Elasticity with Gradient-Disarrangements: A Multiscale Perspective for Strain-Gradient Theories of Elasticity and of Plasticity

Owen, D. R. J.

doi:10.1007/s10659-016-9599-9

Cited by 14 publications

(9 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By taking into account that the SDs theory permits energy to be stored by means of both smooth and non-smooth sub-macroscopic geometrical changes, a free energy function can be assigned as dependent on a pair formed by any combination of kinematic tensors chosen among G, M and ∇g [27]. From an operational point of view, as pointed out in detail in [27] and [28], there are essentially two possible ways to analyze undergoing SDs within a body. A first one is to select a constitutive class, by prescribing constitutive equations for the stresses S \ and S d based on a chosen free energy, then using the relation (12) to restrict the class of admissible processes for the system.…”

Section: Decomposition Of Stresses and Constitutive Assumptionsmentioning

confidence: 99%

Disarrangements and instabilities in augmented one-dimensional hyperelasticity

et al. 2018

Self Cite

View full text Add to dashboard Cite

In the present work, the overall nonlinear elastic behavior of a 1D multi-modular structure incorporating possible imperfections at the discrete (micro-scale) level, is derived with respect to both tensile and compressive applied loads. The model is built up through the repetition of n units, each one comprising two rigid rods having equal lengths, linked by means of pointwise constraints capable to elastically limit motions in terms of relative translations (sliders) and rotations (hinges). The mechanical response of the structure is analyzed by varying the number n of the elemental moduli, as well as in the limit case of infinite number of infinitesimal constituents, in light of the theory of (first order) Structured Deformations (SDs), that interprets the deformation of any continuum body as the projection, at the macroscopic scale, of geometrical changes occurring at the level of its sub-macroscopic elements. In this way, a wide family of nonlinear elastic behaviors is generated by tuning internal microstructural parameters, the tensile buckling and the classical Euler's Elastica under compressive loads resulting as special cases in the so-called continuum limit -say when n → ∞. Finally, by plotting the results in terms of first Piola-Kirchhoff stress versus macroscopic stretch, it is for the first time demonstrated that such SDs-based 1D models can be helpfully used to generalize some standard hyperelastic behaviors by additionally taking into account instability phenomena and concealed defects.

show abstract

Section: Decomposition Of Stresses and Constitutive Assumptionsmentioning

confidence: 99%

Disarrangements and instabilities in augmented one-dimensional hyperelasticity

et al. 2018

Self Cite

View full text Add to dashboard Cite

show abstract

“…Consequently, the results in the present paper on relaxation in the context of second-order structured deformations capture the influence both of disarrangements and of gradient-disarrangements on relaxed energies and provide the starting point for broadening elasticity with disarrangements to the richer multiscale geometry of second-order structured deformations. Initial steps toward such a broadening have been taken [27] in the context of second-order structured deformations. A physical context of significance -phasetransitions in metals [1,2,24] -provides a setting in which deformations can be approximately piecewise homogeneous at small length scales.…”

Section: 12mentioning

confidence: 99%

Second-Order Structured Deformations: Relaxation, Integral Representation and Applications

Barroso

Matías²,

Morandotti

et al. 2017

Arch Rational Mech Anal

Self Cite

View full text Add to dashboard Cite

ABSTRACT. Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral representation for a relaxed energy functional in the setting of second-order structured deformations. Our derivation covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position); finally, we provide explicit formulas for bulk relaxed energies as well as anticipated applications.

show abstract

“…Those of type (ii) are intended to provide descriptions of membranes such as a sheet of rubber, descriptions of thin plates such as a sheet of metal, and descriptions of fibered thin bodies such as a sheet of paper. There are available a variety of approaches for incorporating disarrangements and for adaptation to the case of thin bodies: examples of refinements of type (i) are mechanical theories of no-tension materials [5,21,28], of granular media [1,24,33], of single and polycrystals [25,36], and of elastic bodies in the multiscale geometrical setting of structured deformations [22,34], while for refinements of type (ii) the method of dimension reduction via Γ-convergence [12,26,27] and the method of dimension reduction via Taylor expansions [20] provide examples.…”

Section: Introductionmentioning

confidence: 99%

Dimension Reduction in the Context of Structured Deformations

Carita¹,

Matías²,

Morandotti

et al. 2018

J Elast

Self Cite

View full text Add to dashboard Cite

Dedicated to our friend and colleague Graça Carita, who left us far too soon.Abstract. In this paper we apply both the procedure of dimension reduction and the incorporation of structured deformations to a three-dimensional continuum in the form of a thinning domain. We apply the two processes one after the other, exchanging the order, and so obtain for each order both a relaxed bulk and a relaxed interfacial energy. Our implementation requires some substantial modifications of the two relaxation procedures. For the specific choice of an initial energy including only the surface term, we compute the energy densities explicitly and show that they are the same, independent of the order of the relaxation processes. Moreover, we compare our explicit results with those obtained when the limiting process of dimension reduction and of passage to the structured deformation is carried out at the same time. We finally show that, in a portion of the common domain of the relaxed energy densities, the simultaneous procedure gives an energy strictly lower than that obtained in the two-step relaxations.

show abstract

Elasticity with Gradient-Disarrangements: A Multiscale Perspective for Strain-Gradient Theories of Elasticity and of Plasticity

Cited by 14 publications

References 35 publications

Disarrangements and instabilities in augmented one-dimensional hyperelasticity

Disarrangements and instabilities in augmented one-dimensional hyperelasticity

Second-Order Structured Deformations: Relaxation, Integral Representation and Applications

Dimension Reduction in the Context of Structured Deformations

Contact Info

Product

Resources

About