1 Constitutive Behavior Based on Crystal Plasticity.- 1 Introduction.- 2 Some Important Realities.- 2.1 Uniaxial Monotonic Deformation.- 2.2 Multiaxial Deformation.- 3 Flow Kinetics.- 3.1 Non-uniform Deformation.- 3.2 Uniform Deformation.- 4 Polycrystal Plasticity.- 4.1 Crystal Plasticity.- 4.2 Averaging over a Polycrystal.- 5 Evolution.- 5.1 Texture Evolution.- 5.2 Substructure Evolution.- 6 Internal Stresses.- 6.1 Two-phase Materials.- 6.2 Single-phase Materials.- 7 Application.- 7.1 Diagnostics.- 7.2 Constitutive Relations.- 8 Summary and Recommendations.- 2 State Variable Theories Based on Hart's Formulation.- 1 Introduction.- 2 The Physical and Phenomenological Bases.- 3 A State Variable Description.- 3.1 Hart's Model for Grain Matrix Deformation.- 3.2 An Extension of Hart's Model to a Multiaxial Loading Case.- 3.3 An Extension of Hart's Model to Transient Deformation.- 3.4 An Extension of the State Variable Description to Grain Boundary Sliding.- 4 The Type of Data Utilized in Determining the Material Parameters.- 5 Materials Tested.- 6 Simulative and Predictive Powers of the State Variable Approach.- 6.1 Schematic Description of the Flow Chart.- 6.2 Simulations.- 6.3 Predictions.- 7 Discussion.- 7.1 The Components of the Flow Stress.- 7.2 Work-hardening.- 7.3 Limitations of the Present State Variable Approach.- 7.4 Future Developments.- Appendix 1.- Appendix 2.- 3 The MATMOD Equations.- 1 Introduction.- 2 Development of the Equations.- 2.1 General Relations Between the Phenomena Addressed and the Types of Equations Required.- 2.2 Physical and Phenomenological Bases for the Equations.- 2.3 Phenomenological Development of the Specific Equations.- 3 Simulations and Predictions.- 3.1 Aluminum (emphasizing strain hardening and strain softening behaviors).- 3.2 Austenitic Stainless Steel (emphasizing solute effects).- 3.3 Zircaloy (emphasizing irradiation effects).- 4 Numerical Integration Methods.- 5 Calculation of the Material Constants.- 6 Summary.- 4 The Mechanical Equation of State.- 1 Yield Criteria.- 1.1 Von Mises Yield Criterion.- 1.2 Other Yield Criteria.- 1.3 Yield Criteria Applicable to Polymers.- 1.4 Yield Criteria Applicable to Metals.- 2 Mechanical Equation of State for Dislocation Creep under Multiaxial Stresses.- 2.1 Some Anticipated Features of the MEOS.- 2.2 Anelasticity: the Delayed Elastic Strain Diagram.- 2.3 Non-recoverable Strain.- 2.4 Remobilisation by Stress Reversal.- 2.5 Multiaxial Strain Rates and the Dislocation Velocity.- 2.6 The Strain-Time Equation.- 2.7 Computer Program that Solves the MEOS.- 5 A Physically Based Internal Variable Model for Rate Dependent Plasticity.- 1 Introduction.- 2 The General Problem.- 2.1 Linear Model.- 2.2 Non-Linear Model.- 3 Proposed New Model.- 3.1 The Kinematic Internal Variable.- 3.2 The Isotropic Internal Variable.- 3.3 Final Equations for the Model.- 3.4 Determination of Constants.- 3.5 Problems with Parameter Determination.- 4 Behavior of the Model.- 6 Review of Unified Elastic-Viscoplastic Theory.- 1 Introduction.- 2 Constitutive Equations.- 2.1 Basic Equations.- 2.2 Evolution Equations.- 2.3 Temperature Dependence.- 3 Interpretation and Evaluation of Material Constants.- 4 Modeling of Metals.- 5 Applications.- 5.1 Finite Element Computer Programs.- 5.2 Finite Difference Computer Programs.- 5.3 Special Problems.- 7 Summary and Critique.- 1 Introduction.- 2 Model by Krieg, Swearengen and Jones.- 3 Model by Miller.- 4 Model by Bodner.- 5 Model by Korhonen, Hannula and Li.- 6 Model by Gittus.- 7 Numerical Difficulties with the Models.- 8 Conclusion.- Appendix A.- Appendix B.

Unified constitutive equations for creep and plasticity

A phenomenological, polycrystalline version of a nonlocal crystal plasticity model is formulated. The presence of geometrically necessary dislocations (GNDs) at, or near, grain boundaries is modeled as elastic lattice curvature through a curl of the elastic part of the deformation gradient. This spatial gradient of an internal state variable introduces a length scale, turning the local form of the model, an ordinary differential equation (ODE), into a nonlocal form, a partial differential equation (PDE) requiring boundary conditions. Small lattice elastic stretching results from the presence of dislocations and from macroscopic external loading. Finite deformation results from large plastic slip and large rotations. The thermodynamics and constitutive assumptions are written in the intermediate configuration in order to place the plasticity equations in the proper configuration for finite deformation analysis.

A Nonlocal Phenomenological Anisotropic Finite Deformation Plasticity Model Accounting for Dislocation Defects

Sn-Ag-based solders are susceptible to appreciable microstructural coarsening due to the combined effect of thermal and mechanical stimuli during service and storage. This results in evolution of the creep properties of the solder over time, necessitating the development of a thermo-mechanical history-dependent creep model for accurate prediction of the long-term reliability of microelectronic solder joints. In this paper, the coarsening behavior of Ag3Sn and Cu6Sn5 precipitates in ball grid array-sized joints of Sn-3.8Ag-0.7Cu solder attached to Ni bond-pads with four different thermo-mechanical histories is reported. Because of the substantial numerical superiority of Ag3Sn over Cu6Sn5, it was inferred that the evolution of mechanical properties during aging is controlled largely by the coarsening of Ag3Sn. An effective diffusion length (x) for Ag diffusion in Sn was defined, and it is shown to adequately describe the thermo-mechanical history dependence of Ag3Sn particle size. The shear creep behavior of these joints was experimentally characterized, and the entire creep data were fitted to a unified model combining exponential primary creep and power-law steady state creep. The parameter x was then incorporated into the creep equation to produce a unified creep model, which can adapt to thermo-mechanical history-dependent microstructural coarsening in the solder. Predictions using this creep law show very good agreement with experimental creep data for several different test and microstructural conditions.

Microstructurally Adaptive Model for Primary and Secondary Creep of Sn-Ag-Based Solders

Among the several sets of unified constitutive equations described in this volume, the MATMOD (standing simply for MATerials MODel) equations probably attempt to cover the broadest set of phenomena. These are enumerated in Table 1. Like many of the other unified approaches, the MATMOD equations treat most of the phenomena within the categories of ‘creep’ and ‘plasticity’; however, the equations are also designed to predict many aspects of cyclic deformation and, more generally, multiaxial non-proportional straining. The manner in which the equations represent solute drag effects (dynamic strain aging) is unique. This breadth does come at the expense of accuracy, given our present incomplete knowledge of the ‘first principles’ physical laws governing dislocation-controlled straining.

The MATMOD Equations

The creep of polycrystalline copper containing a dispersion of silica particles has been investigated at stresses below 1 MN m−2, and at temperatures from 820 to 1120 K. Previous work has shown that pure copper of similar grain size deforms by Coble creep in this regime. The introduction of silica particles is shown to inhibit creep. It is found that the deformation characteristics can be represented by the operation of two coupled processes, one of which can be identified as Coble creep. It is suggested that the other process is the motion of grain-boundary dislocations controlled by the rate at which they can bypass particles. Using this approach it is shown that this can lead to an apparent threshold stress for creep with characteristics similar to those observed by other workers.

F. von Gemmingen

Papers

Paper 30: Strain-Time, Rate-Stress and Rate-Temperature Relations during Large Deformations in Creep