A note on the inertia and strain-rate effects in the tam and calladine model

The dynamic behaviour of a simple plate structure subjected to an axial impact is studied using an elastic-plastic model which takes into account inertia effects and the influence of material strain-rate sensitivity. The entire time-dependent deformation process, including elastic unloading and plastic reloading is obtained. The predictions for the absorbed energy are in reasonable agreement with the corresponding experimental results reported by Tam and Calladine and the detailed behaviour provides some further insight into the dynamic plastic buckling of structural elements.     

Dynamic effects on cavitation instabilities in solids

Summary. Unstable growth of a cavity in a solid subjected to a suddenly applied, remote stress field was investigated for an elastic-plastic material, including the effects of both inertia and a strain-rate sensitive yield criterion. Both spherically-symmetric and axisymmetric loading was considered. It was found that for both types of loading, the critical load for cavitation in the quasi-static case would cause unstable cavity growth in the dynamic analysis case, and the long-time response was cavity expansion at a constant rate. Cavity growth in a material governed by power-law creep was also considered. Here, it was found that unlimited cavity growth occurs at any level of stress for both spherically-symmetric and axisymmetric loading.     

Effect of yield surface vertex on failure of ductile materials

This paper is concerned with the lengthening of a solid rod or tubular specimen along the principal axis about which it is being twisted, this aspect of the elastic-plastic deformation of a material being referred to as the Poynting-Swift effect. The Swift effect is the counterpart in the plastic range of deformation to the Poynting effect in the elastic range of deformation. Central to any study of the Swift effect is the problem of identifying an appropriate yield criterion. A generalised isotropic yield criterion is formulated in such a way that it can be applied to materials which satisfy the von Mises yield criterion, or some modified, but continuously difierentiable form of it, and to materials which satisfy a piece-wise continuous yield condition such as the Tresca yield criterion. The choice of the constitutive equation describing the purely elastic deformation behaviour determines the initial yield function. In this context, the constitutive equation of a simple elastic material is only compatible with von Mises yield criterion, a conclusion which applies also to classical infinitesimal theory. An attempt is made to generalise the constitutive equation of a simple elastic material to give a constitutive equation which is compatible with the proposed generalised isotropic yield criterion. This approach introduces an additional term into the constitutive equation which is quadratic in the stress. The two loading coefficients associated with the stress loading function are assumed to be deriveable from the generalised isotropic yield criterion which is now assumed to hold over the entire range of deformation, and in this context is referred to as the stress intensity function. It is a matter of observation that the proposed constitutive equation describes the total, post-yield, elastic-plastic response to simple loading. Consideration is given to the application of the Poynting-Swift effect to the failure of ductile materials using the proposed constitutive equation.     

Influence of isotropic strain hardening on plane strain dynamic growth in elastic-plastic materials

In this paper, dynamic crack growth in an elastic-plastic material is analysed under mode I, plane strain, small-scale yielding conditions using a finite element procedure. The material is assumed to obey J₂ incremental theory of plasticity with isotropic strain hardening which is of the power-law type under uniaxial tension. The influence of material inertia and strain hardening on the stress and deformation fields near the crack tip is investigated. The results demonstrate that strain hardening tends to oppose the role of inertia in decreasing plastic strains and stresses near the crack tip. The length scale near the crack tip over which inertia effects are dominant also diminishes with increase in strain hardening. A ductile crack growth criterion based on the attainment of a critical crack tip opening displacement is used to obtain the dependence of the theoretical dynamic fracture toughness on crack speed. It is found that the resistance offered by the elastic-plastic material to high speed crack propagation may be considerably reduced when it possesses some strain hardening.     

A finite element analysis of plane strain dynamic crack growth in materials displaying the Bauschinger effect

In this paper dynamic crack growth in an elastic-plastic material is analyzed under mode I plane strain small-scale yielding conditions using a finite element procedure. The main objective of this paper is to investigate the influence of anisotropic strain hardening on the material resistance to rapid crack growth. To this end, materials that obey an incremental plasticity theory with linear isotropic or kinematic hardening are considered. A detailed study of the near-tip stress and deformation fields is conducted for various crack speeds. The results demonstrate that kinematic hardening does not oppose the role of inertia in decreasing the plastic strains and stresses near the crack tip with increase in crack speed to the same extent as isotropic strain hardening. A ductile crack growth criterion based on the attainment of a critical crack opening displacement at a small micro-structural distance behind the tip is used to obtain the dependence of the theoretical dynamic fracture toughness with crack speed. It is found that for any given level of strain hardening, the dynamic fracture toughness displays a much more steep increase with crack speed over the quasi-static toughness for the kinematic hardening material as compared to the isotropic hardening case.     

Dynamic asymptotic mode III crack tip field in rate dependent materials

The asymptotic field at a dynamically growing crack tip in strain-rate sensitive elastic-plastic materials is investigated under anti-plane shear loading conditions. In the conventional viscoplasticity theory, the rate sensitivity is included only in the flow stress. However, it is often found that the yield strength is also affected by previous strain rates. The strain rate history effects in metallic solids are observed in strain rate change tests in which the flow stress decreases gradually after a rapid drop in strain rate. This material behavior may be explained by introducing the rate sensitivity in the hardening rule in addition to the flow rule. The strain-rate history effect is pronounced near the propagating crack where the change of strain rates take place. Effects of the rate dependency in the flow rule and the hardening rule on the crack propagation are analyzed. The order of the stress singularity in the asymptotic field is determined in terms of material parameters which characterize the rate sensitivity of the material. The results show that an elastic sector is present in the wake zone when the rate-dependency is considered only in the hardening rule. Terminal crack propagation speed is determined by applying the critical stress fracture criterion and the critical strain criterion to the asymptotic fields under the small scale yielding condition.     

On stress field near a stationary crack tip

It is well known that the stress and elastic-plastic deformation fields near a crack tip have important roles in the corresponding fracture process. For elastic-perfectly-plastic solids, different solutions are given in the literature. In this work we examine and compare several of these solutions for Mode I (tension), Mode II (shear), and mixed Modes I and II loading conditions in plane strain. By consideration of the dynamic solution, it is shown that the assumption that the material is yielding all around a crack tip may not be reasonable in all cases. By admitting the existence of some elastic sectors, we obtain continuous stress fields even for mixed Modes I and II.     

Temperature and strain rate dependence of the deformation behaviour of poly(paraphenylene benzobisthiazole)

The mechanical properties of PRT films feature an interesting temperature and strain-rate dependence. The elastic modulus and yield behaviour have been studied over a wide temperature range from 30 to 650° C. The onset of a structural reorganization is observed at about 300° C. The dependence of yield stress on strain rate at different temperatures was examined in terms of the Eyring theory of an activated rate process. It was found that the stress activation volume varies with temperature. The overall elastic-plastic behaviour as a function of temperature and strain rate was interpreted in terms of a previously suggested model that incorporates residual stresses in a rigid rod-like polymer. Enhancement of modulus was also observed due to deformation of the films under elevated temperatures, where better molecular orientation and lateral ordering are achieved.     

Pressure-shear stress wave analysis in plate impact experiments

Numerical results are presented for the combined longitudinal and shear wave propagation in an elastic-viscoplastic solid as it occurs in high strain-rate plate impact experiments. Special attention is paid to the initial stage of the impact experiment and the effects of the specimen thickness, elastic impedances of flyer-anvil plates, and viscoplastic properties of materials on the time to reach a homogenous stress and deformation state within a specimen. The simple interpretation of experimental results which assumes a homogenous stress and deformation state within a specimen is found in general to be valid only at a much later time after impact. It is recommended that the measurement of the stress wave profiles should be made at the back face of the specimen rather than at its impact face, and that a pressure-shear stress wave analysis of the plate impact experiment should be performed to evaluate the inertial effects on the early part of experimental recordings. The comparison between measured stress wave profiles and the numerical simulation of the experiment provides a critical assessment of advanced viscoplastic models developed for applications under impact loading.     

About the dynamic uniaxial tensile strength of concrete-like materials

Experimental methods for determining the tensile strength of concrete-like materials over a wide range of strain-rates from 10⁻⁴ to 10² s⁻¹ are examined in this paper. Experimental data based on these techniques show that the tensile strength increases apparently with strain-rate when the strain-rate is above a critical value of around 10⁰-10¹ s⁻¹. However, it is still not clear that whether the tensile strength enhancement of concrete-like materials with strain-rate is genuine (i.e. it can be attributed to only the strain-rate effect) or it involves “structural” effects such as inertia and stress triaxility effects. To clarify this argumentation, numerical analyses of direct dynamic tensile tests, dynamic splitting tests and spalling tests are performed by employing a hydrostatic-stress-dependent macroscopic model (K&C concrete model) without considering strain-rate effect. It is found that the predicted results from these three types of dynamic tensile tests do not show any strain-rate dependency, which indicates that the strain-rate enhancement of the tensile strength observed in dynamic tensile tests is a genuine material effect. A micro-mechanism model is developed to demonstrate that microcrack inertia is one of the mechanisms responsible for the increase of dynamic tensile strength with strain-rate observed in the dynamic tensile tests on concrete-like materials.     

Saturated duration for dynamic structural plastic response and fundamental periods of elastic vibration

The relationship is determined between saturated duration of rectangular pressure pulses applied to rigid, perfectly plastic structures and their fundamental periods of elastic vibration. It is shown that the ratio between the saturated duration and the fundamental period of elastic vibration of a structure is dependent upon two factors: the first one is the slenderness or thinness ratio of the structure; and the second one is the square root of ratio between the Young’s elastic modulus and the yield stress of the structural material. Dimensional analysis shows that the aforementioned ratio is one of the basic similarity parameters for elastic-plastic modeling under dynamic loading.     

FINITE ELEMENT ELASTIC-PLASTIC-CREEP AND CYCLIC LIFE ANALYSIS OF A COWL LIP

Results are presented of elastic, elastic-plastic and elastic-plastic-creep analyses of an actively cooled cowl lip. A cowl lip is part of the leading edge of an engine inlet of proposed hypersonic aircraft and is subject to severe thermal loadings and gradients during flight. Values of stresses calculated by elastic analysis are well above the yield strength of the cowl lip material. Such values are highly unrealistic, and thus elastic stress analyses are inappropriate. The inelastic (elastic-plastic and elastic-plastic-creep) analyses produce more reasonable and acceptable stress and strain distributions in the component. Finally, using the results from these analyses, predictions are made for the cyclic crack initiation life of a cowl lip. A comparison of predicted cyclic lives shows the cyclic life prediction from the elastic-plastic-creep analysis to be the lowest and, hence, most realistic.     

Collapse of periodic planar lattices under uniaxial compression, part II: Dynamic crushing based on finite element simulation

The collapse strength is analyzed for periodic planar lattices under uniaxial compression. In part I, the quasi-static strengths of the lattices are predicted by limit analysis [1]. In contrast to part I, the dynamic crushing is studied by finite element simulations in this part. Under different impact velocities, the deformation modes and the average stress of lattices with various relative densities are studied, whilst the emphasis is put on the inertia effect. It is found that the average stress increases with the impact velocity, the density of the base material and the relative density of the lattice. Therefore, the average dynamic stress can be expressed as the sum of the average static stress and the dynamic enhancement caused by the inertia effect. The average dynamic stress for membrane-dominated lattices is higher than that of bending-dominated lattices, but the difference in the dynamic enhancement is not as significant as that in the static collapse stress. According to the shock wave theory, the empirical formula of a hexagonal lattice under uniaxial compression is modified and generalized to all the interested lattices. The empirical formulas based on the shock wave theory agree well with the FE simulations, especially for the membrane-dominated lattice that has less transverse expansion.     

Axial crushing of thin-walled high-strength steel sections

Quasi-static and dynamic axial crushing tests were performed on thin-walled square tubes and spot-welded top-hat sections made of high-strength steel grade DP800. The dynamic tests were conducted at velocities up to 15 m/s with an impacting mass of 600 kg in order to assess the crush behaviour, the deformation force and the energy absorption. Typical collapse modes developed in the sections and the associated energy absorbing characteristics were examined and compared with previous studies on high-strength steel. A significant difference was observed between the quasi-static and the dynamic crushing tests in terms of the deformation force and impact energy absorption. As this difference is attributed to strain-rate and inertia effects, material tensile tests at elevated strain rates have been carried out. A comparison is made with analytical methods and the response was under-predicted. In addition, numerical simulations of the axial crushing of the thin-walled sections were performed and comparisons with the experimental results were satisfactory. The validated numerical model was used to study the energy absorption capacity of thin-walled sections with variations in the yield strength, sheet thickness, flange width and spot-weld spacing. Structural effectiveness differences have been captured through simulations between spot-welded top-hat sections made of mild steel and high-strength steel.     

Asymptotic deformation and stress fields at the tip of a sharp notch in an elastic-plastic material

The singular elastic-plastic stress, strain and the displacement fields at the tip of a sharp notch for both plane stress and plane strain conditions are investigated analytically. The material is assumed to be governed by the deformation theory of plasticity with linear strain hardening characteristic. Since the elastic strain is retained in the analysis, the final strain and displacement fields can be separated into the elastic and the plastic parts. In the case with zero notch angle, the results reduce to the classical crack problem. The relationship of the amplitude of the near crack tip elastic-plastic field to the elastic far field is obtained. Both mode I and mode II cases are investigated. The mixed mode case is also discussed.     

Macroscopic theory and nonlinear finite element analysis of micromechanics of single crystals at finite strains

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the macroscopic rate-independent and rate-dependent analysis of micromechanics of metal single crystals undergoing finite elastic-plastic deformations which is based on the assumption that inelastic deformation is solely due to crystallographic slip. The formulation relies on a multiplicative decomposition of the material deformation gradient into incompressible elastic and plastic as well as a scalar valued volumetric part. Furthermore, the crystal deformation is described as arising from two distinct physical mechanisms, elastic deformation due to distortion of the lattice and crystallographic slip due to shearing along certain preferred lattice planes in certain preferred lattice directions. Macro- and microscopic stress measures are related to Green’s macroscopic strains via a hyperelastic constitutive law based on a free energy potential function, whereas plastic potentials expressed in terms of the generalized Schmid stress lead to a normality rule for the macroscopic plastic strain rate. Estimates of the microscopic stress and strain histories are obtained via a highly stable and very accurate semi-implicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step, and, furthermore, the development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical simulation of finite strain elastic-plastic tension tests is presented to demonstrate the efficiency of the algorithm.     

基材属性对铝蜂窝共面力学性能的影响

目的研究不同属性的基体材料对铝蜂窝共面压缩力学性能的影响。方法在保持正六边形蜂窝结构参数不变的情况下,改变基材属性,基体材料模型分别选择不同应变强化参数的双线性各向同性强化模型和理想弹塑性模型,建立相关可靠的有限元模型并进行大量的模拟计算。获得相应的变形模式和应力-应变曲线,对曲线进一步处理得到蜂窝共面静动态峰应力,并将结果以图表形式展示并分析。结果随着冲击速度的增加,样品依次出现了"X","V","一"字型3种变形模式,基体材料的应变强化效应使变形趋于均匀化;基体材料的应变强化效应显著增加了蜂窝的静态峰应力,对动态峰应力增量的影响可以忽略,对计算数据处理后得到了应变强化参数与动态峰应力的计算公式。结论基材具有强化特性的蜂窝,其共面静态力学性能优于基材为弹性理想塑性材料模型的蜂窝;在利用数值模拟的方法来研究蜂窝结构共面静态力学行为时,需要考虑基体材料的强化效应。     

砂浆材料SHPB实验及惯性效应的数值模拟研究

进行了砂浆材料在不同应变率下的SHPB实验,拟合实验数据得到了动态强度放大因子DIF随应变率变化的关系曲线。基于实验测得应变率时程曲线,采用简化有限元模型,对实验进行了数值模拟。该文探讨了动态压缩实验中惯性效应产生的原因,并基于数值模拟对本实验中惯性效应对材料动态强度的影响进行了剥离,得到了砂浆材料动态强度放大因子随应变率变化的固有特性曲线,并将该固有特性曲线作为数值模拟中应变率效应的输入,计算结果与实验得到的应力-应变曲线吻合较好。进一步通过比较输入CEB推荐曲线和已有半经验公式的模拟结果,验证了所提出砂浆材料动态强度放大因子固有特性曲线的优越性。     

Constitutive equation for a class of non-simple elastic materials

Summary Initial yield is the upper limit of the purely elastic deformation behaviour of an elasticplastic solid. Thus the choice of the constitutive equation describing the purely elastic deformation behaviour determines the initial yield function. The constitutive equation of a simple elastic material is only compatible with von Mises yield criterion, a conclusion which applies also to the classical infinitesimal theory. A more general form of constitutive equation for an elastic material is formulated by way of the concept of a stress loading function, the proposed constitutive equation being quadratic in the stress. The two loading coefficients associated with the stress loading function are assumed to be deriveable from a generalised isotropic yield criterion which is now assumed to hold over the entire range of deformation, and in this context is referred to as the stress intensity function. The proposed constitutive equation has the same representation in terms of the left Cauchy-Green deformation tensor as that for a simple elastic material. Using the Cayley-Hamilton theorem, this representation is rearranged and expressed in terms of a measure of finite strain which is defined to be one quarter of the difference between the left Cauchy-Green deformation tensor and its inverse. In this way the strain properties of the proposed constitutive equation are formulated by way of the concept of a strain response function. The three response coefficients associated with the strain response function are assumed to be deriveable from a generalised, isotropic, strain intensity function. The predictions of the proposed constitutive equation are considered in the context of the combined stressing of a thin sheet of incompressible material. In this way, it is shown that the proposed constitutive equation is not limited in the same way as the constitutive equation of a simple elastic material.     

On the relation between rigid-plastic and elastic-plastic predictions of response to pulse loading

Using a simple model, comparisons are shown between the predictions of elastic-plastic and rigid-plastic analyses as to the permanent displacements due to pulse loading. Several pulse shapes are considered, and the comparisons are made over a large range of the ratio τ/T (pulse duration time/natural period of the structure). When the rise time of the pulse is not zero, these show a wavy character that was not seen in earlier comparisons of this type, where smaller ranges were used. Thus the two predictions are found to be very close at regular intervals, while the rigid-plastic prediction has negative (unconservative) errors in between, relative to the elastic-plastic one. These errors may be large even though the plastic deformation greatly exceeds the elastic displacement at yield. The ‘energy ratio’ criterion for validity of rigid-plastic analysis must be supplemented, for ‘long’ pulses, by consideration of the effect of pulse duration. The waviness of the elastic-plastic spectral plots is explained in terms of the similar shapes of the ‘dynamic amplification’ spectra of wholly elastic shock analysis.