Abstract A numerical method for nonlinear steady-state wave scattering due to a crack with contact acoustic nonlinearity (CAN) is developed in this article. The in-plane scattering problem is addressed here. The two systems studied are composed of an unbounded elastic solid that includes an interior crack and an elastic half plane that includes a surface-breaking crack. A time-harmonic plane wave is incident, and clapping and rubbing motions on the crack faces are induced as nonlinear phenomena. A harmonic balance-boundary element method (HB-BEM) is utilized to solve the nonlinear steady-state problem. The system of equations obtained by the HB-BEM consists of an elastostatic boundary integral equation (BIE) and frequency-domain elastodynamic BIEs of several different frequencies, which are coupled by nonlinear boundary conditions.To solve the system numerically, the nonlinear boundary conditions are regularized. In the regularized boundary conditions, traction is expressed by a smooth single-valued function with respect to crack-opening displacement. The solution of the steady-state system is tracked with a parameter continuation method to investigate bifurcations. The parameters that vary in the continuation method are the incident frequency and static field variables. To determine the stability of the steady-state solutions, a stability analysis method based on Hill’s infinite determinant is proposed. The stable solutions obtained by the proposed method agree with the conventional time-domain solutions after enough time has elapsed. A 1 / 2 -order subharmonic resonance is observed with bifurcations for particular parameters. The interaction between the local resonant properties due to the crack and the CAN causes subharmonic resonance.

中文翻译：

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具有接触声非线性的内部和表面破裂裂纹稳态波散射的谐波平衡边界元和延拓方法

摘要 本文开发了一种具有接触声学非线性（CAN）的裂纹引起的非线性稳态波散射的数值方法。此处解决了面内散射问题。研究的两个系统由包含内部裂纹的无界弹性实体和包含表面断裂裂纹的弹性半平面组成。时谐平面波入射，裂纹面上的拍击和摩擦运动作为非线性现象被诱导。谐波平衡边界元法（HB-BEM）用于解决非线性稳态问题。通过 HB-BEM 获得的方程组由弹性边界积分方程 (BIE) 和几个不同频率的频域弹性动力学 BIE 组成，它们通过非线性边界条件耦合。为了对系统进行数值求解，对非线性边界条件进行了正则化。在正则化边界条件下，牵引力由关于开裂位移的平滑单值函数表示。用参数延拓法跟踪稳态系统的解以研究分岔。延续方法中变化的参数是入射频率和静态场变量。为确定稳态解的稳定性，提出了一种基于希尔无限行列式的稳定性分析方法。经过足够的时间后，通过所提出的方法获得的稳定解与传统的时域解一致。观察到具有特定参数的分叉的 1 / 2 阶次谐波共振。