In this paper, a new data-driven multiscale material modeling method, which we refer to as deep material network, is developed based on mechanistic homogenization theory of representative volume element (RVE) and advanced machine learning techniques. We propose to use a collection of connected mechanistic building blocks with analytical homogenization solutions which avoids the loss of essential physics in generic neural networks, and this concept is demonstrated for 2-dimensional RVE problems and network depth up to 7. Based on linear elastic RVE data from offline direct numerical simulations, the material network can be effectively trained using stochastic gradient descent with backpropagation algorithm, further enhanced by model compression methods. Importantly, the trained network is valid for any local material laws without the need for additional calibration or micromechanics assumption. Its extrapolations to unknown material and loading spaces for a wide range of problems are validated through numerical experiments, including linear elasticity with high contrast of phase properties, nonlinear history-dependent plasticity and finitestrain hyperelasticity under large deformations.By discovering a proper topological representation of RVE with fewer degrees of freedom, this intelligent material model is believed to open new possibilities of high-fidelity efficient concurrent simulations for a largescale heterogeneous structure. It also provides a mechanistic understanding of structure-property relations across material length scales and enables the development of parameterized microstructural database for material design and manufacturing. cost and accuracy. Analytical micromechanics methods [9,10,1,11,12,13] can be regarded as one type of reduced-order models with high efficiency. However, due to a loss of detailed physics in the microscale, they normally lose accuracy or require extensive model calibrations when irregular complex morphologies, nonlinear history-dependent properties or large deformations are presented. For heterogeneous hyperelastic materials, manifold-learning methods like isomap are used for nonlinear dimensionality reduction of microscopic strain fields [14]. The model reduction of history-dependent plastic materials can be more complex and challenging. Two examples are non-uniform transformation field analysis (NTFA) [15,16] and variants of the principle component analysis [17] or proper orthogonal decomposition (POD) [18,19,20]. However, they usually require extensive a priori simulations for interpolating nonlinear responses, and their extrapolation capability for new material inputs is usually limited, Recently, the self-consistent clustering analysis (SCA) [21,22] has demonstrated a powerful trade-off between accuracy and efficiency in predicting smallstrain elasto-plastic behavior though clustering techniques, and it only requires linear elastic simulations in the offline stage.Meanwhile, current advanced machine learning models (e.g. artificial neural networks and deep learning) hav...

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A three-dimensional microstructure-based finite element framework is presented for modeling the mechanical response of rubber composites in the microscopic level. This framework introduces a novel finite element formulation, the meshfree-enriched FEM, to overcome the volumetric locking and pressure oscillation problems that normally arise in the numerical simulation of rubber composites using conventional displacement-based FEM. The three-dimensional meshfree-enriched FEM is composed of five-noded tetrahedral elements with a volume-weighted smoothing of deformation gradient between neighboring elements. The L 2 -orthogonality property of the smoothing operator enables the employed Hu-Washizu-de Veubeke functional to be degenerated to an assumed strain method, which leads to a displacement-based formulation that is easily incorporated with the periodic boundary conditions imposed on the unit cell. Two numerical examples are analyzed to demonstrate the effectiveness of the proposed approach.On the other hand, the rubber composites in 3D microscopic level often experience excessive strains, which could be in the order of several hundred percent and are difficult to be simulated using the standard FEM. Furthermore, the characteristic of incompressibility in rubber has presented another difficulty in the numerical simulation. The so-called volumetric locking is due to the over-constrained nature in the low-order displacement-based FEM and special numerical formulation has to be employed. Many approaches have been developed to overcome this difficulty. Among some popular methods for hyperelasticity materials are the methods of mixed formulation [4] and enhanced assumed strain formulation [5]. The u/p mixed formulation requires a stable inf-sup pair of spaces for the displacement and pressure, and usually a high-order finite element is recommended. For mixed formulation with continuous pressure approach, additional attention has to be paid to enforce the periodic boundary condition at the pressure nodes on RVE. To counteract the lack of inf-sup stability, low-order pairs are usually supplemented by stabilization or postprocessing procedures [6] that remove spurious pressure modes. A 3D stabilized Petrov-Galerkin formulation using equal-order interpolations for displacement and pressure was adopted by Matouš and Geubelle [7] to model large deformation and particle debonding of rubber composites involving uniaxial loading of unit cell in meso-scale. The basic idea of enhanced strain formulation consists in enriching the space of discrete strains by means of suitable local modes or bubble functions. However, those local modes or bubble functions may not be derived from admissible displacements, which lead to a nonconforming approximation and require consistency error estimates for a proof of robust convergence of the formulation. Furthermore, it was soon discovered that classical enhanced strain formulation suffered from undesirable nonphysical instabilities, especially when applied to strong compression tests [8]. In addi...

The finite element method has been used widely in tire engineering. Most tire simulations using the finite element method are static analyses, because tires are very complex nonlinear structures. Recently, transient phenomena have been studied with explicit finite element analysis codes. In this paper, the authors demonstrate the feasibility of tire cornering simulation using an explicit finite element code, PAM-SHOCK. First, we propose the cornering simulation using the explicit finite element analysis code. To demonstrate the efficiency of the proposed simulation, computed cornering forces for a 175SR14 tire are compared with experimental results from an MTS Flat-Trac Tire Test System. The computed cornering forces agree well with experimental results. After that, parametric studies are conducted by using the proposed simulation.