Rigid Plasticity Research Articles

Effect of strain-softening and strength anisotropy on the undrained bearing capacity of non-associated clay

The bearing capacity of soils is critical in the design and analysis of foundations. The classical solutions derived from methods of characteristics and bound theories are based on the rigid plasticity and associated flow rule, which may not be realistic for natural soils. The undrained shear strength of soft clays exhibits strain-softening and anisotropic behaviour in laboratory tests, which creates complexity when choosing appropriate strength parameters in the conventional design process. In addition, the stress–strain characters are non-associative and path dependent, which also affects the bearing capacity of soils. This paper investigates two classical stability problems (i.e. deeply embedded pile/pipe section and rigid strip footing) using finite-element analysis and the MIT-E3 soil model, and demonstrates the effects of stain-softening and strength anisotropy on the undrained bearing capacity of soils. The computed results are compared with analytical solutions and finite-element limit analyses, as well as those from finite-element analysis using conventional soil models. These findings have strong practical implications in predicting the bearing capacity of foundations and interpreting in situ tests.

Interplay between microstructure, micromechanical, and tensile properties in wire arc additive manufactured nickel-aluminum bronze: As-built and heat treated

The generalized hardness-strength relationship in additively manufactured materials may differ from that observed in cast/wrought materials. To investigate this, the present study focuses on wire-arc additive manufactured (WAAM) Nickel-Aluminum Bronze (NAB) in both its as-built state and three distinct heat-treated conditions. Tensile tests and nanoindentation hardness tests were conducted simultaneously to establish a correlation between hardness and strength. Additionally, a comparison is made between the cast-NAB material and WAAM-NAB in this article. By considering the variations in grain size (coarse vs. fine) and intermetallic compounds in the cast and WAAM-NAB materials (in their as-built and post-heat treated forms), the correlation factor is found to differ based on the specific manufacturing processes and heat treatments employed. This paper aims to develop empirical equations that account for the various influencing factors, including rigid plasticity and uniform plastic work hardening conditions. Moreover, the study reports several indentation-based quantitative variables, such as the plasticity index, elastic recovery, recovery resistance, and wear resistance index of the WAAM-NAB material. By providing a comprehensive analysis of these factors, this research contributes to a better understanding of the hardness-strength relationship and the associated material properties in WAAM-NAB.

An analytical model for the Kelvin–Helmholtz instability in solids

Although Rayleigh–Taylor (RT) and Richtmyer–Meshkov(RM) instabilities in solids have long been confirmed to play a significant role in high-energy-density physics and to serve as tools to diagnose material properties under high-strain and high-strain-rate conditions, the Kelvin–Helmholtz (KH) instability in solids under extreme conditions has yet to be fully investigated. In addition, according to experiments driven by high explosives or by high-intensity lasers, the interface behavior is frequently determined by the KH instability with RT and RM instabilities, as well as by gravity decay waves. Therefore, a comprehensive analysis is required to study the mechanical properties of solids under extreme conditions and the multiple stability mechanisms that affect the interface behavior. This paper proposes a simplified and analytical mathematical model for stability analysis by considering surface tension, viscosity, elasticity, viscoelasticity, and perfectly rigid plasticity as a function of the specific physical conditions. The model is used to examine how the KH instability, RT instability, and gravity decay wave affect the evolution of the interface. We first recover the well-known dispersion relations based on the viscous potential flow method for the KH instability, and in particular investigate the stability threshold with varying shear velocities and in the presence of gravity decay waves. For the KH instability in elastic solids, once the velocity difference exceeds an asymptotic value (i.e., when the effect of elasticity cancels that of shear velocity), the interface always remains unstable for all perturbed wavelengths—a result that departs from the traditional conclusion about the RT instability between elastic solids. Unexpectedly, the viscosity in Kelvin–Voigt materials can enhance the resistance to the hydrodynamic instability, while the elasticity has the opposite effect. The growth rates and the cutoff wave numbers increase gradually as the viscoelastic parameter decreases, and the inhibition of the high-wave number disturbance by the viscoelasticity is correspondingly reduced. In addition, the cutoff wave numbers depend only on densities and the shear modulus of the material and are independent of the viscous properties of the materials. Finally, the evolution of the perturbed amplitude is studied in rigid plastic materials, which offers physical insights of interest to further study the KH instability in viscoplastic and elastoplastic materials. Based on the models developed herein, experiments are proposed for further research into the mechanical properties of materials under extreme conditions.

A finite strain framework for steady-state problems: Hyperelasto-viscoplasticity

A numerical framework for analyzing steady-state elastic–plastic material deformation at finite strains is developed and demonstrated in the present work. The framework is an extension of the original method by Dean and Hutchinson (1980) develop to analyze steady-state crack propagation, under a small strain assumption, in which the history-dependent material response is captured through streamline integration. Steady-state problems are encountered in numerous engineering processes, and the original studies of crack growth have already been extended to include rolling and drawing, though within a small strain framework. However, the investigation of such manufacturing processes, where strains greater than 10% easily develop, requires a finite strain formulation to provide accurate results. The framework proposed in the present work offers an efficient and accurate method to extract the steady-state solution at finite strains without encountering the numerical issues related to traditional Lagrangian procedures. Furthermore, the framework also accounts for elastic unloading compared to many existing numerical steady-state schemes as they are often restricted to rigid plasticity. The new numerical framework employs a hyperelastic material model, in terms of a Neo-Hookean material, combined with an isotropic viscoplastic material behavior. However, the framework is not limited to any specific hyperelastic material model, nor any specific model for the plastic behavior. The finite strain steady-state framework has been verified through comparison to a traditional Lagrangian analysis conducted in ANSYS. The benchmark case constitutes a plane strain drawing process where the thickness of the specimen is reduced by drawing it between two circular cylindrical tools.

Study on Numerical Oscillations of Forming Load Occurring in Forward Open Cold Extrusion

A forward open cold extrusion process is evaluated to reveal the sensitivity of forming load to numerical factors including number of total degrees of freedom, time step size and mesh size. The allowable reduction of area was examined by varying number of finite elements and the total solution steps to determine the reliability, in which the results are affected significantly by the positions of the lowest and highest nodal points in contact with the die. An investigation of the oscillations of the forming load in forward open extrusion revealed a set of critical numerical conditions that minimize unwanted oscillatory behavior. The limits of the material initial radii were obtained, suggesting that rigid plasticity with rigid dies is inappropriate for simulations of forward open cold extrusion with extreme reduction of area. Due to artificial numerical deformation of the rigid zone and die elastic deformation thus each of which has a non-negligible influence on the maximum reduction of area.

Plastic bending of sheet metal with tension/compression asymmetry

The plastic bending of a sheet metal having tension/compression asymmetry was investigated in this work with AZ31B Mg sheets, particularly focused on its deformation behavior. With the assumptions of pure bending and rigid plasticity, analytic expressions were obtained to describe sheet bending under the plane strain condition, taking into account the movement of the neutral surface and the associated deformation histories of longitudinal fibers in the sheet. Mechanical properties associated with bending such as anisotropy, work hardening in tension and compression, and untwinning under compression-tension were characterized considering the intricate hardening nature of AZ31B Mg sheet at room temperature. With the material coefficients characterized, the analytical expressions of bending were solved incrementally by numerical procedures. The computed thicknesses were compared to the thicknesses of V-die bending tests for validation purposes. It was found that the bending formulation reflecting the tension/compression asymmetry of AZ31B Mg sheet predicts the thickness increase at the initial stage of bending numerically, in agreement with the experimental results. Besides, the thickness variation during bending was found to be significantly affected by the asymmetric R-values of AZ31B Mg sheet. These results were discussed in connection with the movement of the neutral surface, as well as the material asymmetry in tension and compression.

The Technology and Mold Design of Super Large Height-diameter Ratio Local Upsetting Forming for Automobile Half Axle

The local upsetting blank forming of super large height-diameter ratio for automobile’s half shaft has been analyzed. Based on the rigid plasticity theory model of super height-diameter ratio local upsetting, the multi-ram forging method was used, and the die was designed as harvard architecture that the blank was clamped by the lower die, and the upper die with cavity structure to improves the stress condition. And the FEM software Deform-3D was used to analyzed the forming. The result showed the uniform region of equivalent strain and no loss of stability or folding defects during the forming.

A shear hardening plasticity model with nonlinear shear strength criterion for municipal solid waste

A shear hardening plasticity model with nonlinear strength criterion and appropriate stress-dilatancy description is presented to numerically study the bearing behavior of landfills. Simulations of drained triaxial tests are performed to illustrate the capability of the model in capturing the mechanical behavior of municipal solid waste. The bearing response and failure mode of a landfill foundation are determined using 2D and 3D finite element simulations. In contrast to the general shear failure mode obtained from rigid plasticity, the failure obtained from the hardening model shows a local mode, and prior to failure, a larger reaction force is produced for the same settlement.

Shear Versus Tensile Failure Mechanisms Induced by Sill Intrusions: Implications for Emplacement of Conical and Saucer‐Shaped Intrusions

AbstractSills, saucer‐shaped sills, and cone sheets are fundamental magma conduits in many sedimentary basins worldwide. Models of their emplacement usually approximate the host rock properties as purely elastic and consider the plastic deformation to be negligible. However, many field observations suggest that inelastic damage and shear fracturing play a significant role during sill emplacement. Here we use a rigid plasticity approach, through limit analysis modeling, to study the conditions required for inelastic deformation of sill overburdens. Our models produce distinct shear failure structures that resemble intrusive bodies, such as cone sheets and saucer‐shaped sills. This suggests that shear damage greatly controls the transition from flat sill to inclined sheets. We derive an empirical scaling law of the critical overpressure required for shear failure of the sill's overburden. This scaling law allows to predict the critical sill diameter at which shear failure of the overburden occurs, which matches the diameters of natural saucer‐shaped intrusions' inner sills. A quantitative comparison between our shear failure model and the established sill's tensile propagation mechanism suggests that sills initially propagate as tensile fractures, until reaching a critical diameter at which shear failure of the overburden controls the subsequent emplacement of the magma. This comparison also allows us to predict, for the first time, the conditions of emplacement of both conical intrusions, saucer‐shaped intrusions, and large concordant sills. Beyond the application to sills, our study suggests that shear failure significantly controls the emplacement of igneous sheet intrusions in the Earth's brittle crust.

Punching shear capacity of reinforced concrete slabs with headed shear studs

Punching shear in slabs is analogous to shear in beams. Despite this similarity, current design codes provide distinctly different methods for the design of shear reinforcement in the two situations. For example, the Eurocode method for beam shear design is founded on the theory of rigid plasticity. To design shear reinforcement in slabs, on the other hand, the engineer must settle for an empirical equation. The aim of the study reported is to demonstrate that it is possible in a simple manner to design shear reinforcement in slabs based on the same rigid-plasticity foundation as for beam shear design. For this purpose, an extension of the upper-bound crack sliding model is proposed. This involves analysis of sliding mechanisms in yield lines developed both within and outside the zone with shear reinforcement. Various types of headed shear studs were considered. The results obtained using the model were compared with a large number of published test results, and satisfactory agreements were found.

Influence of punch face angle and reduction on flow mode in backward and combined radial backward extrusion process

This paper is concerned with the analysis on the flow mode which determines lubrication limit such as stiction onto or sliding over punch face. The main goal of this study is to examine the influence of geometrical parameters such as punch face angle, reduction in area, and the gap height in radial direction in backward and combined radial-backward extrusion process on the flow mode and surface stresses such as sliding velocity, sliding distance and surface expansion. Annealed steel 17Cr3 was selected as a model material, a rigid-plastic material, for simulation, which was conducted using a commercially available FEA tool, Deform 2-D, programmed in a rigid plasticity theory. Change of flow mode during deformation was also investigated to find under which conditions of process parameter adopted the flow mode changes from stiction to transition or transition to sliding. In this paper, sliding velocities were quantitatively analyzed to determine the flow mode.

Some remarks on standard gradient models and gradient plasticity

The standard gradient models, originally introduced in the works of Gurtin (Physica D 1996; 92: 178) and Frémond and Nedjar (Int J Solids Struct 1996; 33: 1083) have been studied intensively in the last few decades for various applications in plasticity, damage mechanics and multi-phase analysis. In this paper, these models are revisited and discussed in relation with some classical descriptions of solids such as plasticity and visco-plasticity. The constitutive equations of such a model and the governing equations for a solid have been initially derived by these authors from an extended virtual work principle. Without this starting point, it is shown here that these equations appear as a generalized Biot equation for the solid and can be obtained directly from the global expression of the energy and dissipation potentials. This result gives the possibility to write the governing equations for standard gradient models of any order. These models also appear as a simple generalization of classical descriptions in solid mechanics. As an example, for a time-independent process such as incremental plasticity, our attention is focused on the constitutive modeling and on the governing equations of the response. It is shown that an elastic regularization can be introduced to avoid some theoretical and numerical difficulties which are well known in the classical theory of rigid plasticity.

Development of Finite Element Analysis Model for Plug Mill Rolling

In the seamless pipe rolling process, the pipe wall thickness is largely determined at the mandrel mill or plug mill. It is possible to obtain the target thickness at these mills by defining the gap of a grooved roll and an inside tool such as a plug. However, the thickness of the free deformation part, which is not in contact with the roll and tool, had generally been estimated by experimental techniques. Although a number of analytical studies of mandrel mill rolling had been reported, few reports had examined plug mill rolling. Therefore, in this research, a finite element analysis model for plug mill rolling was developed by extending the rigid plasticity finite element model "Computational Rolling Mill (CORMILL)." Good agreement between the calculated results and experimental results was obtained for the wall thickness, and it was found that the thickness of the flange part decreases with reduction of the wall thickness at the grooved bottom. These results suggested that the wall thickness distribution of rolled pipes can be controlled by using a suitable inside tool and roll shape in each rolling pass, and the necessary shapes can be obtained by using the newly-developed model.

Plane strain problem in elastically rigid finite plasticity

Summary A theory of elastically rigid finite deformation plasticity emphasizing the role of material symmetry is developed. The fields describing lattice rotation, dislocation density and plastic spin, irrelevant in the case of isotropy, are found to be central to the present framework. A plane strain characteristic theory for anisotropic plasticity is formulated wherein the solutions, as well as the nature of their discontinuities, show remarkable deviation from the classical isotropic slipline theory.

A plasticity model to assess the keying of plate anchors

Suction-embedded plate anchors (SEPLAs) have been developed to answer the growing need for anchors to withstand significant vertical loading. The concept combines the advantage of suction caissons (known penetration depth and location) and ‘drag-embedded' plate anchors (efficiency and low cost). The main issue associated with SEPLAs relates to the keying process, as the anchor is first loaded, and the associated loss of embedment and reduction in capacity. The paper presents a plasticity model developed to predict the trajectory and load development during anchor keying, and up to peak load. Rigid plasticity is assumed, allowing the kinematics of the anchor to be determined from a yield surface and associated plastic potential. The trajectory and performance of a typical SEPLA are predicted using the model, and are compared with results from centrifuge tests and large-deformation finite-element analysis. The anchor loss of embedment ranged from ∼0·2 to 1·5 times the anchor height for loading inclinations between 40° and 90° from the horizontal. The model was used further to calculate the anchor loss of embedment and capacity for varying padeye offsets. Results indicated that the loss of embedment could be reduced significantly by increasing the offset, but at the detriment of the ultimate anchor capacity.

On the Effect of Particle Size Distribution in Cold Powder Compaction

The effect of particle size distribution in powder compaction has been studied using the discrete element method. Both isostatic compaction and closed die compaction are studied together during the entire loading process. Particle rotation and frictional effects are accounted for in the analysis. The particles are, constitutively described by rigid plasticity, assumed to be spherical with the size of the radii that follows a truncated normal distribution. The results show that size distribution effects are small on global compaction properties like compaction pressure if the size distribution is small. Furthermore, the size distribution had no influence at all on the macroscopic behavior at unloading. To verify the model, comparisons were made on two different sets of experiment found in the literature where the particles were of varying sizes. Good agreement was found both on fundamental properties like the average number of contacts per particle and on more important properties from a practical point of view, like the compaction pressure.

A general method to calculate passive earth pressure on rigid retaining wall for all displacement modes

A general analytical method to calculate the passive rigid retaining wall pressure was deduced considering all displacement modes. First, the general displacement mode function was setup, then the hypotheses were made that the lateral passive pressure is linear to the corresponding horizontal displacement and the soil behind retaining wall is composed of a set of springs and ideal rigid plasticity body, the general analytical method was proposed to calculate the passive rigid retaining wall pressure based on Coulomb theory. The analytical results show that the resultant forces of the passive earth pressure are equal to those of Coulomb's theory, but the distribution of the passive pressure and the position of the resultant force depend on the passive displacement mode parameter, and the former is a parabolic function of the soil depth. The analytical results are also in good agreement with the experimental ones.

A new analytical theory for earing generated from anisotropic plasticity

Commercial canmaking processes include drawing, redrawing and several ironing operations. It is experimentally observed that during the drawing and redrawing processes earing develops, but during the ironing processes earing is reduced. It is essential to understand the earing mechanism during drawing and ironing for an advanced material modeling. A new analytical approach that relates the earing profile to r-value and yield stress directionalities is presented in this work. The analytical formula is based on the exact integration of the logarithmic strain. The derivation is for a cylindrical cup under the plane stress condition based on rigid perfect plasticity while force equilibrium is not considered. The earing profile is obtained solely from anisotropic plastic properties in simple tension. The earing mechanism is explained from the present theory with explicit formulae. It has been proved that earing is the combination of the contributions from r-value and yield stress directionalities. From a directionality ( y-axis) vs. angle from the rolling ( x-axis) plot, the earing profile is generated to be a scaled mirror image of the r-value directionality with respect to 90° ( x = 90) and also a scaled mirror image of the yield stress directionality with respect to the reference yield stress ( y = 1). Three different materials (Al–5% Mg alloy, AA 2090-T3 and AA 3104 RPDT control coil) are considered for verification purposes. This approach provides a fundamental basis for understanding the earing mechanism. In practice, the present theory is also very useful for the prediction of the earing profile of a drawn and iron cup and its related convolute cut-edge design for an earless cup.

Crystal plasticity model of shear and kink bands–energetic approach

We present a continuum crystal plasticity model of a lamellar deformation substructure of shear and kink bands. An evolutionary problem for the development of a spontaneous structural inhomogeneity is formulated in the framework of energetic solutions. Conti and Theil proved that in the case of an isothermal single-slip crystal, rigid plasticity with no hardening lamellaea form an optimal microstructure. Moreover, their model predicts the existence of a boundary layer which accommodates the lamellar substructure to displacement boundary conditions. It is suggested that the width of the shear and kink bands is a compromise: the minimization of bulk energy tends to decrease their size, while the energy of the band interfaces or the inner structure of the bands opposes this tendency.

Codf‐evolutions on polycrystalline orientation continua obtained by fast geometric estimates of plastic slip

AbstractThis paper presents a framework for the modeling of evolving crystal orientation distribution functions (codf) and orientation averages of polycrystalline aggregates undergoing finite deformations. The method is designed for fast estimates of the polycrystalline deformation textures in metals. It bases on the Taylor approach of rigid polycrystal plasticity, where the decision of slip‐activity in the local crystalline grains is based on a geometric argument. This approach allows a purely kinematic estimate of codf evolution completely independent of the hardening response of the single crystal grains. The investigation starts with a precise description of a deforming orientation continuum. This concerns the general geometric setup as well as details of the parameterization of rotations in terms of Rodrigues vectors. We then outline Lagrangian and Eulerian geometric settings of crystal fraction conservation laws and discuss the details of their discretization. An important aspect is the definition of orientation averages. This homogenization over the orientation space is an important ingredient of a multiscale modeling of evolving anisotropy. It is shown that the discrete evaluation of orientation averages does not need the explicit computation of codf. The core of the proposed method is a fast update algorithm for rate‐independent rigid crystal plasticity that directly determines the current Rodrigues vectors. We show for several deformation modes of polycrystalline aggregates that the proposed geometric estimates predict the codf evolution and overall anisotropy very well in the range of large strains. The new method is extremely robust and suitable for multiscale methods for the description of evolving anisotropy in polycrystals. Copyright © 2010 John Wiley & Sons, Ltd.