Long-range automaton models of earthquakes: Power-law accelerations, correlation evolution, and mode-switching

We introduce a conceptual model for the in-plane physics of an earthquake fault. The model employs cellular automaton techniques to simulate tectonic loading, earthquake rupture, and strain redistribution. The impact of a hypothetical crustal elastodynamic Green's function is approximated by a long-range strain redistribution law with a r(-p) dependance. We investigate the influence of the effective elastodynamic interaction range upon the dynamical behaviour of the model by conducting experiments with different values of the exponent (p). The results indicate that this model has two distinct, stable modes of behaviour. The first mode produces a characteristic earthquake distribution with moderate to large events preceeded by an interval of time in which the rate of energy release accelerates. A correlation function analysis reveals that accelerating sequences are associated with a systematic, global evolution of strain energy correlations within the system. The second stable mode produces Gutenberg-Richter statistics, with near-linear energy release and no significant global correlation evolution. A model with effectively short-range interactions preferentially displays Gutenberg-Richter behaviour. However, models with long-range interactions appear to switch between the characteristic and GR modes. As the range of elastodynamic interactions is increased, characteristic behaviour begins to dominate GR behaviour. These models demonstrate that evolution of strain energy correlations may occur within systems with a fixed elastodynamic interaction range. Supposing that similar mode-switching dynamical behaviour occurs within earthquake faults then intermediate-term forecasting of large earthquakes may be feasible for some earthquakes but not for others, in alignment with certain empirical seismological observations. Further numerical investigation of dynamical models of this type may lead to advances in earthquake forecasting research and theoretical seismology.

A modified thermal viscoplastic constitutive law involving the effect of temperature rise rate

At high temperature rise rate, the mechanical properties of 10 # steel were determined experimentally in a very wide range of temperature and strain rates. A new constitutive relationship was put forward, which can fit with the experimental results and describe various phenomena observed in our experiments. Meanwhile, some interesting characteristics about the temperature rise rate, strain and strain rate hardening and thermal softening are also shown in this paper. Finally, the reliability of the constitutive law and the correctness of the constitutive parameters were verified by comparing the calculation results with the experimental data.

A new hardening law for strain gradient plasticity

A new hardening law of the strain gradient theory is proposed in this paper, which retains the essential structure of the incremental version of conventional J(2) deformation theory and obeys thermodynamic restrictions. The key feature of the new proposal is that the term of strain gradient plasticity is represented as an internal variable to increase the tangent modulus. This feature which is in contrast to several proposed theories, allows the problem of incremental equilibrium equations to be stated without higher-order stress, higher-order strain rates or extra boundary conditions. The general idea is presented and compared with the theory given by Fleck and Hutchinson (Adv. in Appl. Mech. (1997) 295). The new hardening law is demonstrated by two experimental tests i.e. thin wire torsion and ultra-thin beam bending tests. The present theoretical results agree well with the experiment results.

Scaling relationships in indentation of power-law creep solids using self-similar indenters

We use dimensional analysis to derive scaling relationships for self-similar indenters indenting solids that exhibit power-law creep. We identify the parameter that represents the indentation strain rate. The scaling relationships are applied to several types of indentation creep experiment with constant displacement rate, constant loading rate or constant ratio of loading rate over load. The predictions compare favourably with experimental observations reported in the literature. Finally, a connection is found between creep and 'indentation-size effect' (i.e. changing hardness with indentation depth or load).

Extended self-similar scaling law of multi-scale eddy structure in wall turbulence

The longitudinal fluctuating velocity of a turbulent boundary layer was measured in a water channel at a moderate Reynolds number. The extended self-similar scaling law of structure function proposed by Benzi was verified. The longitudinal fluctuating velocity, in the turbulent boundary layer was decomposed into many multi-scale eddy structures by wavelet transform. The extended self-similar scaling law of structure function for each scale eddy velocity was investigated. The conclusions are I) The statistical properties of turbulence could be self-similar not only at high Reynolds number, but also at moderate and low Reynolds number, and they could be characterized by the same set of scaling exponents xi (1)(n) = n/3 and xi (2)(n) = n/3 of the fully developed regime. 2) The range of scales where the extended self-similarity valid is much larger than the inertial range and extends far deep into the dissipation range,vith the same set of scaling exponents. 3) The extended selfsimilarity is applicable not only for homogeneous turbulence, but also for shear turbulence such as turbulent boundary layers.

Asymptotic analysis based on K-S constitutive law for a rubber wedge compressed by a line load at its tip

In the present paper, a rubber wedge compressed by a line load at its tip is asymptotically analyzed using a special constitutive law proposed by Knowles and Sternberg (K-S elastic law) [J. Elasticity 3 (1973) 67]. The method of dividing sectors proposed by Gao [Theoret. Appl. Fract, Mech. 14 (1990) 219] is used. Domain near the wedge tip can be divided into one expanding sector and two narrowing sectors. Asymptotic equations of the strain-stress field near the wedge tip are derived and solved numerically. The deformation pattern near a wedge tip is completely revealed. A special case. i.e. a half space compressed by a line load is solved while the wedge angle is pi.

Nonlinear Constitutive Behavior Of Ferroelectric Materials

The ferroelectric specimen is considered as an aggregation of many randomly oriented domains. According to this mechanism, a multi-domain mechanical model is developed in this paper. Each domain is represented by one element. The applied stress and electric field are taken to be the stress and electric field in the formula of the driving force of domain switching for each element in the specimen. It means that the macroscopic switching criterion is used for calculating the volume fraction of domain switching for each element. By using the hardening relation between the driving force of domain switching and the volume fraction of domain switching calibrated, the volume fraction of domain switching for each element is calculated. Substituting the stress and electric field and the volume fraction of domain switching into the constitutive equation of ferroelectric material, one can easily get the strain and electric displacement for each element. The macroscopic behavior of the ferroelectric specimen is then directly calculated by volume averaging. Meanwhile, the nonlinear finite element analysis for the ferroelectric specimen is carried out. In the finite element simulation, the volume fraction of domain switching for each element is calculated by using the same method mentioned above. The interaction between different elements is taken into account in the finite element simulation and the local stress and electric field for each element is obtained. The macroscopic behavior of the specimen is then calculated by volume averaging. The computation results involve the electric butterfly shaped curves of axial strain versus the axial electric field and the hysteresis loops of electric displacement versus the electric field for ferroelectric specimens under the uniaxial coupled stress and electric field loading. The present theoretical prediction agrees reasonably with the experimental results.

Energy Principle And Nonlinear Electric-Mechanical Behavior Of Ferroelectric Ceramics

Many experimental observations have shown that a single domain in a ferroelectric material switches by progressive movement of domain walls, driven by a combination of electric field and stress. The mechanism of the domain switch involves the following steps: initially, the domain has a uniform spontaneous polarization; new domains with the reverse polarization direction nucleate, mainly at the surface, and grow though the crystal thickness; the new domain expands sideways as a new domain continues to form; finally, the domain switch coalesces to complete the polarization reversal. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of the ferroelectric material and used to study the nonlinear constitutive behavior of a ferroelectric body in this paper. The principle of stationary total potential energy is put forward in which the basic unknown quantities are the displacement u(i), electric displacement D-i and volume fraction rho(I) of the domain switching for the variant I. The mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total potential energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion established by Hwang et al. [ 1]. Based on the domain switching criterion, a set of linear algebraic equations for determining the volume fraction rho(I) of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. If the volume fraction rho(I) of domain switching for each domain is prescribed, the unknown displacement and electric potential can be obtained based on the conventional finite element procedure. It is assumed that a domain switches if the reduction in potential energy exceeds a critical energy barrier. According to the experimental results, the energy barrier will strengthen when the volume fraction of the domain switching increases. The external mechanical and electric loads are increased step by step. The volume fraction rho(I) of domain switching for each element obtained from the last loading step is used as input to the constitutive equations. Then the strain and electric fields are calculated based on the conventional finite element procedure. The finite element analysis is carried out on the specimens subjected to uniaxial coupling stress and electric field. Numerical results and available experimental data are compared and discussed. The present theoretic prediction agrees reasonably with the experimental results.

Reconsideration On The Role Of The Specific Heat Ratio In Arrhenius Law Applications

Arrhenius law implicates that only those molecules which possess the internal energy greater than the activation energy E-a can react. However, the internal energy will not be proportional to the gas temperature if the specific heat ratio gamma and the gas constant R vary during chemical reaction processes. The varying gamma may affect significantly the chemical reaction rate calculated with the Arrhenius law under the constant gamma assumption, which has been widely accepted in detonation and combustion simulations for many years. In this paper, the roles of variable gamma and R in Arrhenius law applications are reconsidered, and their effects on the chemical reaction rate are demonstrated by simulating one-dimensional C-J and two-dimensional cellular detonations. A new overall one-step detonation model with variable gamma and R is proposed to improve the Arrhenius law. Numerical experiments demonstrate that this improved Arrhenius law works well in predicting detonation phenomena with the numerical results being in good agreement with experimental data.

Study of drag reduction by gas injection for power-law fluid flow in horizontal stratified and slug flow regimes

In this work, the drag reduction by gas injection for power-law fluid flow in stratified and slug flow regimes has been studied. Experimentswere conducted to measure the pressure gradient within air/CMC solutions in a horizontal Plexiglas pipe that had a diameter of 50mm and a length of 30 m. The drag reduction ratio in stratified flow regime was predicted using the two-fluid model. The results showed that the drag reduction should occur over the large range of the liquid holdup when the flow behaviour index remained at the low value. Furthermore, for turbulent gas-laminar liquid stratified flow, the drag reduction by gas injection for Newtonian fluid was more effective than that for shear-shinning fluid, when the dimensionless liquid height remained in the area of high value. The pressure gradient model for a gas/Newtonian liquid slug flow was extended to liquids possessing the Ostwald–de Waele power law model. The proposed model was validated against 340 experimental data point over a wide range of operating conditions, fluid characteristics and pipe diameters. The dimensionless pressure drop predicted was well inside the 20% deviation region for most of the experimental data. These results substantiated the general validity of the model presented for gas/non-Newtonian two-phase slug flows.

Crack propagation in the power-law nonlinear viscoelastic material

An analysis on crack creep propagation problem of power-law nonlinear viscoelastic materials is presented. The creep incompressilility assumption is used. To simulate fracture behavior of craze region, it is assumed that in the fracture process zone near the crack tip, the cohesive stress sigma(f) acts upon the crack surfaces and resists crack opening. Through a perturbation method, i. e., by superposing the Mode-I applied force onto a referential uniform stress state, which has a trivial solution and gives no effect on the solution of the original problem, the nonlinear viscoelastic problem is reduced to linear problem. For weak nonlinear materials, for which the power-law index n similar or equal to 1, the expressions of stress and crack surface displacement are derived. Then, the fracture process zone local energy criterion is proposed and based on which the formulas of cracking incubation time t

Nonlinear fatigue damage model based on the residual strength degradation law

In this paper, a logarithmic expression to describe the residual strength degradation process is developed in order to fatigue test results for normalized carbon steel. The definition and expression of fatigue damage due to symmetrical stress with a constant amplitude are also given. The expression of fatigue damage can also explain the nonlinear properties of fatigue damage. Furthermore, the fatigue damage of structures under random stress is analyzed, and an iterative formula to describe the fatigue damage process is deduced. Finally, an approximate method for evaluating the fatigue life of structures under repeated random stress blocking is presented through various calculation examples.

Energy principle of ferroelectric ceramics and single domain mechanical model

Many physical experiments have shown that the domain switching in a ferroelectric material is a complicated evolution process of the domain wall with the variation of stress and electric field. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of ferroelectric ceramic and used to study the nonlinear constitutive behavior of ferroelectric body in this paper. The principle of stationary total energy is put forward in which the basic unknown quantities are the displacement u (i) , electric displacement D (i) and volume fraction rho (I) of the domain switching for the variant I. Mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion. On the basis of the domain switching criterion, a set of linear algebraic equations for the volume fraction rho (I) of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. Then a single domain mechanical model is proposed in this paper. The poled ferroelectric specimen is considered as a transversely isotropic single domain. By using the partial experimental results, the hardening relation between the driving force of domain switching and the volume fraction of domain switching can be calibrated. Then the electromechanical response can be calculated on the basis of the calibrated hardening relation. The results involve the electric butterfly shaped curves of axial strain versus axial electric field, the hysteresis loops of electric displacement versus electric filed and the evolution process of the domain switching in the ferroelectric specimens under uniaxial coupled stress and electric field loading. The present theoretic prediction agrees reasonably with the experimental results given by Lynch.