The combination of digital image correlation (DIC) and scanning electron microscopy (SEM) enables to extract high resolution full field displacement data, based on the high spatial resolution of SEM and the sub-pixel accuracy of DIC. However, SEM images may exhibit a considerable amount of imaging artifacts, which may seriously compromise the accuracy of the displacements and strains measured from these images. The current study proposes a unified general framework to correct for the three dominant types of SEM artifacts, i.e. spatial distortion, drift distortion and scan line shifts. The artifact fields are measured alongside the mechanical deformations to minimize the artifact induced errors in the latter. To this purpose, Integrated DIC (IDIC) is extended with a series of hierarchical mapping functions that describe the interaction of the imaging process with the mechanics. A new IDIC formulation based on these mapping functions is derived and the potential of the framework is tested by a number of virtual experiments. The effect of noise in the images and different regularization options for the artifact fields are studied. The error in the mechanical displacement fields measured for noise levels up to 5 % is within the usual DIC accuracy range for all the cases studied, while it is more than 4 pixels if artifacts are ignored. A validation on real SEM images at three different magnifications confirms that all three distortion fields are accurately captured. The results of all virtual and real experiments demonstrate the accuracy of the methodology proposed, as well as its robustness in terms of convergence.

This paper focuses on size effects in periodic mechanical metamaterials driven by reversible pattern transformations due to local elastic buckling instabilities in their microstructure. Two distinct loading cases are studied: compression and bending, in which the material exhibits pattern transformation in the whole structure or only partially. The ratio between the height of the specimen and the size of a unit cell is defined as the scale ratio. A family of shifted microstructures, corresponding to all possible arrangements of the microstructure relative to the external boundary, is considered in order to determine the ensemble averaged solution computed for each scale ratio. In the compression case, the top and the bottom edges of the specimens are fully constrained, which introduces boundary layers with restricted pattern transformation. In the bending case, the top and bottom edges are free boundaries resulting in compliant boundary layers, whereas additional size effects emerge from imposed strain gradient. For comparison, the classical homogenization solution is computed and shown to match well with the ensemble averaged numerical solution only for very large scale ratios. For smaller scale ratios, where a size effect dominates, the classical homogenization no longer applies.

This paper presents a homogenization framework for elastomeric metamaterials exhibiting long-range correlated fluctuation fields. Based on full-scale numerical simulations on a class of such materials, an ansatz is proposed that allows to decompose the kinematics into three parts, i.e. a smooth mean displacement field, a long-range correlated fluctuating field, and a local microfluctuation part. With this decomposition, a homogenized solution is defined by ensemble averaging the solutions obtained from a family of translated microstructural realizations. Minimizing the resulting homogenized energy, a micromorphic continuum emerges in terms of the average displacement and the amplitude of the patterning long-range microstructural fluctuation fields. Since full integration of the ensemble averaged global energy (and hence also the corresponding Euler-Lagrange equations) is computationally prohibitive, a more efficient approximative computational framework is developed. The framework relies on local energy density approximations in the neighbourhood of the considered Gauss integration points, while taking into account the smoothness properties of the effective fields and periodicity of the microfluctuation pattern. Finally, the implementation of the proposed methodology is briefly outlined and its performance is demonstrated by comparing its predictions against full scale simulations of a representative example.

Lattice networks with dissipative interactions can be used to describe the mechanics of discrete mesostructures of materials such as 3D-printed structures and foams. This contribution deals with the crack initiation and propagation in such materials and focuses on an adaptive multiscale approach that captures the spatially evolving fracture. Lattice networks naturally incorporate non-locality, large deformations and dissipative mechanisms taking place inside fracture zones. Because the physically relevant length scales are significantly larger than those of individual interactions, discrete models are computationally expensive. The Quasicontinuum (QC) method is a multiscale approach specifically constructed for discrete models. This method reduces the computational cost by fully resolving the underlying lattice only in regions of interest, while coarsening elsewhere. In this contribution, the (variational) QC is applied to damageable lattices for engineering-scale predictions. To deal with the spatially evolving fracture zone, an adaptive scheme is proposed. Implications induced by the adaptive procedure are discussed from the energy-consistency point of view, and theoretical considerations are demonstrated on two examples. The first one serves as a proof of concept, illustrates the consistency of the adaptive schemes and presents errors in energies. The second one demonstrates the performance of the adaptive QC scheme for a more complex problem. AN ADAPTIVE VARIATIONAL QUASICONTINUUM METHODOLOGY FOR LATTICE 175 beam (fibre or yarn) versus that of the network. Second, the formulation and implementation of lattice models is generally significantly easier compared with that of alternative continuum models. Large deformations, large yarn reorientations and fracture are easier to formulate and implement (cf. e.g. the continuum model of Peng and Cao [5] that deals with large yarn reorientations). Thanks to the simplicity and versatility of lattice networks, they are furthermore used for the description of heterogeneous cohesive-frictional materials such as concrete. The reason is that discrete models can realistically represent distributed microcracking with gradual softening, implement material structure with inhomogeneities, capture non-locality of damage processes and reflect deterministic or stochastic size effects. Examples of the successful use of lattice models for such materials are given in [6][7][8][9].As lattice models are typically constructed at the meso-scale, micro-scale or nano-scale, they require reduced-model techniques to allow for application-scale simulations. A prominent example is the Quasicontinuum (QC) method, which specifically aims at discrete lattice models. The QC method was originally introduced for conservative atomistic systems by Tadmor et al. [10] and extended in numerous aspects later on, see for example [11][12][13]. Subsequent generalizations for lattices with dissipative interactions (e.g. plasticity and bond sliding) were provided in [14,15]. In principle, the QC is a numerical p...

The quasicontinuum (QC) method is a numerical strategy to reduce the computational cost of direct lattice computations -in this study we achieve a speed up of a factor of 40. It has successfully been applied to (conservative) atomistic lattices in the past, but using a virtualpower statement it was recently shown that QC approaches can also be used for spring and beam lattice models that include dissipation. Recent results have shown that QC approaches for planar beam lattices experiencing in-plane and out-of-plane deformation require higherorder interpolation. Higher-order QC frameworks are scarce nevertheless. In this contribution, the possibilities of a second-order and third-order QC framework are investigated for an elastoplastic spring lattice. The higher-order QC frameworks are compared to the results of the direct lattice computations and to those of a linear QC scheme. Examples are chosen so that both a macroscale and a microscale quantity influences the results. The two multiscale examples focused on are (i) macroscopically prescribed uniaxial deformation and (ii) macroscopically prescribed pure bending. Furthermore, the examples include an individual inclusion in a large lattice and hence, are concurrent in nature.

Lattice systems and discrete networks with dissipative interactions are successfully employed as meso-scale models of heterogeneous solids. As the application scale generally is much larger than that of the discrete links, physically relevant simulations are computationally expensive. The QuasiContinuum (QC) method is a multiscale approach that reduces the computational cost of direct numerical simulations by fully resolving complex phenomena only in regions of interest while coarsening elsewhere. In previous work (Beex et al., J. Mech. Phys. Solids 64, 154-169, 2014), the originally conservative QC methodology was generalized to a virtual-power-based QC approach that includes local dissipative mechanisms. In this contribution, the virtual-power-based QC method is reformulated from a variational point of view, by employing the energy-based variational framework for rate-independent processes (Mielke and Roubíček, Rate-Independent Systems: Theory and Application, Springer-Verlag, 2015). By construction it is shown that the QC method with dissipative interactions can be expressed as a minimization problem of a properly built energy potential, providing solutions equivalent to those of the virtual-power-based QC formulation. The theoretical considerations are demonstrated on three simple examples. For them we verify energy consistency, quantify relative errors in energies, and discuss errors in internal variables obtained for different meshes and two summation rules.

Lattice networks with dissipative interactions are often employed to analyze materials with discrete micro-or meso-structures, or for a description of heterogeneous materials which can be modelled discretely. They are, however, computationally prohibitive for engineering-scale applications. The (variational) QuasiContinuum (QC) method is a concurrent multiscale approach that reduces their computational cost by fully resolving the (dissipative) lattice network in small regions of interest while coarsening elsewhere. When applied to damageable lattices, moving crack tips can be captured by adaptive mesh refinement schemes, whereas fully-resolved trails in crack wakes can be removed by mesh coarsening. In order to address crack propagation efficiently and accurately, we develop in this contribution the necessary generalizations of the variational QC methodology. First, a suitable definition of crack paths in discrete systems is introduced, which allows for their geometrical representation in terms of the signed distance function. Second, special function enrichments based on the partition of unity concept are adopted, in order to capture kinematics in the wakes of crack tips. Third, a summation rule that reflects the adopted enrichment functions with sufficient degree of accuracy is developed. Finally, as our standpoint is variational, we discuss implications of the mesh refinement and coarsening from an energy-consistency point of view. All theoretical considerations are demonstrated using two numerical examples for which the resulting reaction forces, energy evolutions, and crack paths are compared to those of the direct numerical simulations.

The purpose of this paper is to provide analytical and numerical solutions of the formation and evolution of the localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. An energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto-plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening. Four examples that differ by regularity of the yield stress and stress distributions are presented. Results for the load level, size of the plastic zone, distribution of plastic strain and its spatial derivatives, plastic elongation, and energy balance are constructed and compared to another, previously discussed non-variational gradient formulation.