Singularity analysis of anisotropic multimaterial corners

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/ 10.1023/A:1023937819943 1 SINGULARITY ANALYSIS OF ANISOTROPIC MULTIMATERIAL CORNERS A. Barroso, V. Mantič y F. París School of Engineering, University of Seville Camino de los Descubrimientos s/n, E-41092 Seville, Spain e-mail: [email protected], [email protected].es, [email protected]s.es Abstract Singular stress states induced at the tip of linear elastic multimaterial corners are characterized in terms of the order of stress singularities and angular variation of stresses and displacements. Linear elastic materials of an arbitrary nature are considered, namely anisotropic, orthotropic, transversely isotropic, isotropic, etc. Thus, in terms of Stroh formalism of anisotropic elasticity, the scope of the present work includes mathematically non-degenerate and degenerate materials. Multimaterial corners composed of materials of different nature are typically present at any metal-composite, or composite-composite adhesive joint. Several works are available in the literature dealing with a singularity analysis of multimaterial corners but involving (in the vast majority) only materials of the same nature (e.g. either isotropic or orthotropic). Although many different corner configurations have been studied in literature, with almost any kind of boundary conditions, there is an obvious lack of a general procedure for the singularity characterization of multimaterial corners without any limitation in the nature of the materials. With the procedure developed here, and implemented in a computer code, multimaterial corners, with no limitation in the nature of the materials and any homogeneous orthogonal boundary conditions, could be analyzed. As a particular case, stress singularity orders in corners involving extraordinary degenerate materials are, to the authors' knowledge, presented for the first time. The present work is based on an original idea by Ting (1997) in which an efficient procedure for a singularity analysis of anisotropic non-degenerate multimaterial corners is introduced by means of the use of a transfer matrix. 1. Introduction A great deal of research has been carried out in the last few decades regarding the singularity analysis of linear elastic multimaterial corners, producing many useful results available in the literature at present. Nevertheless, the possibility of including any kind of materials in the corner configuration has been to some extent restricted in these works. Thus, on one hand, multimaterial corners composed exclusively of isotropic materials with almost any kind of boundary conditions were analyzed in a general way by Dempsey & Sinclair (1979, 1981), see also Bogy (1971), Bogy & Wang (1971), Hein & Erdogan (1971). On the other hand, singularity analysis of multimaterial corners composed exclusively of real anisotropic, but not isotropic, materials was performed, e.g., by Delale (1984), Pageu et al. (1995a, 1996), Ting (1997), Chen (1998). The vast majority of these works, assuming generalized plane strain state or plain stress states, make use of the powerful Lekhnitskii-Eshelby-Stroh complex variable formalism of Anisotropic Elasticity: Lekhnitskii (1938), Eshelby et al. (1953), Stroh (1958, 1962) (in the following referred to as Stroh formalism). Note that in this formalism, isotropic materials are considered as mathematically degenerate materials (Ting, 1996b). This concept refers to repeated complex roots of the characteristic equation with an associated number of linearly independent eigenvectors of the fundamental elasticity matrix N smaller than the multiplicity of the repeated complex roots. The structure of the complex variable representations of elastic solutions is 2 different and very cumbersome for mathematically degenerate materials when compared to nondegenerate materials (Ting & Hwu, 1988 and Wang & Ting, 1997). This is the reason why almost all works which apply analytic representations of elastic solutions to perform singularity analysis of anisotropic multimaterial corners include only real anisotropic, monoclinic or orthotropic mathematically non-degenerate materials. The studies by other authors referenced above are very useful in composite design applications, but of limited application if no isotropic material is allowed to be included together with real anisotropic materials. Very few works, all recent (e.g., Lin and Sung, 1998, Poonsawat et al., 1998, 2001) consider particular cases of bimaterial corners involving degenerate (usually isotropic) as well as non-degenerate (usually orthotropic) materials. Lin and Sung consider degenerate materials (isotropic) by taking the limiting process with the use of L'Hospital rule. Poonsawat et al. present some bimaterial cases with isotropic and orthotropic materials including friction at the interface. Nevertheless, a general procedure for an N-material corner and any material nature, including extraordinary-degenerate materials, is still lacking. Using a procedure of anisotropic elasticity for non-degenerate materials, it might be possible to overcome the above explained restrictions and to obtain approximate results, if quasi-isotropic materials are applied as described in what follows. Replacing the isotropic material, in the multimaterial corner studied, by an anisotropic material with elastic constants very similar but not exactly equal to those of the isotropic material, the theoretical basis of the procedure (analytical representation of elastic solution) can be used. Thus the results will be as approximate to the exact ones as the values of the anisotropic elastic constants used are close to the real isotropic ones. Nevertheless, numerical problems related to the possible ill-conditioning of the expressions used in this case can be expected to appear. It is important to notice that the mathematical transition from non-degenerate materials (quasi-isotropic) to degenerate ones (isotropic), leads to a non-continuous definition of the matrices of normalized eigenvectors of N involved in Stroh formalism. Thus, it is possible to expect lesser accuracy when reaching certain limits in the previously outlined procedure. This kind of procedure can be used if no high accuracy is required. It has to be pointed out that the dependence of the error in the results obtained in this way with respect to the difference of the value of the elastic constants is not easy to estimate. Numerical tools like the finite element method can also overcome the above difficulties present in the analytical methods, but usually at the expense of a higher computational effort or lesser accuracy if coarse discretizations are used. The objective of this work is to present and implement in a computer code a general and analytically based procedure to characterize the singular stress field that appears at the tip of an anisotropic linear elastic multimaterial corner, with no limitations on the nature of the material, considering perfect bonding between the materials, and any combination of homogeneous boundary conditions except unilateral friction contact. The Stroh formalism for the degenerate (Ting & Hwu, 1988) cases and extraordinary degenerate (Wang & Ting, 1997) cases of anisotropic materials will be used to this end. The use of the concept of a transfer matrix in the analysis of multimaterial corners with perfect adhesion between the material wedges (similar to that used in an analysis of a multimaterial infinite strip, see Buffler, 1971) greatly simplifies the procedure, reducing the size of the system considered, see Defourny (1988) in isotropic potential problems, Ting (1997) in anisotropic elasticity and Mantič et al. (2002) in anisotropic potential problems. To take advantage of the concept of the transfer matrix, in the sense of the above mentioned works, the transfer matrix for degenerate materials has been obtained in the present work. 3 It has to be mentioned that Desmorat (1996) presented another approach for reducing the size of the linear system considered in the case of bimaterial anisotropic elastic corners, based on a suitable pre-elimination of some variables involved. The results of Desmorat's (1996) and Ting's (1997) work can be considered equivalent when free-free bimaterial corners are studied. However, due to the general scope of Ting's (1997) elegant method, based on the transfer matrix concept, this will be the method developed in the present work to analyze anisotropic multimaterial corners involving any kind of linear elastic material, non-degenerate or degenerate. Following the approach introduced by Mantič et al. (1997), the boundary conditions analyzed in Ting's work (1997), namely free, fixed, and all materials bonded, have been completed by six additional homogeneous boundary conditions in the present work. It has to be stressed that the theoretical results and the computer code developed in the present work will allow an accurate singularity analysis of metal to composite joints, modelled like multi-material corners with the simultaneous presence of orthotropic (or transversely isotropic) and isotropic materials in the same corner, to be performed. This kind of joint is widely used for example in the aerospace industry. 2. Stroh formalism of anisotropic elasticity The Stroh formalism (Ting, 1996a) is a powerful approach for solving anisotropic elasticity problems in which a generalized plane strain state: ui=ui(x1,x2) (i=1,2,3), can be assumed. The Stroh formalism is based on the following eigensystem: )6,...,1(    ξNξ p, (1) where  p and  ξ are respectively the eigenvalues and eigenvectors of the real fundamental elasticity matrix N (6×6) defined as follows:       T 13 21 NN NN N, QRRTNTNRTN   TT 1 3 1 2 1 1,, , (2) Q, R and T only depend on the elastic constants of the material ijks C ( ksijksij C  ), 11kiik CQ , 21kiik CR  and 22kiik CT , (3) and the superscript T denotes the transpose. The eigenvalues  p are complex if the strain energy is positive definite, and the eigenvectors   TTT  baξ , have an important physical meaning, with  a being proportional to the displacement vector and  b being proportional to the traction vector. If  p and  ξ satisfy (1),  p and  ξ are also a solution, where the overbar denotes the complex conjugate. Thus, it is habitual to write: )3,2,1(,,0Im 33      ξξppp , (4) with Im standing for the imaginary part. 4 Equation (1) is only valid when there are three linearly independent eigenvectors  ξ (  =1,2,3). If the tensor of elastic constants ijks C of a particular material gives rise to an eigensystem with less than three linearly independent eigenvectors  ξ, the formalism has to be modified (Ting & Hwu, 1988; Wang & Ting, 1997), equations (1) being transformed in the case of two linearly independent eigenvectors to: 111 ξNξ p, 1212 ξξNξ  p, 333 ξNξ p, ( 21 pp ), (5) and in the case of only one linearly independent eigenvector to: 11 ξNξ p, 122 ξξNξ  p, 233 ξξNξ  p, ( pppp  321 ). (6) In this sense, all linear elastic materials can be classified in relation with the character of the eigensystem of N, as shown in Table 1 (Ting, 1996b, 1999): p1  p2  p3  p1 p1=p2  p3 p1=p2=p3 3 Linearly Independent Eigenvectors Simple (SP) Semisimple (SS) Extraordinary semisimple (ES) Does not exist 2 Linearly Independent Eigenvectors Degenerate or non-semisimple (D1) Degenerate or non-semisimple (D2) 1 Linearly Independent Eigenvectors Extraordinary degenerate or Extraordinary non-semisimple (ED) Table 1. Classification of the fundamental elasticity matrix N. Ting (1996b) proved the existence of materials with an ED matrix N, as well as the impossibility of finding any material with an ES matrix N, unless the strain energy is allowed to be positive semidefinite. In the following sections, the particular relations satisfied by the eigenvalues  p and the associate eigenvectors   TTT  baξ , will be used for the three eigensystems (1), (5) and (6). Depending on the number of linearly independent eigenvectors, the following relations apply: Three linearly independent eigenvectors (N is SP or SS). For (  ξ,p) with  =1,2,3:   0)( 2  aTRRQ pp T, (7)    aRQaTRb )( 1 )( p p p T . 5 Two linearly independent eigenvectors (N is D1 or D2): For ( 11,ξp) and ( 33 ,ξp) equations (7) hold for  =1,3. For ppp  12 and the generalized eigenvector   TTT 222 ,baξ :     12 22)( aRRTaTRRQ TT ppp  , (8)   212 aTRTab p T . One linearly independent eigenvector (N is ED) with pppp  321 : For ( 11,ξp) equations (7) hold with  =1. For ( 22 ,ξp) equations (8). For 3 p and the generalized eigenvector   TTT 333 ,baξ :     123 22)( TaaRRTaTRRQ  TT ppp , (9)   323 aTRTab p T . The solution in terms of displacements, u , and the stress function vector, φ , can be expressed by means of complex (3×3) matrices   321 ,, aaaA  and   321 ,, bbbB  in the following way: GAAGu ~  , (10) GBBGφ ~  , (11) where G can be written in terms of arbitrary functions )(  zf (  =1,2,3) of complex arguments 21 xpxz   , and G ~ in terms of )( 3  zf  (  =1,2,3). For the analysis of stress singularities it is sufficient to take the same function for  =1,2,3, i.e.  qzfzf )()(  and  qzfzf ~ )()( 3 , where  q and  q ~ are arbitrary real or complex constants. Thus, it is possible to write qFG ·  and qFG ~ · ~ ~  with T qqq ),,( 321 q and T qqq ) ~ , ~ , ~ ( ~ 321 q. The structure of F (and F ~ ) depends on the number of linearly independent eigenvectors, as shown below, where the prime denotes differentiation with respect to the complex variable z. SP and SS cases (non-degenerate cases):            )(00 0)(0 00)( 3 2 1 zf zf zf F,          )(00 0)(0 00)( ~ 3 2 1 zf zf zf F. (12) D1 and D2 cases (degenerate cases):             )(00 0)(0 0)()( 3 1 121 zf zf zfxzf F,          )(00 0)(0 0)()( ~ 3 1 121 zf zf zfxzf F, ( 21 zz ). (13) 6 ED case (extraordinary degenerate case):              )(00 )()(0 )()()( 2 2 2 2 1 2 zf zfxzf zfxzfxzf F,              )(00 )()(0 )()()( ~ 2 2 2 2 1 2 zf zfxzf zfxzfxzf F, ( zzzz  321 ). (14) In the analysis of singular stress states presented in this work, the function   zzf )( will be considered, and therefore      qzzfqzzf ~ )(,)( 3 , (15) where  is the characteristic exponent and, as will be seen,  -1 defines the order of stress singularity for 10    . If  is a real number, then F F  ~ and qq  ~ . The orthogonality and closure relations of the Stroh formalism, which also depend on the number of linearly independent eigenvectors, can be written in the following general form: I X X XX    11 , with:       BB AA X and        TT TT AΓBΓ ΓAΓB X1, (16) where I is the identity matrix (6×6), and for the different cases: non-degenerate (SP and SS), degenerate (D1 and D2) and extraordinary degenerate (ED), Γ is expressed as: SP-SS D1-D2 ED            100 010 001 Γ,            100 001 010 Γ,            001 010 100 Γ (17) All linear elastic materials fall inside one of the above mentioned groups (Table 1). Thus, all materials can be studied following the approach of Stroh formalism. Isotropic materials, for example, belong to group D2, with a triple eigenvalue 1 ip and two linearly independent eigenvectors. Following Tanuma (1996), transversely isotropic materials can belong to every group in Table 1, except ED. Transversely isotropic materials can be non-semisimple (D1 or D2), irrespective of the value of their elastic constants in the material coordinate system, with the x3 axis being perpendicular to the x1-x2 tranversely isotropic plane of the material. Simply through the relative position of the material with respect to the coordinate system which defines the generalized plain strain state, the material can be non-degenerate or degenerate, Tanuma (1996). 7 3. Singularity analysis of anisotropic multimaterial corners including non-degenerate materials 3.1. Summary of Ting's procedure. The procedure originally developed by Ting (1997) is an efficient tool for the singular characterization of non-degenerate anisotropic multimaterial corners. An N-material corner, with N homogeneous wedges, is represented in Figure 1. The i-th material wedge occupies the polar sector ii   1, i=1,...,N. 1 i N  0  N  i  i-1 x1 x2 1 i N  0  N  i  i-1 x1 x2 Figure 1. Multimaterial corner. Perfect bonding is considered between material wedges. Fixed or free boundary conditions are considered at the external faces. The possibility without external faces, all materials being bonded, is also considered and is referred to in this work as a closed corner, as opposed to open corner with external faces. Consider a polar coordinate system with the origin at the tip of the corner (Figure 1). Then, equations (10) and (11), together with (15), considered for an homogeneous wedge, can be written in the following condensed form, defining the complex variable: )()sin(cos 21   rprxpxz  : tXZw )(),(   rr , (18) where X is defined in (16) and ),(  rw is       ),( ),( ),(    r r rφ u w,       q q t~,         )(0 0)( )( * *      Z. (19) The diagonal matrix   )(),(),()( 321*   diag in (19)3 is associated to matrix F in (12)1 through the following relation: qF )( *  r, (20) 8 while   )(),(),()( 321*   diag with     )sin(cos)( p , is related to F ~ in (12)2 through qF ~ )( ~ *   r. Ting's procedure makes use of a transfer matrix, a matrix which transfers the displacements and stress function vector components from one edge of the material wedge to the other. If equation (18) is evaluated for the i-th wedge at 1 i  and i  , and t is eliminated, we obtain: tXZw )(),( 11  ii rr   , tXZw )(),( ii rr   ,  ),(),( 1 iii rr  wEw , (21) where, in view of (16),   1 1 1)()(    XZXZE iii   . (22) i E, called the transfer matrix for the i-th wedge, depends on the material properties, on the angles (1i  and i  ) defining the wedge and on the characteristic exponent  . Using the continuity conditions introduced by the hypothesis of perfect bonding between the wedges, ),(),( 1iiii rr   ww )1,...,1(   Ni , and the transfer matrix for each wedge, it is easy to arrive at the following expression, which is in fact the expression of the transfer matrix for the whole multimaterial corner, as it relates the variables between its external faces ( 0  and N  ):                    ),( ),( ),( ),( 01 0 1 )4()3( )2()1(     r r r r NN NN NN NN φ u KK KK φ u, or ),(),( 01  rr NNN wKw , (23) where N K is obtained by the product of the sequence of the successive transfer matrices i E of all the wedges in the corner: 121 ··...·· EEEEK  NNN . (24) In Ting's work, fixed and free boundary conditions can be prescribed at the external faces of the corner, and the case of all materials being bonded is also considered. The following characteristic equations are obtained from (23) for the different combinations of boundary conditions: Free-fixed: 0φu  )()( 01  NN ,  0K  )1( N. (25) Fixed - fixed: 0uu  )()( 01  NN ,  0K  )2( N. (26) Free-free: 0φφ  )()( 01  NN ,  0K  )3( N. (27) Fixed -free: 0uφ  )()( 01  NN ,  0K  )4( N. (28) All bonded: )()(and)()( 0101  uuφφ  NNNN ,  0IK  N. (29) 15 The eigensystem to be solved in this case is shown in (6). The two generalized eigenvectors ),( 222 TTT baξ  and ),( 333 TTT baξ  must satisfy (8) and (9) respectively, while the first eigenvector ),( 111 TTT baξ  must still satisfy equations (7). The structure of matrix F is shown in (14) and the orthogonality relations are modified following (16) and (17)3. Equation (18) for extraordinary degenerate materials, with pppp  321 , can be written as: tXZw ),(),(   rr . (60) Let us define ),(   Z as in (48):       ),,( ),,( ),(    p p Ψ0 0Ψ Z. (61) In comparison with (18), matrix )( *  associated to matrix F in (12) is replaced by ),,(   pΨ, which is associated to matrix F in (14) by the following relation: qΨF ),,(   pr. (62) Following the same procedure as that presented in Section 4, we can use (14), (62) and the definition of ),,( 1  p in (51), to finally write ),,(   pΨ as:               100 ),,(10 ),,()1(),,(1 )(),,( 21 2 1     p pp pΨ, (63) where: )( sin ),,(      p, and     sincos)( p   (64) The transfer matrix for extraordinary degenerate materials takes the following form:   1 1 1),(),(    XZXZE  iii , (65) where:             ),,,( ),,,( ),(),( 1 1 1 1    ii ii ii p p Ψ0 0Ψ ZZ , (66) ),,(),,(),,,( 1 1 1    iiii ppp ΨΨΨ , (67) with: 16                100 ),,(10 ),,()1(),,(1 )(),,( 21 2 1 1     p pp pΨ. (68) ),,,( 1  ii pΨ can be written as:             100 10 1 ),(),,,( 11 K ZKK piiii   , (69) where ),( 1ii  is defined as in (32) and K is defined in the same way as in (57): )()( )sin( 1 1 ii ii K        , (70) and Z is defined as:           )( sin )( sin 2 1 ),,,( 1 1 1 i i i i ii KpZ      . (71) With ),,,( 1  ii pΨ given by (69), the transfer matrix i E (65) can then be computed. 6. Singularity analysis of multimaterial corners including any anisotropic material With the previous results, the transfer matrix can be computed for any wedge material, irrespective of its nature (SP, SS, D1, D2 and ED). A single and robust approach is presented in this section for the singular characterization of any multimaterial corner with perfect adhesion between material wedges and any orthogonal boundary condition or all materials being bonded. x2 1 - orthotropic (non-degenerate) SP or SS material 2 - isotropic D2 material 3 - ED material Free edge Fixed edge Figure 3. Corner with non-degenerate, degenerate and extraordinary degenerate materials. 17 Consider, as an example, the three-material corner represented in Figure 3, involving one nondegenerate orthotropic material (SP or SS material), one isotropic material (degenerate material D2) and one extraordinary degenerate material (ED), with free-fixed boundary conditions at the external faces. The singularity analysis of this corner can be performed following steps 1 to 5 described below. 1) Evaluation of the transfer matrix for each material ( i E), using (22) for 1 E, (52) for 2 E and (65) for 3 E. 2) Evaluation of the corner transfer matrix 3 K (N=3), by the expression in (24), using the matrices i E previously obtained in step 1. 3) Application of boundary conditions, using )( 0  free D and )( Nfixed  D (Table 2) to finally obtain the modified corner transfer matrix 3 ˆ K (43). 4) Evaluation of the determinant of the submatrix )2( 3 ˆ K of 3 ˆ K, which is explicitly presented in (46). In the case of an open corner, in fact only )2( 3 ˆ K has to be evaluated, while in the case of closed corners the whole matrix 3 ˆ K has to be evaluated. (5) Calculation of roots  of the characteristic equation (45) (or (29) in the case of closed corners), which represent the characteristic exponents. Roots  with 1)Re(0    define stress singularity exponents  -1. The characteristic equations can be solved using (for example) Muller's (1956) method, when characteristic exponents can be complex. Once the particular value of the characteristic exponent is obtained, the corresponding angular behaviour of displacements and stresses near the tip of the corner can also be computed. The procedure starts with solution of (44) for the  obtained, then using (40) we obtain ),( 0  rw . With the transfer matrix of the first material, ),( 1  rw can be computed, and in the same way, with each particular transfer matrix, all ),( i rw  i=0,..., N. Once ),( i rw  i=0,..., N are known, the behaviour of stresses and displacements inside each wedge is easy to obtain, using the concept of the transfer matrix between ),( i rw  and ),(   rw , with 1   ii  . Stresses are computed by means of: 2,1 ii   and 1,2 ii  . For closed corners, the linear system to solve is:   0wIK  ),()( 0  r N,  being obtained by means of characteristic equation in (29). Then, using the transfer matrices for each wedge, in just the same way as outlined previously for open corners, displacements and stresses are directly obtained. 7. Numerical examples 7.1. Comparison with previous results obtained by other authors Several results of singularity analysis of different corner configurations, available in the literature, have been used to study first of all the performance of the computational procedure developed in the present work. Some of them are summarized below. 7.1.1. Isotropic materials (D2 degenerate materials): 18 Single isotropic wedge: A comparison of the results obtained for a solid sector of an isotropic material of interior angle º280 01   , with free-free external faces, versus the results of Seweryn (1994) and Vasilopoulos (1988), is shown in Table 3. The antiplane order of stress singularity is also presented (  -1=-0.357143). An excellent agreement is achieved in all digits. Vasilopoulos Seweryn Present work - 0.469604280870 -0.156560431071 - 0.469604 -0.357143 -0.156560 - 0.469604 -0.357143 -0.156560 Table 3. Results (  -1) for a single free-free isotropic corner. Bimaterial isotropic corners: Classical results from Dempsey and Sinclair (1979, 1981) were used. The results obtained for free-free corners (Table 4), and for corners with all materials bonded (Table 5), are all excellent. The mechanical properties of both materials are: E1=30 GPa, 1=0.25, E2=120 GPa, 2=0.31, for results in Table 4 and: E1=30 GPa, 1=0.2, E2=120 GPa, 2=0.3, for results in Table 5. Additional antiplane order of stress singularities are presented in both cases.  1=120º  2=135º  1=120º  2=135º Dempsey & Sinclair Present work - 0.390748 - 0.390748 -0.269076 Table 4. Results (  -1) for a bimaterial free-free isotropic corner.  1=80º  2=(2-80)º  1=80º  2=(2-80)º Dempsey & Sinclair Present work - 0.229549 -0.0742109 - 0.2295490 -0.1916800 -0.0742109 Table 5. Results (  -1) for a closed bimaterial isotropic corner. Three-material isotropic corners: Results from Hein & Erdogan (1971) and Pageau et al. (1994) are compared with those obtained in the present work in Tables 6 and 7. The agreement is excellent, including the second case (Table 7), with complex valued order of stress singularities. Additional antiplane order of stress singularities are presented (  -1=-0.00580724 in Table 6 and  -1=-0.460283 in Table 7). Mechanical properties for the configuration shown in Table 6 are: E1=20 GPa, 1=0.2, E2=10 GPa, 2=0.2, E3=0.01 GPa, =0.2, while in Table 7 E1=10 GPa, 1=0.2, E2=0.01 GPa, 2=0.2, E3=100 GPa, =0.2.  1 =90º  2 =90º  1 =90º  2 =90º Hein & Erdogan Pageau et al. Present work 19 - 0.0226 - 0.0226 -0.0003 - 0.02263280 -0.00580724 -0.00028330 Table 6. Results (  -1) for a closed three-material isotropic corner with º180,º90 321   .  3=180º  1=179º  2=1º  3=180º  1=179º  2=1º Hein & Erdogan Pageau et al.: Present work: -0.4975  0.1014 i -0.4476  0.0887 i -0.447624  0.088750 i -0.460273 Table 7. Results (  -1) for a closed three-material isotropic corner, º180,º1,º179 321   . It has been verified that the singular exponents associated to the antiplane mode in the previous examples agree with those obtained by Mantič et al. (2002) procedure up to six digits or more. 7.1.2. Non-degenerate anisotropic materials: Bimaterial orthotropic free-free corner: Results from Delale (1984) and Chen (1998) were used in this case. An excellent agreement was found with present results, shown in Table 8. Both materials represent the same orthotropic material equivalent to a fiber reinforced plastic with different orientation of fibers. The fibers are placed in the x2-x3 plane, with angles with respect to x2 axis:  1=60º and  2=30º, and the following mechanical properties: E11=163.4 GPa, E22=E33=11.9 GPa, G12=G13=6.5 GPa, G23=3.5 GPa, 12=13=0.3, 23=0.5.  2=180º  1=90º x1 x2  2=180º  1=90º x1 x2 Delale Chen Present work - 0.4229 - 0.422886 -0.380828 -0.047337 - 0.422886 -0.380828 -0.047337 Table 8. Results (  -1) for a free-free bimaterial orthotropic corner. Interface crack in an orthotropic bimaterial configuration: Results from Wang (1984) and Chen & Huang (1997) were available for this case, in which the agreement with the results obtained by the procedure developed in the present work is also very good. The mechanical properties of the materials are those of a typical graphite-epoxy composite, taking the following values when expressed in orthotropic axes: E11=138 GPa, E22=E33=14.5 GPa, G12=G13=G23=5.9 GPa, 12=13=23=0.21. The angle  , see Table 9, is the angle the fiber forms with respect to x3 axis in the x1-x3 plane. x2 Wang Chen & Huang Present work 20 =45º - 0.5 -0.50.03434 i - 0.5 -0.50.0343365146 i - 0.5 -0.50.0343398 i =60º - 0.5 -0.50.02942 i - 0.5 -0.50.0294152218 i - 0.5 -0.50.0294132 i Table 9. Results (  -1) for an interface crack between two orthotropic materials. Three-material orthotropic corners have also been analyzed, a very good agreement being obtained with available results by Chen (1998) and Pageau et al. (1996). In view of the excellent agreement obtained between the results of the procedure developed in this work and results of other authors in all studies presented and also others not presented here, for the sake of brevity, the computer code developed here can be considered successfully verified. 7.2. Isotropic-orthotropic bimaterial corner In every metal to composite or composite to composite adhesive joint, an example of an isotropicorthotropic bimaterial corner, with the simultaneous presence of non-degenerate and degenerate materials, can be found. Paying attention to the corner depicted in Figure 4, the procedure presented in this work can be used without any of the limitations that usually appear in traditional approaches which make use of Stroh formalism. In the vast majority of cases, these limitations are due to the fact that isotropic materials as well as any other anisotropic material wich is mathematically degenerate, cannot be included in the analysis. x 2 x 1 Orthotropic SP material Isotropic D2 material  x 2 x 1 Orthotropic SP material Isotropic D2 material  Figure 4. Bimaterial corner with SP and D2 materials. The materials in the corner (Figure 4), have the following properties: Composite material (orthotropic non-degenerate, SP material): E11=141.3 GPa E22=9.58 GPa, E33=9.58 GPa G12=5.0 GPa, G13=5.0 GPa, G23=3.5 GPa =0.3, =0.3, =0.32 Epoxy adhesive (isotropic, D2 material): E=3 GPa, =0.3 21 In Figure 5, the adhesive angle () varies from 0º to 180º and the fiber reinforced plastic has the fiber oriented in x1 direction. Two order of stress singularities are obtained until an angle of approximately =85º is reached. Starting from this angle, three real order of stress singularities are obtained. Before the adhesive angle reaches 160º, two real roots convert into two complex conjugate roots. Similar results are obtained for two isotropic and two orthotropic free-free corners. Finally, for the interface crack (=180º), a real root of 0.5 and two complex conjugate roots with real part equal to 0.5 are obtained. -0.50 -0.45 -0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0 20 40 60 80 100 120 140 160 180 Re (  -1) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Im (  -1)   = 0º x 1 x 2 x 3  x 1 x 2 x 3  angle () Figure 5. Order of stress singularities (  -1) for a bi-material SP-D2 corner. The angular behaviour of displacements r u,  u and the stress components rr  ,   r,   , calculated using ),(  rw and 2,1 ii   , 1,2 ii  , are presented in Figure 6, for =70º for one of the singularity modes with -1=-0.266941. -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 50 100 150 200 250 angle  u r u   -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 50 100 150 200 250 angle  u r u   -25 -20 -15 -10 -5 0 5 10 0 50 100 150 200 250 angle    rr  r    -25 -20 -15 -10 -5 0 5 10 0 50 100 150 200 250 angle    rr  r    Figure 6. Angular behaviour of displacements and stresses for =70º and -1=-0.266941. 22 It can be observed that the stress components   r and   fulfil the boundary conditions at the external faces (free-free), and that rr  is not continuous at the interface between both materials. Displacements are continuous at the interface but not their slope. 7.3. Corners involving extraordinary degenerate (ED) materials To the authors' knowledge, no results are available for corners involving extraordinary degenerate materials. In fact, these materials have been proved to exist only in recent years (Ting, 1996b). From Ting (1996b), it is known that a particular group of ED materials can be described with the following reduced elastic compliance matrix:                )3(000 1000 000 0000 0001 '1 11     ss, (72) where   2 1 ))1(( 31 , and the following has to be fulfilled: 0 11   s, 01    , 21    and 2 )3(   . For the numerical example presented in this work, the following values have been taken: 1 11   s, 2 1   , 1   and the positive sign in  , resulting in:                  4 5 2 12 1 2 1 000 1000 00201 0000 00101 's. (73) In Figure 7, the order of stress singularities for a single free-free wedge of the above ED material, from =180º to the crack configuration, =360º, is presented. Two real order of stress singularities are obtained until =260º, where a third real singularity solution appears. For the crack configuration (=360º) three real order of stress singularities, with value -0.5, are obtained. The numerical results obtained are presented in Table 10 for some particular values of . In Figure 8, a free-free three-material corner involving a 90º wedge of as SP (orthotropic nondegenerate material), a 90º of a D2 (isotropic material) and a 90º of a ED material is presented. The mechanical properties of the orthotropic and isotropic material are the same as used in Section 7.2, while the ED material has the properties given above, in (73). 23 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 180 200 220 240 260 280 300 320 340 360 angle  Re (-1)  Figure 7. Order of stress singularities (  -1) for a single free-free ED wedge.  =200º -0.15672 -0.128226  =240º -0.374451 -0.269291  =280º -0.472353 -0.351773 -0.161211  =320º -0.497632 -0.422641 -0.378599  =355º -0.499996 -0.490131 -0.488593 Table 10. Numerical results for some particular cases of Figure 7. 1 - orthotropic (non-degenerate) SP or SS material 2 - isotropic D2 material 3 - ED material Free edge Free edge 90º 90º 90º x 2 x 1 1 - orthotropic (non-degenerate) SP or SS material 2 - isotropic D2 material 3 - ED material Free edge Free edge 90º 90º 90º x 2 x 1 Figure 8. Free-free three-material corner with 90º wedges of SP, D2 and ED materials. For this particular configuration, a real (  -1=-0.323997) and two complex conjugate (  -1=- 0.354922±0.152318 i) orders of stress singularity were found. 24 8. Conclusions In the present work, a powerful procedure for the singularity analysis of anisotropic multimaterial corners allowing any kind of linear elastic anisotropic material to be considered has been completed and implemented in a computer code. The transfer matrix for mathematically degenerate materials has been obtained in the framework of Stroh formalism, and explicit expressions have been presented. This allows the original procedure developed by Ting (1997) for the singular characterization of multimaterial anisotropic corners to be completed with the possibility of including degenerate materials in the analysis. As isotropic materials can be considered degenerate materials in Stroh formalism of anisotropic elasticity, the singular analysis is now open to materials of this kind, as well as any other degenerate and extraordinary degenerate material. A Mathematica (Wolfram, 1991) code has been implemented for the calculation of the order of stress singularities, as well as the graphical representation of displacements and stresses. This practical computational tool has shown excellent agreement when comparing with previous results available in the literature, from the single isotropic wedge to the orthotropic three-material corner. The implemented code has been used to analyze a corner with simultaneous presence of both degenerate (isotropic) and non-degenerate (orthotropic) materials, and also corners involving extraordinary degenerate materials which, to the authors' knowledge, are studied for the first time. With the computational tool developed, any bidimensional corner configuration can be characterized, considering perfect bonding at interfaces, and any homogeneous orthogonal boundary condition at outer interfaces of the corner (also all bonded). Note that only power type singularities have been considered in this work. The presence of logarithmic singularities can be analyzed using the well-known approach developed by Dempsey and Sinclair (1979), Dempsey (1995), Ting (1996a), Sinclair (1999) and others. The study carried out is a previous step to calculating stress intensity factors (by means, for instance, of the finite element or boundary element methods). The possibility of studying proposals of failure criteria for adhesively bonded joints, based on fracture mechanics, will then be opened up. 9. Acknowledgements The study has the financial support of the Spanish Ministry of Science and Technology (PROFIT 2001, Project EUREKA !1882) and Ministry of Education and Culture (Projects No. PB98-1118, MAT 2000-1115). The authors gratefully acknowledge helpful information from SACESA. 10. References Bogy, D.B. Two edge-bonded elastic wedges of different materials and wedges angles under surface tractions. Journal of Applied Mechanics 38 (1971) 377-386. Bogy, D.B. and Wang, K.C. Stress singularities at interface corners in bonded dissimilar isotropic elastic materials. International Journal of Solids and Structures 7 (1971) 993-1005. Buffler, H. Theory of elasticity of a multilayered medium. Journal of Elasticity 1 (1971) 125-143. Chen W.H. and Huang, T.F. Stress singularity of edge delamination in angle-ply and cross-ply laminates. Journal of Applied Mechanics 64 (1997) 525-531.