Ductility of interstitial-free steel under high strain rate tension: Experiments and macroscopic modeling with a physically-based consideration

In this paper, an experimental investigation and a constitutive modeling of the mechanical response of an interstitial-free (IF) steel over a wide range of strain rates (from 0.001/s to 750/s) are presented. Tensile tests at relatively high strain rates, exceeding 100/s, are performed at an initial room temperature, using the so-called one bar technique developed on the basis of the Hopkinson bar method. At a high strain rate, a distinct upper yield limit is observed, and the subsequent flow stress increases remarkably. Furthermore, the ductility is reduced significantly in comparison to the case of low strain rate tension. In order to express such a complicated material response of IF steel, we develop a new constitutive model that takes into account effects of a change in the mobile dislocation density and thermal softening. The model can be easily applicable to large-scale engineering computations, because it is macroscopically formulated. We try to reproduce the tensile response including a diffuse neck formation at high strain rates, using the proposed constitutive model and finite element method. The results indicate that a change in the mobile dislocation density, together with thermal softening, has substantial effects on apparent work hardening behavior at high strain rates, although the change in the mobile dislocation density is transcribed at macroscopic scale in the model. Finally, we discuss characteristics of true stress–true strain curves at various strain rates, and their correlation with the plastic instability behavior.     

Creep analysis with a stress range dependent constitutive model

Many materials exhibit the stress range dependent creep behavior. The power law creep observed for a certain stress range changes to the viscous type creep as the stress value decreases. Recently published experimental data for advanced heat resistant steels indicate that the high creep exponent (in the range 7–12) may decrease to the low value of approximately unity within the stress range relevant for engineering structures. The aim of this paper is to analyze the influence of the stress range dependent power-law-viscous creep transition on the behavior of structures at elevated temperature. A constitutive model for the minimum creep rate is introduced to describe both the linear and the power law creep depending upon the stress level. To demonstrate basic features of the stress range dependent creep modeling, several elementary examples from structural mechanics are presented. They include a stress relaxation problem, a beam subjected to pure bending and a pressurized thick-walled cylinder. Based on the uni-axial transition stress the transition value of the external load is estimated such that above this value the power law can be applied. For the loading levels below this value the character of the stress distribution as well as the stress values are essentially influenced by the viscous creep.     

Swelling of an internally pressurized nonlinearly elastic tube with fiber reinforcing

We study the effect of swelling on the mechanical response of fiber reinforced tubes within the context of finite elastic deformation. The fibers themselves do not swell, setting up a competition between the matrix, for which swelling tends to open the tube, and the fibers, for which swelling tends to constrict the tube. Balancing these tendencies in the constitutive response can lead to an internal channel opening that remains relatively constant over a wide range of swelling. Further, the hoop stress on the inner wall in such a situation may be compressive, rather than tensile. Both effects may be advantageous in certain settings, including biological organ systems.     

A finite deformation fractional viscoplastic model for the glass transition behavior of amorphous polymers

The stress response of amorphous polymers exhibits tremendous change during the glass transition region, from soft viscoelastic response to stiff viscoplastic response. In order to describe the temperature-dependent and rate-dependent stress response of amorphous polymers, we extend the one-dimensional small strain fractional Zener model to the three-dimensional finite deformation model. The Eyring model is adopted to represent the stress-activated viscous flow. A phenomenological evolution equation of yield strength is used to describe the strain softening behaviors. We demonstrate that the stress response predicted by the three-dimensional model is consistent with that of one-dimensional model under uniaxial deformation, which confirms the validity of the extension. The model is then applied to describe the stress response of an amorphous thermoset at various temperatures and strain rates, which shows good agreement between experiments and simulation. We further perform a parameter study to investigate the influence of the model parameters on the stress response. The results show that a smaller fractional order results in a larger yield strain while has little effect on the yield stress when the temperature is below the glass transition temperature. For the stress relaxation tests, a smaller fractional order leads to a slower relaxation rate.     

From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models

We propose a construction method of non-homogeneous solutions for the traction problem of an elastic damaging bar. This bar has a softening behavior that obeys a gradient damaged model. The method is applicable for a wide range of brittle materials. For sufficiently long bars, we show that localization arises on sets whose length is proportional to the material internal length and with a profile that is also a material characteristic. From its onset until the rupture, the damage profile is obtained either in a closed form or after a simple numerical integration depending on the model. Thus, the proposed method provides definitions for the critical stress and fracture energy that can be compared with experimental results. We finally discuss some features of the global behavior of the bar such as the possibility of a snapback at the onset of damage. We point out the sensitivity of the responses to the parameters of the damage law. All these theoretical considerations are illustrated by numerical examples.     

Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress

Uniaxial compression stress–strain tests were carried out on three commercial amorphous polymers: polycarbonate (PC), polymethylmethacrylate (PMMA), and polyamideimide (PAI). The experiments were conducted under a wide range of temperatures (−40 °C to 180 °C) and strain rates (0.0001 s⁻¹ up to 5000 s⁻¹). A modified split-Hopkinson pressure bar was used for high strain rate tests. Temperature and strain rate greatly influence the mechanical response of the three polymers. In particular, the yield stress is found to increase with decreasing temperature and with increasing strain rate. The experimental data for the compressive yield stress were modeled for a wide range of strain rates and temperatures according to a new formulation of the cooperative model based on a strain rate/temperature superposition principle. The modeling results of the cooperative model provide evidence on the secondary transition by linking the yield behavior to the energy associated to the β mechanical loss peak. The effect of hydrostatic pressure is also addressed from a modeling perspective.     

Determination of yield stress fluid behaviour from inclined plane test

The aim of this paper is to determine precisely under which conditions an inclined plane can be used as a rheometer, which could represent a practical and rapid technique for various types of industrial or natural viscoplastic coarse suspensions. We first examine its efficiency and relevancy for determining fluid yield stress in a straight way by measuring the deepest fluid layer able to stay on the inclined plane. We have made experiments with different materials (clay-water suspensions) whose yield stress ranged from 35 to 90 Pa, using 1 m long open rectangular channels with a slope ranging from 10 to 30° and a width ranging from 5 to 25 cm. Our procedure involved measuring the final fluid depth far from edges a long time after the end of the slow gravity-induced emptying of a dam placed upstream. The fluid yield stress was also estimated independently by fitting a Herschel-Bulkley model to simple shear rheometry data obtained within a relatively wide shear rate range. A good agreement between inclined rectangular channel tests and independent usual rheometrical tests is obtained even for aspect ratios (flow depth to channel width ratio) as large as 1 when one assumes that, when the fluid has stopped, the side and bottom wall shear stresses are equal to the fluid yield stress. These results prove the efficiency of the inclined plane test for determining yield stress when appropriate experimental precautions are taken for both tests. In addition we examine the possibility of determining the simple shear flow curve of a mud suspension from fluid depth, velocity and discharge measurements of different steady flows in a wide open channel (8 m long; 60 cm wide) equipped with a recirculating system. The results obtained from inclined plane tests are in good agreement with independent rheometrical data (with torsional geometries). However it is technically difficult to cover a wide shear rate range from the inclined plane technique since this requires a rather wide channel flow rate range.     

Modeling of steady and time-dependent responses in filled, uncured, and crosslinked rubbers

Filled polymer systems have been a subject of interest for rheologists for several decades. Their applications range from paints and pigments to high performance composite materials. Presently, there is a lack of complete understanding of the behavior of these materials under varying kinematic and dynamic conditions. Moreover, there is a lack of a comprehensive theory, which can simultaneously describe the rheology of filled rubbers, their chemorheology, and their behavior in the final fully cured state. The present work is aimed at capturing a wide range of rheological (viscoelastic and kinetic) properties of filled rubbers with one set of constitutive/kinetic equations and a flexible relaxation spectrum. The various experiments covered are yield-flow transition in creep, shear start up responses, dynamic behavior in the melt state, and the changes during the cure stage. In the post cure state, the manifestations in Mullins stress softening-hysteresis and recovery, large strain stress relaxations, and dynamic behavior are also demonstrated. Finally, the non-linearities during large strain dynamic deformations, accompanied by non-isothermal, viscoelastic, and structure effects are exemplified. Received: 24 July 2000 Accepted: 13 November 2000     

Photo-induced deformation of active polymer films: Single spot irradiation

Light-activated polymers can undergo complex deformation in response to the combination of mechanical and optical stimuli. These materials are attractive for remote actuation and sensing applications. However, the behavior of such materials subjected to photomechanical patterning is not well understood. In this paper we consider a polymer that operates by photoactivated stress relaxation; at the molecular level, photoinitiation of residual initiator molecules generate free radicals that break and then reform in-chain functionalities of stretched chains in an elastomeric network, which results in macroscopic stress relaxation. We carry out experiments and finite element calculations that demonstrate the sequence of deformation events culminating in the formation of a buckled spot as a result of biaxially stretching the elastomeric film then irradiating a circular region followed by releasing the mechanical constraint. In order to better understand the photomechanics, we analyze a simpler model problem wherein a linear elastic, stress relaxing disk is subjected to (i) radial extension, (ii) irradiation of a concentric circular region, and (iii) release of the applied displacements in (i), which results in deformation and stress redistribution. In the final step, the deformation may transition from planar to buckling out of the plane depending on system parameters. Companion finite element calculations are performed against which our analytical results are in good agreement. Although not directly comparable, the analytic model qualitatively agrees with the experiments. The results of this work provide a useful foundation from which to explore more interesting behavior of periodically photo-mechanically patterned films and other more challenging actuation problems.     

The solid–fluid transition in a yield stress shear thinning physical gel

We present an experimental investigation of the solid–fluid transition in a yield stress shear thinning physical gel (Carbopol^® 940) under shear. Upon a gradual increase of the external forcing, we observe three distinct deformation regimes: an elastic solid-like regime (characterized by a linear stress–strain dependence), a solid–fluid phase coexistence regime (characterized by a competition between destruction and reformation of the gel), and a purely viscous regime (characterized by a power law stress-rate of strain dependence). The competition between destruction and reformation of the gel is investigated via both systematic measurements of the dynamic elastic moduli (as a function of stress, the amplitude, and temperature) and unsteady flow ramps. The transition from solid behavior to fluid behavior displays a clear hysteresis upon increasing and decreasing values of the external forcing. We find that the deformation power corresponding to the hysteresis region scales linearly with the rate at which the material is being forced (the degree of flow unsteadiness). In the asymptotic limit of small forcing rates, our results agree well with previous steady state investigations of the yielding transition. Based on these experimental findings, we suggest an analogy between the solid–fluid transition and a first-order phase transition, e.g., the magnetization of a ferro-magnet where irreversibility and hysteresis emerge as a consequence of a phase coexistence regime. In order to get further insight into the solid–fluid transition, our experimental findings are complemented by a simple kinetic model that qualitatively describes the structural hysteresis observed in our rheological experiments. The model is fairly well validated against oscillatory flow data by a partial reconstruction of the Pipkin space of the material’s response and its nonlinear spectral behavior.     

Experimental and numerical investigations of pressure drop in a rectangular duct with modified power law fluids

Numerical solutions are presented for fully developed laminar flow for a modified power law fluid (MPL) in a rectangular duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian behavior at low shear rates, through a transition region, to power law behavior at higher shear rates. The analysis identified a dimensionless shear rate parameter which, for a given set of operating conditions, specifies where in the shear rate range a particular system is operating, i.e. in the Newtonian, transition, or power law regions. The numerical results of the friction factor times Reynolds number for the Newtonian and power law region are compared with previously published results showing agreement within 0.05% in the Newtonian region, and 0.9% and 5.1% in the power law region. Rheological flow curves were measured for three CMC-7H4 solutions and were found to be well represented by the MPL constitutive equation. The friction factor times Reynolds number values were measured in the transition region for which previous measurements were unavailable. Good agreement was found between experiment and calculation thus confirming the validity of the analysis.     

Reproducibility and reliability of the response from four SPATE systems

The SPATE (stress pattern analysis by thermal emissions) system is currently the standard equipment for thermoelastic stress analysis (TSA). A carefully designed test program that studied the behavior of four independent SPATE systems over an 8-month period is described. The response of each system is compared with the response of the other systems in the study.     

Patterned bilayer plate microstructures subjected to thermal loading: Deformation and stresses

We studied the deformation of a series of gold/polysilicon patterned plate microstructures fabricated by surface micromachining. The patterned plate microstructures were subjected to a uniform temperature change from 100 °C to room temperature that was intended to induce linear and geometrically nonlinear deformation. We used interferometry to measure full-field deformed shapes of the microstructures. From these measurements we determined the spatially-averaged curvature of the deformed microstructures within individual lines and across the entire plate. The deformation response of the patterned plates can be broadly characterized in terms of the average curvature as a function of temperature change and exhibits linear and geometrically nonlinear behavior. We modeled the deformation response of the patterned plates using geometrically nonlinear plate theory with the finite element method. Good agreement was obtained between predictions and measurements for both local curvature variations across lines and for the evolution of curvature of the entire plate with temperature change. Using a generalized plane strain approach with the finite element method we also modeled the spatial dependence of the stress distribution in the lines and substrate. For thick plates, our results agree with those of previous studies, showing a decrease in the von Mises stress in the metal lines with decreasing linewidth. For thinner substrates, though, we find the behavior with linewidth is opposite and there is a critical substrate thickness (about 10 μm for the system in our study) where the behavior with linewidth changes. These results have important implications in the design of patterned structures for micro-electro-mechanical systems (MEMS) applications where films are of comparable thickness to the underlying substrate.     

Rate independent hysteresis in a bi-stable chain

The nontrivial behavior of an elastic chain with identical bi-stable elements may be considered prototypical for a large number of nonlinear processes in solids ranging from phase transitions to fracture. The energy landscape of such a chain is extremely wiggly which gives rise to multiple equilibrium configurations and results in a hysteretic evolution and a possibility of trapping. In the present paper, which extends our previous study of the static equilibria in this system (Puglisi and Truskinovsky, J. Mech. Phys. Solids (2000) 1), we analyze the behavior of a bi-stable chain in a soft device under quasi-static loading. We assume that the system is over-damped and explore the variety of available nonequilibrium transformation paths. In particular, we show that the “minimal barrier” strategy leads to the localization of the transformation in a single spring. Loaded periodically, our bi-stable chain exhibits finite hysteresis which depends on the height of the admissible barrier; the cold work/heat ratio in this model is a fixed constant, proportional to the Maxwell stress. Comparison of the computed inner and outer hysteresis loops with recent experiments on shape memory wires demonstrates good qualitative agreement. Finally we discuss a relation between the present model and the Preisach model which is a formal interpolation scheme for hysteresis, also founded on the idea of bi-stability.     

A method of analysis for end and transitional effects in anisotropic cylinders

We present a rigorous verification study and an extension to an existing semi-analytical finite element formulation for analysis of end and transition effects in prismatic cylinders. End and transition effects in stressed cylinders are phenomena associated with the difference between results that are predicted by the Saint–Venant solutions and the actual point-wise conditions. These differences manifest themselves as self-equilibrated stress states. Notwithstanding certain well-known exceptions (e.g., restrained torsion of open thin-walled sections), such effects in isotropic cylinders are usually confined to a very small neighborhood of a terminal boundary or transition zone, and are typically neglected. For anisotropy, as in the case of most smart/active and composite material systems, they can persist much further into the interior of the structure, and need to be quantified to design geometry transition zones and to fully understand the delamination effects. In the semi-analytical approach, we first discretize the governing equations within the cross-sectional plane of the cylinder. The end-solution fields satisfy the homogeneous form of the resulting semi-analytical system of ordinary differential equations. This leads to an algebraic eigenvalue problem, and an eigenfunction expansion of the stress and displacement fields due to end effects. Unique to the present study, we formulate a procedure to quantify the transitional effects for end-to-end connected cylinders for which the displacement and stress continuity along the transition interface need to be enforced. The semi-analytical approach has several distinct advantages: (i) It is computationally efficient, as only the cross-sectional geometry is discretized; (ii) it can be applied to arbitrary cross-sectional geometries and the most general form of anisotropy; and (iii) it yields direct measures for the decay lengths (or decay rates) of any end-or transition-solution field. Analytical solutions to end-effect problems are scarce. Those that exist are for simple geometry and material constitution. We use these analytical solutions, as well as solutions obtained using three-dimensional finite element models, to verify our approach and to assess its efficiency.     

A THEORETICAL MODEL FOR THERMAL-SENSITIVE MICROGEL WITH PNIPAM CORE AND ELASTIC SHELL

Poly(N-isopropylacrylamide)(PNIPAM) microgels are widely used in drug delivery due to their fast response to temperature.In order to get a better biocompatibility,PNIPAM microgels are typically coated with a layer of biocompatible material,resulting in composite microgels with core-shell structure.In a composite microgel prepared recently,for example,a microsphere of PNIPAM gel is enclosed by a phospholipid membrane,and the composite microgel exhibits a substantial volume transition in response to temperature changes.Here we develop a theoretical model to describe the thermal-responsive behavior of this composite microgel.In particular,we treat the phospholipid membrane as an elastic layer behaving like rubber-like elastomers and adopt the form of the free-energy function for nematic gels(which refer to anther species of thermalsensitive gels whose behavior has been intensively studied) as that for PNIPAM gels.We show that the thermal-responsive behavior of the composite microgel can be markedly influenced by the membrane.By investigating the state of stress on the interface,we further predict that when the coating membrane is stiff and thin,wrinkles are expected to occur on the outer surface of the composite microgel after the volume transition.     

A modified wake oscillator model for vortex-induced vibration of circular cylinders for a wide range of mass-damping ratio

In this paper, the behavior of an elastically mounted cylinder, subjected to vortex-induced vibrations (VIV), is investigated by a low-dimensional model. The classical wake oscillator model, as a standard model, predicts the behavior of the system at high mass-damping ratios but fails in modeling the system at low mass-damping ratios. A modified wake oscillator model is introduced in order to describe the response of the system over a wide range of mass-damping ratios. The results of this new model are compared to experimental results from the literature and shown to be in good agreement. The new model can describe most of the features of vortex-induced vibration phenomenology, such as the Griffin plot and lock-in domains.     

On secondary loops in LAOS via self-intersection of Lissajous–Bowditch curves

When the shear stress measured in large amplitude oscillatory shear (LAOS) deformation is represented as a 2-D Lissajous–Bowditch curve, the corresponding trajectory can appear to self-intersect and form secondary loops. This self-intersection is a general consequence of a strongly nonlinear material response to the imposed oscillatory forcing and can be observed for various material systems and constitutive models. We derive the mathematical criteria for the formation of secondary loops, quantify the location of the apparent intersection, and furthermore suggest a qualitative physical understanding for the associated nonlinear material behavior. We show that when secondary loops appear in the viscous projection of the stress response (the 2-D plot of stress vs. strain rate), they are best interpreted by understanding the corresponding elastic response (the 2-D projection of stress vs. strain). The analysis shows clearly that sufficiently strong elastic nonlinearity is required to observe secondary loops on the conjugate viscous projection. Such a strong elastic nonlinearity physically corresponds to a nonlinear viscoelastic shear stress overshoot in which existing stress is unloaded more quickly than new deformation is accumulated. This general understanding of secondary loops in LAOS flows can be applied to various molecular configurations and microstructures such as polymer solutions, polymer melts, soft glassy materials, and other structured fluids.     

Experimental investigation and modeling of non-monotonic creep behavior in polymers

The deformation behavior of two unfilled engineering thermoplastics, ultra high molecular weight polyethylene (UHMWPE) and polycarbonate (PC), has been investigated in creep test conditions. It has been found that a loading history (prior to the creep test) comprising of loading to a maximum stress or strain value followed by partial unloading to arrive at the target stress value can greatly modify the strain-time behavior. Under such a test protocol, while the expected increase in strain during creep (constant tensile load) is observed, at relatively low creep stresses specimens have also demonstrated a monotonic decrease in strain. In an intermediate stress range, specimens have demonstrated time dependent behavior comprising of a transition from decreasing to increasing strain during creep in tension. This paper presents experimental results to delineate these findings and explore the effect of prior strain rate on the qualitative and quantitative changes in the output (strain-time) behavior. Furthermore, modification of the viscoplasticity theory based on overstress (VBO) model into a double element configuration is introduced. These changes confer upon the model the ability to yield non-monotonic behavior in creep, and supporting simulation results have been included. These changes, therefore, allow the model to simulate strain rate sensitivity, creep, relaxation, and recovery behavior, but more importantly address the issue of non-monotonic changes in creep and relaxation when a loading history involves some degree of unloading.