A Dynamic Damage Model for Uniaxial Compressive Response of

A Dynamic Damage Model for Uniaxial Compressive Response of AD90 Alumina ∗ Ren huilan 1 , a and Li Ping 2 , b 1 State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081,China 2 Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, CAEP, Mianyang 621900,China a b key words: dynamic response ; dispersion ; Lagrange Analysis; damage; tensile wing cra
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  A Dynamic Damage Model for Uniaxial Compressive Response of AD90 Alumina ∗   Ren huilan 1 , a  and Li Ping 2 , b   1 State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081,China 2 Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, CAEP, Mianyang 621900,China a  b  key words:  dynamic response ; dispersion ; Lagrange Analysis; damage; tensile wing crack Abstract. The dynamic response of polycrystalline alumina were investigated in the pressure range of 0-13Gpa by planar impact experiments. Manganin gauges were employed to obtain the stress-time histories. From the free surface particle velocity profiles indicate the dispersion of the “plastic” wave for alumina. Using path line principle of Lagrange Analysis the dynamic mechanical  behaviors for alumina under impact loading are analyzed, such as nonlinear, strain rate dependence, dispersion and declination of shock wave in the material. A damage model applicable to ceramics subjected to dynamic compressive loading is developed. The model is based on the damage micromechanics and established on wing crack nucleation and growth. The results of the dynamic damage evolution model are compared to the experimental results and a good correlation is obtained. Introduction Compared to the other materials such as metals, ceramics tend to be light weight, high compression strength and high Hugoniot elastic limit. These characteristics make them well suited for armor applications. The dynamic mechanical response of ceramic materials has been the subject of considerable research in the past decades [1] . The micromechanical mechanics to the field of damage induced by compressive waves are investigated by the recovery of completely intact specimens. The primary mechanism of damage in ceramic is widely reported to be micro-cracking along grain boundaries. The causes of micro-cracking have been discussed , starting with the initial  processing or the thermal anisotropy of the grains and the glass phase and dislocation pileups at a triple point [2] . This objective of this work is to investigate the dynamic response for AD90 alumina below the  pressure of 13Gpa by planar impact experiment. The results show the dispersing characteristic are very clear when impact loading are between Hugoniot elastic limit and 13Gpa from the measured free surface velocity profile. Dynamic behaviors were discussed using Lagrange Analysis. A micromechanical damage model is developed based on the nucleation and growth of cracks. The  prediction results of the dynamic damage evolution model are compared to the experimental results  by Lagrange Analysis and the good correlation is obtained. 1 Plate impact experiments AD90 alumina is composed of 89.9% AL 2 O 3 , 7.8%SiO 2  and 2.2% adhesive. The density is 3.625g/m 3 . The average grain size is 8um. The mechanical behaviors of the AD90 alumina were investigated by performing planar impact experiment using an 100mm diameter single-stage ∗  Funded by China Natural Science Foundation under grant 10272023   Key Engineering Materials Vols. 340-341 (2007) pp. 289-294online at © (2007) Trans Tech Publications, Switzerland  All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without thewritten permission of the publisher: Trans Tech Publications Ltd, Switzerland, (ID:,20:05:35)  gas-gun at Institute of Fluid Physics, CAEP and Mechanics Institute, China Academic Science. As shown in Fig.1, a flyer plate mounted at the head of the projectile, was used to impact the target assembly consisting of four pieces of AD90 sample embedded with the three manganin gauges. The experimental setup was designed so that a planar-parallel compressive shock-wave  propagates through the target and flyer and the compression wave-profiles are measured with little interference from radial reflected waves. Compressive wave make the target reach the Hugoniot state. When the left-propagated compressive wave arrive at the left free surface of flyer, then reflected (release) wave propagates the target to make the sample unloading from the Hugoniot state. Manganin gauges records the stress histories at different locations of target. Typical stress profiles at different locations are shown in Fig.2. The particle velocity-time history of a specular reflecting surface during an impact can be determined using VISAR. All ceramic sample surfaces were lapped for flatness and parallelism. VISAR effectively measures the rear surface particle velocity from which a shock pressure versus time trace can be obtained. Fig.4 shows free-surface velocity traces obtained on samples at 728m/s and 735m/s, using a 3mm thick Cu flyer plate. From Fig.4 it can be seen clearly that the compression wave-profile of alumina is composed of linear elastic wave for lower pressure and dispersing “plastic”wave for higher pressure. When stress is high to the Hugoniot elastic limit(  HEL σ  ),the particle velocity ramp firstly then reach the  peak pressure fast. In fact microstructure of material begin to slide under the confined pressure and Fig.1 Schematic drawing of experimental set-up Fig.4 Rear free-surface velocity profiles for AD90 alumina -   s   t  r  e  s  s   (   G  p  a   ) time( µ s) Fig.2 Original stress history at different Lagrange locations 2u max 2u HEL w f  =735m/sw f  =728m/s    F  r  e  e   S  u  r   f  a  c  e   V  e   l  o  c   i   t  y   /   (  m   /  s   ) Time( µ s) Fig.3 Schematic showing the set-up for VISAR Engineering Plasticity and Its Applications 290  shear stress ,then induce to the nucleation and growth of micro-cracks from the srcinal micro-flaws and micro-pores in the ceramic, which make stress relaxing or free surface velocity ramping. 2 Dynamic mechanical characteristics Lagrange analysis is developed by Flowes at 70’s, which obtained some mechanical information(particle velocity or stress) using gauges embedded at different locations in the material. Some unknown mechanical parameters are derived through three conservations. Because the constitutive model is not assumed during the process of Lagrange analysis, it will reflect the actual mechanical response of material under dynamic loading. The calculations are processed for measured stress-time histories using    path line method of Lagrange analysis. The curves of strain, particle velocity, specific internal energy, density, specific volume, strain rate and some other mechanical parameters changing with time are achieved  as shown in Fig.5 at impact velocity of 634m/s. Fig.6 shows that the stress-strain curves under different impact velocities. Fig.5(a) Particle velocity-time curve   Fig.5(b) Strain rate-time curve   Fig.5(c)Stress-strain curves at different Lagrange locations   Fig.6 Stress-strain curves at different impact velocities   Some dynamic mechanical characteristic for alumina are summarized based on the curves of mechanical parameters changing with time. (1) because of nucleation and growth of the microscopic cracks under impact loading the stress-strain curve represent nonlinear characteristic Key Engineering Materials Vols. 340-341 291   partially.(2) strain rate dependence; at different impact velocities, the stress-strain curve is different as shown in Fig.6. Peak stress is increasing with the impact velocities.(3)dispersion of shock wave; (4) declination of the breadth and width of shock wave is related to the dissipation of energy, which reflects the action of the shock wave and microstructures of materials. 3 Constitutive model Micro-cracks may nucleate either at in homogeneities such as inclusions and reinforcements or at defects such as micro-cracks and pores within the sintered ceramics under quasi-static or dynamic loading. Many experimental research indicated that nucleation, growth and interaction of micro-cracks are the most dominant forms of damage induced the failure under dynamic loading or shock loading. The effect of damage is assumed to be a reduction in the elastic modulus of the material. For uniaxial strain compressive loadings, the constitutive relations of the ceramic is written as follows ε σ   E  D )1(  −=  (1) Differentiating Eq.(1), then the time-dependant increment of stress is given as ()  E E D D σ ε ε ε  = − + && &&  (2) Based on the observation of microscopy examination, it is stochastic for the development of cracks due to the random orientation of the grains and the formation of a second phase at some interfaces. So damage scalar is measured in terms of a dimensionless parameter , where 3  D Na =  (3) Where  N  is the number of cracks per unit volume which is variable ; a is the crack size which is favorable to grow. With the new micro-cracks nucleating and the micro-cracks extending,  D is due to increase in crack size and number of cracks. For isotropic damage,  D varies from 0 to 1, 0 =  D with corresponding to the virgin material without damage, and1 =  D corresponding to the fractured material. Intermediate values of correspond to the partially damage material. In the general, the phases of damage evolution consist of (1)nucleation of micro-cracks with some initial crack density; (2)growth of micro-cracks; (3)coalescence of the micro-cracks at some critical crack size. The cracks grow by increasing compression accompanied by the nucleation of cracks simultaneously. The nucleation of cracks is assumed to satisfy the Weibull distribution m k  ε  =  (4) where , k m is the material parameters. Differentiating Eq.(3) and Eq.(4),the damage rate is obtained as follows 32132 33 m m  D Na Naa km a k aa ε ε ε  − = + = + & & && &  (5) Substituting Eq.(2) into Eq.(5), the damage rate and strain rate are represented by stress rate as follows 1323 3(1)1 m mm km a E k a a D D D km a ε σ ε ε  − + −=− − & &&   (6a) 123 /31 mm  E k aa D km a σ ε ε ε  + +=− − & &&   (6b) Engineering Plasticity and Its Applications 292


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