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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
in which λ1 and λ2 are the relaxation time and the retardation time, respectively. The most general linear viscoelastic model, which includes both the above forms, is the generalized Maxwell model. It is conve- TABLE 37.2 Common Constitutive Equations Included in the EightConstant Oldroyd Model
Numerical simulation of dynamic responses of basalt fiber reinforced asphalt pavement under moving loads
Published in Sandra Erkens, Xueyan Liu, Kumar Anupam, Yiqiu Tan, Functional Pavement Design, 2016
Peiqing Wang, Fujian Wang, Mingqiang Ke, Zhigang Lu, Houquan Zhang, Zhiyuan Zeng
Laboratory test on basalt fiber reinforced asphalt mixture had been completed, which investigated the mixture performance and were used for pavement structural analysis. Viscoelasticity parameter was obtained through generalized Maxwell model which is connected parallelly by several Maxwell models and the Maxwell model is a series connection of Young model and viscosity coefficient. Rutting numerical simulation was also conducted in ABAQUS and the result was compared to that in laboratory test.
Practical methodology of path-tracing analysis for buckling process in cylindrical shells
Published in Alphose Zingoni, Insights and Innovations in Structural Engineering, Mechanics and Computation, 2016
T. Kobayashi, Y. Mihara, F. Fujii
The generalized Maxwell model is the most general form of the linear model of viscoelasticity. It takes into account that the relaxation does not occur at a single time but at a distribution of times. The coefficients for the generalized Maxwell model were obtained from the master curve via the optimization approach so that the relaxation curve from Equation (6) could be calculated numerically. The relationship between time-domain constants and frequency-do-main constants is expressed as shown in Equation (7) and Equation (8). Using these formula, the numerical constants for the generalized Maxwell model (Equation (6)), which provides a close approximation of the actual measurement data, can be obtained. () Er(t′)=Ee+∑n−1NEnexp(−t′/τn) () E′=Ee+∑n=1Nτn2ω′21+τn2ω′2En () E′′=∑n=1Nτnω′1+τn2ω′2En
Constrained shape optimization of free-form shells considering material creep
Published in Engineering Optimization, 2022
Bingbing San, Haiyun He, Dongming Feng, Ye Qiu, Yanting Huang
The typical feature of creep, as a viscoelastic phenomenon, is time-dependent deformation under sustained applied loads (Kai and Waisman 2015). When the applied load is constant, viscoelastic materials exhibit a unique response that consists of two different components. One is the elastic component, while the other is a progressive and gradual response increasing with time. Therefore, the behaviour of viscoelastic materials can be modelled by combining simple viscous and elastic elements (Marques and Creus 2012). The generalized Maxwell model is one commonly used approach, which describes viscoelastic behaviour as a combination of springs and dashpots arranged in an array of series and parallel clusters. It can be formulated as a Prony series function, such as where E∞ is the long-term elastic modulus, Np is the number of Prony terms; τj and Ej are the relaxation time and elastic modulus of the jth Maxwell branch, respectively, which can be calibrated on the basis of experimental data (Park and Kim 2001); and E(t) describes the relaxation process of Young’s modulus, which can be deemed as the inverse of creep deformation. The quantity of the relaxation function is treated as the viscoelastic equivalent of the Young’s modulus in elastic materials.
Viscoelastic–viscoplastic model for short-fiber-reinforced composites with complex fiber orientation
Published in Mechanics of Advanced Materials and Structures, 2019
M. Nciri, D. Notta-Cuvier, F. Lauro, F. Chaari, Y. Maalej, B. Zouari
The viscoelastic part of the response is modeled using the linear Wiechert model (i.e., generalized Maxwell model) consisting in a linear elastic Hooke element arranged in parallel with a finite number of Maxwell elements. It is to note here that the choice of a linear viscoelastic model is considered to be appropriate in the case of composites with relatively high volume fraction of reinforcement. However, the modeling scheme allows its adaptability to all kind of matrix model of behavior. If needed, the model can therefore theoretically be extended to nonlinear viscoelastic constitutive laws. The viscoelastic response is updated, when necessary, using a viscoplastic scheme. It is worth noting that physical basis are associated with this strain decomposition. Indeed, several micromechanical observations tend to show that in the case of some polymers, e.g., semi-crystalline thermoplatics, the microstructure presents crystalline lamellae and amorphous chains. Viscoelastic and viscoplastic models can be associated with these microstructures assembled in series; e.g., [34], [35]. Modeling of matrix viscoelastic–viscoplastic behavior is presented in the next paragraphs.
Experimental and Numerical Study of a New Proposed Seismic Isolator Using Steel Rings (SISR)
Published in Journal of Earthquake Engineering, 2020
Habibollah Kakolvand, Mohammad Ghazi, Behnam Mehrparvar, Soroush Parvizi
The mechanical behavior model of viscoelastic materials could be represented by springs and dampers in parallel or series, as seen in the models by Maxwell and Kelvin-Voigt, respectively (Brinson and Brinson 2008). The Generalized Maxwell model is the most general form of the linear model for viscoelasticity. It takes into account that the relaxation does not occur at a single time, but at a distribution of times (Abaqus manual 2014). The generalized Maxwell model in Fig. 23, represents the rheological model of the stress relaxation function G (t) using the Prony series for the shear relaxation of the following form: