Rheological equations of state of weak solutions of polymers as rigid ellipsoidal macromolecules

Rheological equations of state are obtained for weak solutions of polymers as rigid ellipsoidal macromolecules, taking into account the rotational Brownian motion of the macromolecules, their energy, and the external force fields (electric and magnetic). As an example the effect of the inertia of the macromolecules on the rheological behavior of the solutions is examined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 125–129, March–April, 1972.     

Interfacial welding of dynamic covalent network polymers

Dynamic covalent network (or covalent adaptable network) polymers can rearrange their macromolecular chain network by bond exchange reactions (BERs) where an active unit replaces a unit in an existing bond to form a new bond. Such macromolecular events, when they occur in large amounts, can attribute to unusual properties that are not seen in conventional covalent network polymers, such as shape reforming and surface welding; the latter further enables the important attributes of material malleability and powder-based reprocessing. In this paper, a multiscale modeling framework is developed to study the surface welding of thermally induced dynamic covalent network polymers. At the macromolecular network level, a lattice model is developed to describe the chain density evolution across the interface and its connection to bulk stress relaxation due to BERs. The chain density evolution rule is then fed into a continuum level interfacial model that takes into account surface roughness and applied pressure to predict the effective elastic modulus and interfacial fracture energy of welded polymers. The model yields particularly accessible results where the moduli and interfacial strength of the welded samples as a function of temperature and pressure can be predicted with four parameters, three of which can be measured directly. The model identifies the dependency of surface welding efficiency on the applied thermal and mechanical fields: the pressure will affect the real contact area under the consideration of surface roughness of dynamic covalent network polymers; the chain density increment on the real contact area of interface is only dependent on the welding time and temperature. The modeling approach shows good agreement with experiments and can be extended to other types of dynamic covalent network polymers using different stimuli for BERs, such as light and moisture etc.     

Dynamics of spatial structure-varying rigid multibody systems

Summary Couplings in machines and mechanisms exhibiting backlash and friction phenomena can be modeled as multibody systems with unilateral constraints and Coulomb friction. The structure of the differential-algebraic equations describing the system depends on the state of the constraints. The contact forces occurring at active constraints are taken into account in the equations of motion as Lagrange multipliers. Additionally, the kinematic conditions of all active constraints are formulated on the acceleration level. Contact and friction laws are sufficient conditions for state transitions of active constraints, and are represented by nonsmooth characteristics. Several formulations, like the linear complementarity problem, and two different nonlinear systems of equations are presented together with their solution method. The theory is applied to a mechanical system containing three-dimensional and coupled unilateral constraints with friction. Received 14 May 1998; accepted for publication 5 January 1999     

Minimum Time Control of Flexible Spacecraft by Hamilton's Principle

Modern space vehicles structure requisites are getting more and more stringent and complex as mission tasks become more sophisticated. This leads to the necessity of developing analysis methods that take into account structure flexibility and the need of reducing manoeuvre time as much as possible. In this work, a method based on the Hamilton Principle in its weak mixed form is developed, in which co-ordinates derivatives do not appear, but only their virtual variations. The proposed formulation is able to take into account system flexibility and saturation constraints on control torques and forces. A non-linear variational condition is obtained, which can be solved by means of a time-finite-element technique to give the minimum-time solutions of the control problem. The solutions for slewing manoeuvres are given, along with a new solution of the distributed optimal control problem.     

Stabilization of a class of switched systems with state constraints

This paper investigates the stabilization problem for a class of switched systems with state constraints in both continuous-time and discrete-time contexts. The state constraints are converted into state saturations by limiting the state in a unit hypercube. An improved average dwell time method is presented to take into account different decay rates of a Lyapunov function related to an active subsystem according to the saturations occurring or not. Sufficient conditions for stability and stabilizability of the switched system with state constraints are derived; meanwhile, the stabilizing state feedback controllers are designed. An application to a longitudinal motion of highly maneuverable aircraft technology (HiMAT) vehicle is given to illustrate the applicability and the effectiveness of the proposed method.     

Multi-scale constitutive modeling of the small deformations of semi-crystalline polymers

A multi-scale constitutive model for the small deformations of semi-crystalline polymers such as high density Polyethylene is presented. Each macroscopic material point is supposed to be the center of a representative volume element which is an aggregate of randomly oriented composite inclusions. Each inclusion consists of a stack of parallel crystalline lamellae with their adjacent amorphous layers.Micro-mechanically based constitutive equations are developed for each phase. A viscoplastic model is used for the crystalline lamellae. A new nonlinear viscoelastic model for the amorphous phase behavior is proposed. The model takes into account the fact that the presence of crystallites confines the amorphous phase in extremely thin layers where the concentration of chain entanglements is very high. This gives rise to a stress contribution due to elastic distortion of the chains. It is shown that the introduction of chains’ elastic distortion can explain the viscoelastic behavior of crystalline polymers. The stress contribution from elastic stretching of the tie molecules linking the neighboring lamellae is also taken into account.Next, a constitutive model for a single inclusion considered as a laminated composite is proposed. The macroscopic stress-strain behavior for the whole RVE is found via a Sachs homogenization scheme (uniform stress throughout the material is assumed).Computational algorithms are developed based on fully implicit time-discretization schemes.     

Macroscopic thermodynamics of porous media

A macroscopic thermodynamical theory of unsaturated porous media is studied. The porous media under consideration contain air, liquid water and its vapour encountering phase changes. To the classical state quantities, viz. temperature and densities, we add the volume fractions. These fractions are submitted to constraints, for instance their values lie between 0 and 1. The constraints are taken into account by the free energies which become non-smooth functions. The constitutive laws are obtained by classical thermodynamics and a careful treatment of the non-smooth functions. The constitutive laws account for experimental properties of unsaturated porous media: hysteresis of the sorption and desorption, capillary suction ...     

The stochastic elastica and excluded-volume perturbations of DNA conformational ensembles

A coordinate-free Lie-group formulation for generating ensembles of DNA conformations in solution is presented. In this formulation, stochastic differential equations define sample paths on the Euclidean motion group. The ensemble of these paths exhibits the same behavior as solutions of the Fokker–Planck equation for the stochastically forced elastica. Longer chains for which the effects of excluded volume become important are handled by piecing together shorter chains and modeling their interactions. It is assumed that the final chain lengths of interest are long enough for excluded-volume effects to become important, but not so long that the semi-flexible nature of the chain is lost. The effect of excluded volume is then taken into account by grouping short self-avoiding conformations into “bundles” with common end constraints and computing average interaction effects between bundles. The accuracy of this approximation is shown to be good when using a numerically generated ensemble of self-avoiding sample paths as the baseline for comparison.     

Optimization of Axisymmetric Membrane Shells Against Brittle Fracture

The questions investigated in this paper are related to an important class of problems of optimal design of structures against brittle fracture. The primary problem of axisymmetric shell optimization under fracture mechanics constraint is formulated as the weight (volume of the shell material) minimization under stress intensity constraints. Considered problems are characterized by incomplete information concerning crack size, crack location and its orientation. Taking into account the factor of incomplete information the paper presents the formulation of optimal shell design problem based on minimax (guaranteed) approach and provides some results of analytical investigation for thin-walled shells with through cracks.     

On the reduction of the normality conditions in equality-constrained optimization problems in mechanics

A large class of problems in mechanics leads to the minimization of an objective function under equality constraints. In fact, inequality constraints can always be transformed into equality constraints by means of slack variables. The classical approach to solve equality-constrained problems relies on Lagrange multipliers, whose first-order normality conditions (FONC) lead to a system of nonlinear algebraic equations. This system of equations involves as many equations as unknowns, composed of the design variables and Lagrange multipliers, and hence, is amenable to a host of solution methods. In this paper, two methods to eliminate the Lagrange multipliers are reported, by which a reduced system of normality conditions is obtained. Reduction is conducted here either symbolically or numerically using an isotropic orthogonal complement L of the Jacobian matrix of the equality constraints. The relations thus resulting are cast into what is termed the dual form of the FONC. When the problem allows for symbolic calculations, a semi-graphical approach is applied, which leads to the global optimum of the problem at hand. However, the main novelty of the paper lies in an algorithm that returns the stationary points of a constrained optimization problem without requiring the closed-form expressions of the dual form of the FONC. Moreover, numerically efficient and stable procedures are given for the intermediate computational steps. The application of this algorithm is demonstrated with three examples from mechanics.     

A multi level approach to synthesis of planar mechanisms

A multi level approach to synthesis of planar mechanisms is presented. The approach covers both structural and dimensional synthesis of planar rigid body mechanisms containing revolute and translational joints. The synthesis is based on four different criteria. Firstly the type of mechanism is chosen with a view to get the simplest mechanism that satisfactorily fulfills the remaining three criteria. Two of these criteria are formulated as constraints on the kinematic behavior and the total area occupied by the mechanism, respectively. The fourth criteria is simply the desired minimization of the reactive forces/moments that appear in the mechanism. The desired kinematic behavior is based on a finite number, typically 1, ..., 6, of points in time (positions of the mechanism) where the position and orientation of up to two output bodies may be prescribed. The constraints on occupied areas are labelled territory constraints and formulated as a number of restricted areas (boxes). A synthesis is automatically performed at five levels. At the first level the structure of the mechanism is decided. At the second level initial dimensions for the given type of mechanism are found by random checking. At the third level the constraints on the kinematic behavior is fulfilled. At the fourth level the territory constraints are taken into account and, finally, at the fifth level the minimization of reactions is carried out. The entire approach has been implemented in a software package SYNMEC that runs on PCs and constitutes a way of performing the synthesis of a mechanism that is general and flexible with respect to both the type of mechanism that may be synthesized as well as the desired behavior upon which the synthesis is based.     

Effect of combined polymers on the loss of efficiency caused by mechanical degradation in drag reducing flows through straight tubes

The mechanical molecular degradation in drag reducing flows is a huge problem in the effort to improve the efficiency of drag reducers, which is clearly increased when a combination of materials is used. Here, we analyze mixtures of three kinds of water-soluble polymers: Poly (ethylene oxide) (PEO), Polyacrylamide (PAM), and Xanthan Gum (XG). Two kinds of mixtures are tested: (a) PAM and XG; (b) PEO and XG. The synergy between the polymers is clearly noticeable. The values of the drag reduction (DR) obtained by the polymer–polymer combination was larger than that observed for a single polymer in a solution with the same total concentration of the mixture. Our tests are conducted in straight tubes where the total pressure is fixed. The values of DR are computed step-by-step, as the total amount of solution pass through the system. In doing so, we carefully took into account the loss of efficiency caused by the turbulent flow in the straight tubes. It is quite clear that the degradation of the flexible polymers (PEO and PAM) is delayed in the mixtures. In other words, besides the increase in the DR, the flexible polymers are more resistant when in the presence of the rigid one (XG). Such observation is the main conclusion of this work.     

Mechanics of mechanochemically responsive elastomers

Mechanochemically responsive (MCR) polymers have been synthesized by incorporating mechanophores – molecules whose chemical reactions are triggered by mechanical force – into conventional polymer networks. Deformation of the MCR polymers applies force on the mechanophores and triggers their reactions, which manifest as phenomena such as changing colors, varying fluorescence and releasing molecules. While the activation of most existing MCR polymers requires irreversible plastic deformation or fracture of the polymers, we covalently coupled mechanophores into the backbone chains of elastomer networks, achieving MCR elastomers that can be repeatedly activated over multiple cycles of large and reversible deformations. This paper reports a microphysical model of MCR elastomers, which quantitatively captures the interplay between the macroscopic deformation of the MCR elastomers and the reversible activation of mechanophores on polymer chains with non-uniform lengths. Our model consistently predicts both the stress–strain behaviors and the color or fluorescence variation of the MCR elastomers under large deformations. We quantitatively explain that MCR elastomers with time-independent stress–strain behaviors can give time-dependent variation of color or fluorescence due to the kinetics of mechanophore activation and that MCR elastomers with different chain-length distributions can exhibit similar stress–strain behaviors but very different colors or fluorescence. Implementing the model into ABAQUS subroutine further demonstrates our model's capability in guiding the design of MCR elastomeric devices for applications such as large-strain imaging and color and fluorescence displays.     

Optimal design of a beam subject to bending: a basic application

The minimisation of both the mass and deflection of a beam in bending is addressed in the paper. To solve the minimisation problem, a multi-objective approach is adopted by imposing the Fritz John conditions for Pareto-optimality. Constraints on the maximum stress and elastic stability (buckling) of the structure are taken into account. Additional constraints are set on the beam cross section dimensions. Three different cross sections of the beam are analysed and compared, namely the hollow square, the I-shaped and the hollow rectangular cross sections. The analytical expressions of the Pareto-optimal sets are derived. As expected, the I-shaped beam exhibits the best compromise in structural performance, which is related on the particular loading considered.     

Relaxational interactions and viscoelasticity of polymer melts. Part I: model development

Rheological equations of state for the concentrated solutions and melts of high polymers are derived by applying a structural approach. The dynamics of a macromolecule are considered on the basis of the fundamental model of the polymer chain, e.g., the bead-spring model. The drag forces describing correlations of motion macromolecules are determined by means of the relaxation equations. The oscillatory shearing flow of the melts is studied on the basis of the equations derived. Expressions for the dynamic modulus and relaxation times are determined. As can be judged from the form of the dependence of the dynamic modulus on frequency, the relaxation time distribution is the same as in real materials. The relaxation spectrum of high polymers has a terminal zone with abnormally long relaxation times.     

A non-linear model for the dynamics of open cross-section thin-walled beams—Part II: forced motion

The discrete equations developed in Part I are here used to analyze the non-linear dynamics of an inextensional shear indeformable beam with given end constraints. The model takes into account the non-linear effects of warping and of torsional elongation. Non-linear 3D oscillations of a beam with a cross-section having one symmetry axis is examined. Only terms of higher magnitude are retained in the equations, which exhibit quadratic, cubic and combination resonances. A harmonic load acting in the direction of the symmetry axis and in resonance with the corresponding natural frequency, is considered. Steady-state solutions and their stability are studied; in particular the effects of non-linear warping and of torsional elongation on the response are highlighted.     

On the derivation of structural models with general thermomechanical prestress

The vibrating behaviour of thin structures is affected by prestress states. Hence, the effects of thermal prestress are important research subjects in view of ambient vibration monitoring of civil structures. The interaction between prestress, geometrically non-linear behaviour, as well as damping and its coupling with the aforementioned phenomena has to be taken into account for a comprehensive understanding of the structural behaviour. Since the literature on this subject lacks a clear procedure to derive models of thin prestressed and damped structures from 3D continuum mechanics, this paper presents a new derivation of models for thin structures accounting for generic prestress, moderate rotations and viscous damping. Although inspired by classical approaches, the proposed procedure is quite different, because of (i) the definition of a modified Hu–Washizu (H-W) functional, accounting for stress constraints associated with Lagrange multipliers, in order to derive lower-dimensional models in a convenient way; (ii) an original definition of a (mechanical and thermal) strain measure and a rotation measure enabling one to identify the main terms in the strain energy and to derive a cascade of lower-dimensional models (iii) a new definition of “strain–rotation domains” providing a clear interpretation of the classical assumptions of “small perturbations” and “small strains and moderate rotations”; (iv) the introduction of a pseudo-potential with stress constraints to account for viscous damping. The proposed procedure is applied to thin beams.