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.

Stress Analysis for a Nonlinear Viscoelastic Compressible Cylinder

1970 ◽  
Vol 37 (4) ◽  
pp. 1127-1133 ◽  
Author(s):  
E. C. Ting
Keyword(s):  
Large Deformations ◽  
Finite Deformations ◽  
Kernel Functions ◽  
Nonlinear Viscoelastic ◽  
Stress Strain Relation ◽  
Incompressible Materials ◽  
Small Deformations ◽  
Compressible Cylinder ◽  
Elastic Casing ◽  
Finite Difference Techniques

Real solids are not incompressible, although many viscoelastic materials which undergo large deformations show only small changes in volume under ordinary loading conditions. This paper is concerned with a pressurized isotropic viscoelastic hollow cylinder bonded to an elastic casing in which, during a finite deformation, the dilatational change in any element of the cylinder is a small quantity. The analysis is based in part upon the theory of small deformations superposed on finite deformations. Numerical calculations are evaluated by using finite-difference techniques and assuming particular forms of kernel functions in the stress-strain relation. The results for compressible and incompressible materials are compared.

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.

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.

Nonlinear Creep Buckling of Some Simple Structures

1967 ◽  
Vol 34 (3) ◽  
pp. 651-658 ◽  
Author(s):  
N. C. Huang
Keyword(s):  
Cross Section ◽  
Analytical Solutions ◽  
Material Properties ◽  
Numerical Solutions ◽  
Direct Compression ◽  
Nonlinear Creep ◽  
Lateral Buckling ◽  
Nonlinear Viscoelastic ◽  
Buckling Behavior ◽  
Creep Buckling

In this paper, the creep buckling behavior of shallow Mises trusses and two-hinged sinusoidal low arches with sandwich cross section is studied. The material properties are assumed to be nonlinear viscoelastic and obeying the power creep law. The purpose of this study is to find the critical times, critical deflections, and postbuckling deflections of these structures. For Mises trusses, snap-through may be induced by direct compression or lateral buckling of the members. Analytical solutions are obtained. In the arch problem, both symmetrical and antisymmetrical bucklings are considered. Collocation method is used in the analysis, and numerical solutions are obtained.

Stress Analysis for a Nonlinear Viscoelastic Cylinder With Ablating Inner Surface

1970 ◽  
Vol 37 (1) ◽  
pp. 44-47 ◽  
Author(s):  
E. C. Ting
Keyword(s):  
Stress Analysis ◽  
Solid Propellant ◽  
Firing Condition ◽  
Real Situation ◽  
Nonlinear Viscoelastic ◽  
Incompressible Materials ◽  
Inner Surface ◽  
One Step ◽  
Technical Interest ◽  
Short Time

An approximate constitutive equation for nonlinear viscoelastic incompressible materials under small finite deformation and for short time ranges has been derived by Huang and Lee [1]. The resulting equation is applied to solve the problem of a pressurized viscoelastic hollow cylinder bonded to an elastic easing. This problem is of notable technical interest in solid propellant stress analysis, since it is a close model to represent a cylindrical propellant grain in a solid fuel rocket under firing condition. The method used by Huang and Lee is appropriate for numerical calculations when the boundaries of the cylinder are nonablating. To consider one step closer to the real situation, the inner surface is often assumed to be ablating and hence time-dependent. It is then the purpose of the present paper to extend the analysis to a cylinder with moving inner surface.

scholarly journals A Modified Fractional Maxwell Numerical Model for Constitutive Equation of Mn-Cu Damping Alloy

Materials ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 2020
Author(s):  
Baoquan Mao ◽  
Rui Zhu ◽  
Zhiqian Wang ◽  
Yuying Yang ◽  
Xiaoping Han ◽  
...  
Keyword(s):  
Constitutive Equation ◽  
Strain Rate ◽  
Constitutive Relation ◽  
Constant Strain Rate ◽  
Strain Curve ◽  
Maxwell Model ◽  
Nonlinear Viscoelastic ◽  
Fractional Maxwell Model ◽  
Hysteretic Characteristics ◽  
Nonlinear Elastic

To better describe its constitutive relation, we need a new constitutive equation for an important nonlinear elastic material, Mn-Cu damping alloy. In this work, we studied the nonlinear and hysteretic characteristics of the stress-strain curve of the M2052 alloy with the uniaxial cyclic tensile test with constant strain rate. The strain rate and amplitude correlations of M2052 resembled those of nonlinear viscoelastic material. Therefore, we created a new constitutive equation for the M2052 damping alloy by modifying the fractional Maxwell model, and we used the genetic algorithm to carry out numerical fitting with MATLAB. By comparing with the experimental data, we confirmed that the new constitutive equation could accurately depict the nonlinear constitutive relation and hysteretic property of the damping alloy. Taken together, this new constitutive equation for Mn-Cu damping alloy based on the fractional Maxwell model can serve as an effective tool for further studies of the constitutive relation of the Mn-Cu damping alloys.

scholarly journals A Study on Stable Regularized Moving Least-Squares Interpolation and Coupled with SPH Method

2020 ◽  
Vol 2020 ◽  
pp. 1-28
Author(s):  
Hua Jiang ◽  
Yunsai Chen ◽  
Xing Zheng ◽  
Shanqin Jin ◽  
Qingwei Ma
Keyword(s):  
Least Squares ◽  
Numerical Solutions ◽  
Kernel Functions ◽  
Moving Least Squares ◽  
Sph Method ◽  
Particle Distributions ◽  
Particle Hydrodynamics ◽  
Support Domain ◽  
Accuracy Result ◽  
Sph Simulation

The smoothed particle hydrodynamics (SPH) method has been popularly applied in various fields, including astrodynamics, thermodynamics, aerodynamics, and hydrodynamics. Generally, a high-precision interpolation is required to calculate the particle physical attributes and their derivatives for the boundary treatment and postproceeding in the SPH simulation. However, as a result of the truncation of kernel function support domain and irregular particle distribution, the interpolation using conventional SPH interpolation experiences low accuracy for the particles near the boundary and free surface. To overcome this drawback, stable regularized moving least-squares (SRMLS) method was introduced for interpolation in SPH. The surface fitting studies were performed with a variety of polyline bases, spatial resolutions, particle distributions, kernel functions, and support domain sizes. Numerical solutions were compared with the results using moving least-squares (MLS) and three SPH methods, including CSPH, K2SPH, and KGFSPH, and it was found that SRMLS not only has nonsingular moment matrix, but also obtains high-accuracy result. Finally, the capability of the algorithm coupled with SRMLS and SPH was illustrated and assessed through several numerical tests.

Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

2018 ◽  
Vol 28 (4) ◽  
pp. 828-856 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Fluid Flows ◽  
Kernel Functions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type

Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

Dimensional changes during shear without normal tractions (the Poynting effect) in nonlinear viscoelastic fiber-reinforced solids

2019 ◽  
Vol 25 (3) ◽  
pp. 582-596
Author(s):  
Alan Wineman
Keyword(s):  
Constitutive Equation ◽  
Numerical Solutions ◽  
Elastic Fiber ◽  
Time Dependent ◽  
Nonlinear Material ◽  
Fiber Reinforced ◽  
Dimensional Changes ◽  
Fiber Reinforced Materials ◽  
The Matrix ◽  
Poynting Effect

When a rectangular block of a nonlinear material is subjected to a simple shearing deformation, specific normal tractions are required to ensure that the distances between the faces of the block, i.e. its dimensions, do not change. This work investigates the time-dependent dimensional changes during shear in the absence of these normal tractions (the Poynting effect) that occur in a block composed of an incompressible nonlinearly viscoelastic fiber-reinforced solid. The material is modeled using the Pipkin–Rogers nonlinear single integral constitutive equation for viscoelasticity. This constitutive equation is used because (1) it exhibits the essential features of nonlinear viscoelasticity; (2) it is straightforward to include the material symmetry restrictions due to the reinforcing fibers. A system of nonlinear Volterra integral equations is formulated for the dimensional changes in the block. Numerical solutions are presented for the case when the standard reinforcing model for nonlinearly elastic fiber-reinforced materials is incorporated in the Pipkin–Rogers constitutive framework. The results illustrate how the time-dependent dimensional changes depend on the fiber orientation and the viscoelastic properties of the fibers relative to those of the matrix.

Sign in / Sign up

Export Citation Format

Share Document