scholarly journals Modeling Step—Strain Relaxation and Cyclic Deformations of Elastomers

2002 ◽  
Vol 75 (2) ◽  
pp. 333-345
Author(s):  
A. R. Johnson ◽  
T. Chen ◽  
J. L. Mead
Keyword(s):  
Least Squares Method ◽  
Large Strain ◽  
Strain Relaxation ◽  
Nonlinear Least Squares ◽  
Friction Stress ◽  
Maxwell Model ◽  
Constitutive Theory ◽  
Relaxation Data ◽  
Nonlinear Least Squares Method ◽  
Step Strain

Abstract Data for step—strain relaxation and cyclic compressive deformations of highly viscous short elastomer cylinders are modeled using a large strain rubber viscoelastic constitutive theory with a rate—independent friction stress term added. In the tests, both small and large amplitude cyclic compressive strains, in the range of 1% to 10%, were superimposed on steady state compressed strains, in the range of 5% to 20%, for frequencies of 1 and 10 Hz. The elastomer cylinders were conditioned prior to each test to soften them. The constants in the viscoelastic—friction constitutive theory are determined by employing a nonlinear least-squares method to fit the analytical stresses for a Maxwell model, which includes friction, to measured relaxation stresses obtained from a 20% step—strain compression test. The simulation of the relaxation data with the nonlinear model is successful at compressive strains of 5%, 10%, 15%, and 20%. Simulations of hysteresis stresses for enforced cyclic compressive strains of 20%±5% are made with the model calibrated by the relaxation data. The predicted hysteresis stresses are lower than the measured stresses.

Large Strain Viscoelastic Constitutive Models for Rubber, Part II: Determination of Material Constants

1995 ◽  
Vol 68 (2) ◽  
pp. 230-247 ◽  
Author(s):  
Claudia J. Quigley ◽  
Joey Mead ◽  
Arthur R. Johnson
Keyword(s):  
Finite Element ◽  
Large Strain ◽  
Strain Relaxation ◽  
Small Strain ◽  
Material Behavior ◽  
Prony Series ◽  
Relaxation Data ◽  
Material Constants ◽  
Tension And Compression ◽  
Step Strain

Abstract A method for determining material constants in large strain viscoelastic materials was demonstrated for a highly saturated nitrile rubber. Material constant selection was based on viscoelastic stress relaxation data at small and large strains, under both tension and compression, and was constrained to assure Drucker stability. Assuming that the viscoelastic strain energy function was both time and strain separable, a Prony series was constructed for the time dependent material constants. For comparison, four different Prony series were developed from collocation methods and a nonlinear regression analysis, each separately based on either large or small tensile strain relaxation data. In addition, a final Prony series was constructed from dynamic data. These Prony series were included in this comparison to judge their ability to predict both large and small strain material behavior. Finite element analyses of large and small step-strain relaxation tests and a single cycle hysteresis loop at large deformations were performed for each set of Prony series. The results were then compared to experimental behavior. The Prony series based on the constrained method accurately predicted step-strain relaxation behavior at all strain levels, for both tension and compression. The finite element results for the other Prony series show that large strain material behavior was best predicted by those Prony series based on large strain material behavior. Similar findings were found for small strain material behavior. The constrained Prony series and the two large strain based Prony series best modeled the experimental hysteresis loop.

scholarly journals MIM and nonlinear least‐squares inversions of AEM data in Barataria basin, Louisiana

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1126-1131 ◽  
Author(s):  
Melissa Whitten Bryan ◽  
Kenneth W. Holladay ◽  
Clyde J. Bergeron ◽  
Juliette W. Ioup ◽  
George E. Ioup
Keyword(s):  
Least Squares ◽  
Water Depth ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Image Method ◽  
Water Conductivity ◽  
Near Surface ◽  
Barataria Basin ◽  
Airborne Electromagnetic Survey ◽  
Nonlinear Least Squares Method

An airborne electromagnetic survey was performed over the marsh and estuarine waters of the Barataria basin of Louisiana. Two inversion methods were applied to the measured data to calculate layer thicknesses and conductivities: the modified image method (MIM) and a nonlinear least‐squares method of inversion using two two‐layer forward models and one three‐layer forward model, with results generally in good agreement. Uniform horizontal water layers in the near‐shore Gulf of Mexico with the fresher (less saline, less conductive) water above the saltier (more saline, more conductive) water can be seen clearly. More complex near‐surface layering showing decreasing salinity/conductivity with depth can be seen in the marshes and inland areas. The first‐layer water depth is calculated to be 1–2 m, with the second‐layer water depth around 4 m. The first‐layer marsh and beach depths are computed to be 0–3 m, and the second‐layer marsh and beach depths vary from 2 to 9 m. The first‐layer water conductivity is calculated to be 2–3 S/m, with the second‐layer water conductivity around 3 to 4 S/m and the third‐layer water conductivity 4–5 S/m. The first‐layer marsh conductivity is computed to be mainly 1–2 S/m, and the second‐ and third‐layer marsh conductivities vary from 0.5 to 1.5 S/m, with the conductivities decreasing as depth increases except on the beach, where layer three has a much higher conductivity, ranging up to 3 S/m.

The Strength Evaluation and Reliability Analysis of Periodic Symmetric Struts Support

2011 ◽  
Vol 462-463 ◽  
pp. 1164-1169
Author(s):  
Jing Xiang Yang ◽  
Ya Xin Zhang ◽  
Mamtimin Gheni ◽  
Ping Ping Chang ◽  
Kai Yin Chen ◽  
...  
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Stress Distribution ◽  
Least Squares ◽  
Reliability Analysis ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Distribution Model ◽  
Different Types ◽  
Nonlinear Least Squares Method

In this paper, strength evaluations and reliability analysis are conducted for different types of PSSS(Periodically Symmetric Struts Supports) based on the FEA(Finite Element Analysis). The numerical models are established at first, and the PMA(Prestressed Modal Analysis) is conducted. The nodal stress value of all of the gauss points in elements are extracted out and the stress distributions are evaluated for each type of PSSS. Then using nonlinear least squares method, curve fitting is carried out, and the stress probability distribution function is obtained. The results show that although using different number of struts, the stress distribution function obeys the exponential distribution. By using nonlinear least squares method again for the distribution parameters a and b of different exponential functions, the relationship between number of struts and distribution function is obtained, and the mathematical models of the stress probability distribution functions for different supports are established. Finally, the new stress distribution model is introduced by considering the DSSI(Damaged Stress-Strength Interference), and the reliability evaluation for different types of periodically symmetric struts supports is carried out.

A NONLINEAR LEAST‐SQUARES METHOD FOR SEISMIC REFRACTION MAPPING—PART I: ALGORITHM AND PROCEDURE

Geophysics ◽  
1972 ◽  
Vol 37 (2) ◽  
pp. 260-272 ◽  
Author(s):  
Leonidas C. Ocola
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Seismic Refraction ◽  
Vertical Plane ◽  
Nonlinear Least Squares ◽  
Inversion Method ◽  
Almost Everywhere ◽  
Isotropic Media ◽  
Time Term ◽  
Nonlinear Least Squares Method

An iterative inversion method (Reframap) based on the kinematic properties of critically refracted waves is developed. The method is based on ray tracing and assumes homogeneous and isotropic media and ray paths confined to a vertical plane through each source‐detector pair. Unlike the earlier Profile or Time‐Term Methods, no restrictions are imposed on interface topography except that it be continuous almost everywhere (in the mathematical sense). As in the preexisting methods, more observations than unknowns are assumed. The algorithm and procedure, on which the Reframap Method is based, generate apparent dips for each source detector pair at the noncritical interfaces from the slope of a least‐squares line approximation to the interface functional in the neighborhood of each refraction point. In turn, the dip and path along the critical refractor is, at every iteration, pairwise approximated by a line through the critical refracting points. The incidence angles are computed recursively by Snell’s law. The solution of the overdetermined, nonlinear multiple refractor time‐distance system of simultaneous equations is sought by Marquardt’s algorithm for least‐squares estimation of critical refractor velocity and vertical thickness under each element.

Analysis of photoacoustic waveforms using the nonlinear least squares method

1992 ◽  
Vol 42 (1) ◽  
pp. 29-48 ◽  
Author(s):  
Jeanne Rudzki Small ◽  
Louis J. Libertini ◽  
Enoch W. Small
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Nonlinear Least Squares Method

Linear least squares method in nonlinear parametric inverse problems

2020 ◽  
Vol 28 (2) ◽  
pp. 307-312
Author(s):  
Leonid L. Frumin
Keyword(s):  
Inverse Problems ◽  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Linear Least Squares ◽  
Estimation Of Parameters ◽  
Discrete Approximations ◽  
Operator Equations ◽  
Transcendental Functions ◽  
Nonlinear Least Squares Method

AbstractA generalization of the linear least squares method to a wide class of parametric nonlinear inverse problems is presented. The approach is based on the consideration of the operator equations, with the selected function of parameters as the solution. The generalization is based on the two mandatory conditions: the operator equations are linear for the estimated parameters and the operators have discrete approximations. Not requiring use of iterations, this approach is well suited for hardware implementation and also for constructing the first approximation for the nonlinear least squares method. The examples of parametric problems, including the problem of estimation of parameters of some higher transcendental functions, are presented.

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