S.K. Maiti

Deformation and energy absorption diagrams for cellular solids

Abstract The mechanical properties of 3 types of cellular solids (flexible, plastic and brittle) have been measured as a function of density. The results are compared with models for the stiffness, strength and densification; and constitutive laws are developed. Data and models for each type of cellular solid are combined to develop mechanism-mode maps which summarise the properties in a single diagram; this understanding, in turn, allows the construction of energy absorption diagrams for classes of foams. Natural cellular materials fit the same pattern; maps are presented, as an example, for wood. The maps help in design and in the selection of the optimal foam for a given load-bearing or energy-absorbing application.

Detection of multiple cracks using frequency measurements

Abstract A method for detection of multiple open cracks in a slender Euler–Bernoulli beams is presented based on frequency measurements. The method is based on the approach given by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841], transverse vibration modelling through transfer matrix method and representation of a crack by rotational spring. The beam is virtually divided into a number of segments, which can be decided by the analyst, and each of them is considered to be associated with a damage parameter. The procedure gives a linear relationship explicitly between the changes in natural frequencies of the beam and the damage parameters. These parameters are determined from the knowledge of changes in the natural frequencies. After obtaining them, each is treated in turn to exactly pinpoint the crack location in the segment and determine its size. The forward, or natural frequency determination, problems are examined in the passing. The method is approximate, but it can handle segmented beams, any boundary conditions, intermediate spring or rigid supports, etc. It eliminates the need for any symbolic computation which is envisaged by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841] to obtain mode shapes of the corresponding uncracked beams. The proposed method gives a clear insight into the whole analysis. Case studies (numerical) are presented to demonstrate the method effectiveness for two simultaneous cracks of size 10% and more of section depth. The differences between the actual and predicted crack locations and sizes are less than 10% and 15% respectively. The numbers of segments into which the beam is virtually divided limits the maximum number of cracks that can be handled. The difference in the forward problem is less than 5%.

The fracture and toughness of woods

Crack propagation in various woods has been examined by scanning electron microscopy, and the observations related to measurements of fracture toughness. It is found that the toughness is related in a simple way to the density of the wood, which is explained by a straightforward model. The apparent fracture toughness of wood for cracks that lie normal to the grain is larger, by a factor of about 10, than that for cracks which propagate parallel to the grain. This difference can be explained in terms of the fracture mechanics of very anisotropic solids.

A study of vibration of geometrically segmented beams with and without crack

In this paper, a method of modelling for transverse vibrations of a geometrically segmented slender beam, with and without a crack normal to its axis, has been proposed using the Frobenius technique. There are two segments; one segment is uniform in depth and the other segment has a linearly variable depth. The thickness is uniform along the whole length. In the presence of a crack, the crack section is represented by a rotational spring. Thereby, it is possible to solve both the forward and inverse problems. In the forward problem, the frequencies can be determined by giving the rotational spring stiffness as an input. In the inverse problem, the method can be employed to detect the location and size of a crack by providing the natural frequencies as an input. A number of numerical examples are presented to demonstrate the accuracy of the method. Wherever possible, results have been compared with analytical solutions available in the literature. In the remaining cases, the results are found to be in very good agreement with finite element solutions. In the inverse problems, the error in prediction of crack location is less than 3% and that in size is around 25%.

MODELLING OF VIBRATION OF BEAM IN PRESENCE OF INCLINED EDGE OR INTERNAL CRACK FOR ITS POSSIBLE DETECTION BASED ON FREQUENCY MEASUREMENTS

Abstract In this paper modelling of transverse vibration of a slender beam in the presence of an inclined edge, or internal normal, crack using a rotational spring has been done to enable a possible detection of location of the crack based on the measurement of natural frequencies. The characteristic equation obtained from the vibration analysis of the beam is manipulated to give a relationship between the stiffness of the spring and the location of the crack. Plots of spring stiffness vs crack locations are obtained for the three lowest transverse modes using this relationship. Their common intersection point gives the crack location and the corresponding stiffness. Through a large number of numerical examples it has been shown that in the case of an inclined edge crack the modelling through the rotational spring is feasible for at least small crack angles and the error in the prediction of crack location is less than 4.5%. In the case of the internal crack the performance is satisfactory provided the crack size is more than 20% of the section depth.

Finite element based computation of strain energy release rate by modified crack closure integral

Abstract A modified crack closure integral method with square-root stress singularity elements is given for calculation of strain energy release rate for an in-plane extension of a crack. Case studies are presented to illustrate the improvement in accuracy.

Modelling of transverse vibration of beam of linearly variable depth with edge crack

Abstract In this paper modelling of transverse vibration of a beam of linearly variable depth and constant thickness in the presence of an ‘open’ edge crack normal to its axis has been proposed using the concept of a rotational spring to represent the crack section and the Frobenius method to enable possible detection of location of the crack based on the measurement of natural frequencies. The method can also be used to solve the forward problem. A number of numerical examples are presented involving cantilever beams to show the effectiveness of the method for the inverse problem. The error in the prediction of crack location is less than 2% and size is around 10% for all locations except at the fixed end. Crack sizes 10–50% of section depth have been examined.

Criteria for brittle fracture in biaxial tension

Abstract The criteria of maximum tangential stress, maximum tangential principal stress, maximum tangential strain and strain energy density are applied to the problems of slit and elliptical cracks under remote uniform biaxial tension. The predicted direction of crack extension and the critical load are compared with experimental results reported by other investigators. The unstable crack paths are determined. The four criteria differ in the case of unequal tension; the strain energy density criterion is the least satisfactory. The criteria of maximum tangential strain and strain energy density can be modified to give a good prediction of critical load.

Finite element computation of crack closure integrals and stress intensity factors

Abstract The paper deals with the finite element computation of the crack closure integrals (CCIs) and the stress intensity factors (SIFs) under mechanical and/or thermal loadings, which give rise to forces at nodes close to a crack tip region. Four examples are presented to illustrate the usefulness of the method. In each case the SIFs have been calculated through the CCIs and a direct comparison of displacements. In the first three cases results have been compared with some data available in the literature. The agreement is good. The results based on the CCIs are more accurate than those obtained by the displacement comparison.

EXPERIMENTAL AND FINITE ELEMENT STUDIES ON MODE I AND MIXED MODE (I AND II) STABLE CRACK GROWTH. I, EXPERIMENTAL

Abstract Experimental results on mode I and mixed mode stable crack growth under static loadings through an aluminium alloy (D16AT) are presented. The compact tension type of geometry was employed for both the sets of tests. Data pertaining to load-deflection diagrams, crack opening displacements, crack front geometry, etc., are included. There is a greater spurt of crack growth at the initiation stage in a mixed mode than in mode I. The crack opening angle (COA) remained nearly constant during the whole stable growth. There is a substantial tunneling, the extent of which increases as the extension progresses in both mode I and mixed mode. The tunneling reduces as the ratio a 0 W increases. Because of this tunneling, the COD at a point finite distance behind the crack tip and on the specimen surface is much more than expected. At the maximum load the tunneling is 2 to 3.5 mm in the case of mode I. The crack extends initially almost along a straight line at an angle with the initial crack in a mixed mode. The maximum to initiation load ratio varied in the range 1.50 to 1.75 for the whole range of tests.