An Improved Analytical Model for Time-Dependent Shearing Deformation in Area-Array Interconnects

2004 ◽  
Vol 126 (1) ◽  
pp. 74-81
Author(s):  
S. Shakya ◽  
S. M. Heinrich ◽  
P. S. Lee
Keyword(s):  
Analytical Model ◽  
Elastic Solid ◽  
Step Function ◽  
Shear Displacement ◽  
Constitutive Law ◽  
Time Dependent ◽  
Correction Factors ◽  
Area Array ◽  
Linear Viscoelastic Material ◽  
Frequency Independent

An improved time-dependent analytical model is developed for predicting the maximum shearing displacement in an area-array electronic assembly under global thermal mismatch loading. The thermal loads are assumed to be uniform within the component and substrate, with both step-function and sinusoidal temperature histories being considered. The time-dependent effects in the array’s shear deformation are introduced in an approximate manner by modeling the interconnect material (solder) as a temperature-independent linear viscoelastic material. The viscoelastic constitutive law used for the solder is that of a three-parameter viscoelastic standard solid in distortion and an elastic solid in the hydrostatic mode. In the authors’ previous work the steady-state shear force in the joints was assumed to vary sinusoidally with a frequency-independent amplitude. This assumption has been relaxed in the present study, leading to improved accuracy. All results have been derived as closed-form correction factors to be applied to the easily calculated unconstrained shear displacement to obtain the maximum shear displacement. All the correction factors depend on prescribed geometric and material parameters of the component, substrate, and joints. The results have been presented in the form of dimensionless plots to aid in the analysis or design process, thereby providing convenient alternatives or supplements to time-consuming and expensive finite element analyses of entire assemblies.

An Analytical Model for Time-Dependent Shearing Deformation in Area-Array Interconnects

2000 ◽  
Vol 122 (4) ◽  
pp. 328-334 ◽  
Author(s):  
S. M. Heinrich ◽  
S. Shakya ◽  
J. Liang ◽  
P. S. Lee
Keyword(s):  
Analytical Model ◽  
Correction Factor ◽  
Closed Form Solution ◽  
Shear Loading ◽  
Form Solution ◽  
Constitutive Law ◽  
Time Dependent ◽  
Loading Frequency ◽  
Area Array ◽  
Frequency Correction

An analytical model is developed for predicting the time-dependent shearing displacement in area-array solder interconnects due to global CTE mismatch under thermal cycling. As a first step toward incorporating the creep deformation of the solder, the material is modeled as viscoelastic and temperature-independent. This permits one to invoke the correspondence principle of viscoelasticity to map the authors’ previously derived, closed-form solution for an elastic nonprismatic (concave, convex, or cylindrical) Timoshenko beam under shear loading into the associated viscoelastic solution. This leads to general analytical results for the frequency-dependent shear displacement amplitude in the critical joint. The results are expressed conveniently in terms of a “full-creep correction factor” and a “frequency correction factor,” which explicitly show the effects of the following parameters on the joint deformation: joint shape; array population; array, component, and substrate dimensions; viscoelastic material properties of the interconnect material; elastic properties of the component and substrate materials; and loading frequency. To demonstrate the technique for a particular viscoelastic constitutive law, the solder is assumed to behave elastically under hydrostatic loads and as a viscoelastic Kelvin solid under deviatoric conditions. For this special case the creep portion of the deformation is shown to be dependent upon only two dimensionless parameters: a dimensionless loading frequency and a material- and shape-dependent joint parameter. The results of the study may be useful in identifying design and process modifications that may improve the thermal fatigue life of area arrays. [S1043-7398(00)00404-7]

scholarly journals Time-Dependent Response of a Recycled C&D Material-Geotextile Interface under Direct Shear Mode

Materials ◽  
2021 ◽  
Vol 14 (11) ◽  
pp. 3070
Author(s):  
Fernanda Bessa Ferreira ◽  
Paulo M. Pereira ◽  
Castorina Silva Vieira ◽  
Maria de Lurdes Lopes
Keyword(s):  
Shear Stress ◽  
Stress Relaxation ◽  
Shear Displacement ◽  
Time Dependent ◽  
Shear Behaviour ◽  
Displacement Rate ◽  
Shear Mode ◽  
Direct Shear ◽  
Constant Displacement ◽  
Shear Box

Geosynthetic-reinforced soil structures have been used extensively in recent decades due to their significant advantages over more conventional earth retaining structures, including the cost-effectiveness, reduced construction time, and possibility of using locally-available lower quality soils and/or waste materials, such as recycled construction and demolition (C&D) wastes. The time-dependent shear behaviour at the interfaces between the geosynthetic and the backfill is an important factor affecting the overall long-term performance of such structures, and thereby should be properly understood. In this study, an innovative multistage direct shear test procedure is introduced to characterise the time-dependent response of the interface between a high-strength geotextile and a recycled C&D material. After a prescribed shear displacement is reached, the shear box is kept stationary for a specific period of time, after which the test proceeds again, at a constant displacement rate, until the peak and large-displacement shear strengths are mobilised. The shear stress-shear displacement curves from the proposed multistage tests exhibited a progressive decrease in shear stress with time (stress relaxation) during the period in which the shear box was restrained from any movement, which was more pronounced under lower normal stress values. Regardless of the prior interface shear displacement and duration of the stress relaxation stage, the peak and residual shear strength parameters of the C&D material-geotextile interface remained similar to those obtained from the conventional (benchmark) tests carried out under constant displacement rate.

scholarly journals Diffraction of plane elastic waves by a crack, with application to a problem of brittle fracture

1963 ◽  
Vol 3 (3) ◽  
pp. 325-339 ◽  
Author(s):  
M. Papadopoulos
Keyword(s):  
Elastic Waves ◽  
Scattered Field ◽  
Velocity Potential ◽  
Elastic Solid ◽  
Step Function ◽  
Free Surface Waves ◽  
Dependent Variables ◽  
Smooth Plane ◽  
New Feature ◽  
Set Up

AbstractA crack is assumed to be the union of two smooth plane surfaces of which various parts may be in contact, while the remainder will not. Such a crack in an isotropic elastic solid is an obstacle to the propagation of plane pulses of the scalar and vector velocity potential so that both reflected and diffracted fields will be set up. In spite of the non-linearity which is present because the state of the crack, and hence the conditions to be applied at the surfaces, is a function of the dependent variables, it is possible to separate incident step-function pulses into either those of a tensile or a compressive nature and the associated scattered field may then be calculated. One new feature which arises is that following the arrival of a tensile field which tends to open up the crack there is necessarily a scattered field which causes the crack to close itself with the velocity of free surface waves.

scholarly journals Unsteady deformations of a free liquid surface caused by radiation pressure

2011 ◽  
Vol 682 ◽  
pp. 460-490 ◽  
Author(s):  
B. ISSENMANN ◽  
R. WUNENBURGER ◽  
H. CHRAIBI ◽  
M. GANDIL ◽  
J.-P. DELVILLE
Keyword(s):  
Analytical Model ◽  
Radiation Pressure ◽  
Liquid Surface ◽  
Acoustic Pulse ◽  
Free Liquid Surface ◽  
Time Dependent ◽  
Surface Dynamics ◽  
Fluid Interface ◽  
Numerical Predictions ◽  
Strongly Damped

We present an analytical model of the time-dependent, small-amplitude deformation of a free liquid surface caused by a spatially localized, axisymmetric, pulsed or continuous, acoustic or electromagnetic radiation pressure exerted on the surface. By exactly solving the unsteady Stokes equation, we predict the surface dynamics in all dynamic regimes, namely inertial, intermediate and strongly damped regimes. We demonstrate the validity of this model in all dynamic regimes by comparing its prediction to experiments consisting of optically measuring the time-dependent curvature of the tip of a hump created at a liquid surface by the radiation pressure of an acoustic pulse. Finally, we present a numerical scheme simulating the behaviour of a fluid–fluid interface subjected to a time-dependent radiation pressure and show its accuracy by comparing the numerical predictions with the analytical model in the intermediate and strongly damped regimes.

Thermoelastic Constitutive Equations for Chemically Hardening Materials

1974 ◽  
Vol 41 (3) ◽  
pp. 652-657 ◽  
Author(s):  
Bernard W. Shaffer ◽  
Myron Levitsky
Keyword(s):  
Bulk Modulus ◽  
Strain Rate ◽  
Chemical Reaction ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Elastic Solid ◽  
Time Dependent ◽  
Component Model ◽  
Two Component ◽  
Potential Applications

Thermoelastic constitutive equations are derived for a material undergoing solidification or hardening as the result of a chemical reaction. The derivation is based upon a two component model whose composition is determined by the degree of hardening, and makes use of strain-energy considerations. Constitutive equations take the form of stress rate-strain rate relations, in which the coefficients are time-dependent functions of the composition. Specific results are developed for the case of a material of constant bulk modulus which undergoes a transition from an initial liquidlike state into an isotropic elastic solid. Potential applications are discussed.

Transient Internal Forced Convection Under Arbitrary Time-Dependent Heat Flux

2013 ◽  
Author(s):  
M. Fakoor-Pakdaman ◽  
M. Andisheh-Tadbir ◽  
Majid Bahrami
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Forced Convection ◽  
Analytical Model ◽  
Time Dependent ◽  
Bulk Temperature ◽  
Fluid Temperature ◽  
Flux Boundary Condition ◽  
Arbitrary Time

A new all-time model is developed to predict transient laminar forced convection heat transfer inside a circular tube under arbitrary time-dependent heat flux. Slug flow condition is assumed for the velocity profile inside the tube. The solution to the time-dependent energy equation for a step heat flux boundary condition is generalized for arbitrary time variations in surface heat flux using a Duhamel’s integral technique. A cyclic time-dependent heat flux is considered and new compact closed-form relationships are proposed to predict: i) fluid temperature distribution inside the tube ii) fluid bulk temperature and iii) the Nusselt number. A new definition, cyclic fully-developed Nusselt number, is introduced and it is shown that in the thermally fully-developed region the Nusselt number is not a function of axial location, but it varies with time and the angular frequency of the imposed heat flux. Optimum conditions are found which maximize the heat transfer rate of the unsteady laminar forced-convective tube flow. We also performed an independent numerical simulation using ANSYS to validate the present analytical model. The comparison between the numerical and the present analytical model shows great agreement; a maximum relative difference less than 5.3%.

Electrothermal Blinking Vortices for Chaotic Mixing

2012 ◽  
Author(s):  
Sophie Loire ◽  
Paul Kauffmann ◽  
Paul Gimenez ◽  
Igor Mezić ◽  
Carl Meinhart
Keyword(s):  
Step Function ◽  
Time Dependent ◽  
Chaotic Mixing ◽  
Electric Signal ◽  
Dynamical System Theory ◽  
Mixing Efficiency ◽  
Lab On Chip ◽  
Micro Particle Image Velocimetry ◽  
Phase Signal ◽  
On Chip

Thanks to its favorable reduction scale law, and its easy integration, electrokinetics has emerged over the last fifteen years as one of the major solution to drive flows in fully integrated lab-on-chip. At microscale, an efficient mixing is a keystep which can dramatically accelerate bio-reactions. For thirty years, Dynamical System theory has predicted that chaotic mixing must involve at least 3 dimensions (either time dependent 2D flows or 3D flows). However, in microfluidics, few works have yet presented efficient embedded micromixers. This paper presents experimental and theoretical study of 2D time dependent chaotic mixing using AC electrothermal fluid flows. Experiments and numerical simulations are performed on a top view device and a sideview device. In both devices, a sinusoidal electric signal is applied between 3 interdigitated gold electrodes. A phase signal Vpp = 11V and a ground are switched between the two side electrodes using a step function, whereas the opposite phase signal –Vpp is steadily applied to the center electrode (Figure 1). Flow velocity is measured by micro particle image velocimetry μ PIV. The velocity profile shows a dramatic asymmetry between the two vortices. Therefore, during the switch, vortices overlap, leading to stretching and folding flows required to obtain chaotic mixing (Figure 3 and 4). The experimental measurements validate our electrothermal models based on our previous work [1]. The mixing efficiency of low diffusive particles is studied at multiscale using the mix-variance coefficient (MVC) [2] to evaluate mixing at different scales (Figure 4). To do so, the domain is successively divided in boxes along the x and y direction up to nx and ny boxes, respectively. For each box configuration, average bead concentration is computed. The variance of these concentrations is then evaluated: MVCs=1nxny∑i=1ny∑j=1nxρij-0.52. The result of numerically evaluated MVC in Figure 2 show a dramatic increase of mixing efficiency with blinking vortices compared to steady flow. Theoretical, experimental and simulation results of the mixing process will be presented.

Unified Constitutive Modelling of Nonlinear Time-Dependent Materials

1989 ◽  
Vol 42 (11S) ◽  
pp. S78-S82
Author(s):  
P. G. Glockner ◽  
W. Szyszkowski
Keyword(s):  
Strain Softening ◽  
Strain Tensor ◽  
Damage Function ◽  
Constitutive Modelling ◽  
Constitutive Law ◽  
Time Dependent ◽  
Type Integral ◽  
Characteristic Features ◽  
Stress Dependent ◽  
Semi Empirical

A semi-empirical engineering constitutive law modelling in a unified and continuous manner the main characteristic features of time-dependent materials, including creep, strain softening, relaxation and recovery and tensile brittleness, is briefly reviewed. The model, which contains 13 parameters, is a hereditary single Volterra-type integral representation of material response with stress/strain nonlinearity assumed in the form of a power law, the strain tensor dependent on the entire stress history and the stress-anisotropy/brittleness feature handled by means of a tensile-stress dependent damage function. The capability/versatility of the model is illustrated by examples for several materials.

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