A non-associated flow rule for sheet metal forming

An improved model of material behavior is proposed that shows good agreement with experimental data for both yield and plastic strain ratios in uniaxial, equi-biaxial, and plane-strain tension under proportional loading for steel, aluminum and possibly other alloys. This model is based on a non-associated flow rule in which the plastic potential and yield surface functions are defined by quadratic functions of the stress tensor. The plastic potential aspect of the model is identical to that proposed by Hill for a quadratic anisotropic plastic potential defined in terms of measured r values. The new model differs in that the yield surface, although also defined by a quadratic function of the stress tensor, is defined independently of the plastic potential in terms of measured yield stresses. The model is developed and implemented in an FEM code that is based on a convected coordinate system. Since the associated flow rule, which assumes equivalency between the plastic potential and yield functions, is commonly accepted as a valid law in the theory of plastic deformation of most metals, the arguments for the associated flow rule are also discussed.     

Non-linear dynamics of a mechanical system with a frictional unilateral constraint

We analyze the dynamics of a two-dimensional system constituted by two masses subjected to elastic, gravitational and viscous forces and constrained by a moving frictional mono-lateral surface. The model exhibits a time-varying dynamics capable of reproducing the hopping phenomenon, an unwanted phenomenon observed in many applications such as the motion of a robotic arm on a surface or that of a wiper on a windscreen. The system dynamics, besides being affected by geometrical non-linearities, has a non-smooth nature due to the impact and friction laws involved in the model. The complexity of the resulting equations and of the transition conditions require the problem to be solved numerically. Various periodic motions are found and the effect of varying the system parameters, in particular the friction coefficient, is investigated. Finally, simulations are used to gain some insight the behavior of the windscreen wiper.     

Thermoelastic buckling of steel columns with load-dependent supports

The in-plane elastic buckling of a steel column with load-dependent supports under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or elastic foundations. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain–displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the limits for different buckling modes and the critical temperature of columns with different cases are studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The effect of thermal loading on the buckling of steel columns is significant. Furthermore, the thermal gradient plays a positive role in improving the stability of columns, and the effect of thermal gradients decreases while decreasing the modified slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the initial stiffness coefficient α or the stiffening rate β of thermal restraints will increase the critical temperature.     

Creep of metal torsion-tension members subjected to nonproportionate load changes

The creep behavior of torsion-tension members subjected to nonproportionate loading was studied by testing members made of SAE 1035 steel at 975° F and of copper alloy 360 at 700° F. Creep curves for these members were predicted by an incremental theory using both the strain-hardening rule and the time-hardening rule and a total-strain theory using the incremental-hardening rule. Both incremental theories were based on a creep law which assumed that the total strain in specimens subjected to constant stress was made up of an elastic component and a creep component given by a power function of stress and of time. The total-strain theory was based on isochronous stress-strain curves which were approximated by an arc hyperbolic sine relation. The creep tests were limited to a total duration of 60 min. The nonproportionate loading was obtained by stepped increases of either the axial load, the torque, or both at 30 min. Except for the incremental theory based on the time-hardening rule, good agreement was found between theory and experiment.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

Multi-mechanism anisotropic model for granular materials

By representing the assembly by a simplified column model, a constitutive theory, referred to as sliding–rolling theory, was recently developed for a two-dimensional assembly of rods subjected to biaxial loading, and then extended to a three-dimensional assembly of spheres subjected to triaxial (equibiaxial) loading. The sliding–rolling theory provides a framework for developing a phenomenological constitutive law for granular materials, which is the objective of the present work. The sliding–rolling theory provides information concerning yield and flow directions during radial and non-radial loading. In addition, the theory provides information on the role of fabric anisotropy on the stress–strain behavior and critical state shear strength. In the present paper, a multi-axial phenomenological model is developed within the sliding–rolling framework by utilizing the concepts of critical state, classical elasto-plasticity and bounding surface. The resulting theory involves two yield surfaces and falls within the definition of the multi-mechanism models. Computational issues concerning the solution uniqueness for stress states at the corner of yield surfaces are addressed. The effect of initial and induced fabric anisotropy on the constitutive behavior is incorporated. It is shown that the model is capable of simulating the effect of anisotropy, and the behavior of loose and dense sands under drained and undrained loading.     

Reconcile the perfectly elastoplastic model to simulate the cyclic behavior and ratcheting

Perfectly elastoplastic constitutive model is modified through a smoothing factor introduced by Liu [Liu, C.-S., 2003. Smoothing elastoplastic stress–strain curves obtained by a critical modification of conventional models. Int. J. Solids Struct. 40, 2121–2145]. The new model allows plasticity to happen in a non-zero-measure yield volume in stress space, rather than that of conventional zero-measure yield surface, and within the yield volume the plastic modulus is varying continuously. It endows a specific strain-hardening rule of flow stress and is able to describe the phenomena of strain hardening, cyclic hardening, the Bauschinger effect, mean-stress relaxation, strain ratcheting, out-of-phase hardening, as well as erasure-of-memory. In order to suppress the over prediction of ratcheting we consider a scalar function of smoothing factor, which can simulate the saturation behavior of uniaxial/multiaxial strain ratcheting. These effects are demonstrated through numerical examples. The existence of stress equilibrium point and limiting surface is a natural result without requiring an extra design. Moreover, the non-linear constitutive equations can be converted into a linear system for augmented stress in the Minkowski space, of which the symmetry group is a proper orthochronous Lorentz group SO_o(5, 1). The augmented stress is a time-like vector, moving on hyperboloids inside the cone. When taking the Prager kinematic hardening rule into account we can simulate some cyclic behaviors of SAE 4340 and grade 60 steels within a certain accuracy through the use of only three material constants and a fixed smoothing factor. To simulate the ratcheting behaviors of SS304 stainless steel we allow the smoothing factor to be an exponential decaying function of λ.     

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.     

A model of fracture in reinforced concrete arches based on lumped damage mechanics

A simplified model of cracking and damage in RC circular elements is proposed. The model can be used for the structural assessment of arches and rings. The constitutive equations are based on lumped damage mechanics which is an adaptation of fracture and continuum damage mechanics to the theory of frames with plastic hinges. An arch element is assumed to be the assemblage of an elastic circular component and two inelastic hinges where the main inelastic effects, plastic yielding of the longitudinal reinforcement and concrete cracking, are concentrated. Deformations in the elastic part are assumed to be small but the model may include some geometrically nonlinear effects due to large displacements or rotations of the hinges. The numerical examples presented in the paper show that the model describes correctly the global behavior of two structures including the softening phase.     

Modeling of properties and motions of an inhomogeneous one-dimensional continuum of a complicated Cosserat-type microstructure

We consider an approach to modeling the properties of the one-dimensional Cosserat continuum [1] by using the mechanical modeling method proposed by Il’yushin in [2] and applied in [3]. In this method, elements (blocks, cells) of special form are used to develop a discrete model of the structure so that the average properties of the model reproduced the properties of the continuum under study. The rigged rod model, which is an elastic structure in the form of a thin rod with massive inclusions (pulleys) fixed by elastic hinges on its elastic line and connected by elastic belt transmissions, is taken to be the original discrete model of the Cosserat continuum. The complete system of equations describing the mechanical properties and the dynamical equilibrium of the rigged rod in arbitrary plane motions is derived. These equations are averaged in the case of a sufficiently smooth variation in the parameters of motion along the rod (the long-wave approximation). It was found that the average equations exactly coincide with the equations for the one-dimensional Cosserat medium [1] and, in some specific cases, with the classical equations of motion of an elastic rod [4–6]. We study the plane motions of the one-dimensional continuum model thus constructed. The equations characterizing the continuum properties and motions are linearized by using several assumptions that the kinematic parameters are small. We solve the problem of natural vibrations with homogeneous boundary conditions and establish that each value of the parameter distinguishing the natural vibration modes is associated with exactly two distinct vibration mode shapes (in the same mode), each of which has its own frequency value.     

A three-invariant cap model with isotropic–kinematic hardening rule and associated plasticity for granular materials

In this paper, a three-invariant cap model is developed for the isotropic–kinematic hardening and associated plasticity of granular materials. The model is based on the concepts of elasticity and plasticity theories together with an associated flow rule and a work hardening law for plastic deformations of granulars. The hardening rule is defined by its decomposition into the isotropic and kinematic material functions. The constitutive elasto-plastic matrix and its components are derived by using the definition of yield surface, material functions and non-linear elastic behavior, as function of hardening parameters. The model assessment and procedure for determination of material parameters are described. Finally, the applicability of proposed plasticity model is demonstrated in numerical simulation of several triaxial and confining pressure tests on different granular materials, including: wheat, rape, synthetic granulate and sand.     

Analytical evaluation of the elastic and plastic resistances of double symmetric rectangular hollow sections under axial force and biaxial bending

In the recent codes for the design of steel structures, the elastic–plastic methods of analysis are recognised to provide an efficient estimation of the ultimate resistance of some of these structures. These methods are usually based on some basic hypotheses, such as the creation of plastic hinges in the most stressed cross-sections, for instance.As the development of these plastic hinges depends on the interaction between the internal forces and on the cross-section shape, specific equations are required for the analysis of different types of cross-sections. However, most frequently, these equations are not available, or they are expressed by means of simplified expressions; this is usually the case when biaxial bending is involved.This paper presents new interaction criteria for the analysis of steel rectangular hollow sections subjected to an axial force and biaxial bending moments, at the elastic or the plastic limit states (as long as buckling phenomena are not involved). The plastic interaction criteria are presented, in a first step, for some particular combinations of the internal forces, such as axial loading with bending about a main axis, and biaxial bending without axial loading. Then, the global solution for the simultaneous combination of an axial force and bending moments about both the main axes of inertia are described in detail. All these plastic interaction criteria are compared with the corresponding plastic criteria adopted in the Eurocode 3 (EC3). Some suggestions are presented in order to improve the results given by these EC3 criteria.     

混凝土率型内时损伤本构模型

混凝土是一种典型的率敏感材料,为了更好地描述混凝土结构在动力、冲击荷载作用下的强度和变形特征,本文结合内时理论和损伤理论建立了一种考虑混凝土率效应的内时损伤本构模型。该模型的特点:将混凝土材料的受力软化效应分解为密实状态的塑性效应和由微裂缝扩展引起的刚度退化效应。前者由内时理论来描述,这使该模型摆脱了一般弹塑性模型中屈服面的概念,从而更符合混凝土的变形特性,并且简化了非线性计算过程;后者由损伤理论来描述,根据混凝土的动力试验结果建立了增量型的损伤演变方程,从而使该模型能够较好地反映混凝土的动力特性。最后,应用本文建议的模型对一钢筋混凝土简支梁进行了非线性分析,结果表明:当结构承受快速荷载作用时,应变率对结构的受力性能影响较大,在进行结构分析时必须予以考虑。     

弹体贯穿混凝土数值模拟的改进材料模型

研究混凝土结构在冲击载荷下的力学特性对武器以及防护结构的设计和评估具有重要意义,而合适的材料模型可以更准确地预测混凝土结构的力学行为和破坏模式。因此,本文中提出了一种改进的混凝土塑性损伤材料模型来描述其在冲击载荷下的力学响应。该改进模型考虑了压力-体积应变关系、应变率效应、洛德角效应和塑性损伤累积对混凝土材料力学特性的影响,并引入了一个与损伤相关的硬化/软化函数来描述压缩状态下的应变硬化和软化行为。随后,通过对3个独立的强度面进行线性插值得到了该改进模型的破坏强度面,并采用部分关联流动法则考虑了混凝土材料的体积膨胀特性。最后,开展了单个单元在不同加载条件下和弹体贯穿钢筋混凝土靶的数值模拟,验证了该改进模型的可行性、准确性以及预测性能提升。     

Performance Dependent Model for normal and high strength concretes

A new constitutive formulation, the so-called Performance Dependent Model valid for normal and high strength concretes is presented. The distinctive aspect of the proposed model is the consideration of relevant properties of concrete mix components in the evaluation of the involved material performance or quality at the macroscopic stand point. In this way, the composite features of concrete are appropriately taken into account.The model maximum strength surface is defined by means of the Performance Dependent Failure Criterion proposed by the authors in previous works. Concrete behaviors in pre and post peak regimes are modeled with a non uniform hardening law and an isotropic softening rule, respectively. To realistically reproduce the concrete ductility in pre and post peak regimes under different load scenarios, the hardening and softening laws are defined in terms of the acting confining pressure. Concrete dilatancy behavior is approached by means of a volumetric non associative flow rule. The softening law is embedded in fracture energy concepts for mode I and II types of failure. The model considers two main input material parameters: the uniaxial compressive strength and the performance parameter, a quality index defined in the context of the Performance Dependent Failure Criterion.The proposed constitutive model is able to capture the substantial differences in the failure behavior of normal and high strength concretes as well as of concretes with the same compressive strength but different mix components. The predictive capability of the model is demonstrated in the numerical analyses included in this paper where the numerical predictions are compared with experimental results related to concrete specimens of different qualities and subjected to stress histories under both compressive and tensile regimes.     

Towards the solution of finite plane-strain problems for compressible elastic solids

A new approach to the solution of finite plane-strain problems for compressible Isotropie elastic solids is considered. The general problem is formulated in terms of a pair of deformation invariants different from those normally used, enabling the components of (nominal) stress to be expressed in terms of four functions, two of which are rotations associated with the deformation. Moreover, the inverse constitutive law can be written in a simple form involving the same two rotations, and this allows the problem to be formulated in a dual fashion.For particular choices of strain-energy function of the elastic material solutions are found in which the governing differential equations partially decouple, and the theory is then illustrated by simple examples. It is also shown how this part of the analysis is related to the work of F. John on harmonic materials.Detailed consideration is given to the problem of a circular cylindrical annulus whose inner surface is fixed and whose outer surface is subjected to a circular shear stress. We note, in particular, that material circles concentric with the annulus and near its surface decrease in radius whatever the form of constitutive law within the given class. Whether the volume of the material constituting the annulus increases or decreases depends on the form of law and the magnitude of the applied shear stress.     

Large amplitude free vibration of symmetrically laminated magneto-electro-elastic rectangular plates on Pasternak type foundation

Nonlinear free vibration of symmetrically laminated magneto-electro-elastic rectangular plate resting on an elastic foundation is studied analytically. The plate is considered to be simply supported on all edges. It is also assumed that the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. To model the motion of the plate, the first order shear deformation theory along with the Gauss's equations for electrostatics and magnetostatics are used. Then equations of motion are reduced to a single nonlinear ordinary differential equation which is solved analytically by multiple scales method. The results are compared with the published results and good agreement is found. Some numerical examples are presented to investigate the effects of several parameters on the linear and nonlinear behavior of these plates.     

显式方法精确模拟形状记忆合金循环荷载下的变形行为

本文提出全新的有限弹塑性J2流方程,用来显式、精确地模拟SMAs（形状记忆合金）材料在循环加载-卸载条件下从塑性逐渐转变为伪弹性的变形行为．首先,改进流动法,使得本构方程耦合屈服中心的移动和屈服面的增大,并改进背应力演化方程,使模型可以产生强烈的包辛格效应,从理论上具备模拟SMAs独特变形行为的能力;其次,构造全过程下的统一硬化函数显式表达式,代入本构方程后能得到符合要求的形函数;再次,利用选定的数据点构造统一光滑的上屈服函数,再利用上下屈服应力之间的一种线性关系,推导得到下屈服阶段的形函数;最后,只需要给定一个参数就可以得到单个循环结果,利用拉格朗日插值方法构建参数随循环次数变化的函数,就可以模拟任意循环荷载下的变形行为．通过模型结果和实验数据对比证明新方法的有效性．