutions which are interpreted as curves in a four-dimensional phase space conceived as a union of three-dimensional spaces ($\text{P, h, G}{}^2\text{, z}$) with $\text{G}{}^2\text{=}\text{const}$ treated as a parameter. It is shown that all points in these spaces are regular, so that no singular solutions need to be considered. The existence and uniqueness theorem leads to the conclusion that through every point in phase space there passes one and only one solution-curve.The set of differential equations, treated as a system of algebraic equations of each point of the phase space, determines the components of a rate-of-change vector which are obtained explicitly by Cramer's rule. This vector is tangent to the solution curve. Each solution curve turns downward in $\text{z}$ at some specific elevation $\text{z}{}^1$, and this determines the condition for choking. Choking occurs always when the exit flow velocity at $\text{L = z}{}^1$ is equal to the local velocity of propagation of small plane disturbances of sufficiently large wavelength, that is when the flow rate $\text{G}$ becomes equal to a specified, critical flow rate, $\text{G}{}^1$. (The possible dependence of the sonic velocity on frequency in a real flow is ignored, because it has not been allowed for in the equations of the model under study.) A criterion, analogous to the Mach number, which indicates the presence or absence of choking in a cross section is the ratio $\text{K = G/G}{}^7$ of the mass-flow rate $\text{G}$ to the local critical mass flow rate. $\text{G}{}^7$, $\text{K = 1}$ denoting choking. The critical parameters depend only on the thermodynamic properties of the fluid and are independent of the gravitational acceleration and shearing stress at the wall.The topological characteristics of the solutions allow us to study all flow patterns which can, and which cannot, occur in a pipe of given length $\text{L}$ into which fluid is discharged through a rounded entrance from a stagnation reservoir and whose back-pressure is slowly lowered. The set of flow patterns is analogous to that which occurs with a perfect gas, except that the characteristic numerical values are different. They must be obtained by numerical integration and the influence of gravity must be allowed for.The preceding conclusions are valid for all assumptions concerning the shearing stress at the wall which make if dependent on the state parameters only, but not on their derivatives with respect to $\text{z}$. However, the study is limited to upward flows for which the shearing stress at the wall and the gravitational acceleration are codirectional.