Photoelastic analysis of the singular stress field in a bimaterial wedge

This paper presents an experimental investigation of the singular stress field near the vertex of a bimaterial wedge using a digital photoelastic technique. Special attention is given to the casting of bimaterial wedge specimens and analysis technique for extracting stress intensity factors from photoelastic samples. Different bimaterial wedge specimens are made of two different photoelastic materials bonded through a special casting procedure and loaded in simple tension. A new multiple-parameter method is developed to obtain the stress intensity factor reliably from the isochromatic fringe patterns and the series representation of the stress field at the vertex of the wedge. Experimental results are compared with finite element predictions, and good agreement is observed.     

Phase Shifting Full-Field Interferometric Methods for Determination of In-Plane Tensorial Stress

A new method that combines phase shifting photoelasticity and transmission Coherent Gradient Sensing (CGS) is developed to determine the tensorial stress field in thin plates of photoelastic materials. A six step phase shifting photoelasticity method determines principal stress directions and the difference of principal stresses. The transmission CGS method utilizes a standard four step phase shifting method to measure the x and y first derivatives of the sum of principal stresses. These stress derivatives are numerically integrated using a weighted preconditioned conjugate gradient (PCG) algorithm, which is also used for the phase unwrapping of the photoelastic and CGS phases. With full-field measurement of the sum and difference of principal stresses, the principal stresses may be separated, followed by the Cartesian and polar coordinate stresses using the principal stress directions. The method is demonstrated for a compressed polycarbonate plate with a side V-shaped notch. The experimental stress fields compare well with theoretical stress fields derived from Williams solution for a thin plate with an angular corner.     

Stress near apex of dissimilar material with bilinear behavior

This study presents an analytical solution for the asymptotic stress field near the apex of a wedge composed of dissimilar materials exhibiting elastic and/or plastic deformation that can be described by a bilinear material model. Under the same assumptions, previous investigations resulted in eigenvalue differential equations that were not amenable to analytical integration. The present formulation avoids the numerical integration of such equations. After establishing its validity, the corner of the junction formed between a solder ball and a substrate was considered in order the study the effect of the hardening parameter on the strength of the singular stress field. Also, this formulation provides the exact form of the displacement field, which permits the construction of a global element to capture the correct strength of the singular stress field in regions with material and geometric discontinuities.     

A new photoelastic model for studying fatigue crack closure

The photoelastic analysis of crack tip stress intensity factors has been historically developed for use on sharp notches in brittle materials that idealize the cracked structure. This approach, while useful, is not applicable to cases where residual effects of fatigue crack development (e.g., plasticity, surface roughness) affect the applied stress intensity range. A photoelastic model of these fatigue processes has been developed using polycarbonate, which is sufficiently ductile to allow the growth of a fatigue crack. The resultant stress field has been modeled mathematically using the stress potential function approach of Muskhelishvili to predict the stresses near a loaded but closed crack in an elastic body. The model was fitted to full-field photoelastic data using a combination of a generic algorithm and the downhill simplex method. The technique offers a significant advance in the ability to characterize the behavior of fatigue cracks with plasticity-induced closure, and hence to gain new insights into the associated mechanisms.     

Stress-intensity distributions for corner cracks emanating from open holes in plates of finite width

A series of frozen stress photoelastic tests is carried out in the present investigation. Linear Elastic Fracture Mechanics is used to analyze the problem of a part-through corner crack at an open hole in a plate of finite width loaded in tension. The photoelastic modeling capability of three-dimensional subcritical crack growth problems is assessed. This is achieved by comparing monotonically grown flaw shapes in epoxy models with crack profiles relative to fatigue crack propagation tests in a different material. Photoelastic stress-intensity factor distributions are checked against numerical results obtained for quarter elliptical geometries. The dependence of stress intensity factors on flaw shape is also discussed.     

Wedge corner stress behaviour of bonded dissimilar materials

A contour integral, based on Betti’s reciprocal theorem, is used in conjunction with the finite element method to evaluate the magnitude of the wedge corner stress intensities associated with the higher order terms of the singular stress field near the interface corner of a bi-material joint. It is shown that using a different auxiliary field can eliminate the dependence of the wedge corner stress intensity on the integration path observed by [W.C. Carpenter, Int. J. Fracture 73 (1995) 93–108]. Finite element analysis of a typical joint geometry is used to demonstrate the path-independence of the magnitude of the stress intensities evaluated using the proposed method, and to show the effects of higher order terms on the stress state near the interface corner.     

Corner stress singularities in an FGM thin plate

Describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities. This analysis derives asymptotic solutions for functionally graded material (FGM) thin plates with geometrically induced stress singularities. The classical thin plate theory is used to establish the equilibrium equations for FGM thin plates. It is assumed that the Young’s modulus varies along the thickness and Poisson’s ratio is constant. The eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner. The characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner. The non-homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support. Finally, the effects of material non-homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle.     

The birefringent response of a potential photoelastic material with variable elastic properties

In photoelastic stress analysis, a need sometimes exists for a birefringent material in which the experimenter has the ability to vary its mechanical or elastic properties. It was the intent of this investigation to produce a homogeneous, isotropic solid with variable mechanical properties and which was suitable for photoelastic use. This goal is accomplished through the fabrication of a discontinuous composite where the birefringent constitutents have matched indices of refraction in the unstressed state. The constituents of this composite are solid-glass microspheres embedded in a polyester matrix. Three theories are formulated to define the stress-induced birefringent phenomena found in composites of this nature. Conventional photoelastic techniques were employed to produce data appropriate for comparison to the predictions of these theories.     

Stress field tomography based on 3D photoelasticity

A tomographic method for identification of stress fields based on 3D photoelasticity has been developed. A second order tensor field tomographic method based on the general inverse problem of 3D photoelasticity, previously developed by the authors, is found to be highly sensitive to errors in photoelastic observations. In this study a new tomographic method for stress field with fairly high robustness to errors in photoelastic observations has been developed by introducing both equilibrium condition and linear elasticity to the previously developed general tensor field tomographic method. This new stress field tomographic method expands unknown 3D stress distributions as a linear combination of independent set of basis functions and a new inverse problem is posed: identify the amplitudes of basis functions based on photoelastic observations. Just as the inverse problem of 3D photoelasticity, this newly posed inverse problem is also nonlinear and ill posed. Unlike conventional approaches to 3D photoelasticity, both these nonlinearity and ill-posedness are properly treated using a load incremental approach. Load incremental approach chops the nonlinear solution space into segments with unique solutions by conducting photoelastic observations at sufficiently small increments in external load. Validating both numerically and experimentally, it is shown that this new stress field tomographic method has sufficient robustness against errors in photoelastic observations and is applicable to experimental stress measurements.     

Order of singularity and singular stress field about an axisymmetric interface corner in three-dimensional isotropic elasticity

The spatial axisymmetric problem of an isotropic, elastic, homogeneous body containing an inclusion of a different material with an interface corner point (along a circular contour) and arbitrary joining angles is considered in this paper. It is found that the order of the stress singularity at the interface corner coincides with that of the corresponding plane strain problem, but the dependence of the singular stress field on the material properties depends on both the Poisson ratios (of the two isotropic materials) as well as on the ratio of their shear moduli. The theoretical results have been confirmed by numerical, finite-element-based results in a special bimaterial case.     

Reduction of contact stress by use of relief notches

The photoelastic method is used to investigate the possibility of relieving the large local stresses that develop in the corners of a right angled indenter compressing a semi-infinite body by inducing geometric changes to the indenter/semi-infinite body configuration. It is shown that a circular notch cut along the free edges of the indenter can totally eliminate the large corner stresses. The notch, if placed along the interface edge of the half plane, can reduce the stress concentration, but never eliminate it. The results obtained have wide practical application.     

A study of crack-tip nonlinearities in frozen-stress fields

A mechanical and optical characterization study in a uniaxial field was conducted on a commercial di-phase photoelastic material suitable for stress freezing. Results were used in a plane-strain theory for predicting nonlinear crack-tip behavior with the Prandtl-Reuss equations and a Mises criterion. These predictions were compared with frozen-stress photoelastic results obtained from experiments on a variety of technologically important three-dimensional cracked-body problems. Results indicate substantially greater stiffness or constraint in the nonlinear zone near the crack tip than predicted using uniaxial data. However, the value of the maximum shear stress at the onset of nonlinear behavior was accurately established and was the same for all cases examined.     

Dynamic measurements of initiation toughness at high loading rates

An experimental method is described for measuring the dynamic initiation toughness of a sharp stationary crack. A plane specimen is utilized which consists of a central region 50-mm wide and 200-mm long with integral dog-bone ends. The loading is accomplished by the detonation of four small explosive charges which produce two tensile stress waves upon reflection from the dog-bone ends. The stress waves meet at the midpoint of the specimen and reinforce to produce a relatively large, uniformly stressed region with a very high loading rate. The crack is positioned at the midpoint of the specimen at the location where the reinforcing tensile stress waves meet. A series of photoelastic experiments were conducted using Homalite 100 as the model material to observe, in a full-field view, the arrival of the dilatational waves, the subsequent development of the stress field at the tip of the stationary crack and the initiation of the crack. The isochromatic fringe pattern was also used to determine the instantaneous value of the stress-intensity factorK(t) after the characteristic fringe loops developed in the region near the crack tip. Finally,K(t) was measured using a single strain gage positioned and oriented so that its signal output was proportional toK(t) and independent of the next two higher order terms in the series representation of the strain field. A method was developed to determine the instant of initiation from the strain-time trace. Results obtained from the photoelastic and strain measurements of the dynamic-initiation toughnessK _ID were consistently higher than the static value ofK _IC . Paper was presented at the 1987 SEM Fall Conference on Experimental Mechanics held in Savannah, GA on October 25–28.     

Stress analysis of a wedge disclination dipole interacting with a circular nanoinhomogeneity

This paper investigates the interaction between a wedge disclination dipole and a circular nanoinhomogeneity embedded in an infinite matrix. The Gurtin–Murdoch surface/interface elasticity model is applied to account for the interface stress effect of the nanoinhomogeneity. A closed form solution for the stress fields inside the inhomogeneity and matrix is derived with the complex variable method of Muskhelishvili. The influences of the interfacial and bulk material properties as well as the geometric parameters on the material force of the wedge disclination dipole are systematically discussed. It is found that the interface stress effect may influence the material force of the wedge disclination dipole significantly.     

Single-strand strainwire technique: Comparison with existing methods and a proposed photostrain technique

This paper is the third of three papers evaluating a refined internal strainwire technique. This final paper evaluates the technique by comparing it with two elastic solutions, with a photoelastic solution, and with a new proposed photostrain technique. The problem chosen as the basis of comparison was a plane-stress problem of a plate with a circular hole under uniform tension. The proposed technique is experimental in nature and combines parts of the results of a photoelastic solution with those yielded by a three-wire internal strain-gage-rosette analysis to completely fix the state of stress in the model. The scientific techniques used to compare the three-wire strain technique and photostrain technique are as follows: two elastic solutions, one evaluated at a point and one arrived at by integrating the stress functions over a finite length; a finite-element solution; a photoelastic analysis using the shear-difference technique to separate the principal stresses; and a three-wire-rosette analysis. A comparison is made of the values of principal stresses yielded by these methods.     

Stress-concentration factors for fatigue configurations

Stress concentrations in thin-plate configurations commonly used in fatigue bending tests were determined by photoelastic means. The test configurations were rectangular and tapered plates with a centrally located hole. These configurations were machined from plates constructed by cementing together two sheets of similar photoelastic material. A reflective-type cement was used. In this manner, photoelastic measurements for the bending case were indicative of the average between maximum fiber stress and the stress at the reflective surface. The maximum fiber stress was then computed assuming that plane sections remain plane. Although bending was of primary concern, tension tests were also performed. The measured stress concentrations are compared with available analytical solutions. In the case of bending, the results are compared with infinite-plate solutions since the perforated finite-width plate bending problem has not been solved.     

Suitability of the photoelastic implementation of polymers with material birefringence

Scattered photoelastic techniques would gain considerable momentum through better understanding of the nature and response of the available photoelastic materials. The influence of the prevailing birefringent effect in the state of load-free polymers on the radiated scattered light energy is investigated. Six different photoelastic materials are considered. The impact of material birefringences on the quality of reading the fringe orders of the scattered radiation in a stressed photoelastic medium is explored. Spectral dependence of the modulation of light vectors in acrylic materials (Plexiglas) is illustrated. The advantage of this characteristic as a convenient means of compensation in scattered photoelasticity is pointed out. Acrylic sheets, a relatively cheap transparent polymer, seem to be suitable for scattered photoelastic analysis. The moiré technique might find application in determining fringe orders in a material such as Homalite-100. Replacing the primary beam by a primary sheet saves the cost and effort consumed in the scanning process. Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

光弹性复合材料对径受压圆板应力分析与材料条纹值测定

本文根据 Okubo 的各向异性分析,对纤维增强光弹性复合材料制成的对径受纹压圆板进行了研究;在此基础上提出了采用一个对径受压圆板测定光弹性复合材料三个条值(f_L,f_T,f_(LT))的方法。经与由三棵光弹性复合材料直条试样测定的相应材料条纹值数据比较,发现现在提出的方法更为简便、可靠。