Finite gradient elasticity and plasticity: a constitutive mechanical framework

Bertram, Albrecht

doi:10.1007/s00161-014-0387-0

Cited by 30 publications

(25 citation statements)

References 43 publications

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“…A similar additive decomposition of the intermediate description connection tensor into elastic and plastic parts has been advocated by [15]. A similar additive decomposition of the intermediate description connection tensor into elastic and plastic parts has been advocated by [15].…”

Section: Elastic and Plastic Connection Tensorsmentioning

confidence: 93%

“…Kinematically related approaches have been taken, e.g., by [180,31,62,32,43,15]. These are denoted as the elastic and plastic double-distortions.…”

Section: Supplement 82 Second-order Elasto-plasticity In Euclidean mentioning

confidence: 99%

“…In most of the literature (see, e.g., [31,62,32,44,15]) on second-order elasto-plasticity one of the connection tensors is chosen as the relevant measure of higher-order strain. However, based on the additive decomposition of the second deformation gradients into elastic and plastic double-distortions and the resulting additive decomposition of the connection tensors into elastic and plastic contributions, corresponding covariant elastic and plastic double-strains and double-metrics may be defined.…”

Section: Elastic and Plastic Covariant Double-strain Measuresmentioning

confidence: 99%

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Geometrical Foundations of Continuum Mechanics

Steinmann

2015

Lecture Notes in Applied Mathematics and Mechanics

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Section: Elastic and Plastic Connection Tensorsmentioning

confidence: 93%

“…Kinematically related approaches have been taken, e.g., by [180,31,62,32,43,15]. These are denoted as the elastic and plastic double-distortions.…”

Section: Supplement 82 Second-order Elasto-plasticity In Euclidean mentioning

confidence: 99%

Section: Elastic and Plastic Covariant Double-strain Measuresmentioning

confidence: 99%

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Geometrical Foundations of Continuum Mechanics

Steinmann

2015

Lecture Notes in Applied Mathematics and Mechanics

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“…In geometrically nonlinear hyperelasticity we know that frame-indifference is left-invariance of the energy under SO(3)-action and isotropy is right-invariance under SO(3), i.e. 6 for the energy density W we have…”

Section: Objectivity and Isotropy In Nonlinear Elasticity -The Local mentioning

confidence: 99%

“…The simple contraction concerning the index b in eq. (1.30) is not possible via standard symbolic notation 6. We do not discuss O(3)-invariance for simplicity.…”

mentioning

confidence: 99%

Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy

Münch

Neff

2016

Mathematics and Mechanics of Solids

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For homogeneous higher gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frameindifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invariance condition to some specific linear models. The interesting point is the appearance of finite rotations in transformations of a geometrically linear model. It is shown that when starting with a linear model defined already in the infinitesimal symmetric strain ε = sym Grad[u], the new invariance condition is equivalent to isotropy of the linear formulation. Therefore, it may be used also in higher gradient elasticity models for a simple check of isotropy and for extensions to anisotropy. In this respect we consider in more detail variational formulations of the linear indeterminate couple stress model, a new variant of it with symmetric force stresses and general linear gradient elasticity. 3 tr(X) ½ ∈ sl(3) and we have the orthogonal Cartan-decomposition of the Lie-algebra gl(3) gl(3) = {sl(3) ∩ Sym(3)} ⊕ so(3) ⊕ R·½ , X = dev sym X + skew X + 1 3 tr(X) ½ ,(1.1) simply allowing to split every second order tensor uniquely into its trace-free symmetric part, skew-symmetric 2 Indeed, in Murdoch [39, eq.(16)-(20)] we find the same conclusions regarding isotropy of a material point. 3 This remark refers to eq.(1), Mindlin [37], which reads, in the notation of the present paper, ∂ m 31Here, m is the second-order couple stress tensor and σ 21 − σ 12 = 2 axl( skew σ) represents the skew-symmetric part of the total force stress tensor σ. The equation is given for the planar situation, see also eq.(6.11).

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To consider second gradient continua as constrained microstructured continua à la Germain simplifies numerical analysis of metamaterials

Spagnuolo

2024

Z Angew Math Mech

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In this work we base our analysis on a seminal paper by Paul Germain (1973) in which the principle of virtual work is used to found microstructured continuum mechanics. Recently, it was shown that a particular considered microstructured continuum can be regarded as a second gradient continuum of the kind studied by Germain. To prove how these theoretical ideas can be useful in applications, we present the paradigmatic case of pantographic metamaterials, in which the deformation energy may depend only on placement second gradient being independent on placement first gradient. In this case, the numerical simulations become more efficient by introducing the Lagrange multipliers which are dual in work of the introduced kinematical constraint, so proving that the viewpoint about stress presented originally by Lagrange is the most fruitful one.

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Finite gradient elasticity and plasticity: a constitutive mechanical framework

Cited by 30 publications

References 43 publications

Geometrical Foundations of Continuum Mechanics

Geometrical Foundations of Continuum Mechanics

Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy

To consider second gradient continua as constrained microstructured continua à la Germain simplifies numerical analysis of metamaterials

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