Localization-Induced Band and Cohesive Model1

2000 ◽  
Vol 67 (4) ◽  
pp. 803-812 ◽  
Author(s):  
S. Hao ◽  
W. K. Liu ◽  
D. Qian
Keyword(s):  
Strain Gradient ◽  
Multiple Scale ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Finite Width ◽  
Opening Displacement ◽  
One Dimensional ◽  
Material Length Scale ◽  
Material Length

A localization-induced cohesive model has been proposed for shear band evolution, crack growth, and fracture. Strain gradient theory has been applied to establish the criterion of the onset of localization and the governing equation in the post-bifurcation stage. Analytical solutions in one-dimensional case are used to establish the “traction-separation” law, in which strain gradient and material intrinsic length scale present strong effects. In addition, the solution predicts a finite width for the localization-induced band. It is observed that a larger length scale contributes to the growth of a larger width of localization region and separation for softening materials. The proposed model provides a procedure to establish the fracture toughness analytically since the material length scale is taken into account. From the traction-separation analysis, it is found that damage decreases separation, whereas an increase in material length scale increases the opening displacement; however, the traction-normalized opening displacement curves (with respect to the material length scale) are identical. Based on the methodology of multiple scale analysis in meshfree method, a computational approach has been proposed to enrich the one-dimensional traction-separation law to define fracture. [S0021-8936(00)01104-1]

Study of Small Scale Effects on the Nonlinear Vibration Response of Functionally Graded Timoshenko Microbeams Based on the Strain Gradient Theory

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
R. Ansari ◽  
R. Gholami ◽  
S. Sahmani
Keyword(s):  
Strain Gradient ◽  
Beam Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Scale Parameters ◽  
Linear And Nonlinear ◽  
Material Length Scale ◽  
Material Length

In the current study, the nonlinear free vibration behavior of microbeams made of functionally graded materials (FGMs) is investigated based on the strain gradient elasticity theory and von Karman geometric nonlinearity. The nonclassical beam model is developed in the context of the Timoshenko beam theory which contains material length scale parameters to take the size effect into account. The model can reduce to the beam models based on the modified couple stress theory (MCST) and the classical beam theory (CBT) if two or all material length scale parameters are taken to be zero, respectively. The power low function is considered to describe the volume fraction of the ceramic and metal phases of the FGM microbeams. On the basis of Hamilton’s principle, the higher-order governing differential equations are obtained which are discretized along with different boundary conditions using the generalized differential quadrature method. The dimensionless linear and nonlinear frequencies of microbeams with various values of material property gradient index are calculated and compared with those obtained based on the MCST and an excellent agreement is found. Moreover, comparisons between the various beam models on the basis of linear and nonlinear types of strain gradient theory (SGT) and MCST are presented and it is observed that the difference between the frequencies obtained by the SGT and MCST is more significant for lower values of dimensionless length scale parameter.

Influence of micro-length-scale parameters and inhomogeneities on the bending, free vibration and wave propagation analyses of a FG Timoshenko’s sandwich piezoelectric microbeam

2017 ◽  
Vol 21 (4) ◽  
pp. 1243-1270 ◽  
Author(s):  
Mohammad Arefi ◽  
Ashraf M Zenkour
Keyword(s):  
Wave Propagation ◽  
Free Vibration ◽  
Equations Of Motion ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Governing Equations ◽  
Scale Parameters ◽  
Material Length Scale ◽  
Material Length

In this study, the strain gradient theory is employed to derive governing equations of motion of a functionally graded Timoshenko’s sandwich microbeam resting on Pasternak’s foundation. The microbeam is including a micro-core and two piezoelectric face-sheets on top and bottom. The plate is actuated with applied electric potential at top of piezoelectric face-sheets. The governing equations of motion are derived using Hamilton’s principle and strain gradient theory. After derivation of governing equations of motion, the problem is solved for three classes of analysis including wave propagation, free vibration and bending analysis. The numerical results are presented to reflect the effect of important parameters such as wave number, applied voltage, inhomogeneous index, parameters of foundation and material length-scale parameters on the different responses. The obtained results indicated that changing material length-scale parameters leads to a stiffer structure that increase natural frequencies and decreases transverse deflection and maximum electric potential.

Stability of Vibrating Functionally Graded Nanoplates with Axial Motion Based on the Nonlocal Strain Gradient Theory

2020 ◽  
Vol 20 (08) ◽  
pp. 2050088 ◽  
Author(s):  
J. P. Shen ◽  
P. Y. Wang ◽  
W. T. Gan ◽  
C. Li
Keyword(s):  
Strain Gradient ◽  
Critical Speed ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Length Scale ◽  
Gradient Index ◽  
Axial Motion ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

Considering both the nonlocal scale and material length scale effects, we investigate vibration and stability behaviors of functionally graded nanoplates with axial motion in order to model the two-dimensional nanobelt in nanoengineering. The nonlocal strain gradient theory is applied and the differential nonlocal strain gradient constitutive relation is adopted. Using the physical neutral plane of a functionally graded thin plate, we derive the governing equation of motion for the functionally graded nanoplate with axial motion via Hamilton’s principle, where the kinematic characteristics are introduced into the dynamic behaviors. The governing equation is numerically solved using the Galerkin method. Effects of the nonlocal scale and material length scale parameters, axial velocity, gradient index, biaxial pre-tensions and aspect ratio are discussed. The results demonstrate that complex frequencies of the functionally graded nanoplate with axial motion decrease with an increase of axial velocity in the subcritical region, while the moving nanoplate experiences a divergent instability or flutter instability in the supercritical region. Natural frequency and critical speed decrease with the increase of the nonlocal scale parameter while increase with the increase of the material length scale parameter, reflecting the nonlocal softening and strain gradient hardening mechanisms, respectively. Besides, natural frequency and critical speed increase with the increase of the biaxial pre-tensions and aspect ratio, but decrease with the increase of the gradient index. In particular, the influences of the gradient index and size or weight of functionally graded nanoplates on the critical speed are explored.

On size-dependent bending behaviors of shape memory alloy microbeams via nonlocal strain gradient theory

2021 ◽  
pp. 1045389X2098699
Author(s):  
Bo Zhou ◽  
Zetian Kang ◽  
Xiao Ma ◽  
Shifeng Xue
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Size Dependent ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

This paper focuses on the size-dependent behaviors of functionally graded shape memory alloy (FG-SMA) microbeams based on the Bernoulli-Euler beam theory. It is taken into consideration that material properties, such as austenitic elastic modulus, martensitic elastic modulus and critical transformation stresses vary continuously along the longitudinal direction. According to the simplified linear shape memory alloy (SMA) constitutive equations and nonlocal strain gradient theory, the mechanical model was established via the principle of virtual work. Employing the Galerkin method, the governing differential equations were numerically solved. The functionally graded effect, nonlocal effect and size effect of the mechanical behaviors of the FG-SMA microbeam were numerically simulated and discussed. Results indicate that the mechanical behaviors of FG-SMA microbeams are distinctly size-dependent only when the ratio of material length scale parameter to the microbeam height is small enough. Both the increments of material nonlocal parameter and ratio of material length-scale parameter to the microbeam height all make the FG-SMA microbeam become softer. However, the stiffness increases with the increment of FG parameter. The FG parameter plays an important role in controlling the transverse deformation of the FG-SMA microbeam. This work can provide a theoretical basis for the design and application of FG-SMA microstructures.

USE OF STRAIN GRADIENT THEORY FOR MODELING THE SIZE-DEPENDENT PULL-IN OF ROTATIONAL NANO-MIRROR IN THE PRESENCE OF MOLECULAR FORCE

2013 ◽  
Vol 27 (18) ◽  
pp. 1350083 ◽  
Author(s):  
Y. TADI BENI ◽  
M. ABADYAN
Keyword(s):  
Strain Gradient ◽  
Continuum Theory ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Cross Sectional ◽  
Size Dependent ◽  
Proposed Model ◽  
Molecular Force ◽  
Intrinsic Material Length ◽  
Material Length

Experiments reveal that mechanical behavior of nanostructures is size-dependent. Herein, the size dependent pull-in instability of torsional nano-mirror is investigated using strain gradient nonclassic continuum theory. The governing equation of the mirror is derived taking the effect of electrostatic Coulomb and molecular van der Waals (vdW) forces into account. Variation of the rotation angle of the mirror as a function of the applied voltage is obtained and the instability parameters i.e., pull-in voltage and pull-in angle are determined. Nano-mirrors with square and circular cross-sectional beams are investigated as case studies. It is found that when the thickness of the torsional nano-beam is comparable with the intrinsic material length scales, size effect can substantially increase the instability parameters of the rotational mirror. Moreover, the effect of vdW forces on the size-dependent pull-in instability of the system is discussed. The proposed model is able to predict the experimental results more accurately than the previous classic models and reduce the gap between experiment and previous theories.

Torsional Vibration Analysis of Carbon Nanotubes Based on the Strain Gradient Theory and Molecular Dynamic Simulations

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
R. Ansari ◽  
R. Gholami ◽  
S. Ajori
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Molecular Dynamic ◽  
Strain Gradient ◽  
Torsional Vibration ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Molecular Dynamic Simulations ◽  
Dynamic Simulations

In the current study, the torsional vibration of carbon nanotubes is examined using the strain gradient theory and molecular dynamic simulations. The model developed based on this gradient theory enables us to interpret size effect through introducing material length scale parameters. The model accommodates the modified couple stress and classical models when two or all material length scale parameters are set to zero, respectively. Using Hamilton's principle, the governing equation and higher-order boundary conditions of carbon nanotubes are obtained. The generalized differential quadrature method is utilized to discretize the governing differential equation of the present model along with two boundary conditions. Then, molecular dynamic simulations are performed for a series of carbon nanotubes with different aspect ratios and boundary conditions, the results of which are matched with those of the present strain gradient model to extract the appropriate value of the length scale parameter. It is found that the present model with properly calibrated value of length scale parameter has a good capability to predict the torsional vibration behavior of carbon nanotubes.

scholarly journals A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

2019 ◽  
Vol 40 (12) ◽  
pp. 1695-1722 ◽  
Author(s):  
Lu Lu ◽  
Li Zhu ◽  
Xingming Guo ◽  
Jianzhong Zhao ◽  
Guanzhong Liu
Keyword(s):  
Strain Gradient ◽  
Scale Parameter ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Length Scale ◽  
Proposed Model ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

AbstractIn this paper, a novel size-dependent functionally graded (FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton’s principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.

Resonant Frequency and Sensitivity of a Caliper Formed With Assembled Cantilever Probes Based on the Modified Strain Gradient Theory

2014 ◽  
Vol 20 (6) ◽  
pp. 1672-1681 ◽  
Author(s):  
Mohammad Abbasi ◽  
Seyed E. Afkhami
Keyword(s):  
Resonant Frequency ◽  
Strain Gradient ◽  
Couple Stress ◽  
Contact Stiffness ◽  
Current Model ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
The Difference ◽  
Material Length Scale ◽  
Beam Theories

AbstractThe resonant frequency and sensitivity of an atomic force microscope (AFM) with an assembled cantilever probe (ACP) is analyzed utilizing strain gradient theory, and then the governing equation and boundary conditions are derived by a combination of the basic equations of strain gradient theory and Hamilton’s principle. The resonant frequency and sensitivity of the proposed AFM microcantilever are then obtained numerically. The proposed ACP includes a horizontal cantilever, two vertical extensions, and two tips located at the free ends of the extensions that form a caliper. As one of the extensions is located between the clamped and free ends of the AFM microcantilever, the cantilever is modeled as two beams. The results of the current model are compared with those evaluated by both modified couple stress and classical beam theories. The difference in results evaluated by the strain gradient theory and those predicted by the couple stress and classical beam theories is significant, especially when the microcantilever thickness is approximately the same as the material length-scale parameters. The results also indicate that at the low values of contact stiffness, scanning in the higher cantilever modes decrease the accuracy of the proposed AFM ACP.

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