Variational methods in continuum damage and fracture mechanics

Damage is defined as the loss of material stiffness under loading conditions. This process is intrinsically irreversible and, therefore, dissipative. When the stiffness vanishes, fracture is achieved. In order to derive governing equations, variationalmethods have been employed. Standard variational methods for non-dissipative sys-tems are here formulated in order to contemplate dissipative systems as the onesconsidered in continuum damage mechanics

A 1D continuum model for beams with pantographic microstructure: asymptotic micro-macro identification and numerical results

In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future

Heuristic Homogenization of Euler and Pantographic Beams

Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 596)International audienceIn the present contribution, we address the following problem: is it possible to find a microstructure producing, at the macro-level and under loads of the same order of magnitude, a beam which can be both extensible and flexible? Using an asymptotic expansion and rescaling suitably the involved stiffnesses, we prove that a pantographic microstructure does induce, at the macro-level, the aforementioned desired mechanical behavior. Thus, in an analogous fashion to that of variational asymptotic methods, and following a mathematical approach resembling that used by Piola, we have employed asymptotic expansions of kinematic descriptors directly into the postulated energy functional and a heuristic homogenization procedure is presented and applied to the cases of Euler and pantographic beams

Pantographic beam: a complete second gradient 1D-continuum in plane

International audienceThere is a class of planar 1D-continua which can be described exclusively by their placement functions which in turn are curves in a two-dimensional space. In contrast to the Elastica for which the deformation energy depends on the projection of the second gradient to the normal vector of the placement function, i.e. the material curvature, the proposed continuum does also depend on the projection onto the tangent vector, introduced as the stretch gradient. Thus, the deformation energy takes into account the complete second gradient of the placement function. In such a model, non-standard boundary conditions and more generalized forces such as double forces do appear. The deformation energy of the continuum is obtained by applying a heuristic homogenization procedure to a family of slender discrete pantographic structures constituted by extensional and rotational springs. Within the homogenization process, the overall length of the system is kept fixed, the number of the periodically appearing sub-systems, called cells, is increased, and the stiffnesses are appropriately scaled. For two examples, we numerically compare the family of discrete systems with the continuum. The analysis shows that the continuum represents the behaviour of the discrete system already for a relatively moderate number of cells. In particular, the behaviour of the deformation energy error between the discrete and the continuum models when the number of cells tends to infinity is determined by the homogenization process

Coupled phase field and nonlocal integral elasticity analysis of stress-induced martensitic transformations at the nanoscale: boundary effects, limitations and contradictions

In this paper, the coupled phase field and local/nonlocal integral elasticity theories are used for stress-induced martensitic phase transformations (MPTs) at the nanoscale to investigate the limitations and contradictions of the nonlocal integral elasticity, which are due to the fact that the support of the nonlocal kernel exceeds the integration domain, i.e., the boundary effect. Different functions for the nonlocal kernel are compared. In order to compensate the boundary effect, a new nonlocal kernel, i.e., the compensated two-phase kernel, is introduced, in which a local part is added to the nonlocal part of the two-phase kernel to account for the boundary effect. In contrast to the previously introduced modified kernel, the compensated two-phase kernel does not lead to a purely nonlocal behavior in the core region, and hence no singular behavior, and consequently, no computational convergence issue is observed. The nonlinear finite element approach and the COMSOL code are used to solve the coupled system of Ginzburg–Landau and local/nonlocal integral elasticity equations. The numerical implementation of the phase field-local elasticity equations and the 2D nonlocal integral elasticity are verified. Boundary effect is investigated for MPT with both homogeneous and nonhomogeneous stress distributions. For the former, in contrast to the local elasticity, a nonhomogeneous phase transformation (PT) occurs in the nonlocal case with the two-phase kernel. Using the compensated two-phase kernel results in a homogeneous PT similar to the local elasticity. For the latter, the sample transforms to martensite except the adjacent region to the boundary for the local elasticity, while for the two-phase kernel, the entire sample transforms to martensite. The solution of the compensated two-phase kernel, however, is very similar to that of the local elasticity. The applicability of boundary symmetry in phase field problems is also investigated, which shows that it leads to incorrect results within the nonlocal integral elasticity. This is because when the symmetric portions of a sample are removed, the corresponding nonlocal effects on the remaining portion are neglected and the symmetric boundaries violate the normalization condition. An example is presented in which the results of a complete model with the two-phase kernel are different from those of its one-fourth model. In contrast, the compensated two-phase kernel can generate similar solutions for both the complete and one-fourth models. However, in general, none of the nonlocal kernels can overcome this issue. Therefore, the symmetrical models are not recommended for nonlocal integral elasticity based phase field simulations of MPTs. The current study helps for a better study of nonlocal elasticity based phase field problems for various phenomena such as various PTs

Identification of a geometrically nonlinear micromorphic continuum via granular micromechanics

Describing the emerging macro-scale behavior by accounting for the micro-scale phenomena calls for microstructure-informed continuum models accounting properly for the deformation mechanisms identifiable at the micro-scale. Classical continuum theory, in contrast to the micromorphic continuum theory, is unable to take into account the effects of complex kinematics and distribution of elastic energy in internal deformation modes within the continuum material point. In this paper, we derive a geometrically nonlinear micromorphic continuum theory on the basis of granular mechanics, utilizing grain-scale deformation as the fundamental building block. The definition of objective kinematic descriptors for relative motion is followed by Piola’s ansatz for micro–macro-kinematic bridging and, finally, by a limit process leading to the identification of the continuum stiffness parameters in terms of few micro-scale constitutive quantities. A key aspect of the presented approach is the identification of relevant kinematic measures that describe the deformation of the continuum body and link it to the micro-scale deformation. The methodology, therefore, has the ability to reveal the connections between the micro-scale mechanisms that store elastic energy and lead to particular emergent behavior at the macro-scale

Two-Dimensional Analysis of Size Effects in Strain-Gradient Granular Solids with Damage-Induced Anisotropy Evolution

We analyze in two dimensions the mechanical behavior of materials with granular microstructures modeled by means of a variationally formulated strain-gradient continuum approach based on micromechanics and show that it can capture microstructural-size-dependent effects. Tension-compression asymmetry of grain-assembly interactions, as well as microscale damage, is taken into account and the continuum scale is linked to the grain-scale mechanisms. Numerical results are provided for finite deformations and substantiate previous research. As expected, results show interesting size-dependent effects that are typical of strain-gradient modeling

Investigation of deformation behavior of PETG-FDM-printed metamaterials with pantographic substructures based on different slicing strategies

Based on the progress and advances of additive manufacturing technologies, design and production of complex structures became cheaper and therefore rather possible in the recent past. A promising example of such complex structure is a so-called pantographic structure, which can be described as a metamaterial consisting of repeated substructure. In this substructure, two planes, which consist of two arrays of beams being orthogonally aligned to each other, are interconnected by cylinders/pivots. Different inner geometries were taken into account and additively manufactured by means of fused deposition modeling technique using polyethylene terephthalate glycol (PETG) as filament material. To further understand the effect of different manufacturing parameters on the mechanical deformation behavior, three types of specimens have been investigated by means of displacement-controlled extension tests. Different slicing approaches were implemented to eliminate process-related problems. Small and large deformations are investigated separately. Furthermore, 2D digital image correlation was used to calculate strains on the outer surface of the metamaterial. Two finite-element simulations based on linear elastic isotropic model and linear elastic transverse isotropic model have been carried out for small deformations. Standardized extension tests have been performed on 3D-printed PETG according to ISO 527-2. Results obtained from finite-element method have been validated by experimental results of small deformations. These results are in good agreement with linear elastic transverse isotropic model (up to about εxx=1.2% of axial elongation), though the response of large deformations indicates a nonlinear inelastic material behavior. Nevertheless, all samples are able to withstand outer loading conditions after the first rupture, resulting in resilience against ultimate failure.DFG, 414044773, Open Access Publizieren 2021 - 2022 / Technische Universität Berli

GINGER

In this paper, we outline the scientific objectives, the experimental layout, and the collaborations envisaged for the GINGER (Gyroscopes IN GEneral Relativity) project. The GINGER project brings together different scientific disciplines aiming at building an array of Ring Laser Gyroscopes (RLGs), exploiting the Sagnac effect, to measure continuously, with sensitivity better than picorad/ s, large bandwidth (ca. 1 kHz), and high dynamic range, the absolute angular rotation rate of the Earth. In the paper, we address the feasibility of the apparatus with respect to the ambitious specifications above, as well as prove how such an apparatus, which will be able to detect strong Earthquakes, very weak geodetic signals, as well as general relativity effects like Lense-Thirring and De Sitter, will help scientific advancements in Theoretical Physics, Geophysics, and Geodesy, among other scientific fields.Comment: 21 pages, 9 figure