Photo-orthotropic-elasticity

A method of producing transparent model materials for photo-orthotropic-elastic studies is presented. This material fabricated from glass fibers and a modified polyester matrix exhibits continuous relatively smooth fringe patterns which are linearly related to the state of stress. As such, the heterogenous material can be treated as a homogenous medium with orthotropic properties. Three photoelastic constantsf _L ,f _T andf _LT are necessary to describe the photoelastic response of the orthotropic materials to a general state of stress. Methods are established for predicting these photoelastic constants from the properties of the constituents. These methods are based on stress proportioning between the fibers and the matrix and upon the linear summation of the retardation from each constituent. The relations derived forf _L ,f _T andf _LT were verified experimentally and found to be in close agreement with measured values. A stress-optic law is derived on the basis of stress partitioning between the two constituents in a unidirectionally fiber-reinforced laminate. The adequacy of this stress-optic relation is confirmed by experimental verification. Comparison of this stress-optic relation with the expression advanced by Pih and Knight shows the validity of their initial concepts but the inadequacy of their partitioning functions. Detailed comparison of the stress-optic law with the analog relation proposed by Sampson shows excellent agreement. Indeed, the use of Sampson's stress-optic law is recommended and the law based on stress partitioning is to be considered as a fundamental theoretical proof of the Sampson relation. Finally, the applicability of Sampson's stress-optic law to bidirectionally reinforced materials was confirmed with a thorough experimental verification.     

Mechanical response and texture evolution of AZ31 alloy at large strains for different strain rates and temperatures

In order to study the behavior of material under finite deformation at various strain rates, the responses of AZ31 Mg sheet are measured under uniaxial (tension and compression) and multiaxial (simple shear) loadings along rolling direction (RD), 45° to rolling direction (DD), 90° to rolling direction (TD), and normal to the sheet (ND) to large strains. The material exhibits positive strain rate sensitivity (SRS) at room and elevated temperatures; the SRS is more pronounced at high temperatures and lower strain rates. The r-value of the material under tensile loading at room temperatures is higher in TD at lower strain rate. Texture measurements on several failed specimens are reported under tension and simple shear after finite plastic deformation of about 20% equivalent strain. The as-received material exhibits a strong fiber with equal fractions of grains having the c-axis slightly tilted away from the sheet normal towards both +RD and −RD. Pole figures obtained after tensile loading along the rolling direction (RD) show that the texture of the material strengthens even at low strains, with c-axis perpendicular to the sheet plane and prism planes lining up in a majority of grains. However, the tensile loading axis along TD does not lead to similar texture strengthening; the c-axis distribution appears to be virtually unchanged from the virgin state. The pole figures obtained after in-plane compression along RD brings the c-axes of the grains parallel to the loading direction. The pole figures after simple shear loading show that the c-axis rotates to lie on the sheet plane consistent with a compression axis 45° away on the sheet plane.     

Photoelastic studies of the two-dimensional dynamic stress-optic law

Studies of the two-dimensional dynamic stress-optic law were made using Castolite circular disks under diametral impact load. The dynamic stress-fringe relationship of Castolite was found to be approximately linear. The dynamic fringe values were slightly lower than the static values. In four of the five specimens used the maximum difference was 8.2 percent. Previous work on the dynamic stress-optic calibration was limited to cases of one-dimensional stress.A new dual-beam cathode ray oscilloscope and two dualtrace plug-in preamplifiers were adapted for this work. Taking advantage of symmetry, three strain components at an arbitrary point in the disk and the birefringence at the symmetrical point were recorded in one photograph during a single impact.was formerly associated with International Business Machines Corp., Endicott, N. Y.Paper was presented at 1962 SESA Annual Meeting held in Milwaukee, Wis., on October 24–26.     

A system of modified epoxies for dynamic photoelastic studies of fracture

In order to develop a moderately tough transparent polymer, a three-component modified epoxy was investigated. The polymer system included an epichlorohydrin/bisphenol A epoxy, a polyoxypropyleneamine curing agent and a curing accelerator. Twelve different compositions were prepared and evaluated in a series of static and dynamic tests to determine the material properties important in photoelastic studies of fracture. Static tests showed that the critical strain-energy release rate could be varied from 1.4 to 4.1 lb/in. by changing the constituents in the blending of the epoxy. These results forG _Ic indicate that the modified epoxies are considerably tougher than Homalite 100 (G _Ic =0.33 lb/in.) which is commonly employed as a model material in dynamic photoelastic studies. Dynamic photoelastic tests were conducted with half-plane models in order to determine the dilatational- and distortional-wave velocities,c ₁ andc ₂, as well as dynamic values of the modulus of elasticityE and Poisson's ratio ν at loading times of the order of 10^?5 s. These results indicated that the dynamic modulus of some of the modified epoxies was significantly higher than the static modulus indicating that these polymers are rate sensitive. One of the epoxy materials, Blend No. 3 withG _Ic =2.65 lb/in. (464J/m²), was calibrated dynamically. The material fringe value changed nearly linearly as a logarithmic function of time with an increase of about 100 percent as the loading time decreased from 10⁴ to 10^?4 s. This large variation inf _σ implies that the calibration constant must be adjusted when interpreting dynamic fringe patterns. Two of the epoxy compositions were also characterized in a number of fracture experiments involving crack propagation at velocities ranging from arrest conditions to terminal velocity where branching initiates. Dynamic isochromatic-fringe loops were photographed with a Cranz-Schardin multiple-spark camera. The fringe loops were analyzed to give the instantaneous stress-intensity factorK as a function of crack velocity å. The å vs.K curves appear to be invertedL shapes; however, there appears to be a double branch on the vertical part of theL. Also a slightly higherK is required for accelerating cracks and a lowerK for decelerating cracks. Further investigation is required to identify the basic mechanism involved in this fracture behavior.     

Application of Moiré Interferometry to the Characterization of Orthotropic Materials in the Antisymmetric Configuration using the Virtual Fields Method

Moiré interferometry is an effective full-field deformation measurement technique and has been utilized for mechanical behavior analysis of materials and structures. For isotropic materials, Moiré patterns can be obtained by performing standard tests, such as, tensile and bending tests, to calculate the displacement and strain. Then, the mechanical properties can be characterized. However, standard tests are not sufficient to characterize the mechanical parameters of anisotropic materials due to the complexity of their material properties. Thus, in this work, Moiré interferometry was combined with the Virtual Fields Method to obtain the four in-plane elastic constants (Q₁₁, Q₂₂, Q₁₂, and Q₆₆) of orthotropic materials in the form of a diametrically compressed disk. Firstly, according to finite element method simulation results, optimized parameters can be achieved when the principal direction of the material does not coincide with the loading direction, making the loading configuration antisymmetric. Therefore, Moiré interferometry experiment was simulated to demonstrate the feasibility of measurement in the antisymmetric configuration. Finally, the Q₁₁, Q₂₂, Q₁₂ and Q₆₆ values of a unidirectional carbon fiber composite were measured in a real Moiré interferometry experiment using the proposed method, yielding results that agreed closely with those obtained using the strain gauges.     

A critical review of methods for determining stress-intensity factors from isochromatic fringes

Two-parameter methods of fracture analysis for determining the stress-intensity factor from photoelastic isochromatic-fringe data were critically reviewed. The methods of Irwin, Bradley and Kobayashi, and Smith were developed in detail and differences in the three approaches were noted. Theoretical fringe loops were generated for a crack of length 2a in a semi-infinite plate with biaxial loading. These fringe loops were used to compare the three analysis methods and to determine the accuracy of each method. All three methods give a close estimate of the stress-intensity factor, with the Bradley-Kobayashi differencing procedure providing the most precise estimate ofK. However, if measurement errors become excessive (larger than 2 percent) the differencing procedure magnifies these errors and the original method proposed by Irwin is the recommended approach. The two-parameter methods can be employed to determineK to within ±5 percent, provided the angle of tilt of the isochromatic-fringe loop is 73 ≤θ _m < 139 deg. Ifθ _m is outside this range, the two-parameter methods should not be employed.     

On Pressurized Curvilinearly Orthotropic Circular Disk, Cylinder and Sphere Made of Radially Nonuniform Material

A unified analysis is presented for the elastic response of a pressurized cylindrically anisotropic hollow disk under assumed conditions of plane stress, or a hollow cylinder under plane strain conditions, and a spherically anisotropic hollow sphere, made of material which is nonuniform in the radial direction according to the power law relationship. The solution for a cylinder under generalized plane strain is also presented. Two parameters play a prominent role in the analysis: the material nonuniformity parameter m, and the parameter ?? which accounts for the combined effects of material anisotropy, represented by the specified parameters (??, ??, ??), and material nonuniformity, represented by the parameter m. The radial and circumferential stresses are the linear combinations of two power functions of the radial coordinate, whose exponents (n ₁ and n ₂) depend on the parameters m and ??. New light is added to the stress amplification and shielding under combined effects of curvilinear anisotropy and radial nonuniformity. Different loading combinations are considered, including the equal pressure at both boundaries, and the uniform pressure at the inner or the outer boundary. While the stress state for the equal pressure loading is uniform in the case of isotropic uniform material (m=0, ??=1), and for one particular radially nonuniform and anisotropic material, it is strongly nonuniform for a general anisotropic or nonuniform material. If the aspect ratio of the inner and outer radii decreases (small hole in a large disk/cylinder or sphere), the magnitude of the circumferential stress at the inner radius increases for n ₁>0 (stress amplification), and decreases for n ₁<0 (stress shielding). Both can be achieved by various combinations of the material parameters m, ??, ??, and ??. While the stress amplification in the case of a pressurized external boundary occurs readily, it occurs only exceptionally in the case of a pressurized internal boundary. The effects of material parameters on the displacement response are also analyzed. The approximate character of the plane stress solution of a pressurized thin disk is discussed and the results are compared with those obtained by numerical solution of the exact three-dimensional disk model.     

Moiré interferometry with ±45-deg gratings

Specimen gratings with rulings oriented +45 deg and ?45 deg to they axis can be used to determine theu andv displacement fields, i.e., the displacements in thex andy directions. The fringe patterns are identical to those obtained with (X) and (Y) specimen gratings. The analysis leading to this conclusion is presented. A vector quantity related to the fringe gradient in the interference pattern is defined and found effective for visualizing and describing the image-forming rays. Advantages relate to construction of apparatus for moiré interferometry.     

Heat transfer and skin-friction in a turbulent boundary layer under a non-equilibrium longitudinal adverse pressure gradient

The experimental data on the effect of weak and moderate non-equilibrium adverse pressure gradients (APG) on the parameters of dynamic and thermal boundary layers are presented. The Reynolds number based on the momentum thickness at the beginning of the APG region was Re^** = 5500. The APG region was a slot channel with upper wall expansion angles from 0 to 14°. The profiles of the mean and fluctuation velocity components were measured using a single-component hot-wire anemometer. The friction coefficients were determined using two methods, namely, the indirect Clauser method and the direct method of weighting the lower wall region on a single-component strain-gage balance. The heat transfer coefficients were determined by a transient method using an IR camera. It is noticed that in the pressure gradient range realized the universal logarithmic region in the boundary layer profile is conserved. The values of the relative (divided by the parameters in zero gradient flow at the same value of Re^**) friction and heat transfer coefficients, together with the Reynolds analogy factor, are determined as functions of the longitudinal pressure gradient. The values of the relative friction coefficient reduced to c_f/c_f0 = 0.7 and those of the heat transfer to St/St₀ = 0.9. A maximum value of the Reynolds analogy factor (St/St₀)/(c_f/c_f0) = 1.16 was reached for the pressure gradient parameter β = 2.9.     

Cyclic-stress studies by time-averaged photoelasticity

This paper describes an experimental method whereby the amplitude of cyclic stresses may be readily determined by time-averaged photoelasticity. Using an ordinary polariscope with a monochromatic-light source, ‘time-averaged isochromatics fringes’ are formed if the photographic film in the camera is exposed with an exposure time equal to one or several periods while the photoelastic model is undergoing steady-state cyclic loading. The fringe pattern depicts amplitudes of the oscillating stresses according to the zeroth-order Bessel function. These properties permit the determination of a time-averaged cyclic stress-optic law. It is also possible to use the method to determine time-averaged isoclinics. The method has great potentiality in the study of in-plane vibrations.     

Photoelastic analysis of stresses around oblique holes

A three-dimensional photoelastic analysis was conducted to determine the magnitude and distribution of stresses around oblique holes in a uniaxially loaded plate. The holes were circular and inclined at angles of 45 deg and 60 deg with the faces of the plate. The plate-thickness-to-hole-diameter ratiot/D was 2.40. One end of each hole was blended to the face of the plate through a break radius equal to the radius of the hole. The plate dimensions were sufficiently large to simulate conditions of an infinite plate. The plates were loaded perpendicular to the plane of skewness. Stress distributions were obtained on sections perpendicular to the direction of loading. Results point to two critical areas of stress concentration: one at the acute-angle intersection of the hole and the surface of the plate and the other in the break-radius area. The stress concentrations in the latter area reach values of 4.6 and 6.7 compared to 3.6 and 4.5 at the acute-angle intersection, for the inclination angles of 45 deg and 60 deg, respectively. A simplified analysis used for the break-radius area gave results in agreement with the experiment. Thus, it was shown that break radii in oblique penetrations may have deleterious rather than beneficial effects. Comparison of results for the acute-angle intersection with existing theoretical and experimental values shows a definite and pronounced dependence of the stress-concentration factor on thickness-to-diameter ratio.     

Anisotropic hardening – numerical application of a cubic yield theory and consideration of variable r-values for sheet metal

The first part of this paper contains a polynomial yield condition of third order connected with evolution equations for material tensors of higher orders. They are formulated by formal generalisation of an approach by Danilov. The second part presents a possibility of taking into account the rotation of the yield surface as a result of a variable planar anisotropy (r-value) in sheet metal. This is done by an extension of the evolution equations, based on a quadratic yield function. The corresponding deformation law and the set of evolution equations are numerically integrated for selected loading paths in the subspaces σ₁,σ₂ and σ,τ. Some of the experimentally observed effects, such as the increasing curvature of the yield locus curve in the loading direction or the specific rotation of the yield surface, are correctly reproduced.     

Investigation of the pin joints in composites by the moiré method

A method combining moiré-fringe patterns with strain-gage measurements is applied to the analysis of strain distribution in a double-lap joint specimen under pin load. The specimens are crossplied and woven-fabric graphite/epoxy composites. Fiber orientations for both of the composites are 0 deg/90 deg, and ±45 deg to the load direction. The results indicate that the strain distributions in the specimen are greatly influenced by the fiber orientation. In this experiment, these specimens were collapsed at the pin-hole edge by the pin load. However, the failure load is not always dependent on the fiber orientation.     

Full-field displacement and strain rosettes by moiré interferometry

A cross-line phase-type specimen grating was interrogated by moiré interferometry techniques to produce full-field fringe patterns of three displacement components:U _x ,U _y , andU ₄₅. ForU ₄₅, sensitivity was 1/1700 mm/fringe (1/43,000 in./fr). Closely packed fringes with much information content were obtained. Fringe patterns of strain components? _x ,? _y and? ₄₅ were produced by mechanical shearing (or optical differentiation) of the displacement patterns; a shearing distance of only 0.6 mm (0.025 in.) was used. The rosette method yields complete strain information from these three components of normal strain—which were derived from three direct derivatives of displacement. Consequently, the need for cross-derivatives of displacement, which are highly sensitive to accidental rigid-body rotations, is circumvented.     

Stress singularity due to traction discontinuity on an anisotropic elastic half-plane

In a half-plane problem with x₁ paralleling with the straight boundary and x₂ pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x₁ direction may be denoted by t₁ = [σ₁₁, σ₁₂, σ₁₃]^T. Stress components σ₁₁ and σ₁₃ are of more interests since σ₁₂ is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ₁₁ is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ₁₁ will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ₁₁ and σ₁₃ induced by traction-discontinuity on general anisotropic elastic surfaces is studied. By analyzing the problem of uniform tractions applied on the half-plane boundary over a finite loaded region, exact expressions of the stress components σ₁₁ and σ₁₃ are obtained which reveal that these components consist of in general a constant term and a logarithmic term in the loaded region, while only a logarithmic term exists in unloaded region. Whether the constant term or the logarithmic term will appear or not completely depends on what values of the elements of matrices Ω and Γ will take for a material under consideration. Elements for both matrices are expressed explicitly in terms of elastic stiffness. Results for monoclinic and orthotropic materials are all deduced. The isotropic material is a special case of the present results.     

A summation strain-gage alternative to oblique incidence in photoelastic coatings

A special strain gage (PhotoStress^® Separator Gage, Measurements Group, Inc., Raleigh, NC) designed to measure the sum (? ₁ +? ₂) of the principal strains, is used in conjunction with photoelastic-coating measurements (? ₁ -? ₂) to establish the value of each principal strain (? ₁ and? ₂). The summation strain signal is effectively independent of angular orientation (measurement direction), and by design, the gage negates soldering risks, self-heating, and localized-reinforcement considerations normally associated with strain-gage measurements on plastic parts.     

Characterization of the flow over a cylinder moving harmonically in the cross-flow direction

The flow over a stationary cylinder is self-excited with a specific natural frequency f_N. When the cylinder is moved harmonically in the cross-flow direction, the response of the flow (in terms of the lift force) will contain two frequencies, namely, the natural frequency f_N and the excitation frequency f_E. When f_E is close to f_N, the natural flow response will be entrained by the excitation, and the response will be periodic with frequency f_E, and dynamicists refer to this phenomenon as lock-in or synchronization. When f_E is away from f_N, the flow will be either periodic with a period that is multiple of the excitation period (i.e., period-n) and dynamicists refer to this phenomenon as secondary synchronization or quasiperiodic consisting of two incommensurate frequencies, or chaotic. We use modern methods of non-linear dynamics to characterize these responses and show that the route to chaos is torus breakdown.     

Biaxial testing of [O2/±45] s graphite/epoxy plates with holes

An experimental investigation was conducted to study the behavior under biaxial-tensile loading of [O₂/±45] _s graphite/epoxy plates with circular holes and to determine the influence of hole diameter on failure. The specimens were 40-cm×40-cm (16-in.×16-in.) graphite/epoxy plates of [O₂/±45] _s layup. Four hole diameters, 2.54 cm (1.00 in.), 1.91 cm (0.75 in.), 1.27 cm (0.50 in.) and 0.64 cm (0.25 in.), were investigated. Deformations and strains were measured using strain gages and birefringent coatings. Biaxial tension in a 2∶1 ratio was applied by means of four whiffle-tree grip linkages and controlled with a servohydraulic system. Stress and strain redistributions occur around the hole at a stress level corresponding to localized failure around the 67.5-deg location and nonlinear strain response at the 0-deg location. Maximum measured strains at failure on the hole boundary are higher (approximately 0.016) than the highest ultimate strain of the unnotched laminate (0.010). Two basic patterns of failure were observed: (a) horizontal cracking initiating at points off the horizontal axis and accompanied by extensive delamination of the subsurface ±45 deg plies, and (b) vertical cracking along vertical tangents to the hole and accompanied by delamination of the outer 0-deg plies. The strength reduction ratios are lower than corresponding values for uniaxial loading by approximately 16 percent, although the stress-concentration factor under biaxial loading is lower.     

Stress-optic law in a single crystal and its application to photo-anisotropic elasticity

A general stress-optic law in a plate made of a single crystal with birefringence is developed, the plate has an arbitrary crystallographic direction. From the general stressoptic law, a condition for obtaining stress distributions in the plate under a plane stress state is derived. Some optical and mechanical properties when the plate is used as a photoanisotropic model are also explained. Experiments on silicon beams in pure bending are performed by using an infrared photoelastic method. Experimental results show that the stress-optic law is valid. The optical and mechanical properties of silicon beams are shown in tables.     

The effect of suction or injection on the boundary layer flows induced by continuous surfaces stretched with prescribed skin friction

The boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction is studied for permeable surface. Three major cases are studied for isothermal surface (n=0) stretched corresponding to different dimensional skin friction boundary conditions namely; skin friction at the surface scales as (x ^?1/2) at m=0, constant skin friction at m=1/3 and skin friction scales as (x) at m=1. The constants m and n are the indices of the power law velocity and temperature exponent respectively. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, suction/injection parameter f _w , m and n are studied. The results show that for isothermal surface increasing m enhances the dimensionless heat transfer coefficient for fixed f _w at the suction case and the reverse is true at the injection case. Furthermore, for fixed m, as f _w increases the dimensionless heat transfer coefficient increases. Large enhancements are observed in the heat transfer coefficient as the temperature boundary condition along the surface changes from uniform to linear where the dimensional skin friction is of order (x) at m=1. This enhancement decreases as the suction increases.