FIELD OF THE INVENTION
This invention generally relates to a methodology to predict bow and cross bow formation in flat sheets due to residual stress. More particularly, the invention relates to a process of in-situ measurement of residual stress in flat steel sheets for quantitative prediction of bow and cross bow formation during further processing.
BACKGROUND OF THE INVENTION
Steel manufacturers produce flat products for different industrial applications. Residual stress in the flat product is one of the undesired phenomena in subsequent industrial applications, which impairs not only the product usage but may also cause damages to associated tools or other integrated parts. For instance, plasma / laser cutting tools get damaged during cutting at application stage if the sheets possess residual stress, as it leads to distortion (bow/crossbow) of the sheet. It leads to rejection of material and also adversely affects customer-supplier relationship. Hence, the phenomenon of formation of bow/crossbow is considered to be serious concern by the processing industries. Steel manufacturers and the users need a preventive tool to avoid such incidences. Prediction of bow/crossbow of material before causing the flat products to further applications at user end, is the solution for the problem at hand. Mere residual stress measurement does not reduce the prediction errors. Accordingly, the present invention is directed to an in-situ residual stress measurement technique based on ultrasonic technique to predict formation of bow/crossbow. A relationship between residual stress and bow/crossbow formation has been established in this invention. Based on this evolved relationship, an innovative technique can be used for example, a cross bow formation prediction tool and hence it can assist in Go/No, or Go

decision-making while delivering produced flat products to me end-users.
Residual stress in flat products causes shape related defects and it has been a concern for very long time. Various techniques are known in the art to measure residual stresses. Every technique has its own merits and demerits.
The patent US 6424922 B1, claimed a methodology to measure residual stress in a material using critically refracted longitudinal ultrasonic wave. The system uses a frame to hold a hydraulic piston. The piston is used to apply an adjustable force against the probes. A signal is initiated by a first transducer. The signal is angled against the piece under test so as to create a critically refracted wave along the piece. The delay time to receive the wave at a first and second probe is measured. The delay time correlates to a stress in the piece.
The patent US 4755753 A, claims a system and method for nondestructive testing of a part using eddy current impedance measuring techniques. An eddy current probe is positioned above the surface of the part for measuring an induced eddy current signal and generating an electrical signal representative thereof. A first and second movement signal is generated representative of a first and second scan direction of the probe. Relative movement is effected between the probe and the part, whereby the probe scans the part for producing electrical signals varying as a function of eddy current signal. The electrical signals are converted to mutually perpendicular drive signals representative of the eddy current signature at a corresponding location of the part. The first and second movement signals are combined with the mutually perpendicular drive signals of and second composite signal which varies as a function of the movement of the probe and eddy current signal. The first and second composite signals are applied to a display means

for generating a three-dimensional image representative of irregularities in the part.
The patent US4512170A, claimed a process and apparatus for on-line measurement of unflatness of hot or cold rolled strip products, wherein the longitudinal bending of a shape roll over which the strip passes provides a measure of the unflatness of the strip. Actual values of force, displacement or bending moment near the ends of the shape roll may be compared to theoretically predicted values for a flat product and for product having various out-of-flatness conditions. Preferably, the shape roll is supported at its ends by two pairs of supports, and the measurements at the supports are used in determining roll bending and then strip unflatness. Other conditions such as off-center strip and asymmetric operation of the rolling mill may also be determined from the measurement.
All the above mentioned patents are aimed to determine residual stress in the object except patent US45121770A. Apart from these patents there are literatures available which reports various NDT techniques such as magnetic barkhausen, X-Ray diffraction technique and etc. for the measurement of residual stress in steel sheets. But the problem at hand is about predicting distortion (bow/crossbow) during further processing. Though residual stress in the sheets is the cause of bow formation, residual stress measurement alone cannot be a solution. The patent US45121770A claims the prediction of bow in sheets by measuring unflatness of rolls. This patent is aimed directly towards predicting crossbow formation, but it falls short due to two reasons; 1. After rolling sheets tend to experience uneven cooling which is also one of the sources of residual stress in hot rolled sheets. 2. The methodology involves indirect relationship to predict crossbow formation.

In this patent a methodology to predict crossbow formation in sheets using ultrasonic testing technique has been developed. The methodology involves measurement of residual stress distribution across the sheet and development of a relationship between residual stress distribution and crossbow formation. This measurement methodology and the relationship curve can be used as a tool predict crossbow in steel sheets.
OBJECTS OF THE INVENTION
It is therefore an object of the present invention to propose an in-situ NDT tool to predict bow/crossbow in steel sheets.
Another object of the invention is to initially develop an ultrasonic based residual stress measurement technique to measure the residual stress distribution in the steel sheet.
A still another object of the invention is to establish a relationship between the residual stress and crossbow formation.
A further object of the invention is to predict crossbow formation in the steel sheet based on the proposed NDT tool and the established relationship.
BRIEF DESCRIPTION OF THE ACCOMPANYING DRAWINGS
Fig. 1 - Illustrates Lcr wave generation principle
Fig. 2 - Explains the Lcr probe assembly
Fig. 3 - Explains the tensile testing setup
Fig. 4 - Illustrates the relationship between applied stress (MPa) Vs transit
Time (µs)
Fig. 5 - Illustrates the schematic of measuring differential residual stress

Fig. 6 - Explains the schematic of sample tested for residual stress
measurement
Fig. 7 - Shows the measurement of residual stress in actual samples
Fig. 8 - Shows the crossbow measurement using taper gauge
Fig. 9 - Illustrates the relationship between residual stress distribution and
crossbow
DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT OF THE
INVENTION
1. Design of Lcr probe and determining acousto elastic constant
According to the invention, an ultrasonic Lcr technique for residual stress measurement is applied which starts with the development of probe including the acousto-elastic constant for the material. The developed Acousto-elastic constant is not only material specific but also specific to the Lcr probe design (if it is transit time based).
Design of Lcr Probe:
A Ligase chain reaction (LCR) probe is designed to generate refracted longitudinal waves travelling along the surface of the material. Refraction phenomenon of the ultrasound from perspex for example, a transparent plastic to steel is expressed in Eqn.1

Where θp is the incident angle, θs is the refracted angle, Cp and Cs are the longitudinal wave velocity in Perspex and steel respectively.
Since the Lcr wave should travel along the surface as shown in Fig.1, the refracted angle should be 90°. As per Snell’s law, the incident angle is calculated to be 28° which is called as the first critical angle. At this angle, the transmitted longitudinal wave traverses on the surface and subsurface of few millimetres depth. Lcr wave generally traverses on the surface and

subsurface region for certain distance and later it degenerates into bulk waves. Hence the transmitter to receiver distance cannot be maintained at higher value. Transmitter and Receiver are kept apart by X mm using an acrylic block as shown in Fig. 2. Receiver transducer is also provided at same angle to collect the Lcr waves.
Frequency of the probe has been decided based on the thickness of the sample to be tested. Generating the Lcr waves in thin sheets is difficult, as it may generate guided waves. Hence frequency was chosen corresponding to the thickness of the sample to be tested.
2. Establishing Acousto-Elastic Constant:
A tensile specimen of 50 mm width, 10 mm thick and 350 mm long sample of a given steel grade was attached to the Lcr probe as shown in Fig.2. The tensile specimen was mounted in a Universal tensile testing machine along with the probe. Tensile testing setup is shown in Fig.3.
Gradually tensile load was increased and at regular stress level (10 MPa), the ultrasonic signal was analysed for transit time measurement. Then the transit time was plotted against corresponding stress level as shown in Fig. 4. From the slope of the linear relationship between stress and transit time, acousto elastic constant is determined. The developed acousto-elastic constant is specific to that particular grade material and for the probe also.
3. Methodology to measure residual stress distribution:
It is essential to develop the exact relationship between the residual stress and crossbow, as the invention is aimed to develop a residual stress measurement tool, which can predict cross bow formation in flat rolled sheets. Cross bow formation in HR sheets during further processing at

end-users location, happens as a consequence of release of the residual stresses. Then the material redistributes the residual stress across the thickness and width of the sample. Hence the deformation (cross bow) has a direct relationship with stress distribution in the sample. So the residual stress distribution in the material should be correlated to crossbow formation rather than just residual stress alone. A parameter which represents the quality of non-uniformity of the stress distribution is crucial and in this methodology differential residual stress is used. Residual stress distribution is the stress distribution, measured as differential residual stress along the width of the sample as shown in Fig.5, i.e. it is measured as the difference between maximum and minimum residual stress.
4. Experimental Procedure:
Samples were collected from different parts of a coiled steel sheet, i.e. head, middle and tail portion of the coil. As shown in Fig.6, the full width samples of 1 m long were tested for absolute residual stress using ultrasonic probe at three locations along the width (Centre and edges). Differential stress which represents the stress distribution in a sample is calculated as the difference between maximum and minimum residual stress measured along the width. Fig.7 portrays the actual measurement.
The samples were measured for initial bow using taper gauges as shown in Fig. 8. Then, the sample was cut along the line shown in Fig. 7. Then the sample was again measured for cross bow. Final crossbow was observed to be greater as the residual stress in the sample got relieved during cutting. Difference between initial and final cross bow, is measured as the actual cross bow in the sample due to residual stress. The test was carried out for large number of samples.

5. Results and Discussion
Results of the above experiment are plotted as shown in Fig. 9. It shows a clear relationship between cross bow and residual stress. As stated before, as the sample is cut, the residual stress in the sample gets relieved and redistributed, and simultaneously cross bow formation occurs. So after cutting, the residual stress in the sample get reduced / redistributed. For experimental verification of this phenomenon, after cutting, two samples were collected for measuring the residual stress again. Residual stress in the cut samples are observed to be less as given in Table 1. This proves that the established relationship shown in Fig. 9 can be used as an NDT tool to predict formation of cross bow in HR sheets, which allows making a Go/No Go decision while supplying materials to the end users.

Fig.9 shows a relationship between the residual stress and cross bow, although some deviations are also observed. These deviations can be reduced by increasing no. of measurement along the width of the sample (currently three no. of measurements are taken). Hence the above relationship can be refined further and is used as a master graph for crossbow prediction.

We Claim
1. A system for predicting bow or crossbow in flat rolled steel sheets, the
system comprising:
a reference module, the reference module comprising a master graph between differential residual stress and bow/crossbow developed by testing various samples with different residual stress distribution, the differential residual stress being calculated by detecting residual stress at various points by means of an ultrasonic LCR probe along the width/length of flat rolled steel sheets and finding the maximum difference, the measured differential residual stress being plotted against the corresponding bow/crossbow to obtain the master graph; and
an assessment module coupled to the reference module, the assessment module being configured to assess the differential residual stress of a fresh sample, the assessment module takes the input of the measured differential residual stress of the fresh sample and plotting it over the master graph to predict bow/crossbow.