Two Simple Approximate Methods of Laplace Transform Inversion for Viscoelastic Stress Analysis

Two approximate methods of Laplace transform inversion are given which are simple to use and are particularly applicable to stress analysis problems in quasi-static linear viscoelasticity. Once an associated elastic solution is known numerically or analytically, the time-dependent viscoelastic response can be easily calculated using realistic material properties, regardless of how complex the property dependence of the elastic solution may be. The new feature of these methods is that it is necessary to know only 1) an elastic solution numerically for certain ranges of elastic constants and 2) numerical values of the operational moduli or compliances for real, positive values of the transform parameter. One method utilizes a mathematical property of the Laplace transform, while the other is based on some results obtained from Irreversible Thermodynamics and variational principles. Because of this, they are quite general and can be used with anisotropic and inhomogeneous materials. Two numerical examples are given: As the first one, we calculate the time-dependent strain in a long, internally. pressurized cylinder with an elastic case. The second example consists of inverting a transform which was derived by Muki and Sternberg in the thermo-viscoelastic analysis of a slab and a sphere(1). Both methods were found to provide results which are within the usual engineering requirements of accuracy. Application of the approximate methods to problems in dynamic viscoelasticity is discussed briefly. Supplementing the stress analysis, two techniques for calculating operational moduli and compliances from experimental stress-strain data are discussed and applied. Both can be used with creep, relaxation, and steady-state oscillation data. The most direct one consists of numerically integrating experimental data, while the other is a model-fitting scheme. With this latter method finite-element spring and dashpot models are readily found which fit the entire response.curves. In using these methods to calculate the operational functions employed in the stress analysis examples, we found that model-fitting was the fastest of the two, yet was very accurate

Fundamental Studies Relating to Systems Analysis of Solid Propellants

In this report the groundwork is laid for the proposed work scope which stressed the need for a greater understanding of the solid mechanics of grains. Particular emphasis will be directed toward the multi-axial behavior of thick walled configurations. The work falls naturally into three areas; (1) analysis procedures, (2) material properties, and (3) failure criteria. As a necessary preliminary to treating specific designs, certain material of general applicability must be developed, collected, and summarized. The following sections therefore deal with a general description of viscoelastic analysis and material representation, discussed by contrast with more conventional engineering analysis. By this means a background is established for the collection of elastic design formulas which are included in the second section of the report

Compressive Strength and Failure Time Based on Local Buckling in Viscoelastic

The axial compressive strength and failure time of unidirectional, viscoelastic composites are investigated. Effects of nonlinear shear behavior and fiber misalignment are emphasized because they are important strength-limiting factors in those strongly anisotropic composites which fail by local buckling in the shear mode of deformation. We first describe the basic buckling model and then, neglecting hereditary effects, predict the compressive strengths of an untoughened and a rubber-toughened carbon/epoxy composite. Next, using a nonlinear viscoelastic constitutive equation for shear behavior, failure time for constant load and compressive strength for increasing load history are predicted by a numerical method. Additionally, approximate analytical formulas are developed which enable one to easily estimate buckling response as a function of initial fiber misalignment angle as well as loading and material parameters

A Simple Collocation Method for Fitting Viscoelastic Models to Experimental Data

An easily applied collocation method is discussed for fitting the response of finite-element viscoelastic models to experimental stress-strain curves. It can be used with creep, relaxation, and steady-state oscillation data. The method is illustrated by means of two examples. As the first one, a model is obtained utilizing the dynamic shear compliance of polyisobutylene. In the second example we calculate a model from the tensile relaxation modulus of polymethyl methacrylate. With each case the model's response agreed with the experimental data within graphical accuracy over the entire frequency (or time) scale

A dynamical law for slow crack growth in polycarbonate films

We study experimentally the slow growth of a single crack in polycarbonate films submitted to uniaxial and constant imposed stress. For this visco-plastic material, we uncover a dynamical law that describes the dependence of the instantaneous crack velocity with experimental parameters. The law involves a Dugdale-Barenblatt static description of crack tip plastic zones associated to an Eyring's law and an empirical dependence with the crack length that may come from a residual elastic field

Fundamental Studies Relating to Systems Analysis of Solid Propellants

The earlier progress reports presented some essentials of model representation and a summary of some elastic solutions as preliminary material for viscoelastic analyses of solid propellants under various loading conditions. The present report is a continuation of the above with a brief section on Thermal Distributions, a section called Engineering Analysis, and one on Failure Criteria. The thermal distributions, obtained from heat transfer theory, are required for the thermoelastic formulations of section II. The Engineering Analysis section includes several varied examples to assist in understanding the analysis techniques presented in the other sections. The final section relates to mechanical failure of propellants and presents some preliminary thoughts as to how the study of this important problem area will be conducted

Fundamental Studies Relating to Systems Analysis of Solid Propellants

As in the previous progress reports, the contents in this report have been categorized so as to present a clear picture of their role in contributing to the problem of mechanical failure analysis. The subject of material representation by mechanical failure analysis. The subject of material representation by mechanical models is discussed in Section I, while Section II contains additions to the subject of Elastic Solutions for cylinders. The Engineering Analysis section includes an example of the strain response of an internal star grain to pressure. A damped sinusoid has been assumed for the pressure rise, and the use of stress concentration factors for a star grain is demonstrated. Section V on failure includes some preliminary test results which indicate the feasibility of the cumulative damage concept for composite (polyurethane) propellants, at least in the limited range tested. Recommendations are given which would expand this testing to show how damage accumulates under other conditions such as low temperatures, high strain-rates and with other types of propellant

Subcritical crack growth in fibrous materials

We present experiments on the slow growth of a single crack in a fax paper sheet submitted to a constant force $F$. We find that statistically averaged crack growth curves can be described by only two parameters : the mean rupture time $\tau$ and a characteristic growth length $\zeta$. We propose a model based on a thermally activated rupture process that takes into account the microstructure of cellulose fibers. The model is able to reproduce the shape of the growth curve, the dependence of $\zeta$ on $F$ as well as the effect of temperature on the rupture time $\tau$. We find that the length scale at which rupture occurs in this model is consistently close to the diameter of cellulose microfibrils

Fundamental Studies Relating to Systems Analysis of Solid Propellants

In this report the groundwork is laid for the proposed work scope which stressed the need for a greater understanding of the solid mechanics of grains. Particular emphasis will be directed toward the multi-axial behavior of thick walled configurations. The work falls naturally into three areas; (1) analysis procedures, (2) material properties, and (3) failure criteria. As a necessary preliminary to treating specific designs, certain material of general applicability must be developed, collected, and summarized. The following sections therefore deal with a general description of viscoelastic analysis and material representation, discussed by contrast with more conventional engineering analysis. By this means a background is established for the collection of elastic design formulas which are included in the second section of the report