Ductile fracture in glassy polymers

The optical method of caustics was used to study the highly deformed region at the crack tip of a tension plate made of polycarbonate (PCBA) up to fracture. This polymer was selected for its typical quasi-plastic behaviour. The modified Dugdale-Barenblatt (D-B) model was used in order to take into account the influence of the non-linear region of quasi-plastic behaviour of PCBA. Furthermore, it has also been observed that the tensile specimens of PCBA were curved significantly before fracture and this phenomenon is typical of all ductile materials. Comparison of the caustics formed from reflected and transmitted rays allowed the accurate evaluation of the curvature of the cracked plate during loading. The influence of curvature being excluded, the remaining parts of the caustics were related to the stress distribution at the crack tip. The size and the dimensions of the caustics were fitted into one of the six typical stress configurations in the plastic zones studied by the modified Dugdale-Barenblatt model and the form of stress distribution in the plastic zone for each loading step was determined. It was shown that the modified D-B model is capable of describing the plastic behaviour in cracked plates made of ductile materials. Furthermore, the method of caustics was capable of evaluating the influence of curvature of the cracked plate during loading.     

Symmetric bending of cracked plates studied by caustics

The optical method of reflected caustics was applied to the study of singularities in the stress field appearing at the crack tip of symmetric plates containing a symmetric crack and submitted to bending loads. The method of caustics was used in connection with the most effective plate theories, that is the theories of Kirchhoff-Williams and Reissner-Sih as they apply to problems of cracked plates.     

Constrained zones in a periodic array of collinear equal cracks

Thin plates made of Perspex and containing an array of collinear equal cracks were subjected to a combined mode of loading consisting of biaxial tension and shear at infinity. The thickness variation of the plates was depicted by the reflected shadow method developed by the author. According to this method a coherent monochromatic light beam was diffracted along the thickness when the plate was traversed and partly reflected on the rear face. The reflected part, which was again refracted through the thickness and afterwards emerged from the plate, was retarded according to Neumann's law. The retarded light rays created deviating wave-fronts which followed the Eikonal equation of geometric optics. If these rays were received on a ground-glass screen, they created a bright envelope, a caustic, which was related to the constrained zones surrounding the crack tips of the plate. In this paper the case of an array of collinear equal internal cracks was studied and especially the influence of neighbouring cracks on the shape and size of the caustic of each crack. The dimensions of the envelope are directly related to the stress-intensity factor corresponding to each crack tip which can be derived from the dimensions of the caustic. Experimental evidence with cracked Perspex plates corroborated the theory.     

The topography of the core-region around cracks under modes I, II and III of fracture

The core region, defined as the limiting zone of validity of the various brittle fracture criteria, has been intuitively assumed as a circle. Its experimental determination is based here on a simple asymptotic elastic theory and a verification by the method of reflected caustics. This definition covers the three modes of deformation at the surroundings of the crack tip. For each mode the initial curve of the caustic, which for isotropic elastic materials is always a circle, yields the limits of validity of the linear elastic theory and corresponds to the boundary of the core region. Experimental evidence with profilometers and projected-grating techniques indicates a similar nature of the deformed surfaces adjacent to the crack tip, as this given by the asymptotic analysis theory of stress and displacement distribution around the crack-tip.     

The reflected caustics method for the evaluation of mode III stress intensity factor

The optical method of reflected caustics, which has been applied to the evaluation of stress intensity factors in cracked plates deformed under mode I and II, is extended in this paper to the evaluation of the same factor in cracked plates subjected to mode III deformation. It is shown that the method of reflected caustics is capable of detecting and evaluating this factor, whereas all the other experimental methods, i.e. photoelasticity, holographic interferometry and especially the method of transmitted caustics are incapable to yield this quantity. Based on the first-order approximation of the elastic solution around the crack tip and on Sneddon's formulas, the theory of formation of the reflected caustics is developed and the characteristic geometric properties of this envelope curve are defined. It is shown that this envelope is again a generalized epicycloid, whose characteristic dimensions are directly related to K_III. Experimental evidence with specimens made of optically isotropic materials (plexiglas), elastically loaded, has corroborated the theoretical results.     

Kinetic friction and dynamic elastic contact behaviour of polymers

Friction and contact between a glass hemisphere and plates of various polymers were studied under elastic contact. The contact at various sliding speeds was observed by means of Newton's rings. Although the frictional force was proportional to the area of contact observed, it was necessary to consider the increase in the shearing strength of the adhesive junctions with increasing load W. The area of dynamic contact of low density polyethylene (LDPE) was proportional to W^0.78 in a wide speed range, while the area of static contact was proportional to W^2/3 as expected from Hertz' equation. The shape of the dynamic contact was affected by the retarded elasticity of the polymer and frictional heating. The area of dynamic contact decreased and the shear strength increased in proportion to log v (where v is the speed) provided that frictional heating was negligible. The time dependence of elastic compliance 1/E of LDPE could be obtained by applying Hertz' equation to the dynamic contact observed at various speeds. However, the value of 1/E decreased with decreasing load in sliding. It was shown that the time dependence of elastic compliance plays an important role in determining the area of dynamic contact of polymers.     

粘弹性材料中裂纹增长的仿真研究

针对含裂纹的粘弹性材料结构,根据粘弹性断裂理论分析裂纹增长的条件.建立基于裂纹周围小范围滑动的计算模型,采用Laplace转换关系,可以将粘弹性边界值问题变为弹性边界值问题处理,在所得到的表达式中并不直接含有弹性常数,因此它能较好地描述粘弹性问题.得到描述含裂纹粘弹性材料在不同的时间裂纹增长情况的裂纹增长速率以及蠕变时间解析解.并按照所建立的计算模型对不同的材料性质、不同的裂纹尖端应力强度因子、不同的外载荷对裂纹增长速率及蠕变时间的影响进行探讨.     

TRANSVERSE VIBRATION CHARACTERISTICS OF VISCOELASTIC RECTANGULAR PLATE WITH CRACK

Based on the two-dimensional viscoelastic differential constitutive relation and the thin plate theory, the differential equations of motion of the viscoelastic plate with an all-over part-through crack are established and the expression of additional rotation induced by the crack is derived. The complex eigenvalue equations of the viscoelastic plate with crack are derived by the differential quadrature method, and the 8method is used at the crack continuity conditions. Dimensionless complex frequencies of a crack viscoelastic plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated. The effects of the crack parameter, the aspect ratio and dimensionless delay time of the material on the transverse vibration of the viscoelastic plate are analyzed.     

The effect of deformation anisotropy on the plastic zone at the tip of a crack under biaxial loading

A theoretical analysis for investigating the form and dimension of the plastic zone in the tip of a crack in a plate made from a material with deformation anisotropy is presented. The anisotropy is explained by the hardening process during plastic deformation caused by loading along a straight-line trajectory until a crack is formed in the plate plane. The idea of this method is to take into account the second term in the Williams’ series representation in eigenfunctions for stress components for the 2D case. The contribution of the second term in the Cartesian coordinate system is independent of the distance from the crack tip for the K ₁ − T concept. It is shown that for the case of anisotropy, the dimension of the plastic zone decreases. The stress along the crack’s axis changes the plastic zone significantly.     

带有两个共线裂纹载流薄板的温度场及热应力强度因子

采用复变函数的方法,给出带有两个等长度共线穿透裂纹的无限大载流薄板在瞬间电流作用下,裂纹尖端附近产生的温度场的表达式,得到温度在裂纹尖端的奇异特征.在通入垂直于裂纹的均匀电流作用下,由于裂纹的存在,裂纹尖端出现绕流现象,产生焦耳热,瞬时高温形成点热源.通过点热源在金属薄板内形成的温度场,可以导出裂纹尖端产生的热应力强度因子表达式.将其与外载荷产生的应力强度因子叠加,可建立通电状态下的断裂判据.算例分析表明,点热源在裂纹尖端附近产生的热应力强度因子为负值,可以部分抵消无穷远处施加的拉应力产生的应力强度因子的作用,进而达到止裂的目的.文中建立带有两个等长度共线穿透裂纹载流薄板的热应力强度因子的概念.其研究结果对工程结构实施电磁热效应裂纹止裂具有理论意义和实用价值.     

A conversion method from the E system to the M system in measuring surface roughness

The relations between the surface roughness parameters measured by the envelope datum system (E system) and those measured by the mean line datum system (M system) are analysed theoretically. Experimental relations between the two systems are obtained by calculation from the surface roughness profiles of machined surfaces.

From the theoretical and experimental results it was found that the levelling depth R_P(E) in the E system is proportional to the mean arithmetic deviation R_a(M) in the M system and the ratio R_P(E)/R_a(M) increases with an increase in the tip radius of the skid. The mean roughness value R_a(E) in the E system is larger than R_a(M) by 10% at its maximum.     

含半圆形埋藏裂纹金属构件脉冲放电时应力场分析

对纯弯曲载荷作用下含半圆形埋藏裂纹的金属构件在脉冲放电瞬间的应力场进行了理论分析,运用热传导、非定常热应力及汉克尔变换等理论,推导出应力场分布理论公式。由公式可知,在脉冲放电瞬间,电磁热在裂纹尖端形成热压应力场,压应力场能有效地抑制裂纹的扩展。对脉冲放电前后的含半圆形埋藏裂纹构件进行了超声波检测实验,利用超声波对比法测得脉冲放电后构件裂尖应力值,该应力值有所增大。     

Elastic modulus and thermal stress in coating during heat cycling with different substrate shapes

The elastic modulus of a deposit (E_d) can be obtained by monitoring the temperature (∆T) and curvature (∆k) of a one-side coated long plate, namely, a one-dimensional (1D) deformation model. The aim of this research is to design an experimental setup that proves whether a 1D deformation model can be scaled for complex geometries. The setup includes a laser displacement sensor mounted on a robotic arm capable of scanning a specimen surface and measuring its deformation. The reproducibility of the results is verified by comparing the present results with Stony Brook University Laboratory’s results. The ∆k-∆T slope error is less than 8%, and the E_d estimation error is close to 2%. These values reveal the repeatability of the experiments. Several samples fabricated with aluminum as the substrate and 100MXC nanowire (Fe and Cr alloy) as the deposit are analyzed and compared with those in finite element (FE) simulations. The linear elastic behavior of 1D (flat long plate) and 2D (squared plate) specimens during heating/cooling cycles is demonstrated by the high linearity of all ∆k-∆T curves (over 97%). The E_d values are approximately equal for 1D and 2D analyses, with a median of 96 GPa and standard deviation of 2 GPa. The correspondence between the experimental and simulated results for the 1D and 2D specimens reveals that deformation and thermal stress in coated specimens can be predicted regardless of specimen geometry through FE modeling and by using the experimental value of E_d. An example of a turbine-blade-shaped substrate is presented to validate the approach.     

Analysis of a crack approaching a circular hole in cross-ply laminates under biaxial loading

The problem of a crack approaching a circular hole in cross-ply laminates under uniaxial and biaxial loading is investigated in this paper. The effects of material orthotropy, geometry [R/d and a/d], and loading conditions on crack tip singularity are investigated. The stress intensity factors are obtained by the modified mapping collocation method. The present results for an isotropic infinite plate show good agreement with existing solutions. The results for cross-ply laminates show that the stress intensity factors strongly depend on material orthotropy, geometry, and loading condition. The stress intensity factors for cross-ply laminates exist between those for θ=0° and those for θ=90° in the whole range of crack length and decrease as the percentage of 0° plies increases. In the range of small crack length the stress intensity factors for biaxial tension are higher than those for uniaxial tension. In the range of large crack length the stress intensity factors for uniaxial tension are higher than those for biaxial tension.     

Numerical investigation of the stress field near a crack normal to ceramic-metal interface

Ceramic-metal interfaces are often present in composite materials. The presence of cracks has a major impact on the reliability of advanced materials, such as fiber or particle reinforced ceramic composites, ceramic interfaces and laminated ceramics. The understanding of the failure mechanisms is very important, as is as the estimation of fracture parameters at the tip of the crack approaching an interface and crack propagation path. A cracked sandwich plate loaded with axial uniform normal stress was numerically investigated using plane strain Finite Element Analysis. The numerical results for the singularity orders were compared with the analytical solution. The influences of the material combination and crack length on the radial and circumferential stresses and displacement distributions were investigated. The Stress Intensity Factors were determined based on numerical results using a displacement extrapolation method. The results for the non-dimensional stress intensity factors show that at lower crack lengths the influence of material mismatch is lower, but this influence increases with increasing crack length.     

The determination of coefficient of friction between a cylindrical indenter and a plastically deformed body by caustics

The contact between an elastic cylindrical indenter and a plastically deformed bar is examined. For this study it is assumed that the normal and tangential loads in the contact zone are related by a Coulomb-type relation. In order to determine the coefficient, which relates the normal and tangential loads, the method of reflected caustics was used. For the evaluation of this coefficient a single measurement of an appropriate length has to be made on the caustic, which was formed at the contact area of the two materials. This measurement allows the determination of the coefficient and in the case when the normal distribution of the contact load is known, the tangential load distribution can be readily obtained. The method was applied to the case of an uniformly distributed load, applied to an elastically deformed cylindrical indenter, which deforms the indented bar plastically.     

小范围屈服条件下复合材料裂纹尖端塑性区分析

基于复合材料力学,推导Tsai-Hill强度准则在平面应力和平面应变条件下的一般表达式,得到了小范围屈服条件下,含中心裂纹无限大板Ⅰ型裂纹、Ⅱ型裂纹和Ⅰ/Ⅱ复合型裂纹尖端塑性区的解析解。针对不同裂纹倾角及泊松比 和,对裂尖塑性区进行了计算和分析。结果表明平面应变条件下塑性区范围小于平面应力条件下塑性区范围,参数、和 对复合材料裂尖塑性区范围和形状有明显的影响,不同的参数值得到的塑性区结果差别很大。另外,该解既适用于各向异性复合材料,也适用于各向同性材料。     

Characterisation of crack tip stresses in elastic-perfectly plastic material under mode-I loading

Asymptotic crack tip stress fields are developed for a stationary plane strain crack in incompressible elastic-perfectly plastic material under mode-I loading. Detailed investigations have revealed that in between the two extreme conditions of crack tip constraint, that is, between the fully plastic Prandtl [1] field and the uniform stress field the most general elastic-plastic crack tip fields can be completely described by the 5-sector stress solution proposed in this article. The 3-sector stress field proposed by Li and Hancock [2] and the 4-sector field proposed by Zhu and Chao [3] are subsets of the general elastic-plastic field proposed in this work. This study has revealed that cases arise where the severe loss of crack tip constraint can lead to compressive yielding of crack flank. This particular situation leads to 5-sector stress field. Detailed studies have revealed that, in the most general case of elastic-plastic crack tip fields, the T_π parameter proposed by Zhu and Chao [3] cannot be used as a constraint parameter to represent a unique state of stress at the crack tip. A new constraint-indexing parameter T_CS-2 is proposed, which along with T_p is capable of representing the entire elastic-plastic crack tip stress fields over all angles around a crack tip. Excellent agreement is obtained between the proposed asymptotic crack tip stress field and the full-field finite element results for constraint levels ranging from high to low. It is demonstrated that the proposed constraint parameters are adequate to represent the crack tip constraint arising due to specimen geometry and loading conditions as well as the additional constraint that arises due to weld strength mismatch.     

电磁热效应止裂效果与电流通路尺寸关系的研究

从理论、试验和数值模拟三方面研究利用电磁热效应技术对具有单边裂纹的导体进行止裂时，由于裂纹尺寸不同使得导体中电流通路尺寸不同，从而导致裂纹尖端的温度场、温度梯度场分布状态的不同。理论分析、试验研究和数值模拟结果均表明：由于电流产生的焦耳热源的作用，能够在裂纹尖端处很小的范畴内熔化形成焊口，遏制裂纹的扩展；导体中裂纹的长度（即导体中电流通路尺寸）是影响裂纹尖端温度场和温度梯度场的主要因素。     

Roll-pressure distribution and coefficient of friction in hot rolling by caustics

The optical method of reflected caustics was employed to investigate the variation of the coefficient of friction in the roll gap during strip rolling. Experiments have been carried out on an experimental rolling mill, using lead at room temperature, to obtain quantitative information about pressure distribution and friction, relevant to hot rolling, i.e. when frictional effects are pronounced. The rolls were made of Plexiglas. The normal and tangential pressure distribution and thence the coefficient of friction along the arc of contact at the outer edge of the strip were calculated from the caustic curves formed due to elastic deformation of the rolls, although the rolled strip was deformed plastically.An almost linear decrease of the friction coefficient from a maximum value, at the entry plane to the roll gap. to zero at the neutral plane and then a parabolic, tending to linear, increase of it, towards the exit plane, were predicted, in contradiction to what is up-to-now accepted in the praxis, that the friction coefficient remains constant during hot rolling.