Strain-hardening Modulus Research Articles

Practical strain-hardening material properties for use in deformation-based structural steel design

Through the development of an innovative full cross-section tensile testing method, a programme of experiments was conducted to investigate the influence of average cross-section properties on the constitutive relationships for carbon steel, to validate the use of an elastic linear hardening model in practical design, and to assess the resulting accuracy enhancements to the new deformation-based continuous strength method (CSM) of structural steel design. A total of seventeen full cross-section tensile tests on hot-rolled I-sections, hollow sections and cold-formed hollow sections were performed and these were compared with coupon test data obtained from a supplementary programme of 14 tensile coupon tests and data carefully obtained from the literature. The overall behavioural response of the cross-section tensile tests demonstrated that assuming an elastic, linear hardening material model for the CSM is a reasonable assumption and the previous assumption concerning the magnitude of the strain-hardening modulus, based upon the recommendations of EN 1993-1-5, is overly conservative. A revised suite of material models was presented and was shown to furnish the CSM capacity equations with a higher degree of accuracy when compared against experimental data.

From soft to strong elastomers: the role of additional crosslinkings in copolymer-grafted multiwalled carbon nanotube composite thermoplastic elastomers

Fabricating elastomeric materials is one of the most important branches in current nanotechnology and composite science. The macroscopic properties of the elastomers can be modified by adding fillers through changing the types and degrees of crosslinkings. In this study, copolymer-grafted multiwalled carbon nanotube composite thermoplastic elastomers (CTPEs) have been designed, in which the grafted poly(n-butyl acrylate-co-methyl methacrylate) (P(BA-co-MMA)) random copolymers behave as the soft elastomer, and the homogeneously and discretely dispersed multiwalled carbon nanotubes (MWCNTs) of low amounts provide additional physical crosslinkings, which significantly enhance the mechanical properties of the composite elastomers, including high strain-hardening modulus and much improved elastic recovery. This design concept opens a window to produce high-performance strong composite thermoplastic elastomeric materials from soft ones through the incorporation of rigid fillers of low amounts by a grafting strategy.

Effect of Predamage on the Stress-Strain Relationship of Confined Concrete under Monotonic Loading

Jacketing concrete columns with steel or fiber-reinforced polymer (FRP) has been identified as a highly effective way of restoring and enhancing the strength and ductility of columns damaged by catastrophic events such as earthquakes, blasts, and vehicle impact. However, a stress-strain relationship for damaged and subsequently repaired concrete is not available for design of the repair works. An extensive experimental and analytical investigation was undertaken in this work to study the mechanical properties of repaired concrete columns. Tests were conducted on 102 concrete cylinders involving variations of damage degree, concrete grade, and confinement pressure. The effect of damage on the compressive strength, strain capacity, initial elastic modulus, and strain-hardening modulus of repaired concrete is quantified in terms of a damage index. A stress-strain model is developed for the design of structural repair works. Evaluation of the test data also exhibits that the classic model of confined concrete proposed by a previous study and the classic internal friction theory are qualitatively applicable to failure of concrete materials not only in its original form, but also in the heavily disturbed state. The Hoek-Brown failure criterion is unconservative for heavily damaged concrete.

A method for calculating damage evolution in adiabatic shear band of titanium alloy

A method for calculating the evolution of the local damage variable at the adiabatic shear band (ASB) center was proposed. In the present method, the JOHNSON-COOK model and the nonlocal theory were adopted, and the damage variable formula applicable for the bilinear (linearly elastic and strain-softening) constitutive relation was further generalized to consider the plastic deformation occurring in the strain-hardening stage. Aiming at Ti-6Al-4V, the effect of strain rate on the evolution of the local damage variable at the ASB center was investigated. In addition, a parametric study was carried out, including the effects of strain-hardening exponent, strain rate sensitive coefficient, thermal-softening exponent, static shear strength, strain-hardening modulus, shear elastic modulus, work to heat conversion factor, melting temperature and initial temperature. The damage extent at the ASB center in the radial collapse experiment was assessed. It is found that at higher strain rates the damage in the ASB becomes more serious at the same average plastic shear strain of the ASB.

Microscopic Mechanisms of Strain Hardening in Glassy Polymers

The mechanisms underlying the increase in stress for large mechanical strains of a polymer glass, quantified by the strain-hardening modulus, are still poorly understood. In the present paper we aim to elucidate this matter and present new mechanisms. Molecular-dynamics simulations of two polymers with very different strain-hardening moduli (polycarbonate and polystyrene) have been carried out. Nonaffine displacements occur because of steric hindrances and connectivity constraints. We argue that it is not necessary to introduce the concept of entanglements to understand strain hardening, but that hardening is rather coupled with the increase in the rate of nonaffine particle displacements. This rate increases faster for polycarbonate, which has the higher strain-hardening modulus. Also more nonaffine chain stretching is present for polycarbonate. It is shown that the inner distances of such a nonaffinely deformed chain can be well described by the inner distances of the worm-like chain, but with an effective stiffness length (equal to the Kuhn length for an infinite worm-like chain) that increases during deformation. It originates from the finite extensibility of the chain. In this way the increase in nonaffine particle displacement can be understood as resulting from an increase in the effective stiffness length of the perturbed chain during deformation, so that at larger strains a higher rate of plastic events in terms of nonaffine displacement is necessary, causing in turn the observed strain hardening in polymer glasses.

Deforming glassy polystyrene: Influence of pressure, thermal history, and deformation mode on yielding and hardening

The toughness of a polymer glass is determined by the interplay of yielding, strain softening, and strain hardening. Molecular-dynamics simulations of a typical polymer glass, atactic polystyrene, under the influence of active deformation have been carried out to enlighten these processes. It is observed that the dominant interaction for the yield peak is of interchain nature and for the strain hardening of intrachain nature. A connection is made with the microscopic cage-to-cage motion. It is found that the deformation does not lead to complete erasure of the thermal history but that differences persist at large length scales. Also we find that the strain-hardening modulus increases with increasing external pressure. This new observation cannot be explained by current theories such as the one based on the entanglement picture and the inclusion of this effect will lead to an improvement in constitutive modeling.

Tensile and microbend tests of pure aluminum foils with different thicknesses

A series of systematic tensile and microbend tests was conducted on polycrystalline pure aluminum foils having the same microstructure but with different thicknesses. The thickness was changed by electropolishing. With the reduction of the foil thickness, an increase in yield strength was observed in the tensile tests, while in the microbend tests, an increase in the apparent strain-hardening modulus occurred. Numerical simulations of the microbend tests are performed using a higher-order gradient crystal plasticity theory that accounts for the effect of internal stresses due to the geometrically necessary dislocations. The trend observed in the microbend tests is reasonably reproduced in the simulations.

Coupling effects of spinodal decomposition and crystallization on mechanical properties of polyolefin blends

The influences of preferentially occurred liquid–liquid phase separation (LLPS) and following crystallization processes on the mechanical properties of statistical copolymer blends of poly(ethylene-co-hexene) (PEH) and poly(ethylene-co-butene) (PEB) have been investigated in detail through tensile deformation tests with a relatively high extension rate to avoid the effect of interfacial properties of the blends. Crystallinity and lamellar thickness of the samples are estimated by using the wide-angle X-ray diffraction and small-angle X-ray scattering techniques, respectively. The tensile modulus and yield stress are found to increase with LLPS time up to 6h, but decrease afterwards, under the conditions of temperature of 120°C and isothermal crystallization time of 10min. It is considered that the instantaneous tensile properties are substantially largely affected by the much perfect lamellar structures formed during crystallization with a long time prior LLPS step. This finding is further experimentally substantiated by the scanning electron microscope observation. Whereas the strain-hardening modulus described by a simple neo-Hookean relation increases with LLPS time and reaches a plateau after 6h, which can be accounted for by the cooperation effect between amorphous entanglement density, insensitive to LLPS time, and crystallinity redistribution. The similarity of the results observed on the blends experiencing the spinodal decomposition (SD) process supports that the redistribution of crystallizable components contributes to the tensile stress increase, which is primarily controlled by the development of LLPS process. This simple relationship gives us a new insight of what controls the mechanical properties of the phase separated polymer blends and of how we might be able to predict the mechanical properties of as yet unmixed polymer pairs.

Inverse Temperature Dependence of Strain Hardening in Ultrahigh Molecular Weight Polyethylene: Role of Lamellar Coupling and Entanglement Density

Irradiation of ultrahigh molecular weight polyethylene (UHMWPE) with a dose of 150 kGy by an electron beam can effectively increase the entanglement density in the amorphous phase and has little influence on the properties of the crystalline phase, which provides examples to comparatively investigate the role of lamellar coupling and entanglement density in determining the strain-hardening effect in semicrystalline polymers. The strain-hardening modulus, deduced from the Haward plots of true stress-strain curves, is inversely temperature-dependent and has a sharp transition around 65 degrees C that corresponds to the mechanical alphaI-process of the crystalline phase for both nonirradiated and irradiated samples, irrespective of the entanglement density in the amorphous phase. Lamellar coupling takes more effect in determining the strain-hardening behavior before the mechanical alphaI-process is activated. With further increasing temperature, lamellar coupling becomes weaker and the role of the entangled amorphous phase is gradually presented. However, the same temperature dependence of the strain-hardening modulus in both nonirradiated and irradiated samples indicates that the strain-hardening behavior in semicrystalline polymer is mostly determined by lamellar coupling rather than by entanglement density.

Effects of constitutive parameters on adiabatic shear localization for ductile metal based on JOHNSON-COOK and gradient plasticity models

By using the widely used JOHNSON-COOK model and the gradient-dependent plasticity to consider microstructural effect beyond the occurrence of shear strain localization, the distributions of local plastic shear strain and deformation in adiabatic shear band(ASB) were analyzed. The peak local plastic shear strain is proportional to the average plastic shear strain, while it is inversely proportional to the critical plastic shear strain corresponding to the peak flow shear stress. The relative plastic shear deformation between the top and base of ASB depends on the thickness of ASB and the average plastic shear strain. A parametric study was carried out to study the influence of constitutive parameters on shear strain localization. Higher values of static shear strength and work to heat conversion factor lead to lower critical plastic shear strain so that the shear localization is more apparent at the same average plastic shear strain. Higher values of strain-hardening exponent, strain rate sensitive coefficient, melting point, thermal capacity and mass density result in higher critical plastic shear strain, leading to less apparent shear localization at the same average plastic shear strain. The strain rate sensitive coefficient has a minor influence on the critical plastic shear strain, the distributions of local plastic shear strain and deformation in ASB. The effect of strain-hardening modulus on the critical plastic shear strain is not monotonous. When the maximum critical plastic shear strain is reached, the least apparent shear localization occurs.

Molecular Dynamics Simulation of Uniaxial Deformation of Glassy Amorphous Atactic Polystyrene

Molecular dynamics computer simulations have been carried out of a chemically realistic many-chain nonentangled model of glassy atactic polystyrene under the influence of uniaxial mechanical deformation. Both the initial elastic and the postyield (up to 100% of the deformation) behavior have been simulated. The Poisson ratio, the Young modulus, and the temperature dependence of the yield peak are well reproduced. The simulated strain-hardening modulus is in quantitative agreement with existing experiments. The deformationally induced anisotropy in the global and local segmental orientation is accompanied by an anisotropy of the local translational mobility: the mean-square translational displacement of the individual segments in the direction of the deformation is drastically increased just beyond the yield point as compared to the isotropic sample. The mechanical deformation of a quenched sample leads to an almost complete erasure of the aging history.

Linear strain hardening in elastoplastic indentation contact

Finite-element analyses for elastoplastic cone indentations were conducted in which the effect of linear strain hardening on indentation behavior was intensively examined in relation to the influences of the frictional coefficient (μ) at the indenter/material contact interface and of the inclined face angle (β) of the cone indenter. A novel procedure of “graphical superposition” was proposed to determine the representative yield stress YR. It was confirmed that the concept of YR applied to elastic-perfectlyplastic solids is sufficient enough for describing the indentation behavior of strainhardening elastoplastic solids. The representative plastic strain of εR (plastic) ≈ 0.22 tan β, at which YR is prescribed, is applicable to the strain-hardening elastoplastic solids, affording a quantitative relationship of YR = Y + ε;R (plastic) × EP in terms of the strain-hardening modulus EP. The true hardness H as a measure for plasticity is estimated from the Meyer hardness HM and then successfully related to the yield stress Y as H = C(β,μ) × Y for elastic-perfectly-plastic solids and as H = C(β,μ) × YR for strain-hardening solids, by the use of a β- and μ-dependent constraint factor C(β,μ) ranging from 2.6 to 3.2.

A Theoretical Analysis on the Deformation of Rectangular Plate Under Out-of-Plane Tearing

A theoretical analysis is proposed to predict the out-of-plane tearing response of rectangular plates with two or three legs. The nonlinear differentiate equation for deformation and the double linear strain-hardening constitutive relation for bending moment against curvature are used in the analysis. The external work is assumed dissipated through the bending deformation of the legs and the crack propagation at the tip of the crack between the legs. A good agreement has been achieved between the theoretical predictions and the experimental results for the variation of external force with displacement. It has been found that the variation of bending force with displacement significantly depends on the strain-hardening modulus and the tearing force.

Strength and ductility of HPS flexural members

An experimental and analytical research program was recently completed that examined in detail the parameters affecting the strength and ductility of high-performance steel (HPS) flexural members. HPS is a term used to describe a new class of steels being produced under strictly controlled conditions that have high strength, usually greater than 448 MPa (65 ksi) and exceptional toughness and weldability. The mechanical characteristics of these steels are different from conventional steels, leading to concerns over their use in some structural applications. Under earthquake loading, flexural members are expected to deform inelastically, so members fabricated with HPS steels must possess adequate ductility. This paper discusses the inelastic behavior of welded, I-shaped flexural members fabricated from an HPS steel, HSLA-80, having a nominal yield stress of 550 MPa (80 ksi) and an ultimate strength between 610–690 MPa (90–100 ksi) and compares the results to similar flexural members fabricated from conventional A36 steel. The effects of material properties: yield stress, strain-hardening modulus, yield stress-to-ultimate strength ratio, and strain at ultimate stress; cross-section geometry: flange slenderness, web slenderness, and lateral slenderness; and loading condition: monotonic moment gradient, monotonic uniform moment, and cyclic moment gradient are described from the results of experimental testing and analytical modeling. The results are evaluated against the existing design criteria established in the AISC-LRFD specifications and recommendations are made for revising the specifications.

Gauge-invariant self-organization of strain field fluctuations

The method of laser range scanning profilometry is used to study the evolution of mesoscopic perturbations of strain fields in aluminum, copper, and iron polycrystals during loading. The fractal nature of mesoscopic perturbations of the flow fields with a power-flow increase of the correlation interval ξ=ξ0θ−v (θ is the dimensionless strain-hardening modulus) during loading is established. The fundamental structural scale ξ0 and the index ν lie in the intervals ξ0=30–150 µm and ν=0.27–0.70. The dimensionality of the deformation relief increases in the course of loading from ∼1.05–1.25 at the initial stage to ∼1.35–1.65 for strains from ∼0.30 to 0.35 for all the materials. The results indicate an essentially multilevel nature of plastic deformation and argue for the applicability of renormalization group methods for its description.

Instable modes of in-plane bending of strip metal under unequal compressing

The in-plane bending of a strip metal workpiece under unequal compression can be developed into an advanced precision forming process with high quality, high efficiency, low consumption, and good flexibility for size changing. Moreover, the loading path and stress state of the process make it possible to develop the deformation potentiality of the workpiece to the full and to greatly improve the bending forming limit of the workpiece, which is difficult to achieve by any other conventional process. The forming limit of the workpiece depends on the modes and the conditions of instability of the process. However, up to now, literature on the instability has been scant. This paper reports research into the mechanism, the modes, and the conditions of instability of the process by a comprehensive analysis based on the minimum energy principle of plastic deformation and by an experiment with LF21M strip workpieces on apparatus made by the authors for performing in-plane bending with two conical rollers. The research results show the following: (1) Because the in-plane bending process has a specific three-dimensional compressive stress state and the resistance to instability is determined by the geometry parameters and the property values of the workpiece, there are three new modes of instability of the process, different from those of conventional bending processes. (2) For a given material the original thickness t 0, original width b 0, unequal compression width a z, and the smallest thickness of the formed workpiece t 1, there are the instability parameters of K 1, K 2, K 3, m 1, and m 2, and they satisfy the conditions of 0 < K 1 < K 2 < K 3 and 0 < m 1 < m 2. It is possible for the workpiece to produce external wrinkling when a z/ b 0 ≤ K 1 or t 1/ t 0 ≤ m 1, the turning when K 1 < a z/ b 0 ≤ K 2 and m 1 < t 1/ t 0 ≤ m 2, and the internal wrinkling when K 2 < a z/ b 0 ≤ K 3 and m 1 < t 1/ t 0 ≤ m 2. When a z/ b 0 > K 3 and m 1 < t 1/ t 0 ≤ m 2, the workpiece can perform a stable process of in-plane bending and greatly improve the bending forming limit without any mode of instability. The approximate values of K 1, K 2, K 3, m 1, and m 2 for strip workpieces LF21M alloy (sheet thickness: 1.90 and 1.50 mm) are 0.25, 0.45, 0.60, 0.30, and 0.95, respectively. (3) The greater are the elastic modulus E and the strain-hardening modulus D of the material used, the smaller are the instability values of K 1, K 2, K 3, and m 1, and the greater is the value of m 2. The achievements of this research can serve as a significant guide to the determination of the forming limit and the optimal process parameters for the practice of the relevant processes.

Negative bending resistance of HPS70W girders

This paper investigates the influence of material and geometric parameters on the flexural behavior of I-girders fabricated from Grade 70W high performance steel. Three cross-section geometric parameters are considered: compression flange slenderness ( b fc/2 t fc), web slenderness (2 D cp/ t w), and the ratio of the web depth to the compression flange width ( D/ b fc). Also, three material parameters are studied: the yield ratio ( F y/ F u), the ratio of the strain-hardening and yield strains ( ϵ st/ ϵ y), and the strain-hardening modulus ( E st). The parametric studies are conducted using full non-linear shell finite element models of the hypothetical physical girders. The resulting moment–rotation curves obtained from these simulated experiments are compared with simplified moment–plastic rotation equations that have been developed in previous research, and with current AASHTO specification strength equations.

Local Buckling of W Shape Columns and Beams

An analytical method, previously developed and programmed for computer use, is used to carry out an extensive study of the various parameters affecting the local buckling behavior of W shape columns and beams. The effects of flange and web interaction, residual stresses, plate geometry, and inelastic action are included in the study. Specific parameters investigated include web slenderness, flange slenderness, yield stress, residual stress, strain-hardening modulus, and member length. Local buckling curves of critical loads versus web slenderness are presented for various values of flange slenderness for beams and columns. The results of the study indicate that the present values of flange slenderness = 100 (260 SI) and web slenderness = 225 (670 SI), as presently specified for columns in the Canadian Standard, CAN3-S16.1M78, are inappropriate. Values of flange slenderness = 72 (189 SI) and web slenderness = 300 (788 SI) are indicated. The results also show that the presently specified values of flange slenderness = 100 (260 SI) and web slenderness = 690 (1,810 SI) for Class 3 (non-compact) sections may be too liberal with respect to the flanges and conservative with respect to the web. Values of flange slenderness = 72 (189 SI) and web slenderness = 800 (2,100 SI) are indicated by the study. The effects of all the parameters investigated during the study are shown in the form of local buckling curves as described here.

Properties of Steel for Use in LRFD

Background and rationale is presented for selecting mean values and coefficients of variation of material properties from available experiments for use in first-order probability-based design; the following properties of structural steel are included: elastic moduli, yield stress, strain-hardening modulus.

Dynamic Plastic Buckling of Cylindrical Shells in Sustained Axial Compressive Flow

A theory is postulated to explain the dynamic plastic buckling of cylindrical shells in sustained axial compressive flow. Tube impact experiments are described in which uniform axisymmetric waves were produced. Predicted and experimental wavelengths are in satisfactory agreement. According to the theory presented, wavelengths do not depend strongly on strain-hardening modulus and, based on results for two aluminum alloys, neither do experimental wavelengths. This result is shown to apply also to slow plastic buckling.