Theoretical calculation of the strain-hardening exponent and the strength coefficient of metallic materials

The purpose of the present article is to theoretically calculate the strain-hardening exponent and the strength coefficient of metallic materials. For this purpose, two equations are used. The first one correlates the strain-hardening exponent and the strength coefficient with the yield stress-strain behavior, while the other one correlates the fracture strength and the fracture ductility. From these two equations, the expressions of both the strain-hardening exponent and the strength coefficient are deduced. Theoretical results from the deduced expressions are then compared with test data. Through the comparison of equations and data, if adequate test data are lacking, the deduced expressions can be used to theoretically calculate the strain-hardening exponent and the strength coefficient for metallic materials. The characteristics of the theoretical approach are simple and easy to use. In addition, the theoretical results can be further applied to examine the correctness of the test data.     

Evaluating plastic flow properties by characterizing indentation size effect using a sharp indenter

The strain-hardening exponent, used in describing the plastic flow properties of materials, is evaluated from the characteristic length in the indentation size effect (ISE). A linear relationship is found between the strain-hardening exponent and the log of the ISE characteristic length for Ni and SCM21 (structural steel) samples with different plastic pre-strain values. These results are explained through the Taylor dislocation hardening model and a representative stress–strain approach. A dimensionless function characterizing the plastic deformation using only parameters generally measurable by nanoindentation testing is also proposed. The feasibility of developing a unique dimensionless function is studied for 22 metals.     

Theoretical Estimation to the Cyclic Strength Coefficient and the Cyclic Strain-Hardening Exponent for Metallic Materials: Preliminary Study

The ultimate purpose of the present article is to theoretically estimate the cyclic strength coefficient and the cyclic strain-hardening exponent. For this purpose, the performance parameters of 22 alloys were examined and equations that relate the cyclic strength coefficient and the cyclic strain-hardening exponent to the monotonic tensile ones were developed. Then, by using formulas that express the strength coefficient and the strain-hardening exponent through the four conventional tensile performance parameters, i.e., the yield strength, the ultimate tensile strength, the fracture strength, and the fracture ductility, expressions that describe the cyclic strength coefficient and the cyclic strain-hardening exponent are established. By means of cyclic stress-strain curve, the limitations of the traditional methods of estimating the cyclic strength coefficient and the cyclic strain-hardening exponent are pointed out, and the ability of the new equations at describing the process are established. From the equations the cyclic strength coefficient and the cyclic strain-hardening exponent can be theoretically estimated using the monotonic ones. Furthermore, in the absence of the strength coefficient and the strain-hardening exponent, the cyclic ones can still be obtained from the expressions using the four conventional tensile performance parameters.     

Statistical Modeling of Strain-Hardening Exponent and Grain Size of Nb-Microalloyed Steels Using a Two-Level Factorial Design of Experiment

A factorial design of experiment (DOE) was used to statistically model the strain-hardening exponent and grain size of Nb-microalloyed steel sheets following hot rolling. The objective of the statistical model was to develop a method to simultaneously increase the strain-hardening exponent and refine the grain size of Nb-microalloyed steels by controlling three hot rolling process parameters: roughing temperature (RT), finishing temperature (FT), and coiling temperature (CT). The factorial DOE used two levels for the above temperatures and three replicates to obtain a reliable and precise estimate of the strain-hardening exponent and grain size. Analysis of variance was used to determine the most significant factors (individual parameters and their interactions) affecting the responses and develop appropriate regression equations. The regression equations predicted that optimal formability is obtained under the following conditions: RT = 1150 °C, FT = 800 °C, and CT = 700 °C. Validation of the statistical model using microstructural characterization showed that the predicted value of the grain size was close to the experimental value.     

Formula relating fracture strength and fracture ductility with strength coefficient and strain-hardening exponent

To theoretically calculate the strength coefficient and the strain-hardening exponent with conventional mechanical property parameters, formulas relating them with fracture strength and fracture ductility are studied using test data for ten alloys. The applicability of the traditional formula relating these four material constants is discussed first, and then new formulas are proposed based on the premise that the traditional approach cannot be used. The main conclusions made herein are that only under certain conditions can be traditional formula be used to describe the relationship among fracture strength, fracture ductility, strength coefficient, and strain-hardening exponent; otherwise, a new formula must be used.     

板料成形应变计算机视觉测量中的网格应变计算方法

本文介绍了几种常见应变测量用的网格及其应变计算方法 ,提出了适合板料成形应变计算机视觉测量的基于欧拉法和应变增量理论解析的方网格计算方法。该方法对网格按对角线划分三角形并按变形后的网格建立坐标系统 ,由拉形比和边长的关系直接计算出主应变的方向 ,进而求出真实应变、工程应变和等效应变。     

相变/孪晶诱发塑性的β钛合金加工硬化行为研究

研究了具有相变诱发的塑性变形/孪晶诱发的塑性变形(trunsformationinducedplasticity/twinninginduced plasticity,TWIP/TRIP)效应的Ti-12Mo-5Zr钛合金的加工硬化行为,并利用Hollomon理论分析了该合金的加工硬化指数n变化趋势。根据加工硬化率的演变趋势,该合金的加工硬化过程可分为3个不同阶段:Stage I(ε≈0~0.02),弹-塑性变形过渡阶段;Stage II(ε≈0.02~0.15),加工硬化率θ和硬化指数n逐渐增大阶段;Stage III(ε≈0.15~0.34),加工硬化率θ逐渐减小直到颈缩阶段,该阶段硬化指数n达到稳定值0.34。结合金相显微镜(OM)、透射电镜(TEM)等手段详细研究了该合金在这3个不同加工硬化阶段的显微结构演化过程,探讨了每个阶段的变形机理及其硬化行为。     

A new approach for reverse analyses in depth-sensing indentation using numerical simulation

This paper seeks to present a new approach to reverse analysis in depth-sensing indentation which makes use of numerical simulation. This methodology allows the results of experimental hardness tests acquired with single indenter geometry to be used to determine the plastic properties of materials. Forward and reverse analyses of high deformation three-dimensional numerical simulations of Vickers indentation tests are used to determine different mechanical properties of materials: Young’s modulus, yield stress and strain-hardening exponent. The Vickers indenter used in the numerical simulations is formulated as a rigid body and takes into account the presence of the most common imperfection of the tip, so-called offset. The contact friction between the Vickers indenter and the deformable body is also considered. The forward analysis uses materials with Young’s modulus values from 50 to 600 GPa, yield stress values from 0.3 to 10 GPa and strain-hardening exponents from 0 to 0.6; the Poisson ratio did not vary from 0.3. The representative plastic strain ε_r and the correspondent stress σ_r, as previously defined by other authors [Dao M, Chollacoop N, Vliet KJ, Venkatesh TA, Suresh S. Acta Mater 2001;49:3899], were identified by an independent numerical method. The values of the representative plastic strain ε_r obtained for the Vickers indenter confirm those of the above-mentioned authors, despite showing a slight influence from the Young’s modulus values. The forward study enables the production of a unique plot of the hardness H_IT vs. representative stress σ_r, where both are normalized by the Young’s modulus E. The proposed reverse analysis provides a unique solution to the representative stress σ_r and the strain-hardening exponent, n, given that the Young’s modulus is predetermined from the experimental hardness test. Depending on the material properties, the value of n can be more or less sensitive to the scatter of the experimental results obtained using the depth-sensing equipment, particularly the stiffness of the unloading curve. The validity of the proposed reverse analysis method is checked using three real materials: stamping quality steel (DC 06), stainless AISI 304 steel and BK7 glass.     

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.     

New formula relating the yield stress-strain with the strength coefficient and the strain-hardening exponent

The main purpose of this paper is to search formulas for different metals that relate the yield stress-strain with the strength coefficient and the strain-hardening exponent. For this purpose, the test data of nine alloys were used as basic data and the applicability of Hollomon’s equation at the yield point of the alloy was studied. This paper explores new equations relating the yield stress-strain with the strength coefficient and the strain-hardening exponent. At the same time, the study introduces a new fracture-ductility parameter. The new fracture-ductility parameter not only describes the applicability of the new equations to these alloys, but may also be better at describing the hardening behavior of a metallic material.     

Determination of tensile properties by instrumented indentation technique: Representative stress and strain approach

Tensile properties can be evaluated by defining representative stress and strain with the parameters obtained from instrumented indentation tests using a spherical indenter. The accuracy of this approach depends strongly on how the contact depth is analyzed and how the representative stress and strain are defined. The primary factors influencing the determination of contact depth, pile-up/sink-in and elastic deflection, were quantified by analyzing indentation morphology by finite element simulation; then plastic pile-up/sink-in behavior was formulated in terms of the strain-hardening exponent and the ratio of indentation depth to indenter radius. For the representative strain, the definition by tangent function was determined to be more appropriate for assessing tensile properties based on derived behaviors of the strain-hardening exponent. This approach was experimentally verified by comparing tensile properties of 10 metallic materials from uniaxial tensile tests and instrumented indentation tests.     

An analysis of the eccentric nosing process of metal tubes

In this paper, the finite-element method is used to investigate the cold eccentric nosing process of metal tubes with an eccentric conical die from circular tube billets. A series of simulations on the eccentric tube nosing using the program DEFORM-3D was carried out. The influences of the process parameters such as tube length, tube thickness, die fillet radius, die angle, friction factor, strength coefficient and strain-hardening exponent of the billet material on the critical nosing ratio of the tube are examined.     

用压痕法测量316L钢的残余应力和硬化指数

采用量纲分析方法建立了尖压头压痕的量纲一的函数,基于有限元仿真技术,通过正分析,得到了量纲一的函数的拟合关系式。通过压痕实验和反演方法,可以获得316L钢的残余应力和硬化指数值。该方法具有实施方便、成本低、对工件损伤小等特点,特别适合现场测量。     

Influence of Dual-Phase Microstructures on the Properties of High Strength Grade Line Pipes

The influence of dual-phase microstructures on mechanical properties of X70, X80, and X90 line pipes is investigated. It is found that the line pipes with dual-phase microstructures possess both larger uniform elongation and higher hardening exponent, especially for high grade steel X90. The tensile deformation of dual-phase line pipe does not follow the trend predicted by the Hollomon formula, and a stable strain-hardening exponent is not found. This stress-strain behavior is different from the normal line pipe. In the initial stage of plastic deformation, the strain-hardening capacity of dual-phase line pipe increases rapidly. However, it reaches a stable stage after 2.0% total strain. The dual-phase pipeline steel is composed of soft phase (polygonal ferrite) and hard phase (bainite), and thus the relatively soft ferrite is good for its deformability. Besides, the fraction of large angle grain boundaries in the dual-phase microstructures is greater than that of the normal line pipe, which is proven to be critical for improving the resistance to plastic deformation and crack propagation.     

Strain Hardening and Strain-Rate Sensitivity of an Extruded Magnesium Alloy

The strain-hardening behavior and strain-rate sensitivity of an extruded AZ31B magnesium alloy were determined at different strain rates between 10⁻² and 10⁻⁵ s⁻¹ in relation to the thickness of specimens (2.5 and 4.5 mm). Both the common approach and Lindholm’s approach were used to evaluate the strain-rate sensitivity. The yield strength (YS) and the ultimate tensile strength (UTS) increased, the ductility decreased, and the brittle fracture characteristics increased with increasing strain rate. The thinner specimens exhibited a slightly higher UTS, lower ductility, higher strain-hardening exponent, and strain-hardening rate due to smaller grain sizes. The stage III strain-hardening rate linearly decreased with increasing true stress, but increased with increasing strain rate. In comparison to the common approach, the Lindholm’s approach was observed to be more sensitive in characterizing the strain-rate sensitivity due to larger values obtained. The thinner specimens also exhibited higher strain-rate sensitivity. As the true strain increased, the strain-rate sensitivity decreased.     

Optimization of sheet-metal forming processes using the special-purpose program AUTOFORM

Optimizing the parameters of sheet-metal forming processes requires an efficient and accurate simulation program that is able to handle real-world problems and which can be operated by a non-finite-element specialist. The program AUTOFORM is limited to the sheet-metal forming process, and fits the requirements of this process optimally. Not only is the program's technical overhead reduced to a minimum, but also the program is largely vectorizable and parallelizable and allows the use of specially developed algorithms: the number of computational degrees-of-freedom is reduced by decoupling the determination of the sheet-metal form from the flow of metal within this form. The algorithm is not only capable of taking into account the relevant bending effects but also enables more efficient iterative solvers to be used. In addition to appropriate algorithms for material behaviour, friction, blank-holder and drawbeads, the necessary accuracy is attained by using the implicit method and by adaptive mesh refinement algorithms, which allow a very fine discretization of the geometry. Using AUTOFORM, the forming processes of two real-world automotive body parts and other sheet-metal forming parts have been optimized recently by adapting parameters such as blank geometry, material, time variable blank-holder forces, and the location and geometry of drawbeads.     

Sheet metal forming limit prediction based on plastic deformation energy

For sheet metals, the endurance to fracture under different strain paths may be different. Based on plastic deformation energy, the sheet metal forming limit is calculated, and the relationship model between maximum allowable integral value of the general plastic work criterion and the strain path is built. In addition, the strain-hardening exponent, anisotropy coefficient and the initial thickness of the material are also taken into account to consider their effects on forming limit. In order to simplify the process of parameter determination, only uniaxial tension test is used to calculate the material property parameters and necessary limit strain, and the expression of the criterion is determined finally. Then the limit strains under other strain paths between uniaxial tension to equi-biaxial tension are predicted by the criterion combined with numerical simulation of the forming process. The criterion is also applied to limit strain prediction under bilinear strain path.     

An analysis of load–penetration curves from instrumented indentation

A new empirical method is proposed for analyzing nano-indentation load–displacement curves. This method is based on experimental results and finite element (FEM) calculations reported in the literature. This method proposes to link the indentation–unloading curve to elastic modulus and strain-hardening properties. Through the analysis, elastic modulus, yielding stress and strain hardening properties are derived from the complete indentation load–displacement curve. This new method has then applied to the nano-indentation experiments of 14 different materials in which the elastic modulus ranges from 3 to 650 GPa and the hardness ranges from 0.1 to over 30 GPa. The elastic modulus, hardness and yielding stress derived from the new method agreed well with the literature values. An analysis of the loading curve also shows that there is a transition region, in which of the exponent of the loading curve changes from 1.5 to 2 corresponding to the change of the loads from low to high. This implicates that to avoid this transition region in loading curve, one should keep the load of the nano-indentation larger than approx. 30 mN.     

Investigation on ductile fracture in fine-blanking process by finite element method

In order to continuously analyze the whole fine-blanking process.from the geginning of the operation up to the total rupture of the sheet-metal,without computational divergence, a 3-D rigid visco-plastic finite-element method based on Gurson void model was developed.The void volume fraction was introduced into the finite element method to document the ductile fracture of the sheet-metal.A formulation of variation of the rigid visco-plastic material was presented according to the virtual work theory in which both the effects of equivalent stress and hydrostatic pressure in the deformation process were considered.The crack initiation of the sheet was predicted and the crack propagation was geometrically fulfilled in the simulation by separating the nodes according to the stress state.Furthermore.the influences of different state-variables on the deformation process were also studied.     

板料n值测量方法的研究

在板料冲压成形过程中 ,板料的硬化指数n值具有十分重要的意义 ,因而n值的测量无论对理论研究还是对工程实践都非常必要。因此 ,涌现出众多的硬化指数n值的测量方法。到目前为止 ,常用的测量方法主要有一点法、两点法、阶梯形试件拉伸法等。这些方法各有其优缺点 ,其共同不足就是用这些方法测定的n值反映的只是均匀塑性变形阶段的硬化性能。为此 ,本文基于常规单向拉伸试件在拉断后出现的锥形现象 ,以拉伸过程中的3个特征点 (屈服之后靠近屈服的一点、最大载荷时刻的点和断裂时刻的点 )为基础 ,提出了一种测量n值的新方案———三点法。在此方案中 ,用 3个特征点表征的两个n值分别描述了板料在均匀塑性变形阶段和随后大塑性变形阶段的硬化性能 ,而大塑性变形在板料成形过程中是必不可少的。本文通过对软钢板、紫铜板等进行的试验研究 ,证明了此方案是可行的。而且 ,作为板材成形性重要参数的r值也可以在试验中同时获得