Plane Strain Loading Research Articles

Micromechanical modeling of intraspherulitic deformation of semicrystalline polymers

Semicrystalline polymers often show a spherulitic morphology, consisting of a radial assembly of twisted crystalline lamellae and amorphous layers. A multiscale numerical model is used to investigate the mechanics of intraspherulitic deformation of polyethylene. The model establishes links across the microscopic, the mesoscopic, and the macroscopic levels. Constitutive properties of the material are identified for the crystallographic and amorphous domains. The averaged fields of an aggregate of individual phases, having preferential orientations, form the constitutive behavior of intraspherulitic material. The spherulitic macrostructure is described by finite element models. The macroscopic stress–strain response resembles that of a previous random polycrystalline model. However, the current model includes the geometrical effect of the anisotropic structure within a spherulite, causing strain concentrations in the centers, which spread out in the equatorial region for uniaxial loading conditions and in inclined directions for plane strain loading. The deformations are linked to microstructural processes as interlamellar deformation and intralamellar crystallographic slip.

Modeling the Stress-Strain Relations of Sand in Cyclic Plane Strain Loading

A modeling procedure to simulate stress-strain relations of sand subjected to cyclic loading is proposed. Results from drained plane strain compression, extension, and cyclic loading tests on Toyoura sand are analyzed. The monotonic loading behavior is simulated by the generalized hyperbolic equation to use as the skeleton curves in the simulation of cyclic behavior. To construct hysteretic stress-strain curves based on the skeleton curves, the Masing’s rule is generalized to the proportional rule consisting of the internal and external rules. The drag rule is then introduced to simulate cyclic stress-strain behavior in which the stress amplitude increases at a decreasing rate during cyclic loading with a constant strain amplitude. It is assumed that any plastic shear strain increment taking place in a certain direction drags the whole skeleton curve for loading in the opposite direction towards the direction of the concerned shear strain increment. The measured cyclic stress-strain behavior is well simulated by the proposed method.

Bifurcation analysis of diffuse modes by the subloading surface model with tangential stress rate effect

The tangential-subloading surface model is capable of describing the plastic strain rate due to the rate of stress inside the yield surface and the inelastic strain rate due to the stress rate component tangential to the subloading surface. The diffuse bifurcation modes of the rectangular specimen subjected to the plane strain loading is analyzed by adopting the tangential-subloading surface model, exhibiting the characteristic regimes of the governing equations (elliptic, hyperbolic and parabolic) in this article. In each of these regimes both symmetric and anti-symmetric diffuse bifurcation modes are available. Besides, the tangential strain rate makes the inception of bifurcation modes easier in not only normal-yield but also subyield states.

Modeling the Stress-Strain Relations of Sand in Cyclic Plane Strain Loading

Modeling the Stress-Strain Relations of Sand in Cyclic Plane Strain Loading

Plane strain instability of saturated elasto–plastic soils

The two-phase instability of saturated elasto-plastic soils is investigated under plane strain conditions. This type of material instability is characterised by a non-homogeneous field of solid deformation with vortex-like patterns, which grows rapidly and is accompanied by a non-uniform flow of interstitial water exhibiting source–sink flow patterns. Three simple criteria are established to define the conditions for the onset of two-phase instability within elasto-plastic soils saturated with an incompressible fluid and subjected to plane-strain undrained loadings. The simplest criterion states that the onset of instability almost coincides with the peak deviator stress during undrained loading. This analysis shows that two-phase instability in elasto-plastic soils is different from strain localisation, strain softening and con-stitutive singularities. Two-phase instability is likely to emerge before shear bands in contractant elasto-plastic soils of the Mohr–Coulomb type. The study suggests that two-phase instability may disrupt numerical liquefaction analyses using elasto-plastic soil models, and thus prompts the need for further analytical and experimental investigations.

Effective elastoplastic behavior of ductile matrix composites containing randomly located aligned circular fibers

Based on the micromechanical framework and the second-order transverse effective elastic moduli of fiber-reinforced composites derived by Ju and Zhang [Int. J. Solids Struct. 35 (1998) 941], effective elasto-(visco-)plastic behavior of two-phase unidirectional fiber-reinforced ductile matrix composites (FRDMCs) is studied in detail. The circular fibers are assumed to be elastic and unidirectionally aligned while the matrix phase can be either elastic or plastic, depending on the local stress state and effective yield criteria. Furthermore, the ensemble-averaged stress norm is constructed based on the probabilistic distribution of circular fibers, pairwise fiber interactions and the ensemble-area averaging procedure. Together with the plastic flow rule and hardening law postulated in continuum plasticity, the aforementioned stress norm is employed to characterize the overall yield criteria which determine the elastoplastic behavior of FRDMCs under general transverse plane-strain loading and unloading histories. As a special case, the initial effective yield criteria for incompressible ductile matrix containing many unidirectionally aligned cylindrical voids are also derived. In addition, the overall elasto-viscoplastic behavior of FRDMCs is investigated based on the Duvaut–Lions viscoplasticity. Finally, comparison between our theoretical predictions and the available experimental data for FRDMCs is performed to illustrate the capability of the proposed framework.

Influence of the inclusion shape on the edge effects in a ribbon-reinforced composite

The elastic state and edge-effect zones in a weakly ribbon-reinforced composite under plane strain and uniform loading are determined. It is studied how the shape factor of the ribbon affects the nature of edge effects

Localization of deformation in plane elastic–plastic solids with anisotropic elasticity

Localization of deformation is analyzed in elastic–plastic solids endowed with elastic anisotropy and non-associative flow rules. A particular form of elastic anisotropy is considered, for which the localization analysis can be performed with reference to an elastic–plastic solid endowed with isotropic elasticity but whose normals to the yield function and plastic potential are not coaxial. On the other hand, so far, available analytical solutions for the onset of strain localization in elastic–plastic solids assume isotropic elasticity and coaxial plastic properties. Here, a new analytical solution is presented when the plastic normals are not coaxial but the analysis is restricted to plane strain and plane stress loadings. As an illustration, for a material with transverse elastic isotropy and with pressure-dependent yield surface and plastic potential, this solution provides explicit expressions at the onset of strain localization for the plastic modulus, for the orientation of the shear-band and for the slip mode. The numerical results highlight the importance of the coupled influence of elastic anisotropy and non-associativity on the onset of strain localization.

Perturbed cracks in two dimensions : An integral-equation approach

A nominally straight crack of finite length is subjected to plane-strain loadings. A perturbation method is developed for calculating the stress-intensity factors, based on an asymptotic analysis of the governing hypersingular boundary integral equation for the crack-opening displacement. Known exact results for a shallow circular-arc crack are recovered, correct to second order in the small geometrical parameter. The method will extend to three-dimensional problems as it does not make essential use of two-dimensional techniques.

Elastic-plastic analysis of biaxially loaded center-cracked plates

Center-cracked panels loaded in biaxial tension are examined in this paper. Calibration relations for the J integral and the Q constraint factor are presented for a Ramberg–Osgood power law hardening material under plane stress and plane strain loadings. Two cases are examined: an isolated crack and a periodic array of cracks both under biaxial loading conditions. The latter has previously been studied for plane stress conditions. A number of different J estimation schemes are proposed based on the remote load and displacement and their dependence on geometry, biaxiality, and material properties is discussed. The variation of constraint, as characterised by Q, is also presented for plane stress and plane strain conditions. Simple slip line field solutions are derived for perfectly plastic conditions and the resulting limit load solutions are compared with numerically determined values. Implications for failure of cracked plates under biaxial loading are discussed.

Scale effects in friction of single–asperity contacts. I. From concurrent slip to single–dislocation–assisted slip

A micromechanical dislocation model of frictional slip between two asperities is presented. The model suggests that when the contact radius is smaller than a critical value, the friction stress is constant, of the order of the theoretical shear strength, in agreement with reported atomic force microscope (AFM) friction experiments. However, at the critical value there is a transition beyond which the friction stress decreases with increasing area, until it reaches the second transition where the friction stress gradually becomes independent of the contact size. This is in contrast to previous theories, which assume that the friction stress is always independent of the size. The present model also predicts that the mechanisms of slip are size dependent. Before the first transition, the constant friction stress is associated with concurrent slip without the aid of dislocation motion. The first transition corresponds to the minimum contact size at which a single dislocation loop is nucleated and sweeps through the whole contact interface, resulting in a single–dislocation–assisted (SDA) slip. This mechanism is predicted to prevail for a wide range of contact sizes, from 10 nm to 10 µm in radius for typical dry adhesive contacts; however, there are no available experimental data in this size range. The second transition is found to be caused by the effective Peierls stress which stabilizes the dislocation loop within the contact region, resulting in dislocation pile–ups. Beyond the second transition, slip is assisted by cooperative glide of dislocations in the pile–up. For sufficiently large contacts the mechanism of cooperative glide induces a size–independent friction stress, in agreement with observations in surface force apparatus (SFA) friction experiments. This paper (Part I) addresses the first transition: from concurrent slip to SDA slip. The second transition is analysed in a companion paper (Part II).

Approximate analytical equations for the deformation of an inclusion in a viscoplastic matrix

The strain rate in an inclusion embedded in a matrix undergoing large strains is investigated theoretically. The inclusion is either a spheroid (axisymmetric loading), or an elliptical cylinder (plane strain loading), and both are assumed to be made of incompressible power law viscous materials, including the limit case of perfect plasticity. Localization factors (i.e., normalized strain rates) derived from various approaches (viz, linearization models, variational principle, finite element method) are first compared for a sphere and a circular cylinder. The influence of the inclusion aspect ratio is then investigated by means of the variational approach. Finally, an original and practical purpose of the paper is to propose approximate analytical equations to fit the dependence of the localization factor on the strain rate sensitivity parameter, as well as on the inclusion hardness and aspect ratios.

Bifurcation and shear band propagation in sands

Under simple or combined stresses, homogeneous sand specimens exhibit localization of deformation followed by shear band formation. This phenomenon has been extensively studied under plane strain loading conditions. This paper studies the initiation and propagation of the shear bands that develop in thin long hollow cylinders of sand subjected to a combination of axial torsional and spherical stresses. Digital image processing techniques are used to observe and measure the displacements and the initiation and development of the shear bands. The effects of the inclination of the principal stresses on the axis of symmetry, the relative density, the confining pressure and granulometry are examined. Strain measurements within the shear bands open the door to a more complete description of the mechanical behaviour of the material, in particular beyond the peak, in the so-called strain softening zone. L'application de contraintes simples ou complexes à des échantillons homogènes de sable donne lieu à une localization de la déformation sous forme de bandes de cisaillement. Ce phénomène a été largement étudié dans des conditions de déformations planes. Dans cet article on utilise des cylindres creux de sable soumis à des contraintes axiales, torsionelles, et hydrostatiques pour étudier la formation et la propagation de ces bandes de cisaillement. Des techniques basées sur l'analyse digitale d'images sont utilisées pour observer ainsi que pour mesurer les déplacements et l'initiation des bandes. On examine l'influence de l'inclinaison des contraintes principales sur l'axe de Symmetrie, de la densité relative, de la contrainte hydrostatique, et de la granulometrie. La mesure des déformations a l'intérieur des bandes de cisaillement permet de faire une description plus complète du comportement mécanique du matériau; en particulier au-delà du maximum et dans la zone de ramolissement.

The Influence of a Ductile Interphase on the Overall Elastoplastic Behavior of a Fiber-Reinforced Composite

While there exist various homogenization theories for the plasticity of a fiber-reinforced composite, no such theories have been explicitly developed to account for the influence of a ductile interphase. In this paper a simple scheme is developed for such a purpose. The theory evolved out of the work of Qiu and Weng (1992) and Hu (1996), and bears an identical structure to Ponte Castan˜eda’s (1991) variational procedure and Suquet’s (1995, 1996) modified secant moduli approach. An exact solution under the plane-strain biaxial loading is also developed to assess the accuracy of the theory. It is found that, with either a soft or a hard interphase and with or without work-hardening, the homogenization theory can produce sufficiently accurate results under this condition. The theory is then used to examine the influence of the interphase volume concentration on the anisotropic behavior of the composite under axial tension, transverse tension, axial shear, and transverse shear, with both a soft and a hard interphase. The results indicate that, while the axial tensile behavior is not sensitive to the interphase concentration, the behaviors under other types of loading are greatly affected by its presence, especially when the interphase is softer than the matrix.

Snap-back analysis of fracture evolution in multi-cracked solids using boundary element method

According to the Linear Elastic Fracture Mechanics criteria, a numerical model is developed to simulate the failure evolution of multi-cracked finite plates by means of an incremental loading procedure. A modified crack length control scheme is used in order to analyse such problems depending on one or more independent parameters. The aim is to provide information about a discontinuous response, such as the snap-back instability, which can be highlighted only by a deformation controlled process. The load vs. displacement curve, included possible snap-back branches, is numerically traced by means of a procedure based on the Displacement Discontinuity Boundary Element Method. With reference to finite plates with ordered crack distributions in plane strain loading conditions, the model is applied in order to analyse the effects of the crack interaction on the fracture evolution.

Thin film scratch testing in two dimensions—Experiments and analysis

We have modified the microscratch test to create a near plane strain loading condition. In the Microwedge Scratch Test (MWST), a wedge-shaped diamond indenter tip is drawn along a fine line (i.e., narrow strip of film), while simultaneously being driven into the line. We compare microwedge scratching of zone 1 (voided grain boundaries) and zone T (metallurgical grain boundaries) thin film specimens of sputtered tungsten on thermally grown SiO2. Symptomatic of its weak grain boundaries, the zone 1 film displays three separate crack systems. Because of its superior grain boundary strength, the zone T film displayed only one of these—an interfacial crack system. By correlating fracture phenomena to signature events in the load-displacement curve, we develop governing equations for propagating interfacial cracks, including expressions for strain energy release rate, bending strain, and mode mixity. Grain boundary fracture causes zone 1 films to spall before a stable crack is formed. Zone T films survive the bending strains, and hence adhesions may be inferred from stable crack growth mechanics. We conclude by contrasting and comparing experimental results for plane strain indentation versus plane strain scratching.

A general constitutive theory for linear and nonlinear particulate media with microstructure evolution

This work is concerned with the development of a constitutive theory for composite materials with particulate microstructures, which is capable of predicting, approximately, the evolution of the microstructure and its influence on the effective response of composites under general three-dimensional finitestrain loading conditions, such as those present in metal-forming operations. In its present form, the theory is general enough to be used for linearly viscous, nonlinearly viscous and perfectly plastic composites with randomly oriented and distributed ellipsoidal inclusions (or pores), which, in the most general case, can change size, shape and orientation. In addition, the “shape” and “orientation” of their center-to-center statistical distribution functions can also evolve with the deformation. To illustrate the key features of the new theory in the context of a simple example, an application is carried out for plane-strain loading of two-phase systems consisting of random distributions of aligned rigid particles in a power-law matrix phase. The results show that the evolution of the relevant microstructural variables, as well as the effective response, depend in a complex fashion on the initial state of the microstructure, as well as on the specific boundary conditions. In particular, it is found that the changes in orientation of the particles provide a mechanism analogous to “geometric softening” in ductile single crystals, which can lead to significant changes in the instantaneous hardening rate of the composite. This is shown to have important consequences for the possible onset of shear localization in the composite.

In-plane electronic speckle pattern shearing interferometry

An analysis of the out-of-plane shearing interferometer has been performed which shows that production of in-plane strain partial derivatives is possible, which are not affected by out-of-plane displacement function components. The in-plane data are represented as subtraction correlation fringes. This interferometer employs a single diverging illumination beam and is applicable to object plane stress and plane strain loading conditions. The interferometer was tested and compared using a compact tension crack specimen and the results are correlated with finite element software predictions of strain distributions across modelled specimens. This experimental validation was chosen because we had an existing test rig and finite element models which had been independently verified.

Delamination resistance behaviour in unidirectional graphite-epoxy composites

According to the energy criterion for fracture, crack growth occurs when the crack driving force or the energy release rate is at least equal to the crack resistance which is de®ned as the energy consumed in crack propagation. It is well known that, for a cracked metal structure under the plane-strain loading condition, the crack resistance is independent of the crack length. In the case of plane stress, the crack resistance varies with the crack increment. Although a considerable amount of work has been performed to characterize the resistance curve for isotropic materials [1±3], the application of the resistance curve to composites has only recently been performed. Ratwani and Deo [4] applied the resistance curve approach to characterize the delamination growth of AS4=3501-6 composites under mode I loading. They used double-cantilever-beam (DCB) specimens and obtained the delamination growth resistance curve over a wide range of delamination extensions. They found that the resistance increased monotonically with increasing delamination extension, and the delamination growth resistance curve was path independent. Mai and Hakeem [5] noted a resistance-type fracture behaviour of cellulose ®bre cements. They used grooved DCB specimens and observed that the resistance increased monotonically with increasing crack length to a maximum and then dropped to a steady-state value. Foote et al. [6] developed a theoretical model for the crack growth resistance (KR) curve for a strain-softening material with a power-law stress± displacement relationship. They applied the model to a cellulose±asbestos-®bre-reinforced motar and obtained comparable results with the experimental data. Currently, the delamination resistance curve plays an important role in characterizing the delamination growth for unidirectional laminated composites particularly under mode I loading. However, the delamination resistance behaviour of graphite±epoxy composites is not fully characterized at present. In the present work, delamination resistance curves of unidirectional graphite±epoxy laminated composites were constructed as a function of delamination increment, and the effect of initial delamination length on the resistance curve was investigated. DCB specimens made by using graphite±epoxy prepregs (CU125NS) manufactured by Hankuk Fiber Group were used for the resistance curve tests. The DCB specimen was 08 unidirectional and 24 plies thick (3.8 mm) in which delamination was located at the midplane. Initial delaminations applied were 50 and 90 mm, and the specimen length was 230 mm. The edge of the specimen was painted with a white typewriter correction uid and was marked regularly in order to measure the delamination increments. The delamination length was measured using the travelling microscope. Aluminium blocks were adhesively bonded to the ends of the specimen arms to provide the loading points. The experiments were conducted under room-temperature dry environmental conditions, and a displacement-controlled loading of rate 3.5 mm miny1 was applied. After the initial delamination increased by a certain amount, the applied displacement was reduced and then increased to cause delamination growth. This process was repeated at least ten times until the delamination increased to a length of 170 mm. For each specimen, curves of load, P, against displacement, a, during slow delamination growth were obtained where the delamination length was marked against each loading and unloading line. A typical loading±unloading P±a diagram of initial delamination length 50 mm is shown in Fig. 1. As shown in the ®gure, the fracture behaviour is linear elastic. However, it was found that unloading curves do not return to zero displacement although the amount of residual displacement is very small. The reason is that ®bre bridging occurred behind the delamination tip during delamination growth, which prevents the delamination surfaces from closing up

Finite element analysis of two‐phase instability for saturated porous hypoelastic solids under plane strain loadings

Investigates the effects of two‐phase instability on finite element (FE) solutions for porous hypoelastic solids saturated with an insterstitial fluid. Demonstrates that two‐phase instability creates definite problems to the FE computations of coupled solid‐fluid systems. The eigenvectors of the tangential finite element matrices which are responsible for problems are not artificial, but are the bifurcating modes of physical solutions. The investigation, although limited to the plane strain undrained compression of hypoelastic models, is relevant to the investigation of the two‐phase instability of other materials and boundary value problems, and may lead towards an explanation for numerical problems in soil liquefaction analysis.