scholarly journals Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models

2017 ◽  
Vol 25 (7) ◽  
pp. 1429-1449 ◽  
Author(s):  
Samuel Forest ◽  
Karam Sab
Keyword(s):  
Strain Gradient ◽  
Finite Deformation ◽  
Stress Gradient ◽  
Higher Order ◽  
Second Order ◽  
Stress And Strain ◽  
Dynamical Equations ◽  
Generalized Continua ◽  
Micromorphic Theory ◽  
Gradient Models

Germain’s general micromorphic theory of order [Formula: see text] is extended to fully non-symmetric higher-order tensor degrees of freedom. An interpretation of the microdeformation kinematic variables as relaxed higher-order gradients of the displacement field is proposed. Dynamical balance laws and hyperelastic constitutive equations are derived within the finite deformation framework. Internal constraints are enforced to recover strain gradient theories of grade [Formula: see text]. An extension to finite deformations of a recently developed stress gradient continuum theory is then presented, together with its relation to the second-order micromorphic model. The linearization of the combination of stress and strain gradient models is then shown to deliver formulations related to Eringen’s and Aifantis’s well-known gradient models involving the Laplacians of stress and strain tensors. Finally, the structures of the dynamical equations are given for strain and stress gradient media, showing fundamental differences in the dynamical behaviour of these two classes of generalized continua.

scholarly journals Studies on the Residual Stress and Strain Gradients in Poly-SiGe Nanocantilevers

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Tesleem B Asafa
Keyword(s):  
Strain Gradient ◽  
Stress Gradient ◽  
Local Stress ◽  
Stress And Strain ◽  
Fast Method ◽  
Stress Evolution ◽  
Strain Gradients ◽  
Study Results ◽  
Evolution Study ◽  
Gradient Measurement

One of the fundamental structural requirements for Micro/Nano-ElectroMechanical (M/NEM) devices is low strain gradient. Measurement of strain gradients is time consuming, therefore finding a simple and fast method is necessary. In this paper, a comparative study of the strain gradients in poly-SiGe nanocantilevers measured experimentally and obtained using finite element modelling (FEM) approach is reported.  Arrays of nanocantilevers were fabricated from 100 nm thick poly-SiGe films via lithography. Then, strain gradients were calculated from the tip deflections and cantilevers’ lengths. In the modelling study, similar cantilevers were modelled with COMSOL Multiphysics as superposition of smaller layers in which each layer sustained local stress obtained from stress evolution study. Results showed that the average strain gradients obtained from the experimental and FEM studies differ by ~5% and ~6% for film A and B, respectively with standard deviations lying between ±0.004 and ±0.009/µm. While this study established that stress gradient is responsible for the calculated strain gradient, it also emphasises that both parameters are proportional. Key words: Poly-SiGe, Strain gradient, FEM, COMSOL.

scholarly journals The Influence of the Strain and Stress Gradient in Determining Strain Fatigue Characteristics for Oscillatory Bending

Materials ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 173 ◽  
Author(s):  
Andrzej Kurek ◽  
Justyna Koziarska ◽  
Tadeusz Łagoda
Keyword(s):  
Aluminum Alloys ◽  
Compression Test ◽  
Strain Gradient ◽  
Stress Gradient ◽  
Fatigue Tests ◽  
Model Verification ◽  
Bending Test ◽  
Stress And Strain ◽  
Fatigue Characteristics ◽  
Gradient Surface

In this study, we created a new model to determine strain fatigue characteristics obtained from a bending test. The developed model consists of comparing the stress and strain gradient surface ratio for bending and tensile elements. For model verification, seven different materials were examined based on fatigue tests we conducted, or data available in the literature: 30CrNiMo8, 10HNAP, SM45C, 16Mo3 steel, MO58 brass, and 2017A-T4 and 6082-T6 aluminum alloys. As a result, we confirmed that the proposed method can be used to determine strain fatigue characteristics that agree with the values determined on the basis of a tensile compression test.

scholarly journals Continuum thermomechanics of nonlinear micromorphic, strain and stress gradient media

2020 ◽  
Vol 378 (2170) ◽  
pp. 20190169 ◽  
Author(s):  
Samuel Forest
Keyword(s):  
Strain Gradient ◽  
Stress Gradient ◽  
Continuum Theory ◽  
Research Field ◽  
Irreversible Processes ◽  
Current Debate ◽  
Generalized Continua ◽  
Generalized Standard Materials ◽  
New Research ◽  
Continuum Theories

A comprehensive constitutive theory for the thermo-mechanical behaviour of generalized continua is established within the framework of continuum thermodynamics of irreversible processes. It represents an extension of the class of generalized standard materials to higher order and higher grade continuum theories. It reconciles most existing frameworks and proposes some new extensions for micromorphic and strain gradient media. The special case of strain gradient plasticity is also included as a contribution to the current debate on the consideration of energetic and dissipative mechanisms. Finally, the stress gradient continuum theory emerges as a new research field for which an elastic-viscoplastic theory at finite deformations is provided for the first time. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.

scholarly journals Strain Localization Modes within Single Crystals Using Finite Deformation Strain Gradient Crystal Plasticity

Crystals ◽  
2021 ◽  
Vol 11 (10) ◽  
pp. 1235
Author(s):  
Lei Cai ◽  
Mohamed Jebahi ◽  
Farid Abed-Meraim
Keyword(s):  
Single Crystals ◽  
Crystal Plasticity ◽  
Strain Gradient ◽  
Finite Deformation ◽  
Small Deformation ◽  
Higher Order ◽  
Extended Model ◽  
Kink Bands ◽  
Dissipative Effects ◽  
Slip Bands

The present paper aims at providing a comprehensive investigation of the abilities and limitations of strain gradient crystal plasticity (SGCP) theories in capturing different kinds of localization modes in single crystals. To this end, the small deformation Gurtin-type SGCP model recently proposed by the authors, based on non-quadratic defect energy and the uncoupled dissipation assumption, is extended to finite deformation. The extended model is then applied to simulate several single crystal localization problems with different slip system configurations. These configurations are chosen in such a way as to obtain idealized slip and kink bands as well as general localization bands, i.e., with no particular orientation with respect to the initial crystallographic directions. The obtained results show the good abilities of the applied model in regularizing various kinds of localization bands, except for idealized slip bands. Finally, the model is applied to reproduce the complex localization behavior of single crystals undergoing single slip, where competition between kink and slip bands can take place. Both higher-order energetic and dissipative effects are considered in this investigation. For both effects, mesh-independent results are obtained, proving the good capabilities of SGCP theories in regularizing complex localization behaviors. The results associated with higher-order energetic effects are in close agreement with those obtained using a micromorphic crystal plasticity approach. Higher-order dissipative effects led to different results with dominant slip banding.

scholarly journals A classification of higher-order strain-gradient models – linear analysis

2002 ◽  
Vol 72 (2-3) ◽  
pp. 171-188 ◽  
Author(s):  
H. Askes ◽  
A. S. J. Suiker ◽  
L. J. Sluys
Keyword(s):  
Strain Gradient ◽  
Linear Analysis ◽  
Higher Order ◽  
Gradient Models

scholarly journals Homogenization of an elastic material reinforced by very strong fibres arranged along a periodic lattice

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200620
Author(s):  
Houssam Abdoul-Anziz ◽  
Lukáš Jakabčin ◽  
Pierre Seppecher
Keyword(s):  
Strain Gradient ◽  
Periodic Lattice ◽  
Minimizing Sequences ◽  
Elastic Structures ◽  
Generalized Continua ◽  
Minimization Problems ◽  
Weak Part ◽  
Periodic Graph ◽  
The Matrix ◽  
Gradient Models

We provide in this paper homogenization results for the L 2 -topology leading to complete strain-gradient models and generalized continua. Actually, we extend to the L 2 -topology the results obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288) using a topology adapted to minimization problems set in varying domains. Contrary to (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288) we consider elastic lattices embedded in a soft elastic matrix. Thus our study is placed in the usual framework of homogenization. The contrast between the elastic stiffnesses of the matrix and the reinforcement zone is assumed to be very large. We prove that a suitable choice of the stiffness on the weak part ensures the compactness of minimizing sequences while the energy contained in the matrix disappears at the limit: the Γ-limit energies we obtain are identical to those obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288).

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

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