Rheological-dynamical analogy - Papers

**lisine** · (This post was last modified: 05-22-2012, 02:50 AM by usman.)

Rheological-Dynamical Theory of Visco-Elasto-Plasticity and Fatigue: Part 1

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Rheological-Dynamical Theory of Visco-Elasto-Plasticity and Fatigue: Part 2

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Rheological-Dynamical Analogy: Prediction of damping parameters of hysteresis damper

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**lisine** · 05-21-2012, 12:16 PM

Rheological-Dynamical Theory - Papers

Rheological-Dynamical Theory of Visco-Elasto-Plasticity and Fatigue: Part 1

This paper is concerned with a new proposal regarding the analysis of visco-elastoplasticity and fatigue and is based on rheological-dynamical theory. Due to the analogy between rheological model and dynamical model with viscous damping, it becomes obvious that inelastic response of members is essentially a dynamical problem. An analytical rheological-dynamical viscoelasto- plastic solution of one-dimensional longitudinal continuous vibration under loading and solution for the stress relaxation as unloading have been developed and used to obtain the fatigue limit of thin long bars. Rheologic behavior of the bar can be characterized by one parameter, like in a single-degree-of-freedom spring mass system. In all inelastic strains time rate effects are always present to some degree. Whether or not their exclusion has a significant influence on the prediction of the material fatigue behavior depends upon several factors like: maximum absolute stress in the cycle, coefficient of asymmetry of cycle, creep coefficient, slope of the strain hardening portion of the stress-strain curve, relative frequency and uniaxial yield stress. This paper provides description of dynamic magnification factor, relaxation of stress, stress concentration and the fatigue limit of thin long symmetrical bars.

Rheological-Dynamical Theory of Visco-Elasto-Plasticity and Fatigue: Part 2

The paper deals with the rheological-dynamical analogy in which the three-dimensional stress-strain relations are defined under cyclic variation of stress for Hencky’s total strain theory. In many practical visco-elasto-plastic problems, like as multiaxial fatigue under loading at constant stress amplitude and constant stress ratio, the load-carrying members are subjected to proportional loading. The classical Hencky’s theory has the advantage of mathematical convenience but its disadvantage is that the deformations predicted for the volume element are independent of the loading path. The existing formulations of the constitutive models for metals are mainly based on the Prandtl-Reuss incremental theory of elasto-plasticity, slip theory of plasticity or continuum damage mechanics. They have been shown capable of reproducing satisfactorily most experimental results available for metallic specimens. However, from the theoretical viewpoint little has been said about how these formulations relate to realistic predicting many different inelastic and time dependent problems of two- or threedimensional solids, such as fatigue, discontinuous plastic deformation etc. In this paper, fundamentally new aspect of isochronous constitutive relations for Hencky’s theory, which are dependent of the each loading path, is achieved by systematically introducing RDA concept into the continuum framework. Specific inelastic and fatigue formulation of triaxial state of stress is developed and discussed within the new theoretical tool and related to von Mises plasticity..

Rheological–dynamical analogy: Prediction of damping parameters of hysteresis damper

This research aims to predict the damping parameters of hysteresis damper based on an analytical rheological–dynamical (RDA) visco-elasto-plastic solution of one-dimensional longitudinal continuous vibrations of a bar. A visco-elasto-plastic bar or damper is an energy dissipation device. An attempt is made to estimate quantitatively the influence of material physical parameters of materials on the damping ratio in both the linear visco-elastic analysis and the nonlinear visco-elasto-plastic analysis of damper subjected to external vibration forces. Two types of damping are considered: viscous damping in the case of linear analysis, defined as stiffness and/or mass proportional and, in the case of nonlinear analysis, hysteresis damping caused by inelastic deformations of damper. Owing to the visco-elastic nature of the materials of the damper and the frequency dependence of the viscous damping ratio ξ, it is useful to consider separately the situations arising when ξ is positive (the system is stable) and when it is negative. A negative damping ratio means that the complementary solution of the response would not die away (the system is unstable because of factor eξ · ω · t). In the case of nonlinear analysis, the force–displacement relation is nonlinear, so it is very difficult to predict the actual damping and stiffness coefficients, even if the force–displacement characteristic is simply perfect elasto-plastic. Using the RDA method, which takes into account the rate of release of visco-elasto-plastic energy of the dissipation devices; nonlinear behaviors are linearized, enabling to obtain the equivalent damping and stiffness coefficients and the effective period for the damper.

Rheological–dynamical analogy: Visco-elasto-plastic behavior of metallic bars

This paper presents an application of the rheological–dynamical analogy (RDA) for describing the various aspects of a visco-elasto-plastic behavior of metallic bars related to standard tensile tests. The analogy has been developed on the basis of mathematical–physical analogy between a visco-elasto-plastic rheological model and a dynamical model with viscous damping, and is aimed to be used for the analysis of inelastic deforming of materials and structures. In this presentation, the aim will be to highlight the thermodynamics aspect of proportional stress through hysteretic loop dissipation, oscillation in the stress–strain curve (lower and upper yield point), transition from plasticity range, transition from strain hardening range, and RDA fracture stress of thin long metallic bars. This paper provides description of process of visco-elasto-plastic yielding and numerical example of obtaining isochronous σ–ε diagram of metallic bars using RDA similitude. In order to demonstrate the ability of the RDA modeling technique, the comparison with experimental and numerical results by [ASCE, J. Eng. Mech. 125(12) (1999) 1243] is presented. The presented RDA analysis can be readily used to perform precise shape of isochronous σ–ε diagrams of metallic bars. The RDA isochronous stress–strain diagram is used to predict the loading functions for the material of metallic bars.

Rheological–dynamical analogy: Prediction of buckling curves of columns

This paper is concerned with a new proposal regarding the analysis of inelastic deforming of materials and structures and is based on mathematical–physical analogy between rheological model and dynamical model with viscous damping. Due to the analogy, it becomes obvious that inelastic response of engineering structures is essentially a dynamical problem. The analogy exists for this specific model, and is one out of many examples of the analogies that can be observed in mechanics, as well as between mechanical and electrical (thermal, magnetic, etc.) systems, by virtue of their mathematical descriptions. Generally speaking, the rheological–dynamical analogy (RDA) is derived in order to solve the dynamical problems, but can also be used to explain any statical problems, considering the correspondent limit values of presented mathematical expressions. The paper deals with the buckling problem regarding steel, timber and concrete columns. Using the examples of typical columns, it has been demonstrated that results obtained by RDA are in good accordance with those available in cited references.

Rheological–dynamical analogy: Modeling of fatigue behavior

Present paper relates to the analysis of fatigue and fatigue failure of thin long steel bars with the application of a new rheological model and rheological–dynamical analogy (RDA). The analogy has been developed on the basis of mathematical–physical analogy between rheological model and dynamical model with viscous damping, and is aimed to be used for the analysis of inelastic deforming of materials and structures. In this presentation, the aim will be to highlight different aspects of fatigue behavior and fill the gap that other methods cannot. This paper provides a numerical example of obtaining S–N curves of thin long steel bars using RDA model and description of the hysteretic energy dissipation of material subjected to cyclic stresses. First the axial fatigue setup and the experimental results are discussed. The latter in order to demonstrate the ability of the RDA modeling technique, the comparison with Griffith’s theory of fracture is presented. RDA method for fatigue crack growth rate is proposed and compared with Paris power law. On the basis of the comparisons, the present RDA method could be considered as valid and suitable for modeling of fatigue behavior.

Fatigue crack growth and failure of inelastic rods based on rheological–dynamical analogy

This paper presents a study on how the rheological–dynamical analogy (RDA) can be used to predict new notch stress intensities of inelastic rods, where such intensities can be employed to examine the fatigue crack growth and failure. The analogy stems from the mathematical analogy between the rheological model and the viscous damping dynamical model. Physical basis for the introduction of such an analogy is the elastic wave propagation. Based on this theory, notch-induced fatigue crack growth and failure of rods are quantified by some characterizing parameters namely, fatigue failure frequency, crack depth, fatigue strength, cyclic toughness, crack width, crack opening displacement and cyclic stress intensity factor range. A number of experimental results reported in the literature are used to support the present analysis which indicates that the RDA method provides a possible means for examining fatigue effects.

Rheological-dynamical analogy: Design of viscoelastic and viscoplastic bar dampers

An analytical procedure is developed for computing the dynamic response of bar dampers using the rheological-dynamical analogy. This method is very efficient when applied
to inelastic deformations, because it reduces a material nonlinear problem to a linear dynamical problem, which allows the use of modal analysis. Two types of damping are considered: viscous damping in the case of linear analysis; and hysteretic damping caused by inelastic deformations of the damper. Based on the formulas of the two types of damping, an iterative procedure for the design of viscoelastic and viscoplastic bar dampers is derived.

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