scholarly journals Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior

2020 ◽  
Vol 191-192 ◽  
pp. 434-448 ◽  
Author(s):  
J. Yvonnet ◽  
N. Auffray ◽  
V. Monchiet
Keyword(s):  
Strain Gradient ◽  
Second Order ◽  
Anisotropic Strain

Local material symmetry group for first- and second-order strain gradient fluids

2021 ◽  
pp. 108128652110216
Author(s):  
Victor A. Eremeyev
Keyword(s):  
Symmetry Group ◽  
Strain Gradient ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Mass Density ◽  
Material Model ◽  
Second Order ◽  
Material Symmetry ◽  
Local Material ◽  
Current Mass

Using an unified approach based on the local material symmetry group introduced for general first- and second-order strain gradient elastic media, we analyze the constitutive equations of strain gradient fluids. For the strain gradient medium there exists a strain energy density dependent on first- and higher-order gradients of placement vector, whereas for fluids a strain energy depends on a current mass density and its gradients. Both models found applications to modeling of materials with complex inner structure such as beam-lattice metamaterials and fluids at small scales. The local material symmetry group is formed through such transformations of a reference placement which cannot be experimentally detected within the considered material model. We show that considering maximal symmetry group, i.e. material with strain energy that is independent of the choice of a reference placement, one comes to the constitutive equations of gradient fluids introduced independently on general strain gradient continua.

scholarly journals Constitutive Model of Thin Metal Sheet in Bulging Test Considering Strain Gradient Hardening

2020 ◽  
Vol 26 (4) ◽  
pp. 415-425
Author(s):  
Wei LIANG ◽  
Tieping WEI ◽  
Xiaoxiang YANG
Keyword(s):  
Constitutive Equation ◽  
Size Effect ◽  
Strain Gradient ◽  
Large Deviation ◽  
Thin Sheet ◽  
Second Order ◽  
Thin Metal Sheet ◽  
Brass Sheet ◽  
Bulging Test ◽  
Stress Variations

The study of the size effect was one of the most important subjects in the field of micro-forming. To investigate the stress of the thin sheet in the bulging test with the second order size effect, a constitutive equation considering the strain gradient hardening was proposed. Based on the equation, the stress of the thin sheet during the bulging test was calculated by the finite element method. The bulging tests with various thicknesses of brass sheets and radiuses of punching balls were performed to verify the proposed equation. The results showed that the constitutive equation could capture the stress variations, while the simulation using the constitutive equation from the conventional theory of plasticity showed the results with large deviation from those of the experiment. It was found that the stress was sensitive to the thickness of the sheet and the radius of the punching ball in bulging test of thin brass sheet. The bulging of the thin brass sheet with a thickness below ten times of its material intrinsic length would cause the generation of the geometrically necessary dislocations, which induced the strain gradient hardening. Besides that, the decrement of the punching ball radius would also increase the inhomogeneous deformation and enhance the strain gradient hardening during the thin sheet bulging process. The strain gradient hardening during the thin sheet bulging test was related to the strain of the sheet. The hardening effect of the strain gradient was obvious when the strain was small. The strain gradient hardening should be considered in the thin sheet bulging test with the second order size effect.

Nonlinear Pull-in Instability of Rectangular Nanoplates Based on the Positive and Negative Second-Order Strain Gradient Theories with Various Edge Supports

2020 ◽  
Vol 75 (4) ◽  
pp. 317-331 ◽  
Author(s):  
A. Zabihi ◽  
R. Ansari ◽  
K. Hosseini ◽  
F. Samadani ◽  
J. Torabi
Keyword(s):  
Differential Equation ◽  
Strain Gradient ◽  
Plate Theory ◽  
Second Order ◽  
Analysis Method ◽  
Analytical Approximations ◽  
Thin Plate Theory ◽  
Homotopy Analysis ◽  
Gradient Theories ◽  
The Galerkin Method

AbstractBased on the positive and negative second-order strain gradient theories along with Kirchhoff thin plate theory and von Kármán hypothesis, the pull-in instability of rectangular nanoplate is analytically investigated in the present article. For this purpose, governing models are extracted under intermolecular, electrostatic, hydrostatic, and thermal forces. The Galerkin method is formally exerted for converting the governing equation into an ordinary differential equation. Then, the homotopy analysis method is implemented as a well-designed technique to acquire the analytical approximations for analyzing the effects of disparate parameters on the nonlinear pull-in behavior. As an outcome, the impacts of nonlinear forces on nondimensional fundamental frequency, the voltage of pull-in, and softening and hardening effects are examined comparatively.

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.

Sign in / Sign up

Export Citation Format

Share Document