CONTRIBUTIONS OF THEODORE VON KARMAN TO APPLIED MECHANICS
Major scientific contributions of theodore von karman, including turbulence studies, plastic, deformation, supersonic phenomena, aerodynamics, and the buckling of shells
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Results for “CLASSICAL MECHANICS”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Computation of Hydration Free Energies Using the Multiple Environment Single System Quantum Mechanical/Molecular Mechanical Method
A recently-developed MESS-E-QM/MM method (multiple-environment single-system quantum mechanical molecular/mechanical calculations with a Roothaan-step extrapolation) is applied to the computation of hydration free energies for the blind SAMPL4 test set and for twelve small molecules. First, free energy simulations are performed with classical molecular mechanics force field using fixed-geometry solute molecules and explicit TIP3P solvent, then the non-Boltzmann-Bennett method is employed to compute the QM/MM correction (QM/MM-NBB) to the molecular mechanical hydration free energies. For the SAMPL4 set, MESS-E-QM/MM-NBB corrections to the hydration free energy can be obtained 2 or 3 orders of magnitude faster than fully converged QM/MM-NBB corrections, and, on average, the hydration free energies predicted with MESS-E-QM/MM-NBB fall within 0.10–0.20 kcal/mol of full-converged QM/MM-NBB results. Here, out of five density functionals (BLYP, B3LYP, PBE0, M06-2X, and ωB97X-D), the BLYP functional is found to be most compatible with the TIP3P solvent model and yields the most accurate hydration free energies against experimental values for solute molecules included in this study.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
The Relationship Between Constraint and Ductile Fracture Initiation as Defined by Micromechanical Analyses
The use of small laboratory specimens to predict the integrity of large, complex structures relies on the validity of single parameter fracture mechanics. Unfortunately, the constraint loss associated with large scale yielding, whether in a laboratory specimen because of its small size or in a structure because it contains shallow flaws loaded in tension, can cause the breakdown of classical fracture mechanics and the loss of transferability of critical, global fracture parameters. Although the issue of constraint loss can be eliminated by testing actual structural configurations, such an approach can be prohibitively costly. Hence, a methodology that can correct global fracture parameters for constraint effects is desirable. This research uses micromechanical analyses to define the relationship between global, ductile fracture initiation parameters and constraint in two specimen geometries (SECT and SECB with varying a/w ratios) and one structural geometry (circumferentially cracked pipe). Two local fracture criteria corresponding to ductile fracture micromechanisms are evaluated: a constraint-modified, critical strain criterion for void coalescence proposed by Hancock and Cowling and a critical void ratio criterion for void growth based on the Rice and Tracey model. Crack initiation is assumed to occur when the critical value in each case is reached over some critical length. The primary material of interest is A516-70, a high-hardening pressure vessel steel sensitive to constraint; however, a low-hardening structural steel that is less sensitive to constraint is also being studied. Critical values of local fracture parameters are obtained by numerical analysis and experimental testing of circumferentially notched tensile specimens of varying constraint (e.g., notch radius). These parameters are then used in conjunction with large strain, large deformation, two- and three-dimensional finite element analyses of the geometries listed above to predict crack initiation loads and to calculate the associated (critical) global fracture parameters. The loads are verified experimentally, and microscopy is used to measure pre-crack length, crack tip opening displacement (CTOD), and the amount of stable crack growth. Results for A516-70 steel indicate that the constraint-modified, critical strain criterion with a critical length approximately equal to the grain size (0.0025 inch) provides accurate predictions of crack initiation. The critical void growth criterion is shown to considerably underpredict crack initiation loads with the same critical length. The relationship between the critical value of the J-integral for ductile crack initiation and crack depth for SECT and SECB specimens has been determined using the constraint-modified, critical strain criterion, demonstrating that this micromechanical model can be used to correct in-plane constraint effects due to crack depth and bending vs. tension loading. Finally, the relationship developed for the SECT specimens is used to predict the behavior of circumferentially cracked pipe specimens.
Panontin, Tina L.↗
Geochemical Reaction Mechanism Discovery from Molecular Simulation
Methods to explore reactions using computer simulation are becoming increasingly quantitative, versatile, and robust. In this review, a rationale for how molecular simulation can help build better geochemical kinetics models is first given. We summarize some common methods that geochemists use to simulate reaction mechanisms, specifically classical molecular dynamics and quantum chemical methods and discuss their strengths and weaknesses. Useful tools such as umbrella sampling and metadynamics that enable one to explore reactions are discussed. Several case studies wherein geochemists have used these tools to understand reaction mechanisms are presented, including water exchange and sorption on aqueous species and mineral surfaces, surface charging, crystal growth and dissolution, and electron transfer. The impact that molecular simulation has had on our understanding of geochemical reactivity are highlighted in each case. In the future, it is anticipated that molecular simulation of geochemical reaction mechanisms will become more commonplace as a tool to validate and interpret experimental data, and provide a check on the plausibility of geochemical kinetic models.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Autonomous quantum to classical transitions and the generalized imaging theorem
The mechanism of the transition of a dynamical system from quantum to classical mechanics is of continuing interest. Practically it is of importance for the interpretation of multi-particle coincidence measurements performed at macroscopic distances from a microscopic reaction zone. We prove the generalized imaging theorem which shows that the spatial wave function of any multi-particle quantum system, propagating over distances and times large on an atomic scale but still microscopic, and subject to deterministic external fields and particle interactions, becomes proportional to the initial momentum wave function where the position and momentum coordinates define a classical trajectory. Now, the quantum to classical transition is considered to occur via decoherence caused by stochastic interaction with an environment. The imaging theorem arises from unitary Schrödinger propagation and so is valid without any environmental interaction. It implies that a simultaneous measurement of both position and momentum will define a unique classical trajectory, whereas a less complete measurement of say position alone can lead to quantum interference effects.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Nanoscale Fluid Mechanics and Energy Conversion
Under nanoconfinement, fluid molecules and ions exhibit radically different configurations, properties, and energetics from those of their bulk counterparts. These unique characteristics of nanoconfined fluids, along with the unconventional interactions with solids at the nanoscale, have provided many opportunities for engineering innovation. With properly designed nanoconfinement, several nanofluidic systems have been devised in our group in the past several years to achieve energy conversion functions with high efficiencies. This review is dedicated to elucidating the unique characteristics of nanofluidics, introducing several novel nanofluidic systems combining nanoporous materials with functional fluids, and to unveiling their working mechanisms. In all these systems, the ultra-large surface area available in nanoporous materials provides an ideal platform for seamlessly interfacing with nanoconfined fluids, and efficiently converting energy between the mechanical, thermal, and electrical forms. These systems have been demonstrated to have great potentials for applications including energy dissipation/absorption, energy trapping, actuation, and energy harvesting. Their efficiencies can be further enhanced by designing efforts based upon improved understanding of nanofluidics, which represents an important addition to classical fluid mechanics. Through the few systems exemplified in this review, the emerging research field of nanoscale fluid mechanics may promote more exciting nanofluidic phenomena and mechanisms, with increasing applications by encompassing aspects of mechanics, materials, physics, chemistry, biology, etc.
Chen, X↗
Bond breaking in epoxy systems: A combined QM/MM approach
A novel method to combine quantum mechanics (QM) and molecular mechanics has been developed to accurately and efficiently account for covalent bond breaking in polymer systems under high strain without the use of predetermined break locations. Use of this method will provide a better fundamental understanding of the mechano-chemical origins of fracture in thermosets. Since classical force fields cannot accurately account for bond breaking, and QM is too demanding to simulate large systems, a hybrid approach is required. In the method presented here, strain is applied to the system using a classical force field, and all bond lengths are monitored. When a bond is stretched past a threshold value, a zone surrounding the bond is used in a QM energy minimization to determine which, if any, bonds break. The QM results are then used to reconstitute the system to continue the classical simulation at progressively larger strain until another QM calculation is triggered. In this way, a QM calculation is only computed when and where needed, allowing for efficient simulations. A robust QM method for energy minimization has been determined, as well as appropriate values for the QM zone size and the threshold bond length. Compute times do not differ dramatically from classical molecular mechanical simulations.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
On a Lagrange-Hamilton formalism describing position and momentum uncertainties
According to Heisenberg's uncertainty relation, in quantum mechanics it is not possible to determine, simultaneously, exact values for the position and the momentum of a material system. Calculating the mean value of the Hamiltonian operator with the aid of exact analytic Gaussian wave packet solutions, these uncertainties cause an energy contribution additional to the classical energy of the system. For the harmonic oscillator, e.g., this nonclassical energy represents the ground state energy. It will be shown that this additional energy contribution can be considered as a Hamiltonian function, if it is written in appropriate variables. With the help of the usual Lagrange-Hamilton formalism known from classical particle mechanics, but now considering this new Hamiltonian function, it is possible to obtain the equations of motion for position and momentum uncertainties.
Schuch, Dieter↗
Quantum kinetic modeling of KEEN waves in a warm-dense regime
We report the first fully kinetic, quantum study of kinetic electrostatic electron nonlinear (KEEN) waves, showing that quantum diffraction systematically erodes the classical trapping mechanism, narrows harmonic locking to the fundamental, and hastens post-drive decay. Electrons are evolved with a second-order Strang-split 1D1V Wigner–Poisson solver that couples conservative semi-Lagrangian WENO advection to an analytic Fourier space update for the non-local Wigner term, while ions remain classical. We focus on collisionless dynamics in a weakly coupled regime, providing a controlled baseline before collisional extensions. Short, frequency-tuned ponderomotive pulses drive KEEN formation in a uniform Maxwellian plasma; as the dimensionless quantum parameter H rises from the classical limit to values relevant to warm-dense matter, doped semiconductors, and 2D electron systems, the drive threshold increases, higher harmonics are damped, trapped electron vortices diffuse, and the subplasma electrostatic energy relaxes to a lower stationary level, as confirmed by continuous wavelet analysis. These microscopic changes carry macroscopic weight. Ignition-scale capsules now compress matter to regimes where the electron de Broglie wavelength rivals the Debye length, making classical kinetic descriptions insufficient. By extending KEEN physics into this quantum domain, our results offer a potential diagnostic of non-equilibrium electron dynamics for next-generation inertial-confinement designs and high-energy-density platforms, indicating that predictive fusion modeling may benefit from the integration of kinetic fidelity with quantum effects.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Ejection mechanisms in the sublayer of a turbulent channel
The structure of the vorticity field in the viscous wall layer of a turbulent channel is studied by examining the results of a fully resolved direct numerical simulation. It is shown that this region is dominated by intense three-dimensional shear layers in which the dominant vorticity component is spanwise. The advection and reproduction processes of these structures are examined and shown to be consistent with the classical generation mechanism for two-dimensional Tollmien-Schlichting waves. This process is fundamentally different from the usually accepted mechanism involving hairpin vortices.
Jimenez, Javier↗
Rigorous theory of molecular orientational nonlinear optics
Classical statistical mechanics of the molecular optics theory proposed by Buckingham [A. D. Buckingham and J. A. Pople, Proc. Phys. Soc. A 68, 905 (1955)] has been extended to describe the field induced molecular orientational polarization effects on nonlinear optics. In this paper, we present the generalized molecular orientational nonlinear optical processes (MONLO) through the calculation of the classical orientational averaging using the Boltzmann type time-averaged orientational interaction energy in the randomly oriented molecular system under the influence of applied electric fields. The focal points of the calculation are (1) the derivation of rigorous tensorial components of the effective molecular hyperpolarizabilities, (2) the molecular orientational polarizations and the electronic polarizations including the well-known third-order dc polarization, dc electric field induced Kerr effect (dc Kerr effect), optical Kerr effect (OKE), dc electric field induced second harmonic generation (EFISH), degenerate four wave mixing (DFWM) and third harmonic generation (THG). We also present some of the new predictive MONLO processes. For second-order MONLO, second-order optical rectification (SOR), Pockels effect and difference frequency generation (DFG) are described in terms of the anisotropic coefficients of first hyperpolarizability. And, for third-order MONLO, third-order optical rectification (TOR), dc electric field induced difference frequency generation (EFIDFG) and pump-probe transmission are presented.
36 MATERIALS SCIENCE↗
Isothermal fatigue mechanisms in Ti-based metal matrix composites
Stress-controlled isothermal fatigue experiments were performed at room temperature (RT) and 548 C (in argon) on (0)8 SCS6/Ti 15-3 metal matrix composites (MMC's) with 15 and 41 volume percent SCS6 (SiC) fibers. The primary objectives were to evaluate the mechanical responses, and to obtain a clear understanding of the damage mechanisms leading to failure of the MMC's. The mechanical data indicated that strain ranges attained fairly constant values in the stress-controlled experiments at both RT and 538 C, and remained so for more than 85 percent of life. The fatigue data for MMC's with different volume fraction fibers showed that MMC life was controlled by the imposed strain range rather than the stress range. At RT, and at low and intermediate strain ranges, the dominant fatigue mechanism was matrix fatigue, and this was confirmed metallurgically from fractographic evidence as well as from observations of channel type dislocation structures in the matrix of fatigued MMC specimens. Reaction-zone cracks acted as important crack initiating sites at RT, with their role being to facilitate slip band formation and consequent matrix crack initiation through classical fatigue mechanisms. MMC life agreed with matrix life at the lower strain ranges, but was smaller than matrix life at higher strain ranges. Unlike the case of monotonic deformation, debonding damage was another major damage mechanism during fatigue at RT, and it increased for higher strain ranges. At high strain ranges at RT, fractography and metallography showed an absence of matrix cracks, but long lengths of debonds in the outer layers of the SCS6 fibers. Such debonding and consequent rubbing during fatigue is believed to have caused fiber damage and their failure at high strain ranges. Thus, whereas life was matrix dominated at low and intermediate strain ranges, it was fiber dominated at high strain ranges. At 538 C, the mean stain constantly increased (ratchetting) with the number of cycles. At high strain ranges, such ratchetting led to overload failure of the fibers, and debonding of the type at RT was very small. At intermediate strain ranges, fractography showed large areas of matrix cracks. However, in spite of this matrix dominated mechanism, the MMC life at elevated temperatures was significantly less than the matrix fatigue life at all strain ranges. The reason for this difference is still unclear, although metallographic and fractographic evidences suggest that internal crack initiation sites at Mo-ribbons and reaction-zone cracks may have played a critical role, with the former tending to dominate.
Majumdar, Bhaskar S.↗
Crack propagation in silica from reactive classical molecular dynamics simulations
Mechanistic insight into the process of crack growth can be obtained through molecular dynamics (MD) simulations. In this investigation of fracture propagation, a slit crack was introduced into an atomistic amorphous silica model and mode I stress was applied through far–field loading until the crack propagates. Atomic displacements and forces and an Irving–Kirkwood method with a Lagrangian kernel estimator were used to calculate the J –integral of classical fracture mechanics around the crack tip. The resulting fracture toughness (K IC ), 0.76 ± 0.16 MPa√m, agrees with experimental values. In addition, the stress fields and dissipation energies around the slit crack indicate the development of an inelastic region ~30Å in diameter. This is one of the first reports of KIC values obtained from up–scaled atomic–level energies and stresses through the J –integral. In conclusion, the application of the ReaxFF classical MD force field in this study provides the basis for future research into crack growth in multicomponent oxides in a variety of environmental conditions.
36 MATERIALS SCIENCE↗
Truncated dynamics, ring molecules, and mechanical time crystals
Here, we identify circumstances where the effective descriptions of microscopic physical systems leads to a self-consistent reduced dynamics for a truncated subset of the original variables. The effective Hamiltonian involves unusual Poisson brackets that bring in noncommutative geometry. In idealized models of ring molecules, we find time crystal behavior is widespread.
74 ATOMIC AND MOLECULAR PHYSICS↗
An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture
Meshfree discretizations of state-based peridynamic models are attractive due to their ability to naturally describe fracture of general materials. However, two factors conspire to prevent meshfree discretizations of state-based peridynamics from converging to corresponding local solutions as resolution is increased: quadrature error prevents an accurate prediction of bulk mechanics, and the lack of an explicit boundary representation presents challenges when applying traction loads. Herein, we develop a reformulation of the linear peridynamic solid (LPS) model to address these shortcomings, using improved meshfree quadrature, a reformulation of the nonlocal dilatation, and a consistent handling of the nonlocal traction condition to construct a model with rigorous accuracy guarantees. In particular, these improvements are designed to enforce discrete consistency in the presence of evolving fractures, whose a priori unknown location render consistent treatment difficult. In the absence of fracture, when a corresponding classical continuum mechanics model exists, our improvements provide asymptotically compatible convergence to corresponding local solutions, eliminating surface effects and issues with traction loading which have historically plagued peridynamic discretizations. When fracture occurs, our formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. We provide rigorous error analysis and demonstrate convergence for a number of benchmarks, including manufactured solutions, free-surface, nonhomogeneous traction loading, and composite material problems. Finally, we validate simulations of brittle fracture against a recent experiment of dynamic crack branching in soda-lime glass, providing evidence that the scheme yields accurate predictions for practical engineering problems.
42 ENGINEERING↗
Improved semiclassical model for real-time evaporation of matrix black holes
In this paper, we study real-time classical matrix mechanics of a simplified 2 × 2 matrix model inspired by the black hole evaporation problem. This is a step towards making a quantitative model of real-time evaporation of a black hole, which is realized as a bound state of D0-branes in string theory. The model we study is the reduction of Yang–Mills in 2 + 1 dimension to 0 + 1 dimensions, which has been corrected with an additional potential that can be interpreted as a zero-point energy for fermions. Our goal is to understand the lifetime of such a classical bound state object in the classical regime. To do so, we pay particular attention to when D-particles separate to check that the “off-diagonal modes” of the matrices become adiabatic and use that information to improve on existing models of evaporation. It turns out that the naive expectation value of the lifetime with the fermionic correction is infinite. This is a logarithmic divergence that arises from very large excursions in the separation between the branes near the threshold for classical evaporation. The adiabatic behavior lets us get some analytic control of the dynamics in this regime to get this estimate. This divergence is cutoff in the quantum theory due to quantization of the adiabatic parameter, resulting in a long lifetime of the bound state, with a parametric dependence of order log(1/ℏ).
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Shearless effective barriers to chaotic transport induced by even twin islands in nontwist systems
For several decades now it has been known that systems with shearless invariant tori, nontwist Hamiltonian systems, possess barriers to chaotic transport. These barriers are resilient to breakage under perturbation and therefore regions where they occur are natural places to look for barriers to transport. Here we describe a kind of effective barrier that persists after the shearless torus is broken. Because phenomena are generic, for convenience we study the standard nontwist map (SNM), an area-preserving map that violates the twist condition locally in the phase space. The barrier occurs in nontwist systems when twin even period islands are present, which happens for a broad range of parameter values in the SNM. With a phase space composed of regular and irregular orbits, the movement of chaotic trajectories is hampered by the existence of shearless curves, total barriers, and a network of partial barriers formed by the stable and unstable manifolds of the hyperbolic points. Being a degenerate system, the SNM has twin islands and, consequently, twin hyperbolic points. We show that the structures formed by the manifolds intrinsically depend on period parity of the twin islands. For this even scenario the structure that we call a torus free barrier occurs because the manifolds of different hyperbolic points form an intricate chain atop a dipole configuration and the transport of chaotic trajectories through the chain becomes a rare event. This structure impacts the emergence of transport, the escape basin for chaotic trajectories, the transport mechanism, and the chaotic saddle. The case of odd periodic orbits is different: we find for this case the emergence of transport immediately after the breakup of the last invariant curve, and this leads to a scenario of higher transport, with intricate escape basin boundary and a chaotic saddle with nonuniformly distributed points.
classical mechanics↗
Maximal temperature in a simple thermodynamical system
Temperature in a simple thermodynamical system is not limited from above. It is also widely believed that it does not make sense talking about temperatures higher than the Planck temperature in the absence of the full theory of quantum gravity. Here, we demonstrate that there exist a maximal achievable temperature in a system where particles obey the laws of quantum mechanics and classical gravity before we reach the realm of quantum gravity. Namely, if two particles with a given center of mass energy come at the distance shorter than the Schwarzschild diameter apart, according to classical gravity they will form a black hole. It is possible to calculate that a simple thermodynamical system will be dominated by black holes at a critical temperature which is about three times lower than the Planck temperature. That represents the maximal achievable temperature in a simple thermodynamical system.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗