On the stress-optic law for orthotropic-model materials in biaxial-stress fields

The stress-optic law for othotropic-model materials, proposed by Sampson on the basis of a simple analogy to the isotropic-model materials, is examined for biaxial-stress fields. The stress-optic law is reduced to a simple form for special cases. It is also shown that the zero-order isochromatic fringe corresponds to an isotropic state of stress only in the case of balanced laminates. A glass-fiber-reinforced plastic disk with the glass fibers in only one direction is examined under diametral compression photoelastically and by means of strain-gage rosettes, with the loading direction perpendicular and at 45 deg to the reinforcement direction. The fringe order along the horizontal diameter is computed from the simplified stress-optic law making use of stress values from strain-gage readings and compared with the observed fringe order. Based on a fairly good agreement of the fringe orders, it is shown that a circular-disk specimen can be used to calibrate an orthotropic-model material. The three independent material-fringe values,f _L ,f _T ,f _LT , can be found from measurements of the fringe order and the strains at the center of the disk for the three cases of loading perpendicular, parallel and at 45 deg to the reinforcement direction.     

A stress-optic law for photoelastic analysis of orthotropic composites

The feasibility for utilizing transparent filament-resin composites for photoelastic stress analysis was investigated. Satisfactory photoelastic stress patterns were demonstrated in simple models with undirectional and bidirectional fiber orientations. A stress-optic law was formulated, based on the concept that the birefringence components contributed by each component of plane stress are combined according to a Mohr circle of birefringence. Applying this concept, the difference of the physical and optical principal directions was accounted for, and a general method of photoelastic solution for the plane-stress problem in orthotropic sheets was developed. The method of analysis is little more complex than the well-known procedures for isotropic materials, but at least three experimental measurements are required to characterize the optical response of the material to plane stress. Partial confirmation of the proposed stress-optic law was obtained by comparison of the theory to limited experimental data obtained in uniaxial-stress samples. It remains to establish a more positive verification by experiments in a variety of biaxial-stress conditions.     

The interpretation of isoclinics in photo-orthotropic-elasticity

Significant progress has been made in the interpretation of isochromatic fringes in photo-orthotropic-elasticity. However, the isoclinic fringes have not yet been satisfactorily interpreted. While it has been recognized that the isoclinics do not give the principal-stress directions in the composite model, there has been speculation that the isoclinics may give the principal-strain directions. An experimental study of the isoclinic fringes in orthotropic models was undertaken. The model material employed consisted of a transparent, unidirectionally reinforced, E-glass-polyester composite. First, the optical isoclinic parameters were measured in uniaxial stress fields for compression specimens as a function of the fiber-orientation angle. It was observed that, while the optical isoclinic parameter was different from the composite principal-stress angle, it was also different from the composite principal-strain angle. The optical isoclinic parameter was predicted very well by the Mohr circle of birefringence postulated by Sampson. In the second set of experiments, the optical isoclinic parameter was measured in biaxial stress fields by testing a circular disk in diametral compression. The actual state of strain and stress at the center of the disk was measured by means of a rectangular strain-gage rosette. Again it was observed that the optical isoclinic parameter was predicted very well by Sampson's relation. Thus, the isoclinic fringes in orthotropic models can be satisfactorily interpreted and Sampson's scheme can be utilized to obtain the individual principal stresses in orthotropic models.     

Transmission photoelasticity of centrally loaded generally and specially orthotropic beams

Transmission photoelastic patterns for generally and specially orthotropic beams and an isotropic beam are presented. Theoretical isochromatic and isoclinic results for these beams, calculated from classical elasticity stress equations and stress-optic laws, are also presented and compared with experiment. The agreement between the theory and experiment is excellent. For the orthotropic beams, a stressoptic law which accounts for the effects of residual birefringence was used. The residual birefringence observed for the composite used in this study is greater than one fringe order and results from a matrix residual tension which is about one fourth of the resin's ultimate tensile strength. Finally, the influence of such a large residual birefringence on beam and calibration photoelastic data is discussed in detail. Paper was presented at the 1986 SEM Spring Conference on Experimental Mechanics, held in New Orleans, LA on June 8–13.     

Photoelastic determination of mixed-mode stress-intensity factorsK I,KII andK III

On the basis of existing photoelastic methods for the determination ofK _I andK _II, this paper presents an experimental method for determiningK _III with photoelastic data, and a photoelastic method for comprehensively determiningK _I,K _II andK _III under the complex stress condition. A frozen three-dimensional photoelastic model is first used to determineK _I andK _II from the slice perpendicular to the flaw edge. Then, from that slice, a sub-slice is taken to determine the factorK _III. This method is examined by comparison with two test models.     

正交异性动态光弹性方法的几个基本问题的研究

文章对适用于动态研究的正交异性光弹性复合材料进行了分析，详细说明了光弹性复合材料中残余双折射的确定方法；基于静态下Ｈｙｅｒ和Ｌｉｕ应力－光性定律，提出了正交异性动态应力－光性定律，并对正交异性材料的动态力学参数及动态光弹性常数给出了实用的标定方法；最后，利用三个单轴压缩试件（０°，９０°及４５°），采用动态应变测量方法，证实了单轴应力状态下正交异性动态应力－光性定律的正确性。     

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.     

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.     

Characterization of crack-tip strain singularity in brittle-microcracking materials by means of the photoelastic-coating method

The strain fields ahead of crack tips in rock, mortar, and graphite were measured using a photoelastic coating method. A transparent ferroelectric ceramic was used as a coating material to observe the photoelastic fringe pattern. A coating plate of 110–150 μm thick was placed on single-edge-notch specimens, and photoelasticity experiments were conducted in three-point bending under a polarizing microscope. The results show that the principal-strain difference ahead of the crack tip is given by $$\Delta \in = \Delta \in _o [(J/\sigma _{ult} )/r]^m $$ whereσ _ult is the ultimate tensile strength,r is the distance from the crack tip, and9? _o andm are material constants. Based on this observation, the use of theJ _Ic concept in determining the fracture toughness of brittle-microcracking materials is discussed.     

Stability of frictional slipping at an anisotropic/isotropic interface

The stability of steady, quasi-static slip at a planar interface between an anisotropic elastic solid and an isotropic elastic solid is studied. The paper begins with an analysis of anti-plane sliding at an orthotropic/isotropic interface. Friction at the interface is assumed to follow a rate- and state-dependent law. The stability to spatial perturbations of the form exp(ikx₁), where k is the wavenumber and x₁ is the coordinate along which the interface is studied. An expression is derived for the critical wavenumber ∣k∣_cr above which there is stability. In-plane sliding at an anisotropic/isotropic interface is subsequently studied. In this case, slip couples with normal stress changes and a constitutive law for dynamic normal stress changes is adopted. Again a formula for ∣k∣_cr is derived and the results are specialized to the case of an orthotropic/isotropic interface. Numerical plots of the dependence of ∣k∣_cr on the orientation of the orthotropic solid, as well as on the material parameters are provided.     

Computation of thermal fracture parameters for orthotropic functionally graded materials using Jk-integral

This article introduces a computational method based on the J_k-integral for mixed-mode fracture analysis of orthotropic functionally graded materials (FGMs) that are subjected to thermal stresses. The generalized definition of the J_k-integral is recast into a domain independent form composed of line and area integrals by utilizing the constitutive relations of plane orthotropic thermoelasticity. Implementation of the domain independent J_k-integral is realized through a numerical procedure developed by means of the finite element method. The outlined computational approach enables the evaluation of the modes I and II stress intensity factors, the energy release rate, and the T-stress. The developed technique is validated numerically by considering two different problems, the first of which is the problem of an embedded crack in an orthotropic FGM layer subjected to steady-state thermal stresses; and the second one is that of periodic cracks under transient thermal loading. Comparisons of the mixed-mode stress intensity factors evaluated by the J_k-integral based method to those calculated through the displacement correlation technique (DCT) and to those available in the literature point out that, the proposed form of the J_k-integral possesses the required domain independence and leads to numerical results of high accuracy. Further results are presented to illustrate the influences of the geometric and material constants on the thermal fracture parameters.     

A study ofJ-integral of the orthotropic composite material

The relation between J-integral near model I crack tip in the orthotropic plate and displacement derivative is derived in this paper. Meanwhile, the relation between stress intensity factor K _I and displacement is also given in this paper. With sticking film moire interferometry method, the three-point bending beam is tested, thus the values of J-integral and K _I can be obtained from the displacement field, and then the truth of relation formula between J-integral and K _I in the orthotropic composite materials is experimentally verified.     

A study of hysteresis in theK ID -å relation

A photoelastic study of high speed crack propagation in Homalite 100 was conducted to measure hysteresis in the constitutive relation forK _ID -å. The fracture specimen was designed to obtain both crack acceleration and deceleration during a single crack extension. Additional loads perpendicular to the cráck-propagation path were applied at secondary locations to accentuate the magnitude of acceleration-deceleration observed in this specimen. The photoelastic data were analyzed using dynamic stress field equations in conjunction with the least-squares over-deterministic method to obtain the crack-propagation fracture toughness,K _ID . Crack velocity, å, was determined numerically by differentiating a polynomial fitted to the crack length-time data in a leastsquares sense. Experimental results indicate that Homalite 100 does not exhibit significant hysteresis in theK _ID -å relation.     

Several effects of melt viscosity on the nature ofC p —T data aboveT g for acrylate,methacrylate, and hydrocarbon polymers

Using objective computerized statistical procedures, we have examined high precisionC _p data by DSC reported by Wunderlich and Gaur for a series of alkyl acrylate and methacrylate polymers. Although they claimed the data to be linear inT aboveT _g , our results do not support the linear model. One or two endothermic slope changes are revealed aboveT _g in lowT _f polymers (T _f < 20 °C) and at least one exothermic slope change in highT _f polymers (T _f > 20 °C).T _f is the flow temperature of Ueberreiter. Both the first endotherm and the first exotherm occur near (1.22 ± 0.07)T _g , suggesting aT _ll type phenomenon.T _ll varies as \(1/\bar M_n \) . The first exotherm is associated by us with wetting of the DSC pan by molten polymer on the first heating of particulate highT _f polymers. The rate of wetting, and presumably the magnitude of the exotherm, depends in part on the ratio,γ/η, whereγ is surface tension andη is melt viscosity of the molten polymer. Sinceγ is relatively constant, the molecular weight and temperature dependence for rate of wetting resides inη, which depends on \(\bar M_w \) . For \(\bar M_n > > \bar M_c \) , a second exothermic event caused by sintering, and also controlled by η, may be present. The interactive roles of \(\bar M_n ,\bar M_w ,\bar M_w /\bar M_n \) ;M _c (entanglement molecular weight); particle size, and heating rate onC _p —T behaviour are delineated for the first time. LowT _f hydrocarbon polymers, namely atactic polyalphaolefins,C ₃ ,C ₅ ,C ₆ ; PIB; and dienes, PBD and cis-PI, exhibit single or double endotherms. Other results on highT _f polymers showing exothermic effects, notably PS, PnBMA and polyglycidylmethacrylate are cited.     

复合材料光弹性分析的工程方法

本文在推导均衡光弹性复合材料应力(应变)—光定律的基础上,导出了均衡光弹性复合材料的近似应变—光定律。然后加以推广,提出适用于一般光弹性复合材料的近似应变—光定律.精度分析及实验验证表明:作为工程计算,在采用本文推荐的参数时,光弹性复合材料可视作各向同性材料进行光弹性分析。     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.     

Development of a Hybrid Method of Reflection Photoelasticity for Crack Problems in Anisotropic Plates

Reflection type photoelastic experiment can be used more effectively than a transmission type photoelastic experiment especially in industrial fields. Moreover, composite materials have been widely used in engineering applications and structures because of their outstanding advantages which individual isotropic components do not have. The development of these materials requires a promising technique such as reflection photoelasticity to analyze their behaviors in service. Unfortunately, there are few experimental studies based on this technique. Therefore, a hybrid method based on this technique was developed in this research to analyze the fracture behavior of opaque anisotropic materials. The application of this method will help to understand the fracture behaviors of anisotropic materials used in engineering components and structures. The validity of this method was verified by comparison of the results obtained from this method with ones obtained from the hybrid methods for isotropic material on the same isotropic specimen. The reflection type photoelastic experiment for orthotropic materials was then applied to orthotropic plates with a central crack of various inclination angles. Using this hybrid method for anisotropic materials, stress intensity factors and separated stress components were obtained at the vicinity of the crack-tip in orthotropic plates from only the isochromatic fringe patterns of the isotropic coating material.     

A photoelastic study of crack repair

Birefringent coatings have been employed to study the effectiveness of an adhesively bonded repair of a center-cracked tension panel. The repair was one sided, with photoelastic coatings applied to the opposite side. Photoelastic coatings were also applied over the patch. Analysis methods are presented to permit the stress intensity factor to be determined from the isochromatic fringe patterns recorded from both continuous andX- andY-edged coatings. The results showed that the one-sided adhesively bonded patch reduced the stress intensity factor; however, the repair did not markedly change the character of stress distributions. Fringe loops formed near the crack tips for both the cracked and repaired tension panels. The primary difference was in the size of the loops. The reduction inK _I due to repair was smaller than anticipated, but even small improvements in ΔK _I markedly enhance the life of a repaired panel. The Paris power law is used to show the relation between the reduction in ΔK _I and the improvement in the crack growth rateda/dN. Fringe patterns from the birefringent coatings applied to the patch provided a means not only to investigate the stresses in the patch but also to detect the initiation of the local debonding of the adhesive in the neighborhood of the crack. The birefringent coating on the patch is an approach for producing an optically “smart” repair.     

A simple model for dislocation behavior,strain and strain rate hardening evolution in deforming aluminum alloys

In this work, modeling of the stress–strain behavior is carried out using a simple dislocation model. This model uses three variables to characterize the dislocation population: The average forest and mobile dislocation densities, ρ_f and ρ_m, and the average dislocation mean free path L. However, it is shown that within reasonable assumptions, only two of these variables are independent. The mathematical form derived from this dislocation-based model was applied to experimental stress–strain data determined at room temperature for pure aluminum, 3003-O, 2008-T4, 6022-T4, 5182-O and 5032-T4 aluminum alloy sheets. The evolution of the state variables was calculated for these materials from a single stress–strain curve. The average dislocation mean free paths at a strain of 0.5 were compared with TEM observations of dislocation cell sizes or inter-dislocation spacing for specimens deformed equal biaxially with the hydraulic bulge test. A very good agreement was obtained between predictions and experiments.