Constitutive modeling of strain rate effects in nanocrystalline and ultrafine grained polycrystals

We present a variational two-phase constitutive model capable of capturing the enhanced rate sensitivity in nanocrystalline (nc) and ultrafine-grained (ufg) fcc metals. The nc/ufg-material consists of a grain interior phase and a grain boundary affected zone (GBAZ). The behavior of the GBAZ is described by a rate-dependent isotropic porous plasticity model, whereas a rate-independent crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The scale bridging from a single grain to a polycrystal is done by a Taylor-type homogenization. It is shown that the enhanced rate sensitivity caused by the grain size refinement is successfully captured by the proposed model.     

An analytical micro-macro model for textured polycrystals at large plastic strains

An analytical micro-macro model of evolving plastic anisotropy is presented that is suitable for numerical simulation of forming processes. The model is based on the combination of a polycrystal model and different analytical procedures for writing anisotropic plastic potentials, expressing their coefficients in terms of texture coefficients, and updating the texture coefficients as function of the (tensorial) strain increment. The use of a fourth-order dual plastic potential (“C4”) in the analytical micro-macro model is studied, and this use is compared with that of Hill's [1948] yield criterion and also with the usual run of the Taylor model. The coefficients of the C4 potential depend linearly on the texture coefficients, which are updated using a variational polycrystal model. The analytical operation of this updating lies on the method first proposed by Eslinget al. [1984] and is described and checked in some detail. The predictions of the analytical micro-model compare well with measurements of the Lankford coefficient, provided the C4 potential is used. The predicted texture evolution is also in a good experimental agreement: a better one than with the Taylor model, which in some cases, gives a poor updating. The theoretical stress evolution during biaxial or plane-strain tension is experimentally consistent too, although in that case the C4 potential, closer to Taylor's model, makes no improvement as compared with Hill's quadratic criterion.     

Constitutive modeling of plastic-bonded explosives

Typically, elastic and elastic-plastic theory are used in structural-analysis computer programs to model the mechanical behavior of high explosives; these models, however, do not fit the observed behavior of plastic-bonded explosives. This paper discusses the development of an equation-of-state creep model and a linear viscoelastic model for the analysis of these material systems and shows comparisons between experimental results and analytical-model predictions.     

Mesoscale modeling of dynamic compression of boron carbide polycrystals

An anisotropic nonlinear elastic model is advanced for crystals belonging to either of two polytypes of boron carbide ceramic. Crystals undergo transformation to an isotropic, amorphous phase upon attainment of a local state-based criterion associated with a loss of intrinsic stability. The model is implemented using the dynamic finite element method, and is demonstrated on a representative volume consisting of fifty polyhedral grains subjected to uniaxial strain at a uniform high strain rate and shock compression at axial pressures ranging from 10 to 50 GPa. Predicted stress–strain behavior is in close agreement with experimental data. For polycrystals consisting of both polytypes, amorphization initiates at stress levels slightly below the experimental Hugoniot elastic limit, and occurs more readily than observed in experiment. For polycrystals consisting only of the CBC (polar) polytype, amorphization initiates at impact pressures similar to those suggested by experiment. In either case, transformation is promoted by dynamic stress interactions and elastic coefficient mismatch among anisotropic crystals. Results support a previous conjecture that amorphization is related to shear instability and cross-linking of the CBC chain in the polar polytype.     

An explicit formulation for multiscale modeling of bcc metals

Many materials for specialized applications exhibit a body-centered cubic structure; e.g., tantalum, vanadium, barium and chromium. In addition, the successful modeling of body-centered cubic (bcc) metals is a necessary step toward modeling of common structural materials such as iron. Implicit formulations for this class of materials exist [e.g., Stainier, L., Cuitiño, A., Ortiz, M., 2002. A micromechanical model of hardening, rate sensitivity, and thermal softening in bcc crystals. Journal of the Mechanics and Physics of Solids 50 (7), 1511–1545; Kuchnicki, S., Radovitzky, R., Cuitiño, A., Strachan, A., Ortiz, M., 2007. A pressure-dependent multiscale model for bcc metals], but are impractical to resolve large-scale dynamic deformation processes. In this article, we describe a procedure analogous to Kuchnicki et al. [Kuchnicki, S., Cuitiño, A., Radovitzky, R., 2006. Efficient and robust constitutive integrators for single-crystal plasticity modeling. International Journal of Plasticity 22 (10), 1988–2011]. wherein we construct an explicit formulation for the multiscale physics models. This update is based on the model of Kuchnicki et al. (in preparation) using a power law representation for the plastic slip rates. The existing implicit form of the model provides qualitative matching with experiments at quasi-static strain rates. The model is recast in an explicit form and applied first to a high quasi-static strain rate to verify that the two forms of the model return similar predictions for similar input parameters. The explicit model is also applied to several high strain rates, showing that it captures characteristic features observed in experimental tests of high-rate deformations, such as the drop in stress immediately after yield that is present in split Hopkinson pressure bar (SHPB) experiments. This test provides qualitative evidence that the model is suitable for high-strain-rate applications. The utility of the model is further demonstrated by a one-dimensional simulation of a SHPB test. Finally, a test case modeling pressure impact of a Tantalum plate using 600,000 elements is shown. The simulations show that the explicit model is capable of recovering the salient features of the experiments while integrating the constitutive update in a robust manner.     

Constitutive modeling of particle reinforced rubber-like materials

The present study is focused on the constitutive modeling for the mechanical behavior of rubber reinforced with filler particles. A filler-dependent energy density function is proposed with all the continuum mechanics-based necessities of an effective hyperelastic material model. The proposed invariant-based energy function comprises a single set of material parameters for a material subjected to several modes of loading conditions. The model solution agrees well with existing experimental results. Later, the effect of varying concentrations of filler particles in the rubber matrix is also studied.     

Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals

Based on the knowledge of the anisotropy associated with the martensitic transformations obtained from tension/compression experiments with oriented CuAlNi single crystals, a simple constant stress averaging approach is employed to model the SMA polycrystal deformation behaviors. Only elastic and inelastic strains due to the martensitic transformation, variant reorientations in the martensite phase and martensite to martensite transformations in thermomechanical loads are considered. The model starts from theoretical calculation of the stress-temperature transformation conditions and their orientation dependence from basic crystallographic and material attributes of the martensitic transformations. Results of the simulations of the NiTi, NiAl, and Cu-based SMA polycrystals in stress–strain tests are shown. It follows that SMA polycrystals, even with randomly oriented grains, typically exhibit tension/compression asymmetry of the shape of the pseudoelastic σ−ε curves in transformation strain, transformation stress, hysteresis widths, character of the pseudoelastic flow and in the slope of temperature dependence of the transformation stresses. It is concluded that some macroscopic features of the SMA polycrystal behaviors originate directly from the crystallography of the undergoing MT's. The model shows clearly the crystallographic origin of these phenomena by providing a link from the crystallographic and material attributes of martensitic transformations towards the macroscopic σ−ε−T behaviors of SMA polycrystals.     

Constitutive modeling of strain range dependent cyclic hardening

In this paper, a new approach for constitutive modeling of strain range dependent cyclic hardening is proposed by extending the kinematic hardening model based on the critical state of dynamic recovery. It is assumed that isotropic, as well as kinematic, hardening consists of several parts, and that each part of isotropic hardening evolves when the corresponding part of kinematic hardening is in the critical state of dynamic recovery. The extended model is capable of simulating the cyclic hardening behavior in which different characteristics of cyclic hardening appear depending on strain range. The model is verified by simulating the relatively large cyclic straining tests of 304 stainless steel at ambient temperature, in which cyclic hardening does not stabilize before rupture if strain range exceeds a certain value. The model is further verified by predicting the history dependence of cyclic hardening under incremental cyclic loading and the maximum plastic strain dependence of strain hardening in cyclic tension.     

Constitutive modeling of strain-induced softening in swollen elastomers

Under cyclic loading, elastomeric material exhibits strong inelastic responses such as stress-softening due to Mullins effect, hysteresis and permanent set. The corresponding inelastic responses are observed in both dry and swollen rubbers. Moreover, it is observed that inelastic responses depend strongly on the swelling level. For engineering applications involving the interaction and contact between rubber components and solvent, the understanding and consideration of swelling are essential pre-requisites for durability analysis. In this paper, a simple phenomenological model describing Mullins effect in swollen rubbers under cyclic loading is proposed. More precisely, the proposed model adopts the concept of evolution of soft domain microstructure with deformation originally proposed by Mullins and Tobin. The swollen rubbers are obtained by immersing dry ones in solvent until desired degrees of swelling are achieved. Subsequently, their mechanical responses, in particular Mullins effect, under cyclic loading are investigated. These experimental data are used to assess the efficiency of the proposed model. Results show that the model agrees qualitatively well with experiments. Furthermore, the model captures well the fundamental features of strain-induced softening.     

Constitutive modeling of deformation and damage in superplastic materials

The superplastic deformation and cavitation damage characteristics of a modified aluminum alloy are investigated at a temperature range from 500 to 550°C. The baseline alloy is AA5083. Nominally this alloy contains about 4.5% Mg, 0.8% Mn, 0.2% Cr, 0.037% Si, 0.08% Fe and 0.025% Ti by weight. The experimental program consists of uniaxial tension tests and digital image analysis for measuring cavitation. The experiments reveal that evolution of damage is due to both nucleation and growth of voids. A viscoplastic model for describing deformation and damage in this alloy is developed based on a continuum mechanics framework. The model includes the effect of strain hardening, strain rate sensitivity, dynamic and static recovery, and nucleation and growth of voids. The model predictions compare well with the experimental results.     

Constitutive modeling of the dynamic fracture of smooth tensile bars

A recently developed viscoplastic-damage type of constitutive theory for high strain-rate flow processes and ductile fracture is used to model the deformation and fracture of dynamically loaded smooth cylindrical tensile bars. The analysis assumes polycrystalline materials which usually contain microvoids with an average density of the order of 10⁶ per cm³ that are dispersed homogeneously throughout. It is shown that for dynamically imposed loading that produce nominal strain rates ranging between 5 × 10² − 5 × 10³ sec ⁻¹, the inhomogeneous fields of stress and deformation caused by wave propagation and wave reflection induce necking at different locations along the gauge section, depending upon the strain-rate imposed. This occurs without imposition of any geometrical or material irregularity to preposition the location of the necking. The imposed rate of strain is also shown to affect the magnitude of the strain at which necking initiates, as well as the strain required for fracture.     

Constitutive modeling and computational implementation for finite strain plasticity

This paper describes a simple alternate approach to the difficult problem of modeling material behavior. Starting from a general representation for a rate-type constitutive equation, it is shown by example how sets of test data may be used to derive restrictions on the scalar functions appearing in the representation. It is not possible to determine these functions from experimental data, but the aforementioned restrictions serve as a guide in their eventual definition. The implications are examined for hypo-elastic, isotropically hardening plastic, and kinematically hardening plastic materials. A simple model for the evolution of the “back-stress,” in a kinematic-hardening plasticity theory, that is entirely analogous to a hypoelastic stress-strain relation is postulated and examined in detail in modeling a finitely plastic tension-torsion test. The implementation of rate-type material models in finite element algorithms is also discussed.     

高温下混凝土的本构模拟及破坏分析

针对已建立的高温下混凝土中化学-热-水力-力学耦合过程分析的分级数学模型,发展了混凝土的化学-热-水力-力学(CTHM)耦合本构模型。在已有的Willam-Warnke弹塑性屈服准则基础上发展了考虑脱水和脱盐引起的材料损伤及化学塑性软化、塑性应变硬化/软化和吸力硬化的广义Willam-Warnke本构模型,模拟高温下混凝土的材料非线性行为。为保证全局守恒方程的Newton迭代过程的二阶收敛率,导出了非线性化学-热-水力-力学(CTHM)耦合本构模型的一致性切线模量矩阵。数值结果显示了本文所发展的化学-热-水力-力学(CTHM)耦合本构模型在模拟高温下混凝土中复杂破坏过程的能力和有效性。     

Constitutive modeling of ice in the high strain rate regime

The objective of the present work is to propose a constitutive model for ice by considering the influence of important parameters such as strain rate dependence and pressure sensitivity on the response of the material. In this regard, the constitutive model proposed by Carney et al. (2006) is considered as a starting basis and subsequently modified to incorporate the effect of brittle cracking within a continuum damage mechanics framework. The damage is taken to occur in the form of distributed cracking within the material during impact which is consistent with experimental observations. At the point of failure, the material is assumed to be fluid-like with deviatoric stress almost dropping down to zero. The constitutive model is implemented in a general purpose finite element code using an explicit formulation. Several single element tests under uniaxial tension and compression, as well as biaxial loading are conducted in order to understand the performance of the model. Few large size simulations are also performed to understand the capability of the model to predict brittle damage evolution in un-notched and notched three point bend specimens. The proposed model predicts lower strength under tensile loading as compared to compressive loading which is in tune with experimental observations. Further the model also asserts the strain rate dependency of the strength behavior under both compressive as well as tensile loading, which also corroborates well with experimental results.     

Constitutive modeling of orthotropic sheet metals by presenting hardening-induced anisotropy

An essential work on the constitutive modeling of rolled sheet metals is the consideration of hardening-induced anisotropy. In engineering applications, we often use testing results of four specified experiments, three uniaxial-tensions in rolling, transverse and diagonal directions and one equibiaxial-tension, to describe the anisotropic features of rolled sheet metals. In order to completely take all these experimental results, including stress-components and strain-ratios, into account in the constitutive modeling for presenting hardening-induced anisotropy, an appropriate yield model is developed. This yield model can be characterized experimentally from the offset of material yield to the end of material hardening. Since this adaptive yield model can directly represent any subsequent yielding state of rolled sheet metals without the need of an artificially defined “effective stress”, it makes the constitutive modeling simpler, clearer and more physics-based. This proposed yield model is convex from the initial yield state till the end of strain-hardening and is well-suited in implementation of finite element programs.     

Constitutive modeling of fiber composites with a soft hyperelastic matrix

This paper presents an experimental and numerical study of unidirectional carbon fiber composites with a silicone matrix, loaded transversally to the fibers. The experiments show nonlinear behavior with significant strain softening under cyclic loading. The numerical study uses a plane-strain finite element continuum model of the composite material in which the fiber distribution is based on experimental observations and cohesive elements allow debonding to take place at the fiber/matrix interfaces. It is found that accurate estimates of the initial tangent stiffness measured in the experiments can be obtained without allowing for debonding, but this feature has to be included to capture the non-linear and strain-softening behavior.     

Constitutive modeling of shape memory polymer based self-healing syntactic foam

In a previous study, it was found that the shape memory functionality of a shape memory polymer based syntactic foam can be utilized to self-seal impact damage repeatedly, efficiently, and almost autonomously [Li G., John M., 2008. A self-healing smart syntactic foam under multiple impacts. Comp. Sci. Technol. 68(15–16), 3337–3343]. The purpose of this study is to develop a thermodynamics based constitutive model to predict the thermomechanical behavior of the smart foam. First, based on DMA tests and FTIR tests, the foam is perceived as a three-phase composite with interfacial transition zone (interphase) coated microballoons dispersed in the shape memory polymer (SMP) matrix; for simplicity, it is assumed to be an equivalent two-phase composite by dispersing elastic microballoons into an equivalent SMP matrix. Second, the equivalent SMP matrix is phenomenologically assumed to consist of an active (rubbery) phase and a frozen (glassy) phase following Liu et al. [Liu, Y., Gall, K., Dunn, M.L., Greenberg, A.R., Diani J., 2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. Int. J. Plasticity 22, 279–313]. The phase transition between these two phases is through the change of the volume fraction of each phase and it captures the thermomechanical behavior of the foam. The time rate effect is also considered by using rheological models. With some parameters determined by additional experimental testing, the prediction by this model is in good agreement with the 1D test result found in the literature. Parametric studies are also conducted using the constitutive model, which provide guidance for future design of this novel self-healing syntactic foam and a class of light-weight composite sandwich structures.