A non-equilibrium thermodynamic model for viscoplasticity and damage: Two temperatures and a generalized fluctuation relation
Introduction
Extensive industrial utilization of metals and metallic components calls for predictive constitutive models to facilitate an understanding of their behaviour under a wide range of thermal and mechanical loading conditions encountered in myriad applications or response scenarios, e.g. metal forming processes (rolling, drawing, extrusion etc.), high speed machining, high velocity impact, penetration, shear/damage band formation and so on. During these processes, metals typically undergo large plastic deformation with accumulation of damage, culminating in ductile fracture. Consequently, constitutive modelling for such processes must account for plastic deformation, damage as well as fracture, and only a rational coupling of these three may lead to an accurate prediction of the response. Standalone modelling of each of these processes has by itself many intricacies and challenges. For example, a constitutive description of viscoplastic deformation demands an understanding of strain hardening, loading rate sensitivity, effect of temperature, consequences of different physical processes (e.g., motion of dislocations and their interaction, effect of twinning, dynamic recovery, recrystallization) etc. Besides, the modeller has also to grapple with the far-from-equilibrium character of these processes. Spurred by the industrial need, considerable research effort has gone in modelling the mechanical behaviour of metals. This has led to many material models, wherein phenomenology has played a leading role in conforming with the observed stress-strain data. Phenomenological constants, which merely fit the data and yet are often regarded as material parameters, do not typically carry much physical meaning and are hard to determine from experimental observations or small-scale (e.g., molecular dynamics) simulations. For viscoplastic deformation, a popular phenomenological model is due to Johnson and Cook (1983) and it is widely used because of its simplicity and readily available material parameters for several metals of industrial significance. However, the grossly simplified description of the hardening behaviour fails to capture material response accurately over a wide range of strain rate and temperature. An interesting experimental observation, described below, may exemplify the limitations that inhere in many phenomenological plasticity models including Johnson and Cook (1983). The dynamical response of oxygen-free high thermal conductivity (OFHC) copper, when plotted as stress against strain rate at a given strain (specifically at low values of strain), shows an abrupt upturn around the strain rate of s (see Chowdhury et al., 2017a). This extreme strain-rate sensitivity remains unaccounted in most phenomenological models, resulting in erroneous prediction of the stress-strain behaviour. Similarly, an array of ductile damage models, mostly phenomenological and coupling damage with viscoplasticity, has been designed based on the principles of continuum damage mechanics (see for example Lemaitre and Desmorat, 2005; de Sciarra, 2012).
With considerable experimental and theoretical advances on the understanding of material response, it has become possible, in principle, to unravel the specific intricacies of microscopic processes that affect the macroscopic inelastic deformation including viscoplasticity, damage and fracture in metals. On these lines, not a few studies, focussing on physics/micro-mechanics based modelling of viscoplasticity and damage in metals, are available. Thus, it is now established that the macroscopic viscoplastic response in metals is primarily due to the storage and motion of dislocations and much of ductile damage occurs owing to void nucleation, growth and coalescence. We refer to Zerilli and Armstrong (1987); Follansbee and Kocks (1988); Estrin (1996); Voyiadjis and Abed (2005); Gao and Zhang (2012); Krasnikov et al. (2011); Oppedal et al. (2012); Knezevic et al. (2013); Kitayama et al. (2013); Khan and Liu (2016); Djordjevic et al. (2018) for examples of physically motivated models of viscoplastic deformation and Gurson et al. (1977); Tvergaard (1981); Tvergaard and Needleman (1984); Nahshon and Hutchinson (2008); Brünig et al. (2013); Shojaei et al. (2013); Nguyen et al. (2017) for micro-mechanics based damage models. Further, for extreme strain rate response as discussed in the previous paragraph, there are a few physically motivated models of recent origin, e.g. Preston et al. (2003); Austin and McDowell (2011); Hunter and Preston (2015); Luscher et al. (2016). Apart from a more accurate prediction of the constitutive behaviour, these models also align with better design practices wherein parameters with clearer physical meaning admit easier tuning to fit a broad range of experimental data. A concern is however on the thermodynamic consistency of some of these physically guided models – an aspect that remains inadequately disambiguated. Accordingly, even with their predictions validated against certain experimental observations, these models lack a measure of scientific credibility.
While the studies referenced above attempt to model some intricacy or the other in the responses of ductile material, e.g. texture evolution, dislocation-twinning interaction, plasticity of ultra-fine grained metals, effect of shock waves, they generally do not trace these effects to the statistical aspects of the underlying defects. This is true even though some of these studies may very well conform to the laws of thermodynamics, as they are usually stated. Tracing the macroscopic deformation to the statistics of the defect microstructure may not only shed light on the micro-mechanisms, but lead to a consistent derivation of defect kinetics. Keeping this aspect in view, a thermodynamic framework of two temperatures, viz. configurational and kinetic-vibrational, have recently been exploited in Langer et al. (2010) and Chowdhury et al. (2017a, b, 2018) to model viscoplastic response in metals. Rather than using macroscopic phenomenological states and constitutive relations, these models include dislocation density as an internal state variable. Several other relations of microscopic origin, e.g. Orowan's law of equivalent plastic strain rate, Taylor stress etc., also aid in tracing viscoplasticity to its physical roots. A key feature in these models is the notion of configurational entropy arising out of dislocation rearrangement within the material. Chowdhury et al. (2016) has furnished a simplified derivation of this entropy using a statistical mechanics approach. Configurational entropy together with the associated energy defines a configurational temperature that critically influences the dislocation density evolution. In Langer et al. (2010) and Chowdhury et al. (2017a), evolution laws of dislocation density, effective plastic strain rate, two temperatures are derived consistently with the first and second laws of thermodynamics. The fluctuation relation, a more general principle than the second law, is used in Chowdhury et al. (2016) to arrive at the evolution equations of the state variables. These models are shown to faithfully reproduce the experimental response. While Langer et al. (2010) and Chowdhury et al. (2016) report numerical simulations of response under homogeneous deformation only, Chowdhury et al. (2017a) has established the predictive quality of the model for both homogeneous and inhomogeneous dynamic elasto-viscoplastic deformation. A similar framework has been more recently adopted by Das et al. (2018) to model viscoplastic response of thermoplastics.
These two-temperature models are so far limited to describing viscoplasticity only. They are thus inapplicable to model many processes of engineering interest, as discussed above, where viscoplastic deformation is coupled to material degradation via void and micro-crack accumulation. Our present aim is therefore to exploit a similar framework to include material degradation through void formation and micro-cracking. Accumulation of voids and micro-cracks, and their rearrangements in the mesoscale produce configurational entropy and therefore alter its expression. We modify the configuration counting procedure reported in Chowdhury et al. (2016) to take care of this and arrive at an expression for configurational entropy in metals with voids, micro-cracks and dislocations. The twin laws of thermodynamics then guide the derivation of evolution equations for void volume fraction, micro-crack density, dislocation density etc. We focus on an understanding of the elastic strain energy, energy of dislocation and dissipated energy due to formation of voids, micro-cracks and plastic deformation, which is critical in framing the constitutive relations and closing the evolution equations. In the absence of plastic deformation, our model may also be used for brittle damage, typically caused by micro-crack formation and growth. The model performance is numerically assessed through demonstrations with a limited set of simulations on brittle and ductile damage.
Yet another important issue that we intend to clarify is about the splitting of energy and entropy of the thermodynamic system. In previous works, e.g. Langer et al. (2010); Chowdhury et al. (2017a) for metals and Kamrin and Bouchbinder (2014) for amorphous materials, energy and entropy are additively split into two components, one caused by kinetic-vibrational motion of atoms and other by the configurational changes (elastic stretching, defect motion etc.) that occur during the process of deformation. The system itself is thus split into two weakly coupled subsystems – kinetic-vibrational and configurational. Subsequently the notion of two temperatures are introduced based on this splitting of energy and entropy. However, this formalism appears to restrict the applicability of the model only to certain class of materials and responses, e.g. materials with temperature insensitive constitutive parameters, isothermal response. Processes as common as thermo-elasticity cannot be modelled within this scheme. We show that it is not necessary to introduce the splitting a-priori to obtain a thermodynamically consistent theory of plasticity and damage. For completeness, we prescribe a recipe to obtain the correct splitting of energy and entropy verifying the notions of temperature as partial derivative of energy with respect to entropy. Our proposal thus allows for modelling strongly coupled subsystems and not just weakly coupled ones, only for which earlier theories are valid.
While our formulation exploiting the second law modified for the two-temperature system, may be applied over a broad range of strain rates, we do not expect it to be adequate for extremely high strain-rates, for which we present an important modification of the theory based on a fluctuation relation (Seifert, 2005). This modification follows from a generalization of our earlier work on the fluctuation relation applied to thermo-viscoplasticity of metals (Chowdhury et al., 2016). It results is an entirely new procedure for constitutive closure whilst providing valuable insights into short time-scale aspects of the response, e.g. the emergent pseudo-inertial effect in the thermodynamic evolution of states.
We organize the rest of the article as follows. In sections 2 Kinematics, 3 Balance of mass, linear and angular momenta, we briefly describe the kinematics and equations of motion for the deforming continuum. Following this, in section 4 we define the thermodynamic system describing inelastic deformation of the material with defect microstructure comprising of dislocations, micro-voids, micro-cracks etc. The most important contribution of this section is the derivation of an expression of configurational entropy using notions of statistical mechanics. Subsequently, in section 5, we propose the constitutive relations and derive the evolution equations for the internal variables, viz. dislocation density, void volume fraction etc. The special case of brittle damage is also highlighted here. Section 6 presents numerical simulations on brittle and ductile damage. The modified theory based on a generalized fluctuation relation is presented in section 7 before concluding the work in section 8.
Section snippets
Kinematics
Let the body occupying a region in its reference configuration at time deform via processes of elasto-viscoplasticity and damage to attain its spatial configuration at time . Each material point thus deforms to through a smooth invertible deformation map as . The spatial displacement (), velocity (), deformation gradient () and velocity gradient () fields are defined as: , , and , where , and superposed dot denote
Balance of mass, linear and angular momenta
Laws of balance of mass, linear and angular momenta are essential ingredients in continuum modelling. Denoting by density of the matrix material in the reference configuration and by , , densities of the body in the reference, intermediate and current configurations respectively, balance of mass is equivalent to the following equation.Here, is the volume element in the current configuration obtained by deforming under . The reference volume
Thermodynamic system
In arriving at our model for ductile damage, we primarily focus on a system undergoing elasto-viscoplastic deformation along with accumulation of voids and matrix micro-cracking under the influence of externally applied load. The formulation on brittle damage is then obtained as a special case.
In realizing our aim, the thermodynamic description should include state variables describing defect motion/rearrangement, lattice distortion as well as thermal vibration. We choose as the primary state
Constitutive relation
Constitutive closure to the equations of motion is typically accomplished through a stress-strain relationship for a given material. Presently, we base our constitutive theory on the thermodynamic system described in section 4. The goal is to express in terms of the kinematic quantities, internal state variables and material parameters. This exercise also involves defining evolution equations for the dislocation density ρ, total void volume fraction , volume fractions of nucleated voids
Numerical simulations
Using a chosen set of numerical simulations, our aim is to assess the model predictions for both homogeneous and inhomogeneous deformation. While simulations corresponding to homogeneous deformation are done in Matlab®, those for the inhomogeneous case are accomplished by incorporating the proposed material model within the commercial finite element (FE) software ABAQUS 6.14® exploiting the user-defined material subroutine (VUMAT). Simulation of the homogeneous response effectively reduces to
Deformation with extreme strain rate: a stochastic thermodynamics route
Whilst conforming with the two laws of thermodynamics, the theory presented thus far may be inadequate for modelling deformation under extremely high strain rates, wherein finite response variations could occur over very finely resolved time and space scales. This phenomenon may manifest itself in an apparent lack of smoothness of the evolving states, warranting a possible reformulation of the theory based on fluctuation relations and stochastic thermodynamics (Chowdhury et al., 2016). In this
Conclusion
This work is an initial attempt at developing a rationally grounded ductile damage model combining certain physically relevant features of plasticity and material degradation with the principles of non-equilibrium thermodynamics. Specifically, using the notion of a second effective temperature, herein called the configurational temperature, and an expression for configurational entropy due to voids, micro-cracks and dislocations, our model obtains coupled evolution equations for plasticity and
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