On the global stability of two-dimensional, incompressible, elastic bars in uniaxial extension

2009 ◽  
Vol 466 (2116) ◽  
pp. 1167-1176 ◽  
Author(s):  
Jeyabal Sivaloganathan ◽  
Scott J. Spector
Keyword(s):  
Elastic Energy ◽  
Energy Function ◽  
Shear Bands ◽  
Uniaxial Extension ◽  
Two Dimensional ◽  
Stored Energy ◽  
Absolute Minimizer ◽  
Local Bifurcations ◽  
Maximum Value ◽  
Incompressible Hyperelastic Material

When a rectangular bar is subjected to uniaxial tension, the bar usually deforms (approximately) homogeneously and isoaxially until a critical load is reached. A bifurcation, such as the formation of shear bands or a neck, may then be observed. One approach is to model such an experiment as the in-plane extension of a two-dimensional, homogeneous, isotropic, incompressible, hyperelastic material in which the length of the bar is prescribed, the ends of the bar are assumed to be free of shear and the sides are left completely free. It is shown that standard constitutive hypotheses on the stored-energy function imply that no such bifurcation is possible in this model due to the fact that the homogeneous isoaxial deformation is the unique absolute minimizer of the elastic energy. Thus, in order for a bifurcation to occur either the material must cease to be elastic or the stored-energy function must violate the standard hypotheses. The fact that no local bifurcations can occur under the assumptions used herein was known previously, since these assumptions prohibit the load on the bar from reaching a maximum value. However, the fact that the homogeneous deformation is the absolute minimizer of the energy appears to be a new result.

An existence theorem for a two-dimensional nonlinear shell model of Koiter’s type

2018 ◽  
Vol 28 (14) ◽  
pp. 2833-2861 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare
Keyword(s):  
Shell Model ◽  
Energy Function ◽  
Vector Fields ◽  
Elastic Shell ◽  
Existence Theory ◽  
Specific Class ◽  
Two Dimensional ◽  
Stored Energy ◽  
Nonlinear Shell ◽  
Order Part

We propose a minimization problem with a stored energy function that is polyconvex and satisfies all the other assumptions of John Ball’s theorem, while being at the same time well adapted for modeling a nonlinearly elastic shell. By restricting the admissible deformations to be specific quadratic polynomials with respect to the transverse variable, we are able to define a new nonlinear shell model for which a satisfactory existence theory is available and that is still two-dimensional, in the sense that minimizing the corresponding total energy amounts to finding three vector fields defined on the closure of a bounded open subset of [Formula: see text]. The most noteworthy feature of our nonlinear shell model is that the “lowest order part” of its stored energy function coincides, at least formally, with the stored energy function found in Koiter’s model for a specific class of deformations that are to within the first-order identical to the Kirchhoff–Love deformations considered by W. T. Koiter.

NONLINEARLY ELASTIC THIN PLATE MODELS FOR A CLASS OF OGDEN MATERIALS: I

2005 ◽  
Vol 03 (02) ◽  
pp. 195-221 ◽  
Author(s):  
KARIM TRABELSI
Keyword(s):  
Thin Plate ◽  
Energy Function ◽  
Plate Theory ◽  
Deformation Gradient ◽  
Two Dimensional ◽  
Stored Energy ◽  
Nonlinear Materials ◽  
Plate Models ◽  
Elastic Thin Plate ◽  
Nonlinearly Elastic

A new two-dimensional nonlinear membrane plate theory is derived via a formal asymptotic procedure for a family of hyperelastic nonlinear materials proposed by Ciarlet and Geymonat [11], whose stored energy function is polyconvex and becomes infinite, when the determinant of the deformation gradient tends to zero, and can be adjusted to arbitrary Lamé constants.

On Thermodynamics and Kinetic Theory of Ideal Rubber Membranes

1981 ◽  
Vol 48 (2) ◽  
pp. 345-350
Author(s):  
W. Dreyer
Keyword(s):  
Kinetic Model ◽  
Kinetic Theory ◽  
Experimental Evidence ◽  
Energy Function ◽  
Surface Stress ◽  
Arbitrary Shape ◽  
Two Dimensional ◽  
Stored Energy ◽  
Rubber Membrane ◽  
Dimensional Surface

Based on experimental evidence and thermodynamics it will be shown that the stored energy function of an ideal rubber membrane is determined by the entropy alone. The membrane is represented by a two-dimensional surface for the purposes of thermodynamics, and its thickness is taken into account by a scalar parameter so that incompressibility of the membrane can be described. The entropy of the membrane is calculated from a kinetic model and hence results the surface stress as a function of temperature and deformation for arbitrary shape of the membrane.

Dripping instability of a two-dimensional liquid film under an inclined plate

2021 ◽  
Vol 932 ◽  
Author(s):  
Guangzhao Zhou ◽  
Andrea Prosperetti
Keyword(s):  
Liquid Film ◽  
Initial Conditions ◽  
Strong Dependence ◽  
Bond Number ◽  
Quasi Equilibrium ◽  
Two Dimensional ◽  
Inclined Plate ◽  
Maximum Value ◽  
Critical Curvature ◽  
Rayleigh Taylor Instability

It is known that the dripping of a liquid film on the underside of a plate can be suppressed by tilting the plate so as to cause a sufficiently strong flow. This paper uses two-dimensional numerical simulations in a closed-flow framework to study several aspects of this phenomenon. It is shown that, in quasi-equilibrium conditions, the onset of dripping is closely associated with the curvature of the wave crests approaching a well-defined maximum value. When dynamic effects become significant, this connection between curvature and dripping weakens, although the critical curvature remains a useful reference point as it is intimately related to the short length scales promoted by the Rayleigh–Taylor instability. In the absence of flow, when the film is on the underside of a horizontal plate, the concept of a limit curvature is relevant only for small liquid volumes close to a critical value. Otherwise, the drops that form have a smaller curvature and a large volume. The paper also illustrates the peculiarly strong dependence of the dripping transition on the initial conditions of the simulations. This feature prevents the development of phase maps dependent only on the governing parameters (Reynolds number, Bond number, etc.) similar to those available for film flow on the upper side of an inclined plate.

Two-dimensional pressure field and backflow in the annular skirt of vortex gripper

2020 ◽  
pp. 095440622097404
Author(s):  
Jianghong Zhao ◽  
Xin Li
Keyword(s):  
Flow Field ◽  
Pressure Distribution ◽  
Local Stress ◽  
Vortex Chamber ◽  
Physical Contact ◽  
Two Dimensional ◽  
Stress Concentrations ◽  
Suction Force ◽  
Maximum Value ◽  
Gap Height

The vortex gripper is a kind of pneumatic noncontact gripper that does not produce a magnetic field and heat. It can grip a workpiece without physical contact, which avoids any unintentional damage such as mechanical scratches, local stress concentrations, frictional static electricity, and surface stains. This study focused on the two-dimensional pressure distribution field on a workpiece surface under the vortex gripper. Theoretical, experimental, and computational fluid dynamics results were combined to study the backflow phenomenon in the annular skirt, which can decrease the gripper’s suction force after the maximum value is reached. First, the pressure distribution in the annular skirt was theoretically modeled. A comparison with the experimental results showed that increasing the gap height between the gripper and workpiece generates a circumferentially asymmetrical flow field in the skirt. Based on this, it was hypothesized that an airflow in the circumferential direction may exist. The experimental data and simulation results were analyzed under large gap height conditions to observe the backflow in detail and it was found that an uneven pressure distribution with positive and negative pressure regions generated by the uneven flow is the root cause of the backflow. Finally, the effect of the backflow on the flow field in two different flow regions (in the annular skirt and inside the vortex chamber) was analyzed and the reason why the suction force of the vortex gripper has a maximum value was determined.

Generalization of Experimental Data for Compressor Cascades at Low Speeds

1970 ◽  
Author(s):  
V. S. Beknev
Keyword(s):  
Experimental Data ◽  
Pressure Distribution ◽  
Loss Coefficient ◽  
Two Dimensional ◽  
Minimum Loss ◽  
Maximum Value ◽  
Isentropic Exponent ◽  
Loss Coefficients ◽  
Low Speeds ◽  
Drag Ratio

The author compares three different approaches for generalization of experimental data for two-dimensional compressor cascades at low speeds: generalization for maximum value of lift-drag ratio, generalization for maximum cascade quality, and generalization for minimum loss coefficient. Some results given, of comparison for incidence and deviation angles, solidities, and loss coefficients, show the largest difference to be for incidence angles and loss coefficients. Influence of isentropic exponent on the airfoil pressure distribution and cascade losses is considered.

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