Stress separation in the photoelastic study of centrifugal stresses

This technical note refers to the problem of stress separation in the photoelastic analysis of plane models under centrifugal stresses. Two methods are described in order to determine the sum of principal stresses. These methods, which are based on the compatibility equation, reduce the determination of the sum of principal stresses to the solution of a Laplace's or Poisson's equations. As an example of application, the separation of stresses in a rotating disk with two eccentric holes is shown and comparison with the stresses obtained by using the shear-difference method is made.     

Steady-state creep analysis of pressurized pipe weldments by perturbation method

The stress analysis of pressurized circumferential pipe weldments under steady state creep is considered. The creep response of the material is governed by Norton’s law. Numerical and analytical solutions are obtained by means of perturbation method, the unperturbed solution corresponds to the stress field in a homogeneous pipe. The correction terms are treated as stresses defined with the help of an auxiliary linear elastic problem. Exact expressions for jumps of hoop and radial stresses at the interface are obtained. The proposed technique essentially simplifies parametric analysis of multi-material components.     

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.     

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.     

Linear and Nonlinear Algorithms for Stress Separation in Photoelasticity

An experimental-numerical hybrid method for the stress separation in photoelasticity is proposed in this study. In the proposed method, boundary conditions for a local finite element model, that is, tractions along boundaries are inversely determined from photoelastic fringes. Two algorithms are proposed for determining the boundary condition. One is a linear algorithm in which the tractions are obtained by the method of linear least-squares from both principal stress difference and principal direction. Another is the nonlinear algorithm in which the tractions are determined only from the principal stress difference. After determining the boundary conditions for the local finite element model, the stresses can be obtained by finite element direct analysis. The effectiveness is demonstrated by applying the proposed method to a perforated plate under tension and contact problems. Results show that the boundary conditions of the local finite element model can be determined from the photoelastic fringes and then the individual stresses can be obtained by the proposed method. Furthermore, the stresses can be evaluated even if the boundary condition is complicated such as at the contact surface. It is expected that the proposed method can be powerful tool for stress analysis.     

A new analytical-numerical method for calculating interacting stresses of a multi-hole problem under both remote and arbitrary surface stresses

Based on the elementary solutions and new integral equations, a new analytical-numerical method is proposed to calculate the interacting stresses of multiple circular holes in an infinite elastic plate under both remote stresses and arbitrarily distributed stresses applied to the circular boundaries. The validity of this new analytical-numerical method is verified by the analytical solution of the bi-harmonic stress function method, the numerical solution of the finite element method, and the analytical-numerical solutions of the series expansion and Laurent series methods. Some numerical examples are presented to investigate the effects of the hole geometry parameters (radii and relative positions) and loading conditions (remote stresses and surface stresses) on the interacting tangential stresses and interacting stress concentration factors (SCFs). The results show that whether the interference effect is shielding (k <1) or amplifying (k> 1) depends on the relative orientation of holes (α) and remote stresses (σ_x^∞, σ_y^∞). When the maximum principal stress is aligned with the connecting line of two-hole centers and σ_y^∞ <0.5σ_x^∞, the plate containing two circular holes has greater stability than that containing one circular hole, and the smaller circular hole has greater stability than the bigger one. This new method not only has a simple formulation and high accuracy, but also has an advantage of wide applications over common analytical methods and analytical-numerical methods in calculating the interacting stresses of a multi-hole problem under both remote and arbitrary surface stresses.     

Polarimetry for spatiotemporal photoelastic analysis

This paper describes a new photoelastic technique for the spatiotemporal stress analysis. In a polarimeter developed, an elliptically polarized signal beam of light, modulated in state of polarization by two-dimensional principal-stress distributions interferes with a reference beam of light consisting of orthogonal linearly polarized two components. A time-sequential series of two-dimensional interference patterns are received one after another by a MOS video camera, followed by a computer. Of the elliptically polarized signal beam, the orthogonal field components along the directions of the principal stresses in a two-dimensional photoelastic sample can be computed from a recorded interference pattern, which offer the data needer for mapping the spatiotemporal principal-stress distribution over the sample. Not only each of the two orthogonal principal stresses but also the principal-stress difference are mapped in a time-sequential diagram. No use of any movable polarization element such as a rotating analyzer allows us to follow a rapid change in stress distribution within the maximum frame rate 2066 s⁻¹ of the MOS video camera.     

Stresses in a deep beam with a central concentrated load

The stress distribution in deep beams has received the attention of mathematicians an research workers since the early days of Wilson, Stokes and Filon.^{1, 2} A general theoretical solution is not possible and particular solutions of the biharmonic equatiion for the Airy stress function have been worked out to satisfy certain loadings. In the case of a deep beam under a symmetrical pressure on the top edge, solutions by (1) Durant and Garwood,³ (2) Chow, Conway and Winter⁴ have been put forward. Systematic photoelastic experimental work appears to be scarce and the object of this article is to present for comparison with existing theories the results of three photoelastic tests on simply supported deep beams carrying a central point load. The experimental stresses and those obtained from the solutions referred to above agreed reasonably well. The normal stresses σ _x calculated from the Wilson-Stokes theory were in error for the case under consideration (the theory is not valid for (H/L)>0.4), and the simple theory of bending is adequate for the maximum tensile stress provided the span exceeds 1.5 times the depth. Results are summarized in figures showing the magnitude and direction of the principal stresses; magnitudes are in a dimensionless from, hence it is possible to adapt them fior any beam of similar depth to span ratio under the same type of loading.     

On the load transfer from elastic inclusions to an infinite elastic orthotropic plane with cuts

The present paper deals with the problem of load transfer from elastic inclusions to an infinite elastic orthotropic plane with cuts located on one of the principal orthotropy directions. The constitutive system of equations of this problem is derived under the assumption that the inclusions are in a uniaxial stress state. The obtained system consists of a singular integro-differential equation and a singular integral equation for the jumps of the tangential stresses acting on the inclusion shores and for the derivative of the the cut opening function. The behavior of solutions of the system of constitutive equations at the endpoints of the inclusions and cuts is studied, and the solution of this system is constructed by the numerical-analytic discrete singularity method.     

An Integrated Approach to the Separation of Principal Surface Stresses Using Combined Thermo-Photo-Elasticity

A new instrument capable of the full-field separation of principal stresses on the surface of a component is presented. The instrument combines the techniques of thermoelastic stress analysis and reflection photoelasticity in a single optical head, permitting the simultaneous capture of both data from the same point of view. A single strain witness coating is employed for the acquisition of both the thermoelastic and photoelastic data, which is both birefringent under applied stress conditions and opaque at the infrared wavelengths to which the thermoelastic analysis system is sensitive. This enables the combined technique to be performed continuously from the same surface during loading. The performance of the new instrument is validated in the analysis of a classical laboratory specimen of known geometry. Separated stress data from the experiment is compared to simulated data, demonstrating that the accuracy of the stress separation technique is comparable to that of the individual thermoelastic and photoelastic techniques, and it is concluded that combined thermo-photo-elasticity is a powerful tool for the experimental separation of principal surface stresses.     

An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials

An elastic analysis of an internal crack with bridging fibers parallel to the free surface in an infinite orthotropic elastic plane is studied. An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials is presented for analyzing the distributions of stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-homogenous stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is determined by maximum tensile stress, resulting in fiber rupture and hence the crack propagation would occur in a self-similarity manner. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of unidirectional composite materials under the conditions of two moving loads given is obtained, respectively. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be obtained.     

Separation of principal stresses by SOR technique over arbitrary boundaries

A computerized method is presented that generates a grid mesh within the digitized boundary of a photoelastic specimen as it appears in the single viewing through an overhead polariscope. The second-order partial differential equation for the first linear invariant of stress which satisfies the Laplace equation is solved from the boundary values for the digitized domain by the finite-difference method. Connectivity and the weighting functions that are required for the iterative solution of the systems of linear equations are generated from the digitized information along the boundary. Isochromatic values at each nodal point within the boundary are estimated from the digitized fringe patterns by a scanning technique, and the individual values of principal stresses are determined. To enhance convergence, the method of successive over relaxation is applied with an optimum accelerating factor determined in the course of the solution process. The accuracy and the speed of the solution are tested with three different examples. Paper was presented at the 1989 SEM Spring Conference on Experimental Mechanics held in Cambridge, MA on May 28–June 1.     

Asymmetrical dynamic fracture model of bridging fiber pull-out of unidirectional composite materials

An elastic analysis of an internal crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane is studied, and asymmetrical dynamic fracture model of bridging fiber pull-out of unidirectional composite materials is presented for analyzing the distributions of stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-homogenous stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is ascertained by maximum tensile stress, the fiber ruptures and hence the crack propagation should also appear in the modality of self-similarity. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of unidirectional composite materials under the conditions of two increasing loads given is obtained, respectively. In terms of correlative material properties, the variable rule of dynamic stress intensity factor was depicted very well. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be gained.     

Physically and geometrically nonlinear deformation of conical shells with an elliptic hole

The elastoplastic state of conical shells weakened by an elliptic hole and subjected to finite deflections is studied. The material of the shells is assumed to be isotropic and homogeneous; the load is constant internal pressure. The problem is formulated and a technique for numerical solution with allowance for physical and geometrical nonlinearities is proposed. The distribution of stresses, strains, and displacements along the hole boundary and in the zones of their concentration is studied. The solution obtained is compared with the solutions of the physically and geometrically nonlinear problems and a numerical solution of the linear elastic problem. The stress-strain state around an elliptic hole in a conical shell is analyzed considering both nonlinearities __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 69–77, February 2008.     

An analytic separation method for photoelasticity

A method for separating principal stresses in photoelasticity is presented. This method is based upon the series solution of Laplace's equation and the determination of the unknown coefficients arising in this series by a least-squares numerical technique. By selecting an adequate number of terms in the series, the representation of the boundary values of the first stress invariant can be established as accurately as the initial photoelastic data. This form of representation of the first stress invariant at interior points in the region is moe accurate than the boundary values employed.     

An Experimental-Numerical Hybrid Method for Elastic Contact Problems*

ABSTRACT

An experimental-numerical hybrid technique for determining the contact stress distribution between two elastic bodies having both frictionless as well as bonded contact is discussed in this paper. The hybrid method makes use of experimental data collected at a section far from the contact surface and the numerically generated influence coefficients, in terms of the applied unit normal and shear stresses. The experimental data, i.e., the differences in normal stresses and the shear stress, are obtained using photoelastic analysis for the examples illustrated in this paper. When substituted into equations corresponding to the unit normal and shear stress applied in the contact region, this results in a set of algebraic equations which, when solved, allow the contact stress distribution to be obtained. This method is illustrated with examples involving simple and complex geometries of the contacting bodies.     

Photoelastic Analysis of Edge Residual Stresses in Glass by Automated “Test Fringes” Methods

Since the glass is a birefringent material, the analysis of residual stress in glass is usually carried out by means of photoelastic methods. This paper considers the automation of the “test fringes” method which is based on the use of a Babinet compensator or of a beam subjected to bending. In particular, two automated methods are proposed: the first one is based on the use of the centre fringe method in monochromatic light and the second one is based on the use of RGB photoelasticity in white light. The proposed methods have been applied to the analysis of membranal residual stresses in some tempered glasses, showing that they can effectively replace manual methods of photoelastic analysis of residual stresses in glass.     

A note on incremental equations for a new class of constitutive relations for elastic bodies

Some new classes of constitutive relations for elastic bodies have been proposed in the literature, wherein the stresses and strains are obtained from implicit constitutive relations. A special case of the above relations corresponds to a class of constitutive equations where the linearized strain tensor is given as a nonlinear function of the stresses. For such constitutive equations we consider the problem of decomposing the stresses into two parts: one corresponds to a time-independent solution of the boundary value problem, plus a small (in comparison with the above) time-dependent stress tensor. The effect of this initial time-independent stress in the propagation of a small wave motion is studied for an infinite medium.     

A numerical method for separation of stresses in photo-orthotropic elasticity

A numerical method is suggested for separation of stresses in photo-orthotropic elasticity using the numerical solution of compatibility equation for orthotropic case. The compatibility equation is written in terms of a stress parameter S analogous to the sum of principal stresses in two-dimensional isotropic case. The solution of this equation provides a relation between the normal stresses. The photoelastic data give the shear stress and another relation between the two normal stresses. The accuracy of the numerical method and its application to practical problems are illustrated with examples.     

Field fluctuations in a heterogeneous elastic material—an information theory approach

Internal stress and strain fields in disordered elastic solids such as multiphase materials or polycrystals are considered. In order to derive a probability distribution for those random internal fields, the information theory entropy is maximized subject to constraints representing the basic equations of elasticity and certain experimental data. Thus one can find the probability distribution which agrees with all known facts but makes no assertions about the internal fields which cannot be supported by the available information. This approach is in accordance with the formal exact solution of the statistical problem if one has complete microstructural information. In case of incomplete microstructural data, useful approximate solutions can easily be obtained. In particular, the following set of data is sufficiently detailed for the prediction of internal field fluctuations: the average strain, the one-point probability density of the random elastic constants, and the effective (overall) elastic constants. Especially the information supplied by the effective elastic constants plays a major role since it reflects the microstructural topology of the heterogeneous material. One obtains Gaussian probability distributions for stress and strain, which are applied to calculate mean values and fluctuations of stresses in a cemented metal carbide and a zinc polycrystal.