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At V-notched tips in specimens made of quasi-brittle materials a small damaged or plastic zone is evident that cannot not be neglected in terms of dissipated energy and stress state, although it is small. Herein, to predict the failure initiation at the notch tip, we extend the finite fracture mechanics (FFM) coupled criterion, which requires a simultaneously fulfillment of an energy and a stress criteria. In the small damaged zone, a damage model is introduced so to decrease the effective Young's modulus in a power law in terms of the distance to the notch tip in such a way that the stress field remains bounded. It seems particularly suited to quasi-brittle materials, since no diffuse damage can occur. This damage zone is coupled to the FFM criterion to provide the necessary condition for failure initiation. Under the assumption that the damaged zone and the virtual crack extension are small, matched asymptotic expansions are used. It is shown that the damaged zone grows first, proportionally to the square of the applied load and then, above a threshold, a virtual crack of a given length simultaneously satisfies the energy and stress criteria, and failure occurs. The approach allows taking into account varying tensile strength and material toughness in the damaged zone, as may reasonably be expected. Moreover, it is shown that the same coupled stress-energy criterion can directly be applied to quasi-brittle materials by appropriately using the actual material toughness as measured on a cracked specimen. [ABSTRACT FROM AUTHOR]

Abstract: We present a high-order method employing Jacobi polynomial-based shape functions, as an alternative to the typical Legendre polynomial-based shape functions in solid mechanics, for solving dynamic three-dimensional geometrically nonlinear elasticity problems. We demonstrate that the method has an exponential convergence rate spatially and a second-order accuracy temporally for the four classes of problems of linear/geometrically nonlinear elastostatics/elastodynamics. The method is parallelized through domain decomposition and message passing interface (MPI), and is scaled to over 2000 processors with high parallel performance. [Copyright &y& Elsevier]

Abstract: Dynamic von-Kármán plate models consist of three coupled non-linear, time-dependent partial differential equations. These equations have been recently solved numerically [Kirby, R., Yosibash, Z., 2004. Solution of von-Kármán dynamic non-linear plate equations using a pseudo-spectral method. Comp. Meth. Appl. Mech. Eng. 193 (6–8) 575–599 and Yosibash, Z., Kirby, R., Gottlieb, D., 2004. Pseudo-spectral methods for the solution of the von-Kármán dynamic non-linear plate system. J. Comp. Phys. 200, 432–461] by the Legendre-collocation method in space and the implicit Newmark-β scheme in time, where highly accurate approximations were realized. Due to their complexity, these equations are often reduced by discarding some of the terms associated with time derivatives which are multiplied by the plate thickness squared (being a small parameter). Because of the non-linearities in the system of equations we herein quantitatively investigate the influence of these a-priori assumption on the solution for different plate thicknesses. As shown, the dynamic solutions of the so called “simplified von-Kármán” system do not differ much from the complete von-Kármán system for thin plates, but may have differences of few percent for plates with thicknesses to length ratio of about 1/20. Nevertheless, when investigating the modeling errors, i.e. the difference between the various von-Kármán models and the fully three-dimensional non-linear elastic plate solution, one realizes that for relatively thin plates (thickness is 1/20 of other typical dimensions), this difference is much larger. This implies that the simplified von-Kármán plate model used frequently in the literature is as good as an approximation as the complete (and more complicated) model. As a side note, it is shown that the dynamic response of any of the von-Kármán plate models, is completely different compared to the linearized plate model of Kirchhoff–Love for deflections of an order of magnitude as the plate thickness. [Copyright &y& Elsevier]

The von-Kármán nonlinear, dynamic, partial differential system over rectangular domains is considered, and numerically solved using both the Chebyshev-collocation and Legendre-collocation methods for the spatial discretization and the implicit Newmark-β scheme combined with a non-linear fixed point algorithm for the temporal discretization. As the system is non-linear, involving operators of different orders, different timescales, and may contain initial/boundary incompatible conditions at the domain''s corners, it is our aim to highlight some of the difficulties inherent in its numerical treatment using collocation methods. We begin by examining the Chebyshev-collocation scheme considered in [Comput. Meth. Appl. Mech. Engrg. 193 (6–8) (2004) 575]. In that work, filtering was used to stabilize the full von-Kármán system, however the source of the instability was not discussed. We first show that IC/BC incompatibilities are not the culprits and demonstrate empirically that the source of the instability was the numerical treatment of the linear cross-derivative terms found in the in-plane equations. We prove mathematically that although the continuous linear system is well-posed, the Chebyshev-collocation solution of problems involving cross-derivatives and second-order time derivatives is unstable. Instead of adding sufficient filtering to counter the energy growth due to the linear terms, we employ the Legendre-collocation method. We show empirically and prove that the Legendre-collocation scheme for the linear system has the necessary characteristics to remain stable. Nevertheless, the collocation treatment of the non-linear terms still causes the solution to be unstable at long time. We thus return to advocating a filtering solution, but in this case only filtering the non-linear terms to compensate for the inherent aliasing error due to our collocation treatment of these terms. We conclude by comparing the results of filtered Chebyshev-collocation solution, non-linear terms filtered Legendre-collocation solution, and a simplified von-Kármán (solution omitting the inertial terms in the in-plane equations) as presented in (loc. cit). [Copyright &y& Elsevier]

The von-Ka´rma´n non-linear, dynamic, partial differential system over rectangular domains is solved by the Chebyshev-collocation method in space and the implicit Newmark-β time marching scheme in time. In the Newmark-β scheme, a non-linear fixed point iteration algorithm is employed.We monitor both temporal and spatial discretization errors based on derived analytical solutions, demonstrating highly accurate approximations. We also quantify the influence of a common modeling assumption which neglects the in-plane inertia terms in the full von-Ka´rma´n system, demonstrating that it is justified. A comparison of our steady-state von-Ka´rma´n solutions to previous results in the literature and to a three-dimensional high-order finite element analysis is performed, showing an excellent agreement. Other modeling assumptions such as neglecting in-plane quadratic terms in the strain expressions are also addressed. [Copyright &y& Elsevier]

Thermo-mechanical failures may occur in the passivation layer of micro-electronic devices during the fabrication process. These are in form of cracks which initiate at keyhole corners. In order to predict and eventually prevent these cracks a failure criterion is presented, based on an average value of the elastic strain energy in the vicinity of a reentrant corner of any angle. The proposed strain energy density (SED) failure criterion is validated by a test including 24 full size wafers which have been fabricated with different parameters: the interconnects (metal lines) height, the passivation thickness, and the passivating plasma power which was shown to correlate with the mechanical properties of the passivation layer. For each wafer, a FE model has been constructed, and the SED computed. It has been clearly shown, that above the critical value of SEDcr[R=0.15μm]≈1000 [J/m3], all wafers manufactured were cracked. The SED criterion seems to correlate well with the empirical observations, and may be used as a standard tool for the mechanical design of failure free micro-electronic devices. [ABSTRACT FROM AUTHOR]

A simple method for computing the flux intensity factors associated with the asymptotic solution of elliptic equations having a large convergence radius in the vicinity of singular points is presented. The Poisson and Laplace equations over domains containing boundary singularities due to abrupt change of the boundary geometry or boundary conditions are considered. The method is based on approximating the solution by the leading terms of the local symptotic expansion, weakly enforcing boundary conditions by minimization of a norm on the domain boundary in a least-squares sense. The method is applied to the Motz problem, resulting in extremely accurate estimates for the flux intensity factors. It is shown that the method converges exponentially with the number of singular functions and requires a low computational cost. Numerical results to a number of problems concerned with the Poisson equation over an L-shaped domain, and over domains containing multiple singular points, demonstrate accurate estimates for the flux intensity factors. © 1998 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We propose a full 3D benchmark problem for brittle fracture based on experiments as well as a validation in the context of phase-field models. The example consists of a series of four-point bending tests on graphite specimens with sharp V-notches at different inclination angles. This simple setup leads to a mixed mode (I + II + III) loading which results in complex yet stably reproducible crack surfaces. The proposed problem is well suited for benchmarking numerical methods for brittle fracture and allows for a quantitative comparison of failure loads and propagation paths as well as initiation angles and the fracture surface. For evaluation of the crack surfaces image-based 3D models of the fractured specimen are provided along with experimental and numerical results. In addition, measured failure loads and computed load–displacement curves are given. To demonstrate the applicability of the benchmark problem, we show that for a phase-field model based on the Finite Cell Method and multi-level h p -refinement the complex crack surface as well as the failure loads can be well reproduced. • Quantitative full 3D benchmark problem for brittle fracture based on experiments. • Mixed mode (I + II + III) loading results in complex and reproducible crack surfaces. • Validation of a phase-field model for the FCM shows very good correspondence. • Complete experimental and numerical data are provided. [ABSTRACT FROM AUTHOR]

Abstract: The recently introduced Finite Cell Method (FCM) combines the fictitious domain idea with the benefits of high-order Finite Elements. While previous publications concentrated on single-field applications, this paper demonstrates that the advantages of the method carry over to the multi-physical context of linear thermoelasticity. The ability of the method to converge with exponential rates is illustrated in detail with a benchmark problem. A second example shows that the Finite Cell Method correctly captures the thermoelastic state of a complex problem from engineering practice. Both examples additionally verify that, also for two-field problems, Dirichlet boundary conditions can be weakly imposed on non-conforming meshes by the proposed extension of Nitsche’s Method. [Copyright &y& Elsevier]

Abstract: The present work addresses the various ingredients required for reliable finite element simulations of cold isostatic pressing (CIP) of metal powders. A plastic constitutive model for finite deformation is presented and implemented into an explicit finite element (FE) code. The FE implementation is verified so that numerical errors (both temporal and spatial errors) are kept under control. Thereafter, uniaxial die compaction experiments are performed required for determining the material parameters in the constitutive model. Subsequently they are applied for the simulation of a “complex” CIP process. The experimental observations of the complex CIP process were used to validate the overall method by comparing the FE results (final dimensions and average relative density) to the experimental observations. The numerical results (final dimensions and relative density) are in good agreement with the experimental observations. [Copyright &y& Elsevier]