Structure-energy analysis of models of nonlinearly viscoelastic materials with several aging and viscosity functions

We consider generalized one-dimensional Maxwell and Kelvin-Voigt models of viscoelastic materials in which the properties of elastic and viscous elements are determined by the corresponding secant moduli and viscosity coefficients, which are functions of the parameters determined by the deformation process. In contrast to the nonlinear endochronic theory of aging viscoelastic materials (NETAVEM), in which one and the same aging function is used to describe the properties of all elastic elements and one and the same viscosity function is used to describe the properties of all viscous elements [1, 2], it is assumed that the type of these functions is distinct for each elementary model. For the generalized Maxwell and Kelvin-Voigt models under study, we obtain representations of the specific work of internal forces as the sum of four terms of different physical meaning. There representations are similar to those given in [1, 2] for NETAVEM. An example of construction of viscoelasticity constitutive relations containing two aging functions and one viscosity function is given for a material whose properties are sensitive to the strain rate. The simultaneous use of several aging and viscosity functions to describe the properties of structure elements of the model and the use of several components of specific work as arguments of these functions allows us to extend the scope of the models under study.     

Use of structural components of specific work of internal forces for estimating the strength of viscoelastic structures in local stress concentration regions

Material fracture experiments on specimens and structures testify that materials can resist greater stresses in local stress concentration regions than in regions with a nearly homogeneous stress state. Taking this fact into account in design stress analysis permits one to reveal additional structure loading and/or service life margins. One approach aimed at taking into account the increased strength in local stress concentration regions is to use averaged limit characteristics parametrically depending on the characteristic size L of the averaging region. One version of this approach is the concept of “elementary block” of a material [1, 2]. The averaged limit characteristics are determined by an experiment-calculation method involving the analysis of the stress-strain state of a material specimen with a stress concentrator at the time when the specimen attains the limit state preceding macrofracture.In [3], the dependence of the averaged limit separation stresses on the size of the averaging region was determined on the basis of numerical analysis of the singular stress state of the specimen used to determine the standard characteristics of the adhesion strength of a filled polymer material. In the present paper, we generalize the above approach to the case of a viscoelastic material. For the limit characteristics of the material in the local stress concentration region we take the volume-averaged components of the specific work of internal forces [4, 5] (the averaged specific absorbed energy and the averaged specific instantaneously reversible energy). The introduction of two limit energies originates from the fact that, to initiate the process of macrofracture, it is necessary to satisfy the following two conditions simultaneously: the material must be “damaged” sufficiently strongly by the preceding loading, and the “damaged” material must be loaded sufficiently strongly. As an example of determining the material averaged limit energy characteristics in a local stress concentration region, we consider the problem about the strain of a viscoelastic specimen used to determine the standard adhesion strength characteristics. The problem is solved numerically under the following assumptions: the specimen material is assumed to be linearly viscoelastic, and the specific absorbed energy in the stress concentration region is assumed to coincide in magnitude with the specific scattered energy. To estimate the accuracy of the numerical method, we use the solution of the model problem about the action of a plane circular die on a half-space consisting of a linearly viscoelastic incompressible material.     

Determining material true stress–strain curve from tensile specimens with rectangular cross-section

The uniaxial true stress logarithmic strain curve for a thick section can be determined from the load–diameter reduction record of a round tensile specimen. The correction of the true stress for necking can be performed by using the well-known Bridgman equation. For thin sections, it is more practical to use specimens with rectangular cross-section. However, there is no established method to determine the complete true stress–logarithmic strain relation from a rectangular specimen. In this paper, an extensive three-dimensional numerical study has been carried out on the diffuse necking behaviour of tensile specimens made of isotropic materials with rectangular cross-section, and an approximate relation is established between the area reduction of the minimum cross-section and the measured thickness reduction. It is found that the area reduction can be normalized by the uniaxial strain at maximum load which represents the material hardening and also the section aspect ratio. Furthermore, for the same material, specimens with different aspect ratio give exactly the same true average stress–logarithmic strain curve. This finding implies that Bridgmans correction can still be used for necking correction of the true average stress obtained from rectangular specimens. Based on this finding, a method for determining the true stress–logarithmic strain relation from the load–thickness reduction curve of specimens with rectangular cross-section is proposed.     

A ductile damage criterion at various stress triaxialities

The paper discusses the effect of stress triaxiality on the onset and evolution of damage in ductile metals. A series of tests including shear tests and experiments on smooth and pre-notched tension specimens was carried out for a wide range of stress triaxialities. The underlying continuum damage model is based on kinematic definition of damage tensors. The modular structure of the approach is accomplished by the decomposition of strain rates into elastic, plastic and damage parts. Free energy functions with respect to fictitious undamaged configurations as well as damaged ones are introduced separately leading to elastic material laws which are affected by increasing damage. In addition, a macroscopic yield condition and a flow rule are used to adequately describe the plastic behavior. Numerical simulations of the experiments are performed and good correlation of tests and numerical results is achieved. Based on experimental and numerical data the damage criterion formulated in stress space is quantified. Different branches of this function are taken into account corresponding to different damage modes depending on stress triaxiality and Lode parameter. In addition, identification of material parameters is discussed in detail.     

Limiting conditions for compression testing of flat specimens in the split hopkinson pressure bar

The split Hopkinson pressure bar (SHPB) technique is analyzed during the initial stages of loading by means of axisymmetric finite element simulations of dynamic compression tests. Limiting strains as functions of the test parameters such as the specimen diameterd and heighth were found to ensure a one-dimensional stress state and axial stress homogeneity in specimens of elastic-perfectly plastic material. The one-dimensional stress state is necessary and sufficient for accurate test results for flat specimens (h/d≤0.5) and nonflat specimens, respectively, with diameters up to half of the bar diameter. Only very small values of the Coulomb friction constraint (μ≈0.01) seem to be acceptable. The significance of the determined limiting conditions to the more practical case of a rate dependent material is investigated using an elastic-viscoplastic material for the specimen. The stress and strain rate reconstructed from the calculated bar signals (according to the SHPB analysis) are compared with stresses and strain rates averaged over the cross section of the specimen. Well-known inertia corrections improve the results of the SHPB procedure, but errors remain for small strains and highly time dependent strain rates.     

脆性PMMA带板中的动态裂纹传播实验

设计了一种研究裂纹在脆性带板(Strip)中传播行为的实验装置。将矩形无预制裂纹的脆性有机玻璃(PMMA)试件上下边固支,在万能试验机上施加拉伸载荷直到给定幅值,然后在试件侧边中央部位开启一个尖锐裂纹。由于预加的拉伸载荷作用,新开启的裂纹将沿试件中部快速传播,导致整个平板试件断裂。采用印刷在试件表面的导电断裂丝记录裂纹的动态传播历史。设计了一种逻辑电路,可将多达128路电阻丝的断裂信号转换为1路输出信号,使用高速波形记录仪测量该信号,得到裂纹传播距离与时间关系。对不同形状的试件施加不同幅度的预加载荷,完成断裂实验,确定了裂纹的稳定传播速度。通过大量实验,获得了材料中稳定裂纹速度v0与动态裂纹传播能Gc之间的关系。测试结果表明:PMMA的传播能Gc是传播速度v0的递增函数;当裂纹平均的稳定速度超过某一临界值时,出现明显的速度振荡。     

A hybrid technique for the solution of unsteady Maxwell fluid with fractional derivatives due to tangential shear stress

The article describes the unsteady motion of viscoelastic fluid for a Maxwell model with fractional derivatives. The flow is produced by cylinder, considering time dependent quadratic shear stress ft² on Maxwell fluid with fractional derivatives. The fractional calculus approach is used in the constitutive relationship of Maxwell model. By applying Laplace transform with respect to time t and modified Bessel functions, semianalytical solutions for velocity function and tangential shear stress are obtained. The obtained semianalytical results are presented in transform domain, satisfy both initial and boundary conditions. Our solutions particularized to Newtonian and Maxwell fluids having typical derivatives. The inverse Laplace transform has been calculated numerically. The numerical results for velocity function are shown in Table by using MATLAB program and compared them with two other algorithms in order to provide validation of obtained results. The influence of fractional parameters and material constants on the velocity field and tangential stress is analyzed by graphs.     

Design and validation of a new fixture for the shear testing of cellular solids

The design and validation of a new fixture for the shear testing of cellular solids are presented. The fixture is an extended version of a picture-frame shear fixture (EPF) and is suited for comparatively thick rectangular block specimens. The stress state in the specimen is analyzed using a detailed finite element model. The finite element model is based on a 3D CAD model and incorporates friction in the revolute joints. Using specimens with low stiffness, a nearly pure and uniform shear stress state is induced in the specimen. A correction factor for the shear stress is derived which takes into account the friction in the joints and the nonuniformity of the shear stress distribution in the gauge section. The shear response of the polymer foam Rohacell^® 200 WF is determined in order to demonstrate the capabilities of the EPF. The strain state is analyzed by means of digital image correlation and is detected to be very pure and uniform on the specimen’s surface, as predicted by the numerical analysis. The shear modulus obtained with the EPF is in good agreement with the calculated shear modulus derived from tensile tests on the same material. In addition, there is only little scatter of the strength values over the tested specimens which further confirms the accuracy of the new fixture.     

Determination of the material fracture toughness by numerical analysis of 3D elastoplastic dynamic deformation

We develop a technique for calculating the plastic strain and fracture toughness fields of a material by solving dynamical 3D problems of determining the stress-strain state in the elastoplastic statement with possible unloading of the material taken into account. The numerical solution was obtained by a finite difference scheme applied to the three-point shock bending tests of parallelepiped-shaped bars made of different materials with plane crack-notches in the middle. The fracture toughness coefficient was determined for reactor steel. The numerically calculated stress tensor components, mean stresses, the Odquist parameter characterizing the accumulated plastic strain, and the fracture toughness are illustrated by graphs.     

角钢约束混凝土中长柱轴压力学性能试验及承载力计算

为了研究角钢约束混凝土中长柱在轴压荷载作用下的力学性能,以长细比、缀板间距、混凝土强度等级为变化参数,完成了8个试件的静力加载试验。通过试验观察了试件的破坏形态,获取了其极限承载力、刚度、位移延性和耗能系数等力学性能指标;分析了各变化参数对力学性能指标的影响,利用统一强度理论、极限理论和叠加理论对试件的承载力进行计算。研究结果表明:长细比越小越容易发生柱端破坏,长细比加大后易发生柱中破坏;减小缀板间距,试件的极限承载力和变形能力均得到提高。随着混凝土强度等级的提高,试件的极限承载能力和初始弹性刚度得以提高,但位移延性和耗能能力有所降低。采用统一强度理论的计算值略大于试验值,采用极限分析理论和叠加理论的计算值均小于试验值。     

Photoviscoelastic stress analysis of a plate with a central hole

An epoxy resin containing excessive plasticizer was developed and characterized. The material, which deforms viscously at room temperature, has optical properties that depend on stress and strain. A tensile specimen was prepared from the epoxy resin so that the mechanical and optical properties of the epoxy resin could be characterized. The elastic and plastic behavior was determined at 37°C using tensile stresses between 4 and 26 MPa. The birefringence was also recorded as a function of time and stress. From these results, a photoviscoelastic constitutive equation was constructed to describe the dependence of the birefringence on stress and strain. The constitutive equation was then applied to study the deformation of a tensile specimen containing a central circular hole. By using the isochromatic fringes in combination with the isoclinic, the time-dependent variation of the stress field in the specimen was solved.     

A Note on the Direct Determination of the Confining Pressure of Cylindrical Specimens

This note presents a simple approach to the direct determination of the confining pressure, q, for a cylindrical specimen encased in a metallic sleeve. The stress analysis of the problem shows that, for a pressure-insensitive material (e.g. metal), q is the quantity by which the stress level of the confined specimen is elevated with respect to the unconfined. As such, q is directly determined by comparing the results of two tests, one with and the second without confinement. For a pressure-sensitive material, q must be determined independently from a plastic stress analysis of the confining sleeve. Then, the same approach can be applied to determine the pressure sensitivity of the material. The present results greatly simplify testing of confined cylindrical specimens for both pressure-insensitive and sensitive materials.     

环境激励下Benchmark结构损伤识别的试验研究

针对结构健康监测中如何利用在线监测数据进行健康诊断的问题,基于时间序列分析提出了一种新的损伤识别方法.首先,获得结构健康状态的监测数据作为参考状态样本,对各数据样本建立ARMA模型并计算模型残差的方差.然后,将未知状态的监测数据作为待检状态样本,代入已建立的参考状态ARMA模型计算新的残差方差.计算发现,损伤前后两状态模型残差方差存在差异.因而,提出以残差方差之比作为损伤敏感特征,并建立基于F分布的假设检验来辨识结构的状态并预警损伤.最后,以Benchmark结构在环境激励下的试验为例,运用本文方法进行了损伤识别研究.结果表明,基于ARMA模型残差方差的损伤敏感特征可准确地区别结构的健康状态和损伤状态,具备结构在线实时损伤识别的应用价值.     

Equilibrium Paths of a Hyperelastic Body Under Progressive Damage

This paper deals with equilibrium problems in nonlinear dissipative inelasticity, where inelastic effects are produced by the damage of the material. The inelastic constitutive law is obtained by modifying the classical constitutive law for a hyperelastic isotropic material through a damage function. To define this damage function, which allows to measure the effective stress and the dissipated energy, it is first used the Clausius-Duhem inequality, to have the (rate-independent) flow law of the damaged state and then it has been imposed a damage criterion based on an energy approach. After making the constitutive modeling, the boundary-value problem of the Rivlin’s cube, now composed of damaged material, is formulated. The purpose is to analyze a three-dimensional body that, during the deformation process, experiences a progressively increasing damage. Equilibrium branches of symmetric and asymmetric solutions, together to bifurcation points, are computed. Emphasis is placed in investigating how the damage can alter these equilibrium paths with respect to the virgin undamaged case. In particular, the stress reductions caused by damage can give rise to transitions from hardening type to the softening one of the constitutive behavior. These changes can affect the quality of the equilibrium solutions. Accordingly, the analysis is completed by assessing the stability of the solutions. For this aim, the energetic method is extended to damaged materials.     

Intrinsic fatigue curves applied to damage evaluation and life prediction of laminate and fabric material

Modeling of the stochastic fatigue behavior of specimens is made from a knowledge of the intrinsic character of fatigue referred as the intrinsic fatigue curve (IFC). The idea is to predetermine the fatigue behavior of specimens' population such that each sample functions of IFC would correspond to the intrinsic fatigue character of a specimen. The existence proof of IFC is derived from Kolmogorov's theorem for a random function. The form of IFC is obtained for log-normal life distribution functions of specimens subjected to constant-amplitude load histories. Proposed is a linear damage accumulation rule for estimating the life distribution function of a specimen subjected to irregular load histories. For a stationary loading process, the life distribution function is also shown to be log-normal. The relations between the quantile SN curves and IFC are established. The ways of creating IFC from test data and quantile SN curves are illustrated by numerical examples for laminate and fabric material.     

不同尺寸砂岩动态力学性质和应力平衡性的试验研究

采用大直径分离式霍普金森压杆系统获得的不同尺寸试样的实验冲击动态力学参数有差异,因此在直径100 mm压杆上进行了3种直径（50、75和100 mm）和5种长径比（0.4、0.5、0.6、0.8和1.0）的砂岩试样冲击试验,分析了不同尺寸试样应力-应变曲线和应变率曲线随尺寸的变化,提出了用于比较波形对齐重合度的波形叠加系数,并与应力平衡因子共同构建了动态应力平衡性研究体系,由此确定大直径霍普金森压杆试验的试样建议尺寸。同时,利用高速摄影机监测试样的动态破坏状况。结果表明:当长径比相同时,直径75与100 mm岩石试样的动态抗压强度测试值相近,直径50 mm试样具有更明显的长度效应;随着试样直径的增大,应变率曲线从单峰变为双峰;小尺寸试样更易发生轴向劈裂破坏,大尺寸试样受内部应力波叠加影响产生了较大的拉应力,易发生层裂拉伸和轴向劈裂的复合型破坏;对直径75 mm且长径比0.3~0.4的试样,波形对齐后重合度较高,在起始破坏前拥有充足的应力平衡时间,应变率加载效果较好。获得了砂岩试样冲击压缩试验的尺寸效应,可为大直径岩石试样的尺寸选择提供参考。     

SHPB试验中试件的轴向应力均匀性

针对SHPB试验中试件的轴向应力均匀性问题,采用一维弹性波理论,推导了具有任意形状前沿的入射波加载下,试件内应力的时空分布计算公式。以脉冲前沿的上升时间为参数,将矩形、梯形和坡形3种典型的输入脉冲统一表示为梯形波的形式,计算了不同入射波上升时间和不同试件-压杆波阻抗比情况下试件中的应力传播过程,得到了相应的应力均匀度时程曲线以及应力平衡时间。分析了入射波上升时间和试件-压杆波阻抗比对应力平衡时间的影响,得到了一些有意义的认识,为SHPB试验的设计与分析提供了一定的理论依据。     

Identification of the damping properties of rigid isotropic materials by studying the damping flexural vibrations of test specimens

A technique for determining the damping properties of a rigid isotropic material from the experimental data on the damping capacity of elongated cantilever-fixed test specimens due to the internal and external aerodynamic damping is proposed. The following two methods for eliminating the aerodynamic damping component are considered: the extrapolation of the data on the damping capacity of a series of test specimens of different widths to the point corresponding to the zero width and the theoretical-experimental approach. The damping properties of the material are determined by the vibration logarithmic decrement depending on the amplitude of the linear deformation. This dependence is represented by a power polynomial. The polynomial coefficients are determined from the minimum condition of the goal function for the positive logarithmic decrement of the material vibrations. These coefficients are sought at the reference point by repeatedly solving the direct problem of determining the damping capacity of the test specimen from the given damping properties of the material. An example is considered to illustrate the identification of the damping properties of steel St.3.     

Stress analysis of an orthotropic material under diametral compression

The diametral compression test is commonly used to determine the tensile strength of brittle materials. For isotropic materials a simple relation based on specimen geometry and the applied load at failure is used to calculate the tensile strength. Previous to this work the effect of material orthotropy and material orientation on the specimen stress state had not been completely determined. In this study, both isotropic and orthotropic specimens were analyzed using a finite element analysis and experimentally verified by strain gage and photoelastic measurements. Further, this work investigated the effect of the applied load area on the specimen stress state. Results of this work show that there is a significant difference between the theoretical calculations based on the assumption of material isotropy when compared to an orthotropic material. This difference can be as much as 45 percent depending on the degree of orthotropy and the orientation. It was also determined that the applied load area and material orientation significantly influence the specimen stress state. An applied load area of 8 percent of the circumference was found to reduce the stresses in the applied load region.