# Simulation of Three Constitutive Behaviors Based on Nonlinear Ultrasound
## 1 Introduction  
Under cyclic loads or harsh environments, no matter what materials, parts and complete structures will undergo material properties degradation. Continuous degradation leads to predictable and unpredictable problems, sometimes with serious consequences. Therefore, it is necessary to do health testing for structures or materials in practical engineering [1; 2; 3; 4; 5]. The early degradation of the mechanical properties takes up most of the life of the structures. Generally in metals, the damage first appears in the form of dislocation substructures such as veins and persistent slip bands (PSBs), and these accumulate at grain boundaries to produce strain localization and then further produce micro-cracks. This process accounts for 80-90% [6; 7] of the life of the whole structure when the metal material structure is bearing a cyclic load or harsh environment. When the cracks show, the life the of material is almost over, so the testing of the early properties of the materials is particularly important. With the development of ultrasonic non-destructive testing technology, linear ultrasound cannot satisfy the current testing needs, because linear ultrasound technology has no high sensitivity in the early stage of deterioration of material mechanical properties [8]. Studies show that the dislocation substructures and resulting microplastic deformation do not cause large changes in the macroscopic properties such as elastic modulus, sound speed and attenuation of a material [9; 10], thus the changes in the linear ultrasonic quantities are not large enough to be accurately measured with conventional linear ultrasound. The testing method of nonlinear ultrasound has attracted considerable attention in recent years [3; 11; 12; 13; 14; 15; 16]. Ultrasonic waves are highly sensitive to material damage, propagate in all directions,spread quickly and have high penetration into materials [17]. The nonlinear phenomenon manifests itself with second, third and even more harmonic generation: when a sinusoidal ultrasonic wave of finite amplitude at the \(f\) frequency crosses a nonlinear elastic material, harmonics at the \(2f\),\(3f\) (et al.) frequency are generated [18]. In refs [8; 9; 10], experimental results and physical models show that the harmonic amplitude is directly related to material damage. Therefore, it can be expected that the degree of material degradation can be evaluated by measuring the nonlinearity of the ultrasonic wave propagated through the target material. The sources of nonlinearity can be divided into classical acoustic nonlinearity and contact acoustic nonlinearity (CAN). CAN manifests the phenomenon of nonlinearity by introducing a micrometer- or even nanometer-level small crack, which has been done in many simulations [17; 19; 20; 21; 22; 23] and experiments [17; 24; 25; 26]. However, it is known that when the material is under a load or in a severe environment, not only a small area of the material is damaged (e.g., micro-cracks), but also every under-service part of a structure is damaged to different degrees, which is rarely discovered in the current simulation. It is difficult to decide where to set micro-cracks in simulation for the structure under uniform load, but the material or structure must be damaged in service, so it is necessary to study the degradation of the mechanical properties of the material or structure. The work of this manuscript is based on classical acoustic nonlinearity. When the material is undamaged, it complies with the linear constitutive relation. The wave propagation in the material is linear. However, when the material is damaged, the mechanical constitutive behavior no longer obeys the linear relation and the relation is complex and nonlinear [16; 27; 28]. As a result, the distortion of the wave form appears and the harmonics are derived because the constitutive relation is no longer linear. Different materials exhibit different constitutive behaviors when damaged.