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Shear localization is a fundamental and widely observed non-equilibrium phenomenon in metallic materials subjected to impulsive loadings. Despite extensive research over several decades, the evolution of shear bands still remains poorly understood. In this study, we propose a two-stage model to describe the evolution of thermoplastic shear band, where the effects of strain hardening, strain-rate hardening, and thermal softening are involved. Our analysis reveals that the growth of thermoplastic shear band is derived by the competition between momentum diffusion and thermal diffusion processes, which is controlled by the solid-state equivalent of the Prandtl number. By incorporating these factors into our model, we find that the presence of hardening effect retards the evolution of shear bands, resulting in a wider shear band and increased dissipation of energy. Theoretical calculation results of shear bandwidth, shear band strain, propagation velocity, process zone geometry and dimensions, critical dissipation energy, shear band toughness, and shear band spacing are well consistent with available experimental data. Additionally, we explore the potential transition to turbulence controlled by Reynolds number within shear bands and investigate the influence of the Taylor-Quinney coefficient on the evolution of shear bands. These findings are expected to offer valuable insights into the mechanism of shear band evolution.

An extension of the dynamic Smgarorinsky model (DSM) is presented. In the so-called dynamic mixed model (DMM), the Leonard stress (i.e., the resolved part of the Reynolds stress) is no longer parameterized in the test filter range by the Smagorinsky model. Instead, it is modeled by the so-called similarity model, which assumes close self-similarity of the resolved and unresolved flow. This implementation is derived first in general and then specifically for spectral models. In the latter case, spectral cut-off filters allow a simple reformulation of the basic tensor equation of the DSM, requiring only the modification of a particular term to obtain the DMM version of the turbulence model.The DMM is then examined in the Kühlungsborn Mechanistic general Circulation Model (KMCM), a spectral CGM, where, starting from the same initial condition, the flow fields are compared in a model run with the DMM and in a control run with the DSM on day zero and day eight. Although most of the results are comparable, there is, already at day zero, a significant difference at the poles: with the DSM, there are certain locations with large values of the Smagorinsky parameter that are not present in the DMM run. This can be explained mathematically by the fact that the convergence of the associated Legendre polynomials in the spectral representation is worse at the poles. Therefore, numerical errors can occur in the spectral transformation, which cancel out in the DMM due to its formulation, allowing the increase of the time steps.

A discussion of different approaches and solutions of the basic tensor equation within the Dynamic Smagorinsky Model (DSM) suitable for General Circulation Models (GCM) is presented. Particular interest is dedicated to the relationship between various approaches (i.e., the specific formulation of the tensor equation), namely a least-square approach, a time lag approach, and a simple tensor contraction approach, and the impact of the specific solution (i.e., how to solve the equation) on the Smagorinsky parameter cS2$c_S^2$. In addition to the standard solutions, clipped solutions, absolute solutions, and tensor norm solutions are examined.The numerical results are based on calculations from a general circulation model, where the different approaches are applied to provide the turbulent horizontal momentum diffusion. Here, they are examined with focus on two issues: 1) At the beginning of the simulations, the different choices for the tensor equation result in different values for the locally distributed and zonally averaged values of the Smagorinsky parameter. These values show that for the standard solutions almost half of the values of cS2$c_S^2$ are negative, in accordance with known results from isotropic turbulence and leads to unstable simulations. In addition, the tensor norm is related to the absolute solution via the Cauchy-Schwarz inequality. 2) As the simulations proceed, the differences of the Smagorinsky parameter values diminish except for the tensor norm solutions while evolving to a stationary state in a process of self-organization such that they form a group with values comparable to isotropic three-dimensional simulations. In summary, the least-squares and time lag approaches provide reasonable results, while the simple contraction approach fluctuates more. For the solutions, it is discussed whether the clipped or the tensor norm solution is more reasonable.

A generalized formulation of the Dynamic Smagorinsky Model (DSM) is proposed as a versatile turbulent momentum diffusion scheme for Large-Eddy Simulations. The difference to previous versions of the DSM is a modified test filter range that can be chosen independently from the resolution scale to separate the impact of the test filter on the simulated flow from the impact of the resolution. The generalized DSM (gDSM) in a two-dimensional version is validated in a verification study as a horizontal momentum diffusion scheme with the Kühlungsborn Mechanistic General Circulation Model at high resolution (wavenumber 330) without hyperdiffusion. Three-day averaged results applying three different test filters in the macro-turbulent inertial range are presented and compared with analogous simulations where the standard DSM is used instead. The comparison of the different filters results in all cases in similar globally averaged Smagorinsky parameters cS≃0.35$c_S\simeq0.35$ and horizontal kinetic energy spectra. Hence, the basic assumption of scale invariance underlying the application of the gDSM to parameterize atmospheric turbulence is justified. In addition, the smallest resolved scales contain less energy when the gDSM is applied, thus increasing the stability of the simulation.

This study proposes a semi-analytic approximation to the laminar boundary layer growth in a polarized pressure field with temperature gradient represented by the joint Blasius-energy equation. We illuminate that is a probability density function (PDF) approximated by an amended Gaussian PDF with zero mean and standard deviation . This implies a diffusive structure for the molecular momentum conversion as well as the energy flux in the boundary layer. A new limit for the boundary layer edge is also presented. Results suggest an augmented boundary layer when compared to accepted values in the literature. We also reproduce the inverse proportionality of the free stream velocity to the diffusion of both momentum and energy.

Transport and the spread of heat in Hamiltonian one dimensional momentum conserving nonlinear systems is commonly thought to proceed anomalously. Notable exceptions, however, do exist of which the coupled rotator model is a prominent case. Therefore, the quest arises to identify the origin of manifest anomalous energy and momentum transport in those low dimensional systems. We develop the theory for both, the statistical densities for momentum- and energy-spread and particularly its momentum-/heat-diffusion behavior, as well as its corresponding momentum/heat transport features. We demonstrate that the second temporal derivative of the mean squared deviation of the momentum spread is proportional to the equilibrium correlation of the total momentum flux. Subtracting the part which corresponds to a ballistic momentum spread relates (via this integrated, subleading momentum flux correlation) to an effective viscosity, or equivalently, to the underlying momentum diffusivity. We next put forward the intriguing hypothesis: normal spread of this so adjusted excess momentum density causes normal energy spread and alike normal heat transport (Fourier Law). Its corollary being that an anomalous, superdiffusive broadening of this adjusted excess momentum density in turn implies an anomalous energy spread and correspondingly anomalous, superdiffusive heat transport. This hypothesis is successfully corroborated within extensive molecular dynamics simulations over large extended time scales. Our numerical validation of the hypothesis involves four distinct archetype classes of nonlinear pair-interaction potentials: (i) a globally bounded pair interaction (the noted coupled rotator model), (ii) unbounded interactions acting at large distances (the coupled rotator model amended with harmonic pair interactions), (iii) the case of a hard point gas with unbounded square-well interactions and (iv) a pair interaction potential being unbounded at short distances while displaying an asymptotic free part (Lennard–Jones model). We compare our findings with recent predictions obtained from nonlinear fluctuating hydrodynamics theory.