Showing papers on "Photoelasticity published in 1981"

Photoelastic and Electro-Optic Properties of Crystals

Abstract: 1. Photoelasticity of Crystals. Introduction.- 1.1. Discovery of the Phenomenon of Photoelasticity.- 1.2. Mathematical Formulation and Neumann's Constants. Pockels' Contribution.- 1.3. A Brief Historical Survey.- 1.3.1. Amorphous Solids.- 1.3.2. Cubic Crystals.- 1.3.3. Uniaxial and Biaxial Crystals.- 2. Mathematical Tools, Tensor Properties of Crystals, and Geometrical Crystallography.- 2.1. Linear Transformations.- 2.1.1. Coordinate Transformations.- 2.1.2. Orthogonality Relations.- 2.1.3. The Determinant of the Matrix [?ij] of the Direction-Cosine Scheme.- 2.2. Matrix Algebra.- 2.2.1. Introduction.- 2.2.2. Matrix Algebra and Coordinate Transformations.- 2.2.3. Some Common Types of Matrices.- 2.2.4. Orthogonal Matrix.- 2.2.5. Matrix Operators and Transformation of Tensor Components.- 2.2.6. The Diagonalization of a Matrix.- 2.3. Vectors and Their Transformation Laws.- 2.3.1. Vector Components and Coordinate Transformations.- 2.3.2. Transformations of Coordinate Differences.- 2.3.3. Transformation Law of Vectors.- 2.4. Tensor Nature of Physical Properties of Crystals and the Laws of Transformation of Cartesian Tensors.- 2.4.1. Concept of a Tensor Property and Some Examples of Tensor Properties.- 2.4.2. Transformation Law of Cartesian Tensors.- 2.4.3. Physical Properties and Crystal Symmetry.- 2.5. Crystal Symmetry and Geometrical Crystallography. The 32 Point Groups.- 2.5.1. The 32 Crystallographic Point Groups: Their Symmetry Elements and Some Examples of Crystals.- 2.5.2. Some Symmetry Operations and Their Representation by Symbols.- 2.5.3. The 32 Crystallographic Point Groups in the Schonflies Notation. Geometric Derivation.- 2.6. Symmetry Operations and Their Transformation Matrices.- 2.7. Symmetry Elements of the 32 Point Groups.- 2.7.1. Symmetry Elements of the 32 Point Groups.- 2.7.2. Comments on the 32 Crystallographic Point Groups and Their Symmetry Elements as Listed in Tables 2.3 and 2.5a.- 2.8. Neumann's Principle and Effect of Crystal Symmetry on Physical Properties.- 3. Pockels' Phenomenological Theory of Photoelasticity of Crystals.- 3.1. Introduction.- 3.2. Phenomenological Theory, Stress-Optical and Strain-Optical Constants in Four- and Two-Suffix Notations qij and pij Matrices for the 32 Crystallographic Point Groups.- 3.2.1. The Assumptions Forming the Basis of Pockels' Theory.- 3.2.2. Mathematical Formulation of Photoelasticity in Terms of qijkl and pijkl.- 3.2.3. Mathematical Formulation of Photoelasticity in Terms of qij and pij.- 3.2.4. Crystal Symmetry and the Number of Photoelastic Constants.- 3.3. Derivation of the Nonvanishing and Independent Photoelastic Constants for the Various Crystal Classes by Different Methods.- 3.3.1. Classical Method.- 3.3.2. Tensor Method.- 3.3.3. Group Theoretical Method.- 4. Elasticity of Crystals.- 4.1. Introduction.- 4.2. Stress and Strain as Tensors.- 4.2.1. Stress as a Second-Rank Tensor.- 4.2.2. Strain as a Second-Rank Tensor.- 4.3. Hooke's Law.- 4.3.1. Generalized Form of Hooke's Law with Elastic Constants cij and sij and the Matrices of cij and sij for the 32 Point Groups.- 4.3.2. Generalized Form of Hooke's Law with Elastic Constants cijkl and sijkl.- 4.3.3. Interrelation between cijkl and cmn and between sijkl and smn.- 4.4. Experimental Methods of Determining cij and sij Christoffel's Equation and Its Use in Determining cij of Crystals.- 4.5. Ultrasonics.- 4.5.1. Introduction.- 4.5.2. Diffraction of Light by Liquids Excited Ultrasonically.- 4.5.3. Optical Methods of Determining the Ultrasonic Velocities and Elastic Constants of Transparent Solids Employing the Schaefer-Bergmann Pattern, the Hiedemann Pattern, and the Lucas-Biquard Effect.- 4.5.4. Mayer and Hiedemann's Experiments.- 4.5.5. Raman-Nath Theory of Diffraction of Light by Ultrasonic Waves.- 4.5.6. Doppler Effect and Coherence Phenomenon.- 4.6. Brillouin Effect and Crystal Elasticity.- 4.6.1. Introduction.- 4.6.2. Theory of Light Scattering in Birefringent Crystals.- 4.6.3. Concluding Remarks.- 5. Experimental Methods of Determining the Photoelastic Constants.- 5.1. Optical Behavior of a Solid under a Mechanical Stress, and Neumann's Constants.- 5.2. Derivation of Expressions for the Stress Birefringence in Terms of qij for Cubic and Noncubic Crystals.- 5.2.1. Stress Birefringence in Cubic Crystals.- 5.2.2. Stress Birefringence in Noncubic Crystals.- 5.2.3. Tensor Method of Deriving q?ijkl in Terms of qmnop.- 5.2.4. Expression for the Change of Thickness in Terms of sij for an Orthorhombic Crystal for a Specific Orientation.- 5.3. Experimental Determination of qij and pij by Optical Methods.- 5.3.1. Measurement of Stress Birefringence, and Relative Path Retardation.- 5.3.2. Measurement of Absolute Path Retardation by Interferometric Methods.- 5.3.3. Photoelastic Studies of Optically Active Crystals.- 5.4. Dispersion of qij by Spectroscopic Methods.- 5.4.1. Birefringent Compensator for Studying Very Small Changes in Double Refraction.- 5.4.2. Dispersion of the Individual Stress-Optical Coefficients q11 and q12 of Vitreous Silica.- 5.4.3. Interference-Spectroscope Method of Studying the Absolute Photoelastic Coefficients of Glasses and Their Variation with Wavelength.- 5.5. Elliptic Vibrations and Elliptically Polarized Light.- 5.5.1. Composition of Two Rectangular Vibrations Giving an Ellipse: Use of the Senarmont Compensator.- 5.5.2. Photometric Method for the Measurement of Photoelastic Birefringence.- 5.5.3. The Poincare Sphere and Its Application to the Study of the Photoelastic Behavior of Optically Active Crystals.- 5.6. Ultrasonic Methods of Studying the Elasto-Optic Behavior of Crystals.- 5.6.1. Introduction.- 5.6.2. Mueller's Theory.- 5.6.3. Experimental Determination of Pij/pkl by Three Different Methods Due to Mueller.- 5.6.4. Pettersen's Method of Determining Pij/pkl.- 5.6.5. Bragg Diffraction Method of Determining the Individual Values of pij.- 5.6.6. Borrelli and Miller's Method of Determining the pij of Glass.- 5.6.7. Technological Applications of the Acousto-Optic Effect.- 5.7. Brillouin Scattering and Photoelasticity of Crystals.- 6. Atomistic Theory of Photoelasticity of Cubic Crystals.- 6.1. Introduction.- 6.2. Mueller's Theory-A Brief Survey.- 6.3. Effect of Hydrostatic Pressure on the Index of Refraction n The Strain Polarizability Constant ?0.- 6.4. Anisotropy of Rj and ?itj.- 6.5. Thermo-Optic Behavior of Crystals and Photoelastic behavior.- 6.6. Pockels' Photoelastic Groups in Cubic Crystals and Mueller's Theory.- 6.7. Photoelastic Dispersion in Cubic Crystals ?0 as a Function of Crystalline Material, Wavelength of Light, and Temperature.- 6.8. Effect of Elastic Deformation on the Oscillator Strengths and Dispersion Frequencies of Optical Electrons.- 6.9. Temperature Dependence of Stress-Optical Dispersion.- 6.10. Reversal of the Sign of Stress Birefringence in Pure and Mixed Crystals.- 6.10.1. Pure Crystals.- 6.10.2. Mixed Crystals of KCl and KBr.- 6.11. Stress-Optical and Strain-Optical Isotropy in Cubic Crystals.- 6.12. Optic Axial Angle and Its Dispersion in Stressed Cubic Crystals of T and Th Classes.- 7. Piezoelectricity.- 7.1. Introduction.- 7.2. Direct and Converse Piezoelectric Effects.- 7.3. Mathematical Formulation, Piezoelectric Constants dijk in Tensor Notation, and dij in Two-Suffix Notation Relation between dijl and dij.- 7.4. Deduction of the Surviving dijk for Some Crystal Classes by Tensor Method, and the dij Matrices for the 21 Noncentrosymmetric Classes.- 7.5. Concluding Remarks.- 8. Electro-Optic Effects in Crystals: Pockels Linear Electro-Optic and Kerr Quadratic Electro-Optic Effects.- 8.1. Introduction.- 8.2. Demonstration of the Electro-Optic Effects, Linear and Quadratic.- 8.3. Historical Survey.- 8.3.1. Earlier Work.- 8.3.2. More Recent Work.- 8.4. Pockels' Phenomenological Theory of the Linear Electro-Optic Effect in Three- and Two-Suffix Notations, Rijk and rij.- 8.5. Derivation of the Relation between the Linear Electro-Optic Constants of a Crystal: Free and Clamped Constants.- 8.5.1. Discussion: Primary and Secondary Electro-Optic Effects, and Clamped and Unclamped Electro-Optic Coefficients.- 8.5.2. Methods of Obtaining the Primary and Secondary Linear Electro-Optic Effects.- 8.6. Kerr Quadratic Electro-Optic Effect: Pockels' Phenomenological Theory.- 8.7. Crystal Symmetry and the Number of Surviving Linear Electro-Optic Coefficients Rijk and rij and Their Deduction by Tensor Method: rij Matrices for the 21 Noncentrosymmetric Classes.- 8.7.1. Crystal Symmetry and the Surviving Linear Electro-Optic Constants.- 8.7.2. Tensor Method of Deducing the Nonvanishing Independent Rijk.- 8.8. Derivation of the Expressions for ? = f(rij) for Some Typical Crystal Classes and Orientations.- 8.8.1. Cubic System: Classes 23(T) and $$\bar 43m$$ (Td).- 8.8.2. Tetragonal System: Class $$\bar 42m$$ (D2d).- 8.8.3. Trigonal System: Class 32 (D3).- 8.9. Experimental Methods of Determining rij.- 8.9.1. General Description and Application to Some Typical Crystal Classes.- 8.9.2. Some Experimental Methods.- 8.9.3. Methods of Applying the Electric Field to the Crystal Prism.- 8.9.4. Experimental Determination of rij in Some Specific Cases of Crystals.- 8.10. Some Points of Interest on the Use of the Pockels Effect in Crystals, and Half-Wave Voltage V?/2.- 8.11. Some Technological Applications of Pockels Cells (Linear Electro-Optic Devices).- 8.11.1. Use of the Electro-Optic Effect in Technology.- 8.11.2. Some Applications of Electro-Optic Devices.- Author Index.

Stress Distribution of Polymers in Extrusion through a Converging Die

Abstract: An experimental and a theoretical study were carried out to determine the stress distributions of viscoelastic polymeric melts in a converging channel. For the experimental study, two different types of experiment were conducted using converging channels: one was the measurement of wall normal stress with the aid of pressure transducers, and the other was the measurement of stress birefringence with the aid of a circular polariscope, which enabled us to determine both shear stress and normal stress distributions in the channel. It was found that the distribution of wall normal stress goes through a minimum and that the extensional stresses along the centerline of the converging channel determined from the stress birefringence patterns, go through a maximum. For the theoretical study, the Coleman‐Noll second‐order fluid was used to derive theoretical expressions for the wall normal stress, and shear and normal stresses in the converging flow field. It was found that the theoretical analysis corroborates qu...

Dynamic photoelastic investigation of interaction of stress waves with running cracks: A photoelastic study of interaction of stress waves with running cracks is conducted to investigate the influence of crack-wave interaction on crack-propagation behavior and crack branching

Abstract: Dynamic photoelasticity in conjunction with high-speed photography was utilized in experiments to study the interaction of stress waves with a running crack. Experimental data were analyzed to study the effect of wave scattering about a moving crack tip. The results indicated a strong influence of stress waves on crack-propagation behavior and crack branching.

Use of photoelasticity in fracture mechanics

Abstract: A number of transparent amorphous materials which are optically isotropic become optically anisotropic when stressed and exhibit characteristics similar to crystals such as the property of double refraction. Upon unloading the effect disappears. This effect was first observed by Sir David Brewster in 1816 and is known as temporary double refraction. When such materials are loaded and observed in a polarized light field, temporary double refraction produces interference bands known as isochromatics, or stress fringes, each of which represents the locus of points of the same maximum shearing stress in the plane of the model which is normal to the incident light beam. The fringe order, or number, is proportional to the maximum shearing stress. Thus, if we observe a point in a transparent model in a polarized light field as it is loaded, we will observe (with monochromatic light) the change in color from dark to bright to dark, representing one complete optical cycle. The initial dark band would be fringe order zero in a polariscope set for extinction of light, the second fringe order one, then two, three, etc. These bands form contour-like patterns across the model. Closely spaced narrow bands indicate regions of high stress gradients. Broad, widely separated bands indicate regions of low stress gradients. Such effects are shown in Figure 2.1 for a plate with two holes under vertical uniaxial tension.

Stress optic coefficient and stress profile in optical fibers.

Abstract: The stress optic coefficient and stress profile in optical fibers have been determined photoelastically using a polariscope having good reproducibility and high sensitivity. The results of the work presented in this paper indicate that the photoelastic behavior may be different in fibers and in bulk glasses. The photoelastically determined clad compression in strengthened fibers was found to correlate well with the strengthening observed in these fibers using tensile tests.

Some further properties of caustics useful in mechanical applications.

Abstract: The caustics created from reflections of light rays at the vicinity of stress singularities and other stress concentrations in an isotropic and elastic plate submitted to a plane-stress mode of deformation yield important information concerning the order of singularity of the stress intensity or concentration factor at these particular points of a stress field. All information was extracted from particular measurements of the longitudinal or transverse diameters of the caustics, based on basic properties of caustics. In this paper some further properties were found, which are very useful in evaluating the stress field creating the caustic, which make the respective measurements independent of the angle of orientation of the caustic and more reliable than previously.

Quasi-Square Hole with Optimum Shape in an Infinite Plate Subjected to In-Plane Loading,

Abstract: : This paper deals with the optimization of the shape of the corners and sides of a square hole, located in a large plate and subjected to in-plane loads, with the object of minimizing stress concentrations. Appreciable disagreement has been found between the results obtained previously by other investigators. In this paper new tests have been conducted and discrepancies have been corrected. Using an optimization technique, the authors have developed a quasi square shape which introduces a stress concentration of only 2.54 in a uniaxial field, the comparable value for the circular hole being 3. The efficiency factor of the proposed optimum shape is 0.90 whereas the efficiency factor of the best shape developed previously was 0.71. The shape also is developed that minimizes the stress concentration in the case of biaxial loading when the ratio of biaxiality is 1:-1. (Author)

Error analysis of photoelasticity in fracture mechanics

Abstract: An error analysis of the use of photoelasticity in the study of fracture problems is attempted. In particular, it was desired to determine the optimum regions of data collection and to ascertain the sensitivity of the extractedK 1 andK 2 values to errors in such parameters as crack-tip position and fringe location. Experiments were performed on both Mode 1 and Mixed Mode cases and the results compared with the error analysis

Photoelastic investigation of stress wave diffraction about stationary crack-tips

Abstract: Transient phenomena of reflection and diffraction of explosively-generated plane and cylindrical elastic waves at the tips of a finite crack buried at some depth below a free surface are investigated. Dynamic photoelasticity has been used as a means for visualizing the highly complex interaction process between stress waves and cracks. Primary and secondary diffraction are studied and sequences of isochromatic fringe patterns and their associated wave front reconstruction patterns are presented. Fracture mechanics aspects of dynamic crack initiation under stress wave loading are discussed.

Photoelastic study of the influence of non-singular stresses in fracture test specimens

Abstract: Improved computational methods have been developed to determine, from photoelastic fracture patterns, those stress field parameters additional to the stress intensity factor, that are associated with different fracture test specimen geometries. The variations with crack tip position of these non-singular terms in Modified-Compact-Tension and Rectangular-Double-Cantilever-Beam specimens have been studied. Results have been utilized to formulate criteria that can be used to quantify the concept of the singularity-dominated zone around a crack tip in specimens of finite dimensions.

An Ultrasonic Technique for Sizing Surface Cracks

Abstract: Rayleigh surface waves are proposed as a non-destructive method to find the depth of surface cracks. The paper describes how dynamic photoelasticity was used to develop an understanding of the subsurface interactions between R-waves and a narrow slot. A frequency analysis of the transmitted wave confirmed that the slot acts as a low pass filter for the high frequency Fourier components of the input wave. It is then shown that the high frequency cut-off in the spectrum of the transmitted wave from broadband ultrasonic surface pulse can be used to determine the depth of surface slots.

Further Studies on Dynamic Crack Curving

Abstract: : The elasto-dynamic stress field surrounding rapidly propagating cracks in thin polycarbonate, double edged crack tension specimens were analyzed by dynamic photoelasticity using a 16-spark gap Cranz-Schardin camera system. Crack curving was observed in two slanted double edged crack specimens and in two offset parallel double edged crack specimens. In another test, the crack ran straight between two symmetrically located twin cracks. Results of these five tests were used to verify the dynamic crack curving criterion by Ramulu, et al., in which a reference distance from the crack tip is incorporated into the maximum circumferential stress of minimum strain energy density criteria. The critical material property for crack curving in this thin polycarbonate sheet was found to be about ro = 0.5 mm.

Application of Faraday's effect in static and dynamic holographic photoelasticity

Abstract: The basic principle of applying Faraday's effect to achieve the separation of fringes in static and dynamic holographic photoelasticity, and a study and application of Faraday's light rotator are described in this paper. It is proposed that Faraday's light rotator be used for automating photoelastic instrumentation for measuring isoclinics and the decimal orders of isochromatic fringes.

Effects of thermal stress on group delay in jacketed single mode fibres

Abstract: The theoretical investigations on the group delay in single mode fibres have been studied by considering the photoelastic effect in addition to the electron polarisability effect on the refractive index of optical fibres. It is found that, although the electron polarisability effect dominates in unjacketed fibre, the photoelastic effect becomes greater with increasing thickness of the jacketing layer.

Synchronization problems in dynamic holographic photoelasticity

Abstract: An analysis of synchronization problems encountered when recording transient-fringe patterns which are dependent on the light source and the method of shock generation is presented. The analysis is specifically designed to be valid on a photoelastic material having either a high or low Young's modulus when the shock generator is an air gun and the light source is a Q-switched ruby laser. Synchronization is performed using integrated circuits in T.T.L. logic which give a triggering order to the ruby laser under the control of the projectile velocity.

The reciprocal character of rayleigh-waves and cracks

Abstract: Utilizing dynamic photoelasticity a dual crack-elastic wave problem is investigated: Initiation of cracks by superimposing Rayleigh-waves and generation of Rayleigh-waves during instationary crack movement.

Simplifications for scattered-light photoelasticity when using the unpolarized incident beam

Abstract: A method is presented for obtaining scettered-light photoelastic data in three-dimensional problems using an unpolarized incident light beam. Using simplifying optical assumptions, the scattered-light observation path is considered tobe a series of half-wave retarders. Data are obtained through rotation of the optical analyzer and translation of the incident light beam with respect to the model. The method is applied to obtain data in problems where the secondary principal directions are: (1) fixed and (2) rotate. Results compare favorably with those obatained using a polarized incident beam.

Experimental evaluation of stress concentration and intensity factors : useful methods and solutions to experimentalists in fracture mechanics

Abstract: 1 Stress concentrations.- 1.1 Introduction.- 1.2 Advantages and disadvantages of stress analysis methods used to determine stress concentrations.- 1.3 Compilation of results.- 1.4 Geometrically non-linear stress concentrations.- 1.5 Stress concentrations in mixed boundary value problems.- 1.6 Stress concentrations in some specific problems.- 1.7 Stress concentrations in three-dimensional problems.- 1.8 Dynamic stress concentrations.- 1.9 Unconventional approaches to the study of stress concentrations.- References.- 2 Use of photoelasticity in fracture mechanics.- 2.1 Introduction.- 2.2 Analytical foundations for cracked bodies.- 2.3 Experimental considerations.- 2.4 Application of the frozen stress method.- 2.5 Summary and conclusions.- References.- 3 Elastic stress intensity factors evaluated by caustics.- 3.1 Introduction.- 3.2 The basic formulas.- 3.3 The equations of caustics.- 3.4 Properties of the caustics at crack tips.- 3.5 The case of birefringent media.- 3.6 The case of anisotropic media.- 3.7 Interacting crack problems.- 3.8 Branched crack problems.- 3.9 Interface crack problems.- 3.10 V-notch problems.- 3.11 Shell problems.- 3.12 Plate problems.- 3.13 Other applications.- 3.14 Discussion.- 3.15 Conclusions.- References.- 4 Three-dimensional photoelasticity: stress distribution around a through thickness crack.- 4.1 Introduction.- 4.2 Hartranft-Sih plate theory.- 4.3 Triaxial crack border stress field.- 4.4 Experimental considerations: specimens and materials.- 4.5 Test procedure: frozen stress technique.- 4.6 Comparison of Hartranft-Sih theory with experiments.- References.- 5 Experimental determination of dynamic stress intensity factors by shadow patterns.- 5.1 Introduction.- 5.2 Physical and mathematical principles of the method.- 5.3 Theoretical analysis of the shadow pattern after Manogg: Mode-I-loaded stationary crack.- 5.4 Validity of the analysis for stationary cracks under dynamic loading.- 5.5 The dynamic correction for propagating cracks.- 5.6 Experimental technique and evaluation procedure.- 5.7 Applications.- 5.8 Concluding remarks.- References.- 6 Experimental determination of stress intensity factor by COD measurements.- 6.1 Introduction.- 6.2 Principle of the interference optical technique.- 6.3 Determination of stress intensity factor from crack opening displacement.- 6.4 Conclusion.- References.- Author's Index.

Pockels’ Phenomenological Theory of Photoelasticity of Crystals

Abstract: The refractive index n, one of the most important optical properties of all transparent media, does not transform like a tensor. However, the dielectric tensor K ij or its reciprocal, the impermeability tensor (1/K) ij , does transform like a second-rank tensor.