A peridynamic theory for linear elastic shells

A state-based peridynamic formulation for linear elastic shells is presented. The emphasis is on introducing, possibly for the first time, a general surface based peridynamic model to represent the deformation characteristics of structures that have one physical dimension much smaller than the other two. A new notion of curved bonds is exploited to cater for force transfer between the peridynamic particles describing the shell. Starting with the three dimensional force and deformation states, appropriate surface based force, moment and several deformation states are arrived at. Upon application on the curved bonds, such states beget the necessary force and deformation vectors governing the motion of the shell. Correctness of our proposal on the peridynamic shell theory is numerically assessed against static deformation of spherical and cylindrical shells and flat plates

Internal noise driven generalized Langevin equation from a nonlocal continuum model

Starting with a micropolar formulation, known to account for nonlocal microstructural effects at the continuum level, a generalized Langevin equation (GLE) for a particle, describing the predominant motion of a localized region through a single displacement degree-of-freedom (DOF), is derived. The GLE features a memory dependent multiplicative or internal noise, which appears upon recognising that the micro-rotation variables possess randomness owing to an uncertainty principle. Unlike its classical version, the new GLE qualitatively reproduces the experimentally measured fluctuations in the steady-state mean square displacement of scattering centers in a polyvinyl alcohol slab. The origin of the fluctuations is traced to nonlocal spatial interactions within the continuum. A constraint equation, similar to a fluctuation dissipation theorem (FDT), is shown to statistically relate the internal noise to the other parameters in the GLE

A non-equilibrium thermodynamic model for viscoplasticity and damage: Two temperatures and a generalized fluctuation relation

Utilizing a non-equilibrium thermodynamic setting that involves two temperatures, we present a model for ductile and brittle damage. The thermodynamic system consists of two interacting subsystems configurational and kinetic-vibrational. While the kinetic-vibrational subsystem describes fast degrees-of-freedom (DOFs) of ordinary thermal vibration, the configurational subsystem includes the slower DOFs pertaining to a slew of configurational rearrangements that characterize elasto-visco-plasticity and damage, e.g. dislocation motion, lattice stretching, void nucleation, void growth and micro-crack formation. Following statistical mechanics, an expression for the entropy of a plastically deforming metal with growing voids and micro-cracks is derived. Subsequent application of the first and second laws of thermodynamics, suitably modified for the two-temperature system, yields coupled evolution rules for dislocation density, void volume fraction, micro-crack density etc. A modified flow rule for dilatant plasticity and evolution equations for the two temperatures are also derived. Even when the two subsystems are strongly coupled, we show that a splitting of energy and entropy is feasible and that the notion of two temperatures conforms with such splitting. We conduct numerical experiments on both brittle and ductile damage to assess the predictive features of the model and validate the results against available experimental evidence. Finally, a generalized fluctuation relation is put forth for deformations with extremely high strain rates. This leads to an entirely new procedure for constitutive closure, providing valuable insights into the emergent pseudo-inertial aspects of the evolving thermodynamic states

A constitutive model for thermoplastics based on two temperatures

Posed within a two-temperature thermodynamic framework, our aim is to propose a unified glass-rubber constitutive model for thermo-rheologically simple thermoplastic polymers. This modelling set-up usually applies to phenomena wherein sub-macroscopic processes involving different time scales occur and accordingly the thermodynamic system may be interpreted as comprising of two subsystems. The configurational subsystem contains the slower states, while the kinetic-vibrational subsystem comprises of the faster moving states. The two subsystems fail to equilibrate within experimental timescales in the glassy regime (low temperature or high strain rate) due to low structural relaxation rates. As transition to the rubbery regime commences at temperatures higher than glass transition or at sufficiently low loading rates, the two subsystems equilibrate within microscopic timescales. The model exploits physically inspired prescriptions for the free energies due to different underlying mechanisms-elastic stretching, localised shear transformations and infra-molecular straightening of chains. A simple temperature dependent formulation for structural relaxation in terms of heat transfer between the subsystems is used to capture transition between these mechanisms. The model is then validated against experimental results of uniaxial compression tests for various strain rates and temperatures establishing its ability to seamlessly transit between the glassy and rubbery regimes. Also demonstrated is the model's efficacy in capturing the key features of physical ageing and mechanical rejuvenation

A modified peridynamics correspondence principle: Removal of zero-energy deformation and other implications

We look for an enhancement of the correspondence model of peridynamics, emphasizing the elimination of zero-energy deformation modes. We propose an approach based on the notion of sub-horizons. The most useful feature of this proposal is the setup which, whilst providing solutions with the necessary stability, deviates only marginally from the original correspondence formulation. A thorough analysis of the sub-horizon based method is furnished based on the well-posedness of integral equations and energy spectrum, which clearly demonstrate a removal of zero energy modes. We also show how other forms of unphysical deformation modes, e.g. material collapse within horizon, jump discontinuities and vanishing energy modes, can be prevented with the present proposal. Finally, a set of numerical simulations are undertaken that attest to the remarkable efficacy of the sub-horizon based approach. (C) 2018 Elsevier B.V. All rights reserved

Fluctuation relation based continuum model for thermoviscoplasticity in metals

A continuum plasticity model for metals is presented from considerations of non-equilibrium thermodynamics. Of specific interest is the application of a fluctuation relation that subsumes the second law of thermodynamics en route to deriving the evolution equations for the internal state variables. The modelling itself is accomplished in a two-temperature framework that appears naturally by considering the thermodynamic system to be composed of two weakly interacting subsystems, viz. a kinetic vibrational subsystem corresponding to the atomic lattice vibrations and a configurational subsystem of the slower degrees of freedom describing the motion of defects in a plastically deforming metal. An apparently physical nature of the present model derives upon considering the dislocation density, which characterizes the configurational subsystem, as a state variable. Unlike the usual constitutive modelling aided by the second law of thermodynamics that merely provides a guideline to select the admissible (though possibly non-unique) processes, the present formalism strictly determines the process or the evolution equations for the thermodynamic states while including the effect of fluctuations. The continuum model accommodates finite deformation and describes plastic deformation in a yield-free setup. The theory here is essentially limited to face-centered cubic metals modelled with a single dislocation density as the internal variable. Limited numerical simulations are presented with validation against relevant experimental data. (C) 2016 Elsevier Ltd. All rights reserved