Strain-softening Materials Research Articles

Unified solution of limit loads of thick wall cylinder subject to external pressure considering strain softening

Based on the unified strength theory [1], a unified strength criterion for strain softening materials, such as concrete or rock, was derived, and the elastic and plastic limit loads of a thick-walled cylinder made of these materials subject to external pressure were also given. In addition, the influence of some factors on the limit loads of such cylinders as the ratio of the external radius to internal radius, r b/r a, the coefficient b, which reflects the effect of medium principal stress and the normal stress of the relevant surface on the material destroy degree, the ratio of tensile strength to compressed strength of the material, α, and the damage variable β were discussed in detail. Some examples were given and some meaningful results were obtained.

Strain softening of materials and Lüders-type deformations

Corona et al (2002 Int. J. Solids Struct. 39 3313–36) and Kyriakides and Miller (2000 ASME J. Appl. Mech. 67 645–54) pointed out that the strain-softening behaviour of materials was necessary for strain localization to occur in the materials, and the Lüders-type deformations were treated as strain localization. In this case, however, the discontinuity of deformation gradient across interfaces between the sharp Lüders bands and other regions outside the Lüders bands was not considered. In the present paper it is shown that the strain-softening behaviour of materials ensures the non-convexity of the stress-work function of the materials, and, further on, two-phase deformations can occur in the materials. The Lüders bands and other regions outside the Lüders bands can be treated as two phases in equilibrium which coexist under the Maxwell stress, and the discontinuity of the deformation gradient across interfaces can be considered. As an illustration, the Lüders-type deformations in fine-grained steel strips under uniaxial compression are analysed, and the Maxwell stress and inclination angle of the Lüders bands are predicted which are consistent with experimental observations.

Unified Solution of Expansion of Cylindrical Cavity of Elastic-Brittle-Plastic Strain-Softening Materials with Different Elastic Modulus in Tensile and Compression

For strain-softening materials with different elastic modulus of tensile compression, two controlling parameters were introduced to take into account the different modulus and strainsoftening properties. By means of twin shear unified strength theory, unified solutions calculating stress and displacement fields of expansion of cylindrical cavity were derived. The effects caused by different elastic modulus in tensile and compression, different models and strain-softening rates on stress and displacement fields, development of plastic zone were analyzed. The results show the ultimate expansion pressure, stress and displacement fields and development of plastic zone vary with the changes of different elastic modular, different models and strain-softening properties.

Rheological and kinematical responses to flow of two-phase rocks

Numerical simulations have been performed to investigate the strain-dependent behaviour of rheological and kinematical responses to flow of two-phase rocks using the commercial finite-difference program FLAC2D. It was assumed that the two phases have Maxwell rheology. Plane strain and velocity boundary condition, which produces a simple shear deformation, were also assumed. Two types of geometries were considered: strong phase supported (SPS) and weak phase supported (WPS). We calculated strain-dependent variations of effective viscosity and partitioning of strain rate, vorticity and kinematic vorticity number during deformation in both SPS and WPS structure models. The results show that the strain-dependent behaviour is largely influenced by the geometry of the composite. SPS models show both strain hardening and strain softening during the simulations, with strain hardening preceding strain softening. A critical shear strain is necessary to begin the strain softening behaviour. Strain hardening and strain softening are accompanied by a reduction and an increase of the partition of strain rate into the weak phase, respectively. On the other hand, WPS models show only weak strain hardening and strain softening, being the strain-dependent behaviour close to a steady state flow. In addition, the following results are obtained on vorticity and kinematic vorticity number; (1) in both SPS and WPS models the partition of vorticity into weak phase increases with progressive shear strain, i.e. the strong phase becomes less rotational, (2) in SPS models weak inclusions changes from sub-simple shear to super-simple shear with progressive strain, whereas the strong matrix changes from super-simple shear to sub-simple shear, (3) in WPS models the strong inclusions with high viscosity contrasts are less rotational but can be in super-simple shear condition to high strains. The observed strain-dependent behaviours have been compared with previous proposed analytical models. The degree of agreement is variable. Balshin and Ryshkewitch–Duckworth models are only applicable to SPS models. Ji-generalized mixture rule model is applicable to both models. The results suggest that polyphase rocks with SPS structure during ductile shear deformation respond as strain softening materials, after an initial strain hardening stage that may drive to the strain localization into the material.

Ellipsis 3D: A particle-in-cell finite-element hybrid code for modelling mantle convection and lithospheric deformation

We have extended the two-dimensional geodynamics finite-element code “Ellipsis” to three-dimensions. Ellipsis uses a hybrid particle in a cell scheme, which combines a fixed mesh of computational points and a dense arrangement of mobile material points. The fixed Eulerian mesh allows very fast computation, performed in Ellipsis via a multigrid iteration method, while the Lagrangian particle reference frame allows the tracking of material interfaces and history-dependent properties such as strain history for strain-softening materials. The method is exceptionally useful in very large deformation analyses, where purely Lagrangian approaches would be severely hampered by the need for remeshing to minimize element distortion. The Gnu Public Licensed Ellipsis3D code lends itself to combined 2D/3D model prototyping, and has proven to be an excellent geodynamics teaching tool for modelling, covering mantle convection, lithospheric extension and plume–lithosphere interaction.

Modeling the effect of mesoscale randomness on concrete fracture

The random mesostructure of concrete has an important influence on the reliability and failure properties of the material. The objective of the proposed model is to create an efficient link between the mesostructure and the mechanical and damage behavior of concrete and related strain-softening materials. Three theoretical techniques comprise the model: cohesive debonding, the moving-window generalized method of cells, and a strain-softening finite element model. The model is calibrated with direct tension experiments geared towards isolating the mechanical behavior of the aggregate–mortar interface. The model makes a good prediction for the mechanical behavior of concrete in tension, particularly when randomness in the cohesive interface properties is taken into consideration.

Numerical experiments of soil nails in loose fill slopes subjected to rainfall infiltration effects

Finite difference analyses were conducted to explore the possible behaviour of soil nails in loose fill slopes which were formed by strain-softening materials. Numerical models simulating a 20 m high 35° loose fill slope were set up. With the use of an undrained strain-softening soil model, the behaviour of a loose fill slope reinforced with soil nails subjected to rainfall infiltration was studied. Results indicated that soil nails together with a proper facing structure may reduce the deformation of a loose fill slope caused by rainfall infiltration and hence maintain the stability of the slope. Tensile forces in the soil nails were mobilised through the effective connections between the facing structure and the nail heads when a potential slide was being initiated. Through this mechanism, the development of a flow slide at shallow depths could be avoided by the pull out resistance between the soil and the nails at deeper locations of the slope. Results also showed that, if the soil nails were not connected to a facing structure or nail heads at the slope surface, a global failure mechanism involving the development of a flow slide could be observed, which was similar to that of a loose fill slope without soil nails.

Localized Necking Criterion for Strain-Softening Materials

The paper presents the development of a localized necking criterion based on the singularity of acoustic tensor. This criterion is applicable to materials exhibiting strain-softening behavior. The tensor form of the criterion is deployed in simple mathematical expressions, based on which the forming limit diagrams (FLDs) of strain-softening materials can be determined. At the left-hand side of a FLD, or the negative strain ratio region, a closed-form expression of localized band inclination is derived as a function of the strain-ratio value. At the right-hand side of a FLD, or the positive strain ratio regions, the localized band is assumed to be perpendicular to major strain according to the MK [Marciniak and Kuczynski (1967)] model. On both sides of the FLD, the localized necking criteria are analytically expressed by elements of tangent modulus matrix. For the sake of illustration of the proposed criterion, a special case of localized necking employing associative and isotropic plasticity is presented. The material chosen for the illustration is AA-6061 at an elevated temperature. The proposed criterion is also applicable to the formability of other metals at high temperatures and other strain-softening materials such as rocks.

Study of bifurcation in the problem of unloading a circular excavation in a strain-softening material

Abstract This paper presents a study of bifurcation in the problem of unloading a circular excavation in a strain-softening rock mass. The analysis focuses on how bifurcation affects the calculation of ground reaction curves for tunnels. To begin with, the most important issues in strain-softening behaviour are discussed, with the concepts of bifurcation and localisation presented by reviewing physical observations of these phenomena in rocks and rock masses. Next, theoretical formulations and numerical approaches are briefly reviewed, and on this basis, a sufficiently good description of phenomena involved in bifurcation can be obtained using standard FDM or FEM codes with sufficiently fine meshes. A series of models of tunnels excavated in strain-softening materials were created and analysed using an FDM-based code, in which bifurcation and localisation could be clearly observed. These simulation results were compared with self-similar results, revealing that the effects of these phenomena are irrelevant when an appropriately large spatial scale, as in tunnels, is considered.

A study of the quadratic molecular stress function constitutive model in simulation

Constitutive models that conform to separable KBKZ specification have been shown to fit steady-state strain hardening rheological data in planar and uniaxial elongational flows, but with inaccuracy in the rate of strain hardening. The single parameter molecular stress function model of Wagner [Rheol. Acta 39 (2000) 97–109] has been shown to accurately fit the rise-rate in experimental data for a number of strain hardening and strain-softening materials. We study this models accuracy against the well characterised IUPAC LDPE data, and present a method for full implementation of this model for flow solution which is suitable for incorporating into existing separable KBKZ software. A new method for particle tracking in arbitrarily aligned meshes, which is efficient and robust, is given. The quadratic molecular stress function (QMSF) model is compared to existing separable KBKZ based models, including one which is capable of giving planar strain hardening; the QMSF is shown to fit experimental rheological and contraction flow data more convincingly. The issue of ‘negative correction pressures’ notable in some Doi–Edwards based models is addressed. The cause is identified, and leads to a logical method of calculation which does not give these anomalous results.

Analogue and numerical modeling of normal fault patterns produced due to slip along a detachment zone

In this paper, we investigate normal fault patterns produced by the sliding motion along a gently dipping normal fault by using analogue model tests and numerical modeling. The motivation for this study was offered by microseismic test data that indicate the existence of an active low-angle shear zone at a depth of 9–11 km in the extensional region of high seismic activity of the Gulf of Corinth (Greece). Both modeling techniques seem to support the hypothesis that the system of high-angle normal faults that are responsible for the final asymmetrical graben formation initiate at the tip of the active basal detachment nearest to the free surface. The normal faults propagate upwards with progressive sliding of the inclined basal plane, resulting in a first phase of symmetrical graben configuration that is delimited by a main synthetic fault and an antithetic fault forming a Rankine zone. Subsequent sliding on the inclined base induces a family of secondary antithetic normal faults, which are responsible for the asymmetry of the failure pattern and the diffusive character of deformation in that area. Shear deformation is more intense and localized along the synthetic normal fault than along the antithetic faults. Elaboration on the analogue test results has led to the phenomenological relations among four main parameters that describe the geometry of grabens, namely, (i) the width and (ii) the maximum subsidence of the graben, (iii) the dip angles of the conjugate normal faults, and (iv) the amount of sliding along the low-angle normal fault. However, analogue models do not produce the system of synthetic faults that is observed in the Gulf of Corinth. The effects of both friction angle variation along the detachment base and of the constitutive behavior of the model material on the configuration of the final structural pattern were also studied with a series of numerical continuum models. It was found that (a) the fault pattern of the Gulf of Corinth may be reproduced with either a strain-softening material with low elastic modulus or a constant strength material, and (b) two consecutive grabens, such as those of Gulfs of Corinth and Evia, may also be reproduced by an appropriate combination of variation of dip and frictional properties along the hypothesized detachment zone.

Ground response curves for rock masses exhibiting strain‐softening behaviour

AbstractA literature review has shown that there exist adequate techniques to obtain ground reaction curves for tunnels excavated in elastic‐brittle and perfectly plastic materials. However, for strain‐softening materials it seems that the problem has not been sufficiently analysed. In this paper, a one‐dimensional numerical solution to obtain the ground reaction curve (GRC) for circular tunnels excavated in strain‐softening materials is presented.The problem is formulated in a very general form and leads to a system of ordinary differential equations. By adequately defining a fictitious ‘time’ variable and re‐scaling some variables the problem is converted into an initial value one, which can be solved numerically by a Runge–Kutta–Fehlberg method, which is implemented in MATLAB environment.The method has been developed for various common particular behaviour models including Tresca, Mohr–Coulomb and Hoek–Brown failure criteria, in all cases with non‐associative flow rules and two‐segment piecewise linear functions related to a principal strain‐dependent plastic parameter to model the transition between peak and residual failure criteria.Some particular examples for the different failure criteria have been run, which agree well with closed‐form solutions—if existing—or with FDM‐based code results. Parametric studies and specific charts are created to highlight the influence of different parameters.The proposed methodology intends to be a wider and general numerical basis where standard and newly featured behaviour modes focusing on obtaining GRC for tunnels excavated in strain‐softening materials can be implemented. This way of solving such problems has proved to be more efficient and less time consuming than using FEM‐ or FDM‐based numerical 2D codes. Copyright © 2003 John Wiley & Sons, Ltd.

Comparison of integral-type nonlocal plasticity models for strain-softening materials

The paper analyzes and compares a number of softening plasticity models regularized by nonlocal averaging. To highlight the fundamental properties and gain insight into the regularizing effect of various formulations, the localization problem is examined in the one-dimensional setting. It is shown that some of the theoretically appealing formulations are not genuine localization limiters, and that a localized plastic zone of nonzero measure is obtained only with softening laws that take into account the effect of both the local and the nonlocal cumulative plastic strain. The evolution of the plastic zone is studied, and formulations suitable for the description of the entire deformation process up to the complete loss of cohesion are identified. The effect of boundaries on the shape of the plastic strain profile and on the dissipated energy is analyzed. Attention is also paid to the thermodynamic aspects of nonlocal plasticity, especially to the consistent extension of the concept of generalized standard materials to nonlocal continua.

Dynamic thermo-poro-mechanical analysis of catastrophic landslides

In this paper a dynamic analysis is presented of the early stages of an earth slide, considering two mechanically coupled substructures: (a) the rapidly deforming shear band at the base of the slide, and (b) the accelerating (rotating) rigid body. This is done by exploiting the concepts of frictional rate softening and thermo-poro-mechanical softening of the soil, and by modifying the friction-circle method of Taylor (1948) . First the equations are summarised that govern the phenomenon of heat-generated pore pressures inside a rapidly deforming shear band, and then the estimates for the various material parameters that enter the governing equations are critically discussed. In particular an attempt is made to adjust these material parameters to existing experimental data and to the properties of the clay soil responsible for the Vaiont slide of October 1963. The numerical analysis of the governing equations for the two mechanically coupled substructures leads to the following conclusions. The accelerated motion is caused by material strain-softening, which in turn feeds further material strain-rate softening inside the shear band. Owing to the speed of the initiated process, heat is trapped inside the shear band, which leads rapidly to a pore-pressure explosion in it owing to thermoplastic collapse of the clayey gouge, leading rapidly to total loss of strength and to uninhibited sliding motion on a frictionless base.

Fluid jet erosion as a non-linear fracture process: a discussion

Fluid jets are increasingly used to process and machine strain-softening materials. Typical applications are drilling, cutting, fragmentation, hydrodemolition and 3D-machining. Strain-softening materials, such as cement-based composites, ceramics, most rocks, ice and solidified impurities, are quasi-brittle behaving materials characterized by a so-called fracture process zone. Therefore, linear elastic fracture mechanics do not suitably cover these materials. Based on a literature review, it is also found that conventional strength parameters, such as compressive strength, cannot describe the resistance of this group of materials against fluid jet erosion, and that the erosion process is strongly determined by the size of the aggregate (inclusions) in the materials. Also, structure and behavior of non-linear behaving materials are characterized. An SEM-study performed on specimens of concrete, mortar, and hardened cement paste eroded by waterjets with velocities up to 470m/s clearly illuminates features of quasi-brittle behavior, including grain bridging, microcracking and crack branching. It is concluded that a non-linear fracture resistance parameter may be suitable to estimate the erosion resistance of the materials. It is found that the characteristic length of a material is a very suitable resistance parameter. An already existing approximate empirical formula for estimating this length is used, and a very good relationship between the estimated values and the volumetric erosion rate taken from published results for 20 different types of concrete could be obtained. Finally, the application of the proposed resistance parameter for fracturing and drilling rocks and refractory ceramics by plain and abrasive-laden waterjets is checked by discussing results found in the literature.

Slope Stability Analysis Using Finite Element Model for Mixed Type of Failure in Landslides

A finite element model for the mixed type of failure in landslides is presented with special reference to: (1) the progressive failure in strain softening materials and (2) the renewed movement with residual strength along the base layer. Strain softening elasto-plastic model in which the effect of shear band formation can be taken into cosideration is incorporated for first-time slides. Seepage analysis is performed using boundary condition simple and suitable for a long slope in order to evaluate the seepage force. Solution of collapse analysis of actual landslides based on the experiment result of undisturbed soil samples is presented.

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain-softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C1 continuity element. The required higher-order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill-posedness caused by strain-softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post-peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear-band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values. Copyright © 2001 John Wiley & Sons, Ltd.

Bifurcation in the numerical simulation of softening mechanisms

Non-linear simulation of strain-softening materials is often impeded by the presence of singular points encountered along the equilibrium path. Available numerical techniques aimed at the treatment of singular points are presented. Emphasis is placed on bifurcation analysis. Two numerical examples of structural bifurcation are presented. In the first example, a direct tension test on a notched concrete bar is simulated, displaying a symmetric bifurcation. In the second example, a direct tension test on a tapered concrete bar is simulated, displaying a non-symmetric bifurcation. The bifurcation analysis techniques introduced are shown to be effective. The need for systematic bifurcation analysis in the non-linear simulation of strain-softening materials is concluded.

ハイブリッド型応力法一次元有限要素による軟化特性を考慮したコンクリート部材の弾塑性破壊解析

Hybrid stress finite elements based on the beam-column theory to trace elasto-plastic and fracture behavior are developed for concrete member with strain-softening characteristics, wherein the damages are point-wise evaluated. Also, determination procedure of incremental load-displacement path including the instability phenomena due to the softening, such as snap-through and snap-back, is presented, and their adaptability and availability are discussed. Numerical and parametric test results, for the cantilever beams composed of idealized strain-softening material and the simply supported concrete beams, wherein are employed the tangential stress-strain relation using the concept of fracture energy, are given with the experimental ones, and show that the present approach have the excellent promising potentiality to reveal the localization of fracture zone and the size effect on nominal flexural strength of concrete.

Nonlocal constitutive models with gradients of internal variables derived from a micro/macro homogenization procedure

Finite element simulations involving strain softening materials or finite strains are faced with problems of loss of uniqueness and strain localization. If localization occurs, the size of the region where the strains concentrate is determined by the mesh size and no convergence can be obtained with mesh refinement. A non-local approach has been proposed recently by Andrieux et al. [15–16] in order to solve the problem. With this approach, spatial gradients of internal variables are incorporated in the macroscopic constitutive equations via a homogenization procedure. In this paper, this new formulation is investigated and numerical algorithms are presented for the onedimensional case. The results of the numerical simulations demonstrate that the size of the strain localization zone cannot be arbitrarily small anymore.