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For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

In this paper, we study the existence and uniqueness of almost automorphic solutions for non-autonomous linear and nonlinear differential equations in a Banach space, when the linear equation admits an exponential dichotomy.

Micromechanics-based facture models have been proved to predict ductile fracture in steel structures with good accuracy. The stress modified critical strain (SMCS) model and the void growth model (VGM) are suitable to predict fracture initiation under monotonic loading. In order to calibrate the parameters in these models for the largely used Q345 steel in China, material tests, scanning electron microscope tests and finite element analyses were conducted. The test specimens were made from base metal, deposit metal and heat affected zone of Q345 steel to investigate crack initiation in welded connections. The results of this paper can be applied to predict ductile fracture in welded steel connections under monotonic loading.