We consider an infinite chain of particles with nonlinear elastic and dissipative nearest neighbors interactions. Assuming the existence of a traveling front between uniformly compressed (or stretched) states, we obtain jump conditions relating the wave speed and limiting particle velocities to the relative displacements at infinity. Using this result, we characterise compression fronts in chains of touching beads, for viscoelastic contact laws that include a nonlinear elastic force (generalised Hertz contact) and viscous dissipation. We compute compression fronts numerically for the generalised Kuwabara–Kono model in which the viscous contact force is proportional to the derivative of the elastic force, without precompression of the chain. Steady fronts are obtained both as the end result of the compression of one end of the chain and using a shooting method which provides numerically exact traveling waves. Depending on the magnitudes of contact damping and strain applied downstream, we obtain either overdamped (monotonic) or underdamped (oscillatory) compression fronts. To explain this transition and approximate the front profiles, we consider a continuum limit valid when the exponent of the contact nonlinearity is close to (and above) unity. Using multiscale expansions, we formally derive two different amplitude equations for long waves, a Burgers equation with logarithmic nonlinearity, and a logarithmic Korteweg–de Vries (KdV)–Burgers equation for small contact damping. Both models possess traveling front solutions that are in good agreement with the front profiles computed numerically in the granular chain. The analysis of the logarithmic KdV–Burgers equation allows one to approximate the critical damping corresponding to the transition from underdamped to overdamped fronts.

我们考虑具有非线性弹性和耗散最近邻相互作用的无穷粒子链。假设在均匀压缩（或拉伸）状态之间存在行进前沿，我们获得了将波速和极限质点速度与无穷远处的相对位移相关的跳跃条件。使用此结果，我们可以描述接触珠链中的压缩前沿，其粘弹性接触定律包括非线性弹力（广义赫兹接触）和粘性耗散。我们为广义的Kuwabara-Kono模型计算了压缩前沿，其中粘性接触力与弹性力的导数成比例，而无需对链进行预压缩。作为链条一端压缩的最终结果，以及使用提供数值精确的行波的射击方法，可以获得稳定的前锋。根据下游施加的接触阻尼和应变的大小，我们可以得出过阻尼（单调）或欠阻尼（振荡）压缩前沿。为了解释这种过渡并近似前轮廓，我们认为连续非线性极限在接触非线性指数接近（或高于）单位时有效。使用多尺度展开，我们正式得出了两个不同的长波振幅方程，一个具有对数非线性的Burgers方程，以及一个用于小接触阻尼的对数Korteweg-de Vries（KdV）-Burgers方程。两种模型都具有行进式前端解，这与颗粒链中数值计算出的前端轮廓非常吻合。对数KdV-Burgers方程的分析使人们可以近似估计临界阻尼，该临界阻尼对应于从欠阻尼前缘到过度阻尼前缘的过渡。