Real Material Behavior Research Articles

Analytical solution of the Bagley Torvik equation by Adomian decomposition method

The fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PIλDμ controller for the control of dynamical systems, etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also described by differential equations of fractional order. The solution of the differential equation containing fractional derivative is much involved. Instead of application of the existing methods, an attempt has been made in the present analysis to obtain the solution of Bagley–Torvik equation [R.L. Bagley, P.J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, ASME J. Appl. Mech., 51 (1984) 294–298; I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999] by the relatively new Adomian decomposition method. The results obtained by this method are then graphically represented and then compared with those available in the work of Podlubny [I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999]. A good agreement of the results is observed.

Inclusion of Real Material Behaviour in the Numerical Model for Stringer Stiffened Cylinders

Inclusion of Real Material Behaviour in the Numerical Model for Stringer Stiffened Cylinders

The influence of non-dissipative quantities in kinematic hardening plasticity

A kinematic hardening plasticity model valid for finite strains is presented. The model is based on the well-known multiplicative split of the deformation gradient into elastic and plastic parts. The basic ingredient in the formulation is the introduction of a locally defined configuration—a centre configuration—which is associated with a deformation gradient that is used to characterize the kinematic hardening behaviour. The non-dissipative quantities allowed in the model are found when the plastic and kinematic hardening evolution laws are split into two parts: a dissipative part, which is restricted by the dissipation inequality, and a non-dissipative part, which can be chosen without any thermodynamic considerations. To investigate the predictive capabilities of the proposed kinematic hardening formulation, necking of a bar is considered. Moreover, to show the influence of the non-dissipative quantities, the simple shear problem and torsion of a thin-walled cylinder are considered. The numerical examples reveal that the non-dissipative quantities can affect the response to a large extent and are consequently valuable and important ingredients in the formulation when representing real material behaviour.

Importance of heat transfer phenomena during DSC polymer solidification

In this work, solidification of semi-infinite polymer slabs was modelled accounting for heat transfer and phase change. Transient one-dimensional energy balance was numerically solved, coupled with a suitable crystallization kinetic model, coming from literature [1]. To this purpose a generalized code was adopted, which was proposed in a previous communication, together with its preliminary validation [2]. Solidification runs were simulated both under isothermal conditions and under slow cooling rates, comparable to the ones attainable in standard differential scanning calorimetry (DSC). The output DSC signals were simulated, and it is shown that traditional analysis, usually performed on these signals in the frame of crystallization kinetics studies, can give correct results for isothermal tests, but can give suggestions consistently far from real material behaviour during cooling ramps.

Impact tension of sheet metals - Effect of initial specimen length

It is well known that a Specimen for impact testing of materials must be optimized concerning its dimensions. The main reason is to reduce strain gradients due to the effects of elastic-plastic wave propagation. On the other hand, when Split Hopkinson Bar (SHB) is applied for tension test, the net displacement of the specimen ends is very limited, usually from 2.0 to 3.0 mm. Thus, to reach maximum strain 0.5 the specimen length must be reduced to dimensions from 4.0 mm to 6.0 mm. Consequently small diameter, or lateral dimensions in case of flat specimen, must be applied to assure one-dimensional deformation. Such small lengths substantially perturb the real material behavior to be determined. So the main motivation of this study was to perform a systematic analysis, numerical and analytical, to find differences in behavior of short and long specimens loaded in impact tension. The FE code Abaqus/Explicit has been used to simulate several specimen lengths from 10 mm to 40 mm, and several velocities from 10 m/s to 100 m/s.

Symmetric State-Space Method for a Class of Nonviscously Damped Systems

Multiple-degree-of-freedom linear nonviscously damped systems are considered. It is assumed that the nonviscousdampingforcesdependonthepasthistoryofvelocitiesviaconvolutionintegralsoverexponentiallydecaying kernel functions. The traditional state-space approach, well known for viscously damped systems, is extended to such nonviscously damped systems using a set of internal variables. Suitable numerical examples are provided to illustrate the proposed approach. I. Introduction V ISCOUSdampingisthemostcommonmodelforthemodeling of vibration damping in linear systems. This model, e rst introduced by Rayleigh, 1 assumes that the instantaneous generalized velocities are the only relevant variables that determine damping. Viscous damping models are used widely for their simplicity and mathematical convenience, even though the behavior of real structural materials is, at best, poorly mimicked by simple viscous models.Forthisreason,itiswellrecognizedthat,ingeneral,aphysically realistic model of damping will not be viscous. Damping models in whichthedissipativeforcesdependonanyquantityotherthantheinstantaneous generalized velocities are nonviscous damping models. Mathematically, any causal model that makes the energy dissipation functional nonnegative is a possible candidate for a nonviscous damping model. Clearly, a wide range of choice is possible, either based on the physics of the problem or by selecting a model a priori and e tting its parameters from experiments. Here, we will use a particular type of damping model that is not viscous, and throughout the paper the terminology nonviscous damping will refer to this specie c model only. Possiblythemostgeneralwaytomodeldampingwithinthelinear range is to use nonviscous damping models that depend on the past history of motion via convolution integrals over kernel functions. 2 The equations of motion of an N-degree-of-freedom linear system with such damping can be expressed by

Imperfect Bifurcation in Structures and Materials: Engineering Use of Group-Theoretic Bifurcation Theory. Applied Mathematical Sciences, Vol 149

5R19. Imperfect Bifurcation in Structures and Materials: Engineering Use of Group-Theoretic Bifurcation Theory. Applied Mathematical Sciences, Vol 149. - K Ikeda (Dept of Civil Eng, Tohoku Univ, Aoba Sendai, 980-8579, Japan) and K Murota (Dept of Math Informatics, Univ of Tokyo, Tokyo, 113-0033, Japan). Springer-Verlag, New York. 2002. 411 pp. ISBN 0-387-95409-0. $69.95.Reviewed by J Petrolito (Sch of Sci and Eng, La Trobe Univ, PO Box 199, Bendigo, Vic 3550, Australia).Stability theory is of fundamental importance in structural engineering, and there is a large body of literature in the field. Much of the theory is directed toward the prediction of buckling or bifurcation loads for ideal structures. However, the role of imperfections is crucial for real structures, particularly for those that are sensitive to these effects. The current book is a graduate-level text that presents an overview of imperfections and the prediction of the initial post-buckling response of a system. The treatment is predominately analytical, rather than numerical. Hence, the problems treated are relatively simple since the analysis of most practical systems requires numerical techniques such as the finite element method. The book is divided into three parts and 15 chapters. Each chapter includes a range of problems and a summary to consolidate the material. The theory is complemented by nearly 200 references from the field. The first chapter provides an overview of the problems that are considered in the book. This chapter, although brief, could be used as an upper-undergraduate level introduction to the field. The first part of the book introduces the basic theory and derives the imperfection sensitivity laws for simple critical points. The theory is also linked to the classification system from chaos theory. The discussion includes procedures to identify critical imperfections and the role of probability theory for modeling random imperfections. Part 1 concludes with a chapter that links the theory to the behavior of real structures and materials. Two restrictions of the theory should be noted. Firstly, the loading is assumed to be described by one parameter only. Hence, the theory is not applicable to non-proportional loading, which occurs frequently in practice. Secondly, only the initial post-buckling response is considered. Other techniques need to be used if the complete load-deflection response of the system is required. Part 2 extends the theory to the analysis of systems with multiple critical points. This is inherently more difficult and the mathematical demands on the reader increase considerably. In particular, extensive use is made of group theory. Although the basics of this theory are covered, readers will probably need to supplement this with material from standard texts in the field to provide an adequate background to follow this part. Part 3 is primarily focused on modeling bifurcation in materials, including both metals and soils. This part of the books also relies on group theory. In summary, Imperfect Bifurcation in Structures and Materials provides an extensive range of material on the role of imperfections in stability theory. It would be suitable for a graduate-level course on the subject or as a reference to research workers in the field.

Five-parameter fractional derivative model for polymeric damping materials

Fractional derivative models offer a powerful tool to describe the dynamic behaviour of real viscoelastic materials. A version of the fractional derivative models characterized by five parameters is presented and investigated in this paper in order to describe asymmetrical loss factor peak and the high-frequency behaviour of polymeric damping materials. The speculative derivation of the model constitutive equation containing time derivatives of stress and strain of different orders is given. The model behaviour is investigated in the frequency domain, the physical meaning of the model parameters is defined and constraints on the parameter values are made. It is shown that the asymmetry of loss peak and the high-frequency behaviour of the model are governed by the difference between the order of time derivatives of stress and strain. Moreover, it is shown that this difference is related to the high-frequency limit value of the loss factor. The model is fitted to experimental data on some polymeric damping materials to verify its behaviour.

The Influence of Non-Dissipative Quantities in Kinematic Hardening Plasticity

A kinematic hardening plasticity model valid for finite strains is presented. The model is based on the well-known multiplicative split of the deformation gradient into an elastic and a plastic part. The basic ingredient in the formulation is the introduction of locally defined configurations center configurations- which are associated with deformation gradients that are used to characterize the kinematic hardening behavior. One of the aspects of the model investigated here is found when the plastic and kinematic hardening evolution laws are split into two parts: a dissipative part, which is restricted by the dissipation inequality, and a non-dissipative part, which can be chosen without any thermodynamical considerations. To investigate the predictive capabilities of the proposed formulation, the simple shear problem and torsion of a thin-walled cylinder are considered. In the numerical examples it turns out that the non-dissipative quantities affect the response to a large extent and are consequently valuable ingredients in the formulation when representing real material behavior (Less)

Finite-element method for the determination of the surface stress field from the real microstructure of anisotropic materials

A method is described for determining the surface stress distribution from the real microstructure of materials. The major advantages of the analyses, including the determination of anisotropic mechanical and crystallographic properties, are identified. The calculation involves the development of image analysis processing and the implementation in the finite-element code Cast3M of a new procedure for meshing the real microstructure and material models capable of describing the complex behaviour of real materials. The method is applied to the case of austenoferritic stainless steels.

A molecular-dynamics simulation study of the influence of attractive dispersion interactions on the phase behavior of rigid bead-necklace molecules

We have performed a molecular-dynamics simulation study of the influence of attractive dispersion interactions on the phase behavior of an ensemble of rigid bead–necklace molecules by gradually scaling the bead–bead interaction from a purely repulsive potential to a full Lennard-Jones potential. The system consisted of 600 bead–necklace molecules each composed of 11 interaction centers (beads). For all values of the attractive potential investigated the system manifested a nematic phase and a smectic A phase and both the isotropic–nematic and nematic–smectic A transitions were found to be weakly first order. The Gibbs–Duhem integration method was employed to trace the isotropic–nematic and nematic–smectic A(P,T) coexistence curves over the range of attractive interactions. With increasing attraction, the temperature–pressure range over which each phase was stable deceased to the advantage of the more ordered coexisting phase with a consequent narrowing of (P,T) space for the mesophases. Systems with weak attractions manifested transition strengths (entropy of transition) for the isotropic–nematic and nematic–smectic A transitions that were only weakly dependent upon temperature–pressure, while systems with stronger attractions exhibited transition strengths that decreased with increasing temperature. The phase behavior and thermodynamics of the isotropic–nematic and nematic–smectic A phase transitions for the systems with relatively strong dispersion interactions were found to be more representative of the behavior of real liquid crystalline materials than systems dominated by repulsive interactions.

On finite linear viscoelasticity of incompressible isotropic materials

The assumption of incompressibility as a kinematic constraint condition leads to consequences, which are of physical interest in view of a thermodynamically consistent material modelling. Some of these consequences are discussed within the concept of finite linear viscoelasticity. We present two natural possibilities to generalise the familiar Maxwell-model to finite strains; both tensor-valued differential equations are integrated to yield the present Cauchy stress as a functional of the relative Piola or Green strain history. Both types of Maxwell-models are related to a free energy functional in the sense that the dissipation inequality is satisfied. The stress and energy functionals are generalised to incorporate arbitrary kernel functions of relaxation; the only restriction for thermodynamic consistency is that the relaxation functions have a negative slope and a positive curvature. The linear combination of the two types of energy functionals can be understood to be a generalisation of the Mooney-Rivlin model to viscoelasticity. The concrete representation of relaxation functions, motivated from a finite series of Maxwell-elements in parallel, implies a Prony series, corresponding to a discrete relaxation spectrum. A compact notation to express a continuous relaxation spectrum is provided by the concept of a derivative of fractional order. In particular, differential equations of fractional order lead to relaxation functions coming very close to the relaxation behaviour of real materials. As an example, the evolution equation of a fractional Maxwell-model is solved, leading to a relaxation function of the Mittag-Leffler type. An essential advantage is that a rather small number of material constants is involved in this model, while the accuracy of its prediction is quite good. Physically nonlinear effects, such as process-dependent viscosity and relaxation behaviour can be incorporated into finite linear viscoelasticity, if the natural time is replaced with a material-dependent time scale. The rate of the material-dependent time then depends on further internal variables to account for the influence of the process history; this kind of proceeding is fully compatible with the entropy inequality. Temperature-dependence is included, starting from a multiplicative decomposition of the deformation gradient into mechanical and thermal parts. The resulting constitutive theory has a rather simple structure and offers a considerable degree of freedom to allow for physically important phenomena. The numerical simulations illustrate a small selection of these possibilities.

A Tale of Two Phase Diagrams

Two theoretical phase diagrams resulting from different approximate treatments of the Mott transition problem, capture the physics of different regimes (strong and weak frustration) of the strong correlation problem. We describe how they give physical insights into the behavior of real materials. We outline recent methodological developments, which hold great promise of a more unified and more quantitative treatment of strongly correlated electron systems.

Homogenized constitutive equations for linear viscoelastic unidirectional composites with growing transverse cracks

Three-dimensional, thermomechanical constitutive equations and an associated correspondence principle are developed for a linear viscoelastic, unidirectional composite with growing damage. The type of damage modeled is transverse cracking, which is commonly observed in laminates consisting of unidirectional fiber layers. The equations allow for aging. First, the constitutive equations without damage are summarized. They are then modified to allow for time-dependent damage by means of a correspondence principle. This principle relates elastic and viscoelastic solutions in the time-plane; the familiar correspondence principle based on the Laplace transform is not applicable because of the aging and crack growth. Some existing, simplified micromechanical models for elastic media are extended to viscoelasticity and then used to demonstrate that the requirements of the correspondence principle are quite-well satisfied for realistic material behavior. The correspondence principle is developed in the Appendix. It generalizes a previously-developed principle for plane stress in fiber-composites to three-dimensional conditions, and is not limited to the specific type of damage covered in the body of the paper.

Reliability-based computational model for material development and structural design

This paper discusses a methodology to aid in the design and to predict the reliability of metallic components using probabilistic microstructural fatigue models. The capabilities are aimed at predicting the variation in fatigue life based on the statistical variation of the microstructure of the material. Material parameters at metallic grain level are used along with fundamental physics-based models to predict the damage as it accumulates. The methods use a virtual prototyping technique relying on computer simulation of real material behaviour. The computer simulates many "identical" components, but uses a different sample of material microstructure for each simulation. The microstructure of each simulated component is properly sampled from the specified range of material parameters. Each of the components is then virtually tested using computer models to simulate real-world usage conditions. The virtual testing allows the "gathering" of data on thousands or even millions of components.

Low-frequency vibrational modes and static vibrational hyperolarizabilities of long-chain molecules: polyenes and polyacetylene

A model for calculating vibrational frequencies of Transverse Acoustical Modes (TAM) of all-trans polyenes of any length is presented. Based on the results obtained, the relevance of TAM modes in determining static vibrational polarizability and second vibrational hyperpolarizability of very long polyenes (polyacetylene) in real materials is discussed. The major role of the Effective Conjugation Coordinate (ECC) in determining the second vibrational hyperpolarizability is also presented and compared to that of the TAM modes.The results obtained point out the limitations of calculations on isolated molecules when the behaviour of real materials in the low-frequency domain is under investigation.

Grüneisen parameters and isothermal equations of state

The Gruneisen parameter (fl is of considerable importance to Earth scientists because it sets limitations on the thermoelastic properties of the lower mantle and core. However, there are several formulations of the Gruneisen parameter in frequent use which nor only give different values for gamma at ambient pressure but also predict a varying dependence of gamma as a function of compression. The Gruneisen parameter is directly related to the equation of state (EOS), yet it is often the case that both the form of gamma and the EOS are chosen independently of each other and somewhat arbitrarily. In this paper we have assessed some of the more common definitions of the Gruneisen parameter and the EOS, and have applied them to a test material. OF the EOS considered, when compared against ab initio compressional data for hcp-Fe as our exemplar, we find that the fourth order logarithmic and Vinet relations describe the material with the highest accuracy. Of the expressions for gamma considered it has been suggested, on theoretical grounds, that the modified free-volume formulation should be expected to give the most realistic description of the thermoelastic behavior of a material. However, when we use the fourth order logarithmic EOS to obtain the compressional behavior of the various Gruneisen parameters, we find that there is, in fact, poor agreement between the modified free-volume formulation and the Mie-Gruneisen parameter obtained directly from ab initio free energy calculations on hcp-Fe. We conclude that none of the analytical forms of gamma are sufficiently sophisticated to describe the thermoelastic behavior of real materials with great accuracy, and care must therefore be taken when attempting to model the thermoelastic behavior of solids to ensure that the appropriate gamma (ideally obtained from experiments or ab initio calculations) and equations of state are used.

On the volumetric deformation of reconstituted soils

This paper reviews the phenomenon of volumetric hardening, which is a common feature of the mechanical behaviour of many geo-materials. Three different material idealizations have been proposed to describe this hardening, and the paper contains the corresponding mathematical formulation. These idealizations vary in their complexity and hence their ability to capture different aspects of real material behaviour. Any of the three postulates can be implemented into most constitutive models. As a demonstration of their capabilities, the postulates have been implemented into the well-known modified Cam Clay model, and computations are made with the resulting new constitutive models. It is seen that the new models can successfully represent important features of soil behaviour such as plastic yielding associated with loading inside the current virgin yield surface, the loosening or densifying of granular soils caused by shearing, and the accumulation of both volumetric and distortional deformation caused by repeated drained loading over a large number of cycles. Copyright © 2000 John Wiley & Sons, Ltd.

Low velocity impact behavior of glass/epoxy cross‐ply laminates with different fiber treatments

AbstractThis paper deals with the influence of the fiber/matrix adhesion quality on the impact behavior of cross‐ply glass/epoxy laminates. Glass fibers with two different treatments (one to promote and one to prevent adhesion to the matrix) were embedded in epoxy matrix systems and subjected to low velocity impacts at energies below perforation energy. It will be shown that the laminates with good fiber/matrix adhesion are significantely more damage resistant than the plates with poor adhesion. It will be pointed out that the composites with the more brittle matrix system show the lower damage resistance. For all materials, the absorbed energy correlates well with the amount of impact induced damage. Furthermore, an experimental/mathematical model is introduced that gives the possibility of predicting the maximum deflection and maximum force for any given impact energy. Only one experiment is needed for each material to identify the model parameters. These parameters describe the real material behavior and can be used as characteristic values to classify materials with respect to their impact stiffness and damage resistance.

Actual three-dimensional stresses in notches, crack tips and lamination planes

A strange dichotomy exists in the analysis of stresses in local effects and in adhesively bonded parts. It has been known in aeronautical engineering for at least 60 years that the stresses in plates with notches or cracks, or in plates subjected to localized loads, are strongly three-dimensional. Nevertheless, it has been common to invoke the notion of the so-called generalized plane stress state , and to consider the two-dimensional treatment of the stress states in regions of notches, cracks or lamination planes to be completely satisfactory. As a result, many analytical solutions in stress analysis are not testable—the errors are both in the magnitude, up to 30%, and even in the sign. Two problems have developed, one practical, the other aesthetic. First, for obvious reasons, modern technology needs analytical solutions, or computer programs, which simulate reliably the actual physical occurrence. Second, any analytical solution which ignores one or more basic features of the real behavior of materials and bodies under consideration, or is incompatible with some underlying assumptions, or violates some principles of mechanics, is not only insufficiently reliable but cannot be considered elegant. Using Popper's terminology, such a solution is not in the category of a physical theory but is rather in the category of a metaphysical theory. The paper presents selected examples of empirical evidence on the actual, mostly three-dimensional, stresses in notches, cracks, beams with local effects, and in components of adhesively bonded joints . The usefulness of the renormalization procedures for singular solutions is discussed, and more comprehensive and realistic notions of the stress concentration factor in plates are suggested.