Michael K. Neilsen

Bifurcations in elastic-plastic materials

Abstract When strain-softening, elastic-plastic materials are loaded into the plastic regime they often exhibit deformations that are localized into small regions at some point in the loading process. This intense localized deformation limits the formability of materials and will often quickly lead to failure with continued loading. Localized deformation is often associated with satisfaction of the classical discontinuous bifurcation criterion. Here we propose that the loss of strong ellipticity criterion should be used in place of the classical discontinuous bifurcation criterion as a necessary condition for localization. The application of the strong ellipticity criterion implies that a bifurcation mode associated with loss of positive definiteness of the symmetric part of the acoustic tensor must be identified rather than a mode associated with the first zero eigenvalue of the acoustic tensor itself. The eigensystem for the symmetric part of the tangent modulus tensor is obtained for several different plasticity models. This eigensystem provides information about deformation modes associated with both diffuse and discontinuous bifurcations. Material properties, boundary conditions and body geometry are all shown to affect the diffuse and localized deformation modes that are generated. Numerous experimental observations of necking and localization in metal specimens subject to various boundary conditions are explained with the proposed approach.

Microstructurally Based Finite Element Simulation on Solder Joint Behaviour

The most commonly used solder for electrical interconnects in electronic packages is the near eutectic 60Sn‐40Pb alloy. This alloy has a number of processing advantages(suitable melting point of 183°C and good wetting behaviour). However, under conditions of cyclic strain and temperature (thermomechanical fatigue) the microstructure of this alloy undergoes a heterogeneous coarsening and failure process that makes the prediction of solder joint lifetime complex. A finite element simulation methodology to predict solder joint mechanical behaviour, that includes microstructural evolution, has been developed. The mechanical constitutive behaviour was incorporated into the time‐dependent internal state variable viscoplastic model through experimental creep tests. The microstructural evolution is incorporated through a series of mathematical relations that describe mass flow in a temperature/strain environment. The model has been found to simulate observed thermomechanical fatigue behaviour in solder joints.

The sandia fracture challenge: Blind round robin predictions of ductile tearing

Existing and emerging methods in computational mechanics are rarely validated against problems with an unknown outcome. For this reason, Sandia National Laboratories, in partnership with US National Science Foundation and Naval Surface Warfare Center Carderock Division, launched a computational challenge in mid-summer, 2012. Researchers and engineers were invited to predict crack initiation and propagation in a simple but novel geometry fabricated from a common off-the-shelf commercial engineering alloy. The goal of this international Sandia Fracture Challenge was to benchmark the capabilities for the prediction of deformation and damage evolution associated with ductile tearing in structural metals, including physics models, computational methods, and numerical implementations currently available in the computational fracture community. Thirteen teams participated, reporting blind predictions for the outcome of the Challenge. The simulations and experiments were performed independently and kept confidential. The methods for fracture prediction taken by the thirteen teams ranged from very simple engineering calculations to complicated multiscale simulations. The wide variation in modeling results showed a striking lack of consistency across research groups in addressing problems of ductile fracture. While some methods were more successful than others, it is clear that the problem of ductile fracture prediction continues to be challenging. Specific areas of deficiency have been identified through this effort. Also, the effort has underscored the need for additional blind prediction-based assessments.

Characteristics of Creep Damage for 60 Sn-40 Pb Solder Material

This paper presents a viscoplasticity model taking into account the effects of change in grain or phase size and damage on the characterization of creep damage in 60Sn-40Pb solder. Based on the theory of damage mechanics, a two-scalar damage model is developed for isotropic materials by introducing the free energy equivalence principle. The damage evolution equations are derived in terms of the damage energy release rates. In addition, a failure criterion is developed based on the postulation that a material element is said to have ruptured when the total damage accumulated in the element reaches a critical value. The damage coupled viscoplasticity model is discretized and coded in a general-purpose finite element program known as ABAQUS through its user-defined material subroutine UMAT. To illustrate the application of the model, several example cases are introduced to analyze, both numerically and experimentally, the tensile creep behaviors of the material at three stress levels. The model is then applied to predict the deformation of a notched specimen under monotonic tension at room temperature (22 C). The results demonstrate that the proposed model can successfully predict the viscoplastic behavior of the solder material.

Discontinuous bifurcation states for associated smooth plasticity and damage with isotropic elasticity

Abstract For many constitutive equations the tangent tensor consists of a rank one modification to the isotropic elasticity tensor with a total of two elasticity parameters and one parameter describing the current state of inelasticity. For small deformations, general expressions are derived for the loss of ellipticity, the corresponding normal to the bifurcation plane and the mode of discontinuous bifurcation for the velocity gradient. If the principal basis of an evolution tensor is used, the current stress or strain state is characterized by two additional parameters. The small number of material and state parameters makes it feasible to use contour plots to iilustrate all possible combinations that can provide a discontinuous bifurcation. These bifurcation maps can be used to illustrate the bifurcation properties of a particular plasticity or continuum damage constitutive model. Conversely, the bifurcation maps can be used in conjunction with experimental data on bifurcation features to assist in the development of constitutive equations that provide the correct failure criterion for a given material under all possible stress paths.

Capturing the influence of surface constraints in small and thin samples using polycrystalline plasticity theory

A rate-dependent, single-crystal plasticity model for face-centred cubic crystal structures has been implemented into a large strain elastic - plastic, finite-element code to examine the mechanical influence of the reduced surface constraints of relatively small polycrystalline aggregates. The implemented model simulates deformation of a polycrystal composed of cubic grains where each grain is a single finite element. Mechanical constraint is varied by changing (a) specimen thickness and (b) specimen volume, relative to grain size. Numerical uniaxial tensile tests have been performed to a strain level of 0.01. Direct and statistical examination of the model results revealed the reduced flow stress of grains at specimen surfaces, edges and corners. The results of these simulations are in good agreement with previous experimental studies which suggest that 5 - 10 grains across the minimum dimension of a structure are necessary to approximate true continuum polycrystalline response.

Ductile tearing predictions with Wellman’s failure model

Predictions for the Sandia National Laboratories fracture challenge (Boyce et al. in Int J Fract 2013) were generated using a transient dynamic finite element code with a multi-linear elastic plastic failure model developed by Wellman (Simple approach to modeling ductile failure. Sandia National Laboratories, Albuquerque 2012). This model is a conventional, rate independent, von Mises plasticity model for metals with user-prescribed hardening as a function of equivalent plastic strain. In addition to conventional plasticity, this model has empirical criteria for crack initiation and growth. Ductile tearing predictions generated with this model were found to be in good agreement with experimental measurements and observations.

On the Strain Rate- and Temperature-Dependent Tensile Behavior of Eutectic Sn–Pb Solder

The present study examines the thermomechanical strain-rate sensitivity of eutectic 63Sn–37Pb solder over a broad range of strain-rates from 0.0002 s–1 to 200 s–1 , thus encompassing failure events between 1 h and 1 ms, at temperatures ranging from −60 °C to + 100 °C. A newly developed servohydraulic tensile method enabled this broad range of strain-rates to be evaluated by a single technique, eliminating ambiguity caused by evaluation across multiple experimental methods. Two solder conditions were compared: a normalized condition representing a solder joint that has largely stabilized ∼30 days after solidification and an aged condition representing ∼30 years at near-ambient temperatures. The tensile behavior of both conditions exhibited dramatic temperature and strain-rate sensitivity. At 100 °C, the yield strength increased from 5 MPa at 0.0002 s–1 to 42 MPa at 200 s–1 , while at −60 °C, the yield strength increased from 57 MPa at 0.0002 s–1 to 71 MPa at 200 s–1 . The room temperature strain rate-dependent behavior was also measured for the lead free SAC396 alloy. The SAC alloy exhibited thermal strain-rate sensitivity similar to Sn–Pb over this temperature and strain-rate regime. Microstructural characterization using backscatter electron imaging and electron backscatter diffraction showed distinct, morphological changes of the microstructure for different thermomechanical conditions as well as some systematic changes in the crystallographic texture. However, very little intergranular rotation was observed over the range of thermomechanical conditions, suggesting the dominance of a grain boundary sliding (GBS) deformation mechanism. Finally, a recently developed unified-creep-plasticity constitutive model for solder deformation was found to describe the observed behavior with much higher fidelity than the common Johnson–Cook model.

Unified Creep Plasticity Damage (UCPD) Model for Solder

A unified creep plasticity damage (UCPD) model for Sn-Pb and Pb-free solders was developed and implemented into finite element analysis codes. The new model will be described along with the relationship between the model’s damage evolution equation and an empirical Coffin-Manson relationship for solder fatigue. Next, two significant developments were needed to model crack initiation and growth in solder joints. First, an ability to accelerate the simulations such that the effects of hundreds or thousands of thermal cycles could be modeled in a reasonable amount of time was needed. This was accomplished by applying a user prescribed acceleration factor to the damage evolution; then, damage generated by an acceleration factor of cycles could be captured by the numerical simulation of a single thermal cycle. Second, an ability to capture the geometric effects of crack initiation and growth was needed. This was accomplished by replacing material in finite elements that had met the cracking failure criterion with very flexible elastic material. This diffuse crack modeling approach with local finite elements is known to generate mesh dependent solutions. However, introduction of an element size dependent term into the damage evolution equation was found to be effective in controlling mesh dependency. Finally, experimentally observed cracks in a typical solder joint subjected to thermal mechanical fatigue are compared with model predictions.© 2011 ASME

Characterization of Mechanical Behavior of Polyurethane Foams Using Digital Image Correlation

Polyurethane foams have good energy absorption properties and are effective in protecting sensitive components from damages due to impact. The foam absorbs impact energy by crushing cells and undergoing large deformation. The complex deformation of the foam needs to be modeled accurately to simulate the impact events. In this paper, the Digital Image Correlation (DIC) technique was implemented to obtain the deformation field of foam specimens under compression tests. Images of foam specimen were continuously acquired using high-speed cameras. The full field displacement and strain at each incremental step of loading were calculated from these images. The closed-cell polyurethane foam used in this investigation was nominal 0.32 kg/m^3 (20 pcf). In the first experiment, cubic specimens were compressed uniaxially up to 60%. The full-field displacements and strains obtained using the DIC technique provide detailed information about the inhomogeneous deformation over the area of interest during loading. In the second experiment, compression tests were conducted for a simple foam structure - cubic foam specimens with a steel cylinder inclusion. The strain concentration at the interface between steel cylinder and foam was studied to simulate the deformation of foam in a typical application. In the third experiment, the foam was loaded from the steel cylinder during the compression. The strain concentration at the interface and the displacement distribution over the surface were compared for cases with and without a confinement fixture to study the effects of confinement. These experimental results demonstrate that the DIC technique can be applied to polyurethane foams to study the heterogeneous deformation. The experimental data is briefly compared with the results from modeling and simulation using a viscoplastic model for the foam.Copyright