Finite viscoelasticity and viscoplasticity of semicrystalline polymers

Observations are reported on low-density polyethylene in uniaxial tensile and compressive tests with various strain rates and in tensile and compressive relaxation tests with various strains. A constitutive model is developed for the time-dependent response of a semicrystalline polymer at arbitrary three-dimensional deformations with finite strains. A polymer is treated as an equivalent network of chains bridged by junctions (entanglements between chains in the amorphous phase and physical cross-links at the lamellar surfaces). Its viscoelastic behavior is associated with separation of active strands from temporary junctions and merging of dangling strands with the inhomogeneous network. The viscoplastic response is attributed to sliding of junctions between chains with respect to their reference positions. Constitutive equations are derived by using the laws of thermodynamics. The stress–strain relations involve 6 material constants that are found by matching the observations.      

Uniqueness theorems for the entropy in any differential body of complexity one

By using in an essential way a certain condition of mutual physical equivalence between admissible response functions for the heat flux, in a previous paper uniqueness theorems were proved for the response functions of the internal energy and of the equilibrium stress, in connection with differential bodies of complexity 1. It was then pointed out that the equality expressing the vanishing of the static internal dissipation uniquely determines the rate of entropy variation in terms of the rate of the internal energy variation and of the equilibrium stress. This paper shows, in a threefold manner, that the last result also holds if one does not impose the condition of physical equivalence. The first proof uses the assumption that the response functions are Euclidean invariant. The second proof uses (i) the weaker assumption of Galilean invariance and (ii) a greater degree of smoothness of the response function for the internal energy. Both of these proofs use an axiom postulating the possibility of putting the body in contact with a vacuum. The third proof of the uniqueness property for the entropy is independent of the isolation axiom and uses the assumptions of the second proof. Whereas any of the first two proofs is a consequence of the uniqueness theorem for the internal energy-proved here by using the afore-mentioned axiom-the third proof does not depend on this theorem. Rather, disregarding the above isolation axiom, it implies that uniqueness of the entropy is compatible with non-uniqueness of both the stress and internal energy.

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Sommario In un precedente articolo, utilizzando in maniera essenziale una certa condizione di mutua equivalenza fisica tra ammissibili funzioni costitutive per il flusso di calore, si dimostrano teoremi di unicità per le funzioni costitutive di energia interna e di stress statico in corpi di tipo differenziale e complessità 1. Inoltre si osserva che, di conseguenza, la uguaglianza esprimente l'annullarsi della dissipazione interna statica permette di determinare univocamente la velocità di variazione dell'entropia in termini della velocità di variazione dell'energia interna e dello stress statico. Nel presente lavoro si dimostra, in triplice maniera, che l'ultimo risultato vale anche se non si impone la accennata condizione di equivalenza fisica. Nella prima dimostrazione si usa l'ipotesi di invarianza Euclidea per le funzioni costitutive. Nella seconda si usa (i) la più debole ipotesi di invarianza Galileiana e (ii) un maggiore grado di regolarità per la funzione costitutiva dell'energia interna. In entrambe le dimostrazioni si usa un assioma che postula la possibilità di porre in contatto il corpo con il vuoto. La terza dimostrazione della proprietà di unicità per l'entropia, che usa le medesime ipotesi della seconda, è indipendente da tale assioma. Mentre nelle prime due dimostrazioni si richiama il teorema il teorema di unicità per l'energia interna-qui dimostrato con l'assioma di isolamento-la terza dimostrazione non dipende da tale teorema. Invece, se non si assume l'assioma di isolamento con il vuoto, l'unicità dell'entropia risulta compatibile con certe non-unicità per stress ed energia interna.

     

Identification of viscoelastic properties by means of nanoindentation taking the real tip geometry into account

Motivated by recent progress in viscoelastic indentation analysis, the identification of viscoelastic properties from nanoindentation test data taking the real tip geometry into account is presented in this paper. Based on the elastic solution of the indentation problem, the corresponding viscoelastic solution is obtained by the application of the method of functional equations. This general solution, which accounts for the real geometric properties of the indenter tip, is specialized for the case of a trapezoidal load history, commonly employed in nanoindentation testing. Three deviatoric creep models, the single dash-pot, the Maxwell, and the three-parameter model are considered. The so-obtained expressions allow us to determine viscoelastic model parameters via back calculation from the measured load–penetration history. The presented approach is illustrated by the identification of short-term viscoelastic properties of bitumen. Hereby, the influence of loading rate, maximum load, and temperature on the model parameters is investigated.     

Constitutive equations for non-affine polymer networks with slippage of chains

A model is derived for isothermal three-dimensional deformation of polymers with finite strains. A polymer fluid is treated as a permanent network of chains bridged by junctions (entanglements). Macro-deformation of the medium induces two motions at the micro-level: (i) sliding of junctions with respect to their reference positions that reflects non-affine deformation of the network, and (ii) slippage of chains with respect to entanglements that is associated with unfolding of back-loops. Constitutive equations are developed by using the laws of thermodynamics. Three important features characterize the model: (i) the symmetry of relations between the elongation of strands and an appropriate configurational tensor, (ii) the strong nonlinearity of the governing equations, and (iii) the account for the volumetric deformation of the network induced by stretching of chains. The governing equations are applied to the numerical analysis of extensional and shear flows. It is demonstrated that the model adequately describes the time-dependent response of polymer melts observed in conventional rheological tests.     

Finite viscoelasticity of filled rubber: experiments and numerical simulation

Summary  Constitutive equations are derived for the viscoelastic behavior of particle-re-inforced elastomers at isothermal deformation with finite strain. A filled rubber is thought of as a composite medium where inclusions with high and low concentrations of junctions between chains are randomly distributed in the bulk material. The characteristic length of the inhomogeneities is assumed to be small compared to the size of the specimen and substantially exceed the radius of gyration for macromolecules. Inclusions with high concentration of junctions are associated with regions of suppressed mobility of chains that surround isolated clusters and/or the secondary network of filler. Regions with low concentration of junctions arise during the preparation process due to a heterogeneity in the spatial distribution of the cross-linker and the filler. With reference to the concept of transient networks, the time-dependent response of an elastomer is attribute d to thermally activated rearrangement of strands in the domains with low concentration of junctions. Stress–strain relations for particle-reinforced rubber are developed by using the laws of thermodynamics. Adjustable parameters in the constitutive equations are found by fitting experimental data in tensile relaxation tests for several grades of unfilled and carbon black-filled rubber. It is demonstrated that even at moderate finite deformations (with axial elongations up to 100%), the characteristic rate of relaxation is noticeably affected by strain. Unlike glassy polymers, where the rate of relaxation increases with longitudinal strain, the growth of the elongation ratio results in a decrease in the relaxation rate for natural rubber (unfilled or particle-reinforced). The latter may be explained by (partial) crystallization of chains in the regions with low concentration of junctions. Received 16 October 2001; accepted for publication 25 June 2002 Present address: A. D. Drozdov Department of Production, Aalborg University, Fibigerstraede 16, DK-9220 Aalborg, Denmark We would like to express our gratitude to Dr. K. Fuller (TARRC, UK) for providing us with rubber specimens and to Prof. P. Haupt and Dr. S. Hartmann (University of Kassel, Germany) for sending their experimental data. We are indebted to Mr. G. Seifritz for his assistance in performing mechanical tests. ADD acknowledges stimulating discussions with Prof. N. Aksel (University of Bayreuth, Germany).     

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.     

Photomechanics of light-activated polymers

Light-activated polymers are an exciting class of modern materials that respond mechanically when irradiated by light at particular wavelengths. While details of the mechanisms that connect the optical excitation to mechanical behavior are complex and differ from material to material, there is sufficient commonality among them to permit the development of a generalized modeling framework to describe the photomechanics. The features shared by light-activated polymers involve light interacting with the material, which triggers photochemical reactions that alter the structure of the crosslinked polymer network. Many such structural alterations result in an evolution of the polymer network, and subsequent macroscopic deformation. When this process is appropriately executed it can enable a photomechanical shape-memory effect. In this paper, we develop a three-dimensional finite-deformation modeling framework to describe the photomechanical response of light-activated polymer systems. This framework integrates four coupled phenomena that contribute to macroscopic photomechanical behavior: photophysics, photochemistry, chemomechanical coupling, and mechanical deformation. The chemomechanical coupling consists of chemically induced structural alterations of the crosslinked network that result in subsequent deformation. We describe this behavior through a decomposition of the crosslinked network into two components consisting of an original network and a photochemically altered network; both evolve during photomechanical deformation. The modeling framework presented in this paper is sufficiently general that it is applicable to light-activated polymer systems that operate with various mechanisms in each of the four areas. Using this modeling approach, we develop constitutive models for two recently developed light-activated polymer systems [Lendlein, A., Hongyan, J., Junger, O., Langer, R., 2005. Light-induced shape-memory polymers. Nature 434 (7035) 879; Scott, T.F., Schneider, A.D., Cook, W.D., Bowman, C.N., 2005. Photoinduced plasticity in crosslinked polymers. Science 308 (5728) 1615]. For the material developed by Scott and his co-workers we validate our model by measuring and numerically simulating photo-induced stress relaxation and bending deformation and obtain good agreement between measurements and predictions. Finally, we use the model to study the effects of photomechanical parameters (applied strain magnitude, irradiation time and intensity, and photoabsorber concentration) and the behavior of the network evolution rule on the material response.     

Selection of Optimal Parameters of Multilayer Plates at Nonstationary Loading

This investigation features two approaches to optimal design of multilayer plates at impulse loading. The first approach deals with minimization of stresses in the plate layers by selecting the elastic properties of the fillers. The second approach is based on a twolevel synthesis procedure applied to the problem of optimal design of multilayer plates with a minimum mass. The effectiveness of these approaches is illustrated by numerical examples.Sommario. L'analisi è tesa alla caratterizzazione di due procedure di progettazione ottimale di piastre multistrato ad azioni impulsive. La prima procedura concerne con la minimizzazione dello sforzo negli strati della piastra selezionando le proprietà elastiche dei riempimenti.II secondo approccio e basato su di una procedura a due livelli di sintesi applicata al problema del disegno ottimo di piastre multistrato con una massa minima. L'efficacia di questi approcci e illustrata tramite esempi numerici.     

New trends in the analysis of masonry structures

The modern theory of masonry structures has been set up on the hypothesis of no-tension behaviour, with the aim of offering a reference model, independent of materials and building techniques employed. This hypothesis gives rise to inequalities which have to be satisfied by the stress tensor components and, as a dual aspect, to the kinematic behaviour characteristics of media which can be classified as lying between solids and fluids: the structure of the masonry material consists of particles reacting elastically only when in contact. An examination of the plane-stress problem leads us to define, within the prescribed domain under admissible loads, three different subdomains with null, regular, or non-regular principal stress tensors, respectively. As the boundaries of such subdomains are not known a priori, the problem can be classified as a free boundary value problem. The analysis concerns mainly the subdomains where the stress tensor is non-regular; and a non-regularity condition det =0 is added to the equilibrium equations. This condition makes the stress problem isostatic and leads to a violation of Saint-Venant's compliance conditions on strains. Hence there is a need to introduce a strain tensor, not related to the stress tensor, which can be decomposed into an extensional component and a shearing component; we prove that such strains, of the class ^c, are similar to those of the theory of plastic flow. From the point of view of computational analysis the anelastic strains are considered as given distortions; they are computed by means of the Haar-Kármán principle, modified for computational purposes by an idea of Prager and Hodge.

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Sommario La moderna teoria delle strutture murarie, fondata sulla rigorosa non reagenza a trazione del materiale, ha lo scopo di fornire un modello di riferimento indipendente sia dalle caratteristiche del materiale sia dalle techniche costruttive impiegate. L'ipotesi di non reagenza a trazione si traduce in disuguaglianze che le componenti del tensore di stress devono verificare; dualmente il comportamento caratteristico cinematico può esser classificato di confine, come del resto la stessa statica, tra solidi e fluidi: la struttura ipotizzata del materiale muratura consiste di particelle che reagiscono solo se sono in contatto. L'esame del problema piano porta a definire all'interno del dominio di definizione tre differenti tipi di sub-regioni in cui lo stress è nullo, canonico, o singolare. Poiché le frontiere di queste sub-regioni non sono note a priori il problema può anche essere classificato di frontiera libera. L'analisi concerne fondamentalmente la sub-regione in cui il tensore è non regolare, perché deve verificare anche la condizione det =0. Ciò rende isostatico il problema e conduce anche alla violazione della condizione di integrabilità delle deformazioni. Questo passaggio può essere superato introducendo un tensore di deformazioni a tensioni nulle che si può decomporre in una componente estensionale ed in una componente di scorrimento; si dimostra che queste deformazioni sono equivalenti a quelle che intervengono nella Teoria del flusso plastico. Dal punto di vista computazionale le deformazioni anelastiche sono considerate come distorsioni impresse determinate attraverso il principio di Haar-Kármán modificato, per le techniche computazionali, su idee di Prager e Hodge.

     

Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck and Hutchinson and their generalization

We discuss the physical nature of flow rules for rate-independent (gradient) plasticity laid down by Aifantis and by Fleck and Hutchinson. As a central result we show that:

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the flow rule of Fleck and Hutchinson is incompatible with thermodynamics unless its nonlocal term is dropped. If the underlying theory is augmented by a general defect energy dependent on γ^p and ∇γ^p, then compatibility with thermodynamics requires that its flow rule reduce to that of Aifantis.

We establish this result (and others) within a general framework obtained by combining a virtual-power principle of Fleck and Hutchinson with the first two laws of thermodynamics—balance of energy and the Clausius-Duhem inequality—under isothermal conditions.     

Invariance of the Equilibrium Equations of Finite Elasticity for Compressible Materials

For homogeneous, isotropic, compressible nonlinearly elastic materials, a wide class of strain-energy density functions are obtained that leave the equations of equilibrium invariant under simple scaling transformations of the material and spatial coordinates. These strain-energy densities are homogeneous functions of the principal stretches. Several illustrative examples of particular strain-energies are provided. For axisymmetric problems, the invariance discussed here ensures that the equations of equilibrium can be solved by quadratures and thus often leads to analytic solutions in parametric or closed-form. Mathematics Subject Classifications (2000) 74B20, 74G55.     

单向纤维增强复合材料中纤维断裂及其发展

纤维增强复合材料中某根纤维断裂后，断口作为裂纹向何处发展？它可以向纤维和基体的界面发展形成界面脱粘，也可向基体发展，造成基体开展，从而殃及邻近纤维。另外，一根纤维的断裂会在其邻近纤维中造成应力集中。本文采取轴对称边界元法对这些问题进行仔细研究。本文假定纤维在基体中成六角形分布，即每根纤维周围有六根纤维，均匀地分布在以该纤维为中心的圆周上。     

Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers

Shape memory polymers (SMPs) are polymers that can demonstrate programmable shape memory effects. Typically, an SMP is pre-deformed from an initial shape to a deformed shape by applying a mechanical load at the temperature T_H>T_g. It will maintain this deformed shape after subsequently lowering the temperature to T_L<T_g and removing the externally mechanical load. The shape memory effect is activated by increasing the temperature to T_D>T_g, where the initial shape is recovered. In this paper, the finite deformation thermo-mechanical behaviors of amorphous SMPs are experimentally investigated. Based on the experimental observations and an understanding of the underlying physical mechanism of the shape memory behavior, a three-dimensional (3D) constitutive model is developed to describe the finite deformation thermo-mechanical response of SMPs. The model in this paper has been implemented into an ABAQUS user material subroutine (UMAT) for finite element analysis, and numerical simulations of the thermo-mechanical experiments verify the efficiency of the model. This model will serve as a modeling tool for the design of more complicated SMP-based structures and devices.     

Analysis of Two Collinear Cracks in a Piezoelectric Layer Bonded to Two Half Spaces Subjected to Anti-plane Shear

In this paper, the behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces is investigated by a new method for the impermeable crack face conditions. The cracks are parallel to the interfaces in the mid-plane of the piezoelectric layer. By using the Fourier transform, the problem can be solved with two pairs of triple integral equations. These equations are solved using the Schmidt method. This process is quite different from that adopted previously. Numerical examples are provided to show the effect of the geometry of the interacting cracks and the piezoelectric constants of the material upon the stress intensity factor of the cracks.     

Mechanics of soft active materials with phase evolution

This paper studies the mechanics of soft active materials where the actuation is generated due to the formation of phases that are stress-free at the moment of their creation and therefore experience no deformation in the associated configuration. Phase formation is a continuous time-dependent process, which results in individual phases forming at different times and in different configurations of the material body, and thus it is coupled with mechanical deformation. Subsequent deformation of the material body results in individual phases experiencing different states of deformation and the overall material response results from the combined responses of the individual phases weighted by their respective volume fractions. Therefore, a great challenge in modeling the mechanics of soft active materials with evolving phases is to track the deformation and evolution of individual phases formed at different times and in different configurations. In this paper, a generalized one-dimensional model framework is presented to address this challenge. However, this model proves to be computationally inefficient. In response, an effective phase model is developed that tracks the combined deformation histories of new phases through a single, effective deformation. Both the general and effective phase models are evaluated with two fundamentally distinct phase evolution rules for three common mechanical problems: extension, stress relaxation, and creep. The first evolution rule represents a discrete transition from one phase to another while the second rule corresponds to a general transition from several phases into one phase. The effective phase model demonstrates excellent agreement with the generalized theory for all three mechanical problems considered under both types of evolution rules.     

Integral and differential constitutive equations for entangled polymers with simple versions of CCR and force balance on entanglements

The theory of Doi and Edwards for entangled polymers has been recently modified for the case of fast flows to account for convective contributions to molecular dynamics. The flow-induced relative motion between neighboring chains removes constraints and speeds up relaxation. Convective constraint release (CCR) may thus explain why the shear stress is seen to approach a plateau at high shear rates instead of decreasing as predicted by the basic theory. In slow flows, as well as in step strain, another discrepancy between theory and observations can be found in the normal stress ratio in shear Ψ=−N₂/N₁. The theoretical value for Ψ at low deformations is 1/7 whereas measured values for well-entangled systems are systematically larger. We have recently considered the possibility that this discrepancy arises because force balance requirements at the entanglement nodes are ignored in the classical theory. Accordingly, we have proposed a change in the orientational tensor Q. Here, we sum up on these recent findings by proposing single-relaxation-time constitutive equations of the integral or rate type incorporating those concepts in a simple way. Such equations should be suitable for numerical simulation of complex flows. Received: 1 January 2000 Accepted: 8 August 2000     

Modeling the response of filled elastomers at finite strains by rigid-rod networks

Summary  A constitutive model is developed for the isothermal response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains bridged to permanent junctions. A strand between two neighboring junctions is thought of as a sequence of rigid segments connected by bonds. In the stress-free state, a bond may be in one of two stable conformations: flexed and extended. The mechanical energy of a bond in the flexed conformation is treated as a quadratic function of the local strain, whereas that of a bond in the extended conformation is neglected. An explicit expression is developed for the free energy of a network. Stress–strain relations and kinetic equations for the concentrations of bonds in various conformations are derived using the laws of thermodynamics. In the case of small strains, these relations are reduced to the constitutive equation for the standard viscoelastic solid. At finite strains, the governing equations are determined by four adjustable parameters which are found by fitting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several filled and unfilled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, filler content, crosslink density and temperature on the adjustable constants. Received 3 January 2001; accepted for publication 12 July 2001