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Influence of geometric non-linearity on stress-intensity factors
En,&mring Fracture Mechanics Vol. 16, No. 3, pp. 415-423, 1982 Printed in Great Britain.
R. S. ALWARI and K. N. RA~ACHAN~RAN NA~BISSAN~ Departmentof AppliedMechanics,Indian lnstitute of Technology,Madras600036,India ~~t-The inffueuceof geometricnon-linea~ty on the s~ess-intensity factor for D centraItycracked plate subjectedto u~forml~ distributedioad is studied.The bendingand stretchingstress-intensityfactors have been derived by strain energy release rate technique. It is found that the bending stress-~nteos~~ factor varies in a nor&ear fashion as load increasesfor large deffectionsof the plate and the resulting in-plane stretchingof the plate introducesa stretchingstress-intensityfactor.
NOTATIONS s~ess-intensityfactor in bending stress-intensjtyfactor in s~tching Young’smodulus plate thickness Poisson’sratio Cartesianco-ordinates polar co-ordinates displacementsin X,y, z directions stress resultants in bending stress resultants in stretching transverseshear KirchhofYsshear semi-cracklength plate dimensions rt.d.l. on the pfate
E/2(1f V)moduiusof rigidity strain energy due to bending strain energy due to stretching strain energy releaserate in bending strain energy releaserate in stretcbjng mass densitiesin L y, z directions dampingfactors in x, y, z directions residuesof equationsof equilibriumin X,y, z directionsrespectively time increment dynamicrelaxation
THEINVESTIGATION regarding the strength of plate or a she11with through crack of finite length is of great practicai interest in areas like nuclear and aerospace engineering. Knowledge regarding the nature of the elastic stresses near the crack tip is essential for the study of the strength of structures with flaws, since these are the stresses which are respansibie for possible crack propagation. The characteristics of stresses at the crack tip can be determined using the concept of stress intensity factors which are dependent upon load, geometry and in the case of plates and shells, on Poisson’s ratio. Regarding plate and shell type of structures. it was Williamsfl] who applied linear fracture mechanics for a cracked plate in bending and showed that the stresses possess singularity of the order of Ily’r. Duncan-Mama, Sandar, Erdogan and Ratwani using linear thin shell theory have shown that the stresses in shells also have a singularity of the order of I/d/r. All these investigations are based upon the classical theory of bending derived using Kirchhoff assumptions, Recentfy, in view of the three dimensional character of crack tip stresses, higher order plate theories have been applied to bending of cracked plates by Sih et GE.@,31. AlI the above investigations are confined to the application of linear fracture mechanics in the sense that no non-linearities come into piay either in the constitutive equations or in the tprofessor & Head, ElasticityLaboratory. $ResearchScholar.
415
R. S, ALWAR and K. N. ~A~A~~AN~~AN
NA~~~~~A~
equilibrium equations. The object of the present investigation is to determine the influence of geometric non-linearitieson the stress intensity factors, Ttis common k~~iedge that when the detection in a plate is sn~cjentiy large that is of the order of plate thickness significant non-linear terms are introduced in the differential equations, which is due to non-linear strain displacement relations and also interaction between the inplane and bending forces. To the authors’ knowledge, there has been no investigation till now to find out the influence of such non-linearitieson the stress intensity factors connected with inplane stretching forces or out of plane bending moment and shear forces. Such_problems are of great practical interest in thin walled components ivhich are commonly used in Aerospace s~uctures. LARGE DEFLR~T~~NPLATR ~~~ATI~N~ The equations for the large amplitude motion of plates can be written as foItows:
The stress resultants in the above equations can be expressed in terms of displacements as follows.
Influenceof geometricnon-linearityon s~ess-intensityfactors
417
Taking ho, I and E as representative thickness, width and Young’s modulus respectively, the following relations are used for non-dimens~onalising the above equations
t = 12~(P~~~O)* t:
(W
Bar above the letter indicates the nob-dimensjona~ quantities. introducing the above relations into eqns (l)--(9), the following non-dimension& equations are obtained. 2
2
%+%-2$5+N+
where
418
R. S, ALWAR and K. N. ~AMA~HAND~N
NAM~~SSAN
The bar above the ~~a~~i~ies in the above e~ua~~o~~ have been dropped for convenience.
~~cie~cy of dynamic relaxation method in sofving plate and she11problems both for small and large detection cases is well established~5-81.The method consists in writing down the equations of motion pertaining to the problem with proper inertia and damping terms. By adjusting the value of the damping terms and by a step by step catenation of these equations, expressed in explicit unite-difference form, a steady-state static solution is achieved. The fo~mniation is linear in time and co~seqnentiy the procedure of integration requires only a direct substitution of the value of the function at a previous iteration in the time step, to calculate the new value of the function. The procedure altows the non-~jn~arterms to be included in a simple and s~aightforward manner. This enabtes the non-linear problems to be solved with equa1ease, A fast convergence to the static vahtes can be obtained by a proper choice of damping coefficients and mass densities. The dynamic relaxation technique requires much less core space in the computer, compared to other numerical methods. This method, therefore, has been chosen for the solution of the present problem. Expressing all the derivatives in eqns (11~(19) in central-difflerenceform and Fe-arranging the terms of eqns (11~(133,the equations of dynamic equilibrium can be written to suit the a~p~icatiunof dynamic relaxation technique. As an example eqn (12)can be written as follows.
where
Subscripts i, j in the above equation represent the nodal coordinate of the point and II represents the time step. Dot above the quantity stands for derivative with respect to time, From eqn (20),the velocity ii!;’ is obtained as follows
~qnatjon (21) relates the velocity at time n -t 1 to the velocity at time n and the values of stress resultants at time (n + (112)).Using the velocity at time (n + l), the displacement at time (n + l(112))is calculated as follows
assuming the initial values of the stress-resu~~nts, d~spIa~meuts and velocities to be zero at all nodes, the vefocity rl and displacement i( at ah nodes for the subsequent time step can be obtained from eqn (21)and (22). d, u, 9 and w at ali nodes can be similarlyobtained for the new time step from eqns (It) and (13).It is to be noted that the initial value of L, is not zero due to the presence of the load term 4 and this helps the process to continue from an assumed initial zero values for the velocities, displacements and stress-resultants, The stress-resultants cork responding to the new values of displacements are calculated from eqns (14) to (19). These values of stress-resultants are used as the initial conditions to calculate the velocities and displacements at the next time step. The procedure is repeated till the damped steady-state solution is obtained. A proper value for the time step At has to be chosen to obtain convergence.
InfluenCe of geometric non-Iinearity on st~ss-intensity factors
419
BOUNDARYCO~ITIONS The boundary conditions applied on the cracked edge alone are discussed. The classical plate bending theory, being of the fourth order, will admit only two boundary conditions in bending and two in stretching to be satisfied on a free edge. These four boundary conditions can be written for an edge x = 1as follows: (23)
M, = 0
(24)
N, = 0
(25)
Nxy= 0.
Wf
Using relations (4)-(9), boundary conditions (24)-(26) and using the finite-difference formulae, the following relations for the displacements at the fictitious point (1) (Fig. 2) can be obtained. 2 W$
w3-Y =2wo--
uj = Q-AX
z.q = ~3 t
!+z
(> AY
402
{
(uq -
-
(W2-2WQfW4)
04) +
AY
(w -
w312 t
4 w2 -
(27) %I2
4@y12
Wn)*
1
u2)Ax+; (w.,- wZ)(w, - wJ}/Ay.
(28) (29)
Equation (27) is solved first to obtain w1and this value is used in solving eqns (28) and (29) to get u1 and z.+.Since the equations of dynamic equilibrium have to be applied on the crack edge, the stress-resultants M,, MXY,N, and NXYare also to be determined at the fictitious point (1). M,,, is obtained from the condition (23), and I’$.,,, N,,, and N,,, are determined by parabolic extrapolation,
RJlLATIONBETWEEN&AND& There are practical difficulties in having extremely fine meshes near the crack-tip in finite-difference formulation. Hence the technique of strain energy release rate is adopted for the determination of stress-intensity factors. Being an overall phenomenon, the mesh size near the crack tip has less influence on the strain energy release rate. The relation between the stress-intensity factor and the strain energy release rate in bending can be easily established for pure bending.
AY
_.. ‘FICTITIOUS I
~
FREE
EDGE
x =I
Fig. 2. Finitedifference mesh points on cracked edge.
POINT
---w
x
Y
-.--
x
i
Y
The vertical de~ectjon close to the crack tip for symmetrical ~~d~~g in classical theory is
given byfr]
The bending moment in the ~~~~bbo~rhoodof the crack tip is given by
The work done wherr the crack closes along a small d~stao~ is given by
Therelation between bl and & is given by I;r]
~~b~t~t~ti~for bl-from (33)and ca~cuiat~ngthe work done pet unit area of crack closure we get
Mode II st~css-~~te~s~t~ factor does not exist for the symmetrical case.
Influence of geometric non-linearity on stress-intensity factors
421
The relation between the stress-intensity factor and the strain energy release rate in stretching in plane stress is given by (35) Equations (34) and (35) are derived by equating the strain energy release rate to the work done by the singular term in the stress distribution and the corresponding displacement, for an infinitesimal crack extension/closure. It may be observed that the above expressions for stress distribution and displacements are based on small strain displacement theory. The large displacement problem if looked into by way of stress resultants, is nothing but a combination ofout-of-plane forces like bending and shear forces, and in-plane stresses as in the case of plane stress problem. Hence it is assumed that the singular term for stress resultants and strains may have the same angular distribution and order of singularity even in the case of large displacements. Most works to-dateHO, 1I] also indicate that the form of the stress held near the crack front is atways the same, only exception occurring when the classical plate or shell theories, involving Kirchhoff’s boundary conditions are involved. It is also seen that analysis using the displacement function based on Kirchhoff theory, in conjunction with shear deformation theory are found to give good results{l2]. Although this is not a rigorous justification for assuming eqns (30) and (31) to be valid for the case of large displacements, the present work is the first step towards getting results which reveal the influence of geometrical non-linearity on stress intensity factor. Further work has to be done wherein a rigorous solution for the crack-tip stress distribution using large displacement field equations should be obtained. The internal energy stored in the plate due to bending and stretching are separately calculated for various crack lengths. Least square fitting is adopted and the energy release rates are obtained for bending and stretching of the plate and the stress-intensity factors are calculated from (34) and (35). All computations have been done with double precision in IBM 370/l% to reduce the noise effect. NUMERICAL RESULTS AND DISCUSSION Different mesh sizes were adopted in the finite difference formulation to test convergence of the results. A 20 x 20 mesh size has been found to be adequate and the results in the linear range for a square plate with fixed edges and subjected to u.d.1 agree well with those given in Ref. [9]. Near critical damping is introduced by suitably choosing the damping factors for all the cases so that a steady-state solution is obtained for a small number of time steps. All the derivatives at the crack-tip where singularity exists, have been approximated by the derivatives at the neighbouring point.
LINEAR
ANALYSIS
0 AIMED
S LOO
0 PRESENT
Fig. 4. Variation of K, with u/w. Linear analysis.
422
R. S. ALWAR and K. N. RAMACHANDRAN NAMBISSAN
31 F
LOAD
-
Kb-NON-LINEAR
---
Kb-LINEAR
0
Kb
P
Kt
P
2J=qw~lDh-
Fig. 5. Variation of & and 4 with load. Non-linear analysis.
-)(b
I
__,p----~---“__--~~,QQ _+__-+---+---_c__ +___+.---+----P----p--_
-WXl
Fig. 6. Variation of Kb and X1 with a/w. Non-linear analysis.
A square plate fixed on all edges, having a central through crack, subjected to a uniformly dis~buted Ioad is considered. The plate dimensions are W = 1 = 20.32 x lo-* M (8”); h = 2.54 x IO-’ M (0.1”) and 4 = 6894.72 N/M* (1.0 psi). a/w ratio varies from 0.3 to 0.7. The results obtained are given in Fig. 4. As can be seen there is good agreement for the stress-intensity factor &, with those of Ref. [9]. The solutions have extended to the non-linear range by increasing the load progressively. and K/@/h * w - da) have been plotted The non-dimensional values of &/(Wh * w * da)
Influence of geometric non-linearity on stress-intensity factors
423
for a/w = 0.3, 0.4, 0.5, 0.6 and 0.7 and non-dimensional load values of (q - w4/Dh) = 300, 400, 500, 600 and 700 in Fig. 6. It can be seen from Fig. 5 that there is significant difference between the values of K,, obtained from linear and non-linear analyses. This difference between the linear and non-linear results increases with increase in a/w. This may be due to the fact that for the same load and for higher value of a/w, the deflection will be larger resulting in greater influence of geometric non-linearity. It is evident from the figure that Kb and K, both increase in a non-linear fashion with increase in external loads. It can be seen from Fig. 6 that the value of Kb increases to its maximum when a/w is around 0.6 and then falls down which may be due to clamp end effect. This happens for all the loads considered in the analysis. Similarly the value of K, reaches its maximum when a/w is around 0.55 and subsequently, the magnitude of EC,decreases. Further analysis should be done in order to combine the influence of shear deformation and geometric non-linearity in order to get a more realistic picture of the influence of geometric non-linearity on stress-intensity factor. REFERENCES [II M. L. Williams, The bending stress distribution at the base of a stationary crack. J. Appl. Mech. Series E, 82, 78-82 (1961). [2] R. J. Hartranft and G. C. Sih, Effect of plate thickness on the bending stress distribution around through cracks. J. cache. Phys. 47,276291 (1968). (3) R. J. Hartranft and G. C. Sih, An approximate 3-D theory of plates with application to crack problems. fnf. 1. Eagng Sci, 8,7f l-729 (1970). [4] G. C. Sih, P. C. Paris and F. Erdogan, Crack-tip Stress-intensity factors for plane extensions and plate bending problems. J. Appt. Mech. 83, 3O6-312(1962). [S] A. S. Day, An introduction to dynamic relaxation. Engineer, London, 219, 219-221(1965). [6] K. R. Rushton, Large deflection of variable thickness plates. Int. J. Mech. Sci. 10, 723-735(1968). [7] A. C. Cassel, Cylindrical shell analysis by dynamic relaxation. Proc. Inst. Ciu. Engrs. 39, 75-84 (1968). [SJ R. S. Alwar and N. R. Rao, Large elastic deformation of clamped, skewed plates by dynamic relaxation. Contput. Structures 4, 381-398(1974). [9] Jalees Ahmed and Francis T. C. Loo, Solution of plate bending problems in Fracture Mechanics using a speciaiised finite element technique. ~~gj~ee~~g F~~c~~~e Mechunjc~T II, 661-673(1979). (l(t) R. J. Hartranft and G. C. Sih, The use of Eigen function expansions in the general solution of Three-Dimensional Crack Problems. J. Muths and Mechanics 19, 123-138(1969). [l I] M. K. Kassir and G. C. Sih, Three-dimensional stress distribution around an elliptical crack under arbitrary loadings. J. Appl. Mech. 33,601-611 (1966). [12] G. Yagawa and T. Nishioka, Finite Element Analysis of Stress Intensity Factors for plane extensions and plate bending problems. ht. .I Numerical Meth. Engng 14, 727-740(1979). (Received 1 June 1981;received for publication 26 June 1981)
R. S. ALWARI and K. N. RA~ACHAN~RAN NA~BISSAN~ Departmentof AppliedMechanics,Indian lnstitute of Technology,Madras600036,India ~~t-The inffueuceof geometricnon-linea~ty on the s~ess-intensity factor for D centraItycracked plate subjectedto u~forml~ distributedioad is studied.The bendingand stretchingstress-intensityfactors have been derived by strain energy release rate technique. It is found that the bending stress-~nteos~~ factor varies in a nor&ear fashion as load increasesfor large deffectionsof the plate and the resulting in-plane stretchingof the plate introducesa stretchingstress-intensityfactor.
NOTATIONS s~ess-intensityfactor in bending stress-intensjtyfactor in s~tching Young’smodulus plate thickness Poisson’sratio Cartesianco-ordinates polar co-ordinates displacementsin X,y, z directions stress resultants in bending stress resultants in stretching transverseshear KirchhofYsshear semi-cracklength plate dimensions rt.d.l. on the pfate
E/2(1f V)moduiusof rigidity strain energy due to bending strain energy due to stretching strain energy releaserate in bending strain energy releaserate in stretcbjng mass densitiesin L y, z directions dampingfactors in x, y, z directions residuesof equationsof equilibriumin X,y, z directionsrespectively time increment dynamicrelaxation
THEINVESTIGATION regarding the strength of plate or a she11with through crack of finite length is of great practicai interest in areas like nuclear and aerospace engineering. Knowledge regarding the nature of the elastic stresses near the crack tip is essential for the study of the strength of structures with flaws, since these are the stresses which are respansibie for possible crack propagation. The characteristics of stresses at the crack tip can be determined using the concept of stress intensity factors which are dependent upon load, geometry and in the case of plates and shells, on Poisson’s ratio. Regarding plate and shell type of structures. it was Williamsfl] who applied linear fracture mechanics for a cracked plate in bending and showed that the stresses possess singularity of the order of Ily’r. Duncan-Mama, Sandar, Erdogan and Ratwani using linear thin shell theory have shown that the stresses in shells also have a singularity of the order of I/d/r. All these investigations are based upon the classical theory of bending derived using Kirchhoff assumptions, Recentfy, in view of the three dimensional character of crack tip stresses, higher order plate theories have been applied to bending of cracked plates by Sih et GE.@,31. AlI the above investigations are confined to the application of linear fracture mechanics in the sense that no non-linearities come into piay either in the constitutive equations or in the tprofessor & Head, ElasticityLaboratory. $ResearchScholar.
415
R. S, ALWAR and K. N. ~A~A~~AN~~AN
NA~~~~~A~
equilibrium equations. The object of the present investigation is to determine the influence of geometric non-linearitieson the stress intensity factors, Ttis common k~~iedge that when the detection in a plate is sn~cjentiy large that is of the order of plate thickness significant non-linear terms are introduced in the differential equations, which is due to non-linear strain displacement relations and also interaction between the inplane and bending forces. To the authors’ knowledge, there has been no investigation till now to find out the influence of such non-linearitieson the stress intensity factors connected with inplane stretching forces or out of plane bending moment and shear forces. Such_problems are of great practical interest in thin walled components ivhich are commonly used in Aerospace s~uctures. LARGE DEFLR~T~~NPLATR ~~~ATI~N~ The equations for the large amplitude motion of plates can be written as foItows:
The stress resultants in the above equations can be expressed in terms of displacements as follows.
Influenceof geometricnon-linearityon s~ess-intensityfactors
417
Taking ho, I and E as representative thickness, width and Young’s modulus respectively, the following relations are used for non-dimens~onalising the above equations
t = 12~(P~~~O)* t:
(W
Bar above the letter indicates the nob-dimensjona~ quantities. introducing the above relations into eqns (l)--(9), the following non-dimension& equations are obtained. 2
2
%+%-2$5+N+
where
418
R. S, ALWAR and K. N. ~AMA~HAND~N
NAM~~SSAN
The bar above the ~~a~~i~ies in the above e~ua~~o~~ have been dropped for convenience.
~~cie~cy of dynamic relaxation method in sofving plate and she11problems both for small and large detection cases is well established~5-81.The method consists in writing down the equations of motion pertaining to the problem with proper inertia and damping terms. By adjusting the value of the damping terms and by a step by step catenation of these equations, expressed in explicit unite-difference form, a steady-state static solution is achieved. The fo~mniation is linear in time and co~seqnentiy the procedure of integration requires only a direct substitution of the value of the function at a previous iteration in the time step, to calculate the new value of the function. The procedure altows the non-~jn~arterms to be included in a simple and s~aightforward manner. This enabtes the non-linear problems to be solved with equa1ease, A fast convergence to the static vahtes can be obtained by a proper choice of damping coefficients and mass densities. The dynamic relaxation technique requires much less core space in the computer, compared to other numerical methods. This method, therefore, has been chosen for the solution of the present problem. Expressing all the derivatives in eqns (11~(19) in central-difflerenceform and Fe-arranging the terms of eqns (11~(133,the equations of dynamic equilibrium can be written to suit the a~p~icatiunof dynamic relaxation technique. As an example eqn (12)can be written as follows.
where
Subscripts i, j in the above equation represent the nodal coordinate of the point and II represents the time step. Dot above the quantity stands for derivative with respect to time, From eqn (20),the velocity ii!;’ is obtained as follows
~qnatjon (21) relates the velocity at time n -t 1 to the velocity at time n and the values of stress resultants at time (n + (112)).Using the velocity at time (n + l), the displacement at time (n + l(112))is calculated as follows
assuming the initial values of the stress-resu~~nts, d~spIa~meuts and velocities to be zero at all nodes, the vefocity rl and displacement i( at ah nodes for the subsequent time step can be obtained from eqn (21)and (22). d, u, 9 and w at ali nodes can be similarlyobtained for the new time step from eqns (It) and (13).It is to be noted that the initial value of L, is not zero due to the presence of the load term 4 and this helps the process to continue from an assumed initial zero values for the velocities, displacements and stress-resultants, The stress-resultants cork responding to the new values of displacements are calculated from eqns (14) to (19). These values of stress-resultants are used as the initial conditions to calculate the velocities and displacements at the next time step. The procedure is repeated till the damped steady-state solution is obtained. A proper value for the time step At has to be chosen to obtain convergence.
InfluenCe of geometric non-Iinearity on st~ss-intensity factors
419
BOUNDARYCO~ITIONS The boundary conditions applied on the cracked edge alone are discussed. The classical plate bending theory, being of the fourth order, will admit only two boundary conditions in bending and two in stretching to be satisfied on a free edge. These four boundary conditions can be written for an edge x = 1as follows: (23)
M, = 0
(24)
N, = 0
(25)
Nxy= 0.
Wf
Using relations (4)-(9), boundary conditions (24)-(26) and using the finite-difference formulae, the following relations for the displacements at the fictitious point (1) (Fig. 2) can be obtained. 2 W$
w3-Y =2wo--
uj = Q-AX
z.q = ~3 t
!+z
(> AY
402
{
(uq -
-
(W2-2WQfW4)
04) +
AY
(w -
w312 t
4 w2 -
(27) %I2
4@y12
Wn)*
1
u2)Ax+; (w.,- wZ)(w, - wJ}/Ay.
(28) (29)
Equation (27) is solved first to obtain w1and this value is used in solving eqns (28) and (29) to get u1 and z.+.Since the equations of dynamic equilibrium have to be applied on the crack edge, the stress-resultants M,, MXY,N, and NXYare also to be determined at the fictitious point (1). M,,, is obtained from the condition (23), and I’$.,,, N,,, and N,,, are determined by parabolic extrapolation,
RJlLATIONBETWEEN&AND& There are practical difficulties in having extremely fine meshes near the crack-tip in finite-difference formulation. Hence the technique of strain energy release rate is adopted for the determination of stress-intensity factors. Being an overall phenomenon, the mesh size near the crack tip has less influence on the strain energy release rate. The relation between the stress-intensity factor and the strain energy release rate in bending can be easily established for pure bending.
AY
_.. ‘FICTITIOUS I
~
FREE
EDGE
x =I
Fig. 2. Finitedifference mesh points on cracked edge.
POINT
---w
x
Y
-.--
x
i
Y
The vertical de~ectjon close to the crack tip for symmetrical ~~d~~g in classical theory is
given byfr]
The bending moment in the ~~~~bbo~rhoodof the crack tip is given by
The work done wherr the crack closes along a small d~stao~ is given by
Therelation between bl and & is given by I;r]
~~b~t~t~ti~for bl-from (33)and ca~cuiat~ngthe work done pet unit area of crack closure we get
Mode II st~css-~~te~s~t~ factor does not exist for the symmetrical case.
Influence of geometric non-linearity on stress-intensity factors
421
The relation between the stress-intensity factor and the strain energy release rate in stretching in plane stress is given by (35) Equations (34) and (35) are derived by equating the strain energy release rate to the work done by the singular term in the stress distribution and the corresponding displacement, for an infinitesimal crack extension/closure. It may be observed that the above expressions for stress distribution and displacements are based on small strain displacement theory. The large displacement problem if looked into by way of stress resultants, is nothing but a combination ofout-of-plane forces like bending and shear forces, and in-plane stresses as in the case of plane stress problem. Hence it is assumed that the singular term for stress resultants and strains may have the same angular distribution and order of singularity even in the case of large displacements. Most works to-dateHO, 1I] also indicate that the form of the stress held near the crack front is atways the same, only exception occurring when the classical plate or shell theories, involving Kirchhoff’s boundary conditions are involved. It is also seen that analysis using the displacement function based on Kirchhoff theory, in conjunction with shear deformation theory are found to give good results{l2]. Although this is not a rigorous justification for assuming eqns (30) and (31) to be valid for the case of large displacements, the present work is the first step towards getting results which reveal the influence of geometrical non-linearity on stress intensity factor. Further work has to be done wherein a rigorous solution for the crack-tip stress distribution using large displacement field equations should be obtained. The internal energy stored in the plate due to bending and stretching are separately calculated for various crack lengths. Least square fitting is adopted and the energy release rates are obtained for bending and stretching of the plate and the stress-intensity factors are calculated from (34) and (35). All computations have been done with double precision in IBM 370/l% to reduce the noise effect. NUMERICAL RESULTS AND DISCUSSION Different mesh sizes were adopted in the finite difference formulation to test convergence of the results. A 20 x 20 mesh size has been found to be adequate and the results in the linear range for a square plate with fixed edges and subjected to u.d.1 agree well with those given in Ref. [9]. Near critical damping is introduced by suitably choosing the damping factors for all the cases so that a steady-state solution is obtained for a small number of time steps. All the derivatives at the crack-tip where singularity exists, have been approximated by the derivatives at the neighbouring point.
LINEAR
ANALYSIS
0 AIMED
S LOO
0 PRESENT
Fig. 4. Variation of K, with u/w. Linear analysis.
422
R. S. ALWAR and K. N. RAMACHANDRAN NAMBISSAN
31 F
LOAD
-
Kb-NON-LINEAR
---
Kb-LINEAR
0
Kb
P
Kt
P
2J=qw~lDh-
Fig. 5. Variation of & and 4 with load. Non-linear analysis.
-)(b
I
__,p----~---“__--~~,QQ _+__-+---+---_c__ +___+.---+----P----p--_
-WXl
Fig. 6. Variation of Kb and X1 with a/w. Non-linear analysis.
A square plate fixed on all edges, having a central through crack, subjected to a uniformly dis~buted Ioad is considered. The plate dimensions are W = 1 = 20.32 x lo-* M (8”); h = 2.54 x IO-’ M (0.1”) and 4 = 6894.72 N/M* (1.0 psi). a/w ratio varies from 0.3 to 0.7. The results obtained are given in Fig. 4. As can be seen there is good agreement for the stress-intensity factor &, with those of Ref. [9]. The solutions have extended to the non-linear range by increasing the load progressively. and K/@/h * w - da) have been plotted The non-dimensional values of &/(Wh * w * da)
Influence of geometric non-linearity on stress-intensity factors
423
for a/w = 0.3, 0.4, 0.5, 0.6 and 0.7 and non-dimensional load values of (q - w4/Dh) = 300, 400, 500, 600 and 700 in Fig. 6. It can be seen from Fig. 5 that there is significant difference between the values of K,, obtained from linear and non-linear analyses. This difference between the linear and non-linear results increases with increase in a/w. This may be due to the fact that for the same load and for higher value of a/w, the deflection will be larger resulting in greater influence of geometric non-linearity. It is evident from the figure that Kb and K, both increase in a non-linear fashion with increase in external loads. It can be seen from Fig. 6 that the value of Kb increases to its maximum when a/w is around 0.6 and then falls down which may be due to clamp end effect. This happens for all the loads considered in the analysis. Similarly the value of K, reaches its maximum when a/w is around 0.55 and subsequently, the magnitude of EC,decreases. Further analysis should be done in order to combine the influence of shear deformation and geometric non-linearity in order to get a more realistic picture of the influence of geometric non-linearity on stress-intensity factor. REFERENCES [II M. L. Williams, The bending stress distribution at the base of a stationary crack. J. Appl. Mech. Series E, 82, 78-82 (1961). [2] R. J. Hartranft and G. C. Sih, Effect of plate thickness on the bending stress distribution around through cracks. J. cache. Phys. 47,276291 (1968). (3) R. J. Hartranft and G. C. Sih, An approximate 3-D theory of plates with application to crack problems. fnf. 1. Eagng Sci, 8,7f l-729 (1970). [4] G. C. Sih, P. C. Paris and F. Erdogan, Crack-tip Stress-intensity factors for plane extensions and plate bending problems. J. Appt. Mech. 83, 3O6-312(1962). [S] A. S. Day, An introduction to dynamic relaxation. Engineer, London, 219, 219-221(1965). [6] K. R. Rushton, Large deflection of variable thickness plates. Int. J. Mech. Sci. 10, 723-735(1968). [7] A. C. Cassel, Cylindrical shell analysis by dynamic relaxation. Proc. Inst. Ciu. Engrs. 39, 75-84 (1968). [SJ R. S. Alwar and N. R. Rao, Large elastic deformation of clamped, skewed plates by dynamic relaxation. Contput. Structures 4, 381-398(1974). [9] Jalees Ahmed and Francis T. C. Loo, Solution of plate bending problems in Fracture Mechanics using a speciaiised finite element technique. ~~gj~ee~~g F~~c~~~e Mechunjc~T II, 661-673(1979). (l(t) R. J. Hartranft and G. C. Sih, The use of Eigen function expansions in the general solution of Three-Dimensional Crack Problems. J. Muths and Mechanics 19, 123-138(1969). [l I] M. K. Kassir and G. C. Sih, Three-dimensional stress distribution around an elliptical crack under arbitrary loadings. J. Appl. Mech. 33,601-611 (1966). [12] G. Yagawa and T. Nishioka, Finite Element Analysis of Stress Intensity Factors for plane extensions and plate bending problems. ht. .I Numerical Meth. Engng 14, 727-740(1979). (Received 1 June 1981;received for publication 26 June 1981)