A constitutive model for the deformation induced anisotropic plastic flow of metals

A mathematical framework is established for the equations governing inelastic deformation under multi-dimensional stress states and for the associated evolution equations of the internal state variables. The formulation is based on a generalization of the Prandtl-Reuss flow law. In the evolution equations for the inelastic state variables that control plastic flow, it is assumed that part of the rate of change is isotropic and the remaining part varies according to the sign and orientation of the current rate of deformation vector. This leads to a minimum of twelve components of the internal state tensor which represents resistance to inelastic deformation. In this manner, both initial and load history induced plastic anisotropy can be modeled. A specific set of equations for anisotropic plastic flow is developed consistent with the inelastic state variables.     

Numerical investigation of dynamic shear bands in inelastic solids as a problem of mesomechanics

The main objective of the present paper is to discuss very efficient procedure of the numerical investigation of the propagation of shear band in inelastic solids generated by impact-loaded adiabatic processes. This procedure of investigation is based on utilization the finite element method and ABAQUS system for regularized thermo-elasto-viscoplastic constitutive model of damaged material. A general constitutive model of thermo-elasto-viscoplastic polycrystalline solids with a finite set of internal state variables is used. The set of internal state variables is restricted to only one scalar, namely equivalent inelastic deformation. The equivalent inelastic deformation can describe the dissipation effects generated by viscoplastic flow phenomena. As a numerical example we consider dynamic shear band propagation in an asymmetrically impact-loaded prenotched thin plate. The impact loading is simulated by a velocity boundary condition, which are the results of dynamic contact problem. The separation of the projectile from the specimen, resulting from wave reflections within the projectile and the specimen, occurs in the phenomenon. A thin shear band region of finite width which undergoes significant deformation and temperature rise has been determined. Shear band advance, shear band velocity and the development of the temperature field as a function of time have been determined. Qualitative comparison of numerical results with experimental observation data has been presented. The numerical results obtained have proven the usefulness of the thermo-elasto-viscoplastic theory in the investigation of dynamic shear band propagations.     

Anisotropic damage coupled to plasticity: Modelling based on the effective configuration concept

A model framework for anisotropic damage, that is kinetically coupled to inelastic deformations, is outlined. Key ingredients are the concept of the effective undamaged configuration and the assumption of energy equivalence. The pertinent rate equations for the chosen internal variables are integrated using the standard fully implicit scheme, and the Jacobian matrix of the resulting non‐linear incremental constitutive problem is used to compute the algorithmic tangent stiffness in straightforward fashion. Numerical results showing the effect of deformation‐induced anisotropy conclude the paper. Copyright © 2002 John Wiley & Sons, Ltd.     

Parallel sensitivity analysis for efficient large-scale dynamic optimization

An efficient parallel algorithm for the computation of parametric sensitivities for differential-algebraic equations (DAEs) with a focus on dynamic optimization problems is presented. A speedup of about 4 can be obtained for process models of more than 13500 DAEs and 75 parameters employing 8 processor cores in parallel using a Windows based system. The algorithm obtains its efficiency by decoupling the sensitivity equations from the state equations of the DAE. Furthermore, the costly Jacobian matrices are computed separately by other processes. The computational effort for a combined state and sensitivity integration can almost be reduced to the computational effort of the pure state integration, which is the theoretical limit of the suggested approach.     

Modeling of strengthening and softening in inelastic nanocrystalline materials with reference to the triple junction and grain boundaries using strain gradient plasticity

The work presented here provides a generalized structure for modeling polycrystals from micro- to nano-size range. The polycrystal structure is defined in terms of the grain core, the grain boundary and the triple junction regions with their corresponding volume fractions. Depending on the size of the crystal from micro to nano, different types of analyses are used for the respective different regions of the polycrystal. The analyses encompass local and nonlocal continuum or crystal plasticity. Depending on the physics of the region dislocation-based inelastic deformation and/or slip/separation is used to characterize the behavior of the material. The analyses incorporate interfacial energy with grain boundary sliding and grain boundary separation. Certain state variables are appropriately decomposed into energetic and dissipative components to accurately describe the size effects. This new formulation does not only provide the internal interface energies but also introduces two additional internal state variables for the internal surfaces (contact surfaces). One of these new state variables measures tangential sliding between the grain boundaries and the other measures the respective separation. Additional entropy production is introduced due to the internal subsurface and contacting surface. A multilevel Mori–Tanaka averaging scheme is introduced in order to obtain the effective properties of the heterogeneous crystalline structure and to predict the inelastic response of a nanocrystalline material. The inverse Hall–Petch effect is also demonstrated. The formulation presented here is more general, and it is not limited to either polycrystalline- or nanocrystalline-structured materials. However, for more elaborate solution of problems, a finite element approach needs to be developed.     

Finite element analysis of time-dependent inelastic deformation in the presence of transient thermal stresses

A finite element formulation for the solution of time-dependent inelastic deformation problems for metallic structures, in the presence of transient thermal stresses, is presented in this paper. A rate formulation of the equations is used and any of a number of recently proposed combined creep-plasticity constitutive models with state variables can be adopted to describe material behaviour. The computer program developed can solve planar (plane strain and stress) and axisymmetric problems. Using one of the above-mentioned constitutive models, numerical results are presented for several illustrative problems, and comparisons of results, using either the quasi-steady or the unsteady diffusion equation for the determination of the temperature field, are carried out.     

Full-field sensitivity analysis through dimension reduction and probabilistic surrogate models

Computational mechanics models often are compromised by uncertainty in their governing parameters, especially when the operating environment is incompletely known. Computational sensitivity analysis of a spatially distributed process to its governing parameters therefore is an essential, but often costly, step in uncertainty quantification. A sensitivity analysis method is described which features probabilistic surrogate models developed through equitable sampling of the parameter space, proper orthogonal decomposition (POD) for compact representations of the process’ variability from an ensemble of realizations, and cluster-weighted models of the joint probability density function of each POD coefficient and the governing parameters. Full-field sensitivities, i.e. sensitivities at every point in the computational grid, are computed by analytically differentiating the conditional expected value function of each POD coefficient and projecting the sensitivities onto the POD basis. Statistics of the full-field sensitivities are estimated by sampling the surrogate model throughout the parameter space. Major benefits of this method are: (1) the sensitivities are computed analytically and efficiently from regularized surrogate models, and (2) the conditional variances of the surrogate models may be used to estimate the statistical uncertainty in the sensitivities, which provides a basis for pursuing more data to improve the model. Synthetic examples and a physical example involving near-ground sound propagation through a refracting atmosphere are presented to illustrate the properties of the surrogate models and how full-field sensitivities and their uncertainties are computed.     

On the Fractional Order Model of Viscoelasticity

Fractional order models of viscoelasticity have proven to be very useful for modeling of polymers. Time domain responses as stress relaxation and creep as well as frequency domain responses are well represented. The drawback of fractional order models is that the fractional order operators are difficult to handle numerically. This is in particular true for fractional derivative operators. Here we propose a formulation based on internal variables of stress kind. The corresponding rate equations then involves a fractional integral which means that they can be identified as Volterra integral equations of the second kind. The kernel of a fractional integral is integrable and positive definite. By using this, we show that a unique solution exists to the rate equation. A motivation for using fractional operators in viscoelasticity is that a whole spectrum of damping mechanisms can be included in a single internal variable. This is further motivated here. By a suitable choice of material parameters for the classical viscoelastic model, we observe both numerically and analytically that the classical model with a large number of internal variables (each representing a specific damping mechanism) converges to the fractional order model with a single internal variable. Finally, we show that the fractional order viscoelastic model satisfies the Clausius–Duhem inequality (CDI).     

Multi-axial thermo-mechanical analysis of power plant components from 9–12% Cr steels at high temperature

The aim of this paper is to analyze local changes of stress and strain states in a power plant component under a transient thermal environment. A robust constitutive model is developed to describe inelastic behavior of advanced 9–12% Cr heat-resistant steels at high temperature and in a multi-axial stress state. The model includes the constitutive equation for the inelastic strain rate tensor, the evolution equation for a tensor-valued state variable to reflect hardening/recovery processes and two evolution equations for two scalar-valued variables that characterize softening and damage states. The model is calibrated against experimental creep curves and verified for inelastic responses under different isothermal and non-isothermal loading paths. Steam temperature and loading profiles that correspond to an idealized start-up, holding and shut-down sequence of a power plant component are assumed. To estimate the thermal fields, transient heat transfer analysis is performed. The results are applied in the subsequent structural analysis using the developed inelastic constitutive model. The outcome is a multi-axial thermo-mechanical fatigue loop which can be used for damage assessment.     

Analytical investigation of crack tip fields in viscoplastic materials

A macroscopic stationary crack in viscoplastic materials is considered under mode I creep loading conditions. Typical representations of constitutive laws with internal variables. (back stress) which can be derived from a scalar potential function are used to model the inelastic material behaviour. It is shown that, in the limit of high stresses, the constitutive equation adopt the form of Norton's law thus leading to a singular field of the HRR-type at the crack tip, if the material functions for the constitutive equations are represented by power laws. The amplitude of the crack tip field can be evaluated using a crack tip integral. It reduces to the well-known C ^* expression at the crack tip and includes additonal domain integrals which ensure the independence of the choice of the integration contour and the area enclosed around the crack tip in regions where primary creep and linear eleastic effects cannot be neglected. The paper concentrates on results characterising the crack tip fields which can be derived analytically. Numerical aspects focused upon the right modelling of the crack tip zone, the range of validity of the crack tip field and the calculation of the crack tip parameter and the creep zones will be discussed in a subsequent paper.     

Fault detection and identification in dynamic systems with noisy data and parameter/modeling uncertainties

A probabilistic approach is presented which can be used for the estimation of system parameters and unmonitored state variables towards model-based fault diagnosis in dynamic systems. The method can be used with any type of input–output model and can accommodate noisy data and/or parameter/modeling uncertainties. The methodology is based on Markovian representation of system dynamics in discretized state space. The example system used for the illustration of the methodology focuses on the intake, fueling, combustion and exhaust components of internal combustion engines. The results show that the methodology is capable of estimating the system parameters and tracking the unmonitored dynamic variables within user-specified magnitude intervals (which may reflect noise in the monitored data, random changes in the parameters or modeling uncertainties in general) within data collection time and hence has potential for on-line implementation.     

THE INFLUENCE OF THE HARDENING STATE ON TIME DEPENDENT DAMAGE AND ITS CONSIDERATION IN A UNIFIED DAMAGE MODEL

Abstract— A model is presented for the prediction of the lifetime of metals in the high-temperature range under arbitrary variable multiaxial load. The definition of an internal variable for damage in continuum damage mechanics is adopted, which allows indirect measurement of damage via the deformation behaviour. To acquire some knowledge of damage evolution, damage is measured in two ways during uniaxial strain controlled cyclic tests: (a) a change of the modulus of elasticity and (b) a decrease of the peak stress. Surprisingly, both methods lead to results which are in good agreement. A new damage law is then developed (with reference to known models and lifetime rules) which is a modification of the creep damage law of Rabotnov that is extended by a dependence on the inelastic strain rate instead of the dependence on internal variables to take into account the hardening state. Uniaxial as well as multiaxial formulations of the new damage model (Inelastic Strain Rate Modified (ISRM) model) are presented.
The parameters of the ISRM model are determined with a view to applying them to AISI 316 L(N) austenitic steel. Some of the parameters are derived from standard creep experiments. To determine further parameters, the ISRM model is applied to uniaxial cyclic tests. Both failure behaviour and damage evolution are well described.     

A logarithmic‐exponential backward‐Euler‐based split of the flow rule for anisotropic inelastic behaviour at small elastic strain

A basic aspect of modern algorithmic formulations for large‐deformation hyperelastic‐based isotropic inelastic material models is the exponential backward‐Euler form of the algorithmic flow rule in the context of the multiplicative decomposition of the deformation gradient. Advantages of this approach in the isotropic context include the exact algorithmic fulfilment of inelastic incompressibility. The purpose of this short work is to show that such an algorithm can be formulated for anisotropic inelastic models as well under assumption of small elastic strain, i.e. for metals. In particular, the current approach works for both phenomenological anisotropy as well as for crystal plasticity. The major difference between the current and previous approaches lies in the fact that the elastic rotation is reduced algorithmically to a dependent internal variable, resulting in a smaller internal variable system. Copyright © 2006 John Wiley & Sons, Ltd.     

Cradle-to-grave simulation-based design incorporating multiscale microstructure-property modeling: Reinvigorating design with science

We propose a new paradigm for design that incorporates scientifically oriented research directly and feasibly into engineering design practice. The goal is to use this simulation-based tool earlier in design to achieve more optimized components and systems. The method to accomplish this bridge of science and engineering is by using thermodynamically constrained internal state variables that are physically based upon microstructure-property relations. When the microstructure-property relations are included in the internal state variable rate equations, history effects can be captured. Hence, the cradle-to-grave notion arises. The method to help determine the appropriate microstructure-property relations for the internal state variables is through a multiscale modeling methodology which includes experimentation. As such, scientifically oriented research occurs in the multiscale methodology, and the engineering design practice employs the cradle-to-grave internal state variable model. An example of the multiscale methodology is presented in terms of a cast A356 aluminum alloy used in automotive design, and an example of the cradle-to-grave simulation based design is presented in terms of a stamped product used in a crash scenario.     

A simplified canonical form algorithm with application to porous metal plasticity

A canonical form for the representation of material constitutive equations within standard finite element codes has been developed which is deceptive in its simplicity. Substitution of different equation systems is trivial and a consistent material Jacobian may be obtained automatically. The process is essentially numerical and does not involve the difficult algebraic manipulation associated with more traditional approaches. It is nevertheless exact because partial derivatives are derived analytically and simple because calculation of these derivatives is the only algebraic manipulation required. The remainder of the process is generic. In this paper, the algorithm is simplified further. A much more detailed and transparent explanation of the key to the method is given, namely calculation of the consistent Jacobian. A trivial modification also extends the method to plane stress. The original algorithm was validated for a simple material model. The new form is used to implement the significantly more difficult modified Gurson model for porous metal plasticity with hydrostatic yield dependence. It is tested for single element cases involving total collapse and also used to simulate necking of a notched cylindrical bar. Copyright © 2005 John Wiley & Sons, Ltd.     

Gradient theory from the thermomechanics point of view

This work demonstrates how gradients of internal state variables can be used in a set of internal variables when the thermomechanics of internal state variables is utilised. This is done by introducing a thermal internal variable called specific dissipative entropy. The gradient term in a material model can be used to avoid localisation. A material model showing Hookean deformation and creep is evaluated. Damage affects the Hookean response. The damage evolution equation contains a Laplacian of damage being introduced to avoid localisation of damage and mesh-dependence of a finite element solution. The material model satisfies the Clausius–Duhem inequality.     

Towards a theory of granular plasticity

A theory of granular plasticity based on the time-averaged rigid-plastic flow equations is presented. Slow granular flows in hoppers are often modeled as rigid-plastic flows with frictional yield conditions. However, such constitutive relations lead to systems of partial differential equations that are ill-posed: they possess instabilities in the short-wavelength limit. In addition, features of these flows clearly depend on microstructure in a way not modeled by such continuum models. Here an attempt is made to address both short-comings by splitting variables into ‘fluctuating’ plus ‘average’ parts and time-averaging the rigid-plastic flow equations to produce effective equations which depend on the ‘average’ variables and variances of the ‘fluctuating’ variables. Microstructural physics can be introduced by appealing to the kinetic theory of inelastic hard-spheres to develop a constitutive relation for the new ‘fluctuating’ variables. The equations can then be closed by a suitable consitutive equation, requiring that this system of equations be stable in the short-wavelength limit. In this way a granular length-scale is introduced to the rigid-plastic flow equations.     

Incorporation of mean stress effects into the micromechanical analysis of the high strain rate response of polymer matrix composites

The results presented here are part of an ongoing research program, to develop strain rate dependent deformation and failure models for the analysis of polymer matrix composites subject to high strain rate impact loads. A micromechanics approach is employed in this work, in which state variable constitutive equations originally developed for metals have been modified to model the deformation of the polymer matrix, and a strength of materials based micromechanics method is used to predict the effective response of the composite. In the analysis of the inelastic deformation of the polymer matrix, the definitions of the effective stress and effective inelastic strain have been modified in order to account for the effect of hydrostatic stresses, which are significant in polymers. Two representative polymers, a toughened epoxy and a brittle epoxy, are characterized through the use of data from tensile and shear tests across a variety of strain rates. Results computed by using the developed constitutive equations correlate well with data generated via experiments. The procedure used to incorporate the constitutive equations within a micromechanics method is presented, and sample calculations of the deformation response of a composite for various fiber orientations and strain rates are discussed.     

Towards a theory of granular plasticity

A theory of granular plasticity based on the time-averaged rigid-plastic flow equations is presented. Slow granular flows in hoppers are often modeled as rigid-plastic flows with frictional yield conditions. However, such constitutive relations lead to systems of partial differential equations that are ill-posed: they possess instabilities in the short-wavelength limit. In addition, features of these flows clearly depend on microstructure in a way not modeled by such continuum models. Here an attempt is made to address both short-comings by splitting variables into ‘fluctuating’ plus ‘average’ parts and time-averaging the rigid-plastic flow equations to produce effective equations which depend on the ‘average’ variables and variances of the ‘fluctuating’ variables. Microstructural physics can be introduced by appealing to the kinetic theory of inelastic hard-spheres to develop a constitutive relation for the new ‘fluctuating’ variables. The equations can then be closed by a suitable consitutive equation, requiring that this system of equations be stable in the short-wavelength limit. In this way a granular length-scale is introduced to the rigid-plastic flow equations.