Stress and finite-strain analysis of a circular ring under diametral compression

Using a moiré, large-strain analysis method, a complete solution is shown in this paper of the fields of strain and stress for a circular ring subjected to diametral compression between two flat platens. The isotheticsu andv, obtained using 1000-lines-per-inch gratings, were differentiated photographically by the shifting technique (moiré-of-moiré) to determine ?u/?x, ?v/?y, ?u/?y and ?v/?x. Using the exact finite strain-displacement relationship, the Eulerian strains ? _x ^E , ? _y ^E and γ _xy ^E were computed. From these, the principal Eulerian strains were obtained. These results were verified with the isochromatics obtained from a large-deformation photoelasticity analysis. The ring was made of a polyurethane rubber which exhibits a linear relationship between natural strain and a newly introduced concept of “natural stress”. The Eulerian strains were converted to natural strains, and from these natural stresses were computed using the newly developed concept. Results are presented graphically for the whole field of the ring.     

Moiré-fringe multiplication by means of filtering and a wave-front reconstruction process

Two principal problems associated with the practical application of moiré fringes are to obtain sufficient sensitivity for measuring small strains and to develop a simple and inexpensive technique for engraving lines on the surface of a model. This paper deals with simple solutions to both problems. It is shown that the maximum number of fringes that can be observed for a given model is independent of the gratings utilized and depend only on the geometry of the employed optical system. Examples of moiré patterns corresponding to the equivalent of 6000 lines per inch and patterns of the derivatives of the displacements corresponding to the same number of lines illustrate the paper.     

Investigation of shrinkage-induced stresses in stereolithography photo-curable resins

An experimental study has been undertaken to investigate the shrinkage characteristics of acrylic-based and epoxy-based stereolithography (SL) photopolymer resin systems after they have been laser cured and post-cured under ultraviolet (UV), and thermal exposure. The induced residual stresses and strains were determined by the shadow moiré and the hole-drilling strain-gage methods. Out-of-plane displacements (warpage) of acrylic-based post-cured resin plates were recorded by means of the shadow moiré method and correlated to the shrinkage strains by theoretical analysis. The induced residual stresses in the epoxy-based cylindrical resin specimens were determined from strains of three-element strain-gage rosettes of the blind-hole drilling method. Results are presented for the shrinkage stresses and strains for both material systems as a function of the post-curing process (UV, thermal). It was found that the shrinkage strains in the acrylic-based photopolymer resin were of considerable magnitude, while thermal post-curing resulted in higher shrinkage stresses for both material systems. The values of the shrinkage stresses compare well with those of the existing literature.     

The micro-connection of the second-order moiré fringes and its order determination,and the determination of the boundary strains

The micro-connection method for determining the centre lines of second-order moiré fringes presented in this paper can enhance the accuracy in measuring strain fields, and generally can determine the absolute order of the second-order moiré fringes. The strain data obtained from curved beam experiment are in good accordance with theory. The relationship between the second-order moiré fringe and the strains at the specific points of the specimen is derived. Hence a reciprocate shift method is presented for determining strain distributions in the non-overlapping region of the shifted moiré patterns, (usually in the region of specimen where no second-order moiré fringes occur is the boundary region).     

Optical differentiation of geometrical patterns

Two simple optical methods are presented which can perform direct, real-time differentiation of geometrical patterns by shifting. The first one gives uniform shifting in one direction and the resulting moiré fringes are curves of constant partial derivative along the same direction, and the second renders moiré fringes of constant gradient modulus of the function represented by the pattern. Only incoherent light is used with the technique, but applications with coherent light are also envisaged. As an application, the paper presents both types of differentiation of a shadow-moiré pattern.     

Dynamic moiré methods for the bending of plates

Two approaches are proposed to extend the Ligtenberg’s reflective moiré method to dynamic-bending problems of plates. Partial slope contours of two orthogonal directions can be obtained simultaneously with the proposed schemes. Vibrating modes of plates can also be obtained by the methods.     

A general framework for identification of hyper-elastic membranes with moiré techniques and multi-point simulated annealing

This paper presents a hybrid procedure for mechanical characterization of hyper-elastic materials based on moiré, finite element analysis and global optimization. The characterization process is absolutely general because does not require any assumption on specimen geometry, loading or/and boundary conditions.The novel experimental approach followed in this research relies on a proper combination of intrinsic moiré and projection moiré which allows 3D displacement components to be measured simultaneously and independently using always the same experimental setup and just one single camera. In order to properly compare experimental data and finite element predictions, 3D displacement information encoded in moiré patterns which are relative to the deformed configuration taken by the specimen are expressed in the reference system of the unloaded state.A global optimization algorithm based on multi-level and multi-point simulated annealing which keeps memory of all best records generated in the optimization is used in order to find the unknown material properties through the minimization of the Ω functional built by summing over the differences between displacements measured experimentally and those predicted numerically.Feasibility, efficiency and robustness of the proposed methodology are demonstrated for both isotropic and anisotropic specimens subject to increasing pressure loads: a natural rubber membrane and a glutaraldehyde treated bovine pericardium patch, respectively. Remarkably, the results of the characterization process are in very good agreement with target data independently determined. For the isotropic specimen, the maximum error on hyper-elastic constants is less than 1% and the residual error on displacements is less than 3.5%. For the anisotropic specimen, the maximum error on material properties is about 3.5% while the residual error on displacements is less than 3%. The identification process fails or becomes less reliable if “local” displacement values are considered.     

Determination of thermal strains by moiré interferometry

An existing method is extended to measure thermal strain distributions on an absolute basis. Free thermal expansion and stress-induced deformations are separated, allowing the determination of coefficients of expansion, stress-induced strains, normal stresses and shear stresses. The method is applicable to many steady-state and transient thermal-strain problems. A crossed-line grating is replicated on the specimen at elevated temperature. A zero-expansion mold is used for the replication, so that the grating frequency at the replication temperature can be retained to null the moiré interferometer at room temperature. When the specimen is viewed in the moiré interferometer, the fringe patterns reveal theU andV displacements induced by any change from the replication temperature. In addition, carrier fringe techniques are introduced for steady-state problems to subtract off the free thermal expansion and produce fringe patterns of the stress-induced deformations themselves. The method is demonstrated by analysis of a bimaterial plate subjected to a uniform change of temperature. Paper was presented at the 1988 SEM Fall Conference on Experimental Mechanics held in Indianapolis, IN on November 6–8.     

MEASUREMENT OF NON-UNIFORM RESIDUAL STRESSES BY COMBINED MOIRE INTERFEROMETRY AND HOLE-DRILLING METHOD： THEORY, EXPERIMENTAL METHOD AND APPLICATIONS

Hole-drilling method is one of the most convenient methods for engineering residual stress measurement. Combined with moiré interferometry to obtain the relaxed whole-field displacement data, hole-drilling technique can be used to solve non-uniform residual stress problems, both in-depth and in-plane. In this paper, the theory of moiré interferometry and incremental hole-drilling (MIIHD) for non-uniform residual stress measurement is introduced. Three dimensional finite element model is constructed by ABAQUS to obtain the coefficients for the residual stress calculation. An experimental system including real-time measurement, automatic data processing and residual stresses calculation is established. Two applications for non-uniform in-depth residual stress of surface nanocrystalline material and non-uniform in-plane residual stress of friction stir welding are presented. Experimental results show that MIIHD is effective for both non-uniform in-depth and in-plane residual stress measurements. The project supported by the FRAMATOME ANP     

The magic of carrier fringes in moiré interferometry

Practical applications in which carrier fringes are used with moiré interferometry for strain measurements are presented. Examples illustrate how moiré carrier fringes are applied to obtain the desired data in complex laminated composite specimens. In many cases, carrier fringes permit extraction of much more detailed information, with procedures that are easier and more accurate than those using loadinduced fringes alone. The fringe vector for carrier fringes is introduced and its application to the interpretation of fringe patterns is explained. In moiré interferometry, the carrier fringes are produced easily by adjustments of optical elements that control the virtual reference grating.     

The governing equations for moiré interferometry and their identity to equations of geometrical moiré

The equations prescribing the gradient and inclination of fringes in moiré interferometry are derived from the basic laws of diffraction and interference. A vectorial representation of three-dimensional diffraction employs incidence and emergence vectors in the plane of the grating; the representation is especially well suited for this type of analysis. The corresponding equations for geometrical moiré are derived by a remarkably direct vectorial method. The analyses prove that the patterns of moiré interferometry and geometrical moiré are governed by identical relationships. Paper was presented at 1985 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 9–14, 1985.     

Effect of modulus ratio on stress near a discontinuous fiber

The effect of the fiber to matrix modulus of elasticity ratio varying from 1.0 to 200 was investigated for a two-dimensional plane-stress composite configuration having a simulated fiber volume fraction of 0.45 and containing a discontinuous fiber. Uniaxial loading parallel to the fibers was considered. Two independent techniques were used: moiré strain analysis and finite-element analysis. Displacements were measured from four experimental models by utilizing optical fringe-multiplication techniques. The finite-element method yielded stresses which agreed closely with those obtained from the experimental analysis. Matrix stress-concentration factor near the discontinuous fiber was found to increase rapidly with increasing modulus ratio, reaching a value of 20 for a modulus ratio of 200. The finite-element method was shown to be a valuable tool for micromechanical stress analysis of composite materials, and the accuracy of strain analysis by moiré-fringemultiplication techniques was demonstrated for problems containing sever strain gradients.     

Analysis and synthesis of a moire photo-optical system

an analysis based on the modulation-transfer function (MTF) is made of the photo-optical systems used to produce and record moiré fringe patterns. The low-pass filtering capabilities of a real lens and film are demonstrated. The influence of this filtering action on the largest strain that can be measured for a given amount of initial mismatch is discussed. Simple techniques of optical low-pass filtering are presented that increase the quality of moiré patterns that are to be processed automatically.     

A moiré microscope for finite deformation micro-mechanical studies

A new version of a moiré microscope is presented that embodies the theory of optical moiré interferometry. To interrogate the deformed specimen grating, the device uses a transmission diffraction grating that allows for a simple and quick change of the virtual reference grating vector without disturbing the optical alignment of the other components in the optical train. To analyze deformation from the acquired moiré interference fringe patterns, the displacement light-itensity moiré optical law introduced by Sciammarella is revisited. The analysis of deformation is consistent with the continuum principles of finite deformation and can readily be used to obtain micro-mechanical quantities of interest such as the local strains, stretches and rotations.     

Thermal stresses at high temperatures in stainless-steel rings by the moiré method

The research presented in this paper is a continuation of previous work published by one of the authors. Improvements in the techniques of recording and processing moiré data in thermal-strain fields have extended the maximum temperature at which useful information can be obtained from 400 to 1600° F. It has been found that the strains obtained experimentally do not depart very much from the values predicted by the theory of elasticity, although the stresses at some points of the model exceed the yield limit at the corresponding temperature. These results prove that the moiré method can be used with advantage to obtain experimental information in an area where experimental data are scarce if not nonexistent.     

On the limitations of interferometric methods in three-dimensional photoelasticity

The photoelastic response of tinree different epoxy materials was measured. Tensile strips, beams, disks and spheres were loaded and the deformations “frozen in.” Slices from these models were used to calibrate and analyze the absolute retardation of the materials. It was found that the absolute retardation of light vibrating in a given principal direction was directly proportional to the principal strain in that direction and to the deviatoric components of stress in that direction. This result was a consequence of a unique value of the ratio of the absolute-retardation constants and is contrary to published suggestions that absolute-retardation methods could be used to separate normal stresses in the interior of stress-frozen models.     

Sharpening and multiplication of moiré fringes

Moiré fringe sharpening is produced when the widths of clear and opaque bands in a pair of super-imposed screens are complementary. Fringe multiplication is produced when the frequency of lines in the reference screen exceeds the frequency in the specimen screen by an integral factor. The number of moiré fringes then formed corresponds to the greater frequency. The conditions of formation of these moiré patterns are presented and demonstrated. Intensity distribution across projected images of moiré fringes is discussed.     

An electron moiré method for a common SEM

In the electron moiré method, a high-frequency grating is used to measure microscopic deformation, which promises significant potential applications for the method in the microscopic analysis of materials. However, a special beam scanning control device is required to produce a grating and generate a moiré fringe pattern for the scanning electron microscope (SEM). Because only a few SEMs used in the material science studies are equipped with this device, the use of the electron moiré method is limited. In this study, an electron moiré method for a common SEM without the beam control device is presented. A grating based on a multi-scanning concept is fabricated in any observing mode. A real-time moiré pattern can also be generated in the SEM or an optical filtering system. Without the beam control device being a prerequisite, the electron moiré method can be more widely used. The experimental results from three different types of SEMs show that high quality gratings with uniform lines and less pitch error can be fabricated by this method, and moiré patterns can also be correctly generated.The project supported by the National Natural Science Foundation of China (10662005) and JSPS fellowship in Japan. The English text was polished by Keren Wang.     

The moiré method—A review

The paper focuses on the moiré phenomenon as a tool of experimental mechanics. The properties of moiré patterns are outlined. The application of these properties to the measurement of displacements of the points of a surface (intrinsic moiré), contours or deflections (projection moiré) and slopes (reflection moiré) is discussed. Observation methods, recording methods, data-processing techniques are outlined and practical aspects are stressed. Sensitivities and precisions that have been achieved are reviewed. Finally, some typical applications to problems in areas of interest are briefly described.