Three dimensional stress analysis of a cracked plate in the presence of residual stress field

Engineering Fracture Mechanics Printed in Great Britain.

Vol. 34,

No. 4. pp. 861-881, 1989

0013-7944/89 $3.00 + 0.00 0 1989 Pergamon Press plc.

THREE DIMENSIONAL STRESS ANALYSIS OF A CRACKED PLATE IN THE PRESENCE OF RESIDUAL STRESS FIELD A. MISHRA,t

M. C. SINGHI and GEETA AGNIHOTRIt

TDepartment of Mechanical Engineering, Indian Institute of Technology, Delhi, New Delhi, India IDepartment

of Mechanical Engineering, The University of Calgary, Alberta, Canada

Abstract-A three dimensional finite element model is developed to study the influence of residual stress field on stress state of a single edge cracked structural steel plate subjected to an applied nominal stress of known magnitude. The nature and magnitude of residual stresses superposed on the applied nominal stress are idealized to present the residual stress state induced by the grinding process. The magnitudes of all the six components of stresses are estimated in the presence of residual stress field near and away from the crack front. The variation of stresses around the crack front with the thickness of the cracked plate for different loading conditions is presented in this paper.

NOMENCLATURE

I: PI I r t, TH % 0, w W

crack length length of the plate determinant of transformation matrix [J] length of the side of an element radial distance from crack tip thickness of the plate components of displacement in x, y and z directions, respectively width of the plate

WI, Wj Wk3W, Wq

x, Y, z l-

A

da,

e

Gaussian weights or weighting factors global Cartesian coordinate system surface area differential strain increment angle measured from crack plane Poisson’s ratio of an isotropic material total potential energy functional applied nominal stress residual stress stress increment volume strain components in global directions local curvilinear coordinate system natural coordinates of nodes stress components in global directions nodal displacement vector total strain vector initial strain vector total stress vector initial stress vector nodal force vector vector of surface traction displacement vector strain-displacement matrix elasticity matrix for an isotropic material Jacobian matrix element stitfness matrix matrix of linear differential operator matrix of shape function 861

862

A.MISHFU efal. INTRODUCTION

ENGINEERINGcomponents which are subjected to severe operating conditions usually have their functional surfaces finished by the grinding process. It has been observed that the heat generated during grinding on the wheel workpiece contact zone produces some defects in the material such as the burn cracks, variation of hardness and the induction of residual stresses on subsurface layers of the workpiece. In the case of grinding tensile residual stresses of high order of magnitude has been reported by Field and Kahles[l, 21. When the residual tensile stresses at the surface are added to the maximum tensile stresses due to service loads, the resultant may be substantially greater than the expected stresses and ‘premature failure’ may occur. Development of cracks over the ground surface leading to untimely failure of the component has been reported by many investigators[3,4]. During the past twenty five years Linear Elastic Fracture Mechanics (LEFM) has emerged as the technology where by the engineer predicts fast fracture in brittle instances and tracks cyclic growth even in home ductile materials. The basis of this technology lies in the identifi~tion of the classical singular stresses and strains that occur at a crack tip in plane elasticity[S-9]. While the approach within its captions is well understood in two dimensions it is less so in three dimensions. Moreover, configurations occur quite often in practice that are inherently three dimensional. For example, the intersection of a crack with a surface especially in the presence of shallow residual stress distribution. It has long been recognised that in plates of finite thickness, the stress field at the vicinity of stress raisers is three dimensional in nature. Works on three dimensional analysis are few in literature[l~l4] owing to the complexity of mathematics and much larger computer storage requirements and obviously longer computer time. Most of the analyses in presence of crack do not include the effect of residual stresses on the stress pattern. For the design of critical components the effect of residual tensile stresses on the stress field of a cracked plate is undesirable because of its detrimental effect. From the review of the literature it appears that no attempt is made to study the influence of residual stresses induced by the grinding process on stress pattern of a cracked plate. In this paper a three dimensional finite element model is proposed to study the effect of residual stress field due to a particular condition of grinding on the stress state of an edge cracked structural steel plate subjected to an applied nominal stress of known magnitude. The nature and magnitude of residual stresses are idealized to represent the residual stress state of a plate finished by the grinding process. Linear elastic crack analysis by FE&f The solution of the crack problem presented here is obtained by applying the finite element variational approach to the problem posed by the equilibrium, compatibility and constitutive equations and conditions of loading. The minimum potential energy variational principle is the basis of the finite element displa~ment approach used here[l5, 16,171. The potential energy functional of displacement field is expressed as

and is a minimum when {u> equals the exact distribution. In this, {e) is the stress vector, (c.>is the strain vector and (T) is the surface traction on the external surface of the body. Sz is the domain which has boundary I”. If the displa~ment inte~olation function of the body is represented as (N) and the set of nodal displacement vector denoted as (61, the assumed displacement at any point within the element is given by @I = ~~H~1.

(2)

j&1= tm%

(3)

The strain vector can be found as

where [B] is the strain-displacement matrix. The stress can be obtained as {e] = mw~~

(4)

A crackedplate in a residualstressfield

863

The discretized form of eq. (1) for the potential energy is given by

The exact solution to this discretized continuum problem is that displacement which minimizes II, so that the solution is consistent with the choice of continuous function [N] and is given by

Equation (6) is known as the stiffness equation in finite element analysis. The stiffness equation (6) may be rewritten as (7) where [Kj is a n x n symmetric positive definite stiffness matrix, (6) is a vector of n unknown displacements and (q f is a vector of n known forces. Thus from eqs (6) and (7) the element stiffness matrix is written as

and the nodal force vector (q) is given as

(4) =

J:V’lTINldr.

(9

CHOICE OF ELEMENTS AND SHAPE FUNCTIONS Solid elements

Solid elements enable the solution of problem in three dimensional theory of elasticity. The basic solid elements are merely three-dimensional generali~tion of the planer elements. Tetrahedron and brick elements arc more popular amongst three dimensional solid elements. The division of a space volume into individual tetrahedra sometimes present difficulties of visualization and could easily lead to errors in nodal numbering. A more convenient subdivision of space is into brick element. The basic brick can be obtained with more complex type shape functions. In the present analysis brick element which is a counterpart of the planer rectangle is considered, It is easy to locate the origin of coordinates at the centroid of the brick element and these coordinates can conveniently be expressed in non-dimensional form with coordinates {, q, c. Due to the enormity of the three dimensional problem, higher accuracy is required which motivates to choose brick element based on quadratic order displacement. Comparing the efficiency of solution, 20-noded brick element is found superior to 8-noded element in most of the cases[l8]. r~~~ararnetric representation of brick element The three dimensional isoparametric form of the brick element (parent and transformed) of quadratic order is shown in Fig. 1. For the quadratic order distorted brick, the eight point used to define a face must lie in a plane. The quadratic element has three nodes along the edge. As in the two dimensional case, it is necessary to evaluate the element stiffness matrices and nodal force vector for this element[l5, 19,201. The appropriate shape functions for the parent element in terms of serendipity coordinates nodes

(, q and l are given by the following relationships: For corner

Ni=~(l-~~)tl+rlo)(l+G>tro+?o+ro-2).

(10)

For typical mid-side nodes at & = 0, qi = f 1, [ = f 1: 4 =

ftl -

<*Xl + tla)(l + (01

(11)

A. MISHRA

864

et

al.

where &I=

tti;

'lo=

Vii;

CO=

where &, vi and ci are the coordinates of transformed For corner nodes
lC

elements.

vi= +lg

ci= +l

and for mid-side nodes ri=O,

?ji=fl,

(i=+l

&=&l,

rji=o,

&=&l,

‘li=+l,

l,=&l &=O.

The shape functions for 20-noded quadratic isoparametric N, =$(l --Ml

-rlW

+0(-C

brick element are given as

-rl+C

-2)

Nz =$(I - V2)(1 - 5)(1 + 0 N j-8(1

-00

+?)(I

+0(-r

+V +r -2)

N4 = a<1 - 52)U + rl)(l + 0 Ns =

f(l + 0U

+ rt)(l + UC5 + ? + C - 2)

N~=+(1-t12)(1+0(l+0 N, = iU+ NB=

- tl)(l + 0(5 - v + C - 2)

W

t(l -

C2)(l - <)(l + 0

Ns = :U - C’)(l-

<>
N,,

=

$U

-

12)(1

-

t)(l +

tt)

N,,

=

a(1

-

12)(1

+

5)(1+

~1

N,,

=

:(I

-

12)(1

+

W

Nn=:(l-63(1

h=$(l

-

~1

-rt)(l--5X-t

--v’lU

-rl

-C

-2)

+tl

-r

-2)

rl -

C -

2)

-t)U--0

N,,=~(1-5)(1+tt)(l-r)(-5+?-r-2)

N,b=$(1--‘)(1+tl)(l--r)

Nl7

=$(l

+w

+vM

N,,

=

:
-

tt’)(l+

N,,

=

$(I

+

W

N20

=

t
-

t2)(1

-Jx+t

W

-

rl)(l-

-

rl)(l-

-

0

Ott

-

0.

(12)

A typical shape function Ni, and its dependence on 4, q and 4’may be denoted as N, = N,K rl, 0. The serendipity coordinates

(13)

within the element are given as

5 = <(X,Y,2) rl

=tf(x,Ysz)

c=5(x,r9z)

(14)

A cracked plate in a residud stress field

865

where x, y and z are the global coordinates. The total derivatives of a typical shape function Nj are given by anr, enjoy ~N~~~ a~~~2 -~--+---_-_ ax ag: ay at az at de

aiv, aN,aY _--aNiax --

Tq-

aN,az -ax atj + aY aq + a2 aq

aNi aN, ax -=-_+--+__, ax ay ac

aNiay aY ay

aNi a2 a2 at:

(15)

Rewriting eq. (15) in matrix form

aNi

_-z Now a 3 x 3 Jacobian matrix arises and with the help of x = C Njxj Y I: z: NjYj z L=z: Nj5 the Jacobian matrix [J3 may be written as

1 (17)

where xj, yj and z, denote the global coordinates ofjth node. The summation in eq. (17) is made over 20 nodes for quadratic order elements. By premultiplying eq. (16) by the inverse of the Jacobian matrix the desired derivatives of shape fiction with respect to global coordinates are obtained:

arv,

aNi

z

ay

aNi

ai%

-5

art

a4

alv,

az

ay

Equation (18) is useful because it may be used to replace derivatives of the form H&Ix, aN,/ay and aN~/az with expressions involving aNi/a~, aN~/a~ and aNi/aC. The displacement components (a, a, W) within the element can be expressed in terms of shape function as Zi= jE, N;t<, rl, Ckj

(19)

866

A. MISHRA

et al.

The strain components within the element are derived from eq. (18) and are written as

(20) For a particular three-dimensional using eq. (18) as

elastic problem the matrix[B] for any ith node can be set up

alv,

O Odr PI =

alv,

alv,

o

alv,

alv,

o

aNi

ayax 0 aNi

-z

(21)

azay

ax

Numerical integration of brick element

The use of numerical integration technique makes the isoparametric element of practical utility. The element volume dx dy dz of a brick element may be written in terms of d< dq d[ as dxdydz=IJ]dTdqd[

(22)

where ]J] is the determinant of the transformation matrix [J]. Using eq. (22) the element stiffness matrix [K] for isoparametric brick element int he serendipity domain is written in the form as

WI = _+,I_+*I+’Pl’PltBlIJI -1 sss

dt dtl K

(23)

If the integrand in eq. (23) is denoted as (24)

then (25)

A cracked plate in a residual stress field

x PARENT

lRANSFORME0

861

ELEMENT

ELEMENT

Fig. 1. 3D quadratic isoparametric brick elements in local Cartesian and natural coordinate system.

In eq. (25) a scalar integrand is implied because the integral of a matrix is simply the matrix of integral. The most obvious way of obtaining the integral in the serendipity domain for the isoparametric brick element is by Gauss-Legendre quadrature. Gauss quadrature is a numerical integration method that allows the sampling points to be chosen such that the best possible accuracy may be obtained. By applying Gauss-Legendre quadrature to eq. (25) for m, n and p sampling points in <, v and c direction the stiffness matrix [K] can be integrated and is written as [K] = i

i

k=lj=li=l

2

wiwjwk(ti,

Vj,

Ck)

(26)

where wi, wj and wk are referred as weights or weighting factors. For no loss of convergence, 2 x 2 x 2 order of Gauss point integration scheme is adequate for brick element[21]. Therefore a 20-noded brick element with 2 x 2 x 2 order of integration is used in the development of three dimensional finite element program. Singularity in brick element Barsoum[22,23] obtained the singularity in three dimensional prism with four mid-side nodes at the quarter-points by degenerating the cube with one face (e = - 1) collapsed as shown in Fig. 2. He has shown that l/& singularity is embodied in such a collapsed quarter-point quadratic isoparametric brick element because of the relationship r=f(l+#

(27)

ii (a)

PARENT

ELEMENT

(b)

TRANSFORMED

Fig. 2. Collapsed quarter point 3D singularity brick element. EFM 34/4-F

ELEMENT

868

A. MISHRA et al.

where I is the element length along the crack face and r is the radial coordinate defined in Fig. 2. In the present analysis ZO-noded isoparametric brick element is used as the singula~ty element for three dimensional crack analysis. This is achieved by collapsing the brick face with nodes 1, 9, 13, 2, 14, 3, 10 and 15 to crack front line and placing node points 4, 8, 16 and 20 at l/4 from the crack front. Where I is the length of the side of the element joining the crack line as shown in Fig. 2. The CRACK 3D program was modified to incorporate the collapsed element from the brick element. This was possible without any major change in the program which was developed for the purpose. PROBLEM

DESCR~~ON

AND THEORETICAL

CONSIDERATIONS

A rectangular plate with a crack through half of its mid plane is loaded with uniform tractions normal to the crack face applied at opposite ends of the plate. This is sometimes called the single edge notch test specimenf24]. The crack is assumed to be mathemati~lly sharp with a straight crack front in a continue across which mechanical interaction is impossible. The interaction of two crack surfaces is termed as ‘crack front’ for the three dimensional case. Figure 3 shows the dimensions of the plate, the loading conditions and the coordinate system with its origin at the crack front. The material is assumed to be a linear elastic, homogeneous, isotropic solid with Poisson’s ratio 0.3 and Young’s modulus 2.1 x 10’ MN/m’. The effect of body forces is ignored. The loading conditions correspond to ‘mode I’ deformation. Structural steel rectangular plate of finite thickness is selected for the analysis. Plate is assumed to be finished by grinding process which induces the high magnitude of residual stresses[2] and is given by a function a, =

(CT0 &l8X 1 -f(crO)_

when 0 Q t < TH/2 when TH/2 < t < TH,

The nature of residual stresses (a@) is tensile at subsurface of the ground plate followed by compressive stresses. The residual stress pattern in the idealized form is superposed on the applied nominal tensile stress to study the stress-state around the crack front, Subroutine RESLOD is developed which calculates nodal load due to superposition of the idealized residual stress pattern assumed to be pre-existing in the plate due to grinding process. The boundary conditions that are applied to the finite-element mesh are shown in Fig. 4. A uniform traction of 10 MN/m* acting in the positive z-direction is applied to the upper surface of the mesh since this is the load applied to the cracked plate. By taking advantage of the symmetry of the problem, only one-half of the plate is considered. The symmet~ of the problem requires that the nodes on the right hand side of the surface (i.e. on the z = 0 plane with x 2 0) are free to move in x and y directions but are fixed in the z-direction. THREE-DIMENSIONAL

CRACK ANALYSIS IN PRESENCE STRESS FIELD BY FEM

OF RESIDUAL

A master program is developed in various stages and is tested each time with the test model before applying to the actual problem. First a subprogram is developed to set up element {B] matrix and element stiffness matrix is computed by 2 x 2 x 2 Gaussian integration scheme over elemental volume. The computed stiffness matrices are stored in the master stiffness matrix. The nodal displacements are computed by giving nodal forces at the bounda~ nodes in a separate subprogram. After computing nodal displa~ments, stresses are computed by another sub-program,

A rectangular of plate of 140 mm x 16 mm x 4.18 mm size is selected for the analysis by applying the uniformly distributed load in z-direction. The plate has been discretized into six isoparametric brick elements. A three dimensional finite element program CRACK 3D (RESLOD) is developed for 20-noded isoparametric brick element with 2 x 2 x 2 integration scheme. Boundary constraints are provided ‘by fixing the nodes in z-direction. The program was tested for a uniform

869

A cracked plate in a residual stress field UNIFQRM

TRACTION TH

10 MN /m2

z-

OF

lOMN/mZ

IN

DIRECTI~

A!!? I I I

I I I I

.GOmm

CRACK

FACE

mm

I I I

l8mm CRACK

Fig. 3 Single edge cracked plate.

k x1 2

:

FACE

-

0 lSmm-

.. / A

-NODES 2 -

FIXEO

IN THE

DIRECTION

Fig. 4. The boundary conditions that were applied to the finite element mesh.

applied tensile stress of 10 MN/m2 at the boundary of a plate without crack. The magnitudes of stress components o;, , ayu, rxy, zp and r, are found to be 10V6MN/m’ approximately except for the value of 0, which is 10 or 9.99998 MN/m2 at all the integrating points in all the six elements of the plate. This was expected since the applied nominal stresses in z-direction at the boundary is 10 MN/m2. Test problem with crack A through-the-thickness single edge cracked plate of size 140 mm x 16 mm x 4.18 mm having crack length to width ratio equal to 0.5 was selected as a test problem for the analysis of stress state around the crack front. The mesh advents for three dimensional stress analysis using singularity elements [25] at crack front and isoparametric brick elements away from the crack front have been shown in Fig. 5. The mesh consists of 40 brick elements and 273 nodes. An uniform tensile stress of 10 MN/m2 acting in the positive z-direction is applied to the upper surface of the mesh. To prevent a rigid body motion of the mesh, point “0’ on the crack front is chosen to be the origin (X = Y = 2 = 0) of the coordinate system as shown in Fig. 5. The numerical values of all the six components of stresses have been obtained by the computer program developed. Numerical results of stress components in presence of crack show the deviation in results from that without crack. At the boundary in z-direction a, is found to be 10 or 9.99998 MN/m2 which is approximately equal to the applied nominal stress of 10 MN/m’ in z-direction. This proves that the Finite Element Program which is developed for this problem and is used for the computation of ~r~d~ensional stress field in the case of linear elastic crack analysis is accurate. The sequence of compu~tion followed by this program is shown in Fig. 6. Stress state

ofa single edge cracked plate in the presence ofresidual stress field

The mesh arrangement around the crack front shown in Fig. 7 is used for the computation of stresses. The structural steel plate considered for the analysis is assumed to be finished by the grinding process. It is brought to ambient condition before applying nominal stresses. The residual stresses due to grinding is investigated analytically and experimentally by many

870

A. MISHRA et al.

TOTAL NO OF ELEMENTS.40 TOTAL

NO OF NODES

NO OF NOOES ELEMEN7 :20

:273

PER

TOTAL DEGREE OF FREEDOM 5 619

h i 140 mm wtl6

mm

t :4.16rnrn

Fig. 5. 3D finite element mesh of cracked plate showing crack front elements, boundary conditions and restraints.

researchers[2,26-291 in the past. They have reported high magnitudes of tensile residual stresses at the subcutaneous layers of ground surface followed by compressive stresses. For the effective method of treatment (i.e. FEM) to be convenient to use for three dimensional crack problem it is suitable to idealize the residual stress field into elementary case. Therefore superposed residual stress field is idealized, as shown in Fig. 8 to make the proposed three dimensional finite element model tractable. The idealized residual stress field is superposed over the nominal tensile stress field of 10 MN/m* on the boundary of the plate. The boundary conditions as discussed earlier are also applied in presence of residual stress field. The computer program CRACK 3D is modified to incorporate the effect of residual stress field. A subroutine RESLOD is developed for this purpose and is included in the main program. The flow diagram of the complete three-dimensional finite element program CRACK 3D (RESLOD) for evaluation of stress state of cracked continuum in presence of residual stress field is shown in Fig. 6. This program is used to compute the stress state near and away from the crack front. All the six components of stresses are computed for the applied nominal stresses of 10 and 20 MN/m2 in presence of idealized residual stress field which is kept constant.

A cracked plate in a residual stress field

ARE

ALL

ELEMENtS

,

871

OVER

YES

STflR t FOR WA0

ARE

ALL

ELEMENTS

OVER

+VES BAKWAD

Fig. 6. Flow diagram for crack 3D (RESLOD).

RESULTS

AND DISCUSSIONS

Residual stresses due to grinding process under the following conditions have been considered for assuming the maximum magnitude of idealized residual stress: wheel grade A 46 K 8 V, down feed 0.05 mm/pass and wheel speed range 914.4-1066.8 m/mm. It should be mentioned here that residual stress intensities and distributions are closely related to other material parameters. Also it is known that on changing the grinding conditions such as speed, feed and depth of cut, varied magnitudes and nature of residual stress pattern can be obtained. However, in the present analysis the residual stresses induced due to grinding process under the above mentions ~nditions have been considered for deciding the maximum magnitude of tensile residual stress (a&,,, of the assumed idealized residual stress field. Stress state of elements 1 and 5 in presence of residual stress for stepped loading Figure 9 shows the elastic stress distribution in the vicinity of the crack front at a distance of 0.44 mm from the crack front throu~-the-~ckness of the plate in presence of residual stress field induced due to grinding process. The analysis presents the magnitude and nature of all the six components of stresses, i.e. o,, dyY,a,, , zxy, zuz, z,, for the applied nominal tensile stress (a) of 10 MN/m*. The maximum magnitude of residual stress (a,,)- which has been superposed on the applied nominal stress as shown in Fig. 8 is 20 MN/m*. The results pertain to the stress state of the elements 1 and 5 along nodes 1,2,3,4 and 5 of the crack front. The magnitude of stress shows that a,, has a rna~rn~ value of 156.45 MN/m2 at a thickness of 1.6 mm. Q,,,,and a, have ~ximum values of 66 MN/m* and 58 MN/m* at a thickness of 1.6 mm and 2.5 mm, respectively. These maximum values are in the region of mid thickness of the elements 1 and 5. However u, has shown the maximum tensile value of 168 MN/m* at a thickness of 3.7 mm whereas at this thickness a,, and bYvare compressive. In the mid-thickness all the components of shearing stresses are negative.

872

A. MISHRA et al.

Fig. 7. Mesh arrangement around crack front for 3D crack analysis.

At 2/3 depth of the specimen all stress components are positive in nature except 2, which is 50 MN/m2 negative. Figure 10 presents the stress distribution at a distance of 1.96 mm from the crack front which is away from the crack front. The stress field in comparison to the first case, i.e. near the crack front has changed drastically in magnitude as well as in nature. The maximum value of a,, is 50.8 MN/m2 and a,, and auv have become compressive. The nature and magnitude of stresses have changed because away from the crack front the effect of stress raiser has considerably reduced.

I ITHICKNESS

(mm) L

Fig. 8. Idealized model of the residual stress field due to grinding process.

A crackedplate in a residual stress field APPLIED

NOMINAL

STRESS

Q

(qdmox

873

* 10 MN /ItI2 a 20MN/m*

41

TH..4.lO,a/w.O.S,h/w.5.75 LOELEMENTS ELEMENT

5 273

NOS.

INTEGRATION

300

0

0Xx

0

3Y

0

Qzr

. A

TYZ

X

rzx

NOOES

1,s PTS.

3,1

‘xv

1 6 ANALYSIS-OF ELEMENTS 1 AND 5

200 F

Fig. 9. Stress distribution at a distance of 0.44 mm from the crack front with residual stresses.

Results obtained in the absence of residual stress field under the similar conditions of loading and specimen geometry are shown in Figs 11 and 12. Figure 11 depicts the stress state near the crack front. Comparing Figs 9 and 11 it is seen that the presence of residual stress has significantly changed the stress situation in the continuum considered. All the components of stresses have changed considerably. For example, a,, which is in the direction of applied load perpendicular to the crack front has the magnitude of 80 MN/m2 at a distance of 3.74 mm without residual stress whereas 0, at the same position in presence of residual stress field shows stress value of 168 MN/m2. Other values of stress components have also shown changes in magnitude and nature. Considering APPLIED

VI.1 4.18,

NOMINAL

a/w

STRESS

I 0.5,

40 ELEMEMTS ELEMENT NOS

r

: 10 MN / m2 i 20MN/m2

l%’

mox

h/w

: 5.75

6 27 3 NOOES 1,s INTEGRATION

PTS.7,)

OF ELEMENTS AN0 5

z

- 20

THICKNESS

lmm 1 -

: ‘, - 40 t

Fig. 10. Stress distribution at a distance of I .96 mm from the crack front with residual stresses.

874

A. MISHRA et al. APPLIED

NOMINAL

TH*L.l(lr

a,b

40

ELEMENS

ELEMENT

STRESS

V

h/w

z 8.75

I 0.5, b 273

110 MN/m2

NODES

NOS. 1,s

INTEGRATION

PTS.

7,s

60 t

I-

ANALYSIS

6 OF ELEMENS 1 AND S

-

20

>

I8

z is

0

c THICKNESS

(mm

I4

Fig. 11. Stress distribution at a distance of I .96 mm from the crack front without residual stresses.

the nature of residual stresses due to grinding process which consists of high magnitude of tensile stresses at sub-surfaces followed by compressive stresses, the results obtained have shown considerable infiuence of residual stresses on stress pattern around the crack front. Figure 12 shows the stress pattern away from the crack front in the absence of residual stress field. The maximum value of 6, in the mid-thickness is 31.5 MN/m2. In Fig. 10 6, at the mid-thickness is APPLIED TH =4.18,

NOMINAL a,b,

40 ELEMENTS ELEMENT INTEGRATtON

0

%x

6

3V

STRESS

= 0.5, b

NOS.

h/w 213

0

I 10 MN /m2

: 8.75

NODES

1,s

PTS .3,1

0

ANALYSIS

OF ELEMEN TS 5

I AND

-200

t

Fig. 12. Stress distribution at a distance of 0.44 mm from the crack front without residual stresses.

A cracked plate in a residuai stress field

875

31 MN/m2. The mid-point in Fig. 10 presents the case where residual stresses have little effect because this is the transition zone and at this thickness the tensile residual stresses change to compressive residual stresses. It may be concluded from Figs 9, 10, 11 and 12 that the value of a, near the crack front and away from the crack front at the mid zone of the specimen show no effect of residual stresses due to the transition nature of residual stress pattern. All the components of stresses have shown considerable variation near and away from the crack front through the thickness of the plate. Figures 13 and 14 depict the stress dist~b~tion around the crack front for applied nominal stress of 20 MN/m* and maximum residual stress (G&_,~also of 20 MN/m2. The objective was to know the stress state under the increased loading condition. Some changes in the stress pattern are observed for the stress state of c~,,. The magnitudes of stress components 6, and oY,have increased due to higher value of applied nominal stress but the nature of these curves is the same as in Fig. 9. tx., T,,~and z, have also changed in magnitude but no appreciable change in nature has been observed. Results have also been obtained for the stress state at a distance of 1.96 mm from the crack front. This represents the stress distribution away from the crack front. Comparing Fig. 10 and Fig. 14 it may be concluded that the nature of stress state does not change away from the crack front. Only the magnitudes have changed due to higher applied load. Stress state of elements 2 and 6 in presence of residual stress for stepped loading

In Figs 15 and 16 the stress state of an edge cracked plate subjected to an applied nominal stress of 10 MN/m2 in presence of residual stress field [(a@),, = 20 MN~rn~ for the elements 2 and 6 is presented. The stress values are at a distance of 0.44 mm and 1.96 mm as in the previous case. The stress values have shown changes in magnitude and nature in comparison with the stress state of elements 1 and 5. The results in higher applied nominal stress values of 20 MN/m2 have also been presented in Figs 17 and 18 for the residual stress (Q~),,,~equal to 20 MN/m’. Appreciable variations in stress state have been observed. From these figures it is observed that the stress state of elements 1 and 5 is appreciably different in magnitude and nature than the stress state of elements 2 and 6. This is because of the positional change of these elements in the continuum. A$PLtED

STRES$U z 20MN/m2

NOMINAL

(%hX = 0.53

tH.r-C&i,o/w

6 273

40 ELEMENTS ELEMENT

NOS.

tNlEGRAflON 0

@xx

0.

UYY

cl

Qz

h&

: 2a~~~m2

: 6.75

HOOF.5

I,5

PTS.

3,i

CRACK FRONT ’

ANALYSIS

OF ELEMENT

1 AND

THICKNESS

(mm)

5

-

Fig. 13. Stress dist~~ution at a distance of 0.44 mm from the crack front with residual stresses.

876

A. MISHRA et al. APPLIED

NOMINAL

TH. z 4*16,0/,.,

:0.5,

40 ELEMENTS ELEMENT

h/,

41

c * 20 MN/m2

: 6.75

6 273 NODES

NOS.

litlEGRAllON 0 =xX 0

STRESS

1,5

PTS.

7,5

=w

ANALYSIS 100

I

OF ELEMENT 1 AND 5

50

“E I

0

ii ii! t

- 50

-100

THICKNESS

L

Fig. 14. Stress distribution at a distance of 1.96 mm from the crack front with residual stresses.

Stress state of elements 3 and 7 in presence of residual stress for stepped loading

Figures 19 and 20 show the stress distribution for elements 3 and 7 at a distance of 0.44 mm and 1.96 mm. The applied nominal stress at the boundary is 10 MN/m2 and the residual stress is equal to 20 MN/m’. Stresses shown for elements 3 and 7 pertain to integrating points 3 (~oolmax and 1 of these elements. Maximum tensile stress 6, is 75 MN/m2 in the mid thickness of the plate APPLIED

NOMINAL

STRESS

0

ldm_ lH.:4.16,

~/VI:

0.5,

4OELEMENTS ELEMENT

NOS-

INTEGRATION 0

pxx

0

=vv

0

ez

.

TX,

h/w

6 273

I 6.75

NODES

216

PTS.

3,t

150 2 ANQg,

* s I ;I

-50

THICKNESS

(mm

I-

Fig. 15. Stress distribution at a distance of 0.44 mm from the crack front with residual stresses.

811

A cracked plate in a residual stress field

APPLIED

NOMINAL STRESS

Q : 10 MN/m2

(o-&,,,~: lN.aW6,0h,,

x 0.5,

4OEl.EMENlS ELEMENT

hlw

20MNlm2

:6.75

6 273 NOOES NOS.

INTEGRATION

CRACK f RON1

2,6

PlS.

7.5

orx

0 0

3v @iz

0 .

rw

I

4.0 4.16 I ,

4.n

0

THICKNESS (m WI1

-

-40

-60

Fig. 16. Stress distribution at a distance of 1.96 mm from the crack front with residual stresses.

and at a distance of 3.74 mm it shows the value of 100 MN/m’. At this thickness O, is also equal to 100 MN/m2. The value of aY,,is only 12 MN/m2. Howe&, at this thickness 2, shows negative stress value of 125 MN/m2. Due to compressive residual stresses other stress components such as APPLIED

NOMINAL

STRESS (5

TH. i 4.18,

a/w

40 ELEMENTS ELEMENT

= 20 MN/m2 : 20 MN/m2

i 0.5, h/w i 6.15 6

273 NODES

NOS. 2,6

INTEGRATION

x

d

Jnox

PTS’

3,l

rzx

CRACK, FRONT

450

r “‘k-=-\

ANAWS21iA0$

E6LEMENTS

150

t .16 -

0

“E f ;;-lS0 w”

THICKNESS

( mm 1 -

E In -300

Fig. 17. Stress distribution at a distance of 0.44 mm from the crack front with residual stresses.

A. MISHRA et al.

878 APPLtEO

NOMINAL

STRESS

D

6 TH. s 4.18,

ah

4OELEMENTS ELEMENT

0

%x =YY

0

%2t

. A

‘;Y

h/w

273

NOS.

INtEGRAlION

d

a 0.5, L

I 11.75

NODES t

2r6

PTE

* 29 MN /m2

‘mox ii 20 MN/m2

7~5

TV ALYStS OF ELEMENTS 2 AN0 6

too

t

50

T

.18

P 2 iti t

0

-50 THICKNESS

(fmf

-

Fig. 18. Stress distribution at a distance of 1.96 mm from the crack front with residual stresses. i7 xx, Qyy, fJzz7z,, and xx,,are negative in nature at the thicknesses of 2.53 mm. Only zuzhas shown positive stress of 12 MN/m?. Figure 20 presents the stress state away from the check front. The value of CJ, is considerably reduced and the maximum magnitude is 32 MN/m2 tensile. At this depth the shearing component rzx is 11 MN/m’ negative. These two stress components cr,, and z, are tensile and negative, respectively, throu~out the thickness. Other stress ~m~nents such as a,, crvv,zxu and ~~~have

APPLIED

NOMINAL

STRESS

0

~bc’nmx 1H.a 4.18,

i+,v

40 ELEMENTS ELEMEM

NOS.

INTEGRATION

15

0

@xx

0 0

Qvy @zZr

l

rxy

A

ryt

Y

rzx

s 0.5,

h fw

L 273

NODES

3.7

PTS.

3,l

: 10 MN/m2 .20MN/m2

* 8.75

CRACK FR)Nl

ANALYSIS

OF ELEMENTS PAN0 7

THICKNESS(mml-

Fig. 19. Stress distribution at a distance of 0.44 mm from the crack front with residual stresses.

A cracked plate in a residual stress field APPLIED

NOMINAL

STRESS

d

879

a 10 MN/m’

(%I TH. .4.lG,a/w

* 0.5,

hb

40 ELEMENTS

L 273

NODES

ELEMENT

NOS. 3,7

INTEGRATION 0 A

3Y

0

ez

.

PTS.

7,5

%

TV

ANALYSIS OF ELEMENT 3AND7

4Gl-

E

-20

I-

THICKNESS

(ItIm)

-

Fig. 20. Stress distribution at a distance of 1.96 mm from the crack front with residual stresses.

shown variations in nature and magnitude throughout the thickness in comparison with the results presented in the Fig. 19. To study the stress behaviour near and away from the crack front the magnitude of applied nominal stress has been increased to 20 MN/mZ. The same idealized residual stress field with maximum magnitude of tensile residual stress (a o) maxequal to 20 MN/m2 has been superposed as in the previous case. The stress values for the elements 3 and 7 have been considered for the integration points 3 and 1. Figures 21 and 22 show the nature and magnitude of all the stress components at a distance of 0.44 mm and 1.96 mm from the crack front. Not a single integrating point for the elements 3 and 7 have gone beyond yield at this load. The maximum tensile stress cr, of 162 MN/m2 has been obtained at a thickness of 1.65 mm. o,, has shown maximum compressive stresses of 162 MN/m2 at a thickness of 2.53 mm. by,,, o,,, t,,, and t, have shown negative stresses of lower magnitudes at this thickness. These results are comparable in nature as obtained for applied nominal stress of 10 MN/m2 which are shown in Fig. 19. The higher applied nominal stress in this case gives higher magnitudes of stress components. Figure 22 describes the stress state away from the crack front and has shown remarkable change in the nature and magnitude of a,, which is 145 MN/m2 at the thickness of 2.5 mm. At the thickness of 3.74 mm a,, gives the value of 67 MN/m2 compressive. These two values of a,, are showing reverse trend in comparison to the nature of a,. near the crack front. This shows that away from the crack front stress values have less effect of the stress raiser than near the crack front.

CONCLUSIONS From this study it is concluded that it is possible to calculate all the six components of stresses around the crack front through-the-thickness of an edge cracked plate in the presence of residual stress field. Results of the analysis have shown that the idealized residual stresses due to grinding process which consists of tensile and compressive stresses through-the-thickness of the ground plate have influenced the stress distribution in magnitude and nature around the crack front. The influence of residual stresses is pronounced in the region near the crack front. It has also influenced the stress values away from the crack front in magnitude as well as in nature throughout-the-

880

A. MISHRA et al. APPLIED

NOMINAL

lH.a4.18,

O/W I 0.5,

40 ELEMENTS ELEMENTS

oxr

b

CYY

0 .

*z

A Y

h/f,,

L 273 NOS.

INTEGRATION 0

STRESS

o- : 20 MN/m2

a (1.75

NODES

3.7

PTS.

3,l

TV rYz rzx

300

: u

-150

i THICKNESS

(mm)

-

-300

Fig. 21. Stress distribution at a distance of 0.44 mm from the crack front with residual stresses. APPLIED

NOMINAL

STRESS

0

: 20 MN/m2

(~1,~: 1H.z

4.18,

a,b

:

40 ELEMENTS ELEMENT

h/w

b 273

NOS.

INTEGRATION 0

0.5,

:

20MN/m2 6.75

NODES

3,7

PTS.

7,s

=xX

ANALYSIS

OF ELEMENTS 3 AND

THICKNESS

7

(mm)-

Fig. 22. Stress distribution at a distance of 1.96 mm from the crack front with residual stresses.

thickness of the plate. The increased magnitude of applied nominal stresses have also influenced the stress distributions near and away from the crack front. REFERENCES [l] Michael Field, Surface integrity in conventional and non-conventional machining. Seminar on advancements machine tools and production trends, The Pennsylvania State University, Vol. 10 (1969).

in

A cracked plate in a residual stress field

881

[2] M. Field, J. F. Kahles and Cammet, A review of measuring methods for surface integrity. Annals of CZRP 21, 219 (1972). [3] J. Frisch and E. G. Thomson, Residual grinding stresses in mild steel. Trans. ASME 74, 337-346 (1951) [4] W. R. Osgood, Residual stresses in metals and metal construction. Prepared for Ship Structure Committee, Reinhold Publishing Corporation, New York (1954). [S] H. Liebowitz and J. Eftis, Engng Fracture Mech. 3 (1971). [6] P. Stanley, Fracture Mechanics in Engineering Practice. Applied Science, London (1977). [A D. Broek, Elementary Engineering Fracture Mechanics. Noordhoff, Lcyden (1974). [8] S. T. Rolfe and J. M. Barsoum, Fracture and fatigue control in structures, in Appficution of Fracture Mechunics. Prentice-Hall, Englewood Cliffs, New Jersey (1977). [9] D. R. J. Gwen and A. J. Fawkes, Engineering Z+acture ~echunics:N~erieul Methods and Applications. Pineridge Press, Swansea, U.K. (1983). [lo] 0. Yagawa and T. Nisioka, Three dimensional finite element analysis for through wall crack in thick plate. Znt. J. Numer. Meth. Engng 12, 1295-1310 (1978). 111) D. M. Tracey, Finite elements for three dimensional elastic crack analysis. Nucl. Ettgtrg Design 26, 282-290 (1974). [12] W. S. Burton, G. El. Sinclair, J. S. Solecki and J. L. Swedlow, On the implications for LEFM of the three dimensional aspects in some crack/surface intersection problem. Znt. J. Fracture Mech. 2!5, 3-32 (1984). [13] H. G. delorenzi, On the energy release rate and the J-integral for 3-D crack configurations. Znt.J. Fracture 19,183-193 (1982). [14] R. S. Barsoum et al., Analysis of through cracks in cylindrical shells by quarter point elements. Znt. J. Fracture 15, 259-280 (1979). [lS] A. R. IngrafIea and C. Manu, Stress intensity factor computation in three dimensions with quarter-point elements. Znt. J. Numer. Meth. Engng 15, 1427-1445 (1980). [16] D. C. Drucker, Variational principles in the math~ati~l theory of plasticity. Proe. Symp. Appl. Muth, Vol. 8, C&&s of variations and its ~plication, pp. 7-22 (1958). 117) K. J. Bathe, Finite Element Procedures in Engineering AnaZysis. Prentice-Hall, New Jersey (1982). [18] R. H. Gallagher, Finite Element Analysis: F~~entuls. Prentice-Hall, Englewood Cliffs, New Jersey (1975). [19] 0. C. Zienkiewicz, Finite EZementMethod, The third expanded and revised edn. Tata McGraw Hill, New Delhi (1986). [20] Frank L. Stasa, Applied Finite EZementAnalysis for Engineers. CBS International Editions, CBS College Publishing, CBS Publishing Japan Ltd, Japan (1986). [21] C. E. Inglis, Stresses in a plate due to the presence of cracks and sharp comers. Trans. Inst. Naval Architects 55,219-241 (1913). [22] R. S. Barsoum, A degenerate solid elements for linear fracture analysis of plate bending and general shells. Znt. J. Numer. Merh. Engng 10,551-564 (1976). [23] R. S. Barsoum, Application of quadratic isoparametric finite elements in linear fracture mechanics. Znt. J. Fracture (Reports of Current Research) 10, (1974). [24] D. P. Rooke and D. J. Cartwright, Compendium of Stress Zntensity Fuctors, pp. 84. Procurement Executive, Ministry of Defence, London, HMSO (1976). 1251 M. L. Williams, On the stress distribution at the base of a stationary crack. J. appi. Mech. 24, 109 (1957). f26] A. Misbra and T. Prasad, Residual stresses due to a moving heat source. Znt. J. Mech. Sci. 27, 571-581 (1985). [271 F. Sinha, T. Prasad and A. Mishra, Influence of residual stresses on fracture behaviour: An experimental study. Engng Fracture h?ech. 21, 1113-I 118 (1985). {28] G. R. Leverant, B. S. Langer, A. Yuen and S. W. Hopkins, Surface residual stresses surface topography and the fatigue behaviour of Ti-6Al-4V. Met&. Truns. lOA, 251-257(1979). [29] J. P. Bruner, 0. N. Benjamin and D. M. Bench, Analysis of residual thermal and loading stresses in a B33 wheel and their relationship to fatigue damage. J. Engng Industry 89, 249-258 (1967). (Received 21 October 1988)