It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating varie...It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating variety of dynamic behaviors are produced. For some families the behaviors are monotonic in the parameter, while in others they are not [3]. The question is what sort of conditions on a one parameter family will ensure this monotonicity of the behavior with the parameter？ The answer is unknown and will not be given here. What we do instead is to investigate certain geometric-dynamic-combinatorial consequences of assuming that the family has this monotonicity. Specifically, using tools of symbolic dynamics, state space is ＂course grained＂ with a finite alphabet. We decompose a non-invertible map into nonlinear but invertible pieces. From these invertible pieces, we form inverse maps via composition along words. Equations of motion are developed for both forward and inverse orbits （in both the variables of state space and the parameter）, and an equation relating forward and inverse motions at fix-points is exhibited. Finally, we deduce a list of conditions, each of which is equivalent to monotone behavior. One of these conditions states that simple parity characteristics of words correspond to definite dynamics near fixed-points and vice versa.展开更多
Naturally deposited soils are always found in the complex three-dimensional stress state.Constitutive models developed for modeling the three-dimensional mechanical behavior of soils should obey the basic laws of ther...Naturally deposited soils are always found in the complex three-dimensional stress state.Constitutive models developed for modeling the three-dimensional mechanical behavior of soils should obey the basic laws of thermo-mechanical principles.Based on the incremental dissipation function,a new deviatoric shift stress is derived and then introduced into the existing constitutive models to describe the yield behavior in the deviatoric plane for geomaterials.By adopting the proposed shift stress,the relationship between dissipative stress tensors and true stress tensors can be established.Therefore,the threedimensional plastic strain can be calculated reasonably through the associated flow rule in the three-dimensional dissipative stress space.At the same time,three methods that are conventionally adopted for generalizing constitutive models to model the three-dimensional stress-strain relationships are examined under the thermo-mechanical framework.The TS(transformed stress)method is shown to obey the thermo-mechanical rules and the TS space adopted in TS method is actually a translational three-dimensional dissipative stress space.However,it is illustrated that the other two approaches,the method of using failure criterion directly and the method of using g()function,violate the basic rules of thermo-mechanical theories although they may bring convenience and simplicity to numerical analysis for geotechnical engineering.Comparison between model predictions and experimental data confirms the validity of the proposed three-dimensional dissipative stress space.展开更多
文摘It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating variety of dynamic behaviors are produced. For some families the behaviors are monotonic in the parameter, while in others they are not [3]. The question is what sort of conditions on a one parameter family will ensure this monotonicity of the behavior with the parameter？ The answer is unknown and will not be given here. What we do instead is to investigate certain geometric-dynamic-combinatorial consequences of assuming that the family has this monotonicity. Specifically, using tools of symbolic dynamics, state space is ＂course grained＂ with a finite alphabet. We decompose a non-invertible map into nonlinear but invertible pieces. From these invertible pieces, we form inverse maps via composition along words. Equations of motion are developed for both forward and inverse orbits （in both the variables of state space and the parameter）, and an equation relating forward and inverse motions at fix-points is exhibited. Finally, we deduce a list of conditions, each of which is equivalent to monotone behavior. One of these conditions states that simple parity characteristics of words correspond to definite dynamics near fixed-points and vice versa.
基金supported by the National Natural Science Foundation of China (Grants Nos. 11072016,51179003,11272031,51209002)
文摘Naturally deposited soils are always found in the complex three-dimensional stress state.Constitutive models developed for modeling the three-dimensional mechanical behavior of soils should obey the basic laws of thermo-mechanical principles.Based on the incremental dissipation function,a new deviatoric shift stress is derived and then introduced into the existing constitutive models to describe the yield behavior in the deviatoric plane for geomaterials.By adopting the proposed shift stress,the relationship between dissipative stress tensors and true stress tensors can be established.Therefore,the threedimensional plastic strain can be calculated reasonably through the associated flow rule in the three-dimensional dissipative stress space.At the same time,three methods that are conventionally adopted for generalizing constitutive models to model the three-dimensional stress-strain relationships are examined under the thermo-mechanical framework.The TS(transformed stress)method is shown to obey the thermo-mechanical rules and the TS space adopted in TS method is actually a translational three-dimensional dissipative stress space.However,it is illustrated that the other two approaches,the method of using failure criterion directly and the method of using g()function,violate the basic rules of thermo-mechanical theories although they may bring convenience and simplicity to numerical analysis for geotechnical engineering.Comparison between model predictions and experimental data confirms the validity of the proposed three-dimensional dissipative stress space.
