On the dependence of standard and gradient elastic material constants on a field of defects

Abstract

In this work, we consider a strain gradient elasticity theory with an extended number of field variables: the displacement vector and an additional scalar field defining the internal micro-deformation. The total internal energy of the model depends on the strain, the micro-deformation function, their gradients, and the coupling. The considered model can be treated as gradient/micromorphic. Moreover, the micro-deformation field can be treated as a field of scalar defects distributed along the medium. Based on analytic (one-dimensional) solutions of uniform/non-uniform deformation of the rod, we introduce (i) an apparent stiffness and (ii) an apparent length scale parameter. Subsequently, we provide a variant of continuum-on-continuum homogenization by equating tip displacements for the gradient/micromorphic medium and an equivalent strain gradient one. Elongation of the gradient/micromorphic rod is therefore equated with the corresponding elongation of the equivalent strain gradient rod, whose behavior is characterized by the apparent material constants. Subsequently, the non-dimensional coupling number is identified with a damage parameter. It is shown that, on the one hand, the apparent stiffness of the rod is reduced when such parameter increases. On the other hand, the apparent length scale parameter (i.e. the apparent second gradient elastic coefficient) increases when the damage parameter increases. Therefore, it is shown that the presence of defects in a second gradient linear elastic material may increase its apparent strain gradient behavior.

Are higher-gradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures?

“…In Germain [12], micromorphic media of order one were derived in detail, and subsequently the equations for the general micromorphic medium were presented. For interested readers, we refer to the works of Toupin [15], Eringen [16], Misra and Poorsolhjouy [17], Eremeyev [18] and Solyaev et al [19] for further details.…”

Three-point bending test of pantographic blocks: numerical and experimental investigation

“…De Angelo et al [28] investigated the behavior of metallic pantographic sheets. In a number of papers, researchers have worked on the identification of constitutive parameters for second gradient materials (see Giorgio [29], Placidi et al [30], Yang et al [31], De Angelo et al [32], Nejadsadeghi et al [33], Solyaev et al [34], and Turco [35]). In a few studies, the 3D deformation of pantographic structures has been investigated in the context of second gradient modeling, for example in the studies presented by Steigmann and dell’Isola [36], Giorgio et al [37, 38], and Scerrato et al [39].…”

A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory

“…Open problems and challenges that could be tackled in the next future include: (i) a careful analysis of the stiffness parameters used to characterize the elastic response of the whole beam in the largedeformation regime, and in general when de Saint-Venant estimate of stiffness parameters does not apply; such parameters should be related to the material constitutive parameters of the material constituting the meso-beams such as the Young and tangential moduli, and to geometrical parameters of the beams cross-section, as the area, the shear correction factor, and the moment of inertia; (ii) the development of functionally graded materials, meaning those materials having stiffness parameters which are varying along the beam axis; an extended campaign of numerical simulations might unveil new and exotic mechanical behaviors, see [48][49][50][51]; (iii) the development of continuum models, as those developed and exploited, e.g., in [52][53][54][55][56][57][58][59][60], aimed at describing for large displacements systems with many discrete elements of the type presented here; besides being useful in unveiling so-called emerging phenomena, continuum models could help in identifying stiffness parameters; (iv) the exploitation of the presented approach to provide a validation and insight into new and existing approaches for the extension of stability theory in classical elastic media to micromorphic, strain-gradient [61,62], and Cosserat media, see, e.g., [12,[63][64][65][66][67][68][69][70]; (v) the extension of the presented approach to problems where dynamics effects are non-negligible, see, e.g., [71][72][73], like those studied in the active control of vibrations [74]; (vi) the validation of continuum approaches to the study of plane and curved structures moulded as, e.g., shells and tubes, see [75][76][77...…”