Elastic-plastic Loading Research Articles

Pile Capacity in Nonhomogeneous Softening Soil

Closed form solutions for vertically loaded piles in elastic-plastic, nonhomogeneous soil have been established recently using an ideal elastic-plastic load transfer model. However, in some cases, the load transfer from piles to the surrounding soil shows softening behaviour. In this paper, the established solutions are extended to account for this behaviour using a simplified strain softening soil model. The solutions are recast in non-dimensional forms, and expressed via modified Bessel functions. The effect of non-homogeneity and the degree of softening on pile capacity is explored.

Thermomechanical characterization of NiTiCu and NiTiSMA actuators: influence of plastic strains

The focus of this study is the thermomechanicalcharacterization and comparison between two different shapememory alloys (SMAs) quantifying the effect of plastic strain onthe transformation characteristics of SMA actuators. In thisstudy, the thermomechanical response and transformationcharacteristics of a NiTiCu and a NiTi SMA are studied as afunction of the induced plastic strain for four differentloading paths: (1) an elastic-plastic loading of the austeniticphase, (2) a stress-induced martensitic phase transformation, (3)an elastic-detwinning-plastic loading of the martensitic phaseand (4) a thermally-induced phase transformation under a constantapplied stress. Each loading path is repeated multiple times,with an incremental change of the total applied strain, todetermine the effect of accumulated plastic strain on the phasetransformation characteristics of the two SMA material systems.The effects of plastic strain are quantified by measurements ofrecoverable strain during a thermal cycle under zero appliedstress, measurements of heat of transformation from adifferential scanning calorimeter and microstructuralevaluations.

Adhesive contact of elastic–plastic spheres

We examine the process of decohesion of two adhering elastic–plastic spheres following mutual indentation beyond their elastic limit. In the first instance, it is assumed that during unloading to the point of pull-off, the deformation is predominantly elastic. First, elastic–plastic loading is modelled by a fine grid finite element analysis revealing that, for elastic–perfectly plastic materials, the contact pressure at the end of loading p o is approximately uniform. Next, the contact size and pressure distribution during elastic unloading from a uniform pressure, in the absence of adhesion, is found by the method of rigid punch decomposition. The pressure distribution approaches that of Hertz as the contact size approaches zero. The distribution of adhesive traction has been found in two ways. First, using the singular traction distribution of linear elastic fracture mechanics, and second, using a step cohesive law, whereby a constant adhesive stress σ o acts between surfaces separated by less than δ o , while the surfaces separated by more than δ o , are traction-free. The cohesive zone solution depends on two non-dimensional parameters, while the asymptotic singular solution depends on one non-dimensional parameter only. The limit where the singular model provides a good approximation for the more accurate cohesive zone solution is defined. Finally, a decohesion map is deduced, which divides the parameter space into the regions where decohesion process is governed by different physical mechanisms. The deduced mechanisms are compared with the existing experimental data.

Bauschinger Effect Design Procedures for Autofrettaged Tubes Including Material Removal and Sachs’ Method

Autofrettage is used to introduce advantageous residual stresses into pressure vessels and to enhance their fatigue lifetimes. The Bauschinger effect serves to reduce the yield strength in compression as a result of prior tensile plastic overload and can produce lower compressive residual hoop stresses near the bore than are predicted by “ideal” autofrettage solutions (elastic/perfectly plastic without Bauschinger effect). A complete analysis procedure is presented which encompasses representation of elastic-plastic uniaxial loading material behavior and of reverse-loading material behavior as a function of plastic strain during loading. Such data are then combined with some yield criterion to accurately predict elastic-plastic residual stress fields within an autofrettaged thick cylinder. Pressure for subsequent reyielding of the tube is calculated. The numerical procedure is further used to determine residual stress fields after removal of material from inside diameter (i.d.) and/or outside diameter (o.d.), including the effects of any further plasticity. A specific material removal sequence is recommended. It is shown that Sachs’ experimental method, which involves removing material from the i.d., may very significantly overestimate autofrettage residual stresses near the bore. Stress ranges and stress intensity factors for cracks within such stress fields are calculated together with the associated fatigue lifetimes as such cracks propagate under cyclic pressurization. The loss of fatigue lifetime resulting from the Bauschinger effect is shown to be extremely significant.

On the Determination of Fracture Energy Using the Peel Test

Abstract This paper presents an improved model for the prediction of the adherend plastic dissipation in the peel test, thereby allowing the determination of the critical fracture eaergy. The model, which is based on earlier work, takes into account the effects of adhesive shear stresses, and a bilinear stress-strain response for the flexible adherend. Treating the flexible adherend as a beam-on-elastic foundation, expressions for the normal and shear foundation constants due to the compliance at the root of the adherend under elastic-plastic loading are derived using the deformation theory of plasticity. Application of the model to experimental peel data shows that it gives an improved estimate of the critical fracture energy.

Geometry effect of welded joints on failure assessment curves

Abstract In the present work effect of weld width parameter, h c , on the J -integral and the failure assessment diagram (FAD) has been investigated. The FAD in this studies is based on the CEGB R6 procedure option 3. In the investigation center-cracked tensile specimens with mismatched welds were used. Crack size parameter, a W , was equal to 0.1 and 0.5. The values of h c was changed from 0.1 to 0.5. Weld strength mismatch factor, M , was selected to be 0.8, 1.0 and 1.2. The base material investigated was A508C13 steel. Numerical computation was performed using an ABAQUS two-dimensional elastic-plastic finite element analysis code. The results have indicated that the weld width parameter has evident influence on the J -integral and the FAD in the elastic-plastic loading condition. With the increase in the weld width parameter, the J -integral of undermatched joints are increased, while the J -integral of overmatched joints are decreased. In the meantime, with the increase of h c , the safety region of the FAD is reduced for the undermatched joints and expanded for the overmatched joints. Moreover, the results display the effects of strength mismatch and crack depth on the FAD. The safety zone of the FAD clearly expands for the overmatched joints and contracts for the undermatched joints. No matter what mismatched joints or homogeneous materials, the safety margin of the FAD is expanded for the specimens with deep crack. Thus, the weld geometry should be considered, when one aims to construct the expression of J -integral estimation or to carry out the integrity assessment of welded structures using the FAD.

A Nonlinear Multi-Domain Thermomechanical Stress Analysis Method for Surface-Mount Solder Joints—Part II: Viscoplastic Analysis

This is part II of a two-part paper presented by the authors for thermomechanical stress analysis of surface mount interconnects. A generalized multi-domain Rayleigh Ritz (MDRR) stress analysis technique has been developed to obtain the stress and strain fields in surface-mount solder joints under cyclic thermal loading conditions. The methodology was first proposed in Part I by the authors and results were presented for elastic-plastic loading (Ling et al., 1996). This paper extends the analysis for viscoplastic material properties. The solder joint domain is discretized selectively into colonies of nested sub-domains at locations where high stress concentrations are expected. Potential energy stored in the solder domain and in the attached lead and Printed Wiring Board (PWB) is calculated based on an assumed displacement field. Minimization of this potential energy provides a unique solution for the displacement field, consequently, stress and strain distribution. The MDRR technique was demonstrated to provide reasonable accuracy for elastic deformation (Ling and Dasgupta, 1995) and for time-independent elastic-plastic deformation (Ling and Dasgupta, 1996) for solder joints under cyclic thermal loading conditions. A piecewise linear incremental loading technique is used to solve the nonlinear elastic-plastic problem. The focus in the current paper is primarily on time-dependent viscoplastic deformation of the solder joints. Full field elastic, plastic, and viscoplastic analyses are performed, and the stress, strain hysteresis loops are obtained. Results are presented for a J-lead solder joint as an illustrative example.

Elasto-plastic rotational capacity of welded symmetrical moment connections

A study is conducted on the moment-rotation characteristics of welded beam-to-column symmetrical moment connections. The study focuses on the effect of column axial load and beam depth on the moment-rotation behavior of the connection. Numerical experiments using nonlinear finite element analysis are conducted where a special incremental version of a deformation theory is incorporated into the formulation in order to yield information both at imminent buckling of the web and at ultimate elastic-plastic equilibrium load.

Representation of cyclic hardening and softening properties using continuous variables

Abstract Our aim is the modeling of cyclic hardening, cyclic softening, cyclic mean stress relaxation, and additional nonproportional cyclic hardening. We do so by means of hardening functionals for back stress and yield stress without employing additional memory surfaces. Rather, we suppose all quantities to evolve simultaneously during elastic-plastic loading in a continuous manner. The basic idea is to formulate evolution equations for the hardening variables, which are of the “hardening/dynamic recovery” format with respect to a transformed arc length. The corresponding transformation is influenced by continuously evolving parameters, measuring strain amplitude and nonproportionality during the recent process history. Although the resulting, model has a very simple structure, it is capable of describing the basic phenomena under quite general loading conditions.

Elastic unloading, neutral loading, and plastic loading in the theory of materials with elastic range

Abstract A plasticity index is an evolution indicator that allows one to predict, in a situation of potential plastic flow, whether plastic loading, instead of elastic unloading or neutral loading, will actually occur. It is shown in this article how a notion of plasticity index arises within the theory of materials with elastic range.

Fatigue crack growth in consideration of mean stress and damage accumulation

A fatigue crack growth rate in consideration of fatigue damage accumulation and the mean stress on fatigue material elements ahead of a crack tip are derived using both Miner's rule and the Manson-Coffin relationship. The theoretical equation of fatigue crack growth rate is shown in terms of J-integral range, size of a fatigue material element, fatigue damage parameter and stress ratio on the fatigue material element in front of the crack tip, etc. The stress ratio of the crack tip fatigue material element varies with the applied stress ratio and fraction of plastic component of J-integral range. As the fraction of plastic component of J-integral range increases, the fatigue crack growth rate increases. The theory formally predicts the experimental phenomenon that under elastic-plastic fatigue loading the shorter crack grows faster than the longer crack for the same J-integral range. The theoretical fatigue crack growth rates are compared with previous experimental results.

OPTIMIZATION OF STRUCTURAL FRAMES WITH ELASTIC AND PLASTIC CONSTRAINTS

Abstract Structural design methods that employ optimization have traditionally followed one of two approaches: optimal elastic design which considers constraints on elastic stresses; and, optimal plastic design, involving ultimate load constraints. This study addresses elastic and plastic design, incorporated into an iterative scheme for checking both linear elastic stresses and plastic collapse load factors. Structural optimization of steel frames is achieved through material reallocation, which is governed by a generalized stress parameter in those cases where elastic design controls, and by a reduced cost vector from a linear program when the plastic collapse load factor controls. Regression analysis techniques among member properties are employed to reduce the number of independent design variables and to incorporate buckling and axial toad effects for the elastic design. The linear programming optimization procedure is implemented in a computer system developed for the minimum-weight design of steel ...

Numerical method for elastic-plastic waves in cracked solids, Part 1: anti-plane shear problem

The paper deals with a numerical method combining a characteristic-based scheme and Zwas' method in order to solve the hyperbolic PDE's of elastic and elastic-plastic anti-plane shear waves in two space dimensions. First, the need of new physically reasonable numerical methods for stress waves in solids is demonstrated by numerical applications to problems with impulsive loading, where defects of some standard methods are shown. Then, the new secon-dorder accurate method is derived. A suitable procedure to model the elastic-plastic behaviour of materials by simple waves is included. The capability of the methods is demonstrated by application to several examples. Additionally, for comparison with numerical results, a similarity solution for the semi-infinite crack undergoing an elastic-plastic shock loading is derived in the appendix.

A Riemann solver and a second-order Godunov method for elastic-plastic wave propagation in solids

By use of three basic paths of elastic-plastic loading, a Riemann solver is established for the one-dimensional combined longitudinal and torsional stress wave problem of the thin-walled elastic-plastic tube. Based on this, a second-order Godunov method is developed for the numerical computation of elastic-plastic waves. Finally the numerical method is applied to some basic problems with known exact solutions and the comparison of results is made.

EFFECT OF NOTCH STRESS FIELD AND CRACK CLOSURE ON SHORT FATIGUE CRACK GROWTH

Abstract— The growth rate of a short fatigue crack that is partly or wholly embedded within the notch plastic zone, is affected by the extent and intensity of the elastic‐plastic notch stress field and closure effect. The notch stress—strain field and plastic zone were analysed by the Finite Element Method (FEM). The growth rate and the closure curve for a short fatigue crack emanating from the notch root were measured. Based on the experimental and numerical analyses, a modified Linear Elastic Fracture Mechanics (LEFM) parameter is proposed for a short through‐thickness crack emanating from a notch root under elastic—plastic loading conditions.

Evaluation of throughwall crack pipes under displacement controlled loading

Tearing modulus solutions are developed for flawed throughwall pipes subjected to displacement controlled loading. Two cases of loading were considered: (1) a displacement controlled bending loading, and (2) a displacement controlled axial tension loading. A revised version of the EPRI J- integral estimation scheme is used in the development of the solutions. These solutions can be used for the entire range of elastic-plastic loading, from linear elastic, contained yielding, to large scale yielding of the crack section. Experimental data from pipes in bending were used to assess the accuracy of the compliant loading solutions. The evaluations were performed using elastic plastic J- integral ( J) and tearing modulus ( T) analysis methods. These solutions are shown to have good accuracy when used to predict the experimental results. The methodology and procedure can also be applied to part-throughwall cracks. These solutions have application to the leak before break fracture mechanics analyses.

Loading Rate Method for Pile Response in Clay

A method is proposed to predict the behavior of single piles in cohesive soil subjected to vertical loads applied at various rates. The elementary simple shear model is made rate-dependent by introducing the time raised to a negative power n. This viscous exponent n is shown to vary from 0.02-0.08 with a 0.05 average. Correlations with undrained shear strength, water content, plasticity, the liquidity index and overconsolidation ratio are shown based on 152 laboratory tests. Integration of the elementary model leads to rate-dependent elastic-plastic or hyperbolic load transfer curves. A program to predict the head load-head movement response of the pile using these rate-dependent loadtransfer curves is described. The validity of the method is first addressed by examining the physical reasons for the rate-dependent properties of clays, then by using the results of 62 rate-dependent pile load tests and by examining a series of applications of the new method and, finally, by presenting the results of a parametric study.

An assessment of failure probability in elastic-plastic conditions where stable tearing has occurred

An analysis is presented for calculating the failure probability of a structure subject to general elastic-plastic loading and where the fracture mode is ductile. The main statistical variables considered are fracture toughness, flow stress and defect size. The analysis is based on a development of the R6 methodology of defect assessment. The concept of a maximum load locus is developed from the failure assessment line. The maximum load locus is used to predict those combinations of materials properties and defect size that might combine to predict structural failure after some stable crack growth. The method of determining the maximum load locus is described and some examples presented. The maximum load locus is used to estimate the failure probability on the first loading of a structure. It is also shown that the maximum load locus can be used to estimate the failure probability of a proof loaded structure where, in general, there may be a change in the dominant failure mode between proof and fault loadings. Detailed examples are presented to illustrate the analysis.

J-Integral Estimation Procedures

This paper presents efficient procedures for estimating the J-integral, a parameter for characterizing stable crack growth under inelastic conditions. Two different formulations are shown. The basis of the first approximate J-integral formula is the familiar extension of the elastic strain energy release rate computation for elastic-plastic loads. This extension is particularly suited for contained plasticity. With continued yielding, when the global load-deflection curve exhibits nonlinear behavior (i.e., net section yielding commences), this first J-approximation formula deviates from the actual J-integral. A second approximate formula derived here can be used at this point which is based on the definition that J-integral can be interpreted as the potential energy difference between two identically loaded specimens having crack sizes that differ infinitesimally. This latter formula is exact for linear loading and rigid-plastic material response, and is shown to predict J accurately in the elastic-plastic range of loading. Several standard cracked geometries for which documented J-integral solutions exist were applied to test the estimation procedures. This technique is very general and can be applied to virtually any structure or continuum. And no special assumption of material hardening, such as Ramberg Osgood, is necessary. These estimation techniques are significantly easier and more economical to use than a conventional finite element procedure, which models details of a cracked geometry.

Ductile Tearing Instability of a Center-Cracked Panel of an Elastic-Plastic Strain Hardening Material

An analysis for crack instability in an elastic-plastic strain hardening material is presented which utilizes the J-integral and the tearing modulus parameter, T. A center-cracked panel of finite dimensions with Ramberg-Osgood material representation is analyzed for plane stress as well as plane strain. The analysis is applicable in the entire range of elastic-plastic loading from linear elastic to full yield. Crack instability is strongly influenced by the elastic compliance of the system, the conditions of plane stress or plane strain, and the hardening characteristics of the material. Numerical results indicate that if crack stability is ensured in a plane strain situation, then under the same circumstances a geometrically identical but plane stress panel will be stable.