Prandtl-Reuss Model Research Articles

Residual stresses calculation in a thermoelastoplastic torus after cooling

The present study deals with the boundary value problems under toroidal symmetry conditions. The residual stresses after cooling (unloading) in an elasto-plastic material are calculated. Throughout the paper the conventional Prandtl-Reuss model is generalised and used. The solution to the problem of hollow torus cooling under a temperature gradient is obtained and discussed. Analytical solutions, as an approximation of complete boundary value problem, describing residual deformations and stresses under conditions of toroidal symmetry are constructed and discussed.

Computer Exploration of the Ice Samples Strength Using Different Numerical Methods

The exploration of the Arctic region is a very challenging and important problem today. One of the problems, connected with the Arctics’ investigation, is the ice strength change and measurement during the whole year. In this paper we present the results of the ice strength study using different numerical methods on the unstructured and structured grids. We carry out the computations of the ice strength using the discontinuous Galerkin method on unstructured grids and the Prandtl–Reuss model. For this, we compute the model, imitating the interaction between an ice field and a vertical bearing. In addition, we calculate the ice strength after the uniaxial compression and bending using the discontinuous Galerkin method. Then, we present the result strength distribution in the ice sample, computed with the help of the grid-characteristic method on the structured grids. The results demonstrate the possibility of the application of the grid-characteristic method on structured grids to the problem of the ice strength study.

A Developed Damage Constitutive Model for Circular Steel Tubes of Reticulated Shells

The aim of this paper is that the precise description of damage behavior is crucial to well catch the mechanical behavior of structures in the dynamic numerical simulation, and address the issue on the coupled plastic-damage constitutive model for circular steel tubes of reticulated shells under severe earthquake. Continuum Damage Mechanics (CDM) constitutive model established by Lemaitre is reviewed at the beginning. Then, an improved damage model for circular steel tubes of reticulated shells is developed based on Lemaitre’s model by replacing the original damage evolution law with a new one suitable for circular steel tubes. In addition, we introduce the stress update process. In this procedure, the well-known operator split strategy, which leads to the standard elastic predictor/return mapping algorithm, is adopted to solve the evolution problem of the improved model. Exploiting user-defined material subroutine, the implementation of the model is achieved within software ANSYS using BEAM189 element. Finally, the dynamic response of reticulated shells under severe earthquake are numerically simulated with the proposed model and with the conventional Prandtl-Reuss model, respectively. The comparison results show that the consideration of material damage accumulation, on the one hand, may change the failure mode of reticulated shells from dynamic instability to strength failure; on the other, may reduce the dynamic ultimate load obviously. This consideration should be taken into account when conducting nonlinear dynamic analysis of reticulated shells.

Explicit nonlinear and plastic analysis model of steel–concrete–steel sandwich panels

This paper presents an explicit nonlinear and plastic analysis model (ENPAM) for steel–concrete–steel (SCS) panels. This model considers the behavior of concrete under biaxial stresses and the post-yield stress-strain relationship of steel plates. A simplified measure is utilized for the simulation of concrete. The elastic–plastic stiffness of steel plates is derived from the Prandtl–Reuss model. ENPAM fully embodies the stress redistribution during the plastic flow after steel plate yielding. The model's effectiveness is demonstrated based on comparisons with experimental data obtained for SCS panels subjected to pure shear and shear with a small compression. A finite element model is also established to verify the effectiveness of ENPAM for the load cases of shear with large compression.

Analysis of Polymethylmethacrylate Destruction on High-Speed Loading

A system of equations is presented for describing the shock-compressed state of polymethylmethacrylate in the approximation of uniaxial deformation. With the use of the Prandtl–Reuss model, the mechanism of transition of the brittle failure of polymethylmethacrylate to the brittle-plastic one at high speeds of the striker and high times of its effect on this mechanism has been determined.

A revision of results for standard models in elasto-perfect-plasticity theory

We consider two most studied standard models in the theory of elasto-plasticity in arbitrary dimension $$d\ge 2$$ , namely, the Hencky model and the Prandtl–Reuss model subjected to the von Mises condition. There are many available results for these models—from the existence and the regularity theory up to the relatively sharp identification of the plastic strain in the natural function/measure space setting. In this paper we shall proceed further and improve some of known estimates in order to identify sharply the plastic strain. More specifically, we rigorously improve the integrability of the displacement and the velocity (which was known only under a nonnatural assumption that the Cauchy stress is bounded), show the BMO estimates for the stress and finally also the Morrey-like estimates for the plastic strain. In addition, we shall provide the whole theory up to the boundary. As an immediate consequence of such improved estimates, we provide a sharper identification of the plastic strain than that known up to date. In particular, in two dimensional setting, we show that the plastic strain can be point-wisely characterized in terms of the stresses everywhere although the stress is possibly discontinuous and thus the natural duality pairing in the space of measures could be violated.

Discontinuous galerkin method for numerical simulation of dynamic processes in solids

This paper examines the application of the discontinuous Galerkin method for deformation and destruction problems of elastic and elastoplastic bodies and combined problems of elasticity and acoustics. It proposes a 2-sided crack model in the problems of destruction, an account of the elastoplastic rheology in the Prandtl-Reuss model, implementation of the dynamic contact of bodies, an algorithm for the joint solution of linear systems of acoustics and elasticity, and in particular, the issues of shelf seismic prospecting, a comparison of the responses for the model of saturated fluid and infinitely thin cracks, and a modeling of perturbations from underwater objects. The method has been implemented to find the wave picture in heterogeneous media and also the solution of deformation problems by the use of high performance computation systems.

Non-periodic homogenization of infinitesimal strain plasticity equations

We consider the Prandtl-Reuss model of plasticity with kinematic hardening, aiming at a homogenization result. For a sequence of coefficient fields and corresponding solutions , we ask whether we can characterize weak limits u when as . We assume neither periodicity nor stochasticity for the coefficients, but we demand an abstract averaging property of the homogeneous system on reference volumes. Our conclusion is an effective equation on general domains with general right hand sides. The effective equation uses a causal evolution operator Σ that maps strains to stresses; more precisely, in each spatial point x, given the evolution of the strain in the point x, the operator Σ provides the evolution of the stress in x.

Numerical modeling of elastoplastic flows by the Godunov method on moving Eulerian grids

The paper proposes a numerical method for calculating elastoplastic flows on adaptive Eulerian computational grids. Elastoplastic processes are described using the Prandtl-Reuss model. The spatial discretization of the Euler equations is carried out by the Godunov method on a moving grid. In order to improve the accuracy of the scheme, piecewise linear reconstruction of the grid functions is employed using a MUSCL-type interpolation scheme generalized to unstructured grids. The basic idea of the method is to split the system of governing equations into a hydrodynamic and an elastoplastic component. The hydrodynamic equations are solved by an absolutely stable explicit-implicit scheme, and the constitutive equations (elastoplastic component) are solved by a two-stage Runge-Kutta scheme. Theoretical analysis is performed and analytical solutions are obtained for a one-dimensional model describing the structures of a shock wave and a rarefaction wave in an elastoplastic material in the approximation of uniaxial strains. The proposed method is verified by the obtained analytical solutions and the solutions calculated using alternative approaches.

Development and validation of a thermo-mechanical finite element model of the steel quenching process including solid–solid phase changes

A computational thermo-metallographic and thermoelastoplastic model for the analysis of the quenching process is developed and validated. The diffusive transfor-mations are modeled according to the Johnson–Mehl–Avrami–Kolmogorov model and the Scheil’s additivity rule. Two different models are investigated for the non-diffusive transformation—the Koistinen–Marburger model and the Yu model. A large displacement formulation is assumed for the deformation analysis, modeling the plastic behavior of the material according to the Prandtl–Reuss model. Two different bilinear hardening models—the isotropic and the kinematic hardening model—are used and compared. The model allows to evaluate the transient stress and strain distributions during the quenching process, the final phases and hardness distributions, and to predict the residual stress and the final deformation of the processed part. A good agreement between computational results and reference data is found

Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

This paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusionhere we denote by σ the stress tensor, by ε the linearized strain tensor, by B(x) the compliance tensor and by ∂ϕ(·, x) the subdifferential of a convex function ϕ(·, x). This relation accounts for elasto-viscoplasticity, including a nonlinear version of the classical Maxwell model of viscoelasticity and the Prandtl—Reuss model of elastoplasticity.The constitutive law is coupled with the equation of continuum dynamics, and well-posedness is proved for an initial- and boundary-value problem. The function ϕ and the tensor B are then assumed to oscillate periodically with respect to x and, as this period vanishes, a two-scale model of the asymptotic behaviour is derived via Nguetseng's notion of two-scale convergence. A fully homogenized single-scale model is also retrieved, and its equivalence with the two-scale problem is proved. This formulation is non-local in time and is at variance with that based on so-called analogical models that rest on a mean-field-type hypothesis.

Regularity of stresses in Prandtl-Reuss perfect plasticity

We study the differential properties of solutions of the Prandtl-Reuss model. We prove that in dimensions n = 2, 3 the stress tensor has locally square-integrable first derivatives: $$\sigma\in L^\infty([0,T];W^{1,2}_{{\rm loc}}(\Omega;{\mathbb M}^{n{\times}n}_{{\rm sym}}))$$ . The result is based on discretization of time and uniform estimates of solutions of the incremental problems, which generalize the estimates in the case of Hencky perfect plasticity. Counterexamples to the regularity of displacements and plastic strains in the quasistatic case are presented.

Homogenization of some models of hysteresis

This paper deals with the behaviour of the classical Prandtl–Reuss model of elasto-plasticity in the limit of rapidly oscillating parameters. Two- and single-scale effective models are illustrated.

Modelling of the post-localization behaviour in tube hydroforming of low carbon steels

In tube hydroforming, some local thinning can be observed on industrial components. When low carbon steels are concerned, the process is stopped broadly after reaching the plastic instability (strain localization) while having an acceptable part. In numerical simulation, the classical elastic–plastic models reach their limits when the plastic instability occurs, the process simulation becomes very sensitive to numerical parameters which leads to poor prediction of post-localization behaviour. Consequently, a full understanding of post-localization is important for the formability prediction purpose. In this work, various numerical simulations were carried out on both tensile and hydroforming tests. The classical Prandtl–Reuss model is first investigated and the obtained results clearly show that this model is not appropriate to predict the post-localization behaviour. Then, some regularization techniques such as material rate dependence and the introduction of gradient-based internal variable in the constitutive relations were evaluated. In the light of the obtained results, the rate-dependent model seems to be well adapted to predict the post-localization behaviour in tube hydroforming simulation. Thus, the rate-dependent approach was adopted and the basic model was enhanced to fit some experimental data from tensile tests. Finally, the hydroforming Corner Fill test was simulated and the numerical results were compared with measurements to validate the proposed approach. This comparison shows that the rate-dependent model greatly improves the accuracy of the predictions.

The contraction ratios of perfect elastoplasticity under biaxial controls

We studied contraction ratios, one rate form and one total form, of the Prandtl–Reuss model under combined axial and torsional controls. In the transition point of elasticity and plasticity, the rate form contraction ratio may undergo a discontinuous jump, which, depending on the control paths and initial stresses, may be positive, zero, or negative. For the total form contraction ratio, no similar jump phenomenon is observed in the elasticity–plasticity transition point. Depending on initial stresses both ratios may be greater than 1/2 . In the simulations of the axial–torsional strain control tests, the hoop and radial strains are not known a priori and hence can not be viewed as inputs. This greatly complicates the constitutive model analyses because the resulting differential equations become highly non-linear. To tackle this problem, we devise a new parametrization of the axial and shear stresses, deriving a first order differential equation to solve for the parameter variable, with which the consistency condition and initial conditions are fulfilled automatically. For mixed controls, the responses can be expressed directly in terms of the parameter without solving the first order differential equation. In particular, when control paths are rectilinear exact solutions can be obtained.

Numerical Study of the Brinnel Test of Elastoplastic Indentation

In this paper, we present the numerical study of the elastoplastic contact without friction between a deformable spherical punch and a deforrnable support, using the finite element method. We describe the elastoplastic behaviour using the Prandtl-Reuss model (isotropic work hardening) associated with the von Mises yield criterion. The contact problem leads to a non-linear system we solve by applying loading increments. We carry out computations in the work hardening and perfectly plastic cases. Our results are presented in the form of stresses distribution and evolution of the yield zone and deformed shape. Then we study the influence of work hardening on the pressure distribution and on the deformed shape. Moreover, we observe the sinking-in and piling-up phenomena in the work hardening and perfectly plastic cases. All these results are in conformity, on one hand with those published by Johnson, and Sinclair et al. and on other hand with those of Biwa et al.

Simulation of the cold-roll forming of circular tubes

The approach developed in a previous paper for the simulation of the cold-roll forming of elasto-plastic materials is applied here for circular tubes. The main idea is to consider the sheet to be transformed as a thin shell and its middle surface as a Coons patch. Since the strains and rotations are finite, the material is supposed to follow a constitutive rate equation. The Prandtl–Reuss model, including the von Mises yield criterion, strain hardening, and the normality flow rule, is used. In order to integrate the elasto-plastic constitutive equations, a radial return scheme is adapted. The minimization of the plastic power rate provides an optimal shape of the sheet between the rolls as well as an optimal velocity field. This method gives a very fast simulation of the cold-roll forming process.

A simulation of cold-roll forming for elastoplastic materials

A kinematical approach was proposed in a previous paper for predicting the optimal shape and the deformed length of a rigid-plastic metal sheet during cold-roll forming. Because the elastic effects are important in this kind of process, the method has been extended here to elastoplastic materials. In the new formulation, the sheet is still considered as a thin shell and its middle surface is described as a Coons patch depending on one geometrical parameter. Moreover, the material now satisfies a constitutive rate equation in which the corotational rate of stress is used. The Prandtl-Reuss model, including the von Mises yield criterion and the normality flow rule, is used. In order to integrate the elastoplastic constitutive equations, a radial return scheme is adapted so that the plastic power rate is calculated, using a Gauss method. Its minimization gives the optimal shape for a strain hardening elastoplastic material, as well as the optimal velocity field. This approach has been implemented on a workstation and, as for rigid-plastic materials, it gives a very fast simulation of the cold-roll forming process.

Finite Expansion of an Infinitesimal Void in Elastic-Plastic Materials under Equitriaxial Stress

Sudden growth of an infinitesimal void to a finite size under equitriaxial tension is studied for elastic-plastic materials via a bifurcation approach. The analysis employs the Prandtl-Reuss model with finite deformation taken into account, for both strainhardening and perfectly plastic solids. Expressions for critical stress and strain levels for finite void growth, namely, cavitation limits, are obtained in the form of integrals involving material parameters and hardening characteristics. Numerical results for the critical values and post-cavitation behavior are demonstrated for power-law hardening elastic plastic materials, and the influence of hardening exponents as well as elastic compliance is discussed in detail.

Shape optimization of an elasto-perfectly plastic body

Within the range of Prandtl-Reuss model of elasto-plasticity the following optimal design problem is solved. Given body forces and surface tractions, a part of the boundary, where the (two-dimensional) body is fixed, is to be found, so as to minimize an integral of the squared yield function. The state problem is formulated in terms of stresses by means of a time-dependent variational inequality. For approximate solutions piecewise linear approximations of the unknown boundary, piecewise constant triangular finite elements for stress and backward differences in time are used. Convergence of the approximations to a solution of the optimal design problem is proven. As a consequance, the existence of an optimal boudary is verified.