We consider evolutionary PDE inclusions of the form

where \(\mathscr {R}_1\) is a positively 1-homogeneous rate-independent dissipation potential and \(W_0\) is a (generally) non-convex energy density. This work constructs solutions to the above system in the slow-loading limit \(\lambda \downarrow 0\). Our solutions have more regularity both in space and time than those that have been obtained with other approaches. On the “slow” time scale we see strong solutions to a purely rate-independent evolution. Over the jumps, we obtain a detailed description of the behavior of the solution and we resolve the jump transients at a “fast” time scale, where the original rate-dependent evolution is still visible. Crucially, every jump transient splits into a (possibly countable) number of rate-dependent evolutions, for which the energy dissipation can be explicitly computed. This, in particular, yields a global energy equality for the whole evolution process. It also turns out that there is a canonical slow time scale that avoids intermediate-scale effects, where movement occurs in a mixed rate-dependent/rate-independent way. In this way, we obtain precise information on the impact of the approximation on the constructed solution. Our results are illustrated by examples, which elucidate the effects that can occur.

我们考虑以下形式的演化PDE包含

其中\（\ mathscr {R} _1 \）是一个与速率无关的正1均匀耗散电势，而\（W_0 \）是一个（通常）非凸的能量密度。这项工作在缓慢加载限制\（\ lambda \ downarrow 0 \）中构造了上述系统的解决方案。与其他方法相比，我们的解决方案在空间和时间上都更具规律性。在“缓慢的”时间尺度上，我们看到了一种纯粹的与速率无关的演进的强大解决方案。在跃迁上，我们获得了解决方案行为的详细描述，并在“快速”时间尺度上解决了跃迁瞬变，在原始时间范围内，速率相关的演化仍然可见。至关重要的是，每个跃变瞬变都会分解为（可能是可计数的）多个速率相关的演化，为此可以显式计算出能量耗散。特别是，这在整个进化过程中产生了全球能源平等。结果还表明，存在一种规范的慢时标，可以避免中间标度的影响，在这种情况下，运动以依赖于速率/独立于速率的混合方式发生。通过这种方式，我们获得有关近似对构造解决方案影响的精确信息。实例说明了我们的结果，阐明了可能发生的影响。