Cauchy Continuum Research Articles

Homogenization of a Cauchy continuum towards a micromorphic continuum

The micromorphic theory of Eringen and Mindlin, including special cases like strain gradient theory or Cosserat theory, is widely used to model size effects and localization phenomena. The heuristic construction of such theories based on thermodynamic considerations is well-established. However, the identification of corresponding constitutive laws and of the large number of respective constitutive parameters limits the practical application of such theories.In the present contribution, a closed procedure for the homogenization of a Cauchy continuum at the microscale towards a fully micromorphic continuum is derived including explicit definitions of all involved generalized macroscopic stress and deformation measures. The boundary value problem to be solved on the microscale is formulated either for using static or kinematic boundary conditions. The procedure is demonstrated with an example.

Extended Hill’s lemma for non-Cauchy continua based on the simplified strain gradient elasticity theory

Hill's lemma for the Cauchy continuum has been playing an important role in micromechanics. An extended version of Hill's lemma for non-Cauchy continua is formulated using the simplified strain gradient elasticity theory (SSGET), which contains only one material length scale parameter and can account for the microstructure-dependent strain gradient effect. As a corollary of the extended Hill's lemma, the Hill–Mandel macro-homogeneity condition for non-Cauchy continua is obtained along with the general forms of kinematically and statically admissible boundary conditions that are required for constructing an energetically equivalent homogeneous comparison material. Based on these general forms, four sets of uniform boundary conditions are identified, which are implementable in material tests and can be directly used in homogenization analyses of heterogeneous materials. It is shown that when the strain gradient effect is suppressed, the extended Hill's lemma recovers the classical Hill's lemma for the Cauchy continuum and the extended Hill–Mandel condition reduces to its classical counterpart.

Reflection and transmission of elastic waves in non-local band-gap metamaterials: A comprehensive study via the relaxed micromorphic model

In this paper we propose to study wave propagation, transmission and reflection in band-gap mechanical metamaterials via the relaxed micromorphic model. To do so, guided by a suitable variational procedure, we start deriving the jump duality conditions to be imposed at surfaces of discontinuity of the material properties in non-dissipative, linear-elastic, isotropic, relaxed micromorphic media. Jump conditions to be imposed at surfaces of discontinuity embedded in Cauchy and Mindlin continua are also presented as a result of the application of a similar variational procedure. The introduced theoretical framework subsequently allows the transparent set-up of different types of micro-macro connections granting the description of both (i) internal connexions at material discontinuity surfaces embedded in the considered continua and, as a particular case, (ii) possible connections between different (Cauchy, Mindlin or relaxed micromorphic) continua. The established theoretical framework is general enough to be used for the description of a wealth of different physical situations and can be used as reference for further studies involving the need of suitably connecting different continua in view of (meta-)structural design. In the second part of the paper, we focus our attention on the case of an interface between a classical Cauchy continuum on one side and a relaxed micromorphic one on the other side in order to perform explicit numerical simulations of wave reflection and transmission. This particular choice is descriptive of a specific physical situation in which a classical material is connected to a phononic crystal. The reflective properties of this particular interface are numerically investigated for different types of possible micro-macro connections, so explicitly showing the effect of different boundary conditions on the phenomena of reflection and transmission. Finally, the case of the connection between a Cauchy continuum and a Mindlin one is presented as a numerical study, so showing that band-gap description is not possible for such continua, in strong contrast with the relaxed micromorphic case.

Micromorphic continua: non-redundant formulations

The kinematics of generalized continua is investigated and key points concerning the definition of overall tangent strain measure are put into evidence. It is shown that classical measures adopted in the literature for micromorphic continua do not obey a constraint qualification requirement, to be fulfilled for well-posedness in optimization theory, and are therefore termed redundant. Redundancy of continua with latent microstructure and of constrained Cosserat continua is also assessed. A simplest, non-redundant, kinematic model of micromorphic continua, is proposed by dropping the microcurvature field. The equilibrium conditions and the related variational linear elastostatic problem are formulated and briefly discussed. The simplest model involves a reduced number of state variables and of elastic constitutive coefficients, when compared with other models of micromorphic continua, being still capable of enriching the Cauchy continuum model in a significant way.

Thermal and chemical effects in shear and compaction bands

Strain localization zones in the form of shear bands or compaction bands in geomaterials are observed across scales from sub-millimetric (grain size) to kilometric scale (geological structures). Triggering and evolution of such narrow zones of localized deformation depend on many factors. The mechanical behavior of geomaterials is central for the formation of such zones. However, thermal, pore-pressure and chemical effects play a crucial role in shear and compaction banding. Temperature increase and activation of chemical reactions such as mineral dehydration, carbonate decomposition, as well as dissolution/precipitation control the triggering and the evolution of localized deformation zones. Moreover, the inherent heterogeneous microstructure of geomaterials plays a significant role during strain localization. The purpose of this paper is to provide a review of recent research regarding the effects of temperature, pore-pressure, chemical reactions and microstructure on strain localization in geomaterials. Examples have been taken in relation with seismic slip and with compaction banding. Strain localization is treated as an instability from a homogeneous deformation state. Different types of instabilities may (co-) exist depending on different multi-physical couplings and micro-mechanisms. Finally, a comparison of rate dependent Cauchy continuum and rate independent generalized continua (Cosserat continuum) is made. This comparison leads to an analog expression for the instability condition and the thickness of the localized zone.

ON DISCRETE STRUCTURE OF GEOLOGIC MEDIUM AND CONTINUAL APPROACH TO MODELING ITS MOVEMENTS

This paper discusses the structure of a geologic medium represented by accessible lithified rocks and provides an overview of methods used to describe its movements. Two basic opinions are considered in the framework of the discussion: (1) an initially homogeneous and continuous geologic medium acquires the structure composed of blocks in the process of the geologic medium’s deformation/destruction/degradation, and (2) a geologic medium is composed of blocks (and often has hierarchic, active, energy-saturated features), and the continuity model is thus not valid for describing the geologic medium’s deformation. Proponents of the first point of view actively apply the standard or modified continuum model of a solid deformed body (SDB) in estimations of the stress-strain state, but the input parameters of this model do not contain any information on discreteness in principle. Authors who support the second opinion, either explicitly or implicitly assume that the block structure of the geologic medium, which is detectable by geological methods, makes a direct and unambiguous impact on all other mechanical properties of the geologic medium and, above all, on the nature of its movements. Based on results obtained by interpreting the data collected in our long-term field studies of rock fracturing, mathematical processing of GPS-measurements, and theoretical models, we agree with the concept of the geologic medium’s block structure, but argue that the geologic block-structure property is not acquired but congenital. Regarding sedimentary rocks, it means that the discrete structure has been already embodied in the rock before sediment lithification, regardless of the intensity of macroscopic deformations. A discrete structure is the form of the geologic medium existence and a cause of the congenital anisotropy of the geologic medium’s strength characteristics. Due to subsequent deformation of the geologic medium, some elements of the structure can be manifested more clearly, and the structure itself can become more complex due to secondary effects. At the same time, the structure of geologic blocks is not directly manifested in the spatial and temporal features of the recent movements in the geologic medium. However, it is not an obstacle to developing continuum models of such movements in the same way as, for instance, the adequacy of the continuum general theory of relativity is not denied in view of the discrete-hierarchical structure of the Universe. The key requirements to a model include its applicability, testability, confirmability/deniability of its predictions in the investigated space-time scale, and compliance with conservation laws. This paper briefly discusses the most important aspects of the continuum approach based on the concept of an effective continuous medium and, above all, the Cauchy continuum model, envisaging that the dynamic response of the medium in spatial descriptions is given only by the Cauchy symmetric stress tensor (T). In more general continuum models of the medium (such as moment, micropolar, micromorphic and other models), the dynamics of the medium may be characterized by asymmetric tensors of force and couple stresses. This paper refutes the unjustified criticism of the continuum model as such criticism is rooted in the mistaken identification of quite special assumptions or ways of setting the problems with the general principles of the continuity model. Special attention is given to critical comments received from supporters of the active geologic medium concept. The paper considers actual difficulties encountered in studies using the continuity model, specifically in coordinated descriptions of the medium containing mobile defects, as well as the medium that is subject to deformation due to movements on its structure, while this structure is hierarchical at any scale level, down to the zero level (which, in particular, concerns the fractal structure). Discussed are causes of some widespread misunderstandings and mistakes in the geoscience literature, as well as the occurrence of conditions facilitating the revival of Aristotle ideas and preNewton concepts in geology, which repeatedly gain the upper hand over the modern ideas of classical physics. The paper considers the problem of reconstruction of stresses in the geologic medium from in-situ kinematic indicators, specifically from irreversible slips. Attention is drawn to the fact that the currently dominating approach ‘imposes’ a priori speculative rules on the geologic medium, such as a relationship between stresses (to be estimated) and the slip directions. Under this approach, the conservation laws are inevitably ignored, which makes it impossible to interpret the obtained results in terms of stress. Under the alternative approach proposed earlier by the author of this paper, the conservation laws being taken into account allows not only to reconstruct the stress tensor field, but also to judge on the geologic medium rheology. It is concluded that rejecting the continuum approach a priori, with a reference to the geologic medium discreteness, is at least unconstructive.

Research on the deformation of a confined aquifer based on Cosserat continuum mechanics

Abstract. Recent monitoring of land subsidence and soil deformation indicates a new phenomenon where excessive and continuous deformation occurs in the sandy aquifers in Shanghai and the Su-Xi-Chang region of China. It is hard to explain factors contributing to this phenomenon with traditional Cauchy continuum mechanics in which low normal stress in the ground could not cause such large deformation. Steep hydraulic gradient would be formed in the aquifer if groundwater is pumped from densely distributed wells, and shear stresses would develop then. Accumulated shear stress could then lead to deformation of the aquifer or even land subsidence. Accumulated shear stress due to the drawdown of groundwater level is one of the main factors that contribute to deformation within an aquifer. Traditional Cauchy continuum mechanics cannot consider this shear stress because of the hypothesis of equal shear stress in the aquifer unit. Cosserat continuum mechanics can be applied to analyse the mechanism of aquifer deformation controlled by accumulated shear stress by considering the scale effect and the asymmetric distribution of shear stress in the aquifer unit.

Computation of the effective mechanical response of biological networks accounting for large configuration changes

The asymptotic homogenization technique is involved to derive the effective elastic response of biological membranes viewed as repetitive beam networks. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes in the nonlinear regime, reflecting the influence of the geometrical and mechanical micro-parameters of the network structure on the overall response of the equivalent continuum. Biomembranes networks are classified based on nodal connectivity, so that we analyze in this work 3, 4 and 6-connectivity networks, which are representative of most biological networks. The individual filaments of the network are described as undulated beams prone to entropic elasticity, with tensile moduli determined from their persistence length. The effective micropolar continuum evaluated as a continuum substitute of the biological network has a kinematics reflecting the discrete network deformation modes, involving a nodal displacement and a microrotation. The statics involves the classical Cauchy stress and internal moments encapsulated into couple stresses, which develop internal work in duality to microcurvatures reflecting local network undulations. The relative ratio of the characteristic bending length of the effective micropolar continuum to the unit cell size determines the relevant choice of the equivalent medium. In most cases, the Cauchy continuum is sufficient to model biomembranes. The peptidoglycan network may exhibit a re-entrant hexagonal configuration due to thermal or pressure fluctuations, for which micropolar effects become important. The homogenized responses are in good agreement with FE simulations performed over the whole network. The predictive nature of the employed homogenization technique allows the identification of a strain energy density of a hyperelastic model, for the purpose of performing structural calculations of the shape evolutions of biomembranes.

A micromechanical approach for the Cosserat modeling of composites

The present paper deals with the homogenization problem of periodic composite materials, considering a Cosserat continuum at the macro-level and a Cauchy continuum at the micro-level. Consistently with the strain-driven approach, the two levels are linked by a kinematic map based on a third order polynomial expansion. Because of the assumed regular texture of the composite material, a Unit Cell (UC) is selected; then, the problem of determining the displacement perturbation fields, arising when second or third order polynomial boundary conditions are imposed on the UC, is investigated. A new micromechanical approach, based on the decomposition of the perturbation fields in terms of functions which depend on the macroscopic strain components, is proposed. The identification of the linear elastic 2D Cosserat constitutive parameters is performed, by using the Hill–Mandel technique, based on the macrohomogeneity condition. The influence of the selection of the UC is analyzed and some critical issues are outlined. Numerical examples for a specific composite with cubic symmetry are shown.

Investigation into subsidence hazards due to groundwater pumping from Aquifer II in Changzhou, China

This paper presents an investigation into increased deformation of Aquifer II caused by groundwater pumping from the aquifer in Changzhou, China. As groundwater levels of aquifers have been decreasing in recent decades due to uncontrolled water pumping, land subsidence is becoming a serious geohazard in Changzhou. Based on recently reported field data, the compression of aquitards has not increased compared to that of aquifers with the same scale of layer thickness. The Cosserat continuum model was adopted to analyse the observed phenomenon in this study. A classic Cauchy continuum model is also used for comparison. The comparison between these two models indicates that the proposed approach can interpret the increased deformation well, and the classic Cauchy continuum model underestimates the aquifer deformation as it does not consider shear displacement and macro-rotation. A discussion on the relationship between the groundwater level in the aquifer and subsidence is then undertaken. The results show that the severity of the annual subsidence is correlated with the variation in groundwater level in Aquifer II. To mitigate the subsidence hazards, countermeasures should be adopted to avoid the shear stress in aquifers which results from the high hydraulic gradient, by the appropriate allocation of pumping wells and by restricting groundwater withdrawal volume from each pumping operation.

Micromorphic homogenization of a porous medium: elastic behavior and quasi-brittle damage

Today it is well known that the classical Cauchy continuum theory is insufficient to describe the deformation behavior of solids if gradients occur over distances which are comparable to the microstructure of the material. This becomes crucial e.g., for small specimens or during localization of deformation induced by material degradation (damage). Higher-order continuum approaches like micromorphic theories are established to address such problems. However, such theories require the formulation of respective constitutive laws, which account for the microstructural interactions. Especially in damage mechanics such laws are mostly formulated in a purely heuristic way, which leads to physical and numerical problems. In the present contribution, the fully micromorphic constitutive law for a porous medium is obtained in closed form by homogenization based on the minimal boundary conditions concept. It is shown that this model describes size effects of porous media like foams adequately. The model is extended toward quasi-brittle damage overcoming the physical and numerical limitations of purely heuristic approaches.

Buckling analysis through a generalized beam model including section distortions

A geometrically nonlinear beam model suitable for describing complex 3D effects due to non-uniform warpings including non-standard in-plane distortions of the cross-section is presented and applied to the buckling analysis of beams. Each section is endowed with a corotational frame where statics and kinematics are described using a refined linear elastic model which exploits a semi-analytical solution of the Cauchy continuum problem based on a FEM discretization of the cross-section. The stress field in this way is fully 3D, allowing both the exact recovery of the standard Saint Venant solution and the consideration of some additional relevant strain modes of the cross-section that are evaluated in a simple and effective way. Numerical results are presented and compared with 3D shell reference solutions obtained by using the commercial code ABAQUS.

Tetrahedral Finite Element with Rotational Degrees of Freedom for Cosserat and Cauchy Continuum Problems

AbstractThis study formulates a simple tetrahedral finite element equipped with rotational degrees of freedom that can be used effectively to solve problems for both Cosserat and Cauchy continua. The formulation makes no assumption regarding the symmetry of the stress tensor, and such symmetry is achieved at convergence for the Cauchy problems. The numerical implementation of the new element is straightforward, and numerical tests demonstrate second-order accuracy in the case of both Cosserat and Cauchy elasticity.

Derivation of microstructured continua from lattice systems via principle of virtual works: the case of masonry-like materials as micropolar, second gradient and classical continua

The description of the mechanical behaviour of brick/block masonry through equivalent continua is presented here as a paradigmatic example of the problem of gross modelling of discontinuous and heterogeneous materials as continua with microstructure. The approaches reported in the literature differ for the way identification of the continuum is carried out or the nature of the continuum itself. In this paper, continuous models equivalent to rigid particle systems with free or constrained rotations are derived within the general framework of the principle of virtual work. In particular, an integral equivalence procedure is used to derive micropolar, second gradient and classical models. The non-classical models have in the field equations non-standard kinematic and static descriptors accounting for the presence of the material internal structure. The differences in the material responses of the various continua are identified referring to their internal work formulas. For the reference material, it is shown that, unlike the Cauchy continuum, both micropolar and second gradient models are effective in the presence of load and geometrical singularities, which involve significant scale effects on the material response. On the other hand, the second gradient model, as well as the classical model, disregards the role of relative rotation between the local rigid rotation (macrorotation) and the microrotation, which is related to the presence of non-symmetric strains. This circumstance, significant in strongly anisotropic systems, allow us to point out the advantages of the micropolar modelling especially for orthotropic masonry assemblies made of elements of any size. These statements are discussed by means of selected numerical examples of masonry panels differing in size, shape and arrangement, under shear loading conditions.

A second strain gradient elasticity theory with second velocity gradient inertia – Part I: Constitutive equations and quasi-static behavior

A multi-cell homogenization procedure with four geometrically different groups of cell elements (respectively for the bulk, the boundary surface, the edge lines and the corner points of a body) is envisioned, which is able not only to extract the effective constitutive properties of a material, but also to assess the “surface effects” produced by the boundary surface on the near bulk material. Applied to an unbounded material in combination with the thermodynamics energy balance principles, this procedure leads to an equivalent continuum constitutively characterized by (ordinary, double and triple) generalized stresses and momenta. Also, applying this procedure to a (finite) body suitably modelled as a simple material cell system, in association with the principle of the virtual power (PVP) for quasi-static actions, an equivalent structural system is derived, featured by a (macro-scale) PVP having the typical format as for a second strain gradient material model. Due to the surface effects, the latter model does work as a combination of two subsystems, i.e. the bulk material behaving as a Cauchy continuum, and the boundary surface operating as a membrane-like boundary layer, each subsystem being in (local and global) equilibrium by its own. Further, the applied (ordinary) boundary traction splits into two (response-dependent) parts, i.e. the “Cauchy traction” transmitted to the bulk material and the “Gurtin–Murdoch traction” acting, together with all other boundary tractions, upon the boundary layer. The role of the boundary layer as a two-dimensional manifold enclosing a Cauchy continuum is elucidated, also with the aid of a discrete model. A strain gradient elasticity theory is proposed which includes a minimum total potential energy principle featuring the relevant boundary-value problem for quasi-static loads and its (unique) solution. A simple application is presented. Two appendices are included, one reports the proof of the global equilibrium of the boundary layer, the other is concerned with double and triple stresses. The paper is complemented by a companion Part II one on dynamics. Previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137] are improved and extended.

Compatible model for herringbone bond masonry: Linear elastic homogenization, failure surfaces and structural implementation

A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic masonry strength domains in the case of herringbone masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined.A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively.To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads.When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone bond orientation, (2) masonry texture and (3) mechanical properties adopted for joints.At a structural level, a FE homogenized limit analysis is performed on a masonry dome built in herringbone bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone bond orientation.

A gradient elasticity theory for second-grade materials and higher order inertia

Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the sum of two parts, i.e. the Cauchy traction and the Gurtin–Murdoch traction, whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface (as known from the principles of surface mechanics) whereby the Gurtin–Murdoch traction (incorporating the inertial surface force) plays the role of applied surfacial force density. The body’s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it, a feature that makes it possible to exploit the traction Cauchy theorem within second-grade materials. This means that a second-grade material is formed up by two sub-systems, that is, the bulk material operating as a classical Cauchy continuum, and the thin boundary layer operating as a Gurtin–Murdoch material surface. The classical linear and angular momentum theorems are suitably extended for higher order inertia, from which the local motion equations and the moment equilibrium equations (stress symmetry) can be derived. For an isotropic material featured by four constants, i.e. the Lamé constants and two length scale parameters (Aifantis model), the dynamic evolution problem is characterized by a Hamilton-type variational principle and a solution uniqueness theorem. Closed-form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature. The paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics.

On some properties of 2D spectral finite elements in problems of wave propagation

A computational analysis of plane progressive wave propagation in plane stress body is presented. The initial-boundary value problem of linear elastodynamics of Cauchy continuum is approximated spatially by specially designed multi-node C0 displacement-based isoparametric quadrilateral spectral finite elements. To integrate element matrices the Gauss–Lobatto–Legendre quadrature rule is used. The temporal discretization is carried out by Newmark type algorithm reformulated to accommodate the structure of local element matrices. The developed multi-node spectral elements with Gauss–Lobatto–Legendre nodes are validated by running some statics and dynamics tests to investigate the presence of locking effect and of spurious zero-energy modes. Dynamic tests, dedicated to wave propagation in L-shaped structure, are concentrated on energy propagation through right-hand angle of the construction.

Relaxed loading conditions for higher order homogenisation approaches

AbstractThe present paper deals with the formulation of minimal loading conditions for the application of numerical homogenisation techniques, namely the FE2 methodology. Based on the set of volume averaging rules connecting the heterogeneous micro and the homogeneous macro scale, the minimal constraints on the deformation of a micro volume are derived for a classical Cauchy continuum as well as for a micromorphic continuum theory. For both cases, numerical studies are included highlighting the main aspects of the proposed procedure within the context of small deformations. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua

AbstractWe study plane waves in second gradient solids and their reflection and transmission at plane displacement discontinuity surfaces. The needed extension of the treatment adopted to study plane wave propagation in Cauchy continua is not straightforward and is developed here. In particular, the balance of mechanical energy valid for second gradient continua is deduced. The presented results may be of interest as many boundary layer phenomena can be accounted for by second gradient models. We prove that second gradient elastic moduli may influence, in a measurable manner, how planar waves behave at discontinuity surfaces: the novel results presented here can be the basis of experimental procedures to estimate some among these moduli. We explicitly remark that reflection and transmission coefficients which we have estimated show a significant dependence on frequency, which indeed makes easier to conceive effective measurement methods.