Dissipative Work Research Articles

Boudinage and folding as an energy instability in ductile deformation

AbstractWe present a theory for the onset of localization in layered rate‐ and temperature‐sensitive rocks, in which energy‐related mechanical bifurcations lead to localized dissipation patterns in the transient deformation regime. The implementation of the coupled thermomechanical 2‐D finite element model comprises an elastic and rate‐dependent von Mises plastic rheology. The underlying system of equations is solved in a three‐layer pure shear box, for constant velocity and isothermal boundary conditions. To examine the transition from stable to localized creep, we study how material instabilities are related to energy bifurcations, which arise independently of the sign of the stress conditions imposed on opposite boundaries, whether in compression or extension. The onset of localization is controlled by a critical amount of dissipation, termed Gruntfest number, when dissipative work by temperature‐sensitive creep translated into heat overcomes the diffusive capacity of the layer. Through an additional mathematical bifurcation analysis using constant stress boundary conditions, we verify that boudinage and folding develop at the same critical Gruntfest number. Since the critical material parameters and boundary conditions for both structures to develop are found to coincide, the initiation of localized deformation in strong layered media within a weaker matrix can be captured by a unified theory for localization in ductile materials. In this energy framework, neither intrinsic nor extrinsic material weaknesses are required, because the nucleation process of strain localization arises out of steady state conditions. This finding allows us to describe boudinage and folding structures as the same energy attractor of ductile deformation.

Out-of-equilibrium fluctuations in stochastic long-range interacting systems

For a many-particle system with long-range interactions and evolving under stochastic dynamics, we study for the first time the out-of-equilibrium fluctuations of the work done on the system by a time-dependent external force. For equilibrium initial conditions, the work distributions for a given protocol of variation of the force in time and the corresponding time-reversed protocol exhibit a remarkable scaling and a symmetry when expressed in terms of the average and the standard deviation of the work. The distributions of the work per particle predict, by virtue of the Crooks fluctuation theorem, the equilibrium free-energy density of the system. For a large number N of particles, the latter is in excellent agreement with the value computed by considering the Langevin dynamics of a single particle in a self-consistent mean field generated by its interaction with other particles. The agreement highlights the effective mean-field nature of the original many-particle dynamics for large N. For initial conditions in non-equilibrium stationary states (NESSs), we study the distribution of a quantity similar to dissipated work that satisfies the non-equilibrium generalization of the Clausius inequality, namely, the Hatano-Sasa equality, for transitions between NESSs. Besides illustrating the validity of the equality, we show that the distribution has exponential tails that decay differently on the left and on the right.

Soft tissues store and return mechanical energy in human running

During human running, softer parts of the body may deform under load and dissipate mechanical energy. Although tissues such as the heel pad have been characterized individually, the aggregate work performed by all soft tissues during running is unknown. We therefore estimated the work performed by soft tissues (N=8 healthy adults) at running speeds ranging 2–5ms−1, computed as the difference between joint work performed on rigid segments, and whole-body estimates of work performed on the (non-rigid) body center of mass (COM) and peripheral to the COM. Soft tissues performed aggregate negative work, with magnitude increasing linearly with speed. The amount was about −19J per stance phase at a nominal 3ms−1, accounting for more than 25% of stance phase negative work performed by the entire body. Fluctuations in soft tissue mechanical power over time resembled a damped oscillation starting at ground contact, with peak negative power comparable to that for the knee joint (about −500W). Even the positive work from soft tissue rebound was significant, about 13J per stance phase (about 17% of the positive work of the entire body). Assuming that the net dissipative work is offset by an equal amount of active, positive muscle work performed at 25% efficiency, soft tissue dissipation could account for about 29% of the net metabolic expenditure for running at 5ms−1. During running, soft tissue deformations dissipate mechanical energy that must be offset by active muscle work at non-negligible metabolic cost.

Physical variational principle in dissipative media

The physical variational principle (PVP) is a physical principle which is implied in the thermodynamic laws. For a conservative system, the PVP is implied in the first thermodynamic law and gives the motion equation. But for a dissipative system, PVP is implied in the extended Gibbs equation, which is the result of the first and second thermodynamic laws. The precision of the PVP in a dissipative system is in the same order of the Gibbs equation. The dissipative work and its converted internal irreversible heat are simultaneously included in the PVP to get the governing equation and the boundary condition of the dissipated variables. The “generalized motion equations” including governing equations of the mechanical momentum and thermoelastic, thermal viscoelastic, thermal elastoplastic, linear thermoelastic diffusive and linear electromagnetic thermoelastic materials etc. can be derived by the PVP of dissipative media in this paper. The conservative system is the special case of the dissipative system. Other than the mathematical variational principle, which is obtained by a known governing equation, the PVP is used to deduce the governing equation. The PVPs including the hyperbolic temperature wave equation with a finite phase speed are also discussed shortly.

A cascade pulse tube cooler capable of energy recovery

A pulse tube cryocooler (PTC) cannot work with Carnot efficiency due basically to the expansion work that has to be dissipated thermally at the warm end of the pulse tube, this dissipation is especially phenomenal with high cooling capacity or at high temperatures which reduces the COP and limits the application of PTC above 120K. Therefore, how to recover this amount of dissipated work becomes a critical issue in a high efficient PTC. Here, we proposed a cascade PTC with a built-in transmission tube between stages for energy recovery. The key point of this new configuration is that the acoustic power at the outlet of the primary stage can be recovered through the transmission tube which provides proper phase angle to drive the secondary stage. This idea is verified both theoretically and experimentally. The cooling efficiency can be improved by 33% when the machine works at 233K.

Study on a cascade pulse tube cooler with energy recovery: new method for approaching Carnot

A pulse tube cryocooler (PTC) can not achieve Carnot efficiency because the expansion work must be dissipated at the warm end of the pulse tube. How to recover this amount of dissipated work is a key for improving the PTC efficiency. A cascade PTC consists of PTCs those are staged by transmission tubes in between, these can be a two-stage or even more stages, each stage is driven by the recovered work from the last stage by a well-designed long transmission tube. It is shown that the more stages it has, the closer the efficiency will approach the Carnot efficiency. A two-stage cascade pulse tube cooler consisted of a primary and a secondary stage working at 233 K is designed, fabricated and tested in our lab. Experimental results show that the efficiency is improved by 33% compared with the single stage PTC.

Rate-dependent mechanical behavior of VHB 4910 elastomer

ABSTRACTThis article experimentally studied the large nonlinear deformation of VHB 4910 elastomer by uniaxial tests. The study reveals that the monotonic tensile stress-strain, hysteresis, cyclic stress softening, and multistep stress relaxation of this elastomer exhibit rate-sensitivity. Toughness, failure stress, and failure strain are shown to vary with strain rate. Maximum cyclic stress, hysteresis loss, residual strains in cyclic loading-unloading, and stress relaxation in multistep relaxation tests are also shown to be rate-sensitive. The analytical models are also proposed to predict certain important parameters, such as dissipative work, cyclic stress softening, cyclic residual strain, and relaxation stress in different states of deformation.

Exact work statistics of quantum quenches in the anisotropic XY model.

We derive exact analytic expressions for the average work done and work fluctuations in instantaneous quenches of the ground and thermal states of a one-dimensional anisotropic XY model. The average work and a quantum fluctuation relation is used to determine the amount of irreversible entropy produced during the quench, eventually revealing how the closing of the excitation gap leads to increased dissipated work. The work fluctuation is calculated and shown to exhibit nonanalytic behavior as the prequench anisotropy parameter and transverse field are tuned across quantum critical points. Exact compact formulas for the average work and work fluctuation in ground state quenches of the transverse field Ising model allow us to calculate the first singular field derivative at the critical field values.

Soft Tissue Deformations Contribute to the Mechanics of Walking in Obese Adults.

Obesity not only adds to the mass that must be carried during walking but also changes body composition. Although extra mass causes roughly proportional increases in musculoskeletal loading, less well understood is the effect of relatively soft and mechanically compliant adipose tissue. This purpose of this study was to estimate the work performed by soft tissue deformations during walking. The soft tissue would be expected to experience damped oscillations, particularly from high force transients after heel strike, and could potentially change the mechanical work demands for walking. We analyzed treadmill walking data at 1.25 m·s for 11 obese (BMI >30 kg·m) and nine nonobese (BMI <30 kg·m) adults. The soft tissue work was quantified with a method that compares the work performed by lower extremity joints as derived using assumptions of rigid body segments, with that estimated without rigid body assumptions. Relative to body mass, obese and nonobese individuals perform similar amounts of mechanical work. However, negative work performed by soft tissues was significantly greater in obese individuals (P = 0.0102), equivalent to approximately 0.36 J·kg vs 0.27 J·kg in nonobese individuals. The negative (dissipative) work by soft tissues occurred mainly after heel strike and, for obese individuals, was comparable in magnitude to the total negative work from all of the joints combined (0.34 J·kg vs 0.33 J·kg for obese and nonobese adults, respectively). Although the joints performed a relatively similar amount of work overall, obese individuals performed less negative work actively at the knee. The greater proportion of soft tissues in obese individuals results in substantial changes in the amount, location, and timing of work and may also affect metabolic energy expenditure during walking.

Dynamic memory of a single voltage-gated potassium ion channel: A stochastic nonequilibrium thermodynamic analysis.

In this work, we have studied the stochastic response of a single voltage-gated potassium ion channel to a periodic external voltage that keeps the system out-of-equilibrium. The system exhibits memory, resulting from time-dependent driving, that is reflected in terms of dynamic hysteresis in the current-voltage characteristics. The hysteresis loop area has a maximum at some intermediate voltage frequency and disappears in the limits of low and high frequencies. However, the (average) dissipation at long-time limit increases and finally goes to saturation with rising frequency. This raises the question: how diminishing hysteresis can be associated with growing dissipation? To answer this, we have studied the nonequilibrium thermodynamics of the system and analyzed different thermodynamic functions which also exhibit hysteresis. Interestingly, by applying a temporal symmetry analysis in the high-frequency limit, we have analytically shown that hysteresis in some of the periodic responses of the system does not vanish. On the contrary, the rates of free energy and internal energy change of the system as well as the rate of dissipative work done on the system show growing hysteresis with frequency. Hence, although the current-voltage hysteresis disappears in the high-frequency limit, the memory of the ion channel is manifested through its specific nonequilibrium thermodynamic responses.

Mechanical and energetic consequences of reduced ankle plantar-flexion in human walking.

The human ankle produces a large burst of 'push-off' mechanical power late in the stance phase of walking, reduction of which leads to considerably poorer energy economy. It is, however, uncertain whether the energetic penalty results from poorer efficiency when the other leg joints substitute for the ankle's push-off work, or from a higher overall demand for work due to some fundamental feature of push-off. Here, we show that greater metabolic energy expenditure is indeed explained by a greater demand for work. This is predicted by a simple model of walking on pendulum-like legs, because proper push-off reduces collision losses from the leading leg. We tested this by experimentally restricting ankle push-off bilaterally in healthy adults (N=8) walking on a treadmill at 1.4 m s(-1), using ankle-foot orthoses with steel cables limiting motion. These produced up to ∼50% reduction in ankle push-off power and work, resulting in up to ∼50% greater net metabolic power expenditure to walk at the same speed. For each 1 J reduction in ankle work, we observed 0.6 J more dissipative collision work by the other leg, 1.3 J more positive work from the leg joints overall, and 3.94 J more metabolic energy expended. Loss of ankle push-off required more positive work elsewhere to maintain walking speed; this additional work was performed by the knee, apparently at reasonably high efficiency. Ankle push-off may contribute to walking economy by reducing dissipative collision losses and thus overall work demand.

Nonequilibrium Candidate Monte Carlo Simulations with Configurational Freezing Schemes.

Nonequilibrium Candidate Monte Carlo simulation [Nilmeier et al., Proc. Natl. Acad. Sci. U.S.A. 2011, 108, E1009-E1018] is a tool devised to design Monte Carlo moves with high acceptance probabilities that connect uncorrelated configurations. Such moves are generated through nonequilibrium driven dynamics, producing candidate configurations accepted with a Monte Carlo-like criterion that preserves the equilibrium distribution. The probability of accepting a candidate configuration as the next sample in the Markov chain basically depends on the work performed on the system during the nonequilibrium trajectory and increases with decreasing such a work. It is thus strategically relevant to find ways of producing nonequilibrium moves with low work, namely moves where dissipation is as low as possible. This is the goal of our methodology, in which we combine Nonequilibrium Candidate Monte Carlo with Configurational Freezing schemes developed by Nicolini et al. (J. Chem. Theory Comput. 2011, 7, 582-593). The idea is to limit the configurational sampling to particles of a well-established region of the simulation sample, namely the region where dissipation occurs, while leaving fixed the other particles. This allows to make the system relaxation faster around the region perturbed by the finite-time switching move and hence to reduce the dissipated work, eventually enhancing the probability of accepting the generated move. Our combined approach enhances significantly configurational sampling, as shown by the case of a bistable dimer immersed in a dense fluid.

Reply to the discussion by Utili on “Stability assessment of slopes with cracks using limit analysis”

The author would like to thank the discusser for his interest in the paper. The discusser provided multiple comments, and suggested that clarifications be offered in regard to three areas: (i) calculations of the work rate of the weight of the soil, (ii) failure mechanism with a pre-existing crack, and (iii) collapse mechanism with crack formation. The author’s response comments are given in that order. It was indicated in the paper under discussion (Michalowski 2013) that the equations for calculating the work rate of the soil weight were omitted, because they can be found in multiple papers, including the original paper by Chen et al. (1969). The discusser does not seem to agree with this statement as the shape of the moving mass of a slope with a crack (or cracks) is different from that considered by Chen and others. However, proper application of the original equations makes new derivations entirely unnecessary. The work rate of the soil weight was considered in the paper by first calculating the work rate of the moving “uncracked” block ACOBDA (Fig. 6 of the paper under discussion), and then calculating the work rate for a vertical slope of height CD to the right of the crack (ACDA). Subtracting the latter from the former yields the required work rate. Calculating the two rates does not require any equations beyond those in the paper of Chen et al. (1969). The author did not find it necessary to repeat these equations in his paper. The second issue raised by the discusser relates to the mechanism with an existing crack(s). This part of the discusser’s enquiry is more of a commentary, with an emphasis on some of his own contributions in a recently published paper (Utili 2013). Regarding below-toe failures, indeed, the critical mechanisms for slopes with > 5° (with no pore-water pressure) are all toe mechanisms, but the credit for this should go to Chen and Giger who indicated this clearly in 1971 (Chen and Giger 1971). However, the range where below-toe failure is found critical is wider for “shallow” slopes in the presence of pore-water pressure (Michalowski 1995). The search for the best upper bound to the dimensionless critical height H/c was formulated as a constrained optimization problem (the depth of an existing vertical crack cannot be larger than the critical height of a vertical slope in the same soil). However, in most cases the crack depth constraint did not alter the solution, i.e., the best solution was found away from the constraint. The exceptions were found among slopes subjected to pore-water pressure, which the discusser did not consider. Further comments of the discusser are related to his formulation of the problem (Utili 2013), and they reveal a different, perhaps questionable, philosophy of approach from that of the author. It may be helpful to bring up some of the fundamental aspects of limit analysis. The kinematic theorem of limit analysis only indicates that the integral of the rate of internal (dissipated) work is not smaller than the integral of the work rates of true external forces in any kinematically admissible mechanism. Consequently, the theorem can be used to find an upper bound to an active limit load or, as in the paper under discussion, an upper bound to dimensionless group H/c. However, the theorem cannot be used to find the location (or depth) of an existing crack. The location of an existing crack must be a part of the geometry definition of the problem, just as the inclination of the slope is, and it must be known a priori. However, in his cited paper (Utili 2013), the discusser formulated three different problems: one with an “unspecified location” of the crack, one with unspecified crack depth, and one with both being unknown. The method can only provide information as to where the most adverse location of the crack would be in the kinematically admissible mechanism considered in the analysis. This location was calculated in both Michalowski (2013) and Utili (2013) by making the location of the crack one of the variables in the optimization procedure. This, however, cannot be considered as finding the true location of the crack, just as the mechanism used in the analysis cannot be considered a true failure mechanism. Therefore, in the paper under discussion, the problem was formulated differently: the slopes considered had multiple cracks and, among those pre-existing cracks, there was always one in very close proximity to the most critical location determined by the mechanism used in the solution. This is a subtle difference in formulation from that of the discusser, but it allows one to arrive at the solution without suggesting that the crack location so determined is the true location of a pre-existing crack.

Free energy calculations, enhanced by a gaussian ansatz, for the “chemical work” distribution

The evaluation of the free energy is essential in molecular simulation because it is intimately related with the existence of multiphase equilibrium. Recently, it was demonstrated that it is possible to evaluate the Helmholtz free energy using a single statistical ensemble along an entire isotherm by accounting for the "chemical work" of transforming each molecule, from an interacting one, to an ideal gas. In this work, we show that it is possible to perform such a free energy perturbation over a liquid vapor phase transition. Furthermore, we investigate the link between a general free energy perturbation scheme and the novel nonequilibrium theories of Crook's and Jarzinsky. We find that for finite systems away from the thermodynamic limit the second law of thermodynamics will always be an inequality for isothermal free energy perturbations, resulting always to a dissipated work that may tend to zero only in the thermodynamic limit. The work, the heat, and the entropy produced during a thermodynamic free energy perturbation can be viewed in the context of the Crooks and Jarzinsky formalism, revealing that for a given value of the ensemble average of the "irreversible" work, the minimum entropy production corresponded to a Gaussian distribution for the histogram of the work. We propose the evaluation of the free energy difference in any free energy perturbation based scheme on the average irreversible "chemical work" minus the dissipated work that can be calculated from the variance of the distribution of the logarithm of the work histogram, within the Gaussian approximation. As a consequence, using the Gaussian ansatz for the distribution of the "chemical work," accurate estimates for the chemical potential and the free energy of the system can be performed using much shorter simulations and avoiding the necessity of sampling the computational costly tails of the "chemical work." For a more general free energy perturbation scheme that the Gaussian ansatz may not be valid, the free energy calculation can be expressed in terms of the moment generating function of the "chemical work" distribution.

Phase space gradient of dissipated work and information: A role of relative Fisher information

We show that an information theoretic distance measured by the relative Fisher information between canonical equilibrium phase densities corresponding to forward and backward processes is intimately related to the gradient of the dissipated work in phase space. We present a universal constraint on it via the logarithmic Sobolev inequality. Furthermore, we point out that a possible expression of the lower bound indicates a deep connection in terms of the relative entropy and the Fisher information of the canonical distributions.

Calcium phosphate cement augmentation after volar locking plating of distal radius fracture significantly increases stability

Distal radius fractures represent the most common fractures in adults. Volar locking plating to correct unstable fractures has become increasingly popular. Although reasonable primary reduction is possible in most cases, maintenance of reduction until the fracture is healed is often problematic in osteoporotic bone. To our knowledge, no biomechanical studies have compared the effect of enhancement with biomaterial on two different volar fixed-angle plates. Human fresh-frozen cadaver pairs of radii were used to simulate an AO/OTA 23-A3 fracture. In a total of four groups (n = 7 for each group), two volar fixed-angle plates (Aptus 2.5 mm locking fracture plate, Medartis, Switzerland and VA-LCP two-column distal radius plate 2.4, volar, Synthes, Switzerland) with or without an additional injection of a biomaterial (Hydroset Injectable HA Bone Substitute, Stryker, Switzerland) into the dorsal comminution zone were used to fix the distal metaphyseal fragment. Each specimen was tested load-controlled under cyclic loading with a servo-hydraulic material testing machine. Displacement, stiffness, dissipated work and failure mode were recorded. Improved mechanical properties (decreased displacement, increased stiffness, decreased dissipated work) were found in both plates if the biomaterial was additionally injected. Improvement of mechanical parameters after biomaterial injection was more evident in the Synthes plate compared to the Aptus plate. Pushing out of the screws was noticed as a failure mode only in samples lacking supplementary biomaterial. Injection of a biomaterial into the dorsal comminution zone increases stability after volar locking plating of distal radius fractures in vitro.

Dissipated Work in Nonequilibrium Process and Relative Fisher Information

Phase space gradient of thermodynamic quantities in a nonequilibrium process and those limitations by information theoretical ones are keys to well understand the system. Dissipation and work are two such quantities in thermodynamic operation between equilibrium states and our present focus. Any displacement of a system in phase space from a specific equilibrium state to another equilibrium phase point induced by a dynamics has a counterpart process by reversing time. In a previous study, the dissipated work during the process, which is defined as work done on the system minus the associated (equilibrium) free energy difference was one of the chief interests. Therefore, the operational forward and backward movements make us to expect finding a universal relation in the dissipated work in an averaged way. One such intriguing example was focused in [1] and it was shown that the relative entropy in the sense of Kullback-Leibler between the forward and backward equilibrium distributions is equivalent to the dissipated work in the unit of thermal energy. In this presentation, we provide yet another fundamental relation that holds in the process by focusing on the gradient in the phase space. That is, the relative Fisher information between canonical equilibrium phase densities corresponding to forward and backward processes is intimately related to the gradient of the dissipated work in phase space [2]. Since the Kullback-Leibler relative entropy admits various extensions [3], we can pursue a general relation that links the dissipated work and the notion of distance in statistical mechanics. We also present a universal constraint on the averaged gradient of the dissipated work via the logarithmic Sobolev inequality [2]. A possible expression of the lower bound indicates a deep connection between the relative entropy and the Fisher information [4] of the canonical distributions.

Assessing Numerical Error in Structural Dynamics Using Energy Balance

This work applies the variational principles of Lagrange and Hamilton to the assessment of numerical methods of linear structural analysis. Different numerical methods are used to simulate the behaviour of three structural configurations and benchmarked in their computation of the Lagrangian action integral over time. According to the principle of energy conservation, the difference at each time step between the kinetic and the strain energies must equal the work done by the external forces. By computing this difference, the degree of accuracy of each combination of numerical methods can be assessed. Moreover, it is often difficult to perceive numerical instabilities due to the inherent complexities of the modelled structures. By means of the proposed procedure, these complexities can be globally controlled and visualized in a straightforward way. The paper presents the variational principles to be considered for the collection and computation of the energy-related parameters (kinetic, strain, dissipative, and external work). It then introduces a systematic framework within which the numerical methods can be compared in a qualitative as well as in a quantitative manner. Finally, a series of numerical experiments is conducted using three simple 2D models subjected to the effect of four different dynamic loadings.

Modelling high strain rate phenomena in metal cutting simulation

Chip formation in metal cutting is associated with large strains and high strain rates, concentrated locally to deformation zones in front of the tool and beneath the cutting edge. Furthermore, dissipative plastic work and friction work generate high local temperatures. These phenomena together with numerical complications make modelling of metal cutting difficult. Material models, which are crucial in metal cutting simulations, are usually calibrated based on data from material testing. Nevertheless, the magnitude of strains and strain rates involved in metal cutting are several orders higher than those generated from conventional material testing. A highly desirable feature is therefore a material model that can be extrapolated outside the calibration range. In this study, two variants of a flow stress model based on dislocation density and vacancy concentration are used to simulate orthogonal metal cutting of AISI 316L stainless steel. It is found that the addition of phonon drag improves the results somewhat but the addition of this phenomenon still does not make it possible to extrapolate the constitutive model reliably outside its calibration range.

Comparison of free-energy estimators and their dependence on dissipated work

The estimate of free energy changes based on Bennett's acceptance ratio method is examined in several limiting cases and compared with other estimates based on the Jarzynski equality and on the Crooks relation. While the absolute amount of the dissipated work, defined as the surplus of the average work over the free energy difference, limits the practical applicability of Jarzynski's and Crooks' methods, the reliability of Bennett's approach is restricted by the difference of the dissipated works in the forward and the backward processes. We illustrate these points by considering a Gaussian chain and a hairpin chain which both are extended during the forward and accordingly compressed during the backward protocols. The reliability of the Crooks relation predominantly depends on the sample size; for the Jarzynski estimator the slowness of the work protocol is crucial, and the Bennett method is shown to give precise estimates irrespective of the pulling speed and sample size as long as the dissipated works are the same for the forward and the backward processes as it is the case for Gaussian work distributions. With an increasing dissipated work difference the Bennett estimator also acquires a bias which increases roughly in proportion to this difference. A substantial simplification of the Bennett estimator is provided by the 1/2 formula which expresses the free energy difference by the algebraic average of the Jarzynski estimates for the forward and the backward processes. It agrees with the Bennett estimate in all cases when the Jarzynski and the Crooks estimates fail to give reliable results.