Dynamic Modelling of Some Nonlinear Materials

A nonlinear layer can be a model for a cloud of gas bubbles in a liquid, a crack or split plane in a solid, or contact between two tighted surfaces. Solutions were derived for media under strong load. Numerical calculations based on Preisach-Mayergoyz space description (nonlinear stress-strain relationships typical for solids containing mesoscopic inhomogeneities or defects) have given results like those obtained from LISA (see results by Delsanto). Non-uniform massdistribution of grains immersed into a vibrating fluid create internal forces which is responsible for generation of higher harmonics. Tests on slow dynamics were performed. PULSE PROPAGATION THROUGH NONLINEAR LAYERS The problem of normal incidence of a plane wave on a layer is one of the simplest and most important problems in the acoustics of layered media. It attracts considerable interest for two reasons. First, it is rather simple, and one can look forward to obtaining the solution, which must be in analytical form. Second, the layer can serve as a model of a real nonlinear inclusion such as a bubble cloud in a liquid, or a region inside a solid with a high content of defects, or a resonant cavity in concrete, or a geological structure. The statement of the problem is a plane layer with density ρ0 and sound velocity c0 is plaved between x=0 and x=h. This layer is surrounded by another medium with density ρ1 and sound velocity c1. In the figure below is shown the additional nonlinear response from a monopolar rectangular incident pulse, the parameter a=2c0 t0ρ0/(c1ρ1h). [ 1] In next figure is shown the total (nonlinear) response from an incident negative δ-pulse, the parameter a=2c0 t0ρ0/(c1ρ1h) (the same as above). In this case the nonlinear dependence between density and pressure is ρ(p) = p*/c0 2 ln(1+p/p*). It is seen that the normalized pressure p/p* can not be less than –1 which defines the value of p*. This specific case has a analytical solution: p/p* = [(1-exp(-ap0/p*))exp(-at/t0)]/[1-(1-exp(-ap0/p*))exp(-at/t0)]. NUMERICAL MODELLING OF MESOSCOPIC INHOMOGENEOUS MATERIALS The P-M space phenomenological model [2], describes an assemblage of elastically elements called hysteretic mesoscopic units (HMU) correspond to the elastic bond system in NME materials. The elastic behaviour of these units are compared to atomic elasticity, very complex and difficult to explain. But P-M space describes an assembly containing many units and therefore some simplification can be done about the behaviour of these units. The element will close and open at different lengths depending on if stress is decreasing or increasing. Comparing the stresses when the element is loaded and unloaded in a graph, one gets the P-M space graph. The structure in a material can be considered in different levels. For instance there is an atomic structure and a crystal structure. To calculate the mechanical behaviour of a material is extremely time consuming when considering such small scales as the atomic structure level. The size of the calculation would be enormous and there would also be problems with the definition of several different forces that acts on the atomic structure. But there is also a structure considering the grains in the material, which is called mesoscopic. These grains in the material are defined by the arrangement of the atoms. The atomic arrangement will be exactly the same in one specific grain. But the orientation of the atoms in adjoining grains is different. Because of the difference of the behaviour between the grain and the grain boundary it is of interest to control the size and the density of the grains. Change these parameters and you get a different material property. Reducing the size of the grain there will be more grains and of course more boundary that obstructs the movement of a dislocation. Which leads to an increasing strength of the material. To be able to calculate the wave propagation in a NME material it is necessary to make some assumptions considering the behaviour of the grains. They are: the Young’s modulus is constant for all the grains; deformations are only considered in one dimension, which means that the phenomena of lateral contraction will not be considered; all the grains have the same width; and all the grains are subjected to the same stress at the same time. [3]. The most complex part is to describe the bond system. Many factors influence the behaviour of the bond system and it is impossible to pay attention to every one of these factors. It would demand an enormous amount of computer capacity to be able to consider every influence on the grains [4]. Because of the small forces and the large amount of data that have to be considered, the internal forces and microstructure are assumed to not influence the behaviour of the bond system. The external conditions are also considered to not change the behaviour. The model is based on the macrostructure and the bond system will be influenced from forces that act in a macrostructure. To be able to explain the behaviour of a material one have to explain, or in some way determine, the relationship between external forces and the internal behaviour of the material. The most common and satisfying way of explaining the behaviour of a material is to consider the relation between stress and strain. As has been mentioned previously the grains will be modelled as perfectly elastic springs, which means that the nonlinear mesoscopic elastic behaviour of the model will be introduced in the model through the bond system. Figure 1. How the length of interstice i depends on the traction force F. F1 (i) F2 (i) llow (i) lupp (i)