Viscoplastic Problems Research Articles

A space–time gauge theory for dynamic plasticity: Non-linear micro-inertia and its consequences

Exploiting the invariance of energy density under local translation in space–time, a gauge theoretic model is developed for dynamic viscoplasticity in polycrystalline solids. The invariance is preserved using minimally replaced space–time gauge covariant operators. Translation in time leads to a new definition of gauge covariant temporal derivative; hence the classical notion of velocity is modified. Using a space–time pseudo-Riemannian metric, we derive an evolution equation for the equivalent plastic strain, which is expressed via the compensating fields thus giving it a geometric meaning. Dissipative terms and some higher order temporal and spatial derivatives naturally appear in the model. Also established is a correspondence of the compensating field due to spatial translation with Kröner’s multiplicative decomposition of the deformation gradient. Using this framework, we explicate on the geometric interpretation of certain internal variables often used in classical viscoplasticity models. In order to assess how the theory performs, we carry out numerical simulations of uniaxial and 3D continuum viscoplasticity problems. We specifically explore the role of a nonlinear micro-inertia term in phenomena involving anomalous strain rate sensitivity, e.g. negative strain rate sensitivity, strain rate locking, and in possible oscillations in the stress–strain response.

Continuum damage mechanics of cyclic viscoplasticity using asymptotic numerical method

Conscious efforts on reduction of greenhouse gas emissions have led to an energy transition to renewable energy, however uncertainties of renewable energy production have resulted in higher thermal cycling demands from conventional power plants. Thermal load cycling at high temperature regions of steam turbine components leads to enhanced creep-fatigue damage accumulation. It is well established that such damage mechanism is numerically best predicted by unified constitutive modeling including damage as a variable as per the formalism of continuum damage mechanics at an expense of considerable computational efforts using finite element analysis. In this paper, the non-iterative Asymptotic Numerical Method (ANM), currently limited to partial cycle analysis with linear hardening plasticity model, is proposed for the first time to address cyclic viscoplasticity problems including damage capable of handling multiple cycles and most generalized loading conditions. Regularization techniques additionally necessary to implement loading-unloading-reloading criteria and advanced constitutive models etc. are presented. The constitutive model chosen for the formulation includes the non-linear multiple back stress variable modified Chaboche model to include damage combined with modified Chaboche-Rousselier isotropic hardening model to include damage, power law for viscoplasticity, Lemaitre’s damage potential and Kachanov-Rabotnov’s creep damage law. The method is verified with defined error measures and then applied to two high pressure steam turbine rotors, one with and another without thermal stress relief groove (TSRG) at the inlet under service type loading conditions to study the beneficial effect of the TSRG on creep-fatigue damage evolution. The accumulated errors of the proposed ANM and computational time are compared to a conventional Newton-Raphson solution.

A multifield variational formulation of viscoplasticity suitable to deal with singularities and non-smooth functions

In the present paper a multifield variational formulation of non-smooth viscoplasticity is illustrated which is well suited to deal with multivalued operators and non-smooth functions characterized by singularities. The adopted approach proves to be helpful in describing a geometrical framework of the non-smooth viscoplastic problem and in guiding to enhancements towards other kinds of plastic and viscoplastic material behavior. Alternative equivalent expressions of evolutive laws are discussed. The formulation shows to include Koiter’s theory as a special case. In addition, a multifield variational formulation of non-smooth viscoplasticity is illustrated. A procedure for deriving a parent potential in all the unknown variables is illustrated for non-smooth viscoplasticity problems. From the parent potential a set of associated multifield potentials is furtherly derived in one or more unknown fields and the relevant extremum properties are illustrated so that the non-smooth viscoplasticity structural problem is provided with a complete variational formulation suitable for developments in finite element applications.

Numerical simulation of elastoplastic problems by Brezis–Ekeland–Nayroles non-incremental variational principle

This paper is concerned with the energy-dissipation Brezis–Ekeland–Nayroles variational principle (BEN principle) for the numerical study of quasi-static elastoplastic and viscoplastic problems in small strains. This principle is based on the dissipation potential and its Fenchel transform and allows to have a consistent view of the whole evolution by computing the nonlinear response along the whole time history as a solution of a suitable minimization problem. In the present work, the BEN principle is applied to address the elastic perfectly plastic and viscoplastic thick hollow cylinder subjected to internal pressure. It turns out that the BEN variational formulation is based on a two-field functional, that leads naturally to discretize the displacement and stress fields. We present the detailing of the discretization and the numerical implementation of the minimization problem by using the mixed finite element method which is more efficient to enforce the yield condition. Computational accuracy and efficiency of the BEN principle is assessed by comparing the numerical results with the analytical ones and the simulations derived by the classical step-by-step finite element procedure.

Modeling and Simulation of Stochastic Inverse Problems in Viscoplasticity

In this paper, we have proposed an approach for parameter identification of random field for rigid viscoplastic material. It is assumed that the random field is stationary and Gaussian with known autocovariance function. Karhunen–Loève decomposition has been used to quantify the effects of random inputs. A method is presented that takes into account only the first two statistical moments of the analyzed displacement field, and only two values of searched process are identified—mean value and coefficient of variation in autocovariation function. It has been shown that this approach is desirable when complicated systems are analyzed. The discretization of the governing equations has been described by the finite element method. The sparse grid stochastic collocation method has been used to solve the stochastic direct problem. It is shown that for the described nonlinear equations, the response function due to searched parameters with wide bounds and with reduced number of measurement points has many local extrema and global optimization technique is required. Genetic algorithm has been adopted to compute the functional cost. Numerical example shows the identification problem for compressed cylindrical sample. It is revealed that the key factor determining the convergence of the method is the degree of reduction in the height of the tested sample.

Load identification for viscoplastic materials with some unknown material parameters

To inversely identify the loads applied on structure, sufficient information about the material properties is usually required. In this paper, an innovative and robust inverse approach is proposed for the load identification of a viscoplastic material with some unknown viscoplastic constants, where both of the unknown constants and the applied load are simultaneously identified using two nested inverse algorithms. In this work, the well-known damped Gauss–Newton method is used for the load identification and the Marquardt–Levenberg method for identifying the unknown material parameters. Suitable regularization techniques are also used to obtain more smooth solutions. The Perzyna model is used for description of the viscoplastic behavior. A meshfree method is used for the direct and sensitivity analysis of the viscoplastic problem. In order to show the effectiveness of the proposed method, two different examples are presented. Also, influences of the effective parameters, such as the number and location of sampling points, on the results are investigated too.

Extended formulations of evolutive laws and constitutive relations in non-smooth plasticity and viscoplasticity

Plastic and viscoplastic constitutive behavior is of interest in the mechanical modeling of many composite materials and structures. In this paper evolutive laws and constitutive relations in non-smooth plasticity and viscoplasticity are presented by means of a formulation which takes advantage of the proper concepts required to deal with non-smooth problems in plasticity and viscoplasticity. This adopted framework is endowed with considerable advantages in comparison with other formulations of non-smooth problems. In fact, subdifferential calculus shows to be the proper tool to deal with non-smooth functions and corners in plasticity and viscoplasticity. Plastic and viscoplastic constitutive models are revisited and expressed in subdifferential form by adopting the more general context presented herein that includes Koiter’s theory as a special case. The evolutive equations in plasticity and viscoplasticity are derived as optimality conditions of a suitably defined Lagrangian in a form usually not considered. Consequently, alternative equivalent expressions of the evolution laws and of the loading/unloading conditions are presented and the equivalence among them is described. Furthermore, the present approach shows to be useful for extensions into other types of elastoplastic materials and for clarifications on the relations existing between different constitutive models in non-smooth viscoplasticity.

An improved friction stir shear localization model and applications in understanding weld formation process in alloy Ti-6-4

This paper presents a coupled shear localization model incorporating both pin/workpiece contact and three-dimensional (3D) heat flow environment generated by friction heating between tool shoulder and workpiece. It represents a significant improvement over an early shear localization model presented by the same authors (2014, 2015) which was focused on shear localization process in front of pin such that shear band development can be treated as a one-dimensional (1D) thermal viscoplastic problem, without considering shoulder/workpiece heating effects and contact interactions between pin and workpiece. The current model includes a novel analytical-based 3D transient heat flow solution incorporating detailed spatial resolution regarding heat generation at shoulder/workpiece interface, which provides a transient temperature environment within which shear localization process can now be examined in a greater detail. Also, contact interactions between pin and workpiece are now treated explicitly by introducing a classical contact mechanics solution for describing contact behavior between a cylinder (pin) and cylindrical cavity (surrounding base metal). This refined shear localization model has been shown capable of providing more realistic and reasonable estimations for both temperature and welding torque when compared with available experimental data. A robust numerical procedure has also been developed and implemented for determining optimum welding speed for a given combination of stir tool rotational speed and welding travel speed, which shows a good agreement with experimental results for a number of applications.

Meshfree radial point interpolation method for analysis of viscoplastic problems

A truly meshfree radial point interpolation method (RPIM) for analysis of viscoplastic problems is presented. Without recourse to any background cell, a fast and accurate meshfree integration method is employed for computation of the domain integrals in the RPIM. The effects of the total number of nodes and the parameters of the meshfree method on the accuracy of the solutions are studied too. Moreover, the effects of the viscoplastic material parameters on the results are investigated. To demonstrate the accuracy of the proposed method and provide a comprehensive investigation of the viscoplastic and meshfree parameters, several numerical examples are presented. Excellent agreements are observed through comparing the obtained results with those retrieved by the finite element method.

Finite element procedure for one-dimensional viscoplastic problems of an isotropic slab

This study, based on a modified form of the Gurtin–Anand model (J Mech Phys Solids 53:1624–1649, 2005), investigates the effect of an internal microforce on the flow rule of viscoplastic bodies. A generalized microforce balance is obtained via the virtual power principle, and using the free-energy imbalance to suitably restrict the constitutive relations, a flow rule which accommodates two energetic length scales is obtained. This leads to a nonlinear second-order partial differential equation in the plastic strain. Finite element models for one-dimensional problems arising from a pure shear state of a viscoplastic slab are derived, while solutions for linear rate-dependent and nonlinear small strain-gradient rate-independent problems are presented.

On the solvability of incompressible Stokes with viscoplastic rheologies in geodynamics

AbstractPlasticity/failure is an essential ingredient in geodynamics models as earth materials cannot sustain unbounded stresses. However, many questions remain as to appropriate models of plasticity as well as effective solvers for these strongly nonlinear systems. Here we present some simplified model problems designed to elucidate many of the issues involved for the description and solution of viscoplastic problems as currently used in geodynamic modeling. We consider compression and extension of a viscoplastic layer overlying an isoviscous layer and introduce a single plastic yield criterion which includes the most commonly used viscoplasticity models: von Mises, depth‐dependent von Mises, and Drucker‐Prager. We show that for all rheologies considered, successive substitution schemes (aka Picard iteration) often stall at large values of the nonlinear residual, producing spurious solutions. However, combined Picard‐Newton schemes can be effective for rheologies that are independent of the dynamic pressure. Difficulties arise when solving incompressible Stokes problems for rheologies that depend on the dynamic pressure such as Drucker‐Prager viscoplasticity. Analysis suggests that incompressible Stokes can become ill‐posed when the dependence of the deviatoric stress tensor on dynamic pressure (i.e., ) becomes large. We demonstrate empirically that, in these cases, Newton solvers can fail by introducing spurious shear bands and discuss the consequence of interpreting the results of nonconverged computations. Even for problems where solvers converge, Drucker‐Prager viscoplasticity can produce dynamic pressures that deviate significantly from lithostatic and both the velocity and pressure fields should be evaluated to determine whether solutions are geologically reasonable.

A new algorithm based on Moving Least Square method to simulate material mixing in friction stir welding

In the present work, a high order implicit technique is associated with a meshless method to model material mixing observed in friction stir welding (FSW) process. This new algorithm combines the following mathematical procedures: a time discretization, a space discretization, a homotopy transformation, a perturbation technique and a continuation method. The perturbation technique, after a time discretization, a space discretization, a homotopy transformation, transforms the nonlinear problem into a sequence of linear ones at each time. By comparison to the classical iterative algorithms, the proposed one allows obtaining very large time steps and reducing computation time by minimizing the number of tangent matrix decompositions. The strong formulation is considered to avoid the drawback of numerical integration. The resulting algorithm is well adapted to large deformations in the mixing zone nearly the welding tool. We limit ourselves to bidimensional visco-plastic problems to show the performance of the proposed algorithm by comparison to the classical incremental iterative methods.

Finite element modelling of creep process - steady state stresses and strains

Finite element modelling of steady state creep process has been described. Using an analogy of visco-plastic problem with a described procedure, the finite element method has been used to calculate steady state stresses and strains in 2D problems. An example of application of such a procedure have been presented, using real life problem - cylindrical pipe with longitudinal crack at high temperature, under internal pressure, and estimating its residual life, based on the C*integral evaluation.

A Bermúdez–Moreno algorithm adapted to solve a viscoplastic problem in alloy solidification processes

The aim of this work is to present a computationally efficient algorithm to simulate the deformations suffered by a viscoplastic body in a solidification process. This type of problems involves a nonlinearity due to the considered thermo-elastic-viscoplastic law. In our previous papers, this difficulty has been solved by means of a duality method, known as Bermudez–Moreno algorithm, involving a multiplier which was computed with a fixed point algorithm or a Newton method. In this paper, we will improve the former algorithms by means of a generalized duality method with variable parameters and we will present numerical results showing the applicability of the resultant algorithm to solidification processes. Furthermore, we will describe a numerical procedure to choose a constant parameter for the Bermudez–Moreno algorithm which gives good results when it is applied to solidification processes.

Some theoretical and computational aspects of single‐crystal strain‐gradient plasticity

AbstractVariational formulations are constructed for rate‐independent problems in single‐crystal strain‐gradient plasticity. The framework makes use of the flow rule expressed in terms of a dissipation function. The formulation extends to the finite‐deformation context earlier work on this problem. Provision is made for energetic and dissipative microstresses, and a range of defect energies is accounted for. The minimization problem corresponding to the time‐discrete formulation is derived. Two special cases are then treated: first, for the small‐strain problem with energetic microstresses, results on well‐posedness, convergence of finite element approximations, the associated algorithms, and computational examples, are discussed. Secondly, recent computational work on the the related large‐deformation viscoplastic problem, with both energetic and dissipative effects, is presented and discussed for problems involving ensembles of grains.

Dual formulation of a viscoplastic contact problem with unilateral constraint

We consider a mathematical model which describes the contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the contact is frictionless and is modelled with unilateral constraint. We derive a variational formulation of the model which leads to a history-dependent quasivariational inequality for stress field, associated to a time-dependent convex. Then we prove the unique weak solvability of the model. The proof is based on an abstract existence and uniqueness result obtained in [11].

An Implicit Algorithm Based on Continuous Moving Least Square to Simulate Material Mixing in Friction Stir Welding Process

An implicit iterative algorithm, based on the continuous moving least square (CMLS), is developed to simulate material mixing in Friction Stir Welding (FSW) process. Strong formulation is chosen for the modeling of the mechanical problem in Lagrangian framework to avoid the drawback of numerical integration. This algorithm is well adapted to large deformations in the mixing zone in the neighborhood of the welding tool. We limit ourselves to bidimensional viscoplastic problem to show the performance of the proposed implicit algorithm. The results show that the proposed algorithm can be employed to simulate FSW.

Numerical simulation of finite strain viscoplastic problems

A large strain elastoplastic constitutive model based on hyperelasticity and multiplicative decomposition of deformation gradient tensor is extended to viscous case, in a framework similar to the one that has been proposed by Ponthot in an hypoelastic context. In this way a very useful framework can be obtained, able to deal with both rate dependent and rate independent problems.In this work a review of theoretical details and numerical implementation of the model are discussed. Similarly to what is done in rate independent plasticity, a Newton–Raphson scheme has been used to solve the non linear consistency condition in order to compute the viscoplastic multiplier.A plane strain plate with a central circular hole under tension is simulated in order to test the proposed model. Large deformation effects are considered in all the simulations carried out. Different parameters of the constitutive model are varied in order to study the sensitivity of the proposed algorithm.

Numerical Algorithms for J<sub>2</sub> Viscoplastic Models

An algorithmic scheme is illustrated for the numerical integration of evolutive problems in viscoplasticity. The analysis is restricted for brevity to J2 models in viscoplasticity by presenting a unified treatment of different elasto/viscoplastic constitutive models. An appropriate consistent tangent operator is applied which is suitable to be adopted for general numerical schemes. The treatment can be applied to different constitutive models by simply specializing the flow function. Computational results and examples are also illustrated in order to show the effectiveness of the presented algorithmic procedure.

Regularity of solutions to coercive and self-controlling viscoplastic problems

We prove higher regularity of solutions to viscoplastic problems. The main idea is to use the finite difference method, which allows us to cancel the monotone nonlinearities. Our main assumption is that velocity is in L 2 ( H 1 ) . This is true for all coercive and some special (for example linear self-controlling) non-coercive models.