A coupled temperature and strain rate dependent yield function for dynamic deformations of bcc metals

A coupled temperature and strain rate microstructure physically based yield function is proposed in this work. It is incorporated along with the Clausius–Duhem inequality and an appropriate free energy definition in a general thermodynamic framework for deriving a three-dimensional kinematical model for thermo-viscoplastic deformations of body centered cubic (bcc) metals. The evolution equations are expressed in terms of the material time derivatives of the elastic strain, accumulated plastic strain (isotropic hardening), and the back stress conjugate tensor (kinematic hardening). The viscoplastic multipliers are obtained using both the Consistency and Perzyna viscoplasticity models. The athermal yield function is employed instead of the static yield function in the case of the Perzyna viscoplasticity model. It is found that the static strain rate value, at which the material shows rate-independent behavior, varies with the material deformation temperature. Computational aspects of the proposed model are addressed through the finite element implementation with an implicit stress integration algorithm. Finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program ABAQUS/Explicit [ABAQUS, 2003. User Manual, Version 6.3. Habbitt, Karlsson and Sorensen Inc., Providence, RI] via the user material subroutine coded as VUMAT. Numerical implementation for a simple compression problem meshed with one element is used to validate the proposed model implementation with applications to tantalum, niobium, and vanadium at low and high strain rates and temperatures. The analysis of a tensile shear banding is also investigated to show the effectiveness and the performance of the proposed framework in describing the strain localizations at high velocity impact. Results show mesh independency as a result of the viscoplastic regularization used in the proposed formulation.     

Lie groups in nonequilibrium thermodynamics: Geometric structure behind viscoplasticity

Poisson brackets provide the mathematical structure required to identify the reversible contribution to dynamic phenomena in nonequilibrium thermodynamics. This mathematical structure is deeply linked to Lie groups and their Lie algebras. From the characterization of all the Lie groups associated with a given Lie algebra as quotients of a universal covering group, we obtain a natural classification of rheological models based on the concept of discrete reference states and, in particular, we find a clear-cut and deep distinction between viscoplasticity and viscoelasticity. The abstract ideas are illustrated by a naive toy model of crystal viscoplasticity, but similar kinetic models are also used for modeling the viscoplastic behavior of glasses. We discuss some implications for coarse graining and statistical mechanics.     

Two-temperature continuum thermomechanics of deforming amorphous solids

There is an ever-growing need for predictive models for the elasto-viscoplastic deformation of solids. Our goal in this paper is to incorporate recently developed out-of-equilibrium statistical concepts into a thermodynamically consistent, finite-deformation, continuum framework for deforming amorphous solids. The basic premise is that the configurational degrees of freedom of the material – the part of the internal energy/entropy that corresponds to mechanically stable microscopic configurations – are characterized by a configurational temperature that might differ from that of the vibrational degrees of freedom, which equilibrate rapidly with an external heat bath. This results in an approximate internal energy decomposition into weakly interacting configurational and vibrational subsystems, which exchange energy following a Fourier-like law, leading to a thermomechanical framework permitting two well-defined temperatures. In this framework, internal variables, that carry information about the state of the material equilibrate with the configurational subsystem, are explicitly associated with energy and entropy of their own, and couple to a viscoplastic flow rule. The coefficients that determine the rate of flow of entropy and heat between different internal systems are proposed to explicitly depend on the rate of irreversible deformation. As an application of this framework, we discuss two constitutive models for the response of glassy materials, a simple phenomenological model and a model related to the concept of Shear-Transformation-Zones as the basis for internal variables. The models account for several salient features of glassy deformation phenomenology. Directions for future investigation are briefly discussed.     

Inelastic behavior of an AS4/PEEK composite under combined transverse compressionand shear.Part II: modeling

A micromechanical model is presented for simulating the nonlinearities exhibited by AS4/PEEK composites in shear and transverse compression, their interaction, and their rate dependence at room temperature. The fibers are assumed to be transversely isotropic and to be distributed in a hexagonal pattern in the matrix. The PEEK matrix is modeled as an elastic–powerlaw viscoplastic, isotropic solid through two related models. Model I is the simple J₂–type viscoplasticity; Model II is a rate dependent version of the non-associative Drucker-Prager model. Both models are calibrated so that they reproduce the shear response of the composite. Model II is also calibrated to its transverse compression response. Both models capture the rate dependence of the composite well. Model I is significantly less stiff in transverse compression than the experimental data. However, it does a reasonable job of predicting other aspects of the biaxial experiments and captures the important trends of behavior observed. Model II does better in transverse compression, but shearing in the presence of transverse compression is found to be stiffer than the measured responses. The unit cell model allows us to examine the stresses in the composite, providing an explanation for the lack of interaction between the constant stress and the increasing stress observed experimentally for certain loading paths.     

A simple anisotropic clay plasticity model

An anisotropic clay plasticity constitutive model is extended to include a non-associative flow rule for the successful simulation of the response under undrained loading for some normally consolidated sensitive clays, including possible softening response, without altering otherwise the simple basic structure of the formulation. The model has been developed within the framework of critical state soil mechanics (CSSM) for the triaxial space. The model's structure deviates, in general, from the particular premises of CSSM in regards to a unique critical state line in the space of void ratio and effective pressure, in order to simulate observed experimental data.     

Continuum slip viscoplasticity with the Haasen constitutive model: Application to CdTe single crystal inelasticity

Small strain constitutive equations are developed for the thermomechanical behavior of semiconductor single crystals, including dislocation density as an evolving parameter. The model of Haasen, Alexander and coworkers is modified (extended) to include evolution of coefficients in the definition of internal stress. These account for an evolving dislocation substructure. The resulting model is applied in a continuum slip framework to allow multiple slip orientations. Slip system interaction rules are adapted to include slip system interaction for multiple slip conditions and to suppress secondary slip and dislocation density generation for single slip orientations. The approach is discussed relative to other models for viscoplasticity of single crystals and is examined in the context of thermodynamics with internal state variables. The framework is used to correlate experimental data from compression tests of single crystals of the compound semiconductor CdTe from room temperature to near the melting point. Sensitivity of the model to uncertainties such as initial dislocation density is explored.     

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.     

A modern look on yield stress fluids

A concept of viscoplasticity advanced exactly one century ago by Bingham appears very fruitful because there are many natural and artificial materials that demonstrate viscoplastic behavior, i.e., they are able to pass from a solid to a liquid state under the influence of applied stress. However, although this transition was originally considered as a jump-like phenomenon occurring at a certain stress—the yield stress—numerous subsequent studies have shown that the real situation is more complicated. A long-term discussion about the possibility of flow at low stresses less than the yield stress came to today’s conclusion denying this possibility as being opposite to the existence of the maximal Newtonian viscosity in viscoelastic polymeric fluids. So, there is a contradiction between the central dogma of rheology which says that “everything flows” and the alleged impossibility for flow at a solid-like state of viscoplastic fluids. Then, the concept of the fragile destruction of an inner structure responsible for a solid-like state at the definite (yield) stress was replaced by an understanding of the yielding as a transition extending over some stress range and occurring in time. So, instead of the yield stress, yielding is characterized by the dependence of durability (or time-to-break) on the applied stress. In this review, experimental facts and the new understanding of yielding as a kinetic process are discussed. Besides, some other alternative methods for measuring the yield stress are considered.     

A generalized coupled viscoplastic-viscodamage-viscohealing theory for glassy polymers

A thermodynamic consistent, small-strain, non-unified model is developed to capture the irregular rate dependency included in the strain controlled inelastic responses of polymers at the glassy state. The model is considered as a generalized Frederick-Armstrong-Philips-Chaboche (FAPC) theory proposed by [Voyiadjis and Basuroychowdhury, 1998] and [Voyiadjis and Abu Al-Rub, 2003] which is based on a von Mises and Chaboche isotropic hardening type viscoplasticity formulation. Using the proposed model, different experimental results are simulated and the range of viscoplastic related material constants are obtained through a parametric study. The thermodynamic framework is used to incorporate the effect of coupling between viscodamage and viscohealing phenomena into the inelastic deformation of glassy polymers. This coupling effect is crucial for polymeric based self healing systems in which different damage mechanisms are active and the efficiency of the healing processes are highly dependent on the damage. The computational aspect for general coupled inelastic-damage-healing processes together with the required solution algorithms are elaborated and the inelastic-damage-healing response of a polymeric based self-healing system is simulated. The proposed viscoplasticity theory constitutes a physically consistent approach to model the irregular mechanical responses of glassy polymers and the viscodamage model provides an exquisite predicting tool to evaluate the ductile damage associated with the large inelastic deformation and low cycle fatigue in polymeric based material systems. In conclusion, a well structured viscohealing theory is formulated for polymeric based self healing systems.     

Viscoplastic approach for rate-dependent failure analysis of concrete joints and interfaces

In this work, a new rate-dependent interface model for computational analysis of quasi-brittle materials like concrete is presented. The model is formulated on the basis of the inviscid elastoplastic model by [Carol, I., Prat, P.C., López, C.M., 1997. “A normal/shear cracking model. Interface implementation for discrete analysis”. Journal of Engineering Mechanics, ASCE, 123 (8), pp. 765–773.]. The rate-dependent extension follows the continuous form of the classical viscoplastic theory by [Perzyna, P., 1966. “Fundamental problems in viscoplasticity”. Advances in Applied Mechanics, 9, pp. 244–368.]. According to [Ponthot, J.P., 1995. “Radial return extensions for viscoplasticity and lubricated friction”. In: Proceedings of International Conference on Structural Mechanics and Reactor Technology SMIRT-13, Porto Alegre, Brazil, (2), pp. 711–722.] and [Etse, G., Carosio, A., 2002. “Diffuse and localized failure predictions of Perzyna viscoplastic models for cohesive-frictional materials”. Latin American Applied Research (32), pp. 21–31.] it includes a consistency parameter and a generalized yield condition for the viscoplastic range that allows an straightforward extension of the full backward Euler method for viscoplastic materials. This approach improves the accuracy and stability of the numerical solution. The model predictions are tested against experimental results on mortar and concrete specimens that cover different stress paths at different strain rates. The results in this work demonstrate, on one hand, the capabilities of the proposed elasto–viscoplastic interface constitutive formulation to predict the rate-dependency of mortar and concrete failure behavior, and, on the other hand, the efficiency of the numerical algorithms developed for the computational implementation of the model that include the consistent tangent operator to improve the convergence rate at the finite element level.     

Novel variational formulations for nonlocal plasticity

A nonlocal structural model of softening plasticity is considered in the framework of the internal variable theories of inelastic behaviours of associative type. The finite-step nonlocal structural problem in a geometrically linear range is formulated according to a backward difference scheme for time integration of the flow rule. The related finite-step variational formulation in the complete set of local and nonlocal state variables is recovered. A family of mixed nonlocal variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. The specialization of a mixed variational formulation to existing nonlocal models of softening plasticity, assuming both linear and nonlinear constitutive behaviour, is provided to show the effectiveness of the theory.     

Material instabilities in inelastic saturated porous media under dynamic loadings

Instabilities in inelastic saturated porous media are investigated here for general three-dimensional states under dynamic loadings using a perturbation approach.Under quasi-static conditions, unbounded growth of perturbations is related to the emergence of stationary discontinuities under drained or undrained conditions, while under dynamic conditions, unbounded growth is related either to the emergence of stationary discontinuities (and these are set by drained conditions) or to the appearance of the flutter phenomenon (acceleration waves).For associative behaviour unbounded growth always corresponds to localization under drained conditions and the onset of growth of perturbations occurs here only through divergence growth. It is only for non-associative flow that unbounded growth may correspond to undrained localization in quasi-static conditions and to flutter under dynamic conditions.     

Viscoplasticity with a saturation stress: distinguishing features of the model

Viscoplastic models including a saturation stress are considered. The existence of the saturation stress significantly changes the mathematical structure of solutions near maximum friction surfaces (surfaces where the friction stress is equal to the local shear yield stress). The main features of solutions based on such theories are: (a) sliding must occur at the maximum friction surfaces under certain conditions, (b) the velocity field may be singular in the vicinity of maximum friction surfaces. The objective of the present paper is to study these features of solutions. The mathematical structure obtained is considered to be advantageous for a class of materials and may lead to a convergence of viscoplastic solutions to the corresponding rigid perfectly plastic solutions. It seems that the latter is of importance for the construction of a unified theory that could describe the material behavior in the range from rate-independent plasticity to viscoplasticity. In the present paper, the study of the main features of the model is based on the exact closed-form solution to the problem of flow between two coaxial rotating cylinders. In the case of sliding, in addition to the aforementioned features, the asymptotic behavior of the velocity field in the vicinity of the maximum friction surface is found for a class of constitutive laws.     

Coupled viscoplasticity damage constitutive model for concrete materials

A coupled viscoplasticity damage constitutive model for concrete materials is developed within the framework of irreversible thermodynamics.Simultaneously the Helmholtz free energy function and a non-associated flow potential function are given, which include the internal variables of kinematic hardening,isotropic hardening and dam- age.Results from the numerical simulation show that the model presented can describe the deformation properties of the concrete without the formal hypotheses of yield criterion and failure criteria,such as the volume dilatancy under the compression,strain-rate sen- sitivity,stiffness degradation and stress-softening behavior beyond the peak stress which are brought by damages and fractures.Moreover,we could benefit from the application of the finite element method based on this model under complex loading because of not having to choose different constitutive models based on the deformation level.     

Lattice Boltzmann simulations of viscoplastic fluid flows through complex flow channels

We present the results of lattice Boltzmann (LB) simulations for the planar-flow of viscoplastic fluids through complex flow channels. In this study, the Bingham and Casson model fluids are covered as viscoplastic fluid. The Papanastasiou (modified Bingham) model and the modified Casson model are employed in our LB simulations. The Bingham number is an essential physical parameter when considering viscoplastic fluid flows and the modified Bingham number is proposed for modified viscoplastic models. When the value of the modified Bingham number agrees with that of the “normal” Bingham number, viscoplastic fluid flows formulated by modified viscoplastic models strictly reproduce the flow behavior of the ideal viscoplastic fluids. LB simulations are extensively performed for viscoplastic fluid flows through complex flow channels with rectangular and circular obstacles. It is shown that the LB method (LBM) allows us to successfully compute the flow behavior of viscoplastic fluids in various complicated-flow channels with rectangular and circular obstacles. For even low Re and high Bn numbers corresponding to plastic-property dominant condition, it is clearly manifested that the viscosity for both the viscoplastic fluids is largely decreased around solid obstacles. Also, it is shown that the viscosity profile is quite different between both the viscoplastic fluids due to the inherent nature of the models. The viscosity of the Bingham fluid sharply drops down close to the plastic viscosity, whereas the viscosity of the Casson fluid does not rapidly fall. From this study, it is demonstrated that the LBM can be also an effective methodology for computing viscoplastic fluid flows through complex channels including circular obstacles.     

Modelling cell wall growth using a fibre-reinforced hyperelastic–viscoplastic constitutive law

A fibre-reinforced hyperelastic–viscoplastic model using a finite strain Finite Element (FE) analysis is presented to study the expansive growth of cell walls. Based on the connections between biological concepts and plasticity theory, e.g. wall-loosening and plastic yield, wall-stiffening and plastic hardening, the modelling of cell wall growth is established within a framework of anisotropic viscoplasticity aiming to represent the corresponding biology-controlled behaviour of a cell wall. In order to model in vivo growth, special attention is paid to the differences between a living cell and an isolated wall. The proposed hyperelastic–viscoplastic theory provides a unique framework to clarify the interplay between cellulose microfibrils and cell wall matrix and how this interplay regulates sustainable growth in a particular direction while maintaining the mechanical strength of the cell walls by new material deposition. Moreover, the effect of temperature is taken into account. A numerical scheme is suggested and FE case studies are presented and compared with experimental data.     

On the structure of the theory of viscoplasticity

In various formulations of plasticity, there is evident a structure embracing several features, including inviscidity, a yield condition, and a constitutive inequality. By means of these features the constitutive equations of plasticity are derived. In the present paper we introduce a viscoplastic counterpart of the constitutive inequality of plasticity, and we consider its physical significance. We also present a theory of viscoplasticity having a structure similar to that of plasticity and its relation with the Hohenemser-Prager prototype of viscoplastic constitutive relations is considered.     

Mechanical characterisation of a viscoplastic material sensitive to hydrostatic pressure

The present paper deals with the characterisation of the static mechanical behaviour of an energetic material all along its lifespan. The material behaviour is viscoplastic, damageable and sensitive to hydrostatic pressure. For such materials, existing models have generally been developed in the framework of transient dynamic behaviour. These models are not suitable for a static study. Therefore a specific experimental protocol and an associated model are developed. Characterisation is derived from both uniaxial compressive, tensile tests and triaxial tests. Plastic behaviour is described by means of a parabolic yield criterion and a new hardening law. Non-associated plastic flow rule and isotropic damage complete the model. The performance of the model is illustrated through the simulation of various loading paths.