Corotating or Codeforming Models for Thermoforming: Free Forming

Author(s):  
H. M. Baek ◽  
A. J. Giacomin

Our previous work [J Pol Eng, 32, 245 (2012)] explores the role of viscoelasticity for the simplest relevant problem in thermoforming, the manufacture of cones. In this previous work, we use a differential model employing the corotational derivative [the corotational Maxwell model (CM)] for which we find an analytical solution for the sheet deformation as a function of time. This previous work also identifies the ordinary nonlinear differential equation corresponding to the upper convected Maxwell model (UCM), for which she finds no analytical solution. In this paper, we explore the role of convected derivative by solving this UCM equation numerically by finite difference. We extend the previous work to include sag by incorporating a finite initial sheet curvature. We treat free forming step in thermoforming and find that the convected derivative makes the free forming time unreasonably sensitive to the initial curvature. Whereas, for the CM model, we get a free forming time that is independent of initial sheet curvature, so long as the sheet is nearly flat to begin with. We cast our results into dimensionless plots of thermoforming times versus disk radius of curvature.

1998 ◽  
Vol 120 (3) ◽  
pp. 743-751 ◽  
Author(s):  
G. P. Peterson ◽  
J. M. Ha

The capillary flow along a microgroove channel was investigated both analytically and experimentally. In order to obtain insight into the phenomena, and because the governing equation had the form of a nonlinear differential equation, an analytical solution and approximate algebraic model were developed rather than using numerical methods. Approximating the governing equation as a Bernoulli differential equation resulted in an analytical solution for the radius curvature as the cube root of an exponential function. The axial variation of the radius of curvature profile as determined by this method was very similar to the numerical result as was the algebraic solution. However, the analytical model predicted the meniscus dryout location to be somewhat shorter than either the numerical results or the results from the algebraic solution. To verify the modeling results, the predictions for the axial capillary performance were compared to the results of the experimental investigation. The results of this comparison indicated that the experimentally measured wetted length was approximately 80 percent of the value predicted by the algebraic expression. Not only did the prediction for the dryout location from the algebraic equation show good agreement with the experimental data, but more importantly, the expression did not require any experimentally correlated constants. A nondimensionalized expression was developed as a function of just one parameter which consists of the Bond number, the Capillary number, and the dimensionless groove shape geometry for use in predicting the flow characteristics in this type of flow.


2013 ◽  
Vol 24 (3) ◽  
pp. 437-453 ◽  
Author(s):  
CARLOS ESCUDERO ◽  
ROBERT HAKL ◽  
IRENEO PERAL ◽  
PEDRO J. TORRES

We present the formal geometric derivation of a non-equilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of non-equilibrium statistical mechanics.


2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Liecheng Sun ◽  
Issam E. Harik

AbstractAnalytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.


Materials ◽  
2018 ◽  
Vol 11 (12) ◽  
pp. 2506 ◽  
Author(s):  
Chao Liu ◽  
Yaoyao Shi

Dimensional control can be a major concern in the processing of composite structures. Compared to numerical models based on finite element methods, the analytical method can provide a faster prediction of process-induced residual stresses and deformations with a certain level of accuracy. It can explain the underlying mechanisms. In this paper, an improved analytical solution is proposed to consider thermo-viscoelastic effects on residual stresses and deformations of flat composite laminates during curing. First, an incremental differential equation is derived to describe the viscoelastic behavior of composite materials during curing. Afterward, the analytical solution is developed to solve the differential equation by assuming the solution at the current time, which is a linear combination of the corresponding Laplace equation solutions of all time. Moreover, the analytical solution is extended to investigate cure behavior of multilayer composite laminates during manufacturing. Good agreement between the analytical solution results and the experimental and finite element analysis (FEA) results validates the accuracy and effectiveness of the proposed method. Furthermore, the mechanism generating residual stresses and deformations for unsymmetrical composite laminates is investigated based on the proposed analytical solution.


Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.


2017 ◽  
Vol 82 (5) ◽  
pp. 469-481 ◽  
Author(s):  
Slobodan Zdravkovic

In the present paper we deal with nonlinear dynamics of microtubules. The structure and role of microtubules in cells are explained as well as one of models explaining their dynamics. Solutions of the crucial nonlinear differential equation depend on used mathematical methods. Two commonly used procedures, continuum and semi-discrete approximations, are explained. These solutions are solitary waves usually called as kink solitons, breathers and bell-type solitons.


Vestnik MGSU ◽  
2015 ◽  
pp. 72-83
Author(s):  
Armen Zavenovich Ter-Martirosyan ◽  
Zaven Grigor’evich Ter-Martirosyan ◽  
Tuan Viet Trinh

The article presents the formulation and analytical solution to a quantification of stress strain state of a two-layer soil cylinder enclosing a long pile, interacting with the cap. The solution of the problem is considered for two cases: with and without account for the settlement of the heel and the underlying soil. In the first case, the article is offering equations for determining the stresses of pile’s body and the surrounding soil according to their hardness and the ratio of radiuses of the pile and the surrounding soil cylinder, as well as formulating for determining equivalent deformation modulus of the system “cap-pile-surrounding soil” (the system). Assessing the carrying capacity of the soil under pile’s heel is of great necessity. In the second case, the article is solving a second-order differential equation. We gave the formulas for determining the stresses of the pile at its top and heel, as well as the variation of stresses along the pile’s body. The article is also formulating for determining the settlement of the foundation cap and equivalent deformation modulus of the system. It is shown that, pushing the pile into underlying layer results in the reducing of equivalent modulus of the system.


2019 ◽  
Vol 7 (2) ◽  
pp. 97-102 ◽  
Author(s):  
Miroslav Vasilev ◽  
Galya Shivacheva

This article analyzes the process of changing the concentration of enrofloxacin in blood plasma in dogs after a single intravenous injection of the substance. Three mathematical models are proposed - algebraic and two models, based on a differential equation of first and second order. Identification of their parameters has been performed. Based on Akaike information criterion corrected as the best model was chosen the represented by a second-order differential equation. Three equations are identified and the exact numerical values of their parameters are obtained. For the evaluation and comparison of the three models, Akaike information criterion was used. The best results showed the second-order differential model. It will be used in future developments.


2009 ◽  
Vol 79-82 ◽  
pp. 1205-1208 ◽  
Author(s):  
Cheng Zhang ◽  
Lin Xiang Wang

In the current paper, the hysteretic dynamics of magnetorheological dampers is modeled by a differential model. The differential model is constructed on the basis of a phenomenological phase transition theory. The model is expressed as a second order nonlinear ordinary differential equation with bifurcations embedded in. Due to the differential nature of the model, the hysteretic dynamics of the MR dampers can be linearized and controlled by introducing a feedback linearization strategy.


Sign in / Sign up

Export Citation Format

Share Document