A Linear Compressibility Assumption for the Multiple Integral Representation of Nonlinear Creep of Polyurethane

1970 ◽  
Vol 37 (2) ◽  
pp. 441-448 ◽  
Author(s):  
K. G. Nolte ◽  
W. N. Findley
Keyword(s):  
Integral Representation ◽  
Multiple Integral ◽  
Kernel Functions ◽  
Nonlinear Creep ◽  
Volume Changes ◽  
Creep Tests ◽  
Third Order ◽  
Nonlinear Viscoelastic ◽  
Different Order ◽  
Order Kernel

The assumption that volume changes associated with creep of a nonlinear viscoelastic material are only linearly dependent on the stress history is incorporated into a third-order multiple integral representation. This assumption reduces the number of independent kernel functions in the representation from 12 to 7. The traces of these independent kernels may be determined from two tension, two torsion, and one combined tension and torsion creep tests. Experiments on polyurethane are well represented by this method. The time-dependence of the kernel functions is expressed by time raised to a power with the power differing for different-order kernel functions.

Concerning a Creep Surface Derived From a Multiple Integral Representation for 304 Stainless Steel Under Combined Tension and Torsion

1978 ◽  
Vol 45 (4) ◽  
pp. 773-779 ◽  
Author(s):  
R. Mark ◽  
W. N. Findley
Keyword(s):  
Stainless Steel ◽  
Creep Rate ◽  
Integral Representation ◽  
Power Function ◽  
Multiple Integral ◽  
304 Stainless Steel ◽  
Creep Data ◽  
Creep Tests ◽  
Third Order ◽  
Good Agreement

It is shown that a creep surface, defined in terms of a prescribed creep rate, can be determined from the multiple integral formulation representing the creep data. The creep surface for 304 stainless steel was found to be in good agreement with a Mises ellipse. Observed creep rate vectors for this alloy were found to be normal to a Mises ellipse. These results were obtained from creep tests performed on 304 stainless steel under combined tension and torsion at 593°C (1100°F). Creep strains observed for at least 100 hr were adequately represented by a power function of time, the exponent of which was independent of stress. A third-order multiple integral representation together with a limiting stress below which creep does not occur was employed to describe satisfactorily the constant stress creep data.

scholarly journals Experimental Determination of Some Kernel Functions in the Multiple Integral Method for Nonlinear Creep of Polyvinyl Chloride

1971 ◽  
Vol 38 (1) ◽  
pp. 30-38 ◽  
Author(s):  
K. Onaran ◽  
W. N. Findley
Keyword(s):  
Polyvinyl Chloride ◽  
Viscoelastic Behavior ◽  
Multiple Integral ◽  
Kernel Functions ◽  
Product Form ◽  
Superposition Method ◽  
Nonlinear Creep ◽  
Nonlinear Viscoelastic ◽  
Nonlinear Viscoelastic Behavior ◽  
Accuracy Of Prediction

Kernel functions for mixed-time parameters in the multiple integral representation of the nonlinear viscoelastic behavior of polyvinyl chloride were determined from both two-step tension and two-step torsion creep experiments. First and second-order terms were used for tension and first and third-order terms were used for torsion to describe these kernel functions. Stepdown tests were needed for good accuracy of representation. Accuracy of prediction was good for stepdown but not stepup tests. The product form assumption for these kernel functions and the modified superposition method were also investigated. The latter gave the best overall predictability of the three methods, although the product form was nearly as satisfactory.

scholarly journals Non-linear creep of polypropylene utilizing multiple integral

2021 ◽  
Vol 10 (4) ◽  
pp. 288
Author(s):  
Mahmoud Fadhel Idan
Keyword(s):  
Glass Transition ◽  
Integral Representation ◽  
Elastic Stress ◽  
Multiple Integral ◽  
Kernel Functions ◽  
Time Parameter ◽  
Effect Of Temperature ◽  
Nonlinear Creep ◽  
Linear Creep ◽  
Energy Law

<p>Multiple integral representation (MIR) has been used to represent studying the effect of temperature on the amount of nonlinear creep on the semi- crystalline polypropylene (PP) under the influence of axial elastic stress. To complete this research, the Kernel functions were selected, for the purpose of performing an analogy, and for arranging the conditions for the occurrence of the first, second and third expansion in a temperature range between 20 °C-60 °C, i.e., between the glass transition and softening temperatures, within the framework of the energy law. It was observed that the independent strain time increased non-linearly with increasing stress, and non-linearly decreased with increase in temperature, although the time parameter increased non-linearly with stress and temperature directly. In general, a very satisfactory agreement between theoretical and practical results on the MIR material was observed.</p>

Behavior of Nonlinear Viscoelastic Material Under Simultaneous Stress Relaxation in Tension and Creep in Torsion

1969 ◽  
Vol 36 (1) ◽  
pp. 22-27 ◽  
Author(s):  
J. S. Y. Lai ◽  
W. N. Findley
Keyword(s):  
Stress Relaxation ◽  
Multiple Integral ◽  
Analytical Investigation ◽  
Integral Approach ◽  
Successive Approximations ◽  
Tension Stress ◽  
Creep Tests ◽  
Direct Inversion ◽  
Nonlinear Viscoelastic ◽  
Shearing Strain

An experimental and analytical investigation is presented for simultaneous stress relaxation in tension and creep in torsion of polyurethane in the nonlinear range of stresses. The method employed a multiple integral approach with an assumed product form of kernel function to describe creep behavior. The required constants were determined from pure creep experiments on polyurethane. The tension stress and shearing strain versus time for simultaneous stress relaxation and creep were computed from results of these pure creep tests alone, and the results were compared with experiments on the same polyurethane under simultaneous stress relaxation and creep. In the method of analysis, direct inversion of the equation for creep was used as a first approximation for relaxation. Means for obtaining successive approximations and for accounting for cross effects are described. The second approximation was found to be adequate to describe the observed behavior very satisfactorily.

Nonlinear Viscoelasticity for Short Time Ranges

1966 ◽  
Vol 33 (2) ◽  
pp. 313-321 ◽  
Author(s):  
N. C. Huang ◽  
E. H. Lee
Keyword(s):  
Constitutive Equation ◽  
Numerical Solutions ◽  
Kernel Functions ◽  
Nonlinear Creep ◽  
Experimental Procedure ◽  
Nonlinear Viscoelastic ◽  
Tension Tests ◽  
Incompressible Materials ◽  
Inverse Interpolation ◽  
Short Time

Approximate constitutive equations for nonlinear viscoelastic incompressible materials under small finite deformation and for short time ranges are derived. The error bound of such a constitutive equation is investigated. Nonlinear creep is analyzed on the basis of the proposed equation, and also the problem of a pressurized viscoelastic hollow cylinder bonded to an elastic casing. Numerical solutions, evaluated by assuming particular forms of kernel functions in the constitutive equation, are obtained by means of an inverse interpolation technique, and the effects of nonlinearity of material properties are discussed. An experimental procedure is also proposed for measuring kernel functions from uniaxial tension tests for real materials.

Incompressible and Linearly Compressible Viscoelastic Creep and Relaxation

1974 ◽  
Vol 41 (1) ◽  
pp. 243-248 ◽  
Author(s):  
W. N. Findley ◽  
K. Onaran
Keyword(s):  
Multiple Integral ◽  
Incompressible Material ◽  
Kernel Functions ◽  
Integral Representations ◽  
Creep Data ◽  
Third Order ◽  
Viscoelastic Creep ◽  
The Third ◽  
Small Strains

It is shown that defining an incompressible material as one whose response to stressing or straining is insensitive to volumetric-type changes in strain or stress allows the derivation of incompressible forms for multiple integral representations, through the third order, which have only three kernel functions both in the creep formulation and the relaxation formulation for small strains. Earlier work had yielded four kernel functions in the relaxation and three in the creep formulation. Linearly compressible formulations are also discussed and compared with available creep data.

An Experimental Study of a Nonlinear Viscoelastic Solid in Uniaxial Tension

1969 ◽  
Vol 36 (3) ◽  
pp. 558-564 ◽  
Author(s):  
W. G. Gottenberg ◽  
J. O. Bird ◽  
G. L. Agrawal
Keyword(s):  
Uniaxial Tension ◽  
Constitutive Relation ◽  
Simple Procedure ◽  
Constant Strain Rate ◽  
Integral Form ◽  
Multiple Integral ◽  
Kernel Functions ◽  
Load Response ◽  
Material Functions ◽  
Nonlinear Viscoelastic

The multiple integral form of the constitutive relation for nonlinear viscoelasticity is correlated with experimental results for the case of uniaxial tension of a polymeric material. Special forms of the kernel functions are assumed in which the arguments of these functions are taken in additive form. This permits the development of a simple procedure for determining the material functions from stress-relaxation tests. The resulting constitutive relation, for a particular material, is used to predict the load response to single and consecutive constant strain-rate programs and the results are compared with experimentally obtained data.

scholarly journals Identifying Odd/Even-Order Binary Kernel Slices for a Nonlinear System Using Inverse Repeat m-Sequences

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Jin-yan Hu ◽  
Gang Yan ◽  
Tao Wang
Keyword(s):  
Nonlinear System ◽  
Cross Correlation ◽  
Correlation Method ◽  
Kernel Functions ◽  
Third Order ◽  
Mathematical Properties ◽  
M Sequence ◽  
System Output ◽  
Even Order ◽  
Order Kernel

The study of various living complex systems by system identification method is important, and the identification of the problem is even more challenging when dealing with a dynamic nonlinear system of discrete time. A well-established model based on kernel functions for input of the maximum length sequence (m-sequence) can be used to estimate nonlinear binary kernel slices using cross-correlation method. In this study, we examine the relevant mathematical properties of kernel slices, particularly their shift-and-product property and overlap distortion problem caused by the irregular shifting of the estimated kernel slices in the cross-correlation function between the input m-sequence and the system output. We then derive the properties of the inverse repeat (IR) m-sequence and propose a method of using IR m-sequence as an input to separately estimate odd- and even-order kernel slices to reduce the chance of kernel-slice overlapping. An instance of third-order Wiener nonlinear model is simulated to justify the proposed method.

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