Nonlinear photomechanics

The history of developments in photoplasticity can be traced from the work of Filon at the beginning of this century to the most recent attempts to model large-scale flow in metal-forming processes. The results of some of the studies on strain fields in upset forging and hot rolling are presented.     

Computerized fringe analysis in photomechanics

A generalized method is presented for analyzing the fringe patterns frequently encountered in experimental mechanics. By utilizing digital-image-processing and computer-graphics techniques, a set of menu-driven software is developed for interactively implementing the fringe processing. Tests of this software on the images obtained experimentally by photoelasticity, holographic interferometry and speckle interferometry demonstrate its usefulness. Good agreement between the experimental and theoretical results is established.     

Measurement of elastic-plastic stresses by scattered-light photomechanics

Several birefringent materials were studied for their suitability for use in three-dimensional photoplasticity. This study resulted in the selection of cellulose propionate as model material. Its close match in index of refraction with ordinary mineral oil makes cellulose propionate suitable for scattered-light photomechanics. Viscoelastic behavior of the material is used to simulate elastic-plastic behavior of metals. The stress, strain and optical behavior of the material has been studied under slow loading in finite steps. A successful solution of an elastic-plastic torsion problem was obtained, demonstrating the applicability of the techniques of scattered-light photoplasticity to three-dimensional problems. This experimental method does not require the unloading and slicing of the model, avoiding any errors that might be introduced by residual stresses due to unloading. A major advantage of the method is the use of live loading, which allows the investigation of several load levels with the same model. Stress- and strain-concentration factors for the grooved shaft in torsion showed excellent agreement with Neuber's analytical results. Distributions of shear stress and shear strain across the minimum section of the model were compared to elastic theory. Integration of the shear-stress distributions showed good agreement with the measured values of applied torque.     

Stress-difference elasticity and its application to photomechanics

The two-dimensional elasticity equations have been reformulated in terms of the obtainable photoelastic data and the first stress invariant. The result is a new group of partial-differential equations which can be used to either separate the stresses along a line segment or to generate photoelastic data within a region. The equations can also be used to check the quality of the data used in the shear-difference method. Examples which illustrate the use of these equations are presented and discussed.     

A photomechanics material for elastoplastic stress analysis

The object of this investigation was to develop a technique or method for elastoplastic stress analysis using the optical effects of transparent materials. Of paramount importance was the selection and characterization of a suitable model material. In particular, it was desirable that the material be able to undergo large plastic strains while, at the same time, exhibiting a suitable level of optical response. A mixture of flexible and rigid polyester resins was found suitable, i.e., the mixture exhibited large strains and good optical response. It was found that unload birefringence (fringe order immediately upon removal of load) could be used to determine strain for a uniaxial-stress field. In particular, it provided a means for evaluating stress- and strain-concentration factors. Comparisons with other methods showed that the proposed method was reliable and gave results that are similar to those by other means. The usefulness of the material and method for two- and three-dimensional problems awaits further study.     

A simple holographic interferometer for static and dynamic photomechanics

A particular variation of holographic imaging system is described which, when used as a multiple-exposure holographic interferometer, possesses advantages for applications in static and dynamic photomechanics. Large fields of view can be obtained. Rigid-body motions produced by loading are automatically eliminated. The holograms can be recorded on medium-resolution films which have high sensitivity. Specimens manufactured with readily available materials can be used for the determination of isochromatics and isopachics. An attempt is made to describe this contribution in the background of previous developments in interferometry. Operation of the system is interpreted by showing the equivalence of the holographic interferometer to a combination of two systems presently in use in experimental stress analysis: a Fizeau interferometer and an optical spatial filter. The interpretation of isochromatics and isopachis as moiré phenomena is emphasized. Isochromatics and isopachics are presented as illustrations of the applications of the method to the solution of static-stress problems, and they are used in the solution of some not yet solved dynamic-stress problems. Whole-field static isochromatics obtained as absolute-retardation interference are shown. Also shown are whole-field dynamic isopachics.     

Wave motion in anisotropic media by dynamic photomechanics

Photomechanical techniques are developed for studying wave propagation in anisotropic media such as fiber-reinforced composites and rocks. Moiré fringes in transparent and opaque materials, and isochromatic fringes in transparent composites, are photographed at rates up to over one-million frames/second. A Beckman and Whitley 189 framing camera is utilized with illumination provided by argon flashbombs or a special electronic flash.     

Two systems for automatic reduction of time-dependent photomechanics data

Time-dependent photomechanical model materials will not, in general, exhibit a one-to-one relationship between the instantaneous stress and the relative retardation. Due to the complicated nature of this relationship, it is often more accurate to use the analog method of data reduction. This method consists of reproducing the fringe-order history observed in the model in a tensile specimen of the model material and recording the corresponding stress history. Two servo systems are described in this report which may be programmed to produce a desired fringe-order history in a time-dependent tensile specimen.     

Non-linear wave interactions

Non-linear wave interactions in physical systems are considered and analytical solutions of the governing equations obtained. Stability criteria are developed by obtaining general solutions for the first-order perturbations of the solutions. The problems discussed include various wave resonance interactions including Davydov's alpha-helix and Langmuir waves in a plasma.     

A multiple-pulse ruby-laser system for dynamic photomechanics: Applications to transmitted- and scattered-light photoelasticity

An ultrahigh-speed multiple-frame recording system for two- and three-dimensional dynamic photomechanics has been developed and is described here. The output from a ruby laser is modulated with a Pockels cell to produce a train of short, intense, monochromatic and polarized light pulses. Pulse widths of 50 nsec and repetition rates of up to 170,000 pulses/sec are obtained. These light pulses are synchronized with a “smear camera” and the event to produce a multiple-frame record of the phenomenon. The simplified camera requirements necessary for this purpose are indicated. The system is demonstrated by recording two-dimensional dynamic and scattered-light isochromatic fringe patterns. The capability of multiple recording of scattered-light fringe patterns, achieved here for the first time, has a tremendous potential for three-dimensional dynamic stress analysis. The developed system is also well suited for dynamic moiré, interferometry and holography.     

Non-linear elastic constitutive equations

The non-linear elastic constitutive equations proposed by Kauderer, [1], are a particular case of a class of quasi-linear equations. They cannot give an account of some specific non-linear effects of the Poynting and the Kelvin types. It is more useful in two and three dimensional problems to start with general non-linear forms proposed initially by Brillouin, [2].     

Non-linear theory of shells

A partially non-linear theory of anisotropic shells of uniform thickness is presented. Variational integrals of the stress equations of motion (26) and boundary conditions (27) consistent with simplified strain-displacement relations (9) are obtained from the Hamilton principle. The displacements and deflection are assumed to vary linearly across the thickness of the shell. The transverse shear and transverse normal strains as well as rotatory inertia and thermal effects are included in the analysis. One special case of the final equations of motion is considered.     

Non-linear age-dependent population dynamics

Archive for Rational Mechanics and Analysis -     

Non-linear shallow water theory

The equations controlling the three dimensional motion of a viscous, incompressible fluid in shallow water, are derived for use in any orthogonal coordinate system. Simplified equations for two dimensional motion are also developed. The equations are developed in terms of special coordinate dependent functions.     

Non-linear viscoelasticity of hagfish slime

We report here the first experimental measurements of non-linear rheological material properties of hagfish slime, a hydrated biopolymer/biofiber network, and develop a microstructural constitutive model to explain the observed non-linear viscoelastic behavior. The linear elastic modulus of the network is observed to be for timescales , making it one of the softest elastic biomaterials known. Non-linear rheology is examined via simple shear deformation, and we observe a secant elastic modulus which strain-softens at large input strain while the local tangent elastic modulus strain-stiffens simultaneously. This juxtaposition of simultaneous softening and stiffening suggests a general network structure composed of non-linear elastic strain-stiffening elements, here modeled as finite extensible non-linear elastic (FENE) springs, in which network connections are destroyed as elements are stretched. We simulate the network model in oscillatory shear and creep, including instrument effects from rotational inertia. The network model captures the simultaneous softening of the secant modulus and stiffening of tangent modulus as the model enters the non-linear viscoelastic regime.     

Non-linear Model for Wood Saturation

The paper presents a non-linear model of saturation with fluid of anisotropic capillary porous bodies and the results of experimental investigations of wood saturation with methacrylate. The obtained experimental curves illustrating the distribution of methacrylate in wood samples allow the estimation of material coefficients and verification of the theoretical model. The theoretical model is developed based on the balance equations of mass, momentum, and energy, and the thermodynamics of irreversible processes. The non-linear differential equation, describing the distribution of methacrylate content in wood and its evolution during the saturation is solved numerically. The theoretical curves obtained on the basis of both linear and non-linear models are compared with the experimental data and better agreement between them for the non-linear model is stated.     

Non-linear behavior of heavy stays

The behavior of cable structures, widely employed in several engineering fields, is highly non-linear. A common procedure to treat the cables is to substitute them with straight rods assuring an equivalent stiffness. But classical methods allow to determine satisfactorily the equivalent stiffness only when the cable is sensibly stretched, the cable sag is small and the cable is horizontal, since they generally disregard the effects of the self-weight component along the chord. To study non-linear behavior of inclined cables, we extend here a rational criterion stemming from the virtual work principle, formerly proposed for horizontal cables.