scholarly journals An example of microstructure with multiple scales

1997 ◽  
Vol 8 (2) ◽  
pp. 185-207
Author(s):  
MATTHIAS WINTER
Keyword(s):  
Singular Perturbation ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Convex Function ◽  
Calculus Of Variations ◽  
Multiple Scales ◽  
Young Measure ◽  
Linear Boundary ◽  
Minimizing Sequence ◽  
Martensitic Microstructure

This paper studies a vectorial problem in the calculus of variations arising in the theory of martensitic microstructure. The functional has an integral representation where the integrand is a non-convex function of the gradient with exactly four minima. We prove that the Young measure corresponding to a minimizing sequence is homogeneous and unique for certain linear boundary conditions. We also consider the singular perturbation of the problem by higher-order gradients. We study an example of microstructure involving infinite sequential lamination and calculate its energy and length scales in the zero limit of the perturbation.

Lower semicontinuity of multiple integrals and the Biting Lemma

1990 ◽  
Vol 114 (3-4) ◽  
pp. 367-379 ◽  
Author(s):  
J. M. Ball ◽  
K.-W. Zhang
Keyword(s):  
Integral Representation ◽  
Calculus Of Variations ◽  
Lower Semicontinuity ◽  
Young Measure ◽  
Weak Limit ◽  
Periodic Case ◽  
Quasiconvex Functions ◽  
General Sequence ◽  
Weak Lower Semicontinuity ◽  
Multiple Integrals

SynopsisWeak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.

scholarly journals Nonlinear Free and Forced Vibrations of a Hyperelastic Micro/Nanobeam Considering Strain Stiffening Effect

Nanomaterials ◽  
2021 ◽  
Vol 11 (11) ◽  
pp. 3066
Author(s):  
Amin Alibakhshi ◽  
Shahriar Dastjerdi ◽  
Mohammad Malikan ◽  
Victor A. Eremeyev
Keyword(s):  
Size Effect ◽  
Boundary Conditions ◽  
Frequency Response ◽  
Multiple Scales ◽  
Forced Vibrations ◽  
Response Force ◽  
Nonlinear Frequency ◽  
Large Rotation ◽  
Free And Forced Vibrations ◽  
Stiffening Effect

In recent years, the static and dynamic response of micro/nanobeams made of hyperelasticity materials received great attention. In the majority of studies in this area, the strain-stiffing effect that plays a major role in many hyperelastic materials has not been investigated deeply. Moreover, the influence of the size effect and large rotation for such a beam that is important for the large deformation was not addressed. This paper attempts to explore the free and forced vibrations of a micro/nanobeam made of a hyperelastic material incorporating strain-stiffening, size effect, and moderate rotation. The beam is modelled based on the Euler–Bernoulli beam theory, and strains are obtained via an extended von Kármán theory. Boundary conditions and governing equations are derived by way of Hamilton’s principle. The multiple scales method is applied to obtain the frequency response equation, and Hamilton’s technique is utilized to obtain the free undamped nonlinear frequency. The influence of important system parameters such as the stiffening parameter, damping coefficient, length of the beam, length-scale parameter, and forcing amplitude on the frequency response, force response, and nonlinear frequency is analyzed. Results show that the hyperelastic microbeam shows a nonlinear hardening behavior, which this type of nonlinearity gets stronger by increasing the strain-stiffening effect. Conversely, as the strain-stiffening effect is decreased, the nonlinear frequency is decreased accordingly. The evidence from this study suggests that incorporating strain-stiffening in hyperelastic beams could improve their vibrational performance. The model proposed in this paper is mathematically simple and can be utilized for other kinds of micro/nanobeams with different boundary conditions.

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

1997 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Walter Lacarbonara ◽  
Char-Ming Chin
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Buckling Mode ◽  
Stable Mode ◽  
One To One ◽  
Nonlinear Normal

Abstract Nonlinear normal modes of a buckled beam about its first buckling mode shape are investigated. Fixed-fixed boundary conditions are considered. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate expressions for the nonlinear normal modes are obtained by applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Curves displaying variation of the amplitude with the internal resonance detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess either one stable mode, or three stable normal modes, or two stable and one unstable normal modes. On the other hand, for a one-to-one internal resonance between the first and second modes, two nonlinear normal modes exist. The two nonlinear modes are either neutrally stable or unstable. In the case of one-to-one resonance between the third and fourth modes, two neutrally stable, nonlinear normal modes exist.

On the Existence and the Uniform Decay of a Hyperbolic Equation with Non-Linear Boundary Conditions

2000 ◽  
Vol 24 (2) ◽  
pp. 183-199 ◽  
Author(s):  
M. M. Cavalcanti ◽  
V. N. Domingos Cavalcanti ◽  
J. A. Soriano ◽  
L. A. Medeiros
Keyword(s):  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Linear Boundary ◽  
Uniform Decay ◽  
Non Linear
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