Analysis of shear bands in a dynamically loaded viscoplastic cylinder containing two rigid inclusions

Summary We study plane strain thermomechanical deformations of a hollow circular cylinder containing two rigid non-heat-conducting ellipsoidal inclusions placed on a radial line symmetrically with respect to the center. These inclusions can be viewed as precipitates or second phase particles in an alloy. The material of the cylinder is presumed to exhibit thermal softening, but strain and strain-rate hardening. The impact load applied on the inner surface of the cylinder is modeled by prescribing a radial velocity and zero tangential tractions at material particles situated on the inner surface. Rigid body motion of the inclusion is considered and no slip condition between the inclusion and the cylinder material is imposed.It is found that shear bands initiate from points adjacent to inclusion tips near the inner surface of the cylinder and propagate toward this surface. At inclusion tips near the outer surface of the cylinder, the maximum principal logarithmic strain and the temperature are high and the effective stress is low, but severe deformations there do not propagate outward.     

Initiation and growth of microfractures along adiabatic shear bands in Ti–6AI–4V

Abstract

An experimental study has been made of the manner in which microfractures initiate and grow along adiabatic shear bands formed in the titanium alloy Ti–6AI–4V by the normal impact of hard steel spheres at velocities up to 340 m s^?1. It is suggested that a critical shear strain must be exceeded along the shear bands for microvoids to nucleate, or to cause significant local thermal softening in the bands, leading to the formation of single voids or arrays of voids and smooth-sided cracks when the stress state became predominantly tensile. The final shape of the micro fractures within the shear bands and the morphology of the resultant fracture surfaces are explained in terms of the density of void nucleation sites and the tensile-stress state across the shear bands.

MST/179     

Dynamic shear band development in a thermally softening bimetallic body containing two voids

Summary We study the development of shear bands in a thermally softening viscoplastic prismatic body of square cross-section and containing two symmetrically placed thin layers of a different viscoplastic material and two elliptical voids with their major axes aligned along the vertical centroidal axis of the cross-section. One tip of each elliptical void is abutting the common interface between the layer and the matrix material. Two cases, i.e., when the yield stress of the material of the thin layer in a quasistatic simple compression test equals either five times or one-fifth that of the matrix material are studied. The body is deformed in plane strain compression at an average strain-rate of 5,000 sec^–1, and the deformations are assumed to be symmetrical about the centroidal axes.It is found that in each case shear bands initiate from points on the vertical traction free surfaces where the layer and the matrix materials meet. These bands propagate horizontally into the layer when it is made of a softer material and into the matrix along lines making an angle of ±45° with the vertical when the layer material is harder. In the former case, the band in the layer near the upper matrix/layer interface bifurcates into two bands, one propagating horizontally into the layer and then into other into the matrix material along the direction of the maximum shear stress. The band in the layer near the lower matrix/layer interface propagates horizontally first into the layer and then into the matrix material along the direction of the maximum shear stress. Irrespective of the value of the yield stress for the layer material, a band also initiates from the void tip abutting the layer/matrix interface. This band propagates initially along the layer/matrix interface and then into the matrix material along a line making an angle of approximately 45° with the vertical.     

Development of Plastic Bands in the Process of Shear near the Tips of a Rectangular Slot

We study the development of rectilinear plastic bands from the tips of a rectangular slot of constant width in a perfectly elastoplastic body in shear. The plastic bands are formed along the bisectrices of the angles of the slot but, as the load increases, their orientation gradually changes and, finally, they become oriented along the axis of the slot. The slots of greater width are characterized by larger ranges of loads in which the bands propagate almost rectilinearly.     

Studies on the growth of voids in amorphous glassy polymers

Numerical studies are presented of the localized deformations around voids in amorphous glassy polymers. This problem is relevant for polymer–rubber blends once cavitation has taken place inside the rubber particles. The studies are based on detailed finite element analyses of axisymmetric or planar cell models, featuring large local strains and recent material models that describe time-dependent yield, followed by intrinsic softening and subsequent strain hardening due to molecular orientation. The results show that plasticity around the void occurs by a combination of two types of shear bands, which we refer to as wing and dog-ear bands, respectively. Growth of the void occurs by propagation of the shear bands, which is driven by orientational hardening. Also discussed is the evolution of the local hydrostatic stress distribution between voids during growth, in view of possible craze initiation. © 1998 Kluwer Academic Publishers     

INFLUENCE OF SECONDARY VOID DAMAGE IN THE MATRIX MATERIAL AROUND VOIDS

Abstract— The mechanism of ductile damage caused by secondary void damage in the matrix around primary voids is studied by large strain, finite element analysis. A cylinder embedding an initially spherical void, a plane stress cell with a circular void and plane strain cell with a cylindrical or a flat void are analysed under different loading conditions. Secondary voids of smaller scale size nucleate in the strain hardening matrix, according to the requirements of some stress/strain criteria. Their growth and coalescence, handled by the empty element technique, demonstrate distinct mechanisms of damage as circumstances change. The macroscopic stress-strain curves are decomposed and illustrated in the form of the deviatoric and the volumetric parts. Concerning the stress response and the void growth prediction, comparisons are made between the present numerical results and those of previous authors. It is shown that loading condition, void growth history and void shape effect incorporated with the interaction between two generations of voids should be accounted for besides the void volume fraction.     

Void evolution and coalescence in porous ductile materials in simple shear

The fracture of porous ductile materials subjected to simple shear loading is numerically investigated using three-dimensional unit cells containing voids of various shapes and lengths of the inter-void ligament (void spacing). In shear loading, the porosity reduction is minimal while the void rotates and elongates within the shear band. The strain at coalescence was revealed to be strongly related to the initial void spacing and void shape. It is observed that a transitional spacing ratio for shear coalescence exists with coalescence being unlikely at spacing ratios lower than 0.35. Initially prolate voids are particularly prone to shear coalescence while initially oblate (flat) voids are most resistant to shear failure. The cell geometry becomes sensitive to shear coalescence for increasing void aspect and spacing ratios. In addition, the macroscopic shear stress response becomes independent of the void shape at high spacing ratios while showing a weak dependence on the void shape when the voids are far apart.     

On the dependence of crack surface morphology and energy dissipation on microstructure in ductile plate tearing

The two distinct tearing mechanisms observed in ductile metal plates are the void-by-void advance of the crack tip, and the simultaneous interaction of multiple voids on the plane ahead of the crack tip. Void-by-void crack advance, which leads to a cup-cup crack surface morphology, is the dominant mechanism if the plate contains a low number of small void nucleation sites (i.e., second phase particles). Conversely, a large number and/or size of nucleation sites trigger the simultaneous interaction of multiple voids resulting in a slanted crack. The present work aims to provide further insight into the parameters controlling the mechanisms and energy dissipation of plate tearing by focusing on the shape and, thereby, the orientation of the nucleation sites. The study uses a two-dimensional plane strain finite element domain to model the cross section of a plate, subject to mode I tearing, with discretely modeled, randomly distributed, finite-sized elliptic void nucleation sites. The developed finite element setup can capture the dependence of the crack surface morphology on the microstructure of the plate. The simulation results confirm that cup-cup crack propagation develops by intense plastic straining throughout the thinning region of the plate. Conversely, slanted and cup-cone cracks propagate in thin localized shear deformation bands. The energy dissipation is, therefore, greater for cup-cup cracks. The study demonstrates that the damage-related microstructure has a significant role in determining the overall hardening capacity of a plate, which in turn dictates the tearing mode and energy.

     

Some observations of coarse shear bands in polystyrene

On the surface parallel to the shear direction, coarse bands appear as ridges, usually of isosceles cross-section with a height-to-base ratio of about 1/10. They split and join with thickness conservation but a corresponding change of cross-section. The shear strain along the same band is almost a constant with a most probable value of 1.8 but a spread between 1 and 3 among different bands. The bands terminate by reducing the shear strain to zero or by converting themselves to a bundle of fine bands at obstacles. Upon annealing the height of the band ridge is reduced until it disappears. When a banded surface is polished, the band pattern reappears upon either annealing or recompression. However, in the former case, the bands are valleys with reverse shear strain but in the latter they are ridges with almost no strain. Similar behaviour occurs on the polished cut surfaces in the middle of the specimen after annealing. On the side surface, the bands are wavy and they are steps rather than ridges. They also split and join with thickness conservation and they terminate freely. Based on these characteristics, the propagation of coarse bands can be viewed as the motion of macroscopic Volterra dislocations.     

Ductile shear fracture at the surface of a bent specimen

The development of shear bands at the stretched surface of a bent plate is analysed numerically, based on an approximate continuum model of a ductile porous material. This material model accounts for the nucleation and growth of voids as well as the effect of the yield surface curvature, which is represented by a combination of kinematic hardening and isotropic hardening. An imperfection in the form of an initial surface waviness is assumed, which triggers shear bands at the wave bottoms. The corresponding periodic pattern of shear bands is considered, and the growth of the bands is followed, until shear cracks develop from the void-sheets inside the bands. The delay of localization due to the nonuniform strain field is studied for different versions of the material model. Furthermore, the stability of the uniform growth of several adjacent shear bands is investigated.     

Self-organization behaviors of shear bands in 7075 T73 and annealed aluminum alloy

The self-organization behaviors of multiple adiabatic shear bands (ASBs) in the 7075 T73 aluminum alloy were investigated by means of the thick-walled cylinder (TWC) technique. Shear bands first nucleate at the inner boundary of the aluminum alloy tube and propagate along the maximum shear stress direction in the spiral trajectory. On the cross section of the specimen, shear bands distribute either in the clockwise or the anticlockwise direction. The number of ASBs in the clockwise direction is roughly twice that in the anticlockwise direction. However, the 7075 annealed alloy does not generate any shear band under the same experimental conditions.Numerical simulation with coupled thermo-mechanical analysis was carried out to investigate the evolution mechanism of adiabatic shear bands. Both uniform and non-uniform finite element models were created. The simulation results of the non-uniform model are in better agreement with those of the experiment. In the non-uniform case, the spacing between ASBs is larger than that of the uniform model, and most of the ASBs prefer to propagate in the clockwise direction. For the first time, two types of particles (second phase), hard particles and soft particles, are separately introduced into the metal matrix in the non-uniform model to simulate their effects on the self-organization of ASBs. The soft particles reduce the time required for ASBs nucleation. Stress collapse first occurs at the region where the soft particles are located and most of the ASBs pass through these soft particles. However, ASBs propagate along the paths that are adjacent to the hard particles instead of passing through them. As experimental observations, there is no shear band nucleating in the annealed alloy in simulation. Under the same conditions, the energy barrier for the formation of ASBs in the annealed aluminum alloy is about 2.5 times larger than that in the T73 alloy, which means that the adiabatic shearing is less likely to nucleate in the annealed alloy. This is consistent with the experimental and numerical simulation results.     

Detailed Evolution Mechanism of Interfacial Void Morphology in Diffusion Bonding

Similar diffusion bonding of 1Cr11Ni2W2MoV stainless steel was conducted at different bonding temperatures. The interface characteristics and mechanical properties of joints were examined, and the evolution of interfacial void morphology was analyzed in detail. The results showed that there were four typical interfacial void shapes generating sequentially: the large scraggly voids, penny-shaped voids, ellipse voids and rounded voids. The variation of interfacial void shape was dominated by surface diffusion, while the reduction of void volume was ascribed to the combined effects of plastic flow of materials around voids,interface diffusion and volume diffusion. Owing to the elimination of void from the bonding interface,the sound joint obtained could exhibit nearly full interfacial contact, and present excellent mechanical properties, in which the microhardness and shear strength of joint matched those of base material.     

采用数字图像相关方法的莫来石纤维增强气凝胶复合材料力学试验

基于V型缺口试样双轨剪切法设计了面内剪切试验方案,开展了莫来石纤维增强气凝胶复合材料的室温面内剪切和弯曲性能试验,采用数字图像相关方法对试样表面的位移场和应变场进行测量,并分析了力学行为和破坏模式。结果表明:设计的试验方案可以在测试区域获得均匀的剪切应变场,适用于莫来石纤维增强气凝胶复合材料的面内剪切性能测试。试验获得的面内剪切模量和强度分别为248 MPa和0.95 MPa,弯曲模量和强度分别为294 MPa和2.08 MPa。面内剪切载荷下,试样的裂纹萌生于缺口尖端附近,并沿两缺口连线方向扩展。根据弯曲正应变场的分布特点,发现试样中性层与几何对称面不重合,验证了该材料拉压模量不同的性质。采用数字图像相关方法获得的中性层位置和理论计算值比较接近,相对误差在10%左右。     

On the interaction between two circular voids in a nonlinear elastic solid

Summary We study finite plane strain deformations of an infinite body containing two circular voids and made of a Blatz-Ko material. The body is subjected to either uniform tensile tractions at infinity or a uniform pressure on the void surfaces. In each case, the effect of varying the distance between the void centers on the deformations of the body is analyzed. When the voids are located close to each other, a uniform pressure on the void surfaces results in noncircular deformed voids, and for a fixed value of the pressure, the deformation induced increases as the voids get closer to each other. When the body is subjected to uniform tensile tractions at infinity, say along thex-axis, the voids are deformed into ellipsoids with major axes aligned along thex-axis.     

Effect of defect shape and size on the initiation of adiabatic shear bands

Summary Backman and Finnegan [1] have pointed out that shear bands initiate from a defect such as a second phase particle, microcrack, or a void. The defect has been modeled as a sinusoidal variation in the thickness of the tubular specimens tested in torsion by Chi [2] and Murphy [3]. Here we simulate their torsional tests numerically and consider different shapes and sizes of defects.     

Numerical study on void growth in rate and temperature dependent solids

This paper is concerned about void growth and associated deformation models in porous visco-plastic solids under conditions similar to those found in highly stressed regions ahead of a crack. A plane-strain unit cell containing an initially circular void is examined to simulate the stress states during dynamic fracture of a metal. Two proportional loading rates are prescribed in the two directions of the cell and their ratio is called the “strain biaxiality” expressed in a monotonic relation with stress triaxility. Finite element analysis is performed for the effective stress–strain curves of the porous solids during void growth for a range of initial porosities, strain biaxialities, strain rates and thermal softening coefficients. Numerical results show that the void evolution and the associated non-uniform deformation depend in a complex fashion on these factors. The local zone of high stress concentration which emanates from the void spreads out in the cell to trigger non-uniform deformation and plastic yielding. Subsequently, a small zone with intense plastic strain and heating either expands smoothly near the growing voids or propagates in a specific direction determined by its interaction with the boundary conditions of the cell such as strain biaxility. At low strain biaxiality and for small voids, formation and propagation of zones with intense plastic strain and heating is localized. However, high strain biaxiality leads to rapid uniform expansion of small voids as observed experimentally. It is found that the intense heating zone follows the zone of high plastic strain concentration and diffuses with imposed strain. Thermal softening which reduces the overall stress can be neglected at the early stage of void growth, but it is magnified past the peak stress by accelerating the void growth. But in the long term, the void growth rate is insensitive to thermal softening coefficient. Increasing strain rates can promote void growth and the rate of which tends to be proportional to the eventual strain rate.     

Effects of surface stresses and voids on rayleigh waves in an elastic solid

Rayleigh waves propagating at the plane material boundary of an elastic half-space containing a distribution of voids (vacuous pores) are considered. It is found that the waves are generally dispersive and that the dispersion is caused by both the surface stresses exerted by the boundary and the voids inside. If the body is incompressible, voids have no influence on the motion. In the case of small frequency, the effect of the voids is just to modify the speed of propagation. If the frequency is small and surface stresses are due to a residual surface tension, there exists a critical wavelength at which waves propagate with the speed of pure shear waves and below which the motion is not possible. The critical wave length varies directly with the surface tension and the effect of the voids is to widen the range of wavelengths for which the waves exist.     

Local approach to ductile fracture and dynamic strain aging

Experiments on smooth and notched round specimens on a C–Mn steel used in nuclear industry are performed at different temperatures under quasi-static loadings, revealing dynamic strain aging (DSA). The behavior is highly dependent on temperature and strain rate, and a drop in fracture strain is observed. Fracture surface observation on notched tensile specimens shows classical ductile fracture mechanisms with growth and coalescence of voids. The apparent strain hardening behavior at each temperature and strain rate is taken into account to compute the void growth with the Rice and Tracey model and with a damage law developed from unit cell computations. It is shown that the apparent strain hardening at large strains is of major importance to correctly predict fracture with the Rice and Tracey model, but its influence on the void growth law is of minor importance. In particular, the stress triaxiality ratio within the notch is increased due to the negative strain rate sensitivity. The ductility drop observed in DSA domain is then partly explained, but void nucleation and void growth in presence of strain bands should be included in the fracture modeling of such materials.     

Adiabatic shear bands in α-titanium tube under external explosive loading

The radial pressure explosive experiment is successfully used to investigate the formation of adiabatic shear bands (ASBs) in α-titanium (α-Ti) tube under external explosive loading. The ASBs initiate at the inner surface of α-Ti tube, and most of the shear bands are in spiral form along the cross-section of the tube towards counterclockwise direction. Tip of a shear band propagates along the surface of the maximum shear stress. Four patterns of ASBs such as bifurcation, collection, crossing and N-shape are observed. The developed shear bands are the preferred sites for nucleation, growth and coalescence of microvoids. The nucleation of microvoid is caused by the local hot spots and stress concentration in the shear bands. The coalescence of microvoids forms the crack within ASBs, and when the critical crack length is reached catastrophic fracture occurs.     

Adiabatic shear banding in a bimetallic body

Summary Thermomechanical deformations of a body made of two different materials and under-going simple shearing deformations are studied with the objectives of finding out when and where adiabatic shear bands will initiate and how they will subsequently grow. Each material is modeled as strain and strain-rate hardening but thermally softening. A shear band is presumed to have formed if the introduction of a temperature perturbation centered around the common interface between the two materials results in an eventual localization of the deformation into a region of width considerably smaller than the width of the initial temperature bump. For a fixed set of material properties the effect of the applied overall strain-rate, and for a fixed applied strain-rate the effect of varying the shear modulus, thermal conductivity, and the coefficient of thermal softening of one material relative to the other have been examined. It is found that a shear band forms in the material that softens more rapidly.With 8 Figures