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Advanced School on Sandwich Structures 2008
A. Torres Marques (Editor)
FEUP, Porto, 2008
FULL FIELD EXPERIMENTAL STRESS/STRAIN analysis OF SANDWICH STRUCTURES
J. M. Dulieu-Barton
School of Engineering Sciences
University of Southampton
Highfield, Southampton, SO17 1BJ
e-mail: email@example.com, web page: http://www.soton.ac.uk/ses/people/staff/BartonJM.html
Key words: Sandwich structures, Experimental mechanics, Full-field stress/strain analysis.
Summary. The aim of this lecture is to introduce students to four basic types of experimental stress/strain analysis namely: Photoelastic Stress Analysis, Electronic Speckle Pattern Interferometry (ESPI), Digital Image Correlation (DIC) and Thermoelastic Stress Analysis (TSA). The lecture will cover the underlying physics of each technique and discusses the challenges of applying each technique to sandwich structures. Examples of where the techniques have been used successfully on sandwich structures are cited in the lecture.
The most familiar form of strain measurement is the electrical resistance strain gauge, where strain data are recorded using sensors that are mounted on the surface of an artefact at spatially defined points. The underlying basis is that the strain in the artefact is transferred into the sensor, which responds to the strain in such a way that a measurement can be taken. The strain reading from a strain is from a single point on the structure averaged across the length of the sensing element. In application to sandwich structure there are two key considerations: (i) is the size of the sensing element small enough to give sufficient spatial resolution to capture the strain and (ii) does the mounting of the gauge reinforce the structure. In many situations, e.g. taking readings from areas of high stress gradient or from foam cores strain gauges are not suitable and alternative means of measurement are required.
The techniques described in this lecture are ‘full’ or ‘whole’ field techniques for strain measurement. In contrast to the strain gauge techniques, full-field strain measurement techniques based on optical measurements are non-contact, and with the use of high magnification optics, can produce data that has high spatial resolution. In this lecture, four types of full-field optical techniques will be discussed: photoelastic stress analysis , electronic speckle pattern interferometry (ESPI) , digital image correlation (DIC)  and thermoelastic stress analysis (TSA) . In the case of the first two techniques the measurement is based on the light path interference, while in the third technique the measurement is based on the intensity of the light and the third is based on thermal emissions from a structure under load. A description of each technique is provided in lectures in non-specialist terms. Then the current status of the use of these techniques in applications to sandwich structures is reviewed. The potential for expanding their application range is discussed in terms of quantitative analysis and monitoring damage.
2 pHOTOELASTIC STRESS ANALYSIS
One of the oldest and most useful forms of interferometric measurements for stress analysis purposes is photoelasticity, which involves the observation of fringe patterns caused by stress-induced birefringence . The technique dates back to 1816 when Brewster made some initial observations regarding birefringence. However it was not until 1940s when polymers became available, namely Bakelite, that the technique was applied in an engineering context. The technique involves manufacturing a model of an artefact from a birefringent material, usually a transparent polymer, and applying a representative load to the model. The instrument that is used for photoelasticity is called a ‘polariscope’. A schematic of a conventional crossed plane polariscope is shown in Figure 1. Essentially the loaded model, or replica, is placed in the polariscope as shown. Polarised light passes through the model and is observed through a further polarising plate known as an analyser. The definition of birefringence means that the refractive index of the material is stress dependent and causes retardation in the light passage, which produces fringes in the model that can be directly related to stress. In a plane polariscope there are two forms of observed fringes: the isoclinic fringe which indicates the direction of the stresses and the isochromatic fringes which are directly related to the loci of points of equal stress difference. To conduct quantitative photoelastic stress analysis it is necessary to introduce two further polarised plates, known as quarter wave plates that eliminate the isoclinic fringe. Manual photoelasticity is extremely time consuming, so over the past 10 years digital photoelasticity, using high quality CCD cameras  has enabled rapid fringe analysis. By recording a minimum of three data sets with the polarised plates at different known rotations it is possible to determine the ‘fringe order’ of each isochromatic fringe in the model. To convert these to stress difference the following equation is used:
where 1 and 2 are the principal stresses, n is the fringe order, f is the material fringe value and b is the width of the model.
The quantity f is an optical material property and can be obtained experimentally by applying a known stress difference to the material. This is then multiplied by n to give the stress difference.
Photoelasticity was developed for homogeneous materials before the advent of FEA and was proven very useful in determining stresses in complex components such as gears, turbine discs etc. A technique known as stress freezing in which the model is taken to above its glass transition temperature and cooled slowly under load is used to ‘lock’ the fringes into the material. This enables three dimensional analysis by slicing the model and observing slices from different orientations. There are numerous examples of applications in the literature; however it is difficult to see how the technique might be applied practically to sandwich structures. To make a model it would be necessary to obtain birefringent material of similar modular ratio of that between the face sheets and the foam core, i.e. of the order of 1000. A successful application of this idea is reported in  where the face sheet was modelled using a two part epoxy resin (Araldite B) and the core material was modelled using a polyurethane rubber (Photoflex by Vishay). The sandwich beam was loaded in three point bending. The object of the experiment was to validate the high-order sandwich panel theory (HSAPT) for a stress field induced by a point load. Isochromatic fringe patterns were obtained in the core and face sheet material for different face sheet thickness and these corresponded almost exactly to the results obtained from the HSAPT both local to the loading point and away from the stress concentrations caused by the point load.
Figure 1 Plane polariscope in transmission mode with CCD camera
A simple modification of the set-up shown in Figure 1, results in an approach known as Reflection Photoelasticity , which requires a polymer coating with a reflective backing to be applied to the surface of a structure. Here the strain the structure is transferred into the coating, the polarised light passes through the coating and is reflected back into the analyser and where fringes can be observed. The polymer coating is usually polycarbonate or flexible polyurethane. This technique has been used in  to obtain the strain in foam cored sandwich structure. Some representative data from  is shown in Figure 2 from a specimen loaded in three point bending. The coloured isochromatic fringes can be clearly observed in the figure. However it should be considered that applying a coating to foam core presents a number of challenges such as reinforcement by the coating and reinforcement by the adhesive filling the cells.
Figure 2 Isocromatic fringe pattern from foam cored sandwich structure in region away from the loading points 
3 electronic speckle pattern inteRferometry
When a surface is illuminated by laser light interference occurs between the source and the reflected light causing a speckle pattern . The size of the speckles result from the nature of the light and the aperture it is passed through. In speckle pattern interferometry, a video or TV camera records two light patterns: one from the unstrained condition and the second from the strained condition. The light patterns are compared and the changes in the speckle pattern provide readings that can be processed into strain. Modern systems utilise data collection via a CCD camera into a computer system. The technique is known as Digital Speckle Pattern Interferometry or Electronic Speckle Pattern Interferometry (ESPI) was first coined to describe analogue techniques used in conjunction with data collected by TV methods but is now commonly used to describe digital approaches.
In Figure 3 a schematic of a simple ESPI set-up for measuring out-of-plane deformation is shown. Essentially there are two components in the system . A laser is used to illuminate the material under investigation and provide the speckle on the surface of the object. The laser also provides a reference beam to the CCD camera via the beam splitter. The other component is the interferometer, which is shown in Figure 3 as a prism and lenses. Data from the strained and unstrained condition are combined arithmetically by subtracting two speckle patterns. This results in a series of closely spaced lines appearing in the image that is displayed on the monitor, which are known as fringes. This representation is sufficient for a qualitative representation of the deformations. In order to process the data into quantitative information, multiple data sets are required for the strained condition; these are obtained using the phase shifting device (see Figure 3) that changes the optical characteristics of the ESPI system. A sequence of a minimum of three positions of the phase shifting device allows the data to be processed into digitised data maps of the out-of-plane displacement field in terms of the phase of the light. These can be analysed directly or processed further to provide strain. To obtain in-plane deformation data it is necessary to illuminate the object from at least two directions (to provide deformation data in one direction). Commercial systems are available that allow both in-plane and out-of-plane deformations to be measured; these contain usually contain four-laser illumination systems.
Figure 3 Electronic speckle pattern interferometry
A major advantage of the ESPI technique over photoelasticity is that the technique can be applied directly to the structure. The speckle pattern is a function of the material surface and therefore no reinforcing coatings are required. A good example of the application of ESPI to sandwich structures is given in  where a curved beam made from PVC foam core and steel face sheets was analysed. The foam core was of high density (H130) to facilitate focusing of the ESPI system digital camera. The system used in this work was able to measure in-plane displacements, however only small fields of view could be observed, so an algorithm was developed to join together the data sets. It should also be noted that in this work a special loading frame was used and that there was no transmitted vibration from a servo based loading machine. Furthermore the displacements in the beam were fairly small, less than 0.1 mm. The results showed good agreement with a high order theory and some typical data is shown in Figure 4. The upper image is raw phase data and the lower image is an ‘unwrapped’ phase image that shows a continuous map of the displacement. To obtain strain from this type of data further processing step is required that differentiated the displacements to give strain. This inevitably introduces further uncertainty in the data and in  the displacements were used to compare with the theoretical model.
Figure 4 ESPI data from curved beam 
ESPI has also been used to analyse the effect of potted inserts in sandwich structures . The specimens were aluminium alloy face sheets and aluminium honeycomb core. In this work the optical system was built in-house with only one laser and provided out-of-plane displacements local to the inserts, which were then used to validate FEA.
A similar technique to ESPI is Electronic Speckle Pattern Shearing Interferometry (ESPSI) . Here a further optical device is included in the system that allows data directly related to strain to be collected. An example of an application of this to potted inserts in sandwich structures is given in .
4 Digital image correllation (PHOTOGRAMMETRY)
Digital Image Correlation (DIC)  is a relatively new full-field, non-contact optical technique for obtaining the strain distribution across the surface of a deformed component. The technique has been used successfully for analysis of heterogeneous engineering materials . DIC tracks the surface displacements of deformed structure by recognition of geometrical changes in grey scale distribution of surface patterns before and after a strain has been applied. A single camera system can provide in-plane displacements; with two cameras a 3-D map of displacements can be obtained. Figure 5 shows a schematic of the set up. All that is required for DIC is an unstrained (reference image) and a strained (deformed image) that can be compared to the reference image . The reference image is divided into ‘cells’, of a given number of pixels from 2 x 2 to 1024 x 1024. In Figure 6a, a 2 x 2 cell configuration with no overlap is shown. It can be seen that the undeformed image has a grey scale pattern within the 2 x 2 cell before deformation. After deformation the grey scale pattern is retained except its spatial position has moved. By recognition of the grey scale pattern from the undeformed to the deformed condition the deformation of the specimen can be obtained. In three-dimensional analysis an extra initial step is requires where the out-of-plane deformation is obtained by calculating the distance of the surface under analysis from a calibration plane (usually in the form of plate with a known and regular geometric pattern inscribed on it); this defines the ‘working volume’ and the position of the two cameras relative to each other. Once this procedure has been completed, three-dimensional displacement information is obtained from which the in-plane strains are derived. The processing of the images can be controlled by selecting optimum combinations of the two main processing parameters; cell size and cell overlap. The choice of cell size selection is a compromise between accuracy and spatial resolution; the larger the cell size, the more data there is to average over. A second factor to be considered in the compromise is cell overlap, as shown in Figure 6b. The strain is calculated from the deformation vectors in each cell, or in the case of overlapping data each sub-cell. This is indicated by the gauge length shown in Figure 6a and 6b; the overlap shortens the gauge length. Therefore the increase in spatial resolution shortens the distance over which the strain is measured. This inevitably will create more scatter as the measurement region decreases. For high resolution work high magnification optics are essential; particularly in applications where displacements are low. In foam cores the displacement is high and therefore DIC lends itself better to these applications than ESPI. In applications to sandwich structures the technique is in it infancy, however applications include application to wind turbine blades (DTU Risoe, Denmark, Imperial College, London), ship structural components (DTU) and to aircraft sandwich panels (University of Southampton). An example of an application in the open literature  is on a double cantilever beam sandwich specimen where DIC was used to monitor the deformation in the areas of a crack between the foam core and the face sheets.
Figure 5 Set-up for 3D digital image correlation
Figure 6 Cell size and overlap in DIC
5 thermoelastic stress Analysis
Thermoelastic Stress Analysis (TSA)  is a well established technique for the evaluation of stresses in engineering components. The general layout for TSA is shown in Figure 7. All that is required is an infra-red detector and a means of applying a cyclic load. The most preparation that is required is a single coating of matt black paint to enhance emissivity and data that is directly related to the stress can be collected in a matter of seconds. Hence real time data from structures experiencing damage can be obtained.
Figure 7 Test set-up for TSA
To date most quantitative studies have concentrated on isotropic materials and the underlying theory has recently been summarised in a review . Essentially an infra-red detector is used to measure the small temperature change associated with the thermoelastic effect and is related to the changes in the sum of the principal stresses on the surface of the material, , as follows:
where is the coefficient of linear thermal expansion, T0 is the absolute temperature, is the density and Cp is the specific heat at constant pressure.
The output from the detector is termed the ‘thermoelastic signal’, S, and is related to the changes in the sum of the principal stresses on the surface of the material using the following expression:
where A is a calibration constant.
Equation (3) is valid for any linear elastic, isotropic, homogeneous material, assuming that the thermoelastic temperature change takes place under isentropic conditions. Isentropic conditions are achieved by the application of the cyclic load.
Techniques for obtaining the calibration constant for isotropic materials are well established; some common approaches have been described and assessed in . These involve either measuring the surface strains and relating them to the stresses or using a calibration test specimen with a known stress field. The thermoelastic response from orthotropic composite laminates differs significantly to that from homogeneous isotropic materials. The stress field associated with a typical fibre/matrix composite is essentially discontinuous and on a micro scale cannot be considered homogeneous. The simple thermoelastic theory devised for an isotropic body is not valid for orthotropic composite materials. For orthotropic materials the following equation is used 
where and are the coefficients of linear thermal expansion in the principal material directions and and are the changes in the direct surface stresses in the principal material directions and A* is a further calibration constant.
The earliest application of TSA to sandwich structures was in 2001  when the stresses in foam cored tee-joints were analysed. The materials used in the construction of the joints were cross-linked closed cell PVC foams for the web and flange cores and continuous fibre reinforced epoxy laminates for the skins. The flange core was made from four layers of 25 mm thick Divinycell H130 sheet of nominal density 130 kg/m3 and the web core was made from one layer of 50 mm thick Divinycell H80 sheet, of nominal density 80 kg/m3. The web skins and flange inner skin were made from a 2-ply 1200 g/m2 quadriaxial 0/90/45 E-glass fabric (QE1200) in an epoxy resin matrix. The flange and web were joined directly using an epoxy adhesive (SPX7049/SPX7310). Fillets were formed at the base of the joint from epoxy mixed with a colloidal silica filler. The overlaminate material was a 900 g/m2 45 glass fabric tape (XE900) in an epoxy matrix. A schematic of the joint with the thermoelastic data overlaid is shown in Figure 8. Each component of the joint was calibrated quantitative stress data derived. This was then used to validate an FE model of the joint and the experimental analysis revealed the inadequacy of the 2-D model in representing the behaviour of the joints.
Figure 10 Sandwich tee-joint with thermoelastic data
TSA has also been used to evaluate the hygrothermal ageing of sandwich structures , where it was shown that it was unnecessary to paint the surface of the PVC foam core material. More recently to assess the affect of core junctions on the stress in the face sheets , which demonstrated the importance of obtaining isentropic conditions.
This lecture has introduced four full field techniques for application to sandwich structures. The advantages and limitations of each technique has been discussed, in terms of ease of set-up, spatial resolution, strain resolution and processing details. Some examples of applications have been provided.
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