Multimaterial Corners Research Articles

A semi-analytical matrix formalism for stress singularities in anisotropic multi-material corners with frictional boundary and interface conditions

A new general semi-analytic procedure for the characterisation of singular asymptotic elastic states in the vicinity of the apex of linearly elastic anisotropic multi-material corners, including frictional contact, is developed and tested. The corners can consist of any finite number of homogeneous wedges defined by polar sectors. The variability of configurations covered is enormous as frictional contact can be considered on one or more outer boundary surfaces or interfaces, in addition to a large variety of homogeneous boundary conditions and perfect bonding or frictionless sliding interface conditions between wedges in the corner. The Coulomb rate-independent and dry frictional contact law is assumed. One of the novelties is that, in addition to the singularity exponent λ, the angle ω of the friction tangential stress vector on each frictional contact surface is an a priory unknown to be determined by solving a nonlinear corner eigensystem. The procedure, which considers power-law stress singularities, is based on the Stroh formalism of anisotropic elasticity, assuming generalised plane strain (2.5D) conditions, and on the semi-analytic matrix formalism for wedge transfer-matrices and boundary and interface condition matrices. This makes it, firstly, very suitable for computational implementations, secondly, very efficient especially in cases with several perfectly bonded homogeneous wedges, and, thirdly, very accurate due to its fully semi-analytic nature. The code developed is tested by solving a large variety of examples, comparing the present results with those obtained by solving closed-form corner-eigenequations deduced by previous authors for specific useful practical configurations, confirming the extremely high accuracy of the present code in the computation of λ and ω. The differences observed in some cases with anisotropic materials are explained by the fact that some of the previous authors did not take the true 3D Coulomb friction law into account.

A powerful matrix formalism for stress singularities in anisotropic multi-material corners. Homogeneous (orthogonal) boundary and interface conditions

A computational code based on a semianalytic procedure for the determination of the characteristic exponents and the singular stress and displacement fields in multi-material corners is developed. Linear elastic anisotropic materials under generalized plane strain state are considered. This code is a universal computational tool able to analyze both open and closed (periodic) corners, composed of one or multiple materials with isotropic, transversely isotropic or orthotropic constitutive laws, covering both mathematically non-degenerate and degenerate materials in the framework of Stroh formalism. In multi-material corners, material junctions with perfectly bonded or frictionless sliding interfaces can be studied. The considered homogeneous boundary conditions cover stress free and fixed faces, or faces with some restricted or allowed direction of displacements, defined either in the reference frame aligned with the cylindrical coordinate system or in an inclined reference frame. The code is developed in MATLAB and it is based on the Stroh matrix formalism for anisotropic elasticity, the concept of transfer matrix for single material wedges, and on the matrix formalism for homogeneous (orthogonal) boundary conditions. The comparison of the characteristic exponents obtained by the present code and by the solution of closed-form eigenequations available in the literature, has a two-fold objective, first to exhaustively check the general computational implementation of the matrix formalism presented, and second to check the closed-form expressions of eigenequations for relevant specific cases published in the literature.

Analysis of stress singularity in adhesive joints using meshless methods

Recent years saw a rise in the application of bonding techniques in the engineering industry. This fact is due to the various advantages of this technique when compared to traditional joining methods, such as riveting or bolting. The growth of bonding methods demands faster and more powerful tools to analyze the behavior of products. For that reason, adhesive joints have been the subject of intensive investigation over the past few years. Recently, a fracture mechanics based approach emerged with great potential to evaluate joint behavior, called Intesity of Singular Stress Fields (ISSF), similar to the Stress Intensity Factor (SIF) concept. However, it allows the study of multi-material corners and does not require an initial crack. This approach was not yet tested with meshless methods. The present work intends to fill this gap, resorting to the Radial Point Interpolation Method (RPIM). With this purpose, adhesive joints with four different overlap lengths (LO) bonded with a brittle adhesive were studied. The interface corner's stresses were also evaluated. The predicted strengths were compared with the experimental data to assess the accuracy of the applied methods. In conclusion, the ISSF criterion proved to be applicable to meshless methods, namely the RPIM.

Removing stress singularities in standard tests of composite materials

In many standard tests involving composite material, specially those needing tabs at the ends of the sample, configurations of multimaterial corners, with abrupt changes in geometry and material properties, typically appear giving rise to stress fields which achieve very high stresses and theoretically are unbounded. These stress intensifications may promote premature failures, or simply, alterations of the uniform stress state which is expected to occur in the specimen, and the test strength, calculated as a certain load divided by a certain failure area, may not be representative of the real strength of the material under analysis. In the present work three examples are presented, in which, by means of an addecuate selection of the geometrical parameters defining the test configuration which do not affect the mechanical property to be measured, the stress singularities at these multimaterial corners are removed (or at least reduced to a minimum). The three cases are: a) tests in tension and shear (Iosipescu) in bimaterial coupons, b) the off-axis test for the intralaminar shear strength in composite materials and c) a compression test of thick laminates in composite materials. In two of the three cases experimental results are available to corroborate that the removal of the stress singularities gives rise to higher strength values than those obtained in the original test configurations.

Computational semi-analytic code for stress singularity analysis

Problems of stress singularities in single or multi-material corners have been addressed by many authors over the years. Most of the authors presented closed-form corner-eigenequations for special cases, and often there is no easy way to check if the solution is correct. In this work, we present a general computational tool that can solve many different cases of stress singularity problems for multi-material corners under generalized plane strain. The semi-analytic code is based on the matrix formalism presented in Mantič et al. (1997, 2014); Barroso et al. (2003); Herrera-Garrido et al. (2022) and is developed in MATLAB. The following boundary conditions are implemented: stress-free, fixed, some restricted or allowed direction of displacements (defined either in the reference frame aligned with the cylindrical coordinate system or in an inclined reference frame), or frictional sliding. The following interface condition between two consecutive materials are implemented: perfectly bonded, and frictionless or frictional sliding. The code can analyze both open and closed (periodic) corners, composed of one or multiple materials with isotropic, transversely isotropic or orthotropic (with any orientation) constitutive laws. The code has proven to be a reliable, very accurate, robust and easy-to-use tool, which has been verified by comparing the results computed with those obtained by other authors. A summary of the corner singularity problems solved is presented. The results of the corner singularity analysis obtained by the code can be further used for prediction of crack onset at the corner tip by the Coupled Criterion of Finite Fracture Mechanics and FEM, see García and Leguillon (2012) and references therein.

Singular enrichment for multi-material corners with application to assessing the risk of fracture in semiconductor devices

A computational method for multi-material corners that enables accurate estimations of generalized stress intensity factors with relatively coarse and uniform discretizations is described in this paper. Specifically, explicitly modeled geometries of material junctions, crack tips and debonded interfaces are isoparametrically and hierarchically enriched to construct approximations with the known local behavior. The analysis procedure is termed enriched isogeometric analysis since the enrichments are constructed directly (explicitly) on the parametric spline geometry of the vertex or crack face unlike in procedures where the influence of the enriching entity is implicitly inferred on the underlying mesh. The developed method allows direct extraction of generalized stress intensity factors without needing a posteriori evaluation of path independent integrals for decisions on crack propagation. The numerical implementation is first validated through analysis of a bi-material corner and growth of an inclined crack in a homogeneous solid. Next, a detailed analysis of a representative back end of line structure of a semiconductor chip with multiple singular corners is carried out to identify critical corners susceptible to crack initiation. The developed procedure demonstrates rapid convergence to the solution stress intensity factors with relatively fewer degrees of freedom, even with uniformly refined coarse discretizations.

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

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.

Failure initiation criterion in bonded joints with composites based on singularity parameters and practical procedure for design purposes

Composite to metal adhesively bonded joints generates critical points where geometry and material properties change abruptly. These points (multimaterial corners) are potential locations for failure initiation. There exist proposals predicting the initiation of failure at these multimaterial corners using a Fracture Mechanics approach, in which the Generalized Stress Intensity Factors (GSIFs) at the multimaterial corner control the failure initiation. The calculation of these GSIFs at anisotropic multimaterial corners involves not-straightforward calculations of the stress singularities and characteristic angular functions, numerical modeling of the joint and a careful postprocessing of the results. In this study a parametric Finite Element Analysis has been carried out allowing the generation of plots to calculate the GSIFs for unit values of the axial force, shear force and bending moment at one end of the overlap length. These results allow calculating the GSIFs at the multimaterial corners by a simple beam analysis of the joint, the use of these plots and the application of the superposition principle, for their use in the prediction of failure initiation by means of the singular parameters of the joint. Additionally, experiments have been carried out to propose an explicit failure criterion based on GSIF and Generalized Fracture Toughness values which fit very well with double-lap joint test results.

Simulations of Damage, Crack Initiation, and Propagation in Interlayer Dielectric Structures: Understanding Assembly-Induced Fracture in Dies

Performance enhancement by lowering the dielectric constant of interlayer dielectric (ILD) materials often compromises the mechanical integrity of the dielectric stack. At the present time, fracture in the ILD stacks induced by assembly to either an organic substrate or a die stack (3-D) is an important reliability consideration. These interactions include what is popularly referred to as the chip-package interactions. In this paper, we develop insights on the potential crack initiation site within the ILD, die-substrate geometrical parameters that cause most damage, as well as insights on the manufacturing process that is critical to failure. Towards this end, we utilize analytical models based on classical elasticity theory as well as sophisticated numerical techniques that are capable of nucleating and propagating cracks at arbitrary locations within the structure without remeshing. Specifically, we analytically estimate the strength of singularities at all the possible multimaterial corners in the ILD stack to provide insight on the likely damage nucleation sites for various material configurations in the ILD stack. Two novel numerical approaches are used for fracture simulation. In the first, cracks are modeled as discontinuous enrichments over an underlying continuous behavioral approximation. In the second approach, the underlying material description is enriched with a cohesive damage description whose stiffness is evolved according to a prescribed damage law. Multilevel finite-element models are used to determine the load imposed on the ILD structure by the substrate. Maximum damage induced in the ILD stack by the above load is used as an indicator of the reliability risk. Parametric simulations are conducted by varying ILD material, die size, die thickness, as well as the solder material. Through analytical models of bonded assemblies, we identify groups of relevant dimensionless parameters to relate the numerically estimated damage in ILD stacks to the die/substrate material and geometrical parameters. We demonstrate that the damage in the ILD stack is least when the flexural rigidity of the die is matched to that of the assembled substrate. We also demonstrate that ILD damage is only weakly correlated to shear deformation on the die surface due to assembly. We generalize the above observations into mathematical fits (for use as design rules) relating damage in ILD stacks to ILD material choice, relative substrate flexural rigidity, and die size.

Determination of Generalized Fracture Toughness in composite multimaterial closed corners with two singular terms – Part I: Test proposal and numerical analysis

In the present work a general procedure for the experimental evaluation of Generalized-Fracture-Toughness values at multimaterial closed corners is defined. The lack of symmetries in the stress fields at multimaterial corners with non-isotropic materials makes it extremely difficult to define standard test procedures for Generalized-Fracture-Toughness determination. The proposed procedure is suitable for closed corners (all wedge faces being bonded) having two singular terms with real singularity exponents. The procedure begins by finding the load configurations at which one of the singular terms in the asymptotic series expansion vanishes, allowing the failure to be controlled by the other, non-vanishing, singular term. An example of a typical bimaterial corner in an adhesively bonded joint with composite materials is extensively analyzed and suitable test configurations for the experimental evaluation of the Generalized-Fracture-Toughness values K1C and K2C associated to the singular terms are defined. The experimental part of this work using a novel modified configuration of the Brazilian test will be described in a follow up paper.

A least squares procedure for the evaluation of multiple generalized stress intensity factors at 2D multimaterial corners by BEM

Detailed characterization of linear elastic stress states at corners and crack tips requires knowledge of the stress singularity orders, the characteristic angular functions and the generalized stress intensity factors (GSIF). Typically a high accuracy is found in the literature for the evaluation of the stress singularity orders and characteristic angular functions (numerically computed from analytical expressions in most cases). Nevertheless, GSIF values, evaluated by means of a numerical model using FEM or BEM and usually by postprocessing the results, are often reported with a lower level of confidence. A robust procedure is presented in this work for the evaluation of the GSIF at multimaterial corners. The procedure is based on a simple least squares technique involving stresses and/or displacements, computed by BEM, at the neighborhood of the corner tip. A careful verification of the robustness and accuracy of the procedure using a few benchmark problems in the literature has been carried out. Applications of the procedure developed to the evaluation of GSIFs appearing at corners in metal-composite adhesive lap joints are presented.

Representativity of thermal stresses in designing composite joints based on singular stress states at multimaterial corners

Using adhesives curing at temperature in bonded joints between dissimilar materials (composites and metals) gives rise to residual stresses due to the difference in the value of the thermal expansion coefficients of the adherents. The presence of multimaterial corners in these adhesive joints originates critical points where failure is likely to occur. Numerical results of several double-lap joints of aluminium to different carbon fibre laminates including the thermal stresses during curing are presented. The uniform temperature variation (from curing to room temperature) has been shown, numerically, to have a significant influence on the local stress field in the neighbourhood of these corners. Due to the fact that high stress gradients are developed at these corners in the curing process and stress relaxation effects may occur due to the viscoelastic behaviour of the polymeric adhesive at room temperature, experimental tests have also been carried out to study the influence of uniform temperature fields on the strength of these joints based on the parameters which define the singular stress state at these multimaterial corners. The experiments show that the thermal stresses during curing do not significantly alter the behaviour, type and load failures of the joint.

Representativity of the singular stress state in the failure of adhesively bonded joints between metals and composites

Several critical points, where both geometry and material properties change abruptly, arise in an adhesively bonded lap joint between a metallic and a composite material. These critical points, called multimaterial corners or cross-points, at which the linear theory of elasticity predicts unbounded (singular) stresses, are potential points for failure initiation. In this work, a complete stress characterization at these multimaterial corners has been carried out to analyze, after a preliminary experimental test program, the suitability of the application of the parameters defining the singular stress state in the characterization of the failure of these joints. The comparative analysis of the numerical and experimental results obtained show that the singular stress state controlled by a series of the generalized stress intensity factors is controlling the failure path at these corners.

Computing stress singularities in transversely isotropic multimaterial corners by means of explicit expressions of the orthonormalized Stroh-eigenvectors

Composite materials reinforced by unidirectional long fibers behave macroscopically as homogeneous transversely isotropic linear elastic materials. A general, accurate and computationally efficient procedure for the evaluation of singularity exponents and singular functions characterizing singular stress fields in multimaterial corners involving this kind of material is presented in this paper. To take full advantage of the sextic Stroh formalism of anisotropic elasticity applied to this particular problem, the complete set of explicit expressions of the eigenvalues and eigenvectors of the real 6 × 6 fundamental elasticity matrix N has been deduced for all the non-degenerate and degenerate (repeated roots of the sextic Stroh equation) cases. These expressions will also facilitate further applications of the Stroh formalism to these materials. Several numerical examples of singularity analysis of multimaterial corners appearing in adhesively bonded joints and damaged cross-ply laminates of composite materials are presented.

Singularity parameter determination in adhesively bonded lap joints for use in failure criteria

A procedure for the singularity characterization of anisotropic multimaterial corners which typically appear in adhesively bonded lap joints between metals and composites is presented and implemented in the present work. The characterization in terms of characteristic exponents (stress singularities), characteristic functions and generalized stress intensity factors allows the definition of an oriented test program in order to analyze the suitability of a singularity based failure criterion for structures of this type.

Evaluation of power-logarithmic singularities, T-stresses and higher order terms of in-plane singular stress fields at cracks and multi-material corners

The scaled boundary finite-element method is extended to analyze the in-plane singular stress fields at cracks and multi-material corners. A complete singular stress field is represented semi-analytically as a series of matrix power functions of the radial coordinate originating from the singular point. This method is capable of directly evaluating orders of singularity, stress intensity factors, T-stresses, higher order terms, angular distributions of stresses for each term and, if present, power-logarithmic singularities. In this method, the singular functions are represented analytically and are not evaluated close to the singular point. Numerical examples are calculated to demonstrate the simplicity and to evaluate the accuracy of the scaled boundary finite-element method.

Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.

Singularity analysis of anisotropic multimaterial corners

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.