Roberto Serpieri

A 3D multiscale cohesive zone model for quasi-brittle materials accounting for friction, damage and interlocking

A three-dimensional (3D) two-scale Cohesive Zone Model (CZM), which is based on a multiplane approach and couples damage with friction and interlocking, is presented for analysing crack propagation in quasi-brittle materials along structural interfaces where formation of cracks is expected. The main idea of the 3D multiplane formulation herein exploited is to describe the asperities of the interface in the form of periodic patterns of inclined planes, denominated Representative Interface Elements (RIE). The interaction within each plane of the RIE is governed by the interface formulation proposed by Alfano and Sacco in earlier work. After reporting details of the formulation and of its algorithmic implementation, the sensitivity of the macroscopic mechanical response to the specific selection of the RIE is analysed and reported with a general numerical assessment of the 3D interface mechanical response to monotonic and cyclic loading histories. A fundamental issue addressed in this paper is the identification of optimal RIE patterns with a minimum number of planes capable of providing isotropic in-plane behaviour in response to confined slip tests.

Articular Cartilage Biomechanics Modeled via an Intrinsically Compressible Biphasic Model: Implications and Deviations From an Incompressible Biphasic Approach

Biphasic continuum models have been extensively deployed for modeling macroscopic articular cartilage biomechanics [1,2]. This consolidated theoretical approach schematizes tissue as a mixture of an elastic solid matrix embedded in a fluid phase. In physiological conditions, intrinsic compressibility of each phase is very limited when compared to the whole tissue macroscopic counterpart. Based on such experimental evidence, intrinsic phase compressibility is generally reasonably neglected [3]. Hence, traditionally, cartilage biomechanics models have been developed on the basis of incompressible biphasic formulations [3–5], often referred to as Incompressible Theories of Mixtures (ITM). Alternatively, a more general biphasic model for cartilage biomechanics, accounting for full intrinsic compressibility of phases, may be considered. A consistent theoretical formulation of this type has been recently made available [6,7], hereby referred to as Theory of Microscopically Compressible Porous Media (TMCPM). In the present contribution, a new model for articular cartilage biomechanics, based on TMCPM, was developed. Predictions of this new model, and its deviations from a traditional ITM approach were studied. In particular, deviations between compressible and incompressible theoretical frameworks were investigated with a specific focus on the repercussions on models’ capability of characterizing fundamental tissue properties, such as hydraulic permeability, via established experimental testing procedures.Copyright

Variational Theories of Two-Phase Continuum Poroelastic Mixtures: A Short Survey

A comprehensive survey is presented on two-phase and multi-phase continuum poroelasticity theories whose governing equations at a macroscopic level are based, to different extents, either on the application of classical variational principles or on variants of Hamilton’s least Action principle. As a focal discussion, the ‘closure problem’ is recalled, since it is widespread opinion in the multiphase poroelasticity community that even the simpler two-phase purely-mechanical problem of poroelasticity has to be regarded as a still-open problem of applied continuum mechanics. This contribution integrates a previous review by Bedford and Drumheller, and covers the period from the early use of variational concepts by Biot, together with the originary employment of porosity-enriched kinematics by Cowin and co-workers, up to variational theories of multiphase poroelasticity proposed in the most recent years.

Bond-slip analysis via a cohesive-zone model simulating damage, friction and interlocking

A recently proposed cohesive-zone model which effectively combines damage, friction and mechanical interlocking has been revisited and further validated by numerically simulating the pull-out test, from a concrete block, of a ribbed steel bar in the post-yield deformation range. The simulated response is in good agreement with experimental measurements of the bond slip characteristics in the post-yield range of deformed bars reported in the literature. This study highlights the main features of the model: with physically justified and relatively simple arguments, and within the sound framework of thermodynamics with internal variables, the model effectively separates the three main sources of energy dissipation, i.e. loss of adhesion, friction along flat interfaces and mechanical interlocking. This study provides further evidence that the proposed approach allows easier and physically clearer procedures for the determination of the model parameters of such three elementary mechanical behaviours, and makes possible their interpretation and measurement as separate material property, as a viable alternative to lumping these parameters into single values of the fracture energy. In particular, the proposed approach allows to consider a single value of the adhesion energy for modes I and II.

Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach

This study presents an analysis of the stress-partitioning mechanism for fluid saturated poroelastic media in the transition from drained (e.g. slow deformations) to undrained (e.g. fast deformation) flow conditions. The goal of this analysis is to derive fundamental solutions for the general consolidation problem and to show how Terzaghi’s law is recovered as the limit undrained flow condition is approached. The approach is based on a variational macroscopic theory of porous media (VMTPM). First, the linearized form of VMTPM is expressed in a u–p dimensionless form. Subsequently, the behavior of the poroelastic system is investigated as a function of governing dimensionless numbers for the case of a displacement controlled compression test. The analysis carried out in this study produced two crucial results. First, in the limit of undrained flow, it confirmed that the solutions of the consolidation problem recover Terzaghi’s law. Second, it was found that a dimensionless parameter ( P I ), which solely depends on mixture porosity and Poisson ratio of the solid phase, discriminates two diametrically opposed mechanic responses of the poroelastic system. More specifically, when P I is positive, the solid stress in the mixture first increases and then relaxes to an equilibrium value (stress relaxation). In contrast, when P I < 0 , the solid stress monotonically tenses up to reach the equilibrium value (stress tension).

Variational Macroscopic Two-Phase Poroelasticity. Derivation of General Medium-Independent Equations and Stress Partitioning Laws

A macroscopic continuum theory of two-phase saturated porous media is derived by a purely variational deduction based on the least Action principle. The proposed theory proceeds from the consideration of a minimal set of kinematic descriptors and keeps a specific focus on the derivation of most general medium-independent governing equations, which have a form independent from the particular constitutive relations and thermodynamic constraints characterizing a specific medium. The kinematics of the microstructured continuum theory herein presented employs an intrinsic/extrinsic split of volumetric strains and adopts, as an additional descriptor, the intrinsic scalar volumetric strain which corresponds to the ratio between solid true densities before and after deformation. The present theory integrates the framework of the Variational Macroscopic Theory of Porous Media (VMTPM) which, in previous works, was limited to the variational treatment of the momentum balances of the solid phase alone. Herein, the derivation of the complete set momentum balances inclusive of the momentum balance of the fluid phase is attained on a purely variational basis. Attention is also focused on showing that the singular conditions, in which either the solid or the fluid phase are vanishing, are consistently addressed by the present theory, included conditions over free solid-fluid surfaces.

Variational Multi-phase Continuum Theories of Poroelasticity: A Short Retrospective

This chapter aims at offering a comprehensive overview on the family of two-phase continuum poroelasticity theories whose formulations are based on the application of classical variational methods, or on variants of Hamilton’s Least Action Principle. The reader will be walked through several theoretical approaches to poroelasticity, starting from the early use of variational concepts by Biot, then covering the variational frameworks which employ porosity-enriched kinematics, such as those proposed by Cowin and co-workers and by Bedford and Drumheller, to conclude with the most recent variational theories of multiphase poroelasticity. Arguments are provided to show that, as a widespread opinion in the poroelasticity community, even the formulation of a simplest two-phase purely-mechanical poroelastic continuum theory remains, under several respects, a still-open problem of applied continuum mechanics, with the closure problem representing a crucial issue where important divergencies are found among the several proposed frameworks. Concluding remarks are finally drawn, pointing out the existence of delicate open issues even in the subclass of variational two-phase theories of poroelasticity.

Multiplane Cohesive Zone Models Combining Damage, Friction and Interlocking

The present work describes a number of cohesive zone models (CZMs) developed over the last decade; the models are derived from a simplified approach to the micro-mechanics of the fracture process. The models are able to separately consider damage and frictional dissipation; moreover, the most recent proposed models account also for intelocking and dilatancy. Initially, the model developed by Alfano and Sacco [6], coupling together damage and friction, is reviewed. A damage variable is introduced, evolving from zero for no damage to one when cohesion is lost. The main idea is to assume that friction only acts on the damaged part of the interface. The evolution of damage is governed by a mixed-mode criterion widely used in composite materials. Then, some thermodynamical consideration is presented, which leads in a simplified context to the result that the value of the fracture energy in mode I and II has to be the same [44]. A microstructured interface model is presented, obtained as combination of more inclined planes; this model is named as Representative Multiplane Element (RME) and it shows different fracture energies in mode I and II as result of the interplay between residual adhesion and the frictional slips on the inclined elementary planes, which determines significant frictional dissipation also in pure more II. The RME model is also able to account for interlocking and dilatancy [43]. Numerical applications illustrating the capacity of the proposed models are presented.

Stress Partitioning in Two-Phase Media: Experiments and Remarks on Terzaghi’s Principle

Stress partitioning in multiphase porous media is a fundamental problem of solid mechanics, yet not completely understood: no unanimous agreement has been reached on the formulation of a stress partitioning law encompassing all observed experimental evidences in two-phase media, and on the range of applicability of such a law. This chapter has two main objectives. The first one is to show the capability of the variational poroelastic theory developed in Chaps. 2 and 3 (VMTPM) to systematically and consistently describe stress partitioning in compression tests characterized by different loading and drainage conditions, and for three classes of materials: linear media, media with solid phase having no-tension response, and cohesionless granular media. The second objective is to perform a theoretical-experimental analysis on the range of applicability of the notions of effective stress and effective stress principles, in light of the general medium-independent stress partitioning law derived in Chap. 2 which predicts that the external stress, the fluid pressure and the stress tensor work associated with the macroscopic strain of the solid phase are always partitioned according to a relation formally compliant with Terzaghi’s law, irrespective of the microstructural and constitutive features of a given medium. Herein, the description of boundary conditions with unilateral contact is examined making use of a simple and straightforward extension of the standard formulation of contact in single-continuum problems, employing a set-valued law and a gap function. Next, the modalities of stress partitioning characteristic of Undrained Flow (UF) conditions, corresponding to absence of fluid seepage, are examined in further detail, identifying the possibility to characteristically define in a physically meaningful way, expressly at UF conditions, a stress tensor field of the whole mixture, as a quantity closely related to the concept of total stress tensor field. The systematic study carried out in this chapter allows showing that compliance with the classical statement of Terzaghi’s effective stress principle can be rationally derived as the peculiar behavior of the specialization of VMTPM recovered for cohesionless granular media, without making use of artificial incompressibility constraints. Moreover, it is shown that the experimental observations on saturated sandstones, generally considered as proof of deviations from Terzaghi’s law, are ordinarily predicted by VMTPM. In addition, a rational deduction of the phenomenon of compression-induced liquefaction in cohesionless mixtures is reported: such effect is found to be a natural implication of VMTPM when unilateral contact conditions are considered for the solid above a critical porosity. Finally, a characterization of the phenomenon of crack closure in fractured media is inferred in terms of macroscopic strain and stress paths. Altogether these results exemplify the capability of VTMPM to describe and predict a large class of linear and nonlinear mechanical behaviors observed in two-phase saturated materials. As a conclusion of this study, a generalized statement of Terzaghi’s principle for multiphase problems is proposed.

The Linear Isotropic Variational Theory and the Recovery of Biot’s Equations

In this chapter, the general framework presented in Chap. 2 is specialized to address linear isotropic two-phase poroelasticity. The elastic moduli of the resulting isotropic theory are derived with the special forms achieved by the governing PDEs for hyperbolic and parabolic problems. Next, the hyperbolic system is deployed to analyze the propagation of purely elastic waves. The chapter is concluded with a section dedicated to a comparison between the hyperbolic isotropic equations resulting from the present theory and their counterparts in Biot’s theory. This comparison shows the recovery by the medium-independent VMTPM framework of the essential structure of Biot’s PDEs. This recovery is herein deductively achieved in absence of heuristic statements, proceeding from the consideration of individual strain energies of the solid and fluid phases and from the minimal kinematic hypotheses of Chap. 2. This study is complemented by an analysis of the bounds of the elastic moduli of the isotropic theory, which is undertaken deploying a generalization to the present two-phase context of the Composite Sphere Assemblage homogenization technique by Hashin.