| Adiabatic shear bands play very important roles in the expanding fracture of the explosively driven cylinder and in the collapse of the thick-walled cylinder. Shear bands are the precursors of the fractures. The cracks always propagate along the developed shear bands. In this paper, to simulate the self-organization of multiple shear bands in the expanding cylinder and thick-walled cylinder, a multi-stage probabilistic constitutive model is proposed. The emergences of shear bands are always accompanied by high strain-rate deformation, and most of the metals exhibit a distinct strain rate effect, so a stable and precise algorithm for viscoplastic constitutive relationship is needed to model the behavior of shear bands. Based on the generalized Drucker postulate, an iterative algorithm is proposed.The main contents of the dissertation are as follows:1) Various defects exist inherently within polycrystalline ductile metals, such as micro-cracks, micro-holes, grain boundaries, precipitates and dislocations, etc. These defects can change the local yield stress. Some types of defects, such as microcracks and micro-holes, may reduce the local yield stress. Some types of defects, such as precipitates and dislocations, may improve the local yield stress. The yield stress distribution in a material is non-uniform, but the inhomogeneity of the yield stress is hardly considered in a macroscopic constitutive relationship. The perturbation sources play vital roles in emergence of a shear band, because shear bands are the result of deformation instability. So to model the initiation of multiple shear bands, the inhomogeneity of the yield stress should be considered in the constitutive relationship. A probability factor is introduced into the macroscopic constitutive relationship to describe the inhomogeneity. The probability factor is assumed to obey a Gaussian distribution in the material. The formation of an adiabatic shear band experiences three stages:stable plastic flow, nucleation and a fluid-like stage. The thermal conduction between the material and its environment can be neglected while material suffers high strain-rate deformation, so a modified adiabatic J-C constitutive relationship can be used to describe the mechanical behavior of material in stable plastic flow stage. Due to the nucleation and growth of micro-voids, the damage softening is becoming increasingly important in nucleation stage. The damage softening relies on the stored energy of cold work, and the temperature softening relies on the heat energy, yet the stored energy of cold work and heat energy are both converted from the plastic work. So the effect of damage and temperature softening can be described by a function of plastic strain. A viscous fluid constitutive relationship is used to describe the behavior of material in fluid-like stage. Two criteria are needed in the multi-stage model:one is the instability criterion, which is the transition point from the stable plastic flow stage to nucleation stage; the other is the stress collapse criterion, which is the transition point from the nucleation stage to fluid-like stage. Generally speaking, higher strain rates prompt material to lose instability and fail at lower levels of plastic strain, so strain-rate dependent criteria are adopted.2) The radial return algorithm, which is widely adopted in most of impact dynamic software, is proved to be correct only while (a) the yield criterion is independent on the hydrostatic pressure and (b) the elastic deformation is greatly less than the plastic deformation. The elastic deformation is ignored in the radial return algorithm, so it is just an approximation method. To avoid the errors of the approximation, a new algorithm was proposed by Li et al.[Y.C.Li, et al., Thermo-viscoplastic constitutive relation of damaged materials with application, Explosion and Shock Waves,2004,24(4):289-298.], which was called as explicit precise algorithm. It is found that this algorithm is not absolutely stable. A necessary but not sufficient condition for the numerical stability is deduced. Based on the generalized Drucker postulate, an iterative algorithm was presented. Both of the theoretical and numerical results showed that the iterative algorithm is precise and unconditionally stable. The above algorithms are only applicable for yield criterion which is independent on the hydrostatic pressure, but the hydrostatic pressure has a strong influence on the yield criterion in concrete, soil and metals with high pressure. An iterative algorithm for pressure dependent yield criterion is proposed and is successfully applied in the numerical example of concrete.3) The explosively induced fracture of metallic cylinder under different detonating methods was simulated. The results show that different detonating methods lead to significant differences on the pressure, impulse and acting history of the cylindrical shell. Even in the same detonation method, the load history in different axial locations are quite different. Therefore, in the theoretical analysis and numerical simulation, the influences of detonation method and axial position must be considered. The proposed multi-stage probabilistic constitutive relationship and iterative algorithm are applied to simulate the expanding fracture process of45steel cylindrical shell. Based on the comparison of the simulation results and experimental observation, the failure mode of cylindrical shell, the velocity and fragmentation fragments size were analyzed separately,(a) Failure mode:In the fracture process, the shear bands initiate near the inner surface of the cylinder and propagate outwardly along the maximum shear stress direction. After the shear bands arrive at the outer surface of the cylinder, the slippage continue to develop inside the shear bands, the cracks initiate at the outer surface and propagate inward along the developed shear bands,(b) When the shock wave arrives at the outer surface of the cylinder for the first time, the velocity exhibits a high increase instantaneously. Then the velocity fluctuates for several times. These fluctuations mean that the cylindrical wave propagating between the inner and outer surface can’t be neglected although the thickness of the cylinder is very thin. The simulation shows that the zero hoop stress circle assumed by Taylor isn’t exist. Under the following loading of the explosive products, the cylinder is accelerated gradually and reaches the limit,(c) The thickness and width of the recovered fragments are measured separately. The amount of the recovered fragments is so little that the uncertainty of the measured data is very large. To improve the reliability of the experimental results, the photos taken by high speed camera are also used to analyze the width of the fragments. The model developed by Grady and Kipp predicts the width of the fragments accurately, this means that the assumption of the G-K model is right and can be used to estimate the fragment size in the present problem. The simulation results show that after the shear bands arrive at the outer surface of the cylinder, the thickness of the fragments doesn’t decrease any more. This means that the plastic deformation are concentrated inside the shear band, the material outside the shear band is unloaded.4) The proposed multi-stage probabilistic constitutive relationship and iterative algorithm are applied to simulate the fracture process of thick-walled cylinder. The simulation successfully reproduces the fracture process of thick-walled cylinder. The obtained patterns of multiple shear bands in different stages agree well with the experimental results. The collapse velocity, shear band spacing and propagating velocity of shear band are analyzed separately. By combining with the simulation results, the importance of perturbation growth stage is emphasized at last. |
PDF Full Text Request