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This article presents a semianalytical solution to a half-plane contact problem subjected to an arbitrarily distributed surface traction. The half-plane boundary is treated as a material surface of the Steigmann–Ogden type. Under the assumption of plane strain condition, the problem is formulated by coupling the methods of an Airy stress function and Fourier integral transforms. Stresses and displacements in the form of semi-infinite integrals are derived. A non-classical Flamant solution that is able to simultaneously account for the surface tension, membrane stiffness, and bending rigidity of the half-plane boundary is derived through limit analysis on the half-plane contact problem owing to a uniform surface traction. The fundamental Flamant solution is further integrated for tackling two half-plane contact problems owing to classical contact pressures corresponding to a rigid cylindrical roller and a rigid flat-ended punch. The resultant semi-infinite integrals are integrated by the joint use of the Gauss–Legendre numerical quadrature and the Euler transformation algorithm. Extensive parametric studies are conducted for comparing and contrasting the effects of Gurtin–Murdoch and Steigmann–Ogden surface mechanical models. The major observations and conclusions are two-fold. First, the introduction of either surface mechanical model results in size-dependent elastic fields. Second, the incorporation of the curvature-dependent nature of the half-plane boundary leads to bounded stresses and displacements in the fundamental Flamant solution. This is in contrast to the otherwise singular classical and Gurtin–Murdoch solutions. For all four case studies, the Steigmann–Ogden surface model also results in much smoother displacement and stress variations, indicating the significance of surface bending rigidity in nanoscale contact problems. [ABSTRACT FROM AUTHOR]

In this article, we present a data-driven method for parametric models with noisy observation data. Gaussian process regression based reduced order modeling (GPR-based ROM) can realize fast online predictions without using equations in the offline stage. However, GPR-based ROM does not perform well for complex systems since POD projection are naturally linear. Conditional variational autoencoder (CVAE) can address this issue via nonlinear neural networks but it has more model complexity, which poses challenges for training and tuning hyperparameters. To this end, we propose a framework of CVAE with Gaussian process regression recognition (CVAE-GPRR). The proposed method consists of a recognition model and a likelihood model. In the recognition model, we first extract low-dimensional features from data by POD to filter the redundant information with high frequency. And then a non-parametric model GPR is used to learn the map from parameters to POD latent variables, which can also alleviate the impact of noise. CVAE-GPRR can achieve the similar accuracy to CVAE but with fewer parameters. In the likelihood model, neural networks are used to reconstruct data. Besides the samples of POD latent variables and input parameters, physical variables are also added as the inputs to make predictions in the whole physical space. This cannot be achieved by either GPR-based ROM or CVAE. Moreover, the numerical results show that CVAE-GPRR may alleviate the overfitting issue in CVAE. [ABSTRACT FROM AUTHOR]

Elasticity is a fundamental model in mechanics and material sciences. In the article, we present an ensemble variable-separated multiscale method for elasticity problems in random media. A variable-separated method is utilized to provide a low-rank approximation for random fields. The variable-separated method is integrated with the Generalized Multiscale Finite Element Method (GMsFEM) to obtain an efficient reduced model. To this end, a few local problems are solved for snapshots to construct the multiscale basis functions in GMsFEM. An ensemble method is used to construct stochastic basis functions shared by a series of elasticity equations, and a residual decomposition is adopted to calculate the deterministic physical components for all ensemble members. The ensemble variable-separation is used to solve the local problems of the multiscale basis functions and achieves efficient online-computation for the basis. Then, under the Galerkin projection of the generalized finite element method, the solutions of stochastic elasticity problems are projected onto the low-dimensional space spanned by the online multiscale basis functions. This renders an effective coarse model. The ensemble variable-separated multiscale method is applied to steady-state elastic deformation and elastic wave propagation in random media. A few numerical results are carried out to demonstrate the accuracy and efficiency of the presented method. [ABSTRACT FROM AUTHOR]