INVESTIGATION OF CRACK OPENING IN ISOTROPIC STRAIN HARDENING MATERIAL

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1 INVESTIGATION OF CRACK OPENING IN ISOTROPIC STRAIN HARDENING MATERIAL D. Pustaic ), M. Lovrenic-Jugovic ), H. Wolf ) ) University of Zagre, Faculty of Mechanical Engineering and Naval Architecture, Ivana Lucica 5, Zagre, Croatia ABSTRACT The thin infinite late with an emedded straight crack of a length 2a is considered. The late is uniaxially loaded in its lane with uniformly distriuted continuous load yy = in a direction erendicular to the crack lane. The late material is ductile so the small lastic zones around the crack tis are formed. Also, it is assumed that the late material ossesses a roerty of isotroic strain hardening. The strain hardening of the material is non-linear and it oeys the Ramerg-Osgood relation. The different stages of strain hardening of late material are modeled y varying the arameters α and n in the Ramerg-Osgood's analytical exression. So, it is assumed that the strain hardening exonent n takes the values n = 3, 5, 7,, 25 and. The stress intensity factor (SIF) of the esive stresses K ( a+ r ) is determined y means of Green's functions method. The crack ti and crack centre oening, or any aritrary oint at the crack surface, is determined y means of the dislacement field of the oints lying on the crack surface. The rolem is formulated fully exactly and its solution is looked for y means of commercial software Mathematica. If it is assumed an aearance of small lastic zone around crack ti then it is ossile to find an exact analytical solution exressed in a form of secial gamma-functions Γ( x). If the assumtions aout small lastic zone are not introduced, then it is ossile to give the solution y means of secial hyergeometric functions F αβγ, ; ; z. 2 ( ) KEYWORDS Crack oening, isotroic strain hardening material, small lastic zone, strain hardening exonent, esive stresses, stress intensity factor, method of Green s functions, commercial software Mathematica, gamma-function, hyergeometric function INTRODUCTION The aer deals with an analysis of the dislacements of the oints lying on a surface of straight crack of a length 2a. By means of the calculated dislacements the crack ti oening dislacements (CTOD) and the crack centre oening dislacements (CCOD) are determined. The crack is incororated in a thin infinite late and it is sreaded throughout whole thickness of a late. The late is loaded uniaxially in-lane with uniformly distriuted continuous load yy = in a direction of y-axis, while the crack surface is free of loading. Because a late is thin the lane stress state is assumed, i. e. xx =xx ( xy, ), yy = yy ( xy, ) and xy = xy ( xy, ). There are two axes of symmetry in the late, x and y. The shear stresses xy ( xy, ) at the

2 axes of symmetry equals zero, i.e. ( x,) = and (, y) =. This statement will have as xy a consequence that the normal stresses at those axes will e, at the same time, the rincial stresses: xx ( x,), yy ( x,), xx (, y) and yy (, y). Similar conclusion is valid for a field of the dislacements and it means that it must e: u(, y) =, v(, y) and ux (,), vx (,) =, ut only for x a, while it is vx (,) for a< x< a. As the lastic flow of a material occurs around the crack tis it means that the strain and stress fields will e elastic-lastic nature, excetionally comlex and it is very heavy to descrie them exactly analytical. In this aer we will kee an assumtion that those fields will have a structure of the HRR fields (Hutchinson, Rice, Rosengren). In a case of non-linear elastic late material, similarly to isotroic and non-linear strain hardening material, those fields will have a structure, according to [4], [5] λ+ λ λ = n u r u ( ϕ), ε r ε ( ϕ) and r ( ϕ) () A rank of stress singularity within the crack ti deends on the arameter λ. The value of the arameter λ, for analyzing material, amounts λ = n ( n + ), so, for the structure of the mentioned fields is got ( n ) ( ) r + ( ϕ) and ε r n n + ε ( ϕ). (2) The esive stresses within a lastic zone around crack ti in isotroic strain hardening material will not e constant any more, ut variale, and they will change according to some non-linear law. The non-linear distriution of the esive stresses around the crack ti must have the structure of the fields in accordance with the analytical exression (2). As the exact analytical solution is unknown, one of the ossile aroaches to the rolem is the following. It is ossile to determine the distriution of the esive stresses, for examle, y the finite element method and then that distriution is aroximated with an analytical exression, i. e. with some function, for examle with an exonential function, or with a logarithmic function, or with a hyerolic function and so on. The same aroach was used in the aer [3] and it has shown very well. The authors M.Hoffman and T. Seeger have roosed in their aer [2] the next analytical exression ( ) ( n+ ) x ( ) = r x a (3) for the distriution of the esive stresses. The quantity ( x), in that exression, is a function of two arameters, i.e. the magnitude of the lastic zone around the crack ti r and the strain hardening exonent n. In the article [3] it has een shown that this exression aroximates excellently the distriution of the esive stresses otained y means of the finite element method. The same exression, the authors X. G. Chen, X. R. Wu and M. G. Yan have used in their aer []. In a frame of this aer, instead of the real hysical elastic lunt crack, length of 2a and with a stress singularity within its ti, a fictitious elastic crack, the length of 2= 2( a+ r ) is oserved. The normal stress yy (,) at the ti of that fictitious elastic crack has a final magnitude. The real, hysical lunt crack and the lastic zone around its ti make the shar fictitious elastic crack. A non-singularity stress condition within the ti of fictitious elastic crack is ossile to write analytically in the following way K( a + r ) = K ( a + r ) + K ( a + r ) =. (4) ext The singularity within the ti of a fictitious shar elastic crack, x = = a + r, of the external xy 2

3 load is canceled with the singularity of the esive stresses within the lastic zone. CALCULATION OF DISPLACEMENTS FIELD AROUND THE CRACK TIP BY MEANS OF THE GREEN'S FUNCTIONS On the asis of Dugdale's model the crack ti oening dislacement (CTOD) can e determined on the following way where v D δ D =δext δ = 2v D, (5) denotes vertical dislacement of crack ti. By means of weight functions method (Green's functions) the dislacement v D can e calculated as it follows where 2 2 D( ) v x = Kext( ξ) m( ξ, x)d ξ K( ξ) m( ξ, x)d ξ, E x a m(, ξ x) is a Green's function of stress intensity factor (SIF) and it amounts (6) 2 m(, ξ x) = 2 ξ π( ξ x ), (7) and a a for x < a and a = x for x a. The first integral is easyly calculated y uting in = K ext ( ξ ) = π ξ and the exression (7). So, it is get 2 2 v D.ext ( x ) = K ext ( ξ) m ( ξ, x )dξ= 4 x. E E x The magnitudes of the dislacements within the discrete oints on the crack surface amount: D.ext ( ) = 2, ( ) = 2 v E v a a E and v D.ext ( ) =. The second integral in the D.ext exression (6) gives the dislacements of the oints lying on crack surface from esive stresses. The integral was solved in the aer [] and it can e written D. a E a πe πe a t x 2 4 t 2 v ( x) = K ( ξ) m( ξ, x)d ξ= P( t) I( t) + P( t) x t dt. (9) Within the exression (9) the quantities P(t) and I(t) have the following meaning ( n ( ) + ) n ( ) ( + ) n ( n+ ) Pt () = t ()dt= r t a dt=r n+ n ( t a), () () I() t = ln x + t x t. () For solving the integral (9) the sustitution t = r ξ was introduced which is analogous to sustitution x = r s. It is easyly to notice from the exression () that it is P( a) =, and also, it can e seen, from the exression (), that it is I( ) =, so, it is ossile to simlify the exression (9) and it looks like D. πe a t x t 2 v ( x) = δ ( x) = P( t) x t dt. (2) Including the right side of exression () under a sign of integral and transforming that exression introducing mentioned sustitution it will e got 3

4 n + 2 v ( x) =δ ( x) = r r ( ξ) ( n+ ) n ( n+ ) n ( n+ ) D. π E n ( r ξ):2r x ( r dξ ) = 2 2 r ( s ξ) r ( s ξ ) :2r rξ( 2 rξ) r r ξ n ( n+ ) n + 2r 2 r = x ( ξ) π E n r ( s ) ( s r ) r 2 ξ ξ 2 ξ ξ 2 Finally, if an assumtion aout small lastic zone around the crack ti (S.S.Y. condition) is introduced, then it can e taken r 2 and exression (3) assumes the form d. ξ (3) n ( n+ ) n + ( ξ) 2 vd. ( x) = δ ( x) = r x d. π E n ( s ξ) ξ ξ (4) CRACK TIP OPENING DISPLACEMENT (CTOD) ASSUMING THE SMALL PLASTIC ZONE (S.S.Y.) Presentation of exact analytical solution y means of secial gamma-functions Within the ti of real hysical lunt crack a variale x amounts x= a. When it is included in the sustitution x= r s it follows that the variale s, within that oint, takes the value, i.e. s =. Now, the integral from the equation (4) takes the form n n ( n+ ) ( n ) ( ξ) n+ + 2 ( ) dξ= ξ dξ= ξ ξ dξ (5) ξ ξ ξ and it can e solved analytical exactly. The solution of that tye of integral will e where α β ξ ( ξ) dξ=γ( α+ ) Γ( β+ ) Γ( α+β+ 2) = B( α+, β+ ) Γ( x), (6) stands for gamma-function or Euler's integral of second tye, while Bxy (, ) =Γ( x) Γ( y) Γ ( x+ y) denotes the eta-function or the Euler's integral of first tye. Comaring the integrals within the exressions (5) and (6) we conclude that it is α= 2 and β= ( n + ). It is seen that α+ = 2 and β + = n ( n + ). Now, finally, y comaring with the exression (6), we get the solution of integral (5) ( n+ ) ( ) d ( 2) n ( n ) ( 2) n ( n ) B 2, n ( n ) 2 ξ ξ ξ=γ Γ + Γ + + = +. (7) It needs to e noted that the value of gamma-function amounts Γ ( 2) = π. If so realized value of integral (5) is ut in the equation (4), it is got an exact analytical exression for determination of crack ti oening dislacement from the esive stresses, i.e. 4

5 ( ) ( ) 2 vd. ( a) = δ ( a) = π E n+ n r a ( ) n ( n ) Γ n n+ Γ ( 2) + +. Including the otained solutions () and () in the relation (5) an analytical solution is got for crack ti oening dislacement (CTOD). The gamma-functions aear within that solution and their argument deends on strain hardening exonent n. Furthermore, one more arameter r aears in that analytical solution which is calculated also y means of gammafunctions how it was shown within the aers [3], [4] and [5]. The crack ti oening dislacement δ D ( a) was calculated on the asis of the solutions (), () and (5) in deending of monotonously increasing external load of a late and for the six different values of strain hardening exonent n = 3, 5, 7,, 25 and. The diagram is, in dimensionless form, ηδ D ( a) a = f(, n), shown on the Fig.. What kind of the influence has the strain hardening exonent n on the crack ti oening dislacement δ D ( a) will e anallysed at the chater Review on the otained results. Parameter η means η= E 2. ().4 a) ) c).2. η δ (a).6 n = 3 n = 5 n = 7 n = n = 25 n =.4.2. η δ (a).6 n = 7 analytical numerical.4.2. η δ (a).6 n = analytical numerical Fig. : Deendence of crack ti oening dislacement ηδ D ( a) a on a monotonously increasing external load of a late, for the different values of a strain hardening exonent n a) 2. ) 2. c) 2. n = 3 n = analytical 2.4 numerical 2. n = n = 25 n = analytical numerical η δ ().6.2 η δ ().6 D a.2 η δ () n = 3 n = 5 n = Fig. 2: Deendence of crack centre oening dislacement ηδ D ( x = ) a on a monotonously increasing external load of a late, for the different values of a strain hardening exonent n 5

6 CRACK CENTRE OPENING DISPLACEMENT (CCOD) BY THE ASSUMPTION OF SMALL PLASTIC ZONE AROUND THE CRACK TIP (S.S.Y.) Within the crack centre, a variale x is equal to zero, i.e. x =. When it is included in the sustitution x= r s=, we conclude that the variale s, at that oint, takes the value s= r. As it always is > r, ( = a+ r ), we conclude that the variale s will e greater than one, i.e. s >, for x =. Now, the analytical exression (4) will e equal to n n+ { { ( ) }} n + r =δ = ξ ξ ξ ξ= ( ) 2 vd. () () ( ) r d π E n n + r n n+ { {( ) ( ) }} ( ) = ξ r r ξ ξ π E n ( ) d ξ. Let us introduce the assumtion aout small lastic zone around the crack tis (S.S.Y.). In that case it could e taken that it is r 2 and that memer within the denominator of su integral exression is neglected. The exression (9), now, takes the form ( ) r n n+ n + ( ξ) 2 vd. () = δ () = r dξ= π E n ξ n + r 2 n ( n+ ) = r ξ ξ π E n ( ) d ξ. This tye of integral is same as (5), or (6), resectively, only that the arameters amount: α= 2 and β= n ( n+, ) so it is α + = 2 and ( 2n ) ( n ). can e written that it is ( ) ( ) n ( n ) α (9) (2) and β + = + + Equally, it Γ α+ +β+ =Γ α+β+ 2 =Γ α+ + +β =Γ Hence, the exact analytical solution for crack centre oening dislacement (CCOD) from the esive stresses was given through secial gamma-functions and it looks like ( ) ( ) n ( n ) 2 v () = δ () = π E n+ n r r D. Γ + n n+ Γ Including the otained solutions () and (2) in a relation (5), the analytical solution for the crack centre oening dislacement (CCOD) is otained. An argument of the secial gammafunctions in the solution (2) deends on strain hardening exonent n. Within that solution there are two more hysical quantities r and. The magnitude of lastic zone around the crack ti r also deends on strain hardening exonent n and it, also, can e exressed through the secial gamma-functions as it is shown within the aers [3], [4] and [5]. (2) β OPENING ANY ARBITRARY POINT AT CRACK SURFACE ASSUMING SMALL PLASTIC ZONE AROUND THE CRACK TIP (S.S.Y.) Within the frame of non-linear fracture mechanics it is of interest to analyze the vertical dislacements, or the crack oening, resectively, any aritrary oint at crack surface, i.e. for < x< a. This is esecially imortant to know within an analysis of oening and closing of fatigue crack y its cyclic loading. In order to determine that, we start from the general exression (4) for crack oening from the esive stresses. Let us take the same 6

7 sustitution as reviously, i.e. x= a+ r ( ξ ) = rξ (analogously with x= r s). Consider the discrete oint at crack surface, for examle the one with x= a 2. Within that oint the variale s takes the value s= ( x) r = + a 2r. If we restrict to small lastic zone around the crack ti (S.S.Y. condition), it will e r a. It is seen that the variale s will e greater than one ( s > ), for r a<, while for r a=, the variale s will take the value s = 32. We can conclude that then will e a 2r 2. Under that condition the integral within the solution (4) can e transformed and shown in following form n ( ) ( n+ { ) ( ) n n+ ( s ) } d ( ) ( a 2 ) { ( ) } ξ ξ ξ ξ= ξ ξ ξ + ξ ξ r d. (22) The solution of integral (22) was found y means of commercial software Mathematica, version 5., [9] and it amounts n ( n+ ) { ( ) ( ( a 2r ) ) } ( 2 ) d { ( ) ( 3 2) ( ) a r n n n n } 2F 2, ( 32) + ( +, ) 2 ( + 2 ), n n r a r, ( 23) where 2F 2,, ( 32) + n ( n+, ) 2r ( a+ 2 r ) denotes the hyergeometric function. If the ξ ξ ξ + ξ ξ= π + Γ + + Γ + + solution (23) is ut in the exression (4), the final analytical exression for determination of oening any aritrary oint at crack surface (x= a 2), from the esive stresses, is otained. That exression will e vd. a = δ ( a ) = π( E) ( n+ ) n r ( a+ r) ( a+ r) { n ( n ) ( ) n ( n ) } 2F ( ) n ( n ) r ( a r) 2 ( 2) 3 Γ 32 2,, 32, + + Γ By similar rocedure can e determined the oening of any aritrary oint at the crack surface, < x< a. One should only notice that it is ossile to change the quantity ( a+ 2r ) within the solution (23) with 2r s and 2 r ( a+ 2 r ) = s. So, y means of the relations (4) and (23), it is got ( ) ( ) { ( ) } 2 (24) 2 vd. ( x) =δ ( x) = 2 π E n+ n r s r x (25) Γ + n n+ Γ 32 + n n+ F 2,, 32 + n n+, s. The hyergeometric function F ( ) + n ( n+ ) series exansion as it follows 2 2 2,, 32, s is ossile to develo in a 2 3 αβz α ( +α) β ( +β) z α ( +α )(2 +α) β ( +β )(2 +β) z 4 F( α, β, γ, z) = O( z ). (26) γ 2( γ +γ) 6( γ +γ )(2 +γ) Under an assumtion that the small lastic zone is formed around the crack tis (S.S.Y. condition) it is quite enough to take only first memer of series exansion from the exression (26) and ut it in the solutions (24) and (25), according to [5]. In that case the vertical dislacements of the oints lying on the crack surface will e also small. 7

8 REVIEW ON THE OBTAINED RESULTS The aim of these investigations was to estalish in what manner the isotroic strain hardening of a material influences on the magnitude of the dislacements of the oints lying on the crack surface, or on the crack oening dislacements δ D ( x). An investigation of crack ti oening dislacement (CTOD) and crack centre oening dislacement (CCOD) are of secial interest. By analysing the diagrams it is ossile to conclude: isotroic strain hardening of a material will considerally influences on crack oening dislacement (COD) and how on its ti (CTOD) so on its centre (CCOD), that the crack ti oening dislacement δ D ( a) will e as igger as the strain hardening of a material is smaller, for the same level of external load. Therefore, igger δ D ( a) for igger strain hardening exonent n, that the crack ti oening dislacement δ D ( a), for certain level of external load, will e the largest y the elastic-erfectly lastic material ( n ), the crack centre oening dislacement δ D ( x = ) is not sensitive to a level of isotroic strain hardening of a material (arameter n) at a low level of external load. Namely, it is clearly from the Fig. 2 that all the curves are coincided until aroximately =, 4. That load aroximately corresonds to the limit load at which the small lastic zone around the crack ti will e formed. At the igger loads the crack centre oening dislacement distinctly will deend on a level of isotroic strain hardening of a late material (arameter n) and will e as igger as the strain hardening of a material is smaller. Therefore, igger δ D ( x = ) for igger strain hardening exonent n, y anallysing the magnitude of the dislacements of the oints lying on the crack surface 2 vd( x), according to the solution (25) we conclude that the solution doesn't have to e unamiguous any more. It will deend on how many memers of series exansion we include in a consideration when the hyergeometric function 2F exands in a series, according to (26). To one certain external load of a late it could corresond more different crack oening dislacement (COD). The question is how much are those solutions recise and reliale? If it is taken only the first memer of a series exansion the solution will e unamiguous. CONCLUSION In the analysis of crack oening dislacement (COD) (the arameter of Elastic Plastic Fracture Mechanics - EPFM), in this article, the analytical methods and the commercial software Mathematica were used. The dislacements field around the crack ti in an elastic-lastic continuum, as from an external load of late as well from the esive stresses was determined y means of the method of the Green's functions, exression (6). The stress intensity factor (SIF) of the esive stresses K ( ) is calculated on the same way, [5]. The first and the second way require knowing the esive stresses distriution within a lastic zone. Exactly that is the iggest unknown. In this aer we have assumed that the esive stresses are distriuted according to the analytical exression (3), [], [2]. That exression turned out to e good and very recise aroximation of a real stress distriution around the crack ti. The exression (3) has suffered the many criticisms of the scientific and researching circles. Neverthless, many authors have used it in their investigations, as for examle [], [4], [3], and so on.

9 Within the further investigations it will e suggested to measure the crack oening dislacement (COD) and comare the results with the analytical solutions, in order to estalish how recise and reliale are the results, we have otained y analytical way. Also, it is laned, the investigations extend on the crack within the thick late in which there are the triaxial stress state (state of lane strain) and to investigate what is the kind of influence of stress 33 on the crack oening dislacement (COD) and on the magnitude of lastic zone around the crack ti r. REFERENCES [] Chen, X.G.; Wu, X.R.; Yan, M.G.: Dugdale Model for Strain Hardening Materials Engineering Fracture Mechanics 4 (992) No. 6, [2] Hoffman, M.; Seeger, T.: Dugdale Solutions for Strain Hardening Materials Proceedings of the Worksho on the CTOD Methodology, Geesthacht (95), [3] Pustaic, D.; Lovrenic, M.: Analytical and Numerical Investigation of Crack Oening in Strain Hardening Material Proceedings of The 5 th International Congress of Croatian Society of Mechanics and CD-ROM, Trogir, Croatia (26) [4] Pustaic, D.; Lovrenic-Jugovic, M.: Magnitude of Plastic Zone around the Crack Ti in Isotroic Strain Hardening Material The article reviewed and acceted for ulishing in the journal Strojarstvo, Journal for the Theory and Alication in Mechanical Engineering (29) No. 5 [5] Pustaic, D.; Lovrenic-Jugovic, M.: Aout Unamiguity and Reliaility of the Solution for Plastic Zone Magnitude around Crack Ti in Isotroic Strain Hardening Material Proceedings of The 7 th Euroean Conference on Fracture (ECF 7) and CD-ROM, Editors: Pokluda, J. et all, VUTIUM Brno, Brno, Czech Reulic (2) [6] Pustaic, D.; Stok, B.: Crack Ti Plasticity Investigations using Dugdale Stri Yield Model Aroach Proceedings of The th Euroean Conference on Fracture (ECF ) Editors: Petit, J. et all, (EMAS), Poitiers-Futuroscoe, (996), Vol. I, [7] Pustaic, D.; Stok, B.: Some Critical Remarks on the Dugdale Stri Yield Model for Crack Ti Plasticity Proceedings of The 2 th Euroean Conference on Fracture (ECF 2) Editors: Brown, M.W. et all, (EMAS), Sheffield, (99), Vol. II, [] Stok, B.; Pustaic, D.: On the Influence of the Loads acting Parallel to the Crack Surface on the Crack Ti Oening Dislacement (CTOD) and on the Magnitude of the Plastic Zone around the Crack Ti Strojarstvo, Journal for the Theory and Alication in Mechanical Engineering, 3 (996) No. 2/3, [in Croatian] [9] Wolfram, S.: The Mathematica Book, 4 th ed. Wolfram Media/Camridge University Press, Camridge, 999 Corresonding author: dragan.ustaic@fs.hr 9