NDT.net • April 2004 • Vol. 9 No.04 
BIAXIAL STRESS STATE EVALUATION USING NONDESTRUCTIVE ULTRASONIC TECHNIQUEA. M. AbdelhayAssociate Professors, Production Engineering Department Faculty of Engineering, Helwan University, Helwan, Cairo 11792Egypt. Corresponding Author Contact: Email: Abdelhay1953@Yahoo.Com ABSTRACTThe good mastership of working stresses level in mechanical components and structures is an important factor in engineering industries. Evaluation and monitoring of the stress state of these elements is a time consuming job, beside it involves quite tedious work. Ultrasonic nondestructive techniques offer the unique evaluation capability of establishing macro stresses along a finite path length, as well as gradients near to the surface. The principles of ultrasonic stress measurement of specific applications such as pressure vessels by employing the critically refracted longitudinal ultrasound waves are described. Also, the paper describes ultrasonic monitoring of longitudinal and hoop stress components in thin walled pressure vessel, and the determination of their acoustoelastic coefficients.KEYWORDS: Biaxial stress state, Nondestructive stress evaluation, Ultrasonic technique. Acoustoelastic coefficients, Pressure vessels. 1. INTRODUCTIONVarious nondestructive techniques are available for the measurement of either applied stresses or residual stresses. The nondestructive test method based on Barkhausen emissions [14] from ferromagnetic materials is sensible to changes in microstructural characteristics and the stress state of the material and can be used to evaluate these materials characteristics. Silva and Mansur [4] studied the use of Barkhausen Noise Analysis in detection and measurement of residual stresses in ferromagnetic materials. Their results are compared with the ones obtained from the HoleDrilling StrainGage Method. They found that for low values of the tension and compression stresses applied to special designed ASTM A 515 steel samples, the relation between the rms of the magnetic Barkhausen noise (MBN) value and stress magnitude is approximately linear. In all studied situations, the stress values determined by MBN measurements presented values smaller than those obtained by the HoleDrilling StrainGage Method.On the other hand, eddy current method [5,6] allows to evaluate the state of stress in ferromagnetic material. The given method is used for determining own stress as well as that formed in effect of outside load. The study by Dybiec et al. [5] used the eddy current method to evaluate the state of stress in ferromagnetic material. The given method is used for determining own stress as well as that formed in effect of outside load. The results of stress measurement originating from external loading have shown that within the range of tensile deformations the dependence of indications on the loading is linear, whereas after exceeding the proportionality limit the angle of curve inclination changes. Among the various nondestructive evaluation (NDE) techniques, ultrasonic technique is a versatile tool for investigating the changes in microstructure [7,8], deformation process [9], stress measurements [1014] and mechanical properties of different materials / components [15, 16]. It is due to the fact that, the ultrasonic waves are closely related with the elastic and inelastic properties of the materials. Cost and portability are the two main factors limiting the extensive use of the nondestructive techniques. Ultrasonic instrumentation has an advantage in this aspect because it has the lowest cost among the previously cited methods. The increased utilization of this technique is also due to the availability of different frequency ranges and many modes of vibration of the ultrasonic waves to probe into the macro, micro and submicroic levels. Landa and Plesek [14] presented an excellent study for the evaluation of internal stresses using different ultrasonic techniques. They used compression tests of aluminum prismatic specimen for the evaluation of internal stresses. Resulting differences were less than 10%. In the present paper a new experimental setup is used to study the principal stresses of a pressure vessel under biaxial stress state conditions by using the ultrasonic nondestructive evaluation technique, and also, to determine experimentally the required acoustoelastic coefficients (AEC) used for these stress evaluations. 2. THEORY OF THE WORKThe use of nondestructive methods for detection and measurements of stresses has been increased in the last years, because they do not promote changes in the material under examination and do not interfere in its later use. The ultrasonic methods for detection and evaluation of applied and internal (residual) stresses are based on acoustoelastic effect, basically on stressinduced velocity variations of acoustic waves propagating in a deformed solid. The stress dependence of wave velocities is caused by a finite deformation of a body and, in addition, by a nonlinear stressstrain relationship of the material. From the theory of acoustoelasticity and for an isotropic material, the following linearization equation is cited [14],
To evaluate experimentally working stresses, one needs the value of the acoustoelastic coefficient ( b_{i} ), which can be done through a calibration procedure. These coefficients are determined for a given material, by measuring the propagation speed (or propagation time) of ultrasonic waves versus stress to which the material is submitted. The value of this ratio can be obtained experimentally by calibration. In the present study, one defines an acoustoelastic coefficient b_{i} which represents the value of the curve of the relative speed variation (dV/Vo)versus stress ( s) . Eq. (1) can equivalently be rewritten in terms of times as follows :
t_{0} the ultrasonic flight time in stress free condition. t the ultrasonic flight time in stressed condition Therefore, for a distance ( d ) between the ultrasonic transmitter (T) and the ultrasonic receiver ( R ) is maintained constant (see Fig. 3), measurements are not carried out on the speed itself (Eq.1), but on ultrasonic traveltime ( i.e. time of flight ) Eq. (2); which is directly measurable. 3. EXPERIMENTAL WORKA specially designed and fabricated experimental setup was used for generating a biaxial stress state conditions, and to measure the time of flight for a propagated ultrasound waves. Details of this setup is shown schematically in Fig. 1., and is highlighted as follow.3.1 Test Specimen
3.2 Angle Beam Transducers
Measurements of the time of flight (t_{0} and t) are made using the transmission ultrasonic technique. Two separate and identical angle beam transducers were used as shown in Fig. 3. These contact type ultrasonic transducer are of 2.25 MHz frequency, and 12.5 mm contact diameter. Constant pressures were maintained between transducers and the pressure vessel and to conserve a constant distance (d) between the T and R transducers. White grease was used as a couplant between the different interfaces.
3.2 Biaxial stress and Ultrasound Variable Measurements
E modulus of elasticity of the steel pressure vessel, n Poisson's ratio for the steel pressure vessel.
The analysis of the ultrasonic signal ( Fig. 4) was concentrated on the travel time ( t ) between the initial pulse of the Ttransducer and the received pulse by the Rtransducer for the L_{cr} waves. These flight time values are used to calculated the relative change in velocities for the L_{cr} waves traveled between the fixed locations of the T and R transducers. Accuracy of time reading was 0.1 nS (nano seconds). The calculations were using the following formula:
4. RESULTS AND DISCUSSIONGraphs of Fig. 5 show typical experimental results indicating the relationships between the measured principal stresses and the percent relative change in velocities of the L_{cr} waves under biaxial stress state conditions. The stress  velocity change relationships were represented by linear relationships. Oscillations of experimental results along the ideal straight lines, can be attributed to such factors as microinhomogeneity of the pressure vessel's material, and temperature change resulting from the wave propagation.Using the statistical linear regression, the acoustoelastic coefficient (AEC) for each stressed direction was found. As for the axial direction the AEC was found to be b_{a} =  0.0128 MPa1 with correlation coefficient of 0.95, while for the hoop stresses or the circumferential direction it was b_{h} =  0.0140 MPa1 with correlation coefficient of 0.93.
The AEC value indicates the higher sensitivity of the relative change in ultrasound velocities along the circumferential direction compared to the axial one.
5. CONCLUSIONThe effect of stresses on ultrasonic L_{cr} waves velocity propagating along the two principal stress directions of an element under biaxial stress state was investigated experimentally. The ultrasonic modes considered were the critically refracted longitudinal ultrasonic waves polarized along the axial and circumferential directions of a pressure vessel. The experimental results indicated the existence of acoustoelasticity phenomena. The relative velocity changes of the L_{cr} waves polarized axially and circumferentially were different from each other. The relative changes of the L_{cr} waves velocities were given as functions of the working stresses. The acoustoelastic coefficients (AEC) were obtained from the relationships between the relative changes of the Lcr waves velocities and the working stresses. The absolute values of the AEC of material under biaxial stress states were smaller than those under uniaxial stress state conditions.The above mentioned results indicate that ultrasonic L_{cr} waves exhibit more change in their velocities along the circumferential direction with a relatively greater sensitivity ( b_{h} =  0.0140 MPa1 ) compared to the axial direction. This suggests the possible application of ultrasonic nondestructive technique for the evaluation of the stress state conditions of critically loaded engineering components such as pressure vessels. This kind of study can open the road to the concept of " Ultrasound Strain Gage" idea. Such noncontact type strain gage can avoid the limitations and difficulties of the ordinary electric wire type strain gages. To make this dream possible, systematic research on this area of stress induced ultrasonic velocity changes must be pursed to be reached for an integrated measurement system; which is simple and accurate for stress evaluation nondestructively. ACKNOWLEDGMENTSThe author wishes to express his gratitude to his colleague Prof. Dr. O. M. Dawood for his sincere help and objective discussions. Also, thanks are due to the lab. technician Mr. Farouk Abdelbaki for his assistance in molding the required transducer's perspex wedges used in this work.REFERNCES

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