Constitutive Laws for Ceramics Exhibiting Stress-Induced Martensitic Transformation

1986 ◽  
Vol 78 ◽  
Author(s):  
John C. Lambropoulos
Keyword(s):  
Potential Energy ◽  
Volume Concentration ◽  
Inelastic Strain ◽  
Internal Variable ◽  
Constitutive Laws ◽  
Constitutive Law ◽  
Internal Variables ◽  
Strain Rates ◽  
Applied Stresses ◽  
Special Case

ABSTRACTThe theory of internal variables is used in order to develop multiaxial constitutive laws for ceramics undergoing martensitic stress-assisted transformation, such as partially stabilized zirconia or A12O3-ZrO2. The internal variable is identified with the volume concentration of transformed particles, and we assume that transformation occurs so that the change in potential energy due to the transformation is maximized. When the rate of transformation depends on the applied stresses only through the corresponding change in potential energy, it is shown that the inelastic strain rates are along the normal of a stress function in stress space. The constitutive law depends on all three stress invariants. We further discuss specific stress environments such as crack tip fields, the special case of homogeneous transforming particle distribution, and conditions under which normality is not obeyed.

Reliability Modeling of Lead Free Solder Joints in Wafer-Level Chip Scale Packages

2007 ◽  
Author(s):  
Jie-Hua Zhao ◽  
Vikas Gupta ◽  
Alok Lohia ◽  
Darvin Edwards
Keyword(s):  
Solder Joints ◽  
Inelastic Strain ◽  
Constitutive Laws ◽  
Constitutive Law ◽  
Lead Free ◽  
Lead Free Solder ◽  
Wafer Level ◽  
Reliability Modeling ◽  
Correlation Parameters ◽  
Free Solder

Board level thermomechanical fatigue lifetimes of five different wafer-level chip scale packages (WCSP’s) with lead-free solder joints were studied by both experiment and finite element method modeling. The effect of three different constitutive laws of the lead-free solder, namely Anand viscoplasticity, power law break-down creep and time hardening creep are also investigated for each of the five packages. The fatigue correlation parameters based on the increment of volume-averaged inelastic strain energy density are deduced for each of the corresponding three constitutive laws. It is demonstrated that the relative error of the predicted lifetime for WCSP with lead-free solder joints can be within 10% compared to experiment. It is found that the fatigue correlation parameters depend strongly on the specific constitutive law. Another important finding is that the fatigue correlation parameters depend on the specific package family. It is also demonstrated that when fatigue correlation parameters calibrated for other package families are applied to WCSPs, the error in predicted lifetimes is consistently large.

A Theory for Elastic Potential Energy Change and Applied Stresses Accompanying Phase Transformation

2000 ◽  
Vol 16 (3) ◽  
pp. 169-176
Author(s):  
Huang Hsing Pan
Keyword(s):  
Phase Transformation ◽  
Potential Energy ◽  
Volume Concentration ◽  
Elastic Potential ◽  
Energy Change ◽  
Material System ◽  
Transformation Zone ◽  
Finite Concentration ◽  
Elastic Potential Energy ◽  
Applied Stresses

ABSTRACTA theory is proposed to derive the resultant elastic potential energy change in the systems accompanying phase transformation for a finite concentration of inclusions. Based on the thermodynamic equilibrium, this elastic potential energy due to the interaction of ellipsoidal inclusions is employed to evaluate the critically applied stress that induces phase transformation. It is shown that the material system with the disc inclusions is easier to get the stress-induced transformation than that with other shapes. The inclusions with 3-D random orientation are the most effective one to have toughness increments. The height of the transformation zone strongly depends on the volume concentration and the shape of the inclusions. As compared with the experimental data, the theory is in an acceptable range of accuracy.

scholarly journals Multiscale elastic-viscoplastic computational analysis

2011 ◽  
Author(s):  
Nicolas Relun ◽  
David Néron ◽  
Pierre- Alain Boucard
Keyword(s):  
Domain Decomposition ◽  
Computational Analysis ◽  
Constitutive Laws ◽  
Constitutive Law ◽  
Internal Variables ◽  
Proper Generalized Decomposition ◽  
Efficient Strategy ◽  
Drastic Decrease ◽  
Viscoplastic Models ◽  
Technical Features

The objective of this work is to develop an efficient strategy for quasi-static problems with elastic-viscoplastic constitutive laws. Our approach is based on the multiscale LATIN method for domain decomposition, and particularly on the use of the Proper Generalized Decomposition (PGD) method, which allows a drastic decrease in computation costs. We present the method in its general form applicable to problems with constitutive laws expressed using internal variables; then we discuss the technical features which are necessary in order to deal with elastic-viscoplastic models. We illustrate the method in detail through a onedimensional example using a Chaboche-type elastic-viscoplastic constitutive law.

Reliability Modeling of Lead-Free Solder Joints in Wafer-Level Chip Scale Packages

2010 ◽  
Vol 132 (1) ◽  
Author(s):  
Jie-Hua Zhao ◽  
Vikas Gupta ◽  
Alok Lohia ◽  
Darvin Edwards
Keyword(s):  
Solder Joints ◽  
Inelastic Strain ◽  
Constitutive Laws ◽  
Constitutive Law ◽  
Lead Free ◽  
Lead Free Solder ◽  
Wafer Level ◽  
Reliability Modeling ◽  
Correlation Parameters ◽  
Free Solder

Board-level thermomechanical fatigue lifetimes of five different wafer-level chip scale packages (WCSPs) with lead-free solder joints were studied by both experiment and finite element method modeling. The effect of three different constitutive laws of the lead-free solder, namely Anand viscoplasticity, power law break-down creep, and time-hardening creep are also investigated for each of the five packages. The fatigue correlation parameters based on the increment of volume-averaged inelastic strain energy density are deduced for each of the corresponding three constitutive laws. It is demonstrated that the relative error of the predicted lifetime for WCSP with lead-free solder joints can be within 10% compared with experiment. It is found that the fatigue correlation parameters depend strongly on the specific constitutive law. Another important finding is that the fatigue correlation parameters depend on the specific package family. It is also demonstrated that when fatigue correlation parameters calibrated for other package families are applied to WCSPs, the error in predicted lifetimes is consistently large.

scholarly journals A Uniqueness Result for a Simple Superlinear Eigenvalue Problem

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Michael Herrmann ◽  
Karsten Matthies
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Simulations ◽  
Convolution Operator ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Constitutive Laws ◽  
Parameter Family ◽  
Shape Constraint ◽  
Special Case ◽  
Nonlinear Eigenfunctions

AbstractWe study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein–Rutman arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.

Internal Variable Theory Formulated by One Extended Potential Function

2020 ◽  
Vol 45 (3) ◽  
pp. 311-318
Author(s):  
Qiang Yang ◽  
Zhuofu Tao ◽  
Yaoru Liu
Keyword(s):  
Potential Function ◽  
Internal Variable ◽  
The State ◽  
Internal Variables ◽  
State Variables ◽  
Kinetic Rate ◽  
Variable Theory ◽  
Rate Laws ◽  
Flow Potential ◽  
Internal Variable Theory

AbstractIn the kinetic rate laws of internal variables, it is usually assumed that the rates of internal variables depend on the conjugate forces of the internal variables and the state variables. The dependence on the conjugate force has been fully addressed around flow potential functions. The kinetic rate laws can be formulated with two potential functions, the free energy function and the flow potential function. The dependence on the state variables has not been well addressed. Motivated by the previous study on the asymptotic stability of the internal variable theory by J. R. Rice, the thermodynamic significance of the dependence on the state variables is addressed in this paper. It is shown in this paper that the kinetic rate laws can be formulated by one extended potential function defined in an extended state space if the rates of internal variables do not depend explicitly on the internal variables. The extended state space is spanned by the state variables and the rate of internal variables. Furthermore, if the rates of internal variables do not depend explicitly on state variables, an extended Gibbs equation can be established based on the extended potential function, from which all constitutive equations can be recovered. This work may be considered as a certain Lagrangian formulation of the internal variable theory.

Strain-Rate Effect on the Tensile Behaviour of High Strength Alloys

2011 ◽  
Vol 82 ◽  
pp. 124-129 ◽  
Author(s):  
Ezio Cadoni ◽  
Matteo Dotta ◽  
Daniele Forni ◽  
Stefano Bianchi
Keyword(s):  
Strain Rate ◽  
Rate Sensitivity ◽  
High Strength ◽  
Constitutive Law ◽  
Tensile Behaviour ◽  
Strain Rates ◽  
Cross Sectional ◽  
Split Hopkinson Tensile Bar ◽  
Wide Range ◽  
First Results

In this paper the first results of the mechanical characterization in tension of two high strength alloys in a wide range of strain rates are presented. Different experimental techniques were used for different strain rates: a universal machine, a Hydro-Pneumatic Machine and a JRC-Split Hopkinson Tensile Bar. The experimental research was developed in the DynaMat laboratory of the University of Applied Sciences of Southern Switzerland. An increase of the stress at a given strain increasing the strain-rate from 10-3 to 103 s-1, a moderate strain-rate sensitivity of the uniform and fracture strain, a poor reduction of the cross-sectional area at fracture with increasing the strain-rate were shown. Based on these experimental results the parameters required by the Johnson-Cook constitutive law were determined.

scholarly journals Model reduction by mean-field homogenization in viscoelastic composites. I. Primal theory

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200407 ◽  
Author(s):  
Martín I. Idiart ◽  
Noel Lahellec ◽  
Pierre Suquet
Keyword(s):  
Strain Field ◽  
Mean Field ◽  
Inelastic Strain ◽  
Mathematical Structure ◽  
Schwarz Inequality ◽  
Internal Variables ◽  
Legendre Transform ◽  
Time Step ◽  
Legendre Transforms ◽  
Viscoelastic Composites

A homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44 , 507–529 ( doi:10.1016/j.ijsolstr.2006.04.038 )) is revisited. The scheme relies upon an incremental variational formulation providing the inelastic strain field at a given time step in terms of the inelastic strain field from the previous time step, along with a judicious use of Legendre transforms to approximate the relevant functional by an alternative functional depending on the inelastic strain fields only through their first and second moments over each constituent phase. As a result, the approximation generates a reduced description of the microscopic state of the composite in terms of a finite set of internal variables that incorporates information on the intraphase fluctuations of the inelastic strain and that can be evaluated by mean-field homogenization techniques. In this work we provide an alternative derivation of the scheme, relying on the Cauchy–Schwarz inequality rather than the Legendre transform, and in so doing we expose the mathematical structure of the resulting approximation and generalize the exposition to fully anisotropic material systems.

scholarly journals Extensive CGMD Simulations of Atactic PS Providing Pseudo Experimental Data to Calibrate Nonlinear Inelastic Continuum Mechanical Constitutive Laws

Polymers ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1824 ◽  
Author(s):  
Maximilian Ries ◽  
Gunnar Possart ◽  
Paul Steinmann ◽  
Sebastian Pfaller
Keyword(s):  
Md Simulations ◽  
Multiscale Simulation ◽  
Constitutive Laws ◽  
The Body ◽  
Constitutive Law ◽  
Compression Load ◽  
Body Of Knowledge ◽  
Research Activities ◽  
Tension And Compression ◽  
Atactic Polystyrene

In this contribution, we present a characterization methodology to obtain pseudo experimental deformation data from CG MD simulations of polymers as an inevitable prerequisite to choose and calibrate continuum mechanical constitutive laws. Without restriction of generality, we employ a well established CG model of atactic polystyrene as exemplary model system and simulate its mechanical behavior under various uniaxial tension and compression load cases. To demonstrate the applicability of the obtained data, we exemplarily calibrate a viscoelastic continuum mechanical constitutive law. We conclude our contribution by a thorough discussion of the findings obtained in the numerical pseudo experiments and give an outline of subsequent research activities. Thus, this work contributes to the field of multiscale simulation methods and adds a specific application to the body of knowledge of CG MD simulations.

scholarly journals Inverse Strategies for Identifying the Parameters of Constitutive Laws of Metal Sheets

2016 ◽  
Vol 2016 ◽  
pp. 1-18 ◽  
Author(s):  
P. A. Prates ◽  
A. F. G. Pereira ◽  
N. A. Sakharova ◽  
M. C. Oliveira ◽  
J. V. Fernandes
Keyword(s):  
Finite Element ◽  
Cost Function ◽  
Optimization Algorithm ◽  
Yield Criterion ◽  
Kinematic Hardening ◽  
Constitutive Laws ◽  
Constitutive Law ◽  
Identification Procedure ◽  
Metal Sheets ◽  
The Cost

This article is a review regarding recently developed inverse strategies coupled with finite element simulations for the identification of the parameters of constitutive laws that describe the plastic behaviour of metal sheets. It highlights that the identification procedure is dictated by the loading conditions, the geometry of the sample, the type of experimental results selected for the analysis, the cost function, and optimization algorithm used. Also, the type of constitutive law (isotropic and/or kinematic hardening laws and/or anisotropic yield criterion), whose parameters are intended to be identified, affects the whole identification procedure.

Sign in / Sign up

Export Citation Format

Share Document