Softening Parameter Research Articles

Viscoplastic Regularization of Local Damage Models: A Latent Solution

Local damage models are known to produce pathological mesh dependence in finite element simulations. The solution is to either use a regularization technique or to adopt a non-local damage model. Viscoplasticity is one technique which can regularize the mesh dependence of local damage model by incorporating a physical phenomenon in the constitutive model i.e. rate effects. A detailed numerical study of viscoplastic regularization is carried out in this work. Two case studies were considered i.e. a bar with shear loading and a sheet metal under tensile loading. The influence of hardening / softening parameters, prescribed deformation rate and mesh size on the regularization was studied. It was found that the primary viscoplastic length scale is a function of hardening and softening parameters but does not depend upon the deformation rate. Mesh dependency appeared at higher damage values. This mesh dependence can be reduced by mesh refinement in the localized region and also by increasing the deformation rates. The viscoplastic regularization was successfully used with a local anisotropic damage model to predict failure in a cross die drawing process with the actual physical process parameters.

Post-peak Stress-strain Relationship of Rock Mass Based on Hoek-Brown Strength Criterion

The post-peak mechanical characteristic of rock mass plays a leading role in stability of geotechnical engineering. Based on Hoek-Brown strength criterion and evolutional behavior of strength parameters, choosing major principal strain ɛ1 as strain softening parameter, this paper presents the method of solving post-peak stress-strain relationship. In numerical case, the post-peak stress-strain curves of some rock masses are obtained. The effect of evolutional law of strength parameters σc,m, s and a to post-peak stress-strain curve is discussed, and it's concluded that the greater the residual values of σc,m, s are and the smaller the residual value of a is; the greater the residual strength is and the post-peak stress-strain curve falls more gently.

Study on Grouting Pressure of Splitting Grouting Based on Cylindrical Expansion Considering Large Strain

Anti-seepage reinforcement technology of splitting grouting to improve the stability of dam, has been an effective reinforcement method in the field of dam reinforcement. Based on the extended SMP criterion and stress-dilatancy relation considering large strain, the governing equations of axisymmetric problem in the plane strain condition and the partial differential equations for the boundary-value problem of cavity expansion in frictional cohesive soils were established. Then, the early phase of splitting grouting is regarded as the plane strain question of cylindrical expansion infinite soil. Under initial grouting pressure, the soil body was supposed as ideal elastic mass. However, the soil body was supposed to generate plastic damage considering large strain with the increase of grouting pressure and submit the extended spatial mobilization plane theory. Solutions of radial and hoop stresses and strains around the grouting cavity were obtained by recursive computations. Furthermore, the influence of damage softening parameter, cohesion and friction angle was examined by a parametric study.

A Levenberg-Marquardt algorithm for fi tting σ-w curves from three-point bend tests for plain and fi ber reinforced concretes

This technical note describes the implementation of the Levenberg-Marquardt algorithm for the inverse analysis of three-point bend tests on notched specimens of plain and fiber reinforced concretes for determining softening parameters (σ-w curves). Tests indicated that the algorithm is sufficiently efficient and robust.

Experimental characterization and modelling of the cyclic softening of carbon-black filled rubbers

Several carbon-black filled styrene-butadiene rubbers (SBRs) were submitted to uniaxial tension cyclic tests at large strain for investigating filled rubbers cyclic softening. In order to study the effect of the material composition (amount of fillers and crosslink density) and of the stretch intensity, an original method for cyclic softening characterization is proposed. The softening is seen as an amplification of the stretch in the rubber gum, and a stretch amplification factor is then introduced as the softening parameter. The latter is evaluated for various materials and several levels of cyclic stretch intensity. Material softening is shown to increase with the loading intensity, the amount of carbon-black fillers and the decrease of the crosslink density. In terms of modelling, the introduced softening parameter provides access to the entire stress-stretch responses of the softened material for any cycle. Therefore, a model is written for the description and the prediction of the stress-stretch responses of filled rubbers during proportional cyclic tests.

LS-DYNA MAT54 modeling of the axial crushing of a composite tape sinusoidal specimen

The suitability of a progressive failure material model to simulate the quasi-static crushing of a composite specimen is evaluated. The commercially available material model MAT54 “Enhanced Composite Damage” in LS-DYNA is often utilized to simulate damage progression in dynamic failure simulations because it requires a reduced number of experimental input parameters compared to damage mechanics-based material models. The composite specimen used for the experiments is a semi-circular sinusoid, and is comprised of carbon fiber/epoxy unidirectional prepreg tape. Results show that MAT54 can successfully reproduce experimental results, however the simulation is highly sensitive to changes in model parameters, which are either non-physical (i.e. are purely mathematical expedients), or cannot be measured experimentally. These include element size, contact definition, load–penetration curve, and crush front softening parameter, among others. Therefore, achieving successful simulation results requires extensive calibration of these parameters by trial and error, and a deep understanding of the strengths and challenges of the selected modeling strategy.

Advanced constitutive modelling of shotcrete: Model formulation and calibration

Shotcrete is one of the main support elements for tunnels constructed according to the principles of the New Austrian Tunnelling Method (NATM). In this paper, a sophisticated constitutive model for the complex mechanical and time-dependent behaviour of shotcrete is presented, which is formulated within the framework of elasto-plasticity. Two independent yield surfaces govern the material behaviour in compression and tension under multi-axial loading conditions, which is further controlled by non-linear plastic strain hardening and softening rules following uniaxial stress–strain curves. The main novelty of the model is the introduction of normalised hardening and softening parameters which enable a more realistic simulation of the shotcrete loading history. In addition, cracking of the shotcrete is treated within the smeared crack concept by applying a fracture energy approach as a regularisation technique, which eliminates the dependency of the results on the size of the elements in a finite element mesh. Furthermore, the model takes into account the gradual change of the main material properties of the shotcrete due to cement hydration during the first 28 days after spraying. Other important time effects such as creep, shrinkage and hydration temperature induced deformations at early ages are considered in the current model formulation. This paper aims to describe in detail the developed structure of the constitutive model and model calibration.

Effect of torsional oscillations on the stress–strain behavior of Al–5 wt% Mg alloy

The microstructure and mechanical properties of a precipitation hardenable Al–5 wt% Mg alloy subjected to low frequency torsional oscillations and aging treatments are reported in the present work. The stress–strain response was examined at different frequencies (1.53–3.3 Hz) and at constant shear strain amplitude ( φ = 5.69 × 10 −4). An excessive in the softening was observed by increasing both frequency of the applied oscillations and deformation temperature. The change in the hardening and softening parameters of stress–strain relations was explained in view of spinodal decomposition in Al–Mg systems. The mean value of activation energy was found to be equal to that quoted for precipitate–dislocation intersection.

Temperature dependent flow softening of titanium alloy Ti6Al4V: An investigation using finite element simulation of machining

Titanium alloy Ti6Al4V is the most commonly used titanium alloy in the aerospace and medical device industries due to its superior properties. There has been a considerable amount of research to better understand the serrated chip formation mechanism of titanium alloy Ti6Al4V by using finite element simulation of machining. An accurate representation of the behavior of the material is important in order to obtain reliable results from the finite element simulation. Flow softening behavior has been integrated into the material constitutive models to simulate adiabatic shear bands and serrated chips. Flow softening is usually related to the dynamic recrystallization phenomenon which initiates after a critical temperature. The aim of this study is to investigate the influence of various flow softening conditions on the finite element simulation outputs for machining titanium alloy Ti6Al4V. For this purpose, a new flow softening expression, which allows defining temperature-dependent flow softening behavior, is proposed and integrated into the material constitutive model. The influence of flow softening below the critical temperature, as adopted in recent studies, is also investigated. Various temperature-dependent flow softening scenarios are tested using finite element simulations, and the results are compared with experimental data from the literature. The results showed that the flow softening initiating around 350–500 °C combined with appropriate softening parameters yields simulation outputs that agree well with the experimental measurements.

Finite element modeling of impact, damage evolution and penetration of thick-section composites

Impact, damage evolution and penetration of thick-section composites are investigated using explicit finite element (FE) analysis. A full 3D FE model of impact on thick-section composites is developed. The analysis includes initiation and progressive damage of the composite during impact and penetration over a wide range of impact velocities, i.e., from 50 m/s to 1000 m/s. Low velocity impact damage is modeled using a set of computational parameters determined through parametric simulation of quasi-static punch shear experiments. At intermediate and high impact velocities, complete penetration of the composite plate is predicted with higher residual velocities than experiments. This observation revealed that the penetration-erosion phenomenology is a function of post-damage material softening parameters, strain rate dependent parameters and erosion strain parameters. With the correct choice of these parameters, the finite element model accurately correlates with ballistic impact experiments. The validated FE model is then used to generate the time history of projectile velocity, displacement and penetration resistance force. Based on the experimental and computational results, the impact and penetration process is divided into two phases, i.e., short time Phase I – shock compression, and long time Phase II – penetration. Detailed damage and penetration mechanisms during these phases are presented.

A derivation of the Vogel–Fulcher–Tammann relation for supercooled liquids

The non-Arrhenius Vogel–Fulcher–Tamman relation for the viscosity, η, known for about 8 decades, describes simply one of the most characteristic features of supercooled liquids. It may be written η = η 0 exp U / k T − T 0 . Using the Dyre et al. result demonstrating that U is proportional to the shear modulus, G, and the Interstitialcy Theory of Condensed Matter (ITCM) we derive this relation and obtain T 0 / T g = γ / γ + 1 and U = U 0 / 1 + γ , where U 0 is the interstitialcy diffusion energy at the glass temperature. Here, γ is a fragility softening parameter given by γ = βT g ( dc / dT)| T g . β is the shear susceptibility − d ln G / dc, and c is the interstitialcy concentration. γ is also a fragility parameter ranging from 0 for strong materials to 3 or above for fragile ones.

Modified material constitutive models for serrated chip formation simulations and experimental validation in machining of titanium alloy Ti–6Al–4V

Titanium alloys present superior properties such as high strength-to-weight ratio and resistance to corrosion but, possess poor machinability. In this study, influence of material constitutive models and elastic–viscoplastic finite element formulation on serrated chip formation for modeling of machining Ti–6Al–4V titanium alloy is investigated. Temperature-dependent flow softening based modified material models are proposed where flow softening phenomenon, strain hardening and thermal softening effects and their interactions are coupled. Orthogonal cutting experiments have been conducted with uncoated carbide (WC/Co) and TiAlN coated carbide cutting tools. Temperature-dependent flow softening parameters are validated on a set of experimental data by using measured cutting forces and chip morphology. Finite Element simulations are validated with experimental results at two different rake angles, three different undeformed chip thickness values and two different cutting speeds. The results reveal that material flow stress and finite element formulation greatly affects not only chip formation mechanism but also forces and temperatures predicted. Chip formation process for adiabatic shearing in machining Ti–6Al–4V alloy is successfully simulated using finite element models without implementing damage models.

Assessment of computerized treatment planning system accuracy in calculating wedge factors of physical wedged fields for 6 MV photon beams

Wedge filters are commonly used in external beam radiotherapy to achieve a uniform dose distribution within the target volume. The main objective of this study was to investigate the accuracy of the beam modifier algorithm of Theraplan plus (TPP version 3.8) treatment planning system and to confirm that either the beam hardening, beam softening and attenuation coefficients along with wedge geometry and measured wedge factor at single depth and multiple fields sizes can be the replacement of wedged profile and wedged cross-sectional data or not. In this regard the effect of beam hardening and beam softening was studied with physical wedges for 6 MV photons. The Normalized Wedge Factors (NWFs) were measured experimentally as well as calculated with the Theraplan plus, as a function of depth and field size in a water phantom for 15°, 30°, 45°, and 60° wedge filters. The beam hardening and softening was determined experimentally by deriving the required coefficients for all wedge angles. The TPP version 3.8 requires wedge transmission factor at single depth and multiple field sizes. Without incorporating the hardening and softening coefficients the percent difference between measured and calculated NFWs was as high as 7%. After the introduction of these parameters into the algorithm, the agreement between measured and TPP (V 3.8) calculated NWFs were improved to within 2 percent for various depths. Similar improvement was observed in TPP version 3.8 while calculating NWFs for various field sizes when the required coefficients were adjusted. In conclusion, the dose calculation algorithm of TPP version 3.8 showed good accuracy for a 6 MV photon beam provided beam hardening and softening parameters are taken into account. From the results, it is also concluded that, the beam hardening, beam softening and attenuation coefficients along with wedge geometry and measured wedge factor at single depth and multiple fields sizes can be the replacement of wedged profile and wedged cross-sectional data in the TPS. The study also indicated that by ignoring the beam softening and beam hardening will result in an inaccurate dose to the target volume of the patient.

Fluctuations of non-point mass galaxy distribution

We examine cosmic energy equation for extended galaxy structures on the basis of different models of universe. We also extend the power spectrum and density fluctuations for extended structure by introducing softening parameter both for linear and non-linear regimes. The results are compared with earlier results of point mass structures. It is found that softening parameters introduced in the theory influence the thermodynamic fluctuation theory. Results obtained with spectrum analysis are also compared with Riemannian geometric approach (Ruppeiner in Rev. Mod. Phys. 67:605, 1995) to the galaxy clustering. The singular solutions of thermodynamic fluctuation results can be interpreted on the basis of power spectrum analysis in terms of power index law of two point correlation function.

The variation of work hardening characteristics of Al-5 wt% Mg alloy during phase transition

The aim of this study is to investigate the effect of aging conditions on the stress−strain behavior along with microstructure changes of the Al-5 wt% Mg alloy. Following solid solution treatment and aging of specimens at temperatures ranging from 373 to 573 K for various aging times (1/4 to 4 h), stress−strain tests were performed at different testing temperatures (313−343 K). The work hardening parameters ( σ y, σ f, χ p and Y) were found to decrease continuously with pre-aging times at all aging and testing temperatures, where the softening parameters ( ε f and L) oppose this behavior. The variation in stress−strain parameters with increasing aging temperatures and aging times was explained on the basis of structural transformations taking place in the Al−Mg alloy. A precipitate-dislocation intersections mechanism was assumed as the rate-controlling mechanism for alloy.

Elasto-plastic analysis in conventional tunnelling excavation

A new, rigorous approach for calculating the distribution of stresses and displacements around a circular opening excavated in a strain-softening generalised Hoek–Brown rock mass is presented. As most of the current solutions are relevant to elastic-perfectly plastic and elastic-brittle plastic material, it is worth developing a new analytical-numerical approach to address this end. This paper presents an elasto-plastic solution for the axisymmetrical problem of excavating a circular opening in generalised Hoek–Brown material. In the plastic region it is assumed that all the strength parameters are linear functions of the plastic shear strain. By invoking the Runge–Kutta fourth-order method the problem is solved numerically, and states of stress and displacement in the plastic region are determined. The accuracy and practical application of the proposed approach are shown through several examples. For elastic-perfectly plastic and elastic-brittle plastic cases, the proposed approach shows excellent agreement with closed-form solutions. Similarly, for the strain-softening case, this method is in good agreement with current rigorous methods. The influence of the softening parameter, appearing in the evolution of strength parameters, is examined by the construction of different ground reaction curves, and the influence of different definitions for dilation angle in the plastic region is investigated.

Effect of Austenite Deformation on Recrystallisation Behaviour in an X-70 Microalloyed Steel

In the present study, the effect of austenite deformation on the recrystallisation behaviour in terms of recrystallisation-stop and recrystallisation-limit temperatures (T5% and T95%) of an X70 niobium microalloyed pipeline steel have been investigated by interrupted plane strain compression tests. The extents of recrystallisation are calculated using a modified fractional softening parameter. And the 20% and 60% of fractional softening were correlated to T5% and T95%. Quantitative optical metallography indicates that this method provides for a convenient and reliable experimental measurement of the critical temperatures associated with the recrystallisation of austenite. The recrystallisation kinetics and the precipitation kinetics of Nb(CN) were calculated using two widely applied models. The experimental results from this study suggest that the current model of precipitation kinetics might overestimate the precipitation start time.

Stress softening of multigraft copolymers

The hysteresis behaviour of multigraft (MG) copolymers, with a polyisoprene backbone and polystyrene (PS) side chains, was investigated by applying a modified softening model proposed by Elías-Zúñiga, which uses an approach of Ogden and Roxburgh. The model was combined with the non-affine tube model of rubber elasticity of Kaliske and Heinrich. Four parameters are obtained: chemical and physical cross-link moduli ( G c, G e), the number of statistical segments between two successive entanglements ( n e/ T e) and a softening parameter ( b). The model was proven to be valid by a comparison with other methods evaluating hysteresis behaviour. The characterization of the multigraft copolymers revealed a branch point and molecular architecture dependence of the softening parameter. b was low for tetrafunctional MG copolymers with cylindrical microdomains, and it was further reduced for a spherical morphology and for more complex molecular architectures. The magnitude of b also depends on the PS arm molecular weight for hexa- and tetrafunctional multigraft copolymers.

Sérsic Galaxy with Sérsic Halo Models of Early-type Galaxies: A Tool for N-body Simulations

ABSTRACTWe present spherical, nonrotating, isotropic models of early-type galaxies with stellar and dark-matter components both described by deprojected Sérsic density profiles and prove that they represent physically admissible stable systems. Using empirical correlations and recent results of N-body simulations, all the free parameters of the models are expressed as functions of one single quantity: the total (B-band) luminosity of the stellar component. We analyze how to perform discrete N-body realizations of Sérsic models. To this end, an optimal smoothing length is derived, defined as the softening parameter minimizing the error on the gravitational potential for the deprojected Sérsic model. It is shown to depend on the Sérsic index n and on the number of particles of the N-body realization. A software code allowing the computations of the relevant quantities of one- and two-component Sérsic models is provided. Both the code and the results of the present work are primarily intended as tools to perform N-body simulations of early-type galaxies, where the structural nonhomology of these systems (i.e., the variation of the shape parameter along the galaxy sequence) might be taken into account.

On the width and shape of the corotation region for low-mass planets

We study the coorbital flow for embedded, low mass planets. We provide a simple semi-analytic model for the corotation region, which is subsequently compared to high resolution numerical simulations. The model is used to derive an expression for the half-width of the horseshoe region, x_s, which in the limit of zero softening is given by x_s/r_p = 1.68(q/h)^(1/2), where q is the planet to central star mass ratio, h is the disc aspect ratio and r_p the orbital radius. This is in very good agreement with the same quantity measured from simulations. This result is used to show that horseshoe drag is about an order of magnitude larger than the linear corotation torque in the zero softening limit. Thus the horseshoe drag, the sign of which depends on the gradient of specific vorticity, is important for estimates of the total torque acting on the planet. We further show that phenomena, such as the Lindblad wakes, with a radial separation from corotation of ~ a pressure scale height H can affect x_s, even though for low-mass planets x_s << H. The effect is to distort streamlines and to reduce x_s through the action of a back pressure. This effect is reduced for smaller gravitational softening parameters and planets of higher mass, for which x_s becomes comparable to H.