Micromechanical modeling of linear viscoelastic behavior of heterogeneous materials

Study of effective behavior of heterogeneous materials, starting from the properties of the microstructure, represents a critical step in the design and modeling of new materials. Within this framework, the aim of this work is to introduce a general internal variables approach for scale transition problem in linear viscoelastic case. A new integral formulation is established, based on the complete taking into account of field equations and differential constitutive laws of the heterogeneous problem, in which the effects of elasticity and viscosity interact in a representative volume element. Thanks to Green’s techniques applied to space convolution’s term, a new concentration relation is obtained. The step of homogenization is then carried out according to the self-consistent approximation. The results of the present model are illustrated and compared with those provided by Hashin’s and Rougier’s ones, considered as references, and by internal variables models such as those of Weng and translated fields.     

Micromechanical model for cohesive materials based upon pseudo-granular structure

A micromechanical model for cohesive materials is derived by considering their underlying microstructure conceptualized as a collection of grains interacting through pseudo-bonds. The pseudo-bond or the inter-granular force–displacement relations are formulated taking inspiration from the atomistic-level particle interactions. These force–displacement relationships are then used to derive the incremental stiffnesses at the grain-scale, and consequently, obtain the sample-scale stress–strain relationship of a representative volume of the material. The derived relationship is utilized to study the stress–strain and failure behavior including the volume change and “brittle” to “ductile” transition behavior of cohesive materials under multi-axial loading condition. The model calculations are compared with available measured data for model validation. Model predictions exhibit both quantitative and qualitative consistency with the observed behavior of cohesive material.     

Micromechanical analysis of damage in saturated quasi brittle materials

In this paper, we propose a micromechanical analysis of damage and related inelastic deformation in saturated porous quasi brittle materials. The materials are weakened by randomly distributed microcracks and saturated by interstitial fluid with drained and undrained conditions. The emphasis is put on the closed cracks under compression-dominated stresses. The material damage is related to the frictional sliding on crack surface and described by a local scalar variable. The effective properties of the materials are determined using a linear homogenization approach, based on the extension of Eshelby’s inclusion solution to penny shaped cracks. The inelastic behavior induced by microcracks is described in the framework of the irreversible thermodynamics. As an original contribution, the potential energy of the saturated materials weakened by closed frictional microcracks is determined and formulated as a sum of an elastic part and a plastic part, the latter entirely induced by frictional sliding of microcracks. The influence of fluid pressure is accounted for in the friction criterion through the concept of local effective stress at microcracks. We show that the Biot’s effective stress controls the evolution of total strain while the local Terzaghi’s effective stress controls the evolution of plastic strain. Further, the frictional sliding between crack lips generates volumetric dilatancy and reduction in fluid pressure. Applications of the proposed model to typical brittle rocks are presented with comparisons between numerical results and experimental data in both drained and undrained triaxial tests.     

Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-brittle materials

The aim of this paper is to develop a thermodynamically consistent micromechanical concept for the damage analysis of viscoelastic and quasi-brittle materials. As kinematical damage variables a set of scalar-, vector-, and tensor-valued functions is chosen to describe isotropic and anisotropic damage. Since the process of material degradation is governed by physical mechanisms on levels with different length scale, the macro- and mesolevel, where on the mesolevel microdefects evolve due to microforces, we formulate in this paper the dynamical balance laws for macro- and microforces and the first and second law of thermodynamics for macro- and mesolevel.Assuming a general form of the constitutive equations for thermo-viscoelastic and quasi-brittle materials, it is shown that according to the restrictions imposed by the Clausius–Duhem inequality macro- and microforces consist of two parts, a non-dissipative and a dissipative part, where on the mesolevel the latter can be regarded as driving forces on moving microdefects. It is shown that the non-dissipative forces can be derived from a free energy potential and the dissipative forces from a dissipation pseudo-potential, if its existence can be assured.The micromechanical damage theory presented in this paper can be considered as a framework which enables the formulation of various weakly nonlocal and gradient, respectively, damage models. This is outlined in detail for isotropic and anisotropic damage.     

A continuum damage model for piezoelectric materials

In this paper, a constitutive model is proposed for piezoelectric material solids containing distributed cracks. The model is formulated in a framework of continuum damage mechanics using second rank tensors as internal variables. The Helrnhotlz free energy of piezoelectric mate- rials with damage is then expressed as a polynomial including the transformed strains, the electric field vector and the tensorial damage variables by using the integrity bases restricted by the initial orthotropic symmetry of the material. By using the Talreja＇s tensor valued internal state damage variables as well as the Helrnhotlz free energy of the piezoelectric material, the constitutive relations of piezoelectric materials with damage are derived. The model is applied to a special case of piezoelectric plate with transverse matrix cracks. With the Kirchhoff hypothesis of plate, the free vibration equations of the piezoelectric rectangular plate considering damage is established. By using Galerkin method, the equations are solved. Numerical results show the effect of the damage on the free vibration of the piezoelectric plate under the close-circuit condition, and the present results are compared with those of the three-dimensional theory.     

Coupled viscoplasticity damage constitutive model for concrete materials

A coupled viscoplasticity damage constitutive model for concrete materials is developed within the framework of irreversible thermodynamics.Simultaneously the Helmholtz free energy function and a non-associated flow potential function are given, which include the internal variables of kinematic hardening,isotropic hardening and dam- age.Results from the numerical simulation show that the model presented can describe the deformation properties of the concrete without the formal hypotheses of yield criterion and failure criteria,such as the volume dilatancy under the compression,strain-rate sen- sitivity,stiffness degradation and stress-softening behavior beyond the peak stress which are brought by damages and fractures.Moreover,we could benefit from the application of the finite element method based on this model under complex loading because of not having to choose different constitutive models based on the deformation level.     

A micromechanics based damage model for composite materials

The predictive capacity of ductile fracture models when applied to composite and multiphase materials is related to the accuracy of the estimated stress/strain level in the second phases or reinforcements, which defines the condition for damage nucleation. Second phase particles contribute to the overall hardening of the composite before void nucleation, as well as to its softening after their fracture or decohesion. If the volume fraction of reinforcement is larger than a couple of percents, this softening can significantly affect the resistance to plastic localization and cannot be neglected. In order to explicitly account for the effect of second phase particles on the ductile fracture process, this study integrates a damage model based on the Gologanu–Leblond–Devaux constitutive behavior with a mean-field homogenization scheme. Even though the model is more general, the present study focuses on elastic particles dispersed in an elasto-plastic matrix. After assessing the mean-field homogenization scheme through comparison with two-dimensional axisymmetric finite element calculations, an extensive parametric study is performed using the integrated homogenization-damage model. The predictions of the integrated homogenization-damage model are also compared with experimental results on cast aluminum alloys, in terms of both the fracture strain and overall stress–strain curves. The study demonstrates the complex couplings among the load transfer to second phase particles, their resistance to fracture, the void nucleation mode, and the overall ductility.     

A three-dimensional progressive damage model for fibre-composite materials

This note presents a damage model for fibre-composite materials based in the approach by Matzenmiller et al. [Matzenmiller, A., Lubliner, J., Taylor, R.L., 1995. A constitutive model for anisotropic damage in fiber-composites. Mech. Mater. 20, 125]. In this work, the model is developed in a three-dimensional context with modified formulation for the constitutive law and damage evolution. An orthotropic composite subjected to mixed failure modes is assumed in this development. Its formulation and implementation details are provided.     

Parameter identification for viscoelastic materials

Starting from the general stress-strain relation for a linear Boltzmann-Volterra material, which is in agreement with the principle of inertia, a new identification procedure is proposed. Instead of running one long-range relaxation experiment, following asingle suitably specified deformation history, material characterization is done using the data ofn short relaxation experiments followingn different deformation histories. To interpret these data a direct non-iterative algorithm has been developed. Compared with other methods, for example curve fitting by using Gauss' method, this direct method is numerically stable and allows a simple direct evaluation of the error due to the scattering of experimental data. The method has been applied to the determination of the relaxation times of an unsaturated polyester material.     

Anisotropic damage mechanics for viscoelastic ice

We present a formulation of continuum damage in glacier ice that incorporates the induced anisotropy of the damage effects but restricts these formally to orthotropy. Damage is modeled by a symmetric second rank tensor that structurally plays the role of an internal variable. It may be interpreted as a texture measure that quantifies the effective specific areas over which internal stresses can be transmitted. The evolution equation for the damage tensor is motivated in the reference configuration and pushed forward to the present configuration. A spatially objective constitutive form of the evolution equation for the damage tensor is obtained. The rheology of the damaged ice presumes no volume conservation. Its constitutive relations are derived from the free enthalpy and a dissipation potential, and extends the classical isotropic power law by elastic and damage tensor dependent terms. All constitutive relations are in conformity with the second law of thermodynamics.PACS 83.60.Df, 62.20.Mk     

Interaction-based non-local damage model for failure in quasi-brittle materials

The purpose of this paper is to present a new macroscopic approach to describe the evolving non-local interactions which are produced at the mesoscale during damage and failure in quasi-brittle materials. A new-integral type non-local model is provided where the weight function is directly built from these interactions, and therefore takes into account their evolution during the material failure intrinsically.     

Stress analysis for linear viscoelastic materials

Summary The relations between stress and strain prescribing linear viscoelastic behavior are discussed from the standpoint of application to problems of stress analysis. This consideration involves some important differences from the assessment of linear viscoelastic laws for the representation of material properties only.The use of the differential operator relation between stress and strain is usually most convenient, although the integral operators comprising the creep function or relaxation function can also sometimes be conveniently utilized. Examples of these are given, including the problem of indentation, in which the region of contact of a rigid spherical indentor under constant load can be expressed directly in terms of the creep function. When differential operator relations, which correspond to the usually considered viscoelastic models, are used, it is shown that a consistent application of the theory of delta functions and the associated symbolic differentiation permits initial values to be determined for the evaluation of the resulting differential equations. No mathematical difficulties arise if the loading functions are not smooth, as might be anticipated by the high order time derivatives which appear in the viscoelastic relations. An application of this theory is given which falls outside the scope of theLaplace transform type of analysis. The latter is often used to deal with this kind of problem, but has a restricted field of application.

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Zusammenfassung Die vorliegende Arbeit behandelt die Beziehungen zwischen Spannung und Dehnung in linear viskoelastischen Körpern vom Standpunkt der Spannungsberechnung. Diese Problemstellung geht wesentlich über die Erwägungen, die beim Studium der materiellen Eigenschaften auftreten, hinaus.Die Benützung der Differentialoperatorbeziehung zwischen Spannung und Dehnung ist meist günstiger, obgleich die Integraloperatoren, die aus Kriechfunktionen oder Relaxationsfunktionen bestehen, manchmal auch zweckdienlich benützt werden können. Als Beispiel der Anwendung dieser Integraloperatoren wird das Stempelproblem gelöst, wobei die Ausdehnung des Kontaktbereichs des festen, kugelförmigen Stempels unter konstanter Last direkt durch die Kriechfunktion ausgedrückt wird.Bei der Benützung von Differentialoperator-beziehungen (die den bekannten viskoelastischen Modellen entsprechen) wird gezeigt, daß eine Anwendung der Theorie der Deltafunktionen die Bestimmung der Anfangsbedingungen für die Differentialgleichungen zuläßt. Auch wenn die Belastungsfunktionen nur stückweise glatt sind, treten keine Schwierigkeiten auf, trotz der höheren Ableitungen, die in den viskoelastischen Stoffbeziehungen auftreten. Ein zweites Beispiel zeigt eine Anwendung der Deltafunktionen auf ein Problem, das der Behandlung mit Hilfe derLaplace-Transformation nicht zugänglich ist.

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The results presented in this paper were obtained in the course of research sponsored by the Office of Ordnance Research, U. S. Army under Contract No. DA-19-020-ORD-4550.     

Rheology of polymer blends: non-linear model for viscoelastic emulsions undergoing high deformation flows

Non-linear rheology of a mixture of two viscoelastic immiscible liquids undergoing high deformation flow was considered. Using Grmela's formalism (Grmela 1986, 1989, 1993a, b) and the coarse grained picture given by Onuki (1987) and Doi and Ohta (1991), we have derived a set of highly non-linear time-dependent transport equations that take into account a direct coupling between the rheology and morphology. Breakup, coalescence, and the high deformation of the interface were considered. Models of Doi and Ohta (1991), Lee and Park (1994), and Grmela et al. (1998) are recovered as special cases. The parameters of?the model were given a physical meaning in both shear and elongational flows and the predictions of the model were ¶compared to the predictions of the previous models on the basis of experimental results obtained on two model?blends PDM/PB polydimethylsiloxane/poly(1-butene) and PP/PS polypropylene/polystyrene blends.     

Thermistor analog study of dynamic shear in model viscoelastic materials

An electric transmission line analog for studies of the dynamic plane shear of an incompressible viscoelastic material which has a temperature dependent viscosity is described. Thermistors are used to simulate the temperature dependent viscosity. Criteria are established for the similarity of the line and the material. Experiments analogous to constant rate of deformation studies of an elastic material and a material with a single viscoelastic relaxation time are described. More detailed experiments analogous to the deformation of a viscous material at a constant rate of deformation and at constant stress are also described. These show phenomena analogous to necking and fracture.     

Micromechanical modelling of cyclic plasticity incorporating damage

In this paper a numerical investigation on the possibility to simulate and predict cyclic plastic response incorporating damage has been performed. To this purpose, unit cell and continuum approaches based on porous metal plasticity and continuum damage mechanics (CDM) have been considered. In particular, the porous metal plasticity model of Leblond, Perrin and Devaux (LPD model) and the CDM model developed by Pirondi and Bonora were used. Finite element (FE) simulations were performed for each approach with different degrees of triaxiality and the results are analyzed and compared.     

Micromechanical study of plasticity of granular materials

Plastic deformation of granular materials is investigated from the micromechanical viewpoint, in which the assembly of particles and interparticle contacts is considered as a mechanical structure. This is done in three ways. Firstly, by investigating the degree of redundancy of the system by comparing the number of force degrees of freedom at contacts with the number of governing equilibrium equations; Secondly, by determining the spectrum of eigenvalues of the stiffness matrix for the structure that is represented by the particles and their contacts; Thirdly, by investigating the evolution with imposed strain of the continuum elastic stiffness tensor of the system. It is found that, with increasing imposed strain, the degree of redundancy rapidly evolves towards a state with small redundancy, i.e. the system becomes nearly statically determinate. The spectrum of the system shows many singular and near-singular modes at peak shear strength and at large strain. The continuum elastic stiffness tensor becomes strongly anisotropic with increasing imposed strain and shows strong non-affinity of deformation. The assumption of a constant and isotropic elastic stiffness tensor in elasto-plastic continuum constitutive relations for granular materials is generally incorrect. Overall, the plastic continuum behaviour of granular materials originates from the plastic frictional behaviour at contacts and from damage in the form of changes in the contact network.     

Hysteretic deteriorating model for quasi-brittle materials based on micromechanical damage approach

A damage model, which is based on the stochastic modeling of the microstructures, is developed for the quasi-brittle materials subjected to repeated loading. According to this model, the overall response of the material is represented with a series of micro-elements joined in parallel. A combined model is proposed for the micro-element considering the fracture as well as the hysteretic energy dissipation. To account for the progressive failure, the random fracture strains are assigned to the micro-elements. Therefore the overall parallel bundle is considered as a stationary random field. Then by averaging the microscopic random field, the overall loading, unloading and reloading curves are derived analytically. Two hysteretic rules are derived from the proposed model, and the overall hysteretic deteriorating behaviors could be well reproduced. To demonstrate the validity of the present model, the numerical results are shown against the stochastic simulated curves as well as the experimental data. The present model provides an alternative approach for the efficient modeling of the hysteretic deteriorating behaviors for the quasi-brittle materials.     

A thermodynamic consistent damage and healing model for self healing materials

Thermodynamics of the damage and the healing processes for viscoplastic materials is discussed in detail and constitutive equations for coupled inelastic-damage-healing processes are proposed in a thermodynamic consistent framework. Small deformation state is utilized and the kinematic and the isotropic hardening effects for the damage and healing processes are introduced into the governing equations. Two new yield surfaces for the damage and healing processes are proposed that take into account the isotropic hardening effect. The computational aspect for solving the coupled elasto-plastic-damage-healing problem is investigated, and the mechanical behavior of the proposed polymeric based self healing system is obtained. Uniaxial compression tests are implemented on a shape memory polymer based self healing system and the damage and the healing are captured by measurement of the changes in the modulus of elasticity. It is concluded that the proposed constitutive equations can model the damage and healing effectively and the mechanical behavior of a shape memory polymer based self healing system can be precisely modeled using this formulation.