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 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

Non-classical mechanics and thermodynamics for continuum modelling of solids

This thesis dwells upon several aspects of continuum mechanics and thermodynamics to model elastic and inelastic response of solids. Broadly, the work presented may be categorized into two parts{ one focusing on the development of generalized continuum description in the context of elastic materials and the other on the thermodynamics of dissipative processes whilst considering, in some detail, the physics of deformation too. The first part begins with the proposal for a state-based micropolar peridynamic theory for linear elastic solids. The main motivation is to introduce additional micro-rotational degrees of freedom to each material point and thus naturally bring in the physically relevant material length scale parameters into peridynamics. Non-ordinary type modelling via constitutive correspondence is adopted here to de ne the micropolar peridynamic material. Along with a general three-dimensional model, homogenized one dimensional Timoshenko type beam models for both the proposed micropolar and the standard non-polar peridynamic variants are derived. The efficacy of the proposed models in analyzing continua with length scale effects is established via numerical simulations of a few beam and plane-stress problems. Continuing with our e ort in developing homogenized reduced dimensional models, a state-based peridynamic formulation for linear elastic shells is presented next. The emphasis is on introducing, perhaps for the first time, a general surface based peridynamic model to represent the deformation characteristics of structures that have one geometric dimension much smaller than the other two. A new notion of curved bonds is exploited to model 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 yield the necessary force and deformation vectors governing the motion of the shell. The peridynamic shell theory is numerically assessed against simulations on static deformation of spherical and cylindrical shells and those on at plates. As a transition to the second part of the thesis, our next work shares features of the first part (micropolarity and homogenization) as well as the second (equation with viscous force, i.e., dissipative process). 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, 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. In the second part of the thesis, a series of physically motivated models for dislocation mediated thermoviscoplastic deformation and micro-void mediated ductile damage in metals are proposed. The methodology of modelling brittle damage also constitutes another part of discussion. The models are, in essence, posited in the framework of internal-variables theory of thermodynamics, wherein effective dislocation densities, void volume fractions etc., which assume the role of internal variables, track permanent changes in the internal structure of metals undergoing plastic deformation and damage. The thermodynamic formulation involves a two-temperature description of viscoplasticity and damage that appears naturally if one considers the thermodynamic system to be composed of two weakly interacting subsystems, namely, a kinetic- vibrational subsystem of the vibrating atomic lattices and a con gurational subsystem of the slower degrees-of-freedom of defect motion. While most of the models are proposed satisfying the thermodynamic requirements asserted by the second law, one specific interest, however, has been to explore the possible application of a fluctuation relation that subsumes the second law of thermodynamics en route to deriving the evolution equations for the internal variables. Full- edged three-dimensional continuum formulations, valid for the finite deformation regime, are also set forth. Several numerical exercises, including impact dynamic simulations, are carried out and validated against experimental data

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