A theory for amorphous viscoplastic materials undergoing finite deformations, with application to metallic glasses

This study develops a finite-deformation, Coulomb-Mohr type constitutive theory for the elastic-viscoplastic response of pressure-sensitive and plastically-dilatant isotropic materials. The constitutive model has been implemented in a finite element program, and the numerical capability is used to study the deformation response of amorphous metallic glasses. Specifically, the response of an amorphous metallic glass in tension, compression, strip-bending, and indentation is studied, and it is shown that results from the numerical simulations qualitatively capture major features of corresponding results from physical experiments available in the literature.     

Coupled thermo-mechanical modelling of bulk-metallic glasses: Theory, finite-element simulations and experimental verification

A three-dimensional, finite-deformation-based constitutive model to describe the behavior of metallic glasses in the supercooled liquid region has been developed. By formulating the theory using the principles of thermodynamics and the concept of micro-force balance [Gurtin, M., 2000. On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48, 989-1036], a kinetic equation for the free volume concentration is derived by augmenting the Helmholtz free energy used for a conventional metallic alloy with a flow-defect free energy which depends on the free volume concentration and its spatial gradient. The developed constitutive model has also been implemented in the commercially available finite-element program ABAQUS/Explicit (2005) by writing a user-material subroutine. The constitutive parameters/functions in the model were calibrated by fitting the constitutive model to the experimental simple compression stress-strain curves conducted under a variety of strain-rates at a temperature in the supercooled liquid region [Lu, J., Ravichandran, G., Johnson, W., 2003. Deformation behavior of the Zr-Ti-Cu-Ni-Be bulk metallic glass over a wide range of strain-rates and temperatures. Acta Mater. 51, 3429-3443].With the model calibrated, the constitutive model was able to reproduce the simple compression stress-strain curves for jump-in-strain-rate experiments to good accuracy. Furthermore stress-strain responses for simple compression experiments conducted at different ambient temperatures within the supercooled liquid region were also accurately reproduced by the constitutive model. Finally, shear localization studies also show that the constitutive model can reasonably well predict the orientation of shear bands for compression experiments conducted at temperatures within the supercooled liquid region [Wang, G., Shen, J., Sun, J., Lu, Z., Stachurski, Z., Zhou, B., 2005. Compressive fracture characteristics of a Zr-based bulk metallic glass at high test temperatures. Mater. Sci. Eng. A 398, 82-87].     

Yielding and intrinsic plasticity of Ti–Zr–Ni–Cu–Be bulk metallic glass

Bulk metallic glass with composition Ti₄₀Zr₂₅Ni₈Cu₉Be₁₈ exhibits considerably high compressive yield stress, significant plasticity (with a concomitant vein-like fracture morphology) and relatively low density. Yielding and intrinsic plasticity of this alloy are discussed in terms of its thermal and elastic properties. An influence of normal stresses acting on the shear plane is evidenced by: (i) the fracture angle (<45°) and (ii) finite-element simulations of nanoindentation curves, which require the use of a specific yield criterion, sensitive to local normal stresses acting on the shear plane, to properly match the experimental data. The ratio between hardness and compressive yield strength (constraint factor) is analyzed in terms of several models and is best adjusted using a modified expanding cavity model incorporating a pressure-sensitivity index defined by the Drucker–Prager yield criterion. Furthermore, comparative results from compression tests and nanoindentation reveal that deformation also causes strain softening, a phenomenon which is accompanied with the occurrence of serrated plastic flow and results in a so-called indentation size effect (ISE). A new approach to model the ISE of this metallic glass using the free volume concept is presented.     

A finite-deformation constitutive model for bulk metallic glass composites

The focus of this study is the development of an elastic-viscoplastic, three-dimensional, finite-deformation constitutive model to describe the large deformation behavior of bulk metallic glass (BMG) composite. A macroscopic theoretical formulation is proposed based on thermodynamic considerations to describe the response at ambient temperature and pressure, as well as at different strain rates. A constitutive equation is derived using the principle of thermodynamics and the augmenting of free energy. This is done by assuming that deformation within the constituent phases of the composite is affine; kinetic equations defining plastic shear and evolution of free volume concentration are then derived. The constitutive model is subsequently implemented in a finite-element program (Abaqus/Explicit) via a user-defined material subroutine. Numerical predictions are compared with experimental results from tests on La-based in situ BMG composite (La–Al–Cu–Ni) specimens cast in-house; this demonstrates that the model is able to describe the material behavior observed.     

A thermo-mechanically-coupled large-deformation theory for amorphous polymers in a temperature range which spans their glass transition

Amorphous thermoplastic polymers are important engineering materials; however, their non-linear, strongly temperature- and rate-dependent elastic-viscoplastic behavior is still not very well understood, and is modeled by existing constitutive theories with varying degrees of success. There is no generally agreed upon theory to model the large-deformation, thermo-mechanically-coupled, elastic-viscoplastic response of these materials in a temperature range which spans their glass transition temperature. Such a theory is crucial for the development of a numerical capability for the simulation and design of important polymer processing operations, and also for predicting the relationship between processing methods and the subsequent mechanical properties of polymeric products. In this paper we extend our recently published theory [Anand, L., Ames, N. M., Srivastava, V., Chester, S. A., 2009. A thermo-mechanically-coupled theory for large deformations of amorphous polymers. Part I: formulation. International Journal Plasticity 25, 1474–1494; Ames, N. M., Srivastava, V., Chester, S. A., Anand, L., 2009. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: applications. International Journal of Plasticity 25, 1495–1539] to fill this need.     

Thermally actuated shape-memory polymers: Experiments, theory, and numerical simulations

With the aim of developing a thermo-mechanically coupled large-deformation constitutive theory and a numerical-simulation capability for modeling the response of thermally actuated shape-memory polymers, we have (i) conducted large strain compression experiments on a representative shape-memory polymer to strains of approximately unity at strain rates of 10⁻³ and 10⁻¹ s⁻¹, and at temperatures ranging from room temperature to approximately 30 °C above the glass transition temperature of the polymer; (ii) formulated a thermo-mechanically coupled large-deformation constitutive theory; (iii) calibrated the material parameters appearing in the theory using the stress-strain data from the compression experiments; (iv) numerically implemented the theory by writing a user-material subroutine for a widely used finite element program; and (v) conducted representative experiments to validate the predictive capability of our theory and its numerical implementation in complex three-dimensional geometries. By comparing the numerically predicted response in these validation simulations against measurements from corresponding experiments, we show that our theory is capable of reasonably accurately reproducing the experimental results. As a demonstration of the robustness of the three-dimensional numerical capability, we also show results from a simulation of the shape-recovery response of a stent made from the polymer when it is inserted in an artery modeled as a compliant elastomeric tube.     

Size effects on tensile and compressive strengths in metallic glass nanowires

Shear localization induced brittleness is the main drawback of metallic glasses which restricts their practical applications. Previous experiments have provided insights on how to suppress shear localization by reducing the sample size of metallic glasses to the order of 100 nm. In order to reveal the size effects and associated deformation mechanisms of metallic glasses in an even finer scale, we perform large-scale atomistic simulations for the uniaxial compression and tension of metallic glass nanowires. The simulation results show that, as the diameter of metallic glass samples decreases from 45 nm to 8 nm, the tensile yield strength increases while the compressive yield strength decreases. Homogeneous flow is observed as the governing deformation mechanism in all simulated metallic glass samples, where plastic shearing tends to initiate on the sample surface and propagate into the interior. To rationalize the size dependence of yield strengths, we propose a theoretical model based on the concept of surface stress and Mohr–Coulomb criterion. The theoretical predictions agree well with the simulation results, implying the important role of surface stress on the yielding of MGs below 100 nm. Finally, a discussion about the size effects of strength in metallic glasses at different length scales is provided. Our results suggest that the shear band energy and surface stress might be the two crucial parameters in determining the critical size required for the transition from shear localization to homogeneous deformation in MGs.     

Heat transfer enhancement using Al2O3/water nanofluid in a two-phase closed thermosyphon

A two-phase closed thermosyphon (TPCT) is a device for heat transmission. It consists of an evacuated-close tube filled with a certain amount of working fluid. Fluids with nanoparticles (particles smaller than 100 nm) suspended in them are called nanofluids that they have a great potential in heat transfer enhancement. In the present study, we combined two mentioned techniques for heat transfer enhancement. Nanofluids of aqueous Al₂O₃ nanoparticles suspensions were prepared in various volume concentration of 1–3% and used in a TPCT as working media. Experimental results showed that for different input powers, the efficiency of the TPCT increases up to 14.7% when Al₂O₃/water nanofluid was used instead of pure water. Temperature distributions on TPCT confirm these results too.     

Flow pattern and boiling heat transfer of CO2 in horizontal small-bore tubes

Increasing attention has been focused on carbon dioxide (CO₂) heat pump system where the temperature level is rather low, while the operating pressure is rather high. In this system, the density difference between vapor and liquid becomes rather small, which significantly affects flow patterns. Low surface tension and latent heat also have significant influence on two-phase flow patterns and heat transfer. This paper describes experimental and numerical investigation on flow patterns and heat transfer characteristics of boiling flow CO₂ at high pressure in horizontal small-bore tubes ranging from 1.0 mm to 3.0 mm I.D. Even though the density difference is rather small at high pressure, phase stratification takes place, which leads to the intermittent dryout at the upper wall. So far developed discrete bubble model by the authors for vertical flows is modified so as to include horizontal flow mechanisms. The predicted flow patterns with this new model agree on the whole with the experimental observation.     

A constitutive model of cold drawing in polycarbonates

This paper presents a set of constitutive equations to model cold-drawing (necking) in polycarbonates (PC). The model is based on a representation of cold drawing as a double glass transition, i.e., a transition from a glass into a rubbery state, when a certain yield surface in the stress space is reached, and a transition back to the glassy state upon unloading or when a certain molecular orientation (draw ratio) is achieved. The stretching process in the rubbery state is modeled by a hyperelastic extension of the J₂-flow theory to the finite strain range. An appropriate yield surface and an associative flow rule (defined via the Kuhn–Tucker optimality conditions) are presented to simulate this process in polycarbonates. The isochoric constraint during double glass transition is treated via an exact multiplicative decomposition of the deformation gradient into volume preserving and spherical parts. Numerical constitutive integration algorithm is based on an operator splitting technique where constraint/consistency during inelastic deformation is enforced via return mapping algorithm. Numerical results are presented to demonstrate the correspondence with the experimental data.     

Mode II fracture behavior of a Zr-based bulk metallic glass

Shear band formation and fracture are characterized during mode II loading of a Zr-based bulk metallic glass. The measured mode II fracture toughness, K_IIc=75±4 MPa√m, exceeds the reported mode I fracture toughness by ∼4 times, suggesting that normal or mean stresses play a significant role in the deformation process at the crack tip. This effect is explained in light of a mean stress modified free volume model for shear localization in metallic glasses. Thermal imaging of deformation at the mode II crack tip further reveals that shear bands initiate, arrest, and reactivate along the same path, indicating that flow in the shear band leads to permanent changes in the glass structure that retain a memory of the shear band path. The measured temperature increase within the shear band is a fraction of a degree. However, heat dissipation models indicate that the temperature could have exceeded the glass transition temperature for less than 1 ms immediately after the shear band formed. It is shown that this time scale is sufficient for mechanical relaxation slightly above the glass transition temperature.     

Micron-Scale Residual Stress Measurement by Micro-Hole Drilling and Digital Image Correlation

This paper reports a new technique, namely the incremental micro-hole-drilling method (IμHD) for mapping in-plane residual or applied stresses incrementally as a function of depth at the micron-scale laterally and the sub-micron scale depth-wise. Analogous to its macroscale counterpart, it is applicable either to crystalline or amorphous materials, but at the sub-micron scale. Our method involves micro-hole milling using the focused ion beam (FIB) of a dual beam FEGSEM/FIB microscope. The resulting surface displacements are recorded by digital image correlation of SEM images recorded during milling. The displacement fields recorded around the hole are used to reconstruct the stress profile as a function of depth. In this way residual stresses have been characterized around a drilled hole of 1.8microns. diameter, enabling the profiling of the stress variation at the sub-micron scale to a depth of 1.8 microns. The new method is used to determine the near surface stresses in a (peened) surface-severe-plastically-deformed (S²PD) Zr₅₀Cu₄₀Al₁₀ (in atomic percent, at.%) bulk metallic glass bar. In plane principal stresses of -800 MPa ± 90 MPa and −600 MPa ± 90 MPa were measured, the maximum compressive stress being oriented 15° to the axis of the bar.     

Rotational rheometry of liquid metal systems: Measurement geometry selection and flow curve analysis

In the present study, a rotational measurement technique was used to evaluate viscosities of liquid metals and metallic alloys. Three types of measurement geometries in a high temperature rotational rheometer were evaluated: cone and plate, DIN coaxial, and double concentric cylinder (DCC). The DCC geometry proved to be the most effective. An analytical solution has been presented to evaluate the viscosity as a function of shear rate for DCC geometry. The flow curves and shear viscosities of pure Al, pure Zn and Sn_95.8Ag_3.28Cu_0.92 solder alloy have been evaluated as a function of shear rate and melt superheat temperature. It is proposed that liquid metal systems are non-Newtonian and strongly shear thinning in flow behavior.     

Length scale effects on the shear localization process in metallic glasses: A theoretical and computational study

Some recent experiments on sub-micron and nano-sized metallic glass (amorphous alloy) specimens have shown that the shear localization process becomes more stable and less catastrophic when compared to the response exhibited by large sample sizes. This leads to the discovery that the shear localization process and fracture can be delayed by decreasing sample volume. In this work we develop a non-local and finite-deformation-based constitutive model using thermodynamic principles and the theory of micro-force balance to study the causes for the aforementioned observations. The constitutive model has also been implemented into a commercially available finite-element program by writing a user-material subroutine. With the aid of finite-element simulations, our constitutive model predicts that metallic glass samples have the intrinsic ability to exhibit: (a) the delaying of (catastrophic) shear localization with decreasing sample size, and (b) homogeneous deformation behavior for sample volumes smaller than the shear band nucleus.The cause for the observations listed above is the increasing influence of a non-local interaction stress with decreasing sample volume. This interaction stress has energetic origins and it affects plastic deformation due to the strong coupling between plastic shearing and free-volume generation. Akin to strain-gradient plasticity theory, the role of the interaction stress is to strengthen the material at locations where the defect density/free volume is higher compared to the rest of metallic glass sample.     

Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling

Shape memory polymers (SMPs) can retain a temporary shape after pre-deformation at an elevated temperature and subsequent cooling to a lower temperature. When reheated, the original shape can be recovered. Relatively little work in the literature has addressed the constitutive modeling of the unique thermomechanical coupling in SMPs. Constitutive models are critical for predicting the deformation and recovery of SMPs under a range of different constraints. In this study, the thermomechanics of shape storage and recovery of an epoxy resin is systematically investigated for small strains (within ±10%) in uniaxial tension and uniaxial compression. After initial pre-deformation at a high temperature, the strain is held constant for shape storage while the stress evolution is monitored. Three cases of heated recovery are selected: unconstrained free strain recovery, stress recovery under full constraint at the pre-deformation strain level (no low temperature unloading), and stress recovery under full constraint at a strain level fixed at a low temperature (low temperature unloading). The free strain recovery results indicate that the polymer can fully recover the original shape when reheated above its glass transition temperature (T_g). Due to the high stiffness in the glassy state (T < T_g), the evolution of the stress under strain constraint is strongly influenced by thermal expansion of the polymer. The relationship between the final recoverable stress and strain is governed by the stress–strain response of the polymer above T_g. Based on the experimental results and the molecular mechanism of shape memory, a three-dimensional small-strain internal state variable constitutive model is developed. The model quantifies the storage and release of the entropic deformation during thermomechanical processes. The fraction of the material freezing a temporary entropy state is a function of temperature, which can be determined by fitting the free strain recovery response. A free energy function for the model is formulated and thermodynamic consistency is ensured. The model can predict the stress evolution of the uniaxial experimental results. The model captures differences in the tensile and compressive recovery responses caused by thermal expansion. The model is used to explore strain and stress recovery responses under various flexible external constraints that would be encountered in applications of SMPs.     

Effect of temperature and molecular weight on the interfacial tension of PS/PDMS blends

The imbedded-fiber retraction (IFR) method was used to study the effect of temperature and PDMS molecular weight on the interfacial tension of PS/PDMS blends. The interfacial tension decreased with increasing temperature and analysis of the temperature dependence using a simple linear fit gave –dγ/dT value of 0.058±0.010 dyn/cm-deg. Reported –dγ/dT values of PS/PDMS blends are highly dependent on the molecular weights of the polymers and can have values that are <0, 0, or >0. Our interfacial tension values were independent of the molecular weight of PDMS and this was attributed to the molecular weights studied here being well above the entanglement values of both polymers. However, analysis of interfacial tension data from this work and the literature showed the following empirical relationship between apparent blend molecular weight, M_b, and interfacial tension of PS/PDMS blends with a correlation of 0.94: γ₁₂=γ₀+k₂M_b ^(–2/3), where γ₀=7.3±0.3 dyn/cm; k₂=–517±41 (dyn/cm)(g/mol)^2/3.     

Viscoplastic field near a propagating crack-tip

An elastic-viscoplastic constitutive model is proposed instead of the usual elastoplastic model. It is assumed that when crack-tip is approached the viscosity coefficient lends to zero (=₀r). Asymptotic analysis of the dynamic field near a propagating crack-tip is given, and the uniparameter solution is obtained. The numerical result is given for various Mach number and viscosity coefficient. Based on the asymptotic solution, a fracture criterion is proposed and the stability of crack propagation is discussed.     

Axisymmetric thermocapillary convection in a slowly rotating annular cylindrical container

In a slowly rotating annular cylindrical container the free liquid surface (liquid-gas interface) is subjected to a temperature gradient in radial direction. The temperature dependent surface tension creates a shear stress on the interface which is transmitting a thermocapillary convection in the bulk of the liquid. For constant temperature T ₁ of the inner and T ₂ of the outer wall a steady Marangoni convection takes place, exhibiting a double vortex ring of equal directional flow. For time-oscillatory temperatures of the walls a time-dependent thermocapillary convection appears, which will create on the free liquid surface various wave patterns. They shall, depending on the forcing frequency of the temperature, exhibit resonance peaks. The velocity distribution and the response magnitude inside the container has been determined. Received on 3 September 1997     

Converting industrial waste contact masses into effective multicomponent copper-based catalysts for the Rochow process

In this work, we report a simple and inexpensive approach to synthesize effective multicomponent Cu–Cu₂O–CuO catalysts for the Rochow process from industrial waste contact masses (WCMs). WCMs from the organosilane industry were treated with acid followed by reduction with metallic iron powder. The obtained copper powder was then subjected to controlled oxidation in air at different temperatures, followed by ball milling. The orthogonal array approach was applied to optimize this process, and the stirring speed and pH were found to significantly affect the leaching ratio and copper yield, respectively. When used for the Rochow process, the optimized ternary Cu–Cu₂O–CuO catalyst greatly enhanced the dimethyldichlorosilane selectivity and Si conversion compared with Cu–Cu₂O–CuO catalysts prepared without ball milling, bare Cu catalysts, and Cu–Cu₂O–CuO catalysts with different compositions. This could be attributed to their small particle size and the strong synergistic effect among the multiple components in the catalyst with the optimized composition.