Laplace-Carson Space Research Articles

Isogeometric homogenization of viscoelastic polymer composites via correspondence principle

We present an isogeometric homogenization theory (IGH) for efficiently identifying homogenized and local creep and relaxation response of linearly viscoelastic polymer composites with different microstructural parameters. The principal idea is to construct exact geometric representations of both two- and three-dimensional unit cell microstructures for periodic materials by utilizing multiple conforming NURBS patches that are also employed for the displacement field interpolation function at the local scale. The IGH-based unit cell formulation is then converted to the viscoelastic solution with the Laplace-Carson space parameters via the correspondence principle. Subsequently, we leverage the Zakian formula to reverse the transformed IGH solution and obtain the homogenized creep and relaxation response of the composite in the original time space. The modelling and predictive capabilities of the IGH theory have been extensively validated vis-à-vis the elasticity-based and conventional finite-element homogenization techniques, and the advantages of the proposed technique over the reference techniques were demonstrated.

Micromechanical modelling of the anisotropic creep behaviour of granular medium as a fourth-order fabric tensor

The main objective of the present work is to develop a micromechanics approach to predict the macroscopic anisotropic creep behaviour of granular media. To this end, the linear viscoelastic behaviour of the inter-particle interaction at contact is adopted, and the contact distribution is characterized by a fourth-order fabric tensor in the local scale. Then, fourth-order tensor fabric-based micromechanical approaches based on Voigt and Reuss localization assumptions are applied to granular media in the Laplace–Carson space. With help of the inverse Laplace–Carson transformation of these obtained models, the macroscopic anisotropic creep behaviour of granular media submitted to a constant external loading is examined. Finally, the obtained results by specializing the Burgers model into the obtained models are compared with the numerical simulations in the particle flow code (PFC2D) to illustrate the validation and the accuracy of the analytical models for the macroscopic anisotropic creep behaviour of granular media.

A hierarchical asymptotic homogenization approach for viscoelastic composites

Effective properties of non-aging linear viscoelastic and hierarchical composites are investigated via a three-scale asymptotic homogenization method. In this approach, we consider the assumption of a generalized periodicity in the different structural levels and their characterization through the so-called stratified functions. The expressions for the associated local and homogenized problems, and the effective coefficients are derived at each level of organization by using the correspondence principle and the Laplace-Carson transform. Considering isotropic components and a perfect contact at the interfaces between the constituents, analytical solutions, in the Laplace-Carson space, are found for the local problems and the effective coefficients are computed. An interconversion procedure between the effective relaxation modulus and the effective creep compliance is carried out for obtaining information about both viscoelastic properties. The numerical inversion to the original temporal space is also performed. Finally, we exploit the potential of the approach and study the overall properties of a hierarchical viscoelastic composite structure representing the dermis.

Four-Parameter Viscoelastic Model for the Plates of Uniformly Varying Cross-Section

This study aims to investigate the quasi-static response of linear viscoelastic Kirchhoff plates of uniformly varying cross-section subjected to time-dependent loads. Four-parameter solid model is used for defining the linear viscoelastic material behavior. Through an efficient systematic procedure based on the Gâteaux Differential (GD), a functional has been constructed for the analysis. For the analysis, mixed finite element (MFE) method in transformed Laplace-Carson space is used. For transformation of the solutions obtained in the Laplace-Carson domain to the real time domain, Dubner & Abate (D&A) numerical inverse transform technique is employed.

Analytical and 3D numerical analysis of the thermoviscoelastic behavior of concrete-like materials including interfaces

We investigate in this paper analytically and numerically by means of 3D simulations the viscoelastic behavior of concrete and mortar subjected to creep loading and moderate temperatures at mesoscale. These heterogeneous materials are assumed to be composed of thermoelastic aggregates distributed in a linear thermoviscoelastic matrix; moreover, the Interfacial Transition Zones (ITZ) between aggregates and matrix, whose behavior is also considered as linear thermoviscoelastic, are explicitly introduced. The numerical specimens consist in unstructured periodic meshes containing polyhedral aggregates with various size and shapes randomly distributed in a box. Zero-thickness interface finite elements are introduced between aggregates and matrix to model the ITZ. Macroscopic response and averaged stresses and strains in the matrix and aggregate phases are compared to analytical estimations obtained with classical mean-field approximation schemes applied in the Laplace–Carson space, in which the ITZ are introduced via imperfect interfaces modeled with the Linear Spring Model (LSM). The effects of ITZ thickness, aggregate shape and uniform temperature increase are then studied to evaluate their respective influence on the local and macroscopic creep behavior of mortar and concrete. Globally, it is found that the model response is in relatively good agreement with numerical simulations results, and that as expected while the ITZ do not affect significantly the concrete behavior, they have a non-negligible impact on the mortar one.

Thermoviscoelastic Analysis of Concrete Creep at Mesoscale

Concrete is a heterogeneous material made up at mesoscale of linear elastic aggregates distributed in a mortar matrix whose behavior is time and temperature dependent. The Interfacial Transition Zone (ITZ) between aggregates and matrix also influences the overall behavior. We investigate here analytically and numerically by means of 3D simulations the creep behavior of concrete and mortar subjected to moderate temperatures at mesoscale. The numerical specimens consist in unstructured periodic meshes of polyhedral aggregates with various size and shapes randomly distributed in a box. Specific interface finite elements are introduced between aggregates and matrix to model the ITZ. Both matrix and ITZ are considered as linear thermoviscoelastic materials. Averaged stresses and strains in the matrix and aggregate phases are compared to analytical estimations obtained with classical mean-field approximation schemes applied in the Laplace-Carson space, where the ITZ are introduced via imperfect interfaces modelled with the Linear Spring Model (LSM). The effects of ITZ thickness, aggregate shape and temperature are then studied to evaluate their respective influence on mortar and concrete creep behavior.

Viscoelastic properties of heterogeneous materials: The case of periodic media containing cuboidal inclusions

This paper presents analytical solutions for the effective viscoelastic properties of composite materials based on a homogenization approach. The cases of spherical inclusions and cracks were recently developed. The objective of this paper is to use the same technique to deal with the case of periodic media containing cuboidal inhomogeneities. The viscoelastic behavior of both the matrix and the homogeneous equivalent medium is modeled by the Zener rheological model while inclusions are assumed to be linearly elastic. The viscoelastic Hill tensor required for the calculation of the effective viscoelastic tensors is obtained explicitly in the Laplace-Carson space in terms of Fourier series. The final expressions show that overall behavior depends on the viscoelastic properties of the matrix, the 3D dimensions of the inclusions and the thickness of the matrix layer between two nearby inclusions. Applications to masonry structures are presented to illustrate the theoretical results.

A homogenization approach for effective viscoelastic properties of porous media

The aim of this paper is to model the effective linear non-ageing viscoelastic properties of porous media based on a micromechanical approach. A porous medium is modeled as a mixture of a viscoelastic matrix and pore inclusions. The Generalized Maxwell (GM) viscoelastic model is employed for both the solid matrix and the porous medium. The effective parameters of the viscoelastic GM rheology of the porous medium, which are functions of the porosity and the viscoelastic properties of the solid phase, are derived considering the short and long term behaviors in Laplace–Carson space (LC). They are validated against exact solutions obtained from the inverse LC transform for a simple configuration. The proposed method allows avoiding the complexity of the inverse LC transform in general condition. An application for cement with assumption of spherical pore is considered to illustrate the powerful of this method.

Viscoelastic Plate Analysis Based on Gâteaux Differential

In this study, it is aimed to analyze the quasi-static response of viscoelastic Kirchhoff plates with mixed finite element formulation based on the Gâteaux differential. Although the static response of elastic plate, beam and shell structures is a widely studied topic, there are few studies that exist in the literature pertaining to the analysis of the viscoelastic structural elements especially with complex geometries, loading conditions and constitutive relations. The developed mixed finite element model in transformed Laplace-Carson space has four unknowns as displacement, bending and twisting moments in addition to the dynamic and geometric boundary condition terms. Four-parameter solid model is employed for modelling the viscoelastic behaviour. For transformation of the solutions obtained in the Laplace-Carson domain to the time domain, different numerical inverse transform techniques are employed. The developed solution technique is applied to several quasi-static example problems for the verification of the suggested numerical procedure.

Effective viscoelastic properties of micro-cracked heterogeneous materials

The aim of this paper is to model the effective viscoelastic properties of micro-cracked masonry based on a homogenization approach. The masonry is modeled as a mixture of a micro-cracked viscoelastic mortar matrix and isotropic elastic brick inclusions. Firstly, the effect of micro-cracks on the effective viscoelastic properties of mortar is considered at micro-scale. Secondly, brick inclusions are added in the micro-cracked mortar matrix at mesoscale to simulate effective viscoelastic properties of masonry. The modified Maxwell viscoelastic model is employed for noncracked, micro-cracked mortar matrix and micro-cracked masonry. The use of the same modified Maxwell model of the noncracked mortar to model the viscoelastic properties of the micro-cracked mortar is an approximation. This approximation is carried out in the short- and the long-term behaviors in the Laplace–Carson space. It is validated in transient situation with exact solutions obtained from the inverse Laplace–Carson transform for simple loading conditions. This technique allows avoiding the complexity of the inverse Laplace–Carson transform. The isotropic case with random orientation distribution of open micro-cracks in mortar and with spherical brick inclusions is considered.

Quasi-static and dynamic analysis of viscoelastic plates

In this study, the quasi-static and dynamic behavior of viscoelastic Kirchhoff plates is studied numerically by using the mixed finite element method in transformed Laplace–Carson space. In the transformed Laplace–Carson space, a new functional has been constructed for viscoelastic Kirchhoff plates through a systematic procedure based on the Gâteaux differential. For numerical inversion, the Maximum Degree of Precision (MDOP), Dubner and Abate’s, and Durbin’s transform techniques are employed. The developed solution technique is applied to several quasi-static and dynamic example problems.

A homogenization-enriched viscodamage model for cement-based material creep

We develop in this study a viscodamage model for cementitious materials based on the concept of pseudo-strains introduced by Schapery [1] coupled with the damage model of Mazars [2]. The pseudo-strains approach allows for reformulation of the initial viscoelastic problem as an equivalent elastic one by using a simple correspondence principle. The viscoelastic model results from a homogenization procedure in which the Mori–Tanaka scheme is applied in the Laplace–Carson space for estimating the effective linear viscoelastic bulk and shear moduli of the material. A composite made of a linear viscoelastic matrix described by a generalized Maxwell model and containing distributed spherical elastic inclusions and pores is adopted as a simplified representation of the material under consideration. One interesting feature of our approach is that exact analytical expressions for both bulk and shear moduli in the time space are derived in simple cases where a limited number of Maxwell chains are involved to describe the matrix behavior. The evolutions of the damage variable are governed by an equivalent pseudo-strain calculated from the pseudo-strains. The viscodamage model is applied to the simulation of cement paste basic creep tests available in the literature. After a proper identification of the parameters, an analysis of the results obtained with moderate creep loading reveals that the total strains evolutions are due mainly to the viscous characteristics of the material and to a lesser extent to the damage growth. The effects of the pore volume fraction of the material and of high load-levels leading to the specimen rupture on the creep response are further investigated and discussed.

Temperature field in inhomogeneous strongly anisotropic medium with sources

Using an “on-average exact” asymptotic method in a first approximation we constructed an analytical solution to the problem on the temperature field of heat sources in an anisotropic layer and ambient medium with dominant vertical thermal conduction. Results on temperature calculations in application to oil-gas formations are presented. It is shown that in the Laplace-Carson space, the zero and first expansion coefficients of the exact solution in terms of the asymptotic parameter coincide with those constructed on the base of the proposed method.

Generalized self-consistent scheme for the effective behavior of viscoelastic heterogeneous media: A simple approximate solution

This paper gives a simple approximate solution for obtaining the effective behavior of linear viscoelastic heterogeneous media for the case of elastic inclusions immersed within a viscoelastic matrix. The solution in the Laplace–Carson space is obtained by the Generalized self-consistent model and the simplification is in an explicit expression of the inverse Laplace transform. It is shown that the solution in Laplace–Carson space can be approximated by a convenient rational fraction which is given explicitly as a function of viscoelastic parameters. This provides an easy way to perform the inverse Laplace transform. Examples of typical composites, including possibly void and rigid inclusions, are provided and show that the procedure provides reasonably accurate results. In addition, a complete rheological representation can be provided in some cases for describing the behavior of the effective medium.

Multiscale poro-creep model for cement-based materials

SUMMARY Experimental observations clearly show that the relative humidity (hr) conditions influence significantly the creep behavior of cement-based materials, indicating that the water present within these materials plays a crucial role. This work presents a creep model for hardened cement pastes (HCP), based on a multiscale homogenization approach. It takes into account both free and adsorbed water contained in the porosity and investigates their effects on the HCP macroscopic creep behavior. The calcium silicate hydrate phase is assumed to be linear viscoelastic, and the Mori–Tanaka scheme is applied in the Laplace–Carson space to the composite formed of porosity, calcium silicate hydrate, and the other main hydrated compounds (which behavior is linearly elastic) by making use of the correspondence principle. With this model, estimations of the evolution of the macroscopic creep behavior of HCP submitted to constant external loading are examined under different hr and compared with available experimental data. Finally, a method for implementing the model in a finite element code is proposed, and simulations of standard creep tests are performed to assess its validity. Copyright © 2011 John Wiley & Sons, Ltd.

Homogenization of Linear Viscoelastic Three Phase Media: Internal Variable Formulation versus Full-field Computation

Homogenization of linear viscoelastic heterogeneous media is here extended from two phase inclusion-matrix media to three phase inclusion-matrix media. Each phase obeying to a compressible maxwellian behaviour, this analytic method leads to an equivalent elastic homogenization problem in the Laplace-Carson space. For some particular microstructures, such as the Hashin composite sphere assemblage, an exact solution is obtained. The inversion of the Laplace-Carson transforms of the overall stress-strain behaviour gives in such cases an internal variable formulation. As expected, the number of these internal variables and their evolution laws are modified to take into account the third phase. Moreover, evolution laws of averaged stresses and strains per phase can still be derived for three phase media. Results of this model are compared to full fields computations of representative volume elements using finite element method, for various concentrations and sizes of inclusion. Relaxation and creep test cases are performed in order to compare predictions of the effective response. The internal variable formulation is shown to yield accurate prediction in both cases.

The mixed finite element for quasi-static and dynamic analysis of viscoelastic circular beams

The quasi-static and dynamic responses of a linear viscoelastic circular beam on Winkler foundation are studied numerically by using the mixed finite element method in transformed Laplace-Carson space. This element VCR12 has 12 independent variables. The solution is obtained in transformed space and Schapery, Dubner, Durbin and Maximum Degree of Precision (MDOP) transform techniques are employed for numerical inversion. The performance of the method is presented by several quasi-static and dynamic example problems.

The use of homogenization methods for estimating anisotropic viscosities of composite melts

A model for the prediction of the various viscosity components of an anisotropic two-phase fluid is developed so as to simulate the behaviour of a unidirectional composite with discontinuous fibres during the flow process. The approach considered here is based upon the use of a transversely isotropic viscosity tensor proposed in a previous paper to describe the purely viscous behaviour of a composite melt. The fibre is replaced by a reinforcement inclusion assumed to be purely viscous and isotropic. The viscous law is transposed in a Laplace-Carson space in which the behaviour becomes linearly elastic and is investigated by means of classical homogenization methods. The solutions are then compared with the results of other studies and predictive values obtained from variational bounding methods given previously, which have been applied to a viscous composite cylinder assembly (CCA) which is statistically transversely isotropic. Finally, axial elongational and in-plane shearing viscosities are also compared with published experimental data.