Overview and applicability of residual stress estimation of film-substrate structure

Residual stresses arising from thermal mismatch in layered structures rank among the major causes of mechanical failures in light-emitting diodes, integrated circuits, electronic packages, and micro-electro-mechanical systems. Applying analytical solutions to predict or calculate residual stresses’ magnitude and distribution in multilayer film-substrate system has been widely adopted by many researchers. These researches are based on multilayer theories of film-substrate systems, such as Suhir's formula, Stoney's equation, and extend Stoney's equations. To discuss and distinguish the characteristics of these approaches, finite element analysis numerical solutions and multilayer theory analytical solutions are compared and analyzed. This encompasses the theories’ application spectrum as well as their prediction capability. In addition, this work not only discusses the theories’ property and workability but also demonstrate the feasibility of the finite element method (FEM) and bilayer theories in experiment. The experimental result demonstrates that FEM is a reliable approach in predicting the mechanical behavior of multilayer structures. Hence, when calculating or predicting thin film stress using the aforementioned theories, the methodology proposed in this research can be employed to effectively validate the feasibility of these theories.     

Low velocity impact analysis of sandwich plates with functionally graded face sheets

This article deals with the study of low velocity impact response on sandwich plates with functionally graded face sheets. High-order sandwich plate theory is improved by considering the in-plane stresses of the core that usually are ignored in the analysis of sandwich structures. A new approach is used to reduce the equations of motion from 27 equations to 15 equations and then solving them for both unsymmetric and symmetric sandwich plates. The model is also checked by finite element simulation and by comparing with other references for validation. A parametric study is done for various geometrical and mechanical properties.     

A hybrid finite element formulation of the linear biphasic equations for hydrated soft tissue

A hybrid finite element method has been developed for application to the linear biphasic model of soft tissues. The biphasic model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, one solid and one fluid, and employs mixture theory to derive governing equations for its mechanical behaviour. These equations are time dependent, involving both fluid and solid velocities and solid displacement, and will be solved by spatial finite element and temporal finite difference approximation. The first step in the derivation of this hybrid method is application of a finite difference rule to the solid phase, thus obtaining equations with only velocities at discrete times as primary variables. A weighted residual statement of the temporally discretized governing equations, employing C° continuous interpolations of the solid and fluid phase velocities and discontinuous interpolations of the pore pressure and elastic stress, is then derived. The stress and pressure functions are chosen so that the total momentum equation of the mixture is satisfied; they are jointly referred to as an equilibrated stress and pressure field. The corresponding weighting functions are chosen to satisfy a relationship analogous to this equilibrium relation. The resulting matrix equations are symmetric. As an illustration of the hybrid biphasic formulation, six-noded triangular elements with complete linear, several incomplete quadratic, and complete quadratic stress and pressure fields in element local co-ordinates are developed for two dimensional analysis and tested against analytical solutions and a mixed-penalty finite element formulation of the same equations. The hybrid method is found to be robust and produce excellent results; preferred elements are identified on the basis of these results.     

Mechanical system reliability analysis using a combination of graph theory and Boolean function

A new method based on graph theory and Boolean function for assessing reliability of mechanical systems is proposed. The procedure for this approach consists of two parts. By using the graph theory, the formula for the reliability of a mechanical system that considers the interrelations of subsystems or components is generated. Use of the Boolean function to examine the failure interactions of two particular elements of the system, followed with demonstrations of how to incorporate such failure dependencies into the analysis of larger systems, a constructive algorithm for quantifying the genuine interconnections between the subsystems or components is provided. The combination of graph theory and Boolean function provides an effective way to evaluate the reliability of a large, complex mechanical system. A numerical example demonstrates that this method an effective approaches in system reliability analysis.     

张力膜结构初始形态分析的曲面四边形单元

张力膜结构的初始形状不能随意选择,它必须符合平衡条件和建筑使用要求。根据几何非线性有限元理论,提出张力膜结构初始形态分析的8结点曲面四边形等参单元。通过建立曲线坐标,在应变的线性部分引入法向位移及单元曲率和扭率的影响,推导了张力膜结构的单元刚度矩阵和结点力列阵。采用完全的Newton-Raphson迭代法求解非线性方程组。数值算例表明该单元是一种高效、稳定和可靠的单元。     

模糊随机结构屈曲问题的区间有限元法

利用可靠度理论中的置信区间和模糊集理论中的λ截集方法,可以把结构中的随机参数和模糊参数转化为区间数。这时模糊随机有限元平衡方程转化为区间方程组,因此利用区间运算方法或蒙特卡洛直接抽样法可以求解模糊随机结构屈曲问题,并且得到一个区间解。如果结合结构稳定性理论,在某些特定的情况下,其计算量与求解一个相应的确定性问题有限元法的计算量相当。     

The effect of introducing increased-reliability-risk electronic components into 3rd generation telecommunications systems

In this paper, the dependability of 3rd generation telecommunications network systems is studied. Special attention is paid to a case where increased-reliability-risk electronic components are introduced to the system.The paper consists of three parts: First, the reliability data of four electronic components is considered. This includes statistical analysis of the reliability test data, thermo-mechanical finite element analysis of the printed wiring board structures, and based on those, a field reliability estimate of the components is constructed. Second, the component level reliability data is introduced into the network element reliability analysis. This is accomplished by using a reliability block diagram technique and Monte Carlo simulation of the network element. The end result of the second part is a reliability estimate of the network element with and without the high-risk component. Third, the whole 3rd generation network having multiple network elements is analyzed. In this part, the criticality of introducing high-risk electronic components into a 3rd generation telecommunications network is considered.     

Unified formulation of geometrically nonlinear refined beam theories

By using the Carrera Unified Formulation (CUF) and a total Lagrangian approach, the unified theory of beams including geometrical nonlinearities is introduced in this article. According to CUF, kinematics of one-dimensional structures are formulated by employing an index notation and a generalized expansion of the primary variables by arbitrary cross-section functions. Namely, in this work, low- to higher-order beam models with only pure displacement variables are implemented by utilizing Lagrange polynomial expansions of the unknowns on the cross section. The principle of virtual work and a finite element approximation are used to formulate the governing equations, whereas a Newton-Raphson linearization scheme along with a path-following method based on the arc-length constraint is employed to solve the geometrically nonlinear problem. By using CUF and three-dimensional Green-Lagrange strain components, the explicit forms of the secant and tangent stiffness matrices of the unified beam element are provided in terms of fundamental nuclei, which are invariants of the theory approximation order. A symmetric form of the secant matrix is provided as well by exploiting the linearization of the geometric stiffness terms. Various numerical assessments are proposed, including large deflection analysis, buckling, and postbuckling of slender solid cross-section beams. Thin-walled structures are also analyzed in order to show the enhanced capabilities of the present formulation. Whenever possible, the results are compared to those from the literature and finite element commercial software tools.     

索杆膜空间结构协同形态分析

针对索杆膜空间结构进行了协同形态分析。根据现有的索膜非线性有限元找形理论和索杆体系找力方法,并考虑施工成形过程,提出了一种新的适合于索杆膜空间结构形态分析的求解方法——形态迭代求解法,并编制了相应的程序,实现了算法。最后通过一个算例验证了形态迭代求解法的正确性并得到如下结论:按该文算法得到的找形结果满足力平衡方程、相容方程和建筑师给定的边界条件。设计验算时被动张拉索的预应力值应通过计算得到。该文程序可用于索杆膜空间结构考虑施工成形的设计验算。     

Design sensitivity analysis of non-linear structural systems part I: Theory

A unified approach is presented for design sensitivity analysis of non-linear structural systems that include truss, beam, plane elastic solid and plate components. Both geometric and material non-linearities are treated. Sizing design variables, such as thickness and cross-sectional areas of components of individual members and built-up structures, are considered. A distributed parameter structural design sensitivity analysis approach is used that retains the continuum elasticity formulation throughout the derivation of design sensitivity analysis results. Using this approach and an adjoint variable method, expressions for design sensitivity in terms of design variations are derived in the continuous setting which can be evaluated numerically using analysis results of finite element analysis. Both total Lagrangian and updated Lagrangian formulations in non-linear analysis of solid mechanics are used for design sensitivity analysis. Numerical implementation of design sensitivity analysis results using existing finite element code will be presented in Part II of this paper.     

Combination of finite element and reliability methods in nonlinear fracture mechanics

This paper presents a probabilistic methodology for nonlinear fracture analysis in order to get decisive help for the reparation and functioning optimization of general cracked structures. It involves nonlinear finite element analysis. Two methods are studied for the coupling of finite element with reliability software: the direct method and the quadratic response surface method. To ensure the response surface efficiency, we introduce new quality measures in the convergence scheme. An example of a cracked pipe is presented to illustrate the proposed methodology. The results show that the methodology is able to give accurate probabilistic characterization of the J-integral in elastic–plastic fracture mechanics without obvious time consumption. By introducing an “analysis re-using” technique, we show how the response surface method becomes cost attractive in case of incremental finite element analysis.     

Three-dimensional solid-to-beam transition elements for structural dynamics analysis

Complex structural components such as those encountered in many industrial applications may generally be considered as being composed of shell- or beam-like portions linked to three-dimensional solid continua. When discretized into finite elements, these structures present geometrical and mathematical difficulties at the connections between the different element types since the nodal degrees of freedom allocated to the solid, shell and beam elements are incompatible with each other. The development of specific and reliable transition finite elements is, thus, of outstanding practical importance. This paper presents efficient C⁰ compatible transition elements with a variable number of nodes for modelling solid to beam junctions. Based upon the standard isoparametric solid and beam formulations, the current approach includes the properties of both solids and beams, verifies the basic continuity, smoothness and completeness criteria inherent in the finite element convergence requirements, and avoids the shear locking phenomenon typical of C⁰ elements by using a strain-projection method. Several numerical examples which compare this formulation to analytical and experimental solutions are provided in order to show the applicability and efficiency of this approach.     

A theoretical approach to the interaction between buckling and resonance instabilities

This article deals with the interaction between buckling and resonance instabilities of mechanical systems. Taking into account the effect of geometric nonlinearity in the equations of motion through the geometric stiffness matrix, the problem is reduced to a generalized eigenproblem where both the loading multiplier and the natural frequency of the system are unknown. According to this approach, all of the forms of instabilities intermediate between those of pure buckling and pure forced resonance can be investigated. Numerous examples are analyzed, including discrete mechanical systems with one to n degrees of freedom, continuous mechanical systems, such as oscillating deflected beams subjected to a compressive axial load, as well as oscillating beams subjected to lateral–torsional buckling. A general finite element procedure is also outlined, with the possibility to apply the proposed approach to any general bi- or tri-dimensional framed structure. The proposed results provide a new insight in the interpretation of coupled phenomena such as flutter instability of long-span or high-rise structures.     

Prediction of the behaviour of copper alloy components under complex loadings by electro-thermomechanical coupled simulations

ABSTRACT

Electronic devices must be served with electric power for different reasons. A robust and reliable electric connectivity is often realised by electric connectors. For its leading properties, precipitation hardened copper alloys are widely used for designing connectors with high level mechanical or conductance properties. However, copper alloys show a characteristic stress relaxation under mechanical or thermal loads. Finite element analysis is a standard method to design and optimise components with respect to reliability and performance. Hence, a material model considering the characteristic of the mechanical properties and allowing for the simulation of time and temperature dependent elasto-viscoplastic material behavior was developed at the Fraunhofer IWM. The parameters of the model were determined using tensile and relaxation test data of a C19010 alloy. The material model is applied to electro-thermomechanical coupled finite element simulations of connectors with different load histories. The goal of the simulations is the analysis of the impact of stress relaxation on the mechanical properties of systems over time. From the numerical results with the new model it is shown, how stress relaxation influences the connector clamping forces or contact pressure, respectively, in dependence with time or temperature. The simulation results documents that stress relaxation has to be taken into account in finite element simulations during the designing process of electrical devices.

This paper is part of a Thematic Issue on Copper and its Alloys.     

A Phenomenological Theory and Numerical Procedure for Chemo-Mechanical Coupling Behavior of Hydrogel

Coupling and interaction of multi-physical fields exist in hydrogel consisting of a fluid and a solid under external stimulus. In this paper, a phenomenological theory for chemo-mechanical coupling behavior and finite element formulation are developed, based on the thermodynamic laws. The free energy function is constructed and used to derive the constitutive equations and governing equations for a linear coupling system including a chemical effect. Equivalent integral forms of the governing equations and coupled finite element equations are obtained by a variational principle. Numerical examples demonstrate the interaction of chemical and mechanical effects of hydrogel under external force loadings and chemical stimuli. It is shown that the chemo-mechanical coupling behavior of hydrogel can be described by the theory and numerical method presented in this paper.