Scission induced by circular shear and heat conduction in an elastomeric cylinder

Constitutive equations for the thermo-mechanics of elastomeric materials generally assume that they do not undergo microstructural change. A constitutive theory is discussed here which accounts for such changes arising from continuous scission of macromolecular junctions of elastomeric networks due to deformation and high temperatures and the subsequent cross-linking of molecules into new networks with new reference states. The total stress is the superposition of the stresses in the remainder of the original network and in each subsequently formed network. Each network acts as a temperature-dependent non-linear elastic material. The interaction of this material response with inhomogeneous deformation and temperature fields is studied for finite circular shear of a cylinder. Numerical results illustrate how the mechanical response of the cylinder depends on the temperature dependence of both the scission–cross-linking process and the properties of the elastic networks.     

Saint-Venant torsion analysis of bars with rectangular cross-section and effective coating layers

This paper investigates the torsion analysis of coated bars with a rectangular cross-section. Two opposite faces of a bar are coated by two isotropic layers with different materials of the original substrate that are perfectly bonded to the bar. With the Saint-Venant torsion theory, the governing equation of the problem in terms of the warping function is established and solved using the finite Fourier cosine transform. The state of stress on the cross-section, warping of the cross-section, and torsional rigidity of the bar are evaluated. Effects of thickness of the coating layers and material properties on these quantities are investigated. A set of graphs are provided that can be used to determine the coating thicknesses and material properties so as to keep the maximum von Mises stress on the cross-section below an allowable value for effective use of the coating layer.     

Some analytical solutions for Saint-Venant torsion of non-homogeneous anisotropic cylindrical bars

The Saint-Venant torsion of linearly elastic anisotropic cylindrical bars with solid and hollow cross-section is treated. The shear flexibility moduli of the non-homogeneous bar are given functions of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsion problem of non-homogeneous anisotropic bar is expressed in terms of the torsion and Prandtl's stress functions of the corresponding homogeneous anisotropic bar having the same cross-section as the non-homogeneous bar.     

Lüders bands propagation of 1045 steel under multiaxial stress state

Inhomogeneous plastic deformation of 1045 steel under monotonic loading was experimentally studied. Thin-walled tubular specimens were used in the experiments and custom-made small strain gages were bonded on the specimen surface to characterize the local deformation. Experiments were conducted under tension, torsion, and combined tension–torsion. During the propagation of Lüders bands, the local deformation experienced two-stage deformation: an abrupt plastic deformation stage followed by a slower deformation process. In some area of the gage section of the specimen, a small amount of initial plastic deformation occurred before the Lüders front reached. During the propagation of Lüders bands, multiple Lüders fronts can be formed. Under tension, torsion, and combined tension–torsion with a constant axial load, the Lüders front was approximately parallel to the material plane of maximum shear stress. When the combined axial-torsion followed a proportional fashion, the stress–extensometer strain responses were dependent on the axial/torsional loading ratio, and the Lüders fronts were oriented differently and propagated along the specimen axis at a different velocity. The local strain was inhomogeneous even at the work-hardening stage. The relationships between the equivalent stress and the equivalent plastic strain were found to be practically identical for all the loading cases studied.     

基于对数应力率大应变本构关系的参数标定研究

在所有率型弹塑性本构模型中,只有对数应力率对应的本构模型能够满足自适应准则.基于对数应力率,采用实心圆轴扭转实验,对大应变弹塑性本构模型中的参数标定问题进行了讨论.推导出了考虑Swift效应时端部自由实心圆轴扭转变形的变形率、对数旋率、Kirchhoff应力及Kirchhoff应力的对数应力率.对于等向强化大应变弹塑性本构关系,给出了由实心圆轴扭转实验标定的、基于Kirchhhoff应力对数应力率的本构关系中塑性刚度函数的表达式.分析了扭转圆轴的Swift效应对塑性刚度函数的影响.结果表明,实心圆轴扭转的轴向伸长变形和径向变形对基于对数应力率大应变本构关系中的塑性刚度函数都有影响.当不考虑Swift效应时,所得塑性刚度函数表达式与不考虑Swift效应时基于Jaumann应力率的塑性刚度函数表达式相同.     

An Experimental Facility to Measure the Chemorheological Response of Inflated Elastomeric Membranes at High Temperature

An experimental facility for measuring the time-dependent deformation of an inflated elastomer membrane at elevated temperatures is presented. The facility controls the temperature and inflation volume of a membrane specimen and measures the pressure and deformation field. The finite deformation field is determined by comparing the configuration of the inflated membrane to the initially flat reference configuration, determined by photogrammetry of an array of dots printed on the surface of the membrane. The facility provides an effective method to investigate the changes in mechanical properties due to scission and cross-linking in the context of a simple elastomeric structure that has a distribution of multi-axial deformation states.     

Combined deformation- and temperature-induced scission in a rubber cylinder in torsion

When an elastomeric material is deformed and subjected to temperatures above some characteristic value T_cr (near for natural rubber), it undergoes time and temperature dependent chemical changes consisting of scission and crosslinking of its macromolecular structure. The process continues until the temperature decreases below T_cr. Experiments carried out in uniaxial extension have shown that the chemical changes are independent of stretch ratio within moderate stretches. It is reasonable to expect that the chemical changes would be affected by sufficiently large deformations, an interaction referred to as ‘mechanochemistry’. A kinetic theory of the breakdown of solids by Zhurkov [Kinetic concept of strength of solids, Int. J. Fract. Mech. 1 (1965) 311-323. [15]] attributes this interaction to the lowering of activation energy by mechanical work.In a recent constitutive theory, an expression was developed that relates the chemical kinetics of scission of the original elastomeric network to time, temperature and activation energy. The kinetic theory of Zhurkov suggests a method for modifying this expression to account for the influence of deformation. This is explored in the case of simple shear deformations, such as those occurring during torsion of elastomeric cylinders held at fixed length. Using the approach of Penn and Kearsley [The scaling law for finite torsion of elastic cylinders, Trans. Soc. Rheology 20 (1976) 227-238. [16]], it is shown that experiments in torsion can be used to determine the influence of shear deformations on the chemical kinetics of scission.     

Strain gradient elasticity and stress fibers

Since stress fibers have micro-size dimensions, their biomechanical behavior should demand mechanical models conforming with gradient strain deformation theories. In particular, the torsion and the stretching of stress fibers are discussed into the context of strain gradient elasticity theory and their size effects. It is proven for the torsion problem that the torsion moment varies with the axial length of the bar for constant twist angle, whereas for the simple tension problem, the strain is non-uniform along the stress fiber. The proposed theory is supported by experimental evidence.     

Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars

The Saint-Venant torsion problem of linearly elastic cylindrical bars with solid and hollow cross-section is treated. The shear modulus of the non-homogeneous bar is a given function of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsional problem of non-homogeneous bar is expressed in terms of the torsional and Prandtl's stress functions of homogeneous bar having the same cross-section as the non-homogeneous bar.     

An experimental study of inhomogeneous cyclic plastic deformation of 1045 steel under multiaxial cyclic loading

An experimental study was conducted on the inhomogeneous cyclic plastic deformation of 1045 steel under multiaxial cyclic loading. Thin-walled tubular specimens were used and small strain gages were bonded on the specimen surface to characterize the local deformation. The controlled loading paths included cyclic tension–compression, cyclic torsion, proportional axial-torsion, 90°-out-of-phase axial-torsion, and fully reversed torsion with a constant axial stress. The maximum stress in each experiment was lower than the lower yield stress of the material. It was found that the cyclic plastic deformation within the gage section of the specimen under multiaxial stress state followed the three-stage process that was observed from uniaxial loading, namely, incubation, propagation, and saturation. The plastic deformation was significantly inhomogeneous during the propagation stage, and the inhomogeneity continued through the saturation stage. The duration of each stage and the saturated strains were dependent on the cyclic stress amplitude and the loading path. Multiaxial stress state reduced the incubation stage. With identical equivalent stress magnitude, the nonproportional loading path resulted in the shortest incubation and propagation stages, and the saturated equivalent plastic strain magnitude was the smallest. Although the deformation over the gage section was inhomogeneous, the plastic deformation in a given local area was found to be practically isotropic.     

The use of a virtual configuration in formulating constitutive equations for residually stressed elastic materials

Residual stress is the stress present in the unloaded equilibrium configuration of a body. Because residual stresses can significantly affect the mechanical behavior of a component, the measurement of these stresses and the prediction of their effect on mechanical behavior are important objectives in many engineering problems. Common methods for the measurement of residual stresses include various destructive experiments in which the body is cut to relieve the residual stress. The resulting strain is measured and used to approximate the original residual stress in the intact body. In order to predict the mechanical behavior of a residually stressed body, a constitutive model is required that includes the influence of the residual stress.In this paper we present a method by which the data obtained from standard destructive experiments can be used to derive constitutive equations that describe the mechanical behavior of elastic residually stressed bodies. The derivation is based on the idea that for each infinitesimal neighborhood in a residually stressed body, there exists a corresponding stress free configuration. We refer to this stress free configuration as the virtual configuration of the infinitesimal neighborhood. The derivation requires that the constitutive equation for the stress free material be known and invertible; it is used to relate the residual stress to the deformation of the virtual configuration into the residually stressed configuration. Although the concept of the virtual configuration is central to the derivation, the geometry of this configuration need not be determined explicitly, and it need not be achievable experimentally, in order to construct the constitutive equation for the residually stressed body.The general mathematical forms of constitutive equations valid for residually stressed elastic materials have been derived previously for a number of cases. These general forms contain numerous unknown material-response functions or material constants that must be determined experimentally. In contrast, the method presented here results in a constitutive equation that is an explicit function of residual stress and includes only the material parameters required to describe the stress free material.After presenting the method for the derivation of constitutive equations, we explore the relationship between destructive experiments and the theory used in the derivation. Specifically, we discuss the use of the theory to improve the design of destructive experiments, and the use of destructive experiments to obtain the data required to construct the constitutive equation for a particular material.     

Generalized Poynting Effects in Predeformed Prismatic Bars

We use the Signorini expansion method to determine second-order Saint-Venant solution for an infinitesimally bent and stretched bar. The bar in the unstressed reference configuration is straight, prismatic, isotropic, homogeneous and made of a second-order elastic material. These solutions and those found earlier for a pretwisted bar give generalized Poynting effects. A bar when bent stretches and the elongation is determined by the first and second-order elasticities, area of cross-section, torsional rigidity, bending vector and the inertia tensor. When an infinitesimally twisted bar is deformed, there is a second-order bending deformation even when there is no resultant bending moment applied on the end faces. This revised version was published online in August 2006 with corrections to the Cover Date.     

A non-ordinary state-based peridynamic method to model solid material deformation and fracture

In this paper, we develop a new non-ordinary state-based peridynamic method to solve transient dynamic solid mechanics problems. This new peridynamic method has advantages over the previously developed bond-based and ordinary state-based peridynamic methods in that its bonds are not restricted to central forces, nor is it restricted to a Poisson’s ratio of 1/4 as with the bond-based method. First, we obtain non-local nodal deformation gradients that are used to define nodal strain tensors. The deformation gradient tensors are used with the nodal strain tensors to obtain rate of deformation tensors in the deformed configuration. The polar decomposition of the deformation gradient tensors are then used to obtain the nodal rotation tensors which are used to rotate the rate of deformation tensors and previous Cauchy stress tensors into an unrotated configuration. These are then used with conventional Cauchy stress constitutive models in the unrotated state where the unrotated Cauchy stress rate is objective. We then obtain the unrotated Cauchy nodal stress tensors and rotate them back into the deformed configuration where they are used to define the forces in the nodal connecting bonds. As a first example we quasi-statically stretch a bar, hold it, and then rotate it ninety degrees to illustrate the methods finite rotation capabilities. Next, we verify our new method by comparing small strain results from a bar fixed at one end and subjected to an initial velocity gradient with results obtained from the corresponding one-dimensional small strain analytical solution. As a last example, we show the fracture capabilities of the method using both a notched and un-notched bar.     

Thermo-mechanical large deformation response and constitutive modeling of viscoelastic polymers over a wide range of strain rates and temperatures

A phenomenological one-dimensional constitutive model, characterizing the complex and highly nonlinear finite thermo-mechanical behavior of viscoelastic polymers, is developed in this investigation. This simple differential form model is based on a combination of linear and nonlinear springs with dashpots, incorporating typical polymeric behavior such as shear thinning, thermal softening at higher temperatures and nonlinear dependence on deformation and loading rate. Another model, of integral form, namely the modified superposition principle (MSP), is also modified further and used to show the advantage of the newly developed model over MSP. The material parameters for both models are determined for Adiprene-L100, a polyurethane based rubber. The constants once determined are then utilized to predict the behavior under strain rate jump compression, multiple step stress relaxation loading experiment and free end torsion experiments. The new constitutive model shows very good agreement with the experimental data for Adiprene-L100 for the various finite loading paths considered here and provides a flexible framework for a three-dimensional generalization.     

Aspects on the finite-element implementation of the Gurson model including parameter identification

In this contribution, various aspects on the finite-element implementation of the Gurson model are considered. In particular, a linear representation for the plastic potential is used, which shows superior convergence property in the local iteration procedure compared to the original quadratic representation. The formulation of the model is performed in the spatial configuration based on the multiplicative decomposition of the deformation gradient, and for integration an exponential map scheme is used. A further important aspect is the sensitivity analysis consistent with the underlying integration scheme necessary for minimizing a least-squares functional for parameter identification by use of a gradient-based optimization algorithm. In a numerical example the local convergence behavior for the two versions of the Gurson model, linear and quadratic are compared. Furthermore material parameters are determined by least-squares minimization based on experimental data obtained for an axisymmetric tensile bar for a ferritic steel.     

Finite deformation crack-line fields in a thin elasto-plastic sheet

Finite deformation in the crack-tip zone of plastic deformation is investigated for Mode-I opening of a crack in a thin sheet of elasto-plastic material. The material obeys the von Mises yield criterion in the true stresses, and the stretching tensor satisfies a flow law of the Prandtl-Reuss type. Incompressibility and a state of generalized plane stress are assumed. It is assumed that linearized elasticity applies outside the zone of plastic deformation. On the crack-line between the crack-tip and the elastic—plastic boundary, two distinct regions have been recognized: the near-tip zone and the intermediate region. In the near-tip zone the fields are controlled by the radius of curvature of the blunted crack-tip. Here the stress field has been approximated by classical plane stress results. It has been assumed that the crack-line stresses may be taken as uniform in the intermediate region. In each region, deformation variables have been determined by the use of the constitutive relations, and the results have been matched to the corresponding quantities in the neighboring region(s). In this manner expressions have been constructed for the deformation gradients on the crack-line, in terms of the distance to the crack-tip in the deformed configuration, the yield stress in shear, and the stress intensity factor of linear elastic fracture mechanics.     

The significance of pure measures of distortion in nonlinear elasticity with reference to the Poynting problem

The objective of this paper is to show the significance of expressing the strain energy function in terms of a scalar pure measure of dilatation and a tensor pure measure of distortion, which were essentially introduced by Flory [1]. It is shown that convenient representations for the strain energies of dilatation and distortion, and the pressure and deviatoric Cauchy stress may be recorded in terms of these deformation measures. After specializing to the case of an isotropic material, specific constitutive equations are proposed and the Poynting problem is considered. It is shown that the Poynting effect (extension of a bar in torsion) is significantly influenced by coupling between dilatational and distortional strain energies, which is caused by the dependence of the shear modulus on dilatation.     

薄壁管预扭冲击拉伸实验装置的研制

材料在复合应力下的拉压或扭转行为常常与单轴拉压或者单轴扭转应力下不同,拉压以及扭转常体现出相互影响的特点。而复合应力下的塑性波可以有效地反映材料在复合应力下的本构行为特点。本文通过对霍普金森压杆进行改造,建立了一套薄壁管预扭冲击拉伸的实验装置,可以在薄壁管内产生拉扭耦合塑性波,并对率相关材料304不锈钢进行了薄壁管预扭拉伸实验研究,得到了该材料的拉扭耦合塑性波。结果表明,304不锈钢薄壁管的拉扭耦合塑性波具有明显的耦合快波和耦合慢波的双波结构,并且快波慢波之间没有恒值区间隔。同时也表明,该装置可以实现预期效果,并且可以有效避免薄壁管屈曲的产生。     

Damage characterization of SAE 1020 and 1045 steel under torsion and compression

A damage model is applied to characterize the ductile deformation of SAE 1020 and 1045 steel. Damage is evaluated for thin-walled cylindrical specimens in torsion and solid bar specimens in compression where stress triaxiality enhances crack initiation. Analyzed are the variations of the damage parameter with the average compressive axial strain at the different locations of the solid bar. Initially, stress triaxiality being largest at the center appeared to dominate damage. With increasing strain, pronounced damage tends to occur in the mid-plane at locations closer to the free surface. Change in the aspect ratio of the cylindrical bar specimens also had an effect on the stress triaxiality and hence the damage parameter. Less damage is prediated for slender bars at the same strain level although the difference is small for height to diameter ratio up to 1.86.