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Calculation-experimental method for determining crack resistance of materials under biaxial loading
Engineering Fracture Mechanics 64 (1999) 203±215
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Calculation-experimental method for determining crack resistance of materials under biaxial loading A.V. Boiko*, L.N. Karpenko, A.A. Lebedev, N.R. Muzika, O.V. Zagornyak Institute for Problems of Strength, National Academy of Sciences of Ukraine, 2 Timiryazevskaya str., Kiev, 252014, Ukraine Received 21 May 1997; received in revised form 8 February 1999; accepted 12 June 1999
Abstract The paper presents new equipment and method for studying various aspects of crack resistance of sheet structural materials under biaxial static and cyclic loading. A procedure is proposed and substantiated for processing the test results for two types of specimens with cracks for the cases of brittle and elasto±plastic fracture. Experimental results are presented which illustrated the application of the proposed method. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Sheet materials; Crack resistance; Stress intensity factor; Crack tip opening displacement; Biaxial loading
1. Introduction The paper presents new equipment and method for studying various aspects of structural sheet materials crack resistance under biaxial static and cyclic loading. The idea of the method is in the establishment of the relation between crack resistance parameters and the stress±strain state at the crack tip which may be determined by using the values of displacements on the boundary of some area surrounding it. This approach is the result of practical considerations and it diers from the conventional ones (e.g., see survey [1]) when the stress±strain state at the crack tip is related to the stresses at in®nity. Indeed, if a crack is initiated in a structure, it is unlikely that an area of the material uniform stress state could be found around it, which would correspond to the model of an in®nite plane, due to the presence of various technological stress concentrators and other design factors. At the same time, it is practically * Corresponding author. Fax: +380-7296-1684. 0013-7944/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 0 6 4 - 8
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always possible to measure displacements on the boundary of a comparatively small area surrounding the crack. And in this case in order to predict the crack behaviour in a structure, one should determine the material crack resistance under biaxial loading by the displacements adequate to those occurring in the structure. Thus the feasibility of the realization of the approach proposed is obvious and it predetermines the development of speci®c experimental devices, as well as respective mathematical models for the interpretation of the experimental data. 2. Specimens for biaxial loading A simple method for inducing biaxial tension in the material can be realized by means of a special disk-shaped specimen [2]. Figs. 1 and 2 present a drawing of the specimen and its test
Fig. 1. Disc specimen.
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Fig. 2. Scheme of testing a disc specimen.
loading scheme, respectively. The testing is conducted in the following way. Specimen 1 is placed with its conical surface on the round support 2 and is uniformly loaded along the inner edge of the rim by means of the punch 3. The rim gets buckled to trust with its inner surface against the punch circular recess and the specimen working section 4 is subjected to a uniform biaxial tension. The described principle can be used for testing as-received sheet materials with specimens of simple geometrical shape [3]. For this purpose one should separate the disk-specimen working section and its rim. In this case the rim plays the role of a reusable ®xture of the plane circular specimen 2 (Fig. 3) ®xed by means of pins arranged round the perimeter of the ®xture. In order to reduce the loads applied to the ®xture, the latter is composed of separate sectors made of a high-strength material. During the specimen assembly the sectors are arranged symmetrically with respect to the punch axis owing to centring capability of the punch cylindrical projection. A scheme of the device for testing plane specimens in biaxial tension is shown in Fig. 4. It comprises the aligned punch 1 with cylindrical recess 2 and support 3 with ®xture 4 of specimen 5. The ®xture consists of separate sectors with pins 6 (arranged in parallel to the axis of punch 1) for fastening the specimen with nuts 7. The outer edges of the sectors have conical surface 8 and are arranged with respect to punch 1 so that their inner surface 9 can come into contact with cylindrical surface of recess 2 of punch 1. A kinematic analysis of the plane specimen deformation process is performed which con®rms the correctness of the elaborated experimental method and its applicability in the practice of the materials mechanical testing. In particular, it has been found that under a 3% tensile strain (such a strain makes a crack specimen to reach its limit state for a wide class of structural sheet materials) the bending induced strain in 0.6% of the tensile strain. Besides, the warping of the specimen which is due to the dierence in the ®xture support arms is as small as several thousandths or hundredth of a mm, i.e. is a negligible magnitude.
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Fig. 3. Plane specimen.
3. Testing device for specimens The schemes of specimens loading allow the material testing in uniform biaxial tension to be realized. However, of considerable interest are also the experiments in biaxial tension other than the uniform one. It has been shown earlier that the variation of the ®xture support arm size is an eective means to achieve the condition of biaxial proportional loading with the prescribed biaxiality parameter in the specimen working section. The device [4] for the realization of the above approach is shown in Fig. 5. It comprises aligned punch 1 and support 2 with ®xture 3 of specimen 4 between them. The ®xture is made as a truncated cone consisting of separate sectors 5 with radial arrangement in respect to punch 1 whose tapered surface 6 can come into contact with supporting elements 7. Supporting elements 7 are made of equal height and positioned in guiding recesses 8 and can be moved in radial direction and clamped in position. Guiding recesses 8 for the installation of supporting elements 7 are made along the generating surface 9 of support 2. The quantity of recesses 8 corresponds to the number of sectors 5 in ®xture 3. The working surfaces of the sectors and the support are made equidistant that provides the contact of each supporting element (of equal height) with the tapered surface of sectors 5 of ®xture 3 when the supporting elements are positioned in guiding recesses 8 of support 2 at a dierent distance from the axis of punch 1. The plane of the larger base 11 of ®xture 3 is mating with punch 1 while the smaller base 12 is intended for fastening specimen 4 with bolts 13.
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Fig. 4. A device for testing plane specimens in biaxial tension.
The device works in the following way. Specimen 4 of the as-received sheet material is fastened with bolts 13 to ®xture 3, which is then placed on supporting elements 7 that are positioned in guiding recesses 8 of support 2. For the prescribed displacements distribution on the boundary of the plane specimen working section, the positioning of the supporting elements on the support is determined on the basis of the results of the performed kinematic analysis. In this case the larger the distance ratio l2/l1, the larger the displacements ratio w1/w2, where w1 and w2 are corresponding radial displacements of the specimen working section boundary. Then ®xture 3 together with support 2 are placed between the plates of a press or a pulsator and the loading is carried out by means of punch 1, as a result of which the specimen 4 is subjected to biaxial tension with a speci®ed biaxiality parameter which is assumed to be the w1/w2 ratio.
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Fig. 5. A device for testing plane specimens in biaxial tension.
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4. Calculation schemes Parameters which characterize the loading of a disc (Fig. 1) or plane (Fig. 3) specimen are radial displacements of its working section boundary which can be measured by means of tensometers or other means, e.g., optically. Considering the geometry of the specimen working section, it is feasible to formulate the problem of theoretical determination of the load carrying capacity for a disc or plane specimen with a crack as a problem of the elasticity or plasticity theory for a crack in a round disc with speci®ed radial displacements wr (y ) of the circular boundary points whose distribution is described by the following law: wr y w1 cos 2 y w2 sin 2 y,
1
where y is the polar angle, w1 and w2 are the radial displacements along the abscissa and ordinate axes, respectively, which intersect in the centre of the circle. At small values of w1 and w2, distribution (1) does not practically dier from elliptical. The crack is oriented along abscissa axis. The elasticity theory problem for a circle of radius R with a straight-line central crack of 2 l length with prescribed radial displacements on its boundary (1) is reduced to the following singular integral equation to be solved for the unknown function g(t ) proportional to the derivative of the displacement discontinuity on the crack faces [5]: 1 l g t 1 l dt S to , tg tdt G to , ÿ l < to < l, 2 pi ÿl t ÿ to 2pi ÿl where S to , t
4t 1 k2 7t ÿ 6to 2 t t ÿ to 2t ÿ 3to 2 t2 to t ÿ to 2 ÿ ÿ , R2 ÿ tto k k ÿ 1R2 k k k R2 ÿ tto 3 R2 ÿ tto 2
2m w1 w2 2 b 3b 6b 2 G to ÿ ÿ ÿ t , R k ÿ 1 2 2k kR2 o 2
3
4
b w1 ÿ w2 = w1 w2 , m is the shear modulus, k=3ÿ4n for plane strain and k=(3ÿn )/(1+n ) for the generalized plane stress state, n is the Poisson ratio. The numerical and approximate analytical solutions of Eq. (2) have been obtained [5]. On the basis of the latter, the expression for a dimensionless stress intensity factor at the crack tip is written in the form p p k1 R k ÿ 1 kÿ1 pl 2 1ÿb k 3 ÿ 6l , 5 4k 2m w1 w2 1 Zl2 where l=l/R,
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Z
k3 ÿ k2 7k ÿ 3 : 4k k ÿ 1
6
Comparison of the numerical and approximate analytical calculations reveals that Eq. (5) provides a fairly high accuracy of the calculation at l R 0.5. Fig. 6a gives the dependence of the dimensionless stress intensity factor on the parameter w1/w2 for k=(3ÿn )/(1+n ), n=0.3. Fig. 6b shows the dependence of limiting displacements on the relative crack dimension for dierent values of w1/w2. Here w1 , w2 and Kc are the critical values of w1, w2 and k1 respectively. It is obvious that the tendency to the stable crack growth reduces with an increase in w1/w2. The elasto±plastic problem for a round plate of elastic±ideally plastic material with a central Dugdale's crack [6] of length 2 l with speci®ed radial displacements on the plate boundary (1) is reduced to the following singular integral equation [7]: 1 l1 g t 1 l1 dt S to , tg tdt F to , 7 pi ÿl1 t ÿ to 2pi ÿl1
Fig. 6. Dependences of the dimensionless stress intensity factor on the w1/w2 parameter (a) and of limiting displacements on the relative crack size (b).
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where F to
sp G to , G to ,
l
Functions S(to, t ) and G(to) are de®ned by Eqs. (3) and (4), l1=l+a, a is the plastic zone size at the crack tip determined from the solution of the problem; sp is the material yield strength. The numerical and approximate analytical solutions of Eq. (7) have been obtained [7]. On the basis of the latter, the expression for the plastic zone is written in the form: p p kÿ1 p 2 2 2 1ÿb k 3 ÿ 6l , 8 arccos s l Zs 1 ÿ s 2 4k sp where p 2m w1 w2 = k ÿ 1R,
l l1 =R,
the expressions for b and Z are determiend in accordance with Eqs. (4) and (6), s=l/l1=1ÿa/l1. The crack tip opening displacement related to l1, is de®ned from the formula 2 0 q p 2 ÿ s 1 ÿ t2 j 1 ÿ s j t 1 1 ksp 6 1 1B 2k ÿ 1 p q ÿ s d t1 l ÿ t21 3=2 @t1 ln 4 ÿ bl p k sp p 2m j t1 1 ÿ s2 s 1 ÿ t21 j q p 1 3 1 ÿ t21 ÿ 1 ÿ s2 j C 7 ln q p A 5, j l ÿ t21 1 ÿ s2 j j
where t1=t/l1. The value of d(t1) at the point t1=s (crack tip) is of practical interest: 1 ksp 2 1 2k ÿ 1 p 2 3=2 : 1 ÿ s s ln ÿ bl d s k sp p s 2m
9
10
Comparison of the numerical and approximate analytical calculation reveals that Eqs. (8) and (10) ensure an appreciably high accuracy of the calculations at l < 0.5. Of certain interest for the analysis and interpretation of the experimental results is the distribution of local stresses sx acting along the crack in the region adjacent to the crack tip. As is known [8], the crack model used is valid at stresses sx which in the plastic zone at the crack tip satisfy the condition sx =sy R1:
11
On the basis of the results presented in [7], the expression for local stresses sx is written in the form
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" p 4sp kÿ1 3bl2 2 s 1 ÿ a2 tÿa p arccos s t 1 sx t sp 2p arccos p ÿ 2 2 k 2 kp 1ÿa s t 1 ÿ a2 1 ÿ a2 p # 2l2 sp 1 at ÿ 2 t ÿ t w t 2sp 1 w t a 1 ÿ ats 1 ÿ s2 p dt ÿ p ÿ kp ÿ1 at ÿ 12 p ÿ1 t ÿ t 1 ÿ t2 t 1 ÿ a2 1 ÿ a2 s2
dt p , 1 ÿ t2 ÿ 1 < t < 1, where w(t ) is determined in [7], t= 21 corresponds to points w= 2l1, a=l 2t, the other parameters are described in formulas (7) and (8). We shall use the stresses at the crack tip sx=sx (s ) in condition (11). The calculation error which in all cases has never exceeded 0.1% was veri®ed by comparing the calculation results at dierent accuracies of algebraic approximation of integral Eqs. (2) and (7) [5,7].
5. Experimental results Let us consider some experiments which illustrate the application of the preceding method. Disc specimens (Fig. 1) were used successfully to study the ultimate state of elastic plates weakened by the cracks of complex shape under uniform biaxial tension. Those results are presented completely in [2]. Elastoplastic deformation of plates with cracks in biaxial tension including nonuniform one was studied using plane 1.5 mm thick specimens (Fig. 3) of aluminium alloy. Central linear cuts of length 10, 20 and 30 mm oriented along one of the main axes of the specimen deformation were made by electric-spark technique that made it possible to get the slits with the curvature radius at the tips of 0.03 mm. Mechanical properties of the aluminium alloy were determined using a standard method and were: E = 71,500 MPa, n=0.32, sp=320 MPa. The anisotropy of the material mechanical properties did not exceed 1%. The experiments were performed with the prescribed displacements on the boundary of the specimen working: section in accordance with Eq. (1) for dierent w1/w2 ratios along the main axes of the specimen deformation. The limiting w1 and w2 values of displacements corresponding to the crack growth onset moment were measured on the basis of 2R = 50 mm. The specimen loading rate was 100 N/s. The procedure of the experimental results processing is as follows. First the dependence of the relative crack tip opening displacement D=2md(s )l/[(1+k )sp] on the parameter c(w1+w2)/ R (where c is the dimensionless constant 2m/[(kÿ1)sp]) is determined by computation (see Section 4). Then, using the results obtained we de®ne D which corresponds to the crack
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213
growth onset instant for displacements w1 and w2 determined from the experiment. This makes it possible to ®nd the dependence of the critical value of the absolute crack tip opening displacement dA on the w1/w2 parameter (Fig. 7). These data are useful for the evaluation of the structural element ultimate state within the framework of the conception formulated in the Introduction. The following step is the construction of the calculated dependence of sx/sy (the local stress ratio at the crack tip) on the parameter c(w1+w2)/R. Using these results and the experimental w1 and w2 values, we ®nd the relationship between the critical value of the absolute crack tip opening displacement dA and the sx/sy parameter (Fig. 8). The above dependence characterizes the crack growth resistance of the tested thin-sheet material of the given thickness. 6. Conclusion The proposed method based on the establishment of relationship between the stress±strain state at the crack tip and the displacements on the boundary of the area surrounding it appears to be promising from the standpoint of the maximum approach (as to the type of the stress state) of the laboratory test conditions to real service conditions of a structural material.
Fig. 7. Dependence of the critical crack tip opening displacement on the w1/w2 parameter.
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Fig. 8. Dependence of the crack tip opening displacement on the normal stress ratio at the crack tip.
In addition, the advantage of the equipment developed over the known one is in the possibility of biaxial loading of specimens using the simplest types of testing machines: presses for static loads and pulsators for cyclic loads. This simpli®es the experiment appreciably and makes it cheaper, allows the experiment to be performed in practically any laboratory engaged in mechanical testing of materials.
References [1] Moyer ET, Liebowitz H. Biaxial load eects in the mechanics of fracture. Journ of AERO Soc of India 1984;36:221±36. [2] Boiko AV, Karpenko LN. Cracks in a round plate (specimen and calculation method). Engng Fracture Mech 1984;20:489±99. [3] Lebedev AA, Boiko AV, Muzika NR. A device for testing materials in biaxial tension (in Russian). Author's certi®cate 1572199, USSR, MKI G 01 N3/08. Otkrytiya. lzobreteniya, N22, 273, 1990. [4] Boiko AV. A device for testing materials in biaxial tension (in Russian). Author's certi®cate 1637513, USSR, MKI G 01 N3/08. Otkrytiya. izobreteniya, N22, 193, 1990. [5] Boiko AV. Some eects of biaxial tension of an elastic plate with a crack. Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 1991;N6:142±7 [in Russian]. [6] Dugdale DC. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8:100±4.
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[7] Boiko AV. Some eects of biaxial elastoplastic deformation of a plate with a crack. Problemy Prochnosti 1990;N6:20±7 [in Russian]. [8] Alpa G, Bozzo E, Gambarotta L. Validity limits of the Dugdale model for thin cracked plates under biaxial loading. Engng Fracture Mech 1979;12:523±9.
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Calculation-experimental method for determining crack resistance of materials under biaxial loading A.V. Boiko*, L.N. Karpenko, A.A. Lebedev, N.R. Muzika, O.V. Zagornyak Institute for Problems of Strength, National Academy of Sciences of Ukraine, 2 Timiryazevskaya str., Kiev, 252014, Ukraine Received 21 May 1997; received in revised form 8 February 1999; accepted 12 June 1999
Abstract The paper presents new equipment and method for studying various aspects of crack resistance of sheet structural materials under biaxial static and cyclic loading. A procedure is proposed and substantiated for processing the test results for two types of specimens with cracks for the cases of brittle and elasto±plastic fracture. Experimental results are presented which illustrated the application of the proposed method. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Sheet materials; Crack resistance; Stress intensity factor; Crack tip opening displacement; Biaxial loading
1. Introduction The paper presents new equipment and method for studying various aspects of structural sheet materials crack resistance under biaxial static and cyclic loading. The idea of the method is in the establishment of the relation between crack resistance parameters and the stress±strain state at the crack tip which may be determined by using the values of displacements on the boundary of some area surrounding it. This approach is the result of practical considerations and it diers from the conventional ones (e.g., see survey [1]) when the stress±strain state at the crack tip is related to the stresses at in®nity. Indeed, if a crack is initiated in a structure, it is unlikely that an area of the material uniform stress state could be found around it, which would correspond to the model of an in®nite plane, due to the presence of various technological stress concentrators and other design factors. At the same time, it is practically * Corresponding author. Fax: +380-7296-1684. 0013-7944/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 0 6 4 - 8
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always possible to measure displacements on the boundary of a comparatively small area surrounding the crack. And in this case in order to predict the crack behaviour in a structure, one should determine the material crack resistance under biaxial loading by the displacements adequate to those occurring in the structure. Thus the feasibility of the realization of the approach proposed is obvious and it predetermines the development of speci®c experimental devices, as well as respective mathematical models for the interpretation of the experimental data. 2. Specimens for biaxial loading A simple method for inducing biaxial tension in the material can be realized by means of a special disk-shaped specimen [2]. Figs. 1 and 2 present a drawing of the specimen and its test
Fig. 1. Disc specimen.
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205
Fig. 2. Scheme of testing a disc specimen.
loading scheme, respectively. The testing is conducted in the following way. Specimen 1 is placed with its conical surface on the round support 2 and is uniformly loaded along the inner edge of the rim by means of the punch 3. The rim gets buckled to trust with its inner surface against the punch circular recess and the specimen working section 4 is subjected to a uniform biaxial tension. The described principle can be used for testing as-received sheet materials with specimens of simple geometrical shape [3]. For this purpose one should separate the disk-specimen working section and its rim. In this case the rim plays the role of a reusable ®xture of the plane circular specimen 2 (Fig. 3) ®xed by means of pins arranged round the perimeter of the ®xture. In order to reduce the loads applied to the ®xture, the latter is composed of separate sectors made of a high-strength material. During the specimen assembly the sectors are arranged symmetrically with respect to the punch axis owing to centring capability of the punch cylindrical projection. A scheme of the device for testing plane specimens in biaxial tension is shown in Fig. 4. It comprises the aligned punch 1 with cylindrical recess 2 and support 3 with ®xture 4 of specimen 5. The ®xture consists of separate sectors with pins 6 (arranged in parallel to the axis of punch 1) for fastening the specimen with nuts 7. The outer edges of the sectors have conical surface 8 and are arranged with respect to punch 1 so that their inner surface 9 can come into contact with cylindrical surface of recess 2 of punch 1. A kinematic analysis of the plane specimen deformation process is performed which con®rms the correctness of the elaborated experimental method and its applicability in the practice of the materials mechanical testing. In particular, it has been found that under a 3% tensile strain (such a strain makes a crack specimen to reach its limit state for a wide class of structural sheet materials) the bending induced strain in 0.6% of the tensile strain. Besides, the warping of the specimen which is due to the dierence in the ®xture support arms is as small as several thousandths or hundredth of a mm, i.e. is a negligible magnitude.
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Fig. 3. Plane specimen.
3. Testing device for specimens The schemes of specimens loading allow the material testing in uniform biaxial tension to be realized. However, of considerable interest are also the experiments in biaxial tension other than the uniform one. It has been shown earlier that the variation of the ®xture support arm size is an eective means to achieve the condition of biaxial proportional loading with the prescribed biaxiality parameter in the specimen working section. The device [4] for the realization of the above approach is shown in Fig. 5. It comprises aligned punch 1 and support 2 with ®xture 3 of specimen 4 between them. The ®xture is made as a truncated cone consisting of separate sectors 5 with radial arrangement in respect to punch 1 whose tapered surface 6 can come into contact with supporting elements 7. Supporting elements 7 are made of equal height and positioned in guiding recesses 8 and can be moved in radial direction and clamped in position. Guiding recesses 8 for the installation of supporting elements 7 are made along the generating surface 9 of support 2. The quantity of recesses 8 corresponds to the number of sectors 5 in ®xture 3. The working surfaces of the sectors and the support are made equidistant that provides the contact of each supporting element (of equal height) with the tapered surface of sectors 5 of ®xture 3 when the supporting elements are positioned in guiding recesses 8 of support 2 at a dierent distance from the axis of punch 1. The plane of the larger base 11 of ®xture 3 is mating with punch 1 while the smaller base 12 is intended for fastening specimen 4 with bolts 13.
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Fig. 4. A device for testing plane specimens in biaxial tension.
The device works in the following way. Specimen 4 of the as-received sheet material is fastened with bolts 13 to ®xture 3, which is then placed on supporting elements 7 that are positioned in guiding recesses 8 of support 2. For the prescribed displacements distribution on the boundary of the plane specimen working section, the positioning of the supporting elements on the support is determined on the basis of the results of the performed kinematic analysis. In this case the larger the distance ratio l2/l1, the larger the displacements ratio w1/w2, where w1 and w2 are corresponding radial displacements of the specimen working section boundary. Then ®xture 3 together with support 2 are placed between the plates of a press or a pulsator and the loading is carried out by means of punch 1, as a result of which the specimen 4 is subjected to biaxial tension with a speci®ed biaxiality parameter which is assumed to be the w1/w2 ratio.
208
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Fig. 5. A device for testing plane specimens in biaxial tension.
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209
4. Calculation schemes Parameters which characterize the loading of a disc (Fig. 1) or plane (Fig. 3) specimen are radial displacements of its working section boundary which can be measured by means of tensometers or other means, e.g., optically. Considering the geometry of the specimen working section, it is feasible to formulate the problem of theoretical determination of the load carrying capacity for a disc or plane specimen with a crack as a problem of the elasticity or plasticity theory for a crack in a round disc with speci®ed radial displacements wr (y ) of the circular boundary points whose distribution is described by the following law: wr y w1 cos 2 y w2 sin 2 y,
1
where y is the polar angle, w1 and w2 are the radial displacements along the abscissa and ordinate axes, respectively, which intersect in the centre of the circle. At small values of w1 and w2, distribution (1) does not practically dier from elliptical. The crack is oriented along abscissa axis. The elasticity theory problem for a circle of radius R with a straight-line central crack of 2 l length with prescribed radial displacements on its boundary (1) is reduced to the following singular integral equation to be solved for the unknown function g(t ) proportional to the derivative of the displacement discontinuity on the crack faces [5]: 1 l g t 1 l dt S to , tg tdt G to , ÿ l < to < l, 2 pi ÿl t ÿ to 2pi ÿl where S to , t
4t 1 k2 7t ÿ 6to 2 t t ÿ to 2t ÿ 3to 2 t2 to t ÿ to 2 ÿ ÿ , R2 ÿ tto k k ÿ 1R2 k k k R2 ÿ tto 3 R2 ÿ tto 2
2m w1 w2 2 b 3b 6b 2 G to ÿ ÿ ÿ t , R k ÿ 1 2 2k kR2 o 2
3
4
b w1 ÿ w2 = w1 w2 , m is the shear modulus, k=3ÿ4n for plane strain and k=(3ÿn )/(1+n ) for the generalized plane stress state, n is the Poisson ratio. The numerical and approximate analytical solutions of Eq. (2) have been obtained [5]. On the basis of the latter, the expression for a dimensionless stress intensity factor at the crack tip is written in the form p p k1 R k ÿ 1 kÿ1 pl 2 1ÿb k 3 ÿ 6l , 5 4k 2m w1 w2 1 Zl2 where l=l/R,
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Z
k3 ÿ k2 7k ÿ 3 : 4k k ÿ 1
6
Comparison of the numerical and approximate analytical calculations reveals that Eq. (5) provides a fairly high accuracy of the calculation at l R 0.5. Fig. 6a gives the dependence of the dimensionless stress intensity factor on the parameter w1/w2 for k=(3ÿn )/(1+n ), n=0.3. Fig. 6b shows the dependence of limiting displacements on the relative crack dimension for dierent values of w1/w2. Here w1 , w2 and Kc are the critical values of w1, w2 and k1 respectively. It is obvious that the tendency to the stable crack growth reduces with an increase in w1/w2. The elasto±plastic problem for a round plate of elastic±ideally plastic material with a central Dugdale's crack [6] of length 2 l with speci®ed radial displacements on the plate boundary (1) is reduced to the following singular integral equation [7]: 1 l1 g t 1 l1 dt S to , tg tdt F to , 7 pi ÿl1 t ÿ to 2pi ÿl1
Fig. 6. Dependences of the dimensionless stress intensity factor on the w1/w2 parameter (a) and of limiting displacements on the relative crack size (b).
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where F to
sp G to , G to ,
l
Functions S(to, t ) and G(to) are de®ned by Eqs. (3) and (4), l1=l+a, a is the plastic zone size at the crack tip determined from the solution of the problem; sp is the material yield strength. The numerical and approximate analytical solutions of Eq. (7) have been obtained [7]. On the basis of the latter, the expression for the plastic zone is written in the form: p p kÿ1 p 2 2 2 1ÿb k 3 ÿ 6l , 8 arccos s l Zs 1 ÿ s 2 4k sp where p 2m w1 w2 = k ÿ 1R,
l l1 =R,
the expressions for b and Z are determiend in accordance with Eqs. (4) and (6), s=l/l1=1ÿa/l1. The crack tip opening displacement related to l1, is de®ned from the formula 2 0 q p 2 ÿ s 1 ÿ t2 j 1 ÿ s j t 1 1 ksp 6 1 1B 2k ÿ 1 p q ÿ s d t1 l ÿ t21 3=2 @t1 ln 4 ÿ bl p k sp p 2m j t1 1 ÿ s2 s 1 ÿ t21 j q p 1 3 1 ÿ t21 ÿ 1 ÿ s2 j C 7 ln q p A 5, j l ÿ t21 1 ÿ s2 j j
where t1=t/l1. The value of d(t1) at the point t1=s (crack tip) is of practical interest: 1 ksp 2 1 2k ÿ 1 p 2 3=2 : 1 ÿ s s ln ÿ bl d s k sp p s 2m
9
10
Comparison of the numerical and approximate analytical calculation reveals that Eqs. (8) and (10) ensure an appreciably high accuracy of the calculations at l < 0.5. Of certain interest for the analysis and interpretation of the experimental results is the distribution of local stresses sx acting along the crack in the region adjacent to the crack tip. As is known [8], the crack model used is valid at stresses sx which in the plastic zone at the crack tip satisfy the condition sx =sy R1:
11
On the basis of the results presented in [7], the expression for local stresses sx is written in the form
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" p 4sp kÿ1 3bl2 2 s 1 ÿ a2 tÿa p arccos s t 1 sx t sp 2p arccos p ÿ 2 2 k 2 kp 1ÿa s t 1 ÿ a2 1 ÿ a2 p # 2l2 sp 1 at ÿ 2 t ÿ t w t 2sp 1 w t a 1 ÿ ats 1 ÿ s2 p dt ÿ p ÿ kp ÿ1 at ÿ 12 p ÿ1 t ÿ t 1 ÿ t2 t 1 ÿ a2 1 ÿ a2 s2
dt p , 1 ÿ t2 ÿ 1 < t < 1, where w(t ) is determined in [7], t= 21 corresponds to points w= 2l1, a=l 2t, the other parameters are described in formulas (7) and (8). We shall use the stresses at the crack tip sx=sx (s ) in condition (11). The calculation error which in all cases has never exceeded 0.1% was veri®ed by comparing the calculation results at dierent accuracies of algebraic approximation of integral Eqs. (2) and (7) [5,7].
5. Experimental results Let us consider some experiments which illustrate the application of the preceding method. Disc specimens (Fig. 1) were used successfully to study the ultimate state of elastic plates weakened by the cracks of complex shape under uniform biaxial tension. Those results are presented completely in [2]. Elastoplastic deformation of plates with cracks in biaxial tension including nonuniform one was studied using plane 1.5 mm thick specimens (Fig. 3) of aluminium alloy. Central linear cuts of length 10, 20 and 30 mm oriented along one of the main axes of the specimen deformation were made by electric-spark technique that made it possible to get the slits with the curvature radius at the tips of 0.03 mm. Mechanical properties of the aluminium alloy were determined using a standard method and were: E = 71,500 MPa, n=0.32, sp=320 MPa. The anisotropy of the material mechanical properties did not exceed 1%. The experiments were performed with the prescribed displacements on the boundary of the specimen working: section in accordance with Eq. (1) for dierent w1/w2 ratios along the main axes of the specimen deformation. The limiting w1 and w2 values of displacements corresponding to the crack growth onset moment were measured on the basis of 2R = 50 mm. The specimen loading rate was 100 N/s. The procedure of the experimental results processing is as follows. First the dependence of the relative crack tip opening displacement D=2md(s )l/[(1+k )sp] on the parameter c(w1+w2)/ R (where c is the dimensionless constant 2m/[(kÿ1)sp]) is determined by computation (see Section 4). Then, using the results obtained we de®ne D which corresponds to the crack
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growth onset instant for displacements w1 and w2 determined from the experiment. This makes it possible to ®nd the dependence of the critical value of the absolute crack tip opening displacement dA on the w1/w2 parameter (Fig. 7). These data are useful for the evaluation of the structural element ultimate state within the framework of the conception formulated in the Introduction. The following step is the construction of the calculated dependence of sx/sy (the local stress ratio at the crack tip) on the parameter c(w1+w2)/R. Using these results and the experimental w1 and w2 values, we ®nd the relationship between the critical value of the absolute crack tip opening displacement dA and the sx/sy parameter (Fig. 8). The above dependence characterizes the crack growth resistance of the tested thin-sheet material of the given thickness. 6. Conclusion The proposed method based on the establishment of relationship between the stress±strain state at the crack tip and the displacements on the boundary of the area surrounding it appears to be promising from the standpoint of the maximum approach (as to the type of the stress state) of the laboratory test conditions to real service conditions of a structural material.
Fig. 7. Dependence of the critical crack tip opening displacement on the w1/w2 parameter.
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Fig. 8. Dependence of the crack tip opening displacement on the normal stress ratio at the crack tip.
In addition, the advantage of the equipment developed over the known one is in the possibility of biaxial loading of specimens using the simplest types of testing machines: presses for static loads and pulsators for cyclic loads. This simpli®es the experiment appreciably and makes it cheaper, allows the experiment to be performed in practically any laboratory engaged in mechanical testing of materials.
References [1] Moyer ET, Liebowitz H. Biaxial load eects in the mechanics of fracture. Journ of AERO Soc of India 1984;36:221±36. [2] Boiko AV, Karpenko LN. Cracks in a round plate (specimen and calculation method). Engng Fracture Mech 1984;20:489±99. [3] Lebedev AA, Boiko AV, Muzika NR. A device for testing materials in biaxial tension (in Russian). Author's certi®cate 1572199, USSR, MKI G 01 N3/08. Otkrytiya. lzobreteniya, N22, 273, 1990. [4] Boiko AV. A device for testing materials in biaxial tension (in Russian). Author's certi®cate 1637513, USSR, MKI G 01 N3/08. Otkrytiya. izobreteniya, N22, 193, 1990. [5] Boiko AV. Some eects of biaxial tension of an elastic plate with a crack. Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 1991;N6:142±7 [in Russian]. [6] Dugdale DC. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8:100±4.
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[7] Boiko AV. Some eects of biaxial elastoplastic deformation of a plate with a crack. Problemy Prochnosti 1990;N6:20±7 [in Russian]. [8] Alpa G, Bozzo E, Gambarotta L. Validity limits of the Dugdale model for thin cracked plates under biaxial loading. Engng Fracture Mech 1979;12:523±9.