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Tipping points associated with bifurcations (B-tipping) or induced by noise (N-tipping) are recognized mechanisms that may potentially lead to sudden climate change. We focus here on a novel class of tipping points, where a sufficiently rapid change to an input or parameter of a system may cause the system to 'tip' or move away from a branch of attractors. Such rate-dependent tipping, or R-tipping, need not be associated with either bifurcations or noise. We present an example of all three types of tipping in a simple global energy balance model of the climate system, illustrating the possibility of dangerous rates of change even in the absence of noise and of bifurcations in the underlying quasi-static system.

Oscillatory pumping tests presents a potential advancement in aquifer testing as they minimize hydraulic perturbations and result in no net water ion or injection. However, aquifer property estimates can be less accurate if they are compromised by well head losses during the pumping test. Currently, oscillatory pumping test models ignore these losses. The objective of this study is to develop and evaluate an analytical solution that can interpret oscillatory pumping test data affected by rate‐dependent skin losses. Accordingly, the sensitivity of the drawdown response to small variations in aquifer and well parameters was analyzed. In addition, a previous field aquifer test was analyzed to evaluate the feasibility of the solution for aquifer parameter estimation. The results indicate that neglecting the rate‐dependent skin effect can lead to substantial errors in the determination of aquifer parameters such as transmissivity and storativity. Additionally, the proposed solution effectively reproduces the sharp groundwater level peaks observed in field data, whereas the solution that does not consider the rate‐dependent skin effect greatly underestimates the peak values. Plain Language Summary This study looks at a new method called the oscillatory pump test for measuring aquifer properties. There can be problems with obtaining accurate results when testing aquifer properties, because there can be a loss of water at the top of the well. This study aims to solve this problem by creating a new solution that can account for this loss of water and still yield accurate results. The study found that ignoring this loss of water can lead to significant errors in the estimation of the aquifer properties. The study shows that when the value of the loss of water is over 10,000 min/m3, it can have a substantial effect on the accuracy of the results. Overall, this study improves our ability to estimate aquifer properties using the oscillatory pumping test method. Key Points An analytical solution is developed for oscillatory pumping tests with rate‐dependent skin The effects of rate‐dependent skin on the head response are evident in field data Neglecting rate‐dependent skin can lead to biased estimates of aquifer parameters

There are several models for magnetic hysteresis. Their key purposes are to model magnetization curves with a history dependence to achieve hysteresis cycles without a frequency dependence. There are different approaches to handling history dependence. The two main categories are Duhem-type models and Preisach-type models. Duhem models handle it via a simple directional dependence on the flux rate, without a proper memory. While the Preisach type model handles it via memory of the point where the direction of the flux rate is changed. The most common Duhem model is the phenomenological Jiles–Atherton model, with examples of other models including the Coleman–Hodgdon model and the Tellinen model. Examples of Preisach type models are the classical Preisach model and the Prandtl–Ishlinskii model, although there are also many other models with adoptions of a similar history dependence. Hysteresis is by definition rate-independent, and thereby not dependent on the speed of the alternating flux density. An additional rate dependence is still important and often included in many dynamic hysteresis models. The Chua model is common for modeling non-linear dynamic magnetization curves; however, it does not define classical hysteresis. Other similar adoptions also exist that combine hysteresis modeling with eddy current modeling, similar to how frequency dependence is included in core loss modeling. Most models are made for scalar values of alternating fields, but there are also several models with vector generalizations that also consider three-dimensional directions.

Microbes in the wild face highly variable and unpredictable environments and are naturally selected for their average growth rate across environments. Apart from using sensory regulatory systems to adapt in a targeted manner to changing environments, microbes employ bet-hedging strategies where cells in an isogenic population switch stochastically between alternative phenotypes. Yet, bet-hedging suffers from a fundamental trade-off: Increasing the phenotype-switching rate increases the rate at which maladapted cells explore alternative phenotypes but also increases the rate at which cells switch out of a well-adapted state. Consequently, it is currently believed that bet-hedging strategies are effective only when the number of possible phenotypes is limited and when environments last for sufficiently many generations. However, recent experimental results show that gene expression noise generally decreases with growth rate, suggesting that phenotype-switching rates may systematically decrease with growth rate. Such growth rate dependent stability (GRDS) causes cells to be more explorative when maladapted and more phenotypically stable when well-adapted, and we show that GRDS can almost completely overcome the trade-off that limits bet-hedging, allowing for effective adaptation even when environments are diverse and change rapidly. We further show that even a small decrease in switching rates of faster-growing phenotypes can substantially increase long-term fitness of bet-hedging strategies. Together, our results suggest that stochastic strategies may play an even bigger role for microbial adaptation than hitherto appreciated.

The phase-field model has been employed for fracture analysis over the last decade. One of the main advantages is that the fracture evolution does not depend on any explicit criterion. Good agreement is obtained by comparing the results of simulations and experiments. Most elastomers exhibit both elastic and viscous behavior simultaneously, which yields rate-dependent properties for both the mechanical and fracture responses. In this contribution, a viscoelastic rheological model based on Reese and Govindjee (Int J Solids Struct 35:3455–3482, 1998) is coupled to phase-field modeling to investigate rate-dependent fracture within elastomers. The elastic strain energy potential supposed to evolve fracture is provided by both the equilibrium and non-equilibrium branches. The fracture mechanism is characterized by a volumetric–isochoric split, which specifies a varying driving force in the case of tensile or compressive deformation. Representative numerical examples are studied and related findings and potential perspectives are summarized to close the paper.

Fracture of materials with rate-dependent mechanical behaviour, e.g. polymers, is a highly complex process. For an adequate modelling, the coupling between rate-dependent stiffness, dissipative mechanisms present in the bulk material and crack driving force has to be accounted for in an appropriate manner. In addition, the resistance against crack propagation can depend on rate of deformation. In this contribution, an energetic phase-field model of rate-dependent fracture at finite deformation is presented. For the deformation of the bulk material, a formulation of finite viscoelasticity is adopted with strain energy densities of Ogden type assumed. The unified formulation allows to study different expressions for the fracture driving force. Furthermore, a possibly rate-dependent toughness is incorporated. The model is calibrated using experimental results from the literature for an elastomer and predictions are qualitatively and quantitatively validated against experimental data. Predictive capabilities of the model are studied for monotonic loads as well as creep fracture. Symmetrical and asymmetrical crack patterns are discussed and the influence of a dissipative fracture driving force contribution is analysed. It is shown that, different from ductile fracture of metals, such a driving force is not required for an adequate simulation of experimentally observable crack paths and is not favourable for the description of failure in viscoelastic rubbery polymers. Furthermore, the influence of a rate-dependent toughness is discussed by means of a numerical study. From a phenomenological point of view, it is demonstrated that rate-dependency of resistance against crack propagation can be an essential ingredient for the model when specific effects such as rate-dependent brittle-to-ductile transitions shall be described.

Objective To evaluate the strain‐rate‐dependent viscoelastic properties of the intervertebral disc by in vitro experiments. Method The biomechanical experiments were conducted from September 2019 to December 2019. The lumbar spines of sheep were purchased within 4–6 hours from the local slaughterhouse, and the intervertebral disc samples were divided into three groups. In rupture group, the samples were used to test the mechanical behavior of the intervertebral disc rupture at different strain rates. In fatigue injury group, the samples were used to test the mechanical behavior of fatigue injury on the intervertebral disc under different strain rates. In internal displacement group, the samples were used to test the internal displacement distribution of the intervertebral disc at different strain rates by applying an optimized digital image correlation (DIC) technique. Results Both the yielding and cracking phenomenon occurs at fast and medium loading rates, while only the yielding phenomenon occurs at a slow loading rate. The yield stress, compressive strength, and elastic modulus all increase with the increase of the strain rate, while the yield strain decreases with the increase of the strain rate. The logarithm of the elastic modulus in the intervertebral disc is approximately linear with the logarithm of the strain rate under different strain rates. Both before and after fatigue loading, the stiffness in the loading and unloading curves of the intervertebral disc is inconsistent, forming a hysteresis loop, which is caused by the viscoelastic effect. The strain rate has no significant effect on the internal displacement distribution of the intervertebral disc. Based on the experimental data, the constitutive relationship of the intervertebral disc at different strain rates is obtained. The fitting curves are well coupled with the experimental data, while the fitting parameters are approximately linear with the logarithm of the strain rate. Conclusions These experiments indicate that the strain rate has a significant effect on the mechanical behavior of the intervertebral disc rupture and fatigue injury, while the constitutive equation can predict the rate‐dependent mechanical behavior of lumbar intervertebral disc under flexion very well. These results have important theoretical guiding significance for preventing lumbar disc herniation in daily life. These experiments indicate that the strain rate has a significant effect on the mechanical behavior of the intervertebral disc rupture and fatigue injury, while the constitutive equation can predict the rate‐dependent mechanical behavior of lumbar intervertebral disc under flexion. These results have important theoretical guiding significance for preventing disc herniation

The promising phase-field method has been intensively studied for crack approximation in brittle materials. The realistic representation of material degradation at a fully evolved crack is still one of the main challenges. Several energy split formulations have been postulated to describe the crack evolution physically. A recent approach based on the concept of representative crack elements (RCE) in Storm et al. (The concept of representative crack elements (RCE) for phase-field fracture: anisotropic elasticity and thermo-elasticity. Int J Numer Methods Eng 121:779–805, 2020) introduces a variational framework to derive the kinematically consistent material degradation. The realistic material degradation is further tested using the self-consistency condition, which is particularly compared to a discrete crack model. This work extends the brittle RCE phase-field modeling towards rate-dependent fracture evolution in a viscoelastic continuum. The novelty of this paper is taking internal variables due to viscoelasticity into account to determine the crack deformation state. Meanwhile, a transient extension from Storm et al. (The concept of representative crack elements (RCE) for phase-field fracture: anisotropic elasticity and thermo-elasticity. Int J Numer Methods Eng 121:779–805, 2020) is also considered. The model is derived thermodynamic-consistently and implemented into the FE framework. Several representative numerical examples are investigated, and consequently, the according findings and potential perspectives are discussed to close this paper.

A critical examination is made of two classes of strain gradient plasticity theories currently available for studying micrometre-scale plasticity. One class is characterized by certain stress quantities expressed in terms of increments of strains and their gradients, whereas the other class employs incremental relationships between all stress quantities and the increments of strains and their gradients. The specific versions of the theories examined coincide for proportional straining. Implications stemming from the differences in formulation of the two classes of theories are explored for two basic examples having non-proportional loading: (i) a layer deformed into the plastic range by tensile stretch with no constraint on plastic flow at the surfaces followed by further stretch with plastic flow constrained at the surfaces and (ii) a layer deformed into the plastic range by tensile stretch followed by bending. The marked difference in predictions by the two theories suggests that critical experiments will be able to distinguish between them.

The nonlinear mechanical behavior and temperature sensitivity of HTPB propellant for solid rocket motors were investigated. The rate-dependent mechanical properties of the propellant were examined through a combination of experiments and numerical simulations. Experimental results demonstrate that the tensile mechanical properties of HTPB propellant are rate-dependent at 223 K and 323 K; stresses at a given strain gradually increase with increasing strain rate. By use a generalized nonlinear ZWT intrinsic model, the tensile mechanical behavior of HTPB propellant under a wide range of strain rates was to described. A numerical simulation of a uniaxial tensile test was performed using a UMAT subroutine. The results demonstrate that the model accurately represents the mechanical properties of the HTPB propellant.