Incremental Theory Of Plasticity Research Articles

Estimating the support effect of energy-absorbing rock bolts based on the mechanical work transfer ability

An interaction model is proposed to describe the interaction between the energy-absorbing rock bolt and the rock mass. Based on the plane-strain axial symmetry assumption and the incremental theory of plasticity, the equilibrium equations and compatibility equations of rock mass, as well as the response of the energy-absorbing rock bolt are deduced theoretically. The proposed method was programmed in a Visual Basic environment, and a semi-analytical solution for the coupling model was achieved. The reinforcement mechanism of the energy-absorbing rock bolt in conventional tunneling is clearly demonstrated through an illustrative case study. The reinforcement effect of the energy-absorbing rock bolt under different conditions was estimated quantitatively, and its mechanical work transfer ability is presented. In addition, the validity of the proposed method was verified through numerical simulations. Finally, a number of derivative cases were investigated to reveal the influence of the bolt and rock properties on the reinforcement effect and the bolt work transferred on the rock mass. In the case of higher in-situ stress or low-strength rock mass, the support effect of the energy-absorbing rock bolt is significantly improved, and the bolt absorbs more energy. Increasing the bolt installation density could always be helpful for the stabilization of the surrounding rock mass. However, additional rock-bolt length could hardly affect ground reinforcement because the bolt section embedded in the elastic region of the rock mass could barely help to constrain the elastic displacement release. The bolt should be installed no later than the stage of critical inner pressure, namely when the plastic region occurs.

Numerical investigation of residual thermal stresses in welded joints of heterogeneous steels with account of technological features of multi-pass welding

The paper describes a two-dimensional mathematical model to evaluate stresses in welded joints formed in multi-pass welding of multi-layered steels. The model is based on a system of equations that includes the Lagrange's variational equation of the incremental theory of plasticity and the Biot's variational principle for heat transfer simulation. In the constitutive equations, the changes in the volume which occur as a result of phase transitions can be taken into account. Therefore, the prehistory and impact of thermal processing of materials on macroscopic properties of the medium can be considered.The variational-difference method is used to solve both the heat transfer equation for calculation of the non stationary temperature field and the quasi-static problem of thermoplasticity at each time-step. The two-dimensional problems were solved to estimate the residual thermal stresses (for the case of plane stress or plane strain) during cooling of welds and assessing their impact on strain localization in the heat-affected zone under tensile and compressive loading considering differences in mechanical properties of welded materials.It is shown that at initial stages of the plastic flow, the residual stresses significantly affect the processes of stress concentration and localization of strains in welded joints. To estimate the model parameters and to verify the modeling results, the available experimental data from scientific literature obtained on the basis of the Satoh test for different welding alloys was used.

A fuzzy set plasticity model for cyclic loading of granular soils

A constitutive model for describing the stress–strain behavior of granular soils subjected to cyclic loading is presented. The model is formulated using fuzzy set plasticity theory within the classical incremental plasticity theory framework. A special membership function is introduced to provide an analytical and simple geometrical interpretation to formulate hardening, hysteresis feature, material memory, and kinematic mechanisms without resorting to complicated kinematic hardening formulations. The model can accurately describe cyclic loading, dilatancy, material theory and critical state soil mechanics features effects. Two series of cyclic drained triaxial tests data are considered. The characteristic features of behavior in granular soils subjected to cyclic loading are captured.

An elastoplastic semi-analytical method to analyze the plastic mechanical behavior of buried pipelines under landslides considering operating loads

An elastoplastic semi-analytical method is proposed to deal with the plastic mechanical behavior of buried pipelines subjecting to landslides based on the plane stress condition, considering the inner pressure and the temperature variation. According to the incremental theory of plasticity, the longitudinal, radial and hoop plastic strains are expressed as the corresponding equations that contain the longitudinal and hoop stresses. Considering the effect of the inner pressure and the temperature variation, the longitudinal stress is derived based on the elastoplastic-beam theory, and then the axial force can be determined based on the relation of the internal force and the stress. Combining the differential equation and the boundary condition, the transverse horizontal displacement, the bending strain, axial strain can be obtained. To verify the proposed model, the comparison of the obtained results from the proposed method and the finite element method is performed, with minor deviations of within 5.4%. Additionally, the effects of the inner pressure and the temperature variation on the strain and displacement of pipelines are investigated through parametric studies. The results show that the plastic deformation and strain are much larger when neglecting operating loads; the pipeline develops mainly the longitudinal and radial plastic deformations instead of the hoop plastic deformation.

Cyclic plasticity in three-dimensional notched components under in-phase multiaxial loading at R = −1

In this work a comprehensive theoretical and numerical study on the cyclic elastic–plastic notch stress and strain distributions is carried out. In more detail, the incremental cyclic plasticity theory, already proposed by other authors to determine the actual stress and strain state arising in two-dimensional or axis-symmetric notched components, is extended to the study of three dimensional effects at the tip of rounded notches in plates of finite thickness. The analytical frame is initially validated considering some bi-dimensional problems and later modified to consider the out-of-plane 3D-effects. Different notch geometries and loading conditions are investigated, such as axis-symmetric specimens weakened by a circumferential U-notch, subjected to cyclic torsional and multiaxial (tension–torsion) in-phase loading, and plane specimens with finite thickness, weakened by rounded V-notches and subjected to cyclic mode II loading. A nominal load ratio R=−1 has been taken into account in all cases. Theoretical results based on the incremental cyclic plasticity theory are compared with time-consuming elastic–plastic finite element analyses carried out with a commercial FE code showing a very satisfactory agreement.Finally, a link between the averaged strain energy density (SED) criterion and the area inside the hysteresis loops tied to the different stress and strain components acting at the notch tip has been investigated.

Bounding surface model for soil resistance to cyclic lateral pile displacements with arbitrary direction

The development of a two-surface elastic–plastic bounding surface P–Y model for cyclic lateral pile motions is described. The kinematic-hardening model is applicable to the analysis of pile foundations subjected to loading with arbitrary azimuths relative to the pile axis. The model realistically captures the hysteretic energy damping associated with dynamic loading of subsea foundations through physically correct plastic mechanisms and provides results consistent with those observed in physical tests including cyclic loading. Its performance is demonstrated in element states of stress and in pile foundation analyses. The development based on the incremental theory of plasticity results in more robust solutions than may be obtained using alternative elastic, variable moduli and deformation plasticity formulations.

A probabilistic non-linear finite element analysis for slope stability problem

This paper deals with the slope stability problem by a finite element reliability analysis considering the spatial variation of soil strength parameters. In this work, the spatial variation is described using random field theory. To simplify the implementation of the proposed procedure for slope reliability analysis, elastic perfectly plastic constitutive model describing the soil behaviour using the Mohr coulomb yield criterion is adopted. The application of the proposed approach for the slope reliability analysis is performed using a performance function expressed in terms of stress fields mobilised along a circular failure surface. For this purpose, a computation of the stress gradient based on the incremental theory of plasticity is used. The results of the proposed procedure are checked by a probabilistic method based on the combination of the Bishop's model and the Monte Carlo simulation. The numerical examples have elucidated the efficiency and the validity of the present procedure to slope reliabi...

Three-dimensional effects at the tip of rounded notches subjected to mode-I loading under cyclic plasticity

In this work, a comprehensive theoretical and numerical study on the cyclic elastic–plastic notch stress and strain distributions is carried out. In more detail, the incremental cyclic plasticity theory, already proposed by other authors to determine the actual stress and strain state arising in two-dimensional or axisymmetric notched components, is extended to the study of three-dimensional effects at the tip of rounded notches in plates of finite thickness. The analytical frame is initially validated considering a number of plane problems and later modified to consider three-dimensional effects. Different notch geometries are investigated, such as plane specimens with finite thickness weakened by circular holes, U-notches and rounded V-notches subjected to cyclic mode-I loading. Theoretical results based on the incremental cyclic plasticity theory are compared with time-consuming elastic–plastic finite element analyses carried out with a commercial finite element code showing a very satisfactory agreement. Finally, a link between the averaged strain energy density criterion and the area subtended by the hysteresis loops tied to the different stress and strain components acting at the notch tip has been investigated.

Modifying Duncan–Selig Soil Model for Plasticlike Behavior

The variable-modulus Duncan–Selig soil model is very effective in capturing the nonlinear stress–strain behavior of most soils in loading environments, including the softening effect from increased shear stress as well as the stiffening effect from increased confining pressure. Accordingly, the Duncan–Selig model is used in many geotechnical applications, such as construction of culvert installations with incremental layers of soil, for which soil loading is the dominant condition. However, the Duncan–Selig model is a nonlinear elastic formulation that retraces the same stress–strain path upon unloading. Consequently, the model does not predict residual deformation upon unloading. In contrast, plasticity-based soil models are generally less accurate in simulating the nonlinear loading response of soil samples; however, they inherently include permanent deformation if plastic deformation occurs during loading. This paper introduces modifications to the Duncan–Selig model that result in permanent deformations upon unloading similar to those of advanced plasticity models. Although the modifications are based on concepts from incremental plasticity theory, the modified Duncan–Selig model remains a variable-modulus formulation without the need of defining any additional model parameters or evoking a plastic-flow rule. Thus, the large existing database of Duncan–Selig parameters remains valid for the modified formulation. Most important, the modified Duncan–Selig model is shown to satisfy thermodynamic and continuity requirements and to compare very favorably with triaxial and hydrostatic laboratory test data under load–unload–reload conditions. Finally, the modified Duncan–Selig model is used to simulate the effects of soil compaction for an installation of a long-span corrugated-steel culvert.

Recent Trends in Elastic-Plastic Analysis Using Elastic Solutions

Methodologiesfor predicting elastic-plastic stresses and strains under various loadings are discussed and reviewed. An overview of various approaches for predicting notch tip elastic-plastic stresses and strains are presented. The need for computing elastic-plastic stress strain history and methods evolved during time from simplest models under uniaxial loading to models for addressing multiaxial proportional and non-proportional loadings are discussed and summarized. Analytical approximation methods which use elastic solutions to estimate elastic-plastic behaviour at geometric discontinuities are mainly focused. Methodologies for elastic-plastic stresses and strains under uniaxial loading with plane stress and plane cases are discussed and analyzed. Methodologies for non-proportional loading based on incremental plasticity theories and total deformation theory are thoroughly studied. The research efforts on this topic have been increased, emphasizing the development of more generalized approaches for simulating elastic plastic cyclic stress strain behaviour under complex loading conditions.

Physically appropriate characterization of fatigue crack propagation rate in elastic–plastic materials using the $$J$$ J -integral concept

The current paper discusses the physically correct evaluation of the driving force for fatigue crack propagation in elastic–plastic materials using the $$J$$ -integral concept. This is important for low-cycle fatigue and for short fatigue cracks, where the conventional stress intensity range ( $$\Delta K$$ ) concept cannot be applied. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008) , have derived the $$J$$ -integral for elastic–plastic materials with incremental theory of plasticity, $$J^{\mathrm{ep}}$$ , which is applicable for cyclic loading and/or for growing cracks, in contrast to the conventional $$J$$ -integral. The variation of this incremental plasticity $$J$$ -integral $$J^{\mathrm{ep}}$$ is studied in numerical investigations conducted on two-dimensional C(T)-specimens with long cracks under cyclic Mode I loading. The crack propagates by an increment after each load cycle. The maximum load is varied so that small- and large-scale yielding conditions prevail. Three different load ratios are considered, from pure tension to tension-compression loading. By theoretical considerations and comparisons with the variation of the crack tip opening displacement $$\delta _{\mathrm{t}}$$ , it is demonstrated that the cyclic, incremental plasticity $$J$$ -integral $$\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$$ , which is computed for a contour around the active plastic zone of the growing crack, is physically appropriate to characterize the growth rate of fatigue cracks. The validity of the experimental cyclic $$J$$ -integral, $$\Delta J^{\mathrm{exp}}$$ , proposed by Dowling and Begley (ASTM STP 590:82–103, 1976), is also investigated. The results show that $$\Delta J^{\mathrm{exp}}$$ is correct for the first load cycle, however, not fully appropriate for a growing fatigue crack.

Numerical modelling and parametric analysis of bond strength of masonry members retrofitted with FRP

The paper discusses the bond between fibre-reinforced polymer (FRP) strips and brick masonry, using advanced non-linear simulations of experimental test results. The context of the research is the study of the interfacial behaviour of a brittle substrate (masonry) and a thin elastic addition (the FRP) used as a strengthening system for historical structures. Moreover, to further validate the numerical results, the problem is also approached analytically. In the numerical model, the interface is represented with zero-thickness finite elements, with inelastic behaviour fully formulated using the incremental plasticity theory. The mechanical characterization of the materials involved and the local interfacial behaviour allowed defining the main parameters and the constitutive law to be adopted. Based on the experimental results, the detailed finite element model is calibrated so that it is possible to describe the bond phenomenon at the local level up to failure, identifying the strain and slip distribution, the local bond stress peak and its migration along the bond length for flat and curved layers. Numerical results and analytical predictions show a good agreement with experimental data, allowing to provide a comprehensive discussion on the bond behaviour. Moreover some differences between interfacial behavior of FRP–masonry/concrete joints are highlighted. Finally applications of the model to analyze strengthened structural components (such as masonry arches) are provided.

A new basis for the application of the $$J$$ J -integral for cyclically loaded cracks in elastic–plastic materials

Fatigue crack propagation is by far the most important failure mechanism. Often cracks under low-cycle fatigue conditions and, especially, short fatigue cracks cannot be treated with the conventional stress intensity range \(\Delta K\)-concept, since linear elastic fracture mechanics is not valid. For such cases, Dowling and Begley (ASTM STP 590:82–103, 1976) proposed to use the experimental cyclic \(J\)-integral \(\Delta J^{\exp }\) for the assessment of the fatigue crack growth rate. However, severe doubts exist concerning the application of \(\Delta J^{\exp }\). The reason is that, like the conventional \(J\)-integral, \(\Delta J^{\exp }\) presumes deformation theory of plasticity and, therefore, problems appear due to the strongly non-proportional loading conditions during cyclic loading. The theory of configurational forces enables the derivation of the \(J\)-integral independent of the constitutive relations of the material. The \(J\)-integral for incremental theory of plasticity, \(J^{\mathrm{ep}}\), has the physical meaning of a true driving force term and is potentially applicable for the description of cyclically loaded cracks, however, it is path dependent. The current paper aims to investigate the application of \(J^{\mathrm{ep}}\) for the assessment of the crack driving force in cyclically loaded elastic–plastic materials. The properties of \(J^{\mathrm{ep}}\) are worked out for a stationary crack in a compact tension specimen under cyclic Mode I loading and large-scale yielding conditions. Different load ratios, between pure tension- and tension–compression loading, are considered. The results provide a new basis for the application of the \(J\)-integral concept for cyclic loading conditions in cases where linear elastic fracture mechanics is not applicable. It is shown that the application of the experimental cyclic \(J\)-integral \(\Delta J^{\exp }\) is physically appropriate, if certain conditions are observed.

Investigation on Validity of J-Integral of Edge Crack under Cold Rolling Condition

In this paper the characterization of the edge crack in the strip steel are studied by using numerical simulation method. The developments of the stress and strain near the crack tip are obtained and the value of the J-integral of edge crack under the rolling process is then examined. FE simulation result shows that the J-integral is not always path independent in the whole rolling process. When the crack is far away from the roller, the J-integral is path independent. When the crack enters the cold rolling region, the unload phenomena will occur near the crack tip which cause the incremental theory of plasticity failed and the conservation of the J-integral is not valid any more. The J-integral failure region is then determined by a series of FE simulations.

Micro-structural evolution subjected to combined tension–torsion deformation for pure copper

A finite element simulation and experiment were conducted to investigate the characteristics of combined tension–torsion (CCT) deformation. According to the simulation results, a very obvious strain gradient distribution develops on the cross-sections of specimens. The equivalent strains obtained by simulation and the incremental theory of plasticity are in good agreement. The experimental results indicate that the micro-structural evolution on the cross-section and longitudinal section act similarly; grains at the edge suffer a higher degree of refinement compared to those in the center, which is consistent with the simulated results. The micro-voids proliferate and grow as the torsion turns increase, but the volume of the voids decreases, and voids tends to aggregate when a critical torsion strain is reached. The EBSD investigation indicates that grains in the processed specimens are mainly small-angle grain boundaries, and the misorientations concentrate between 2° and 10°.

A new view on J-integrals in elastic–plastic materials

It is well known that the application of the conventional $$J$$ -integral is connected with severe restrictions when it is applied for elastic–plastic materials. The first restriction is that the $$J$$ -integral can be used only, if the conditions of proportional loading are fulfilled, e.g. no unloading processes should occur in the material. The second restriction is that, even if this condition is fulfilled, the $$J$$ -integral does not describe the crack driving force, but only the intensity of the crack tip field. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008), have derived a $$J$$ -integral, $$J^{\mathrm{ep}} $$ , which overcomes these restrictions: $$J^{\mathrm{ep}} $$ is able to quantify the crack driving force in elastic–plastic materials in accordance with incremental theory of plasticity and it can be applied also in cases of non-proportionality, e.g. for a growing crack. The current paper deals with the characteristic properties of this new $$J$$ -integral, $$J^{\mathrm{ep}}$$ , and works out the main differences to the conventional $$J$$ -integral. In order to do this, numerical studies are performed to calculate the distribution of the configurational forces in a cyclically loaded tensile specimen and in fracture mechanics specimens. For the latter case contained, uncontained, and general yielding conditions are considered. The path dependence of $$J^{\mathrm{ep}} $$ is determined for both a stationary and a growing crack. Much effort is spent in the investigation of the path dependence of $$J^{\mathrm{ep}} $$ very close to the crack tip. Several numerical parameters are varied in order to separate numerical and physical effects and to deduce the magnitudes of the crack driving force for stationary and growing cracks. Interpretation of the numerical results leads to a new, completed picture of the $$J$$ -integral in elastic–plastic materials where $$J^{\mathrm{ep}} $$ and the conventional $$J$$ -integral complement each other. This new view allows us also to shed new light on a long-term problem, which has been called the “paradox of elastic–plastic fracture mechanics”.

The Collapse Load of Plates by Hierarchical C<sup>0</sup>-Plate Element

Although a structural analysis of plates based on the linear elastic theory yields good results for deformations and stresses produced by working loads, it fails to assess the real load-carrying capacities of plates on the verge of yielding. In the case of a limit analysis of plates, the yield line theory is widely used on the basis of the upper bound theorem and theoretically it overestimates the strength of plates. That is why the p-version of the finite element method has been proposed for determining the accurate limit load of plates causing collapse. In this method, the hierarchical Co -plate element for bending of elastic-plastic plates accounting for transverse shear effects has been formulated, and is based on the incremental theory of plasticity and the Reissner-Mindlin plate theory. The numerical results are presented for a variety of rectangular plate problems and are compared to the results obtained by the h-version software ADINA, as well as with the available analytical solutions in literature.

An efficient homogenization method for composite materials with elasto-plastic components

The homogenization problem for elasto-plastic media with arrays of isolated inclusions (matrix composite) is considered. A combination of self-consistent and numerical methods is used for calculation of the overall response of such composites under quasi-static loading. Elasto-plastic properties of the medium and the inclusions are described by the equations of the incremental theory of plasticity with isotropic hardening. For the construction of the average stress–strain relations of the composites, the process of external loading is divided into a sequence of small steps, and the problem is linearized at every step. The self-consistent effective field method allows reducing the homogenization problem at every step to the calculation of stresses and elasto-plastic deformations in a composite cell that contains a finite number of inclusions. The linearized problems are formulated in terms of volume integral equations for the stress or elastic strain field increments in the cell. For the numerical solution, these equations are discretized by Gaussian approximating functions concentrated in a set of nodes that cover the composite cell. For such functions, elements of the matrix of the discretized problems are calculated in explicit analytical forms. If the approximating nodes form a regular grid, the matrix of the discretized problem has Toeplitz’s properties, and the matrix–vector products of such matrices can be calculated by the fast Fourier transform technique. The latter accelerates substantially the process of iterative solution of the discretized problems. The dependencies of the overall stress–strain curves on the number of inclusions inside the cell are studied in the 2D and 3D cases. The inclusions that are stiffer or softer then the matrix are considered. The predictions of the method are compared with the finite element calculations available in the literature.

Numerical solution of the integral equations of elasto-plasticity for a homogeneous medium with several heterogeneous inclusions

The work is devoted to the calculation of stress and strain fields in a homogeneous elasto-plastic medium with a finite number of heterogeneous inclusions. The medium is subjected to an arbitrary external stress field. Elasto-plastic behavior of the medium is described by the equations of the incremental theory of plasticity with isotropic hardening. For the numerical solution, the external stress field applied to the medium is divided on a consequence of small steps, and the problem is linearized at every step. The linearized problem is reduced to the solution of the volume integral equations for the stress field increment inside the inclusions and in the regions involved in the plastic deformations. Then, these equations are discretized using Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problems are calculated in explicit analytical forms. If the approximating nodes compose a regular grid, the matrix of the discretized problem obtains the Toeplitz properties, and the product of such a matrix and a vector can be calculated by the Fast Fourier Transform technique. The latter accelerates essentially the process of iterative solution of the discretized problems. The proposed method is mesh free, and the coordinates of the approximating nodes and elasto-plastic properties of the material at the nodes are the only information required for carrying out the method. Distributions of stresses and plastic strains in the media with isolated inclusions are compared with the finite element calculations. The influence of the number of approximating nodes and the rate of hardening of the material on the convergence of the numerical solutions is analyzed.

On the predictability of elasto-plastic and geometrically non-linear oscillations of beams under harmonic excitation

Vibrations in one plane of beams with fixed ends, vibrating in the geometrically non-linear and elasto-plastic regimes under the influence of harmonic external forces, are studied. A p-version finite element that considers transverse and longitudinal displacements, as well as shear deformation, is employed. The incremental theory of plasticity with isotropic hardening is followed. Numerical methods are employed to solve the differential equations of motion and to carry out integrals where plastic terms exist. The main interest of this work is that “chaotic-like” behavior—in the sense of unpredictable behavior of an apparently deterministic system—is discovered. This behavior is explained by a buckling phenomenon induced by the plastic strains: no longitudinal external force, other than the boundary forces, exists in the cases investigated. Since the plastic strains depend on the past history, the predictions of the long term behavior are also affected by the different predictions on the brief transition phase. An eigenvalue problem is defined to compute eigenvalues that provide an indication that a bifurcation induced by the plastic strains is likely to occur.