On the energy conservation and critical velocities for the propagation of a "steady-shock" wave in a bar made of cellular material

The propagation of shock waves in a cellular bar is systematically studied in the framework of continuum solids by adopting two idealized material models, viz. the dynamic rigid, perfectly plastic, locking （D-R-PP-L） model and the dynamic rigid, linear hardening plastic, locking （D-R-LHP-L） model, both considering the effects of strain-rate on the material properties. The shock wave speed relevant to these two models is derived. Consider the case of a bar made of one of such material with initial length L 0 and initial velocity v i impinging onto a rigid target. The variations of the stress, strain, particle velocity, specific internal energy across the shock wave and the cease distance of shock wave are all determined analytically. In particular the ＂energy conservation condition＂ and the ＂kinematic existence condition＂ as proposed by Tan et al. （2005） is re-examined, showing that the ＂energy conservation condition＂ and the consequent ＂critical velocity＂, i.e. the shock can only be generated and sustained in R-PP-L bars when the impact velocity is above this critical velocity, is incorrect. Instead, with elastic deformation, strain-hardening and strain-rate sensitivity of the cellular materials being considered, it is appropriate to redefine a first and a second critical impact velocity for the existence and propagation of shock waves in cellular solids. Starting from the basic relations for shock wave propagating in D-R-LHP-L cellular materials, a new method for inversely determining the dynamic stress-strain curve for cellular materials is proposed. By using e.g. a combination of Taylor bar and Hopkinson pressure bar impact experimental technique, the dynamic stress-strain curve of aluminum foam could bedetermined. Finally, it is demonstrated that this new formulation of shock theory in this one-dimensional stress state can be generalized to shocks in a one-dimensional strain state, i.e. for the case of plate impact on cellular materials, by simply making proper replacements of the elastic and plastic constants.

Exact thickness-shear resonance frequency of electroded piezoelectric crystal plates

The determination of the precise thickness-shear frequency of electroded crystal plates has practical importance in quartz crystal resonator design and fabrication, especially when the high fundamental thickness-shear frequency has reduced the crystal plate thickness to such a degree that proper consideration of the effect of electrodes is very important. The electrodes effect as mass loading in the estimation of the resonance frequency has to be modified to consider the stiffness of electrodes, as the relative strength is increasingly noticeable. By following a known procedure in the determination of the thickness-shear frequency of an infinite AT-cut crystal plate, frequency equations of crystal plate without and with piezoelectric effect are obtained in terms of elastic constants and the electrode material density. After solving these equations for the usual design parameters of crystal resonators, the design process can be optimized to pinpoint the precise configuration to avoid time-consuming trial and reduction steps. Since these equations and solutions are presented for widely used materials and parameters, they can be easily integrated into the existing crystal resonator design and manufacturing processes.

Spall Strength of Resistance Spot Weld for QP Steel

The spall tests under the plane tensile pulses for resistance spot weld （RSW） of QP980 steel are performed by using a gun system. The velocity histories of free surfaces of the RSWare measured with the laser velocity interferometer system for any reflector. The recovered specimens are investigated with an Olympus GX71 metallographic microscope and a scanning electron microscope （SEM）. The measured velocity histories are explained and used to evaluate the tension stresses in the RSW applying the characteristic theory and the assumption of Gathers. The spall strength （1977 2784MPa） of the RSW for 0,P980 steel is determined based on the measured and simulated velocity histories. The spall mechanism of the RSW is brittle fracture in view of the SEM investigation of the recovered specimen. The micrographs of the as-received QP980 steel, the initial and recovered RSW of this steel for the spall test are compared to reveal the microstructure evolution during the welding and spall process. It is indicated that during the welding thermal cycle, the local martensitic phase transformation is dependent on the location within the fusion zone and the heat affected zone. It is presented that the transformation at high strain rate may be cancelled by other phenomenon while the evolution of weld defects is obvious during the spall process. It may be the stress triaxiality and strain rate effect of the RSW strength or the dynamic load-carrying capacity of the RSW structure that the spall strength of the RSW for QP980 steel is much higher than the uniaxial compression yield strength （1200 MPa） of the rnartensite phase in 0,P980 steel. Due to the weld defects in the center of the I^SW, the spall strength of the RSW should be less than the conventional spall strength or the dynamic load-carrying capacity of condensed structure.

DEVELOPMENT OF EXPERIMENTAL METHODS FOR IMPACT TESTING BY COMBINING HOPKINSON PRESSURE BAR WITH OTHER TECHNIQUES

The split Hopkinson pressure bar（SHPB） technique and the wave propagation inverse analysis（WPIA） technique are both extensively used to experimentally investigate the impact behavior of materials, although neither of them alone provides a fully satisfactory analysis. In the present paper, attention is given to new experimental techniques by incorporating a damagemodified constitutive model into the SHPB technique and combining the Hopkinson pressure bar（HPB） technique with WPIA. First, to distinguish the response due to dynamic constitutive behavior and the response due to dynamic damage evolution, the SHPB method incorporating a damage-modified constitutive model is developed, including an explicit damage-modified Zhu–Wang–Tang model and an implicit damage-modified constitutive model. Second, when the SHPB results become invalid, a method of combining new Lagrange inverse analyses with the HPB technique is developed, including cases of the HPB arranged in front of a long specimen and behind the specimen. As examples of these new methods, typical results are given for nonlinear viscoelastic polymers and concretes considering damage evolution, a super-elastic Ti–Ni alloy with phase transformation and an aluminum foam with shock waves propagating within it.