Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.     

Stress relaxation after a step strain in uniaxial extension of polyisobutylene and polyethylene

The morphology of molten polymeric materials is known to be less sensitive to shear than to extensional deformations. However, it is not easy to characterise molten polymeric materials in simple extensional flows due to the large number of experimental difficulties involved. This has led to the effective absence of a structure-preserving, morphology probing technique similar to the ones commonly found in shear, i.e., the equivalent of stress relaxation and oscillatory experiments. It is the aim of the present work to demonstrate the usefulness of a recently developed experimental technique that enables stress relaxation experiments after a step strain in uniaxial extension to be performed. Results are presented for two model melts (polyisobutylene, PIB, of different molecular weights) and for a series of linear low-density polyethylenes, LLDPE, in which the molecular structure (molecular weight, MW, molecular weight distribution, MWD and degree of long chain branching, LCB) is changed systematically. It is shown that, for both types of materials, stress relaxation experiments in extension yield quantitatively correct results and that this technique is more sensitive to differences in molecular structure than oscillatory experiments in shear.     

A quartz-crystal-embedded split Hopkinson pressure bar for soft materials

A dynamic experimental technique that is three orders of magnitude as sensitive in stress measurement as a conventional split Hopkinson pressure bar (SHPB) has been developed. Experimental results show that this new method is effective and reliable for determining the dynamic compressive stress-strain responses of materials with low mechanical impedance and low compressive strengths, such as elastomeric materials and foams at high strain rates. The technique is based on a conventional SHPB. Instead of a surface strain gage mounted on the transmission bar, a piezoelectric force transducer was embedded in the middle of the transmission bar of a high-strength aluminum alloy to directly measure the weakly transmitted force profile from a soft specimen. In addition, a pulse-shape technique was used for increasing the rise time of the incident pulse to ensure stress equilibrium and homogeneous deformation in the low-impedance and low-strength specimen.     

A New Experimental Technique for the Multi-axial Testing of Advanced High Strength Steel Sheets

This paper deals with the development of a new experimental technique for the multi-axial testing of flat sheets and its application to advanced high strength steels. In close analogy with the traditional tension-torsion test for bulk materials, the sheet material is subject to combined tension and shear loading. Using a custom-made dual actuator hydraulic testing machine, combinations of normal and tangential loading are applied to the boundaries of a flat sheet metal specimen. The specimen shape is optimized to provide uniform stress and strain fields within its gage section. Finite element simulations are carried out to verify the approximate formulas for the shear and normal stress components at the specimen center. The corresponding strain fields are determined from digital image correlation. Two test series are performed on a TRIP-assisted steel sheet. The experimental results demonstrate that this new experimental technique can be used to investigate the large deformation behavior of advanced high strength steel sheets. The evolution of the yield surface of the TRIP700 steel is determined for both radial and non-proportional loading paths.     

Constitutive model of plasticity in finite deformation

An elastoplastic constitutive model for large strains is proposed using three kinds of equi-plastic strain surfaces, a yield surface and two other surfaces from which the power law for stress–strain relation is confined. The prediction by the model gives no oscillatory stress in simple shear, even though the Jaumann stress rate is utilized. Torsion with free- and fixed end conditions are also predicted by the model, which shows good agreement with experimental results of SUS 304.     

A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials

In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials is derived. The evolution equation for the active yield surface with reference to the memory yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function for pressure insensitive and sensitive materials. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress–plastic strain curves are also derived and plotted for materials under cyclic loading conditions based on the three yield functions. In addition, the closed-form solutions for one-dimensional stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function under cyclic loading conditions are summarized and presented. For materials based on the Mises and the Hill anisotropic yield functions, the stress–plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker–Prager and Cazacu–Barlat yield functions, the stress–plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. With these closed-form solutions, the important effects of the yield surface geometry on the cyclic plastic behavior due to the pressure-sensitive yielding or the unsymmetric behavior in tension and compression can be shown unambiguously. The closed form solutions for the Drucker–Prager and Cazacu–Barlat yield functions with the associated flow rule also suggest that a more general anisotropic hardening theory needs to be developed to address the ratcheting effects for a given stress range.     

Determining plastic properties of a material with residual stress by using conical indentation

Instrumented indentation is a popular method for determining mechanical properties in engineering materials. However, there are several shortcomings and challenges involved with correctly interpreting the test results. We propose here a unified method for evaluating instrumented indentation testing conducted on a material that exhibits both strain hardening under yielding and which is subjected to uniform, equi-biaxial residual stresses. The proposed method is based on extensive finite element simulations that relate the parameter-space spanned by Young’s modulus, yield strength, strain hardening and residual stress, to the response from the indentation test. Based on reverse analysis, the proposed method can be used to determine two unknown quantities, such as yield strength and strain hardening. The technique involves utilizing the concept of representative strain and plural indenter-shapes.     

纳米压痕法测磁控溅射铝薄膜屈服应力

为了在考虑残余应力下测量出磁控溅射铝薄膜的屈服应力,提出了一种实验测量方法,通过曲率测试法和球形压头纳米压痕法测出磁控溅射铝薄膜的屈服应力.建立球形压痕力学模型,并用ANSYS对球形压痕进行力学有限元仿真,利用直流磁控溅射技术在硅基上淀积一层l μm厚的铝薄膜,首先通过曲率测试法测量膜内等双轴残余应力,再利用最小二乘曲线拟合法从薄膜/基底系统的球形压头纳米压痕实验数据中提取出铝薄膜的屈服应力,测得磁控溅射铝薄膜的屈服应力为371±89 Mpa.该方法也可以用来研究其他材料的薄膜和小体积材料的力学特性.     

On an elastic-plastic transition zone in cellular metals

Summary A theory of plasticity is proposed for cellular metals to describe their elastic-plastic transition zone at small strain. Under certain conditions, only a plane strain test is necessary to determine the yield surface. The method to derive the elastic–plastic behaviour [14, 15] was originally proposed for classical metals. A simple cubic model of a cellular metal is used to demonstrate the method by the finite element method. Recommendations for the numerical simulation are given. The influence of the relative density and the hardening behaviour of the cell wall material is investigated.     

On the methodologies of stress analysis of composite structures: Part 1: State of the art—phenomenological and physical methodologies

The rapidly increasing technological importance of composite materials and composite structures is leading to the development of new, more advanced models of their actual response to mechanical and thermal loads. This in turn results in the development of new experimental and analytical methods for determination of the mechanical and thermal responses of such structures and materials to various loads. In this respect the reliability and the predictive power of various methods and techniques of stress analysis become very important since all the analytical, experimental and numerical methods used for the determination, prediction and optimization of the actual mechanical responses of composite structures and materials are based on the concepts of strain and stress. Because of the inherently three-dimensional stress and strain states in composite materials and structures and the wide use of viscoelastic polymers as the matrix and some reinforcing fiber materials, a more rigorous type of modelling than had been common in the past is needed of all the involved physical phenomena which influence the strain and stress states at the local and global levels. Also, a more rigorous analysis of practical consequences of the physical and mathematical simplifications is required to assure reliability and accuracy of various methods of stress analysis. The influence of the above-mentioned factors on the reliability and applicability of analytical and experimental procedures is illustrated by examples of actual material responses.Part 2 of this paper presents theories and techniques of three new methods of strain/stress analysis which have been developed on the basis of comprehensive physical models of involved phenomena: the isodyne, strain gradient and thermoelastic effect methods. Presented examples illustrate the efficacy of these methods.     

On the methodologies of stress analysis of composite structures: Part 2: New experimental approaches

The rapidly increasing technological importance of composite materials and composite structures is leading to the development of new, more advanced models of their actual response to mechanical and thermal loads. This in turn results in the development of new experimental and analytical methods for determination of the mechanical and thermal responses of such structures and materials to various loads. In this respect the reliability and the predictive power of various methods and techniques of stress analysis become very important since all the analytical, experimental and numerical methods used for the determination, prediction and optimization of the actual mechanical responses of composite structures and materials are based on the concepts of strain and stress. Because of the inherently three-dimensional stress and strain states in composite materials and structures and the wide use of viscoelastic polymers as the matrix and some reinforcing fiber materials, a more rigorous type of modelling than had been common in the past is needed of all the involved physical phenomena which influence the strain and stress states at the local and global levels. Also, a more rigorous analysis of practical consequences of the physical and mathematical simplifications is required to assure reliability and accuracy of various methods of stress analysis. The influence of the above-mentioned factors on the reliability and applicability of analytical and experimental procedures is illustrated by examples of actual material responses.Part 2 of this paper presents theories and techniques of three new methods of strain/stress analysis which have been developed on the basis of comprehensive physical models of involved phenomena: the isodyne, strain gradient and thermoelastic effect methods. Presented examples illustrate the efficacy of these methods.     

基于数字散斑相关实验测量的材料构型力的计算方法

材料构型力学主要研究材料中的缺陷(夹杂、空穴、位错、裂纹、塑性区等)的构型(形状、尺寸和位置)改变时,所引起的系统自由能的变化。本研究将基于数字散斑相关技术,实验测量材料试件的位移场分布,随后通过材料构型力的定义式,计算求得弹塑性材料中缺陷构型力的分布。其方法概括如下:位移场通过数字图像相关技术测得;应变及位移梯度场利用三次样条拟合获得;线弹性材料应力通过简单线弹性本构方程获取,而塑性材料的表面应力场通过Ramberg-Osgood本构方程计算求得;弹塑性应变能密度分布则由应力-应变曲线数值积分获得。该方法对普通弹性材料或者弹塑性材料均适用,可以用于各种不同的缺陷及缺陷群的材料构型力测量。     

A finite strain constitutive model for metal powder compaction using a unique and convex single surface yield function

Constitutive modelling of metal powder compaction processes is a challenge in view of realistic simulations. To this end, the article under consideration has two objectives: the first goal is to present a new unique and convex single surface yield function for pressure dependent materials, which is also applicable to other areas of granular materials such as soils or concrete. The flexibility is shown at various materials. The yield function is based on a log-interpolation of two known simple yield functions. A convexity proof of the new yield function is provided. The second objective is to propose a new rate-independent finite strain plasticity model for metal powder compaction, which is based on the multiplicative decomposition of the deformation gradient into an elastic and a plastic part with evolution equations for internal variables representing the basic behaviour of powder materials under compaction conditions. These variables are used for the evolution of the yield function in order to represent the compressible hardening behaviour of powder materials. On the basis of the constitutive model, the material parameters are identified at experimental data of copper powder.     

ELASTO-PLASTIC CONSTITUTIVE MODEL OF SOIL-STRUCTURE INTERFACE IN CONSIDERATION OF STRAIN SOFTENING AND DILATION

The behavior of soil-structure interface plays a major role in the definition of soil-structure interaction. In this paper a bi-potential surface elasto-plastic model for soil-structure interface is proposed in order to describe the interface deformation behavior,including strain softening and normal dilatancy. The model is formulated in the framework of generalized potential theory,in which the soil-structure interface problem is regard as a two-dimensional mathematical problem in stress field,and plastic state equations are used to replace the traditional field surface. The relation curves of shear stress and tangential strain are fitted by a piecewise function composed by hyperbolic functions and hyperbolic secant functions,while the relation curves of normal strain and tangential strain are fitted by another piecewise function composed by quadratic functions and hyperbolic secant functions. The approach proposed has the advantage of deriving an elastoplastic constitutive matrix without postulating the plastic potential functions and yield surface. Moreover,the mathematical principle is clear,and the entire model parameters can be identified by experimental tests. Finally,the predictions of the model have been compared with experimental results obtained from simple shear tests under normal stresses,and results show the model is reasonable and practical.     

用于高强度材料的SHPB实验添加垫块法

提出了用于高强度材料的改进的SHPB实验方法添加垫块法,运用数值模拟方法,利用有限元程序LS-DYNA3D分析了添加垫块实验方法的合理性和可行性。根据一维应力波理论,给出了数据处理的修正方法。作为应用实例,采用改进的实验方法对高强度的Al2O3陶瓷材料的动态力学性能进行了研究,得到了比常规方法较高的应变率及应力应变范围的动态应力应变曲线,表明Al2O3陶瓷为应变率相关的非线性弹脆性材料。结果表明,添加垫块实验方法可有效地防止实验中压杆端面的变形,提高试件的应力应变及应变率水平。添加垫块实验方法为在SHPB装置上实现高强度材料的动态实验提供了一种方便实用的途径。     

改进的霍普金森压杆技术在聚氨脂泡沫塑料动态力学性能研究中的应用

用改进的霍普金森杆技术得到了聚氨脂泡沫塑料在动态应力均匀和恒应变率条件下的实验结果。     

铁合金和青铜多孔材料屈服应力的实验研究

对铁合金和青铜多孔材料做静态实验和Taylor冲击实验,得到静态屈服应力和孔隙率以及动态屈服应力和应变率的变化规律,同时将静动态载荷作用下的多孔材料的本构方程与实验结果进行了对比,发现理论分析和实验研究有良好的一致性,表明本文采用的模型基本能满足实际工程应用。     

Analyses of steady quasistatic crack growth in plane strain tension in elastic-plastic materials with non-isotropic hardening

Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J₂, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids,27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J₂ incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J₂ incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.     

Multiaxial yield surface of transversely isotropic foams: Part II—Experimental

A robust understanding and modeling of the yield behavior in solid foams under complex stress states is essential to design and analysis of optimal structures using these lightweight materials. In pursuit of this objective a new custom-built Multi-Axial Testing Apparatus (MATA) is developed to probe the yield surface of transversely isotropic Divinycell H-100 PVC foam under a multitude of uniaxial, biaxial and triaxial strain paths. Experimental yield data produced constitutes the most comprehensive data set ever produced for any foam as it covers the entire spectrum of stress paths from hydrostatic compression to hydrostatic tension. Experimental results reveal that yielding in foams exhibits not only a quadratic pressure dependence, which is widely recognized in literature, but also a significant linear pressure dependence, which has been largely overlooked in previous studies. A new energy-based yield criterion developed for transversely isotropic foams is also validated using the experimental yield data.