A note on Pageau et al. (1994) results for the order of stress singularities in multimaterial junctions

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in International Journal of Solids and Structures, Vol. 191-192, on May 2020, available at: https://doi.org/10.1016/j.ijsolstr.2019.12.019 Copyright 2019 Elsevier. En idus Licencia Creative Commons BY-NC-ND A note on Pageau et al. (1994) results for the order of stress singularities in multimaterial junctions Filipe F. Fornazaria, Daniane F. V icentinia,b, Alberto Barrosoc aGroup of Advanced Research and Development of Infrastructures Department of Transportation, Federal University of Parana Av. Cel. Francisco H. dos Santos, 210. Jd. das Americas CEP 81530-000. Curitiba, PR, Brazil. bCorresponding author:[email protected] cGroup of Elasticity and Strength of Materials School of Engineering, University of Seville Camino de los Descubrimientos, s/n, E-41092. Seville, Spain. Abstract In 1994, Pageau, Joseph and Biggers Jr. evaluated “The order of stress singularities for bonded and disbonded three-material junctions”. Their work represented an important finding on multimaterial corners, since their formulation can be used to study the stress state on these junctions. Despite the relevance, the graphs of some results presented in their paper might lead to an improper interpretation, and consequently to a wrong asymptotic stress representation. This is evidenced and discussed in the present work, in which an analytical algorithm was implemented, allowing to reproduce, to complete and to clarify the original results. The physical interpretation of the divergences indicates, among other observations, that the failure mode associated with the highest stress singularity order in a multimaterial junction may change by only varying the Young’s modulus of one of the materials. Keywords: multimaterial junctions, order of stress singularity, Generalized Stress Intensity Factors, Fracture Mechanics 1. Background Stress singularities are an analytical prediction from the theory of linear elasticity and their study finds applicability in several engineering problems, Preprint submitted to International Journal of Solids and Structures December 16, 2019 as in the analysis of crack propagation through the Stress Intensity Factors (SIFs). Williams (1952) was one of the pioneers in the stress singularities study. Using Airy functions, he investigated angular corners of plates according to the linear elasticity theory. For three boundary conditions (free-free, clamped-free and clamped-clamped), Williams obtained the characteristic equations governing the stresses and displacements in the vicinity of the angular corners. He also extracted the eigenvalues λfor the problem, which would be later called characteristic exponents. These exponents would find direct application in their ability to describe the potential of a given geometry in generating a stress singularity. This potential can be measured by the so-called order of stress singularity δof the problem, sometimes associated with the very concept of the characteristic exponents λ, commonly defined as δ= (1 −λ) [1] with 0 < λ < 1 (singularity condition). The work of Williams (1952) included the solution for λto the classical problem of Fracture Mechanics, i.e., the case of a wedge varying the opening angle between its faces. In a later paper, Williams (1957) used this concept to characterize the stress field in the front of the crack tip (in which λ= 0.5). In the following decades, several authors devoted efforts to the study of the stress singularity order for different problems, among which are included the multimaterial junctions (also called multimaterial corners). The evaluation of the eigenvalues λk(for kvarying from 1 up to n, being n the number of eigenvalues) reached a new level of application when Vasilopoulos (1988) correlated its calculation (from Williams’ work, in 1952) with the Stress Intensity Factors (SIFs) from the classical Fracture Mechanics, formalizing the concept of the Generalized Stress Intensity Factors (GSIFs). Thus, as a generalization of the SIFs, the stress field in the vicinity of a point of singularity, under the Linear Elastic Fracture Mechanics (LEFM), could be now generalized by: σi,j(r, θ) = n X k=1 Kk r1−λkfk ij (θ) (1) where Kkare the GSIFs, rand θare the polar coordinates of the point at which the stresses are evaluated and fk ij(θ) are the characteristic angular functions. The sub index kvaries from 1 to n, depending on the problem, generating nterms in the sum (Eq. 1). It is also important to emphasize 1See, e.g.,Barroso et al. (2012a) 2 that Eq. (1) is a stress representation with variable separation, with the role of the radial distance rand the angular position θbeing clearly identified. Few exceptional cases do not accept this variable separation, some of those being identified in Sinclair (1999). The stress field around the crack tip in the SIFs formulation is given by a sum of three terms, each one corresponding physically to a deformation mode. Following this approach, the stress field for other geometries could also be given by a sum of terms, according to Eq. (1), being each mode associated to an eigenvalue λk. Related with the studies of the order of stress singularity, the definition of the GSIFs would allow foreseeing a fracture failure criterion for several situations aligned with the practice for crack problems, in which the SIF can be compared to a critical value KIc, denominated fracture toughness. The GSIFs can be calculated numerically by a method as the one described in Barroso et al. (2012a). For an example of GSIFs being used more recently in a failure criterion, the reader may consult the works of Barroso et al. (2012b) and Vicentini et al. (2012). For multimaterial corners, studies such as the one of Dempsey and Sinclair (1979) have succeeded in establishing general formulations to obtain the eigenvalues for junctions with an unlimited number of materials converging to one point and several boundary conditions on the faces (or interfaces) between the materials. Nevertheless, as this kind of analytical general solution may not be practical for some design purposes, particular solutions to simpler problems, such as two or three-material closed corners, are quite useful for engineering problems. Dempsey and Sinclair (1981) have particularized their general solution for bimaterial junctions, directly offering the characteristic equations of the problem under different boundary conditions, making the solution implementation in a straightforward computational code. For three-material junctions, a particular solution was presented in the paper by Pageau et al. (1994). They used a general solution proposed by Theocaris (1974), who developed a formulation for the problem expressed in complex potentials, in a different way from that in Williams (1952). The authors particularized a solution for four situations: two and three-material corners, considering perfectly bonded interfaces and considering unbonded interfaces. All solutions considered isotropic materials. This paper concentrates on the formulation for multimaterial closed corners. When implementing a program containing Pageau et al.’s formulation, and trying, first of all, to reproduce their results, some slight differences at 3 some graphs in the original work led to important physical considerations for the stress representation. Thus, Section 2 briefly reviews the formulation and describes its implementation. Results are presented in Section 3 and discussed in Section 4. 2. Implementation of Pageau et al.’s Formulation As the solutions employing the method of Williams (1952, 1957), as well as in Dempsey and Sinclair (1979), the solution in complex powers from Theocaris (1974) leads to a system of 4Nequations (being Nthe number of materials), which express the stresses and displacements of the problem. For a closed three-material corner with perfectly bonded interfaces, Pageau et al. (1994) presented a system of equations in the form: µ2[κ1a11e2iλθ1−λa21e2iθ1−b21] = µ1[κ2a12e2iλθ1−λa22e2iθ1−b22] [a11e2iλθ1+λa21e2iθ1+b21]=[a12e2iλθ1+λa22e2iθ1+b22] µ2[κ1a21e−2iλθ1−λa11e−2iθ1−b11] = µ1[κ2a22e−2iλθ1−λa12e−2iθ1−b12] [a21e−2iλθ1+λa11e−2iθ1+b11]=[a22e−2iλθ1+λa12e−2iθ1+b12] µ3[κ2a12e2iλθ2−λa22e2iθ2−b22] = µ2[κ3a13e2iλθ2−λa23e2iθ2−b23] [a12e2iλθ2+λa22e2iθ2+b22]=[a13e2iλθ2+λa23e2iθ2+b23] µ3[κ2a22e−2iλθ2−λa12e−2iθ2−b12] = µ2[κ3a23e−2iλθ2−λa13e−2iθ2−b13] [a22e−2iλθ2+λa12e−2iθ2+b12]=[a23e−2iλθ2+λa13e−2iθ2+b13] µ3[κ1a11 −λa21 −b21] = µ1[κ3a13eiλθ3−λa23ei(2−λ)θ3−b23e−iλθ3] [a11 +λa21 +b21]=[a13eiλθ3+λa23ei(2−λ)θ3+b23e−iλθ3] µ3[κ1a21 −λa11 −b11] = µ1[κ3a23e−iλθ3−λa13e−i(2−λ)θ3−b13eiλθ3] [a21 +λa11 +b11]=[a23e−iλθ3+λa13e−i(2−λ)θ3+b13eiλθ3] (2) in which µjis the shear modulus of the material j, and κis the Kolosov constant, given as a function of the Poisson coefficient ν, i.e., κ= (3 −4ν) for plane strain state, or κ= (3−ν)/(1+ν) for plane stress state. The angles θjdefine the interfaces between the materials, according to Fig. 1. For a non-trivial solution to exist (to the unknown constants a1j,a2j,b1j and b2j), the determinant of the coefficients matrix must be equal to zero, in 4 Figure 1: Closed three-material corner. order to determine λk. Thus, the characteristic equation can be implemented in the program and the solution for λkcan be obtained. The method of Muller (1956) was chosen to evaluate the eigenvalues of the equation, following Barroso et al. (2003). The solution can be finally obtained by the implementation of the algorithm in a program, here developed in Python 2.7 language (Python Software Foundation). The program’s structure and details about its development are presented in Fornazari (2019)’work. Pageau et al. (1994) used their formulation to plot some graphs, attempted to demonstrate the effect of parameter variation (especially Young’s modulus) in the order of stress singularity of the problem (i.e., to study the effect of the stiffness of the materials related on the roots λk). They focused on the eigenvalues that would generate singularities, presenting only those with Re λ < 1. In this paper, the program implemented was used in order to not only reproduce, but also to complete some of those graphs, plotting one additional root (the first non-singular one). During this preliminary stage, some important remarks concerning some particular results of the original paper were observed. These will be detailed in Section 3. 3. Results In this section, the graphs obtained by Pageau et al. (1994) were reevaluated and reinterpreted, in comparison to the original ones. These graphs considered the relationship between Young’s moduli of two of the materials, 5 varying the ratio between the moduli of another pair of the corner’s materials. In the first case presented here, the authors set E1/E2= 2.0, while E3/E2 varied. Fig. 2a shows the original graph, while Fig. 2b shows the result obtained by using the algorithm developed in this research. Numbers and other marks in red, over the original graphs, were added for the purposes of this paper. In this example, all roots are real, except for the segment 6 in the original graph (Fig. 2a). The imaginary segment 6 was not depicted in Fig. 2b, in order to allow amplifying the zones of interest. In order to remember that λk= 1 is always a root of this kind of problem, this result was included in all graphs here presented, although in most cases this value is related to rigid rotations—for exceptions, see Vasilopoulos (1988)—and, therefore, this term is not normally considered in the form of Eq. (1). Figure 2: Study of the graph (Fig. 9, in the original source) of Pageau et al. (1994). One can notice the exact correspondence for the singular results in this case, while, in addition, the curve F in Fig. 2b represents the λkassociated to the first non-singular term (with values λk>1). Note, however, what occurs in the second simulated case, where E1/E2= 10.0 is set and E3/E2varies. The roots are all real and the results are shown in Fig. 3. Since the graph in Fig. 3b (obtained from the algorithm described in Section 2) was generated with a huge number of points, the roots’ curves with Re λ < 1 clearly show differences between the connections from the original graph (Fig. 3a). Now it is possible to appreciate that the curves AD and BC in Fig. 3b intersect each other, when compared to their correspondent 6 Figure 3: Study of the graph (Fig. 10, in the original source) of Pageau et al. (1994). in Fig. 3a, where no intersection is visible. This observation can yield to a very important interpretation because, at the intersection point from curves AD and BC (Fig. 3b), singularities change their order of importance in the problem. It is also important the fact that the final representation of stresses (Eq. 1) needs not only the order of stress singularity values but also to properly identify the associated characteristic angular functions. The correct identification of each term in Eq. (1) for each order of stress singularity is of major importance to correctly obtain the asymptotic stress field in the neighborhood of the corner tip. The obtained result clearly shows that the singular terms change their position when crossing E3/E2 ∼ =1, when compared with Pageau et al.’s results. Thus, not identifying this fact, would lead to a wrong stress representation. From this point onwards, i.e., for materials in which Young modulus relationship is E3> E2, the previous most singular term becomes less dominant and the reverse occurs when analyzing the other term, which becomes prominent in Eq. (1). Conversely, in the results of Pageau et al. (1994) (Fig. 3a) the singular terms keep their status independently of the materials combination. In a third problem, Pageau et al. (1994) set E3/E1= 10.0 and varied E2/E1. The expanded graph, next to the original, is shown in Fig. 4b, where the imaginary parts of the roots (i.e., results with Re λ > 1) were colored, in order to identify to which real parts they are referring to. Their curves were also identified with the same letters E and H of the real parts, but in a blue tone. By expanding the original graph with a third value of λk in Eq. (1), it became clear that the three curves (AC, BG and FD) intersect 7 themselves at E2/E1= 1.0. Thus, in this case, the original results were only complemented. Figure 4: Study of the graph (Fig. 11, in the original source) of Pageau et al. (1994). Considering that the formulation allows to solve problems of bimaterial corners inclusively (for that, the same parameters are used for materials 2 and 3, and for the angles which define the interfaces), the bimaterial corner graph presented by Pageau et al. (1994) was also re-studied. It is shown in Fig. 5 and, as in Fig. 3, the results show that, unlike the original graph, the segments CF and DE intersect each other for E2/E1 ∼ =10.0. Also, the curves ACF, BH and GDE intersect each other at E2/E1= 1.0. The same comments made for results in Fig. 3 can be applied for this case. Although the values of the orders of stress singularities remain almost the same at both sides of the intersection, it is mandatory to correctly relate each term of Eq. (1) to the correspondent value of the order of stress singularity. 4. Discussion In a practical interpretation from a LEFM point of view, each λkis related to a deformation mode of the problem, i.e., a term of the finite sum expressed in Eq. (1). Thus, the λkwhich generally defines the first mode is the smaller one, indicating more potential to generate stress singularities than the other modes. In the classical crack problem, mode I is called the opening mode and it is associated with normal stresses. Mode II is called the sliding mode and it is associated with in-plane shear stresses. Finally, mode III is called the 8