scholarly journals STRIPE PATTERNS IN A MODEL FOR BLOCK COPOLYMERS

2010 ◽  
Vol 20 (06) ◽  
pp. 843-907 ◽  
Author(s):  
MARK A. PELETIER ◽  
MARCO VENERONI
Keyword(s):  
Stripe Pattern ◽  
Rank One ◽  
Local Direction ◽  
Energy Formula ◽  
Limit Functional ◽  
Pattern Forming ◽  
Material Length Scale ◽  
Material Length ◽  
Jump Set ◽  
Projection Field

We consider a pattern-forming system in two space dimensions defined by an energy [Formula: see text]. The functional [Formula: see text] models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0, 1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter ε is the ratio between the intrinsic, material length-scale and the scale of the domain Ω. We show that in the limit ε → 0 any sequence uε of patterns with uniformly bounded energy [Formula: see text] becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(ε). In the limit the stripes become uniform in width and increasingly straight. Our results are formulated as a convergence theorem, which states that the functional [Formula: see text] Gamma-converges to a limit functional [Formula: see text]. This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional [Formula: see text] is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L2-norm of the divergence of the projection field, or equivalently the L2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures |∇uε| combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and Röger, Arch. Rational Mech. Anal.193 (2009) 475–537, provides the initial estimate and leads to weak measure-function pair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.

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.

Size-dependent free vibration analysis of a three-layered exponentially graded nano-/micro-plate with piezomagnetic face sheets resting on Pasternak’s foundation via MCST

2017 ◽  
Vol 22 (1) ◽  
pp. 55-86 ◽  
Author(s):  
Mohammad Arefi ◽  
Masoud Kiani ◽  
Ashraf M Zenkour
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Modified Couple Stress Theory ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length ◽  
Exponentially Graded

The present work is devoted to the free vibration analysis of elastic three-layered nano-/micro-plate with exponentially graded core and piezomagnetic face-sheets using the modified couple stress theory. To capture size-dependency for a nano-/micro-sized rectangular plate, the couple stress theory is used as a non-classical continuum theory. The rectangular elastic three-layered nano-/micro-plate is resting on Pasternak’s foundation. The present model contains one material length scale parameter and can capture the size effect. Material properties of the core are supposed to vary along the thickness direction based on the exponential function. The governing equations of motion are derived from Hamilton’s principle based on the modified couple stress theory and first-order shear deformation theory. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually the natural frequency is scrutinized for different side length ratio, thickness ratio, inhomogeneity parameter, material length scale, and parameters of foundation numerically.

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.

scholarly journals Numerical Simulations of the Hall–Petch Relationship in Aluminium Using Gradient-Enhanced Plasticity Model

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Yooseob Song ◽  
Jaeheum Yeon ◽  
Byoungjoon Na
Keyword(s):  
Grain Boundaries ◽  
Thermal Activation ◽  
Energetic Material ◽  
Gradient Plasticity ◽  
Dislocation Interaction ◽  
Petch Relation ◽  
Scale Parameters ◽  
Dimensional Finite Element ◽  
Material Length Scale ◽  
Material Length

The Hall-Petch relation in aluminium is discussed based on the strain gradient plasticity framework. The thermodynamically consistent gradient-enhanced flow rules for bulk and grain boundaries are developed using the concepts of thermal activation energy and dislocation interaction mechanisms. It is assumed that the thermodynamic microstresses for bulk and grain boundaries have dissipative and energetic contributions, and in turn, both dissipative and energetic material length scale parameters are existent. Accordingly, two-dimensional finite element simulations are performed to analyse characteristics of the Hall–Petch strengthening and the Hall–Petch constants. The proposed flow rules for the grain boundary are validated using the existing experimental data from literatures. An excellent agreement between the numerical results and the experimental measurements is obtained in the Hall–Petch plot. In addition, it is observed that the Hall–Petch constants do not remain unchanged but vary depending on the strain level.

A Modified Gradient Plasticity Theory for Micro-Bending and Micro-Torsion Size Effects

2004 ◽  
Author(s):  
George Z. Voyiadjis ◽  
Rashid K. Abu Al-Rub
Keyword(s):  
Size Effects ◽  
Scale Parameter ◽  
Plasticity Theory ◽  
Material Behavior ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Gradient Plasticity ◽  
Non Local ◽  
Material Length Scale ◽  
Material Length

The definition and magnitude of the intrinsic length scale are keys to the development of the theory of plasticity that incorporates size effects. Gradient plasticity theory with a material length scale parameter is successfully in capturing the size dependence of material behavior at the micron scale. However, a fixed value of the material length-scale is not always realistic and that different problems could require different values. Moreover, a linear coupling between the local and non-local terms in the gradient plasticity theory is not always realistic and that different problems could require different couplings. A generalized gradient plasticity model with a non-fixed length scale parameter is proposed. This model assesses the sensitivity of predictions in the way in which the local and non-local parts are coupled. The proposed model gives good predictions of the size effect in micro-bending tests of thin films and micro-torsion tests of thin wires.

Stress and free vibration analysis of piezoelectric hollow circular FG-SWBNNTs reinforced nanocomposite plate based on modified couple stress theory subjected to thermo-mechanical loadings

2017 ◽  
Vol 24 (15) ◽  
pp. 3471-3486 ◽  
Author(s):  
Mehdi Mohammadimehr ◽  
S Javad Atifeh ◽  
Borhan Rousta Navi
Keyword(s):  
Natural Frequencies ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Stress Theory ◽  
Material Length Scale Parameter ◽  
Inner Radius ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length

In this article, stresses and free-vibration behaviors of annular circular piezoelectric nanocomposite plate reinforced by functionally graded single-walled boron nitride nanotubes (FG-SWBNNTs) embedded in an elastic foundation based on modified couple stress theory (MCST) are explored. The mechanical properties of FG-SWBNNT-reinforced nanocomposite plate are assumed to be graded in the direction of thickness and estimated through the micro-mechanical approach. The governing equations are obtained using the energy method. The natural frequencies and stresses of FG-SWBNNT-reinforced nanocomposite plate are computed using the differential quadrature method (DQM). An excellent agreement is observed between the obtained results and the results in the literature. Influences of the internal radius to the external radius, the thickness to the internal radius ratio, the material length scale parameter, the functionally graded parameter, temperature changes and elastic coefficients on the natural frequencies and stresses of the hollow circular nanocomposite plate are investigated. The results of this research show that the natural frequencies of the piezoelectric nanocomposite plate increase by increasing the material length scale parameter, the elastic foundation parameters, the ratio of the inner radius to the outer radius, the ratio of the thickness to the inner radius, and decreasing the power index and temperature change. The radial stress of the nanocomposite plate varies proportionally to its mode shape. The results can be employed to design smart structures in micro-electro-mechanical systems (MEMS).

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]

The Flapwise Bending Free Vibration Analysis of Micro-rotating Timoshenko Beams Using the Differential Transform Method

2017 ◽  
Vol 24 (20) ◽  
pp. 4868-4884 ◽  
Author(s):  
Hadi Arvin
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Slenderness Ratio ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Model ◽  
Transform Method ◽  
Differential Transform ◽  
Material Length Scale ◽  
Material Length

The flapwise bending free vibration analysis of isotropic rotating Timoshenko microbeams, including the size effects, is presented in this paper. A nonclassical theory, i.e. the Modified Couple Stress theory, has been employed to include the size effect in the presented formulation. By consideration of the Timoshenko beam assumptions the shear deformation and the rotary inertia effects are taken into account. The Hamilton’s principle is applied to the obtained strain and kinetic energy relations to derive the nonlinear equations of motion and the associated boundary conditions. After nondimensionalization of the equations of motion and the corresponding boundary conditions, the linearized form of the equations of motion and the accompanied boundary conditions are developed. A semi-analytical approach, i.e. the differential transform method, is implemented to achieve the flapping and axial frequencies. The achieved results are validated via comparison with the available results in the literature. The material length scale, shear deformation consideration, rotating speed and the slenderness ratio influences on the natural frequencies are examined. The results demonstrate that the slenderness ratio and the thickness to the material length scale parameter quotient are the dominant indicators in determining the usage of the nonclassical theories against classical theories. On the other hand, the precision in determination of the higher modes frequencies motivates us to implement the Timoshenko beam model instead of the Euler–Bernoulli beam model.

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.

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