analysis of elastic and elastoplastic strains in a …

ANALYSIS OF ELASTIC AND ELASTOPLASTIC STRAINS IN A NOTCHED PLATE

A. ESIN T. H. LAMBERT Department of Mechanical Engineering, University College London

Department of Mechanical Engineering, University College London

The photoelastic and moirC techniques are used to investigate elastic and elastoplastic distribution of stresses and strains around a 45" V notch with a circular base. It is shown that the plastic flow takes place in the highly stressed region; resulting in considerable increase in the strain ratios while lowering the stress ratios. The stress relaxation accompanying the plastic distortion causes significant changes in the general elastic distribution pattern.

INTRODUCTION THIS PAPER DESCRIBES an experimental investigation of the elastic stress and elastoplastic strain distribution around a 45" V notch with a circular base, in a plate subjected to axial tension.

In engineering design, the effect of stress concentration is a major item, as it is hardly possible to conceive designs without introducing stress raisers. The effects of various types of stress concentrations have been studied by mathematical and experimental methods. These are too numerous to mention, but much of the earlier work has been limited to studies within the elastic limit or to the influence of such discontinuities on the fracture strength of materials (1)-(6)*.

The extension of design techniques into the plastic region has opened the way into the study of the effect of plastic straining on stress concentration factors. One of the earliest workers to carry out such an investigation was Box (7). With the advent of techniques to determine plastic strains experimentally, similar investigations, mostly on central holes, have been carried out by different investigators (8) (12) (13). The general conclusion that can be drawn from their findings is that, as the load increases, the strain concentration factor increases to a very large value while the stress-concentration factor decreases to a value near unity.

The large strains occurring in the vicinity of a stress raiser, once the yield point of the material has been passed, must also affect the strain distribution in those regions which remain wholly elastic and it was found of interest to study both the elastoplastic and elastic distributions after local yielding had occurred.

The stress raiser chosen for the investigation was a 45" V notch with a circular base. Such a notch represents a stress raiser of severity between that of a 45" V notch and a deep notch with a circular base. In making this choice, it was believed that the general trend of results could be used to gain some insight into the two extreme cases as well. The M S . of this paper was first received at the Institution of

Mechanical Engineers on 25th January 1966 and in its revised form, as accepted by the Council for publication, on 31st March 1966. 33

* References are given in Appendix 2.

The photoelastic method was used to evaluate the principal stresses around the notch in the elastic region. For the extension of the investigation into the elastoplastic analysis, the moire technique was adopted; although the photoelastic coating technique also could have been used for such a case of contained plastic deformation.

The investigation was confined to a quadrant of radius approximately five times that of the base circle of the notch. The results are presented in the form of plots of elastic and elastoplastic distribution within the quadrant.

Notation Photoelastic constant. Photoelastic fringe order. Subscripts refer to the

order of measurements (0, normal incidence; 1,2, cw and ccw oblique incidence about p; 3, normal incidence after the model is turned 7r/2 in its plane; 4, 5, cw and ccw oblique incidence about q).

Principal stresses in the plane of the model. Pitch of the moirt grid. Radius of the base circle of the notch. Thickness of the model. x and y displacements in the directions of the

transverse and longitudinal axes of the model respectively.

Principal strains in the plane of the model. Strain ratio; ratio of the principal strain at any

point to the average tensile strain of the uniform section.

Stress ratio; ratio of the principal stress and principal stress difference at any point to the average tensile stress across the uniform section. Subscript denotes p, q or (p-q).

EXPERIMENTAL PROCEDURE The photoelastic specimen was machined from cast Araldite B. The main dimensions are given in Fig. la.

A polar grid system with the centre of the base circle as the pole, was scribed around the notches on both faces of the specimen, Fig. 2a.

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A. ESIN AND T. H. LAMBERT

-I

(All dimensions around the notch area are identical for both specimens.)

Fig. 1. Specimens

a The polar grid system scribed on the photoelastic specimen. b The rectangular grid pattern constructed on the moirC photographs.

Fig. 2. Grid systems

Tensile load was applied to the specimen through a whipple tree arrangement and the stresses were frozen. A quadrant with a radius large enough to contain the grid system was removed from the frozen model. It was sliced into three segments to fit the tilting stage. At every node of the grid, one normal and two oblique incidence (45") readings (No, and N,, N2) were taken; similar readings were taken with the slice turned through 90" in its own plane to obtain the second principal stress. The fractional orders, where necessary, were determined by the Senarmont method of compensation.

The specimen for the elastoplastic analysis was machined from 0.175 in thick, aluminium alloy sheet, to specification HS 30 WP. The essential dimensions around the notch area are the same as that of the first specimen, Fig. lb.

An orthogonal grid pattern of 500 lines/in, aligned with the longitudinal and transverse axes, was photoengraved on the specimen.

From the information obtained by the elastic analysis, the first yield at the base of the notch was calculated to take place at a load of 6.6 ton; photographs were taken starting at this load. The tensile load was applied in small increments up to 12-5 ton, and at various stages of the loading the photographs of the moirC patterns for 0" and 90" (with respect to the transverse axis) setting of the master grid were taken. Figs 3a and 3b show the moire patterns when the applied load was 12 ton.

EXPERIMENTAL RESULTS Since the photoelastic specimen was not observed during loading, the fringe value across the uniform section was roughly calculated as 1.21. The true fringe value was determined as 1.204 by averaging the readings taken at every a in along the width of the uniform section.

In the photoelastic separation of the principal stresses, p and q, the 'oblique incidence' technique (9) (10) was

358 J O U R N A L OF S T R A I N A N A L Y S I S V O L I N O 4 1966




a Master grid normal to axis of load. b Master grid parallel to axis of load.

Fig. 3. Moire' fringe patterns at a load of 12 ton

J O U R N A L O F S T R A I N A N A L Y S t S V O L I N O 4 1966 359




used. To satisfy the conditions of compatibility and to minimize the errors due to imperfect setting, etc., the method suggested by Allison (10) was employed, from which the values of p and q are given by

p = - {~N,+~[~(N~+Nz)--(N,+N,)]) 13t f

Q = 13t { dj[$(Ni 4- Nz) - 4(N4 NE)] -2No)

The variations of the stress ratios (A,, A, and A, - ,) were plotted along the polar radii and the arcs of the grid system. Three such plots are shown in Fig. 4. Using these graphs, the complete distribution of elastic stresses around the notch is given in terms of A, and A, in Figs 5 and 6.

The stress-strain diagram for aluminium alloy used in the test is shown in Fig. 7.

From the evidence of the photographs, it was con- sidered that a sufficient number of moire fringes was present at a load of 12 ton to permit the elastoplastic analysis of the contained plastic deformation.

From the sequence of the photographs, the fringe orders were determined and the highest was assigned an arbitrary fringe value, the remaining fringes being identi- fied in descending order.

For every horizontal and vertical line of the grid system, constructed on the moire fringe photographs (Fig. 26)' the curves of fringe order against co-ordinate distance were plotted (Appendix 1). Graphical differentiation of

GRID SPACING- in

Fig. 4. Variation of stress ratios

I I I I 11.90 I I \

360

Fig. 5. The distribution of elastic 'p' stresses around the notch in terms of the stress ratios A,

J O U R N A L O F S T R A I N A N A L Y S I S VOL I N O 4 1966




Fig. 6. The distribution of elastic '9' stresses around the notch in terms of the stress ratios Aa

Fig. 7 . Stress-strain diagram for aluminium alloy ( H S 30 WP) used for elastoplastic analysis

J O U R N A L OF S T R A I N A N A L Y S I S VOL I NO 4 1966 6

these curves was carried out at every point corresponding to a node of the grid system; thus, for every node, a set of four slopes (directional derivatives) was obtained.

The moire patterns were interpreted on the basis of Eulerian strains. Using the relationships for Eulerian strains for large strains and rotations (XI),

EX = 4 1 - 2 g)+(y+(g)a cy = 1- J1-2 ($)+(;)'+($)'

ex, q, and yXv were calculated. A suitable computer programme was prepared to carry

out the operations shown above and to determine the principal strains from the Mohr strain circle.

The lineal strain in the original direction of a unit line is e = e1l2+eZm2+e3n2

in which e is the strain and I , m, n are the direction cosines. The rotation, 0, is given by

ea + Oa = eI2l2 + eZ2m2+ e 3 ~ n 2 The use of the Mohr strain circle is limited by the value

of 0 but even with plastic strains around the base of the

361




notch it was found that no serious error was introduced when plotting the general trend of the strain distribution. To be consistent with the strain values in the stress-strain diagram, the Eulerian principal strains were converted to Lagrangian strains. Due to the scales used in the curves for the directional derivatives and the magnification of the photographs, the computed values have to be multiplied by a factor of 1.55 to determine the principal strains undergone by the specimen.

The computed principal strain values were plotted along every grid line. From these the loci of particular strain values around the notch were determined.

Three such graphs are shown in Fig. 8. The isotenics for different values of p and q strains are in Figs 9 and 10.

DISCUSSION OF THE RESULTS If a structure is to be efficient from a weight-strength point of view, the average stresses must be near the yield point. On account of stress concentration, the stresses near a discontinuity would be several times greater and conse- quently plastic flow must take place.

Under wholly elastic conditions the geometrical distribution of stress and strain in the region of a notch is independent of applied load. Once the yield point of the material is reached, however, plastic flow occurs in the highly stressed region. In order to satisfy the conditions of equilibrium a change in the elastic distribution must occur.

K J I H G F E D C B A "

Fig. 8. Variation of strains along grid lines

Fig. 9. 'p' isotenics around the notch

362 J O b R N A L O F S T R A I N A N A L Y S I S VOL I N O 4 1966




Fig. 10. ‘q’ isotenics around the notch

Compatibility considerations then require deeper penetration of plastic flow and considerable increase in the strains around the discontinuity. For comparison purposes, using the strain values in Fig. 9, the corresponding stresses have been determined from the stress-strain diagram (7, 13), and the stress and strain ratios computed. These show that plastic strain has a greater effect on the strain ratios than on the stress ratios. Dixon (12), from his experimental curves, deduces that the strain concentration factor does not change appreciably until the impending plastic flow takes place. However, when his results are compared with those of Box (7) and of Durelli and Sciammarella (13), for similar types of stress raiser, the definitions used by Dixon seem to underestimate the strain concentration factor. The results of this investigation are therefore, in the general sense, comparable with those of Box and Durelli and Sciammarella. In the region of low strains, the values of the strain and stress ratios approach one another and, below the yield, are identical.

However, when the stress ratios are compared on the basis of Figs 5 and 9, a more interesting situation is re- vealed. The stress ratios are only lower than their elastic values in a small area confined to the base of the notch. Elsewhere in the plastic region the stress ratios tend to increase (beyond the isotenic (6 x 1.55 x and this causes a general increase in the elastic region. This is accounted for by equilibrium conditions which require extra loading of the elastic area to counterbalance the ‘weaker’ plastic portion.

Apart from the changes in the stress and strain factors, the plastic strain causes a distortion in the distribution of p and q stresses, which must be different for different amounts of plastic flow undergone by the specimen. Although this distortion may not produce results which are important as regards strength, nevertheless it is important in so far as it indicates a deformation of the notch contour which is quite different from the elastic case.

In considering the elastoplastic analysis of strains around a stress raiser, it-is extremely difficult to make a rigorous comparison of the results with those of other investigators. These difficulties arise from the necessity to fulfil the conditions of the geometric similarity, and to have the same properties for the materials used. The general trend of the distorted pattern was compared with and found to be in accordance with the theoretical investigation of Garr, Lee and Wang (14).

CONCLUSIONS The experimental results show that when plastic flow occurs in the highly stressed region, stress relaxation accompanies the plastic deformation which results in a decrease in stress ratios in the vicinity of the notch. To maintain equilibrium the stresses in the elastic portions are increased and the stress ratios increase with respect to the elastic ratios.

During the plastic flow of a notched tension specimen, the direction of the maximum shear stress changes (IS) and the elastic distribution is distorted accordingly.

J O U R N A L O F S T R A I N A N A L Y f i S VOL I N O 4 1 9 6 6 363




ACKNOWLEDGEMENTS The authors wish to express their gratitude to Mr R. Fidler and his colleagues at the Central Electricity Generating Board Research Laboratories, Leatherhead, for their help in the preparation of the specimens for the moirk fringe investigation.

Mr A. Esin wishes to thank the United Nations Educational, Scientific and Cultural Organization and the Central Treaty Organization for sponsoring his research at University College London.

A P P E N D I X 1

E L A S T O P L A S T I C A N A L Y S I S

The moirt method was used to extend the analysis into the second stage.

Moire fringes represent a relative displacement with respect to the unstrained (master) grid. The direction of this displacement is normal to the grid lines of the master. Fringe order indicates the order of the relative displacement which is a multiple of the pitch of the master grid provided that the pitches of the strained and the unstrained grid are equal. When the distance between two adjacent fringes is measured in a certain direction, 0, the ratio of the master grid pitch to this length defines the natural strain in that direction.

,,“=p’ . . . . . . (1) le

Let the directions of the orthogonal grid pattern on the specimen coincide with the x and y axes; the displacement vector of any point, W, will be

Sw = ui+wj . . . . . (2) The displacement components u and e, must be continuous func- tions of x and y to ensure that the displacement be geometrically compatible. When the grid lines of the master grid are parallel to the x axis (or y axis), the fringe pattern represents the e, (or u) displacements as a function of x, y . Fig. 11 illustrates this interpretation geometrically. If w = f (x ,y ) is a function of two independent variables, this equation is the equation of a surface. Such a surface can be defined by the projection of its contour lines on a plane and the moirC fringes are the projections of the contour lines of the

P

displacement surface on the xy plane. The contour lines EJ, FK, G L and H M have the intercepts of 3p‘, 2p’, p‘ and 0 on the OP axis respectively. For a given fixed value of y , yo ; e, = f ( x , yo ) is the equation of the curve D M on the surface, formed by the inter- section of the surface and the plane y = yo .

The slope at any point W is 2v t a u = - . . . . . ox (3)

In the same way, for the curve DH (Fig. 1 l), the slope at any point is

iv b . . . . . . . (4) -

Therefore, the directional derivatives of the v surface at any point give the equations (3) and (4).

When the P axis represents the u displacements

are obtained. From the classical mathematical treatments, these directional derivatives furnish the necessary information for the complete determination of plane strains at any point.

The curves D M h d DH are easily obtained from the moirt patterns. The points J, K, L, M on the curve have the co-ordinates (OJ’, 3p’), (OK, 2p’), (OL’, p’) and (0, OM) in that order. The abscissae are determined from the photographs by careful measurements, and the ordinates are the multiples of the master pitch. A smooth curve drawn through these points gives the equivalent of the curve DM. The complete determination of strains at a point requires four of such curves.

A P P E N D I X 2

REFERENCES

(I) TIMOSHENKO, S . Theory of elasticity 1934, Engineering SOC. Monograph (McGraw-Hill, New York).

(2) GENSAMER, M., SAIBEL, E., RANSOM, J. T. and LOWRIE, R. ‘Report on the fracture of metals’, WeZdingJ. 1947 28 (8).

(3) WHEELON, G. A. and BARRETT, St. J. ‘Effect on notches upon limiting strain of high strength aluminium alloys’, 3. Aero. Sci. 19% 13.

(4) FROCHT, M. M. Photoelusticiry 1941 1, 1948 2 (John Wiley?-New,York).

IN x y PLANE

F*. 11. Geometrical interpretation of moirb fringe patterns

364 JOURNAL OF S T R A I N bNALYSl.5 V O L I NO 4 1966




(5) LING, C.-B.

(6) MAUNSELL, G. F.

(7) BOX, W. A.

(8) LAMBERT, T. H. and SNELL, C.

‘Stresses in a notched strip under tension’, Trans. Am. SOC. mech. Engrs 1947 69. 3. Appl. Mech. 275.

‘Stresses in a notched plate under tension’, Phil. Mag. 1936 21, 765.

‘The effect of plastic strains on stress: concentrations’, Proc. SOC. exp. Stress Analysis 1951 8 (2), 99.

‘Effect of yield on the inter- ference between a pin and a plate’, 3. mech. Engng Sci. 1964 6 (1).

(9) DRUCKER, D. C. ‘Photoelastic separation of principal stresses by oblique incidence’, Trans. Am. SOC. mech. Engrs 1945 63.3. Appl. Mech. 1945 156, 1944 125.

‘A least squares solution of the oblique incidence equations’, 3rd International Meeting for Exp. Stress Analysis, Berlin 1966 (Z. Ver. dt. Ing, Dussel- dorf).

(10) ALLISON, I. M.

(11) PARKS, V. J. and DURELLI, A. J. ‘Various forms of the strain displacement relations applied to experimental stress analysis’, Proc. SOC. exp. Stress Analysis 1964 21 (l) , 37.

‘Elastic-plastic strain distribution in flat bars containing holes or notches’, 3. Mech. Phys. Solids 1962 10.

‘Elastoplastic stress and strain distribution in a finite plate with a circular hole subjected to unidimensional load’, Trans. Am. SOC. mech. Engrs 1963 75E.

‘The pattern of plastic deformation in a deeply notched bar with semi- circular roots’, Trans. Am. SOC. mech. Engrs 1956 78. 3. Appl. Mech. 56.

An introduction to plasticity 1959 (Addison- Wesley, Reading, U.S.A.).

(12) DIXON, J. R.

(13) DURELLI, A. J. and SCIAMMARELLA, C. A.

(14) GARR, L. J., LEE, E. H. and WANG, A. J.

(IS) PRAGER, W.

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analysis of elastic and elastoplastic strains in a …

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