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Influence of crack closure for geometrically nonlinear plates bending problem
Engineering Fracture Mechanics Vol. 37, No. 5, pp. 915-920,
OOI3-7944190
1990
$3.00 + 0.00
Pergamon Press plc.
Printed in Great Britain.
INFLUENCE OF CRACK CLOSURE FOR GEOMETRICALLY NONLINEAR PLATES BENDING PROBLEM R. S. ALWAR and S. THIAGARAJAN Department of Applied Mechanics, Indian Institute of Technology, Madras 600 036, India Abstract-Cracked plates subjected to large deformations are analysed using the finite element technique. An important aspect of the cracked plate problem subjected to bending is the crack closure which takes place in the compression zone. This has been considered in this investigation. The numerical results regarding SIF are presented for cracked plates with and without crack closure. The constraint condition is taken in the form of line closure. Total Lagrangian approach is used for the formulation of the problem. It is observed that there is significant nonlinearity in the case of large deformation and the reduction in SIF due to closure is more than in the case of linear analysis.
INTRODUCTION THE BENDING problem of an infinite plate with a through crack was first analysed by Williams[l]a using classical plate theory. The use of Kirchhoff theory led to certain discrepancies in the solution. Knowles and Wang[2] obtained an improved solution for bending of vanishingly thin plate using Reissner’s theory. Later Hartranft and Sih[3], Wang[4] independently solved the plate bending problem introducing the effect of plate thickness on cracktip stress distribution. More refined theories were subsequently developed by Sih et a1.[5-91 and Folias[lO-111 using a 3-D theory of elasticity. All the above theories assume the model crack faces to be stress free. These theories do not take into account the crack face interference due to bending. It is assumed in all the above solutions that the external inplane forces of sufficient magnitude have been present such that crack faces do not close and the effect of extensional and bending be superimposed. Jones and Swedlow[l2] obtained numerical results for stresses and displacements near the cracktip taking into account crack closure phenomena in bending including the effect of plasticity around the cracktip. They employed the finite element technique for solution using Kirchhoff boundary conditions along crack faces. Crack closure could be effected along a line on compression surface. Heming Jr[l3] improved upon the solution for linear elastic case using Mindlin theory again imposing a line closure along the compression side. In both the cases mentioned previously only regular elements were used around the cracktip. Alwar et aZ.[l4, 151 employed 3D finite element analysis using singular elements to introduce area closure. To the authors knowledge no work has been done in the area of crack closure for a geometrically nonlinear problem. SCOPE
OF PRESENT
WORK
An attempt has been made to study the influence of crack closure in the case of plate bending subjected to large deformation. The order of singularity in the case of geometrically nonlinear situation as above is assumed to be the same as that of the linear case. The assumption is based on the premise[l6] that a nonlinear problem if looked into by way of stress resultants is nothing but a combination of inplane and out of plane forces, the order of singularity in each case being r-I/‘. In view of the above assumption, the linear singular element of Barsoum[l7] for cracktip region and geometrically nonlinear element of Bathe[l8] for the surrounding region are employed. In an earlier study by the authors[l9] connected with a geometrically nonlinear fracture problem a similar combination of linear and nonlinear elements has been used. In the present study a similar approach has been employed to investigate the problem of crack closure in the case of large deformation of a cracked plate subjected to bending.
R. S. ALWAR
916
and S. THIAGARAJAN
METHOD OF ANALYSIS In geometric nonlinear analysis, in view of the large deformations, the equilibrium equations must be employed in the current configuration during deformation. The equilibrium of the body at time f +At can be established using principle of virtual work as follows
where ti, the Cauchy’s stress tensor and eij is Almansi strain tensor. R is the virtual work due to external forces. The left hand superscript indicates the configuration in which the quantity occurs and the left hand subscript indicates the configuration with respect to which the quantity is measured. Equation (1) cannot be solved directly since the configuration at time t + At is unknown. A solution can be obtained by referring a11variables to a known configuration. For this purpose, in principle any one of the already calculated configuration could be used. In practice, however, the choice lies essentially between two formulations, namely, Total Lagrangian and updated Lagrangian. In Total Lagrangian the equilibrium configurations at time t = 0 is used. Total Lagrangian formulation is adopted in the present analysis,
TOTAL LAGRANGIAN FORMULATION
1181
The volume integral of Cauchy’s stress times the variation of Almansi strain in eq. (I) can be written in Lagrangian formulation as follows
s
I;** OV
sijs ’ +,*’e,j *d I’
(2)
where ‘+,,Af Sij is the Cartesian component of Second Piola-Kirchhoff stress tensor corresponding to configuration at time t + At but measured in configuration at time 0 and 6 ‘+OA’ ci, is the variation in Green-Lagrange strain tensor in configuration at time t + At referred to configuration at time 0. Since the stresses and strains are unknown liO*’Sij, ‘fOA’cij,for the solution the following incremental decomposition is used
where 6 S,, and Ati, are known 2nd Piola-Kirchhoff and Green-Lagrange strain in configuration at time t , o S,, and 0 c/l are respectively incremental quantities. It can be seen that variation in lfgA’ci, is nothing but variation in ,, tij. 0cij can be written in terms of linear and nonlinear components as 0 Lij =
0 eij +
0 rlij
where 0 ei, and Dvii are incremental linear and nonlinear strains respectively. Incremental linear strain can be written in terms of incremental dispia~ments 0 ??ij=
0 eii
as
I 5 0 uk.iO ukJ
=:(O”iJ+OUjai+
h
uk,iOukj$.
0Uk.i;
ukj)*
In the above equation a comma denotes differentiation and the left subscript denotes the time indicating the configuration in which the coordinate is measured.
Influence of crack closure
The incremental 2nd Piola-Kirchhoff Strain 0 cijr
917
stress 0 S, are related to incremental Green-Lagrange
0 si j =
0 ci
jm 0
6s
where o Ci, is the constitutive tensor. The governing eq. (1) can be written as
s 0
OY
Cijrs0 6,s60 gij Od V+
,J,SijiioqijodV = c+A’R -
:,SijSo eij d V s OV
(5)
which represents a nonlinear equation for incremental displacements ui. The solution for this cannot be obtained directly since they are nonlinear in displacement increments. Approximate solutions can be obtained by making the simplifications namely o~ij = Oeij. This means that in addition to using 6, cij = a0 eij the incremental constitutive relations employed are 0
The approximate
s
0 cijrs
0
ers*
(6)
equilibrium equation to be solved is
o Cijrso
OV
sij =
e, 6 o ersOd V+
s
1 Sijc30qijodV = ‘+“R -
The above equation is solved using modified Newton-Raphson
s
oCij,~Aoe~~Aoeijod~+ Oi’
i Sij6,eijodV.
(7)
s ov
b
scheme. The equation becomes
s I
r+dli,‘$;-‘)&+A’+ o Sij~~~~~OdV = t+A’R s OY OV
‘i’ is the current iteration number. A computer program has been written inco~orating
PROBLEM
‘todv
(8)
the above procedure.
DEF~~~ON
An isotrophic plate containing a crack subjected to uniaxial loading at plate boundaries including the effects of large deformation and crack closure is analysed. The cracked plate geometry and typical mesh is shown in Fig. 1. The displacement of the plate in x, y, z directions are respectively 24, 0, W. The assumed displacement will become
~x,Y,z)= 4%
Y,
~,(x,y)+Za(x,y)
f) = V&Y)
w(x,y,z)=
- ZP(x,Y)
w,(&Yl
where -h /2 < 2 < h 12 is the through thickness coordinate and U, and V, are inplane translational displacements in x, y directions respectively. 61,/3 are rotations about y and x axis and W, is the transverse deflection. The first case designated as no closure case or inte~netration case allows the crack faces to overlap or pass through each other. The crack face is modelled stress free, which means that all moment, shear, axial resultants are zero on crack faces.
918
R. S. ALWAR and S. THIAGARAJAN
Fig. 1. Typical mesh pattern.
-
w/a=
L/a
No
dOSUre
.20
0 =
&O-
2.0-
0
1.0
2.0 w/h
Fig. 2. Variation of KT with w/h for linear analysis.
3.0
919
Influence of crack closure
La--j
-
No
---
Closure
closure
/fP -2w Wla
5 Lia
I 20
Fig. 3. Variation of Kr with w/h for nonlinear analysis.
~‘~ MO
e.o-
&tear
2L
7.0-
&204
--
- non linear
1
co-
‘i.-2W
wla=
j
Lfa
t 20
5.0 -
I,ii r” a%
_
&
2
3.0 -
2.0-
l.O-
w/h
Fig. 4. Comparison of KT values for linear and nonlinear analysis with
closu~.
920
R. S. ALWAR and S. THIAGARAJAN
In the second case designated as the closure case the actual physical situation is modelled such that as bending occurs the crack faces come into contact at the compression surface and do not overlap. The crack surface is modelled using an additional constraint equation to account for the crack closure phenomena. The constrain condition takes the form of line closure i.e. zero z’ displacement on compression side of plate (2 = -h/2 of crack face). The constraint equation is as follows: V,(x, 0, -h/2)
-h/2/3(x,
0) = 0
in the region 0 < x < a. The constraint condition is imposed using the Lagrange multiplier technique. RESULTS AND DISCUSSION
The large deflection analysis of a cracked plate of dimensions which simulate the conditions of an infinite plate subjected to uniaxial bending is studied. Figure 2 shows the variation of combined (bending and stretching) stress intensity factor (Kr) with transverse deflection to thickness ratio (w/h) for linear case. It is observed that there is significant reduction of SIF in the case when closure occurs. Figure 3 shows the variation of Kr with w/h for closure case, and no closure case in the case of nonlinear analysis. It can be observed that there is significant nonlinearity in the case of large deformation and the reduction in SIF due to closure is more than in the case of linear analysis. It is also observed from Fig. 4 that the difference between SF’s obtained by linear closure and nonlinear closure analysis is not significant in the higher range of w/h although it is considerable in the middle range. This may be due to the interplay between membrane effects due to nonlinearity which enhances the stress intensity factor and the closure effect which reduces the SIF. REFERENCES [I] M. L. Williams, The bending stress distribution at the base of stationary crack. J. uppl. Mech. 28, 78-82 (1961). [2] J. K. Knowles and N. M. Wang, On the bending of an elastic plate containing a crack. J. Maths Phys. 39,223-236 (1960). [3] R. J. Hartranft and G. C. Sih, Effect of plate thickness on bending stress distribution around through crack. J. Maths Phys. 47, 276-291 (1968). [4] N. M. Wang, Effect of plate thickness on the bending of an elastic plate containing a crack. J. Muths Phys. 47, 371-390 (1968). [5] G. C. Sih, Bending of a cracked plate with an arbitrary stress distribution across the thickness, J. Engng Industry. Trans. ASME, 92, 350-356 (1970). [6] G. C. Sih, M. L. Williams and J. L. Swedlow, Three Dimensional stress distribution near a sharp crack in a plate of finite thickness. AFML-TR-66-242, Airforce Materials Lab., Write-Patterson Airforce Base (1966). [7] R. J. Hartranft and G. C. Sih, An approximate three dimensional theory of plates with applications to crack problems. Inl. J. Engng Sci. 8, 711-729 (1970). [8] G. C. Sih, A review of three dimensional crack problems for a cracked plate. Int. J Fracture 7, 3941 (1971). [9] G. C. Sih and R. J. Hartranft, Variation of strain energy release rate with plate thickness. fnt. J. Frncfure9, 75-82 (1973). [IO] E. S. Folias, Method of solution of a class of three-dimensional elastostatic problem under mode-1 loading. Int. J. Fracture 16, 335-348 (1980). [I l] E. S. Folias, On three dimensional theory of cracked plates. J. uppl. Mech. 42, 663474 (1975). [12] D. P. Jones, J. L. Swedlow, The influence of crack closure and elastoplastic flow on the bending of a cracked plate. Int. J. Fracture 11, 897-914 (1975). [ 131F. S. Fleming, Jr, Sixth order analysis of crack closure in bending of an elastic plate. Int. J. Fracture 16, 289-304 (1980).
[14] R. S. Alwar and Ramachandran, Influence of crack closure on the stress intensity factor for plates subjected to bending A-3D finite element analysis. Engng Fracture Mech. 17, 323-333 (1983). [I51 K. N. Ramachandran, R. S. Alwar, Influence of crack closure on the stress intensity factor for cylindrical shells subjected to bending. Technology applied to material evaluation and structural design. Proc. Int. Conf. on Fracture Mech., Melbourne (1982). [16] R. S. Alwar, K. N. Ramachandran, Influence of geometric nonlinearity on stress intensity factors. Engng Fracture Mech. 16, 415-424 (1982). [I71 R. S. Barsoum, A degenerate solid element for linear fracture analysis of plate bending and general shells. Int. J. numer. Meth. Engng 10, 551-564 (1976). (181 K. J. Bathe, S. Bolourchi, A geometric and material nonlinear plate and sheel element. Compos. Structures, 112348 (1981). [19] R. S. Alwar, S. Thiagarajan, Combined effect of shear and large deformation on stress intensity factors. Communicated to Int. J. numer. Meth. Engng. [20] K. J. Bathe, E. Ramm and E. L. Wilson, Finite element formulations for large deformation dynamic analysis. Int. J. numer. Meth. Engng. 9, 353-386 (1975). (Received 11 May 1989)
OOI3-7944190
1990
$3.00 + 0.00
Pergamon Press plc.
Printed in Great Britain.
INFLUENCE OF CRACK CLOSURE FOR GEOMETRICALLY NONLINEAR PLATES BENDING PROBLEM R. S. ALWAR and S. THIAGARAJAN Department of Applied Mechanics, Indian Institute of Technology, Madras 600 036, India Abstract-Cracked plates subjected to large deformations are analysed using the finite element technique. An important aspect of the cracked plate problem subjected to bending is the crack closure which takes place in the compression zone. This has been considered in this investigation. The numerical results regarding SIF are presented for cracked plates with and without crack closure. The constraint condition is taken in the form of line closure. Total Lagrangian approach is used for the formulation of the problem. It is observed that there is significant nonlinearity in the case of large deformation and the reduction in SIF due to closure is more than in the case of linear analysis.
INTRODUCTION THE BENDING problem of an infinite plate with a through crack was first analysed by Williams[l]a using classical plate theory. The use of Kirchhoff theory led to certain discrepancies in the solution. Knowles and Wang[2] obtained an improved solution for bending of vanishingly thin plate using Reissner’s theory. Later Hartranft and Sih[3], Wang[4] independently solved the plate bending problem introducing the effect of plate thickness on cracktip stress distribution. More refined theories were subsequently developed by Sih et a1.[5-91 and Folias[lO-111 using a 3-D theory of elasticity. All the above theories assume the model crack faces to be stress free. These theories do not take into account the crack face interference due to bending. It is assumed in all the above solutions that the external inplane forces of sufficient magnitude have been present such that crack faces do not close and the effect of extensional and bending be superimposed. Jones and Swedlow[l2] obtained numerical results for stresses and displacements near the cracktip taking into account crack closure phenomena in bending including the effect of plasticity around the cracktip. They employed the finite element technique for solution using Kirchhoff boundary conditions along crack faces. Crack closure could be effected along a line on compression surface. Heming Jr[l3] improved upon the solution for linear elastic case using Mindlin theory again imposing a line closure along the compression side. In both the cases mentioned previously only regular elements were used around the cracktip. Alwar et aZ.[l4, 151 employed 3D finite element analysis using singular elements to introduce area closure. To the authors knowledge no work has been done in the area of crack closure for a geometrically nonlinear problem. SCOPE
OF PRESENT
WORK
An attempt has been made to study the influence of crack closure in the case of plate bending subjected to large deformation. The order of singularity in the case of geometrically nonlinear situation as above is assumed to be the same as that of the linear case. The assumption is based on the premise[l6] that a nonlinear problem if looked into by way of stress resultants is nothing but a combination of inplane and out of plane forces, the order of singularity in each case being r-I/‘. In view of the above assumption, the linear singular element of Barsoum[l7] for cracktip region and geometrically nonlinear element of Bathe[l8] for the surrounding region are employed. In an earlier study by the authors[l9] connected with a geometrically nonlinear fracture problem a similar combination of linear and nonlinear elements has been used. In the present study a similar approach has been employed to investigate the problem of crack closure in the case of large deformation of a cracked plate subjected to bending.
R. S. ALWAR
916
and S. THIAGARAJAN
METHOD OF ANALYSIS In geometric nonlinear analysis, in view of the large deformations, the equilibrium equations must be employed in the current configuration during deformation. The equilibrium of the body at time f +At can be established using principle of virtual work as follows
where ti, the Cauchy’s stress tensor and eij is Almansi strain tensor. R is the virtual work due to external forces. The left hand superscript indicates the configuration in which the quantity occurs and the left hand subscript indicates the configuration with respect to which the quantity is measured. Equation (1) cannot be solved directly since the configuration at time t + At is unknown. A solution can be obtained by referring a11variables to a known configuration. For this purpose, in principle any one of the already calculated configuration could be used. In practice, however, the choice lies essentially between two formulations, namely, Total Lagrangian and updated Lagrangian. In Total Lagrangian the equilibrium configurations at time t = 0 is used. Total Lagrangian formulation is adopted in the present analysis,
TOTAL LAGRANGIAN FORMULATION
1181
The volume integral of Cauchy’s stress times the variation of Almansi strain in eq. (I) can be written in Lagrangian formulation as follows
s
I;** OV
sijs ’ +,*’e,j *d I’
(2)
where ‘+,,Af Sij is the Cartesian component of Second Piola-Kirchhoff stress tensor corresponding to configuration at time t + At but measured in configuration at time 0 and 6 ‘+OA’ ci, is the variation in Green-Lagrange strain tensor in configuration at time t + At referred to configuration at time 0. Since the stresses and strains are unknown liO*’Sij, ‘fOA’cij,for the solution the following incremental decomposition is used
where 6 S,, and Ati, are known 2nd Piola-Kirchhoff and Green-Lagrange strain in configuration at time t , o S,, and 0 c/l are respectively incremental quantities. It can be seen that variation in lfgA’ci, is nothing but variation in ,, tij. 0cij can be written in terms of linear and nonlinear components as 0 Lij =
0 eij +
0 rlij
where 0 ei, and Dvii are incremental linear and nonlinear strains respectively. Incremental linear strain can be written in terms of incremental dispia~ments 0 ??ij=
0 eii
as
I 5 0 uk.iO ukJ
=:(O”iJ+OUjai+
h
uk,iOukj$.
0Uk.i;
ukj)*
In the above equation a comma denotes differentiation and the left subscript denotes the time indicating the configuration in which the coordinate is measured.
Influence of crack closure
The incremental 2nd Piola-Kirchhoff Strain 0 cijr
917
stress 0 S, are related to incremental Green-Lagrange
0 si j =
0 ci
jm 0
6s
where o Ci, is the constitutive tensor. The governing eq. (1) can be written as
s 0
OY
Cijrs0 6,s60 gij Od V+
,J,SijiioqijodV = c+A’R -
:,SijSo eij d V s OV
(5)
which represents a nonlinear equation for incremental displacements ui. The solution for this cannot be obtained directly since they are nonlinear in displacement increments. Approximate solutions can be obtained by making the simplifications namely o~ij = Oeij. This means that in addition to using 6, cij = a0 eij the incremental constitutive relations employed are 0
The approximate
s
0 cijrs
0
ers*
(6)
equilibrium equation to be solved is
o Cijrso
OV
sij =
e, 6 o ersOd V+
s
1 Sijc30qijodV = ‘+“R -
The above equation is solved using modified Newton-Raphson
s
oCij,~Aoe~~Aoeijod~+ Oi’
i Sij6,eijodV.
(7)
s ov
b
scheme. The equation becomes
s I
r+dli,‘$;-‘)&+A’+ o Sij~~~~~OdV = t+A’R s OY OV
‘i’ is the current iteration number. A computer program has been written inco~orating
PROBLEM
‘todv
(8)
the above procedure.
DEF~~~ON
An isotrophic plate containing a crack subjected to uniaxial loading at plate boundaries including the effects of large deformation and crack closure is analysed. The cracked plate geometry and typical mesh is shown in Fig. 1. The displacement of the plate in x, y, z directions are respectively 24, 0, W. The assumed displacement will become
~x,Y,z)= 4%
Y,
~,(x,y)+Za(x,y)
f) = V&Y)
w(x,y,z)=
- ZP(x,Y)
w,(&Yl
where -h /2 < 2 < h 12 is the through thickness coordinate and U, and V, are inplane translational displacements in x, y directions respectively. 61,/3 are rotations about y and x axis and W, is the transverse deflection. The first case designated as no closure case or inte~netration case allows the crack faces to overlap or pass through each other. The crack face is modelled stress free, which means that all moment, shear, axial resultants are zero on crack faces.
918
R. S. ALWAR and S. THIAGARAJAN
Fig. 1. Typical mesh pattern.
-
w/a=
L/a
No
dOSUre
.20
0 =
&O-
2.0-
0
1.0
2.0 w/h
Fig. 2. Variation of KT with w/h for linear analysis.
3.0
919
Influence of crack closure
La--j
-
No
---
Closure
closure
/fP -2w Wla
5 Lia
I 20
Fig. 3. Variation of Kr with w/h for nonlinear analysis.
~‘~ MO
e.o-
&tear
2L
7.0-
&204
--
- non linear
1
co-
‘i.-2W
wla=
j
Lfa
t 20
5.0 -
I,ii r” a%
_
&
2
3.0 -
2.0-
l.O-
w/h
Fig. 4. Comparison of KT values for linear and nonlinear analysis with
closu~.
920
R. S. ALWAR and S. THIAGARAJAN
In the second case designated as the closure case the actual physical situation is modelled such that as bending occurs the crack faces come into contact at the compression surface and do not overlap. The crack surface is modelled using an additional constraint equation to account for the crack closure phenomena. The constrain condition takes the form of line closure i.e. zero z’ displacement on compression side of plate (2 = -h/2 of crack face). The constraint equation is as follows: V,(x, 0, -h/2)
-h/2/3(x,
0) = 0
in the region 0 < x < a. The constraint condition is imposed using the Lagrange multiplier technique. RESULTS AND DISCUSSION
The large deflection analysis of a cracked plate of dimensions which simulate the conditions of an infinite plate subjected to uniaxial bending is studied. Figure 2 shows the variation of combined (bending and stretching) stress intensity factor (Kr) with transverse deflection to thickness ratio (w/h) for linear case. It is observed that there is significant reduction of SIF in the case when closure occurs. Figure 3 shows the variation of Kr with w/h for closure case, and no closure case in the case of nonlinear analysis. It can be observed that there is significant nonlinearity in the case of large deformation and the reduction in SIF due to closure is more than in the case of linear analysis. It is also observed from Fig. 4 that the difference between SF’s obtained by linear closure and nonlinear closure analysis is not significant in the higher range of w/h although it is considerable in the middle range. This may be due to the interplay between membrane effects due to nonlinearity which enhances the stress intensity factor and the closure effect which reduces the SIF. REFERENCES [I] M. L. Williams, The bending stress distribution at the base of stationary crack. J. uppl. Mech. 28, 78-82 (1961). [2] J. K. Knowles and N. M. Wang, On the bending of an elastic plate containing a crack. J. Maths Phys. 39,223-236 (1960). [3] R. J. Hartranft and G. C. Sih, Effect of plate thickness on bending stress distribution around through crack. J. Maths Phys. 47, 276-291 (1968). [4] N. M. Wang, Effect of plate thickness on the bending of an elastic plate containing a crack. J. Muths Phys. 47, 371-390 (1968). [5] G. C. Sih, Bending of a cracked plate with an arbitrary stress distribution across the thickness, J. Engng Industry. Trans. ASME, 92, 350-356 (1970). [6] G. C. Sih, M. L. Williams and J. L. Swedlow, Three Dimensional stress distribution near a sharp crack in a plate of finite thickness. AFML-TR-66-242, Airforce Materials Lab., Write-Patterson Airforce Base (1966). [7] R. J. Hartranft and G. C. Sih, An approximate three dimensional theory of plates with applications to crack problems. Inl. J. Engng Sci. 8, 711-729 (1970). [8] G. C. Sih, A review of three dimensional crack problems for a cracked plate. Int. J Fracture 7, 3941 (1971). [9] G. C. Sih and R. J. Hartranft, Variation of strain energy release rate with plate thickness. fnt. J. Frncfure9, 75-82 (1973). [IO] E. S. Folias, Method of solution of a class of three-dimensional elastostatic problem under mode-1 loading. Int. J. Fracture 16, 335-348 (1980). [I l] E. S. Folias, On three dimensional theory of cracked plates. J. uppl. Mech. 42, 663474 (1975). [12] D. P. Jones, J. L. Swedlow, The influence of crack closure and elastoplastic flow on the bending of a cracked plate. Int. J. Fracture 11, 897-914 (1975). [ 131F. S. Fleming, Jr, Sixth order analysis of crack closure in bending of an elastic plate. Int. J. Fracture 16, 289-304 (1980).
[14] R. S. Alwar and Ramachandran, Influence of crack closure on the stress intensity factor for plates subjected to bending A-3D finite element analysis. Engng Fracture Mech. 17, 323-333 (1983). [I51 K. N. Ramachandran, R. S. Alwar, Influence of crack closure on the stress intensity factor for cylindrical shells subjected to bending. Technology applied to material evaluation and structural design. Proc. Int. Conf. on Fracture Mech., Melbourne (1982). [16] R. S. Alwar, K. N. Ramachandran, Influence of geometric nonlinearity on stress intensity factors. Engng Fracture Mech. 16, 415-424 (1982). [I71 R. S. Barsoum, A degenerate solid element for linear fracture analysis of plate bending and general shells. Int. J. numer. Meth. Engng 10, 551-564 (1976). (181 K. J. Bathe, S. Bolourchi, A geometric and material nonlinear plate and sheel element. Compos. Structures, 112348 (1981). [19] R. S. Alwar, S. Thiagarajan, Combined effect of shear and large deformation on stress intensity factors. Communicated to Int. J. numer. Meth. Engng. [20] K. J. Bathe, E. Ramm and E. L. Wilson, Finite element formulations for large deformation dynamic analysis. Int. J. numer. Meth. Engng. 9, 353-386 (1975). (Received 11 May 1989)