FILECCIA SCIMEMI, Giuseppe

GIAMBANCO, Giuseppe

SPADA, Antonino

Archivio istituzionale della ricerca - Università di Palermo

ECCM 2010IV European Conference on Computational MechanicsPalais des Congrès, Paris, France, May 16-21, 2010Meso-modeling of heterogeneous structures via interphase modelG. Giambanco1, A. Spada1, G. Fileccia Scimemi11 Department of Structural, Aerospace and Geothecnical Engineering, University of Palermo, Italye-mails: ggiamb@unipa.it, antonino.spada@unipa.it, giuseppe.filecciascimemi@unipa.itAll those structures that are constituted by heterogeneous materials, such as masonry structures andcomposite laminates, exhibit a complex anisotropic behaviour strictly related to the static and kinematicphenomena occurring in each constituent and at their interfaces. The overall macroscopic approach tothe analysis of structures made up of heterogeneous material consists in formulating phenomenologicalconstitutive laws expressed in terms of macroscopic stress and average strain for the equivalent homoge-neous continuum. This way to operate in some cases may be not appropriate to describe the elastic andpost-elastic response of the structures since it requires the introduction of strong simplifications. In par-ticular, most of the nonlinearities of the overall response depend on local phenomena occurring betweencomponents such as debonding, sliding and dilatancy.The meso-modelling approach is more rigorous since it considers the material as a discontinuous assem-bly of units and joints with their own geometry and constitutive properties. The joints in most casesrepresent the weakness areas of the heterogeneous material where fractures appear and propagate. Inthe case of adhesive joints a thin layer of a third material connects two or more units and its thicknessis usually small if compared with the other dimensions of the assembly. In literature, a common way tosimulate the thin joint is by applying the so-called ’zero-thickness interface’.With reference to Fig. 1 the contact layer of thickness h, composed by the third material and by thetwo physical interfaces Σ+ and Σ− collapses to its middle surface Σ where the contact tractions and thediscontinuous displacements represent the static and kinematical quantities regulating the joint response.Therefore, the interface constitutive laws, elastic or inelastic, are relations between the above cited staticand kinematic quantities.However, in many cases the joint response depends also on internal stresses and strains within the thirdmaterial. A typical example is the squeezing effect of the mortar joint interposed between two rigidblocks and subjected to a compression load.In this direction the enhancement of the zero-thickness interface is represented by the interphase model.By employing the term interphase, we shall mean a layer separated by two interfaces from the bulk ma-terial or a multilayer structure with varying properties and several interfaces.Two main assumptions are included into the model. The first one assumes that the fibers inside theinterphase along the z-direction during the deformation process continue to remain rectilinear. The sec-ond one considers a constant strain state along the thickness of the joint because of its small dimension.The kinematical and equilibrium equations are written in matrix form as:εˆ= 1h A1 (u+−u−)+ 12 A2 (u+ +u−)t+ = σˆ · I3 − h2 divσˆ, t− = −σˆ · I3 − h2 divσˆ on Σm · σˆ = 0 on Σ’s boundary.In the equations given above u+, u−, εˆ and σˆ are the displacement vector in the surface Σ+, the dis-1Figure 1: The interphase modelplacement vector in the surface Σ−, the strain and stress vectors in the interphase respectively.Each one of these equations is written as the sum of two components. The first one refers to the interfaceresponse, the second one contains the additive terms obtained introducing internal state of stress andstrain into the formulation.The interphase model has been introduced as a new user element in the open-source finite element pro-gram FEAP [6]. The finite element is characterized by 4-node rectangular shaped and two or three Gaussor Lobatto points are used as quadrature rule. Strain and stresses at the interphase can be separated intointernal and contact components and related to the nodal displacements by using:εˆi = BiUεˆc = BcUwhere Bi and Bc are two matrices containing the derivatives of the form functions, and U is the vec-tor collecting the nodal displacements. The total strains at the integration point is given byεˆ = εˆc + εˆi = BUwhere B = Bc +Bi.Stresses, forces, and elastic stiffness matrix of the element are evaluated in the classical way.Summarizing, in this work a mesoscale interphase model is presented. The constitutive laws of theinterphase are written in terms of internal state of stresses and contact tractions and related kinematicvariables. The model is implemented in a research oriented finite element code. Numerical simulationsare provided to show the main features of the model and novelties introduced with respect to the commoninterface models.References[1] Spada, A., Giambanco, G., Rizzo, P., “Damage and Plasticity at the interfaces in composite materials andstructures,” Computer Methods in Applied Mechanics and Engineering, 198, 3884-3901 (2009).[2] Giambanco, G., Rizzo, S., Spallino, R., “Numerical analysis of masonry structures via interface models”,Computer Methods in Applied Mechanics and Engineering, 190(49-50), 6493-6511 (2001).[3] Alfano, G., Sacco, E., “Combining interface damage and friction in a cohesivezone model”, InternationalJournal for Numerical Methods in Engineering, 68, 542-582, (2006).[4] Giambanco, G., Mróz, Z., “The interphase model for the analysis of joint in rock masses and masonry struc-tures,” Meccanica, 36, 111-130 (2001).[5] Proceedings of the XIX Congresso AIMETA di Meccanica Teorica e Applicata, Spada, A., Fileccia Scimemi,G., Giambanco, G., “The interphase model applied to simulate chemically bonded anchors”, Ancona, 2009.[6] Taylor, R. L., “FEAP - - A Finite Element Analysis Program,” Version 7.5 User Manual, (2005).2

Meso-modeling of heterogeneous structures via interphase model

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Meso-modeling of heterogeneous structures via interphase model

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