scholarly journals A note on the higher order strain and stress tensors within deformation gradient elasticity theories: Physical interpretations and comparisons

2016 ◽  
Vol 90 ◽  
pp. 116-121 ◽  
Author(s):  
Castrenze Polizzotto
Keyword(s):  
Deformation Gradient ◽  
Gradient Elasticity ◽  
Higher Order ◽  
Stress Tensors

A nonlocal operator method for finite deformation higher-order gradient elasticity

2021 ◽  
Vol 384 ◽  
pp. 113963
Author(s):  
Huilong Ren ◽  
Xiaoying Zhuang ◽  
Nguyen-Thoi Trung ◽  
Timon Rabczuk
Keyword(s):  
Finite Deformation ◽  
Gradient Elasticity ◽  
Operator Method ◽  
Higher Order ◽  
Nonlocal Operator

scholarly journals Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

2016 ◽  
Vol 472 (2188) ◽  
pp. 20150755 ◽  
Author(s):  
Samuel Forest
Keyword(s):  
Free Energy ◽  
Energy Function ◽  
Kinematic Hardening ◽  
Deformation Gradient ◽  
Gradient Elasticity ◽  
Gradient Term ◽  
Deformation Model ◽  
Finite Width ◽  
Free Energy Function ◽  
Constitutive Material Model

The construction of regularization operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the Helmholtz free energy function of the constitutive material model. The combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the regularization operator. Within the small strain framework, the choice of a quadratic potential w.r.t. the gradient term provides the widely used Helmholtz operator whose regularization properties are well known: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and finite width localization bands in softening materials. The objective is to review and propose nonlinear extensions of micromorphic and strain/damage gradient models along two lines: the first one introducing nonlinear relations between generalized stresses and strains; the second one envisaging several classes of finite deformation model formulations. The generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling. Two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach: additive split of some Lagrangian strain measure or choice of a local objective rotating frame. Three distinct operators are finally derived using the multiplicative decomposition of the deformation gradient. A new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable.

Modeling the Tensile Size-Dependency in Polymer Nanofiber Elasticity

2011 ◽  
Vol 236-238 ◽  
pp. 2187-2190
Author(s):  
Bo Yuan ◽  
Qun Feng Liu ◽  
Cai Lin ◽  
Xiao Feng Chen
Keyword(s):  
Tensile Test ◽  
Model Prediction ◽  
Strain Gradient ◽  
Surface Effect ◽  
Gradient Elasticity ◽  
Tensile Tests ◽  
Higher Order ◽  
Uniaxial Tensile ◽  
Polymer Nanofiber ◽  
Strain Gradient Model

In this paper, a higher order strain gradient model is constructed to predict this size dependence of the elastic property of nanofibers under uniaxial tensile tests. We can show that the size effects in tensile test can be explained using a new model based on the higher order strain gradient elasticity (HSGE). A series of mechanical testing were performed to verify the model, and good agreement is found between the model prediction and the data obtained in the experiment. Compared with the model prediction based on surface effect (SE), our model can better capture the size effect in tensile test.

Numerical Implementation of Higher-Order Homogenization Problems and Computation of Gradient Elasticity Coefficients

2012 ◽  
Author(s):  
V. Monchiet ◽  
T. H. Tran ◽  
G. Bonnet
Keyword(s):  
Strain Gradient ◽  
Gradient Elasticity ◽  
Local Scale ◽  
Higher Order ◽  
Strain Gradient Elasticity ◽  
Change Of Scale ◽  
Periodic Linear ◽  
Periodic Composite ◽  
Elasticity Tensors ◽  
Homogenization Problems

A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of tridimensional elasticity, but, since the assumption of strict separation of scale is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which, higher-order terms (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. All these terms are then obtained by solving a hierarchy of higher-order elasticity problems with prescribed body forces and eigen-strains whose expression depends on the solution at the lower-order. An energy based micro-macro transition is then proposed for the change of scale and constitutes, in fact, a generalization of the Hill-Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. It is rigorously proved that the macroscopic quantities derived from this homogenization procedure comply with the equations of strain gradient elasticity. As an illustration, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed whose efficiency is shown in the particular case of composites reinforced by long fibers.

Unified higher-order theory of two-phase nonlocal gradient elasticity

Meccanica ◽  
2021 ◽  
Vol 56 (3) ◽  
pp. 607-627
Author(s):  
S. Ali Faghidian ◽  
Esmaeal Ghavanloo
Keyword(s):  
Gradient Elasticity ◽  
Order Theory ◽  
Higher Order ◽  
Two Phase ◽  
Higher Order Theory

scholarly journals An energy residual-based approach to gradient effects within the mechanics of generalized continua

2012 ◽  
Vol 21 (3-4) ◽  
pp. 101-121
Author(s):  
Castrenze Polizzotto ◽  
Aurora A. Pisano
Keyword(s):  
Long Range ◽  
Constitutive Equations ◽  
Gradient Elasticity ◽  
Main Tool ◽  
Higher Order ◽  
Gradient Theory ◽  
Generalized Continua ◽  
Standard Format ◽  
Gradient Effects ◽  
Insulation Condition

AbstractGeneralized continua exhibiting gradient effects are addressed through a method grounded on the energy residual (ER)-based gradient theory by the first author and coworkers. A main tool of this theory is the Clausius-Duhem inequality cast in a form differing from the classical one only by a nonstandard extra term, the (nonlocality) ER, required to satisfy the insulation condition (its global value has to vanish or to take a known value). The ER carries in the nonlocality features of the mechanical problem through a strain-like rate field, being the specific nonlocality source, and a concomitant higher-order long-range stress (or microstress) field. The thermodynamic restrictions on the constitutive equations are determined by the latter inequality with no need for microstress equilibrium equations, whereas the principle of virtual power (PVP) is left in a standard format. The derived state equations include a set of partial differential equations involving the nonlocality-source strain-like quantity and the related long-range stress, as well as the associated higher-order boundary conditions determined by the insulation condition. Second-grade materials within gradient elasticity, gradient plasticity and crystal plasticity, as well as materials with microstructure (micromorphic and Cosserat materials) are considered to derive the pertinent constitutive equations. The proposed ER-based approach to gradient effects is shown to constitute a more straightforward and “economic” way to formulate the relevant constitutive equations than the PVP-based one.

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