On approximate constitutive equations in nonlinear viscoelasticity

Summary In this paper a simplified constitutive equation for nonlinear slightly viscoelastic materials is derived by a method similar to that ofColeman andNoll. This equation is also valid for short times, slow motions, or small deformations superposed on a large steady deformation for all viscoelastic materials. An experimental method for determining the material parameters contained in this constitutive equation for dynamic conditions is also presented.

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Über genäherte Materialgleichungen in der nichtlinearen Viskoelastizität

Zusammenfassung Mittels einer Methode ähnlich der vonColeman undNoll wird eine vereinfachte Materialgleichung für nichtlineare, schwach viskoelastische Stoffe hergeleitet. Diese Gleichung gilt auch für das Kurzzeitverhalten, für langsame Bewegungen, oder für die Überlagerung kleiner Deformationen auf große stationäre Verformungen bei allen viskoelastischen Materialien. Eine experimentelle Methode zur Bestimmung der in der Materialgleichung auftretenden Stoffkonstanten für dynamische Bedingungen wird gleichfalls angegeben.

     

Ginzburg–Landau equations and first and second order phase transitions

In this paper we apply to non-isothermal processes the general Ginzburg–Landau model used in superconductivity and then we extend this framework to superfluidity and to first order phase transitions. Finally a new balance law on the order structure is proposed to obtain the classical Ginzburg–Landau equation.     

Finite element methods for second order differential equations with significant first derivatives

Galerkin finite element methods based on symmetric pyramid basis functions give poor accuracy when applied to second order elliptic equations with large coefficients of the first order terms. This is particularly so when the mesh size is such that oscillations are present in the numerical solution. In the present note asymmetric linear and quadratic basis functions are introduced and shown to overcome this difficulty in an appropriate two point boundary value problem. In particular symmetric quadratic basis functions are oscillation free and highly accurate for a working range of mesh sizes.     

Difference schemes for second order hyperbolic equations

Implicit difference methods for the wave equation in two space variables have been discussed with the help of a stability diagram. The difference methods of intermediate accuracy 0(h⁴+k²) have been determined. A method of order of accuracy 0(h²+k²) with minimum truncation error has also been found.     

Equivalence groups for second order balance equations

The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector fields of an ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations.     

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.     

Modified constitutive equation for quasi-linear theory of viscoelasticity

The constitutive equation of the quasi-linear theory of viscoelasticity is modified with the view to simplify our understanding of the physical meaning of numerous kernel functions. In the resulting equation, the instantaneous and the regular parts have the same structure, and the instantaneous part coincides with the well-known constitutive equation of the nonlinear theory of elasticity.     

The mesoscopic approach and constitutive equations for characterizing mechanical properties of alloys containing second phases

This paper discusses the feasibility of characterizing mechanical properties of materials by using a mesoscopic approach. Based on the mesoscopic system with the dislocation substructure at the core, the equations for flow stress and microhardness of alloys with second phases are proposed. The flow stress equation is experimentally verified. The problems are grouped so as to build the universal, quantitative, and constitutive equations relating the external properties and internal structure of materials.     

A study into first and second order thermal transitions of materials using Spectral-DSC

Thermal and spectral analysis is conducted routinely to characterise a large range of materials and compounds. However, tests are often conducted independently on separate samples where comparison between essentially the same material can provide conflicting results. Simultaneous thermal and spectral measurements have the advantage of being able to directly compare results using the same sample. A novel design of a simultaneous thermal and spectral technique is described along with application examples that highlight the benefits of this technique. The thermal analysis was conducted using Differential Scanning Calorimetry (DSC) and the in situ spectral analysis was conducted using a Fourier Transform Near Infrared (FT-NIR) spectrometer. Two examples are used to illustrate the versatility and potential advantages of the combined thermal and spectral method. Analysis of the first and second order transitions of polyethylene terephthalate (PET) is presented along with the pharmaceutical polymorphic conversion of carbamazepine from Form III to Form I through an isothermal hold at 160 °C.     

Optimal PID controller parameters for first order and second order systems with time delay using a connectionist approach

This article focuses on developing a neural controller based on a proportional integral derivative (PID) law. The main objective is to show that using neural networks represents a better alternative compared with other conventional models that were used in the past to express the tuning parameters as a function of process parameters such as process gain (K _P ), process time constants (τ₁, τ₂, etc.,) and process time delay (θ). A Levenberg-Marquardt backward propagation algorithm is used to get the required PID parameters corresponding to different types of processes. In the present study, the PID parameters for the first order plus time delay (FOPTD) and the second order plus time delay (SOPTD) are obtained. It was observed that very high R ² values between the actual PID parameters and the parameters obtained from the NN were achieved. Furthermore, the performance of PID control systems with FOPTD and SOPTD processes are applied on a number of case studies and compared with the conventional Zeigler-Nichols (Z-N) method of tuning PID controllers. Using the proposed NN controller was found to be efficient and the current method could be of potential use in control systems with gain scheduling also where the controller parameters are tuned according to a continuous gain schedule variable that changes based on different levels of process parameters.     

A constitutive model for thin sheet metal in micro-forming considering first order size effects

Size effects in the minimization make most know-how of traditional macro-forming not suitable for the micro-forming process. Material behaviours greatly vary in micro-forming process with different thickness and grain size. The influence of the first order size effects (size effects I) on flow stress has been studied quantitatively using the surface layer model and a new presented internal grain boundary model. It was discovered that the flow stress of micro-forming can be expressed as a function of t/d (thickness/average grain size) ratio. Then, a new constitutive model considering the size effects I for thin sheet-based micro-forming has been developed combining the Hollomon equation and Hall–Petch relation. The model has also been validated with the experimental results in the literature. A good coincidence is obtained when t/d > 1. However, it has great offset when t/d < 1 due to the orientations of the local individual grains have great influence on flow stress, which can be ignored by the new constitutive model because it only considers the influence of grain size. This investigation gives the support for theoretical and numerical study of micro-forming.     

Rate-type constitutive equations and elastic-plastic materials

This paper is concerned with a class of rate-type constitutive equations, which characterize the behavior of elastic-plastic solids under large deformation. The theory is developed without à priori introduction of the concept of plastic strain and involves only the stress and the total strain as basic ingredients, apart from the thermodynamic variables. The relationship between the results obtained here and those of a more familiar form of a theory of elastic-plastic solids is indicated.     

An Optimal successive overrelaxtion technique for solving second order finite difference equations for triangular meshes

A finite difference method for analyzing two-dimensional steady-state potential flow in a zoned anisotropic material is described. A triangular mesh is used is as to avoid difficulties often occurring in the more familiar rectangular mesh where fractional mesh spaces exist at irregular boundaries. The system of finite difference equations based on the triangular mesh is assembled in a form such that the coefficient matrix is 3-cylic and consistently ordered.^{1, 2} This enables the optimum accelerating factor for successive overrelaxation to be closely estimated. Since the rate of convergence to the solution, and consequently the efficiency of the method, increases rapidly as the accelerating factor tends towards its optimum value, close determination of the optimum is important. The method is particularly suitable for problems having large numbers of nodes. Examples of the method on specific problems are given. The solution accuracy is assessed and has been found acceptable.     

Analysis of creep curve by constitutive equations

Abstract

Creep curves of 0·5Cr–0·5Mo–0·25V steel have been analysed based on a simple power function of time. The long term creep properties, namely, the creep rate and the rupture lifetimes at low stress levels can be very well predicted by this simple power function. The Monkman–Grant relationship can also be obtained based on the predicted values.

MST/1411     

Symmetry restrictions on the constitutive equations for magnetic materials

A group-theoretical procedure is given for the decomposition of magnetic material tensors with intrinsic symmetry into basic symmetry types. This decomposition is used in finding the total number of independent components of magnetic material tensors in linear constitutive equations, and it simplies the determination of polynomial integrity bases in non-linear constitutive equations. Next, a method is devised to obtain the independent components of magnetic material tensors explicitly. Using the isomorphism between a classical point group and the corresponding magnetic point groups together with the property of basic quantities, determination of polynomial integrity bases is made immediate.     

Symbolic derivation of potential based constitutive equations

Structural alloys used in high temperature applications exhibit complex thermomechanical behavior that is inherently time dependent and hereditary, as the current behavior depends not only on current conditions but on the thermomechanical history. Derivation of mathematical expressions (constitutive equations) which describe this high temperature material behavior can be quite time consuming, involved, and error-prone, thus intelligent application of symbolic systems to facilitate this tedious processes can be of significant benefit. Here a computerized package, running under MACSYMA, capable of efficiently deriving potential based constitutive models, in analytical form (involving tensors, partial differentiation, invariants, and the like) is presented. Special purpose utility algorithms are designed and implemented to perform partial differentiation (chain rule), tensor manipulation, case distinction and simplification. Four constitutive theories reported in the literature are utilized to verify implementation accuracy. It is expected that this symbolic package can and will provide a significant incentive to the development of new constitutive theories.