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A theoretical analysis of instability and bifurcation failure phenomena in periodic microstructured nonlinear composite solids embedding discontinuity interfaces
The article delves into the theoretical analysis of instability and bifurcation failure phenomena in periodic microstructured nonlinear composite solids, focusing on the role of discontinuity interfaces. It begins with an introduction to the demand for advanced materials with superior properties in various engineering fields, such as aerospace and transportation. The text then explores the complex failure mechanisms in composite materials, including microscopic buckling, instabilities, and localization of deformation. A significant portion of the article is dedicated to the development of a theoretical framework based on finite strain continuum mechanics, which includes an enhanced cohesive-contact model to simulate decohesion and contact phenomena. The article also discusses the stability and uniqueness conditions associated with the equilibrium problems of composite microstructures. A notable feature is the use of a two degrees-of-freedom (2-DOF) example to illustrate primary stability and bifurcation properties, which is analyzed from both structural mechanics and continuum mechanics perspectives. The article concludes with a discussion on the importance of accurate modeling of cohesive-contact interfaces for predicting critical load levels and understanding the physical mechanisms governing instability. Readers will gain insights into the advanced theoretical approaches and practical considerations for analyzing failure phenomena in composite materials, making this article a valuable resource for professionals in the field.
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Abstract
This paper proposes a novel theoretical study on the onset of failure in finitely deformed periodic nonlinear composite materials because of microscopic instability and bifurcation mechanisms in conjunction with decohesion and contact effects at interfaces between different constituents. Original analytical investigations are firstly carried out on an introductory 2-DOF example highlighting the main features of the examined problem and using a structural mechanics approach. The theoretical setting of the problem is then developed within a finite strain continuum mechanics framework and a nonlinear homogenization formulation is adopted to drive the system along macro-deformation loading paths. The formulation includes a continuum contact mechanics model in conjunction with a class of irreversible cohesive traction–separation laws for treating both unilateral contact constraint and progressive decohesion at discontinuity interfaces. The main equations governing the equilibrium problem of the microstructure in both finite and rate forms are developed, and the relevant issues associated with loss of uniqueness in the rate equilibrium solution together with the instabilities onset are also investigated by developing an exact second-order analysis. The introductory example is then re-examined by using the proposed continuum mechanics formulation and comparisons with simplified cohesive-contact models frequently adopted in the literature are performed. The obtained results show the role played by contact and cohesive mechanisms and the significance of an appropriate modelling of their deformation sensitivity and conditionality nature to perform accurate stability and bifurcation analyses. Strategies to circumvent the complications arising both from cohesive behavior and contact mechanics nonlinearities arising at the interface are also discussed.
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1 Introduction
Many of the current engineering applications and emerging technologies demand advanced materials with unconventional and superior properties in comparison with conventional materials (ceramics, metal, polymers, for instance). For instance, in the context of aerospace, structural and transportation engineering, materials with low density but exhibiting high strength, stiffness, durability and impact resistance are increasingly demanded. By virtue of technological advances, further increased by the recent development of additive manufacturing techniques, it is currently possible to meet such requirements by taking advantage of the use of advanced composite materials, whose combined characteristics are superior to those of the constituent materials owing to an appropriate design of the microstructure geometrical and mechanical properties. These engineered materials are often referred to as metamaterials [1, 2] because they may exhibit exotic and/or improved static, dynamic, and multifunctional characteristics resulting from their hierarchical microstructure rather than their physical composition.
Composite materials, composed by a periodic arrangement of strong and stiff reinforcement phases, in the form of fibers (which in turn may be continuously or discontinuously arranged) or particles (rigid or compliant) surrounded by a soft elastomeric matrix phase and often including material discontinuities (in the form of voids or cracks), represent a significant class of composite materials. Man-made fiber-reinforced and particle-reinforced composites with elastomeric matrix, are frequently used in several advanced engineering applications where multi-field coupled properties (related, for instance, to electric, magnetic and thermic and deformation fields) and capability to sustain large deformations avoiding failure modes are required (see, for instance, [3‐11]). Examples of the above applications are sensors and actuators, energy and vibration absorbers, biomedical components, packaging systems, nanostructured materials, soft robotics components, and reinforced elastomers used for car tires. Multimaterial composites and bioinspired microstructured materials, often fabricated by using the 3D printing methodology [12‐18], represent very recent types of reinforced composite materials undergoing large deformations of increasing interest to material scientists, owing to their remarkable mechanical properties which can be obtained mimicking the material architecture of natural biocomposites. Moreover, many materials found in nature exhibit a fiber-reinforced composite structure at different length scales and experience large deformations [19‐21], such as wood, arterial walls, tendon fascicles, bones and soft biological tissues.
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In the above mentioned advanced applications composite materials must be able to provide their high performances without failure, even when subjected to extreme loading conditions inducing large deformations and fatigue. Failure in reinforced composite materials may occur according to a complex variety of precursor microscopic mechanisms, the most studied ones, mainly in the context of small deformations, being matrix cracking, damage, plasticity, fiber breakage and interface debonding (see [22‐28], for instance). When large deformations are also involved, as in the case of the above mentioned advanced engineering application, additional failure mechanisms may occur, such as microscopic buckling, instabilities and localization of deformation, in combination with micro-cracking and contact phenomena occurring under macroscopic loading conditions involving normal interface compression [29‐37]. Decohesion phenomena at the interface between different micro-constituents may be simulated by introducing cohesive interface constitutive laws which may be formulated in a finite strain context [38‐44].
The prediction of failure in advanced composite materials as a consequence of the above mentioned microscopic mechanisms and of their possible interaction, is of great practical importance owing to the necessity to obtain a realistic modeling and analysis of their nonlinear finite strain behavior and to fully exploit their optimized properties.
A vast amount of literature has been dedicated to understanding how the above micromechanical phenomena may influence the macroscopic or homogenized behavior of such materials in terms of both static and dynamic properties. In the last decades, researchers have devoted much attention to effectively considering these problems by proposing sound and accurate, as well as computationally efficient, numerical modeling approaches based on nonlinear homogenization and multiscale techniques in a large deformation context [45‐49].
An increasing interest has been recently emerged in investigating the dynamic and wave propagation characteristics of periodic microstructured metamaterials subjected to large levels of pre-deformation. In some applications involving custom-designed metamaterial samples that can incorporate controlled defects, the microstructural evolution due to microscopic instabilities, instead of being a symptom of failure, may also result in a method to control the wave attenuation properties of metamaterials [2, 50, 51] or can be exploited to obtain new functionals actuation mechanisms [52, 53].
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In a recent Authors’ work [54] a theoretical investigation on the combined effects of fiber/matrix decohesion and microscopic instability phenomena on the macroscopic behavior of periodic microstructured materials has been carried out restricted to special macro-deformation paths and contact conditions leading to a unified treatment of instabilities and bifurcation events and numerically demonstrated for layered composites. The present work represents a generalization of the above-mentioned study to arbitrary macro-deformation loading paths leading to general nonlinearities in both interface contact conditions and cohesive response and it provides an in-depth theoretical analysis of the related stability and uniqueness issues with all the necessary mathematical developments.
In light of the above considerations, the present paper presents a novel bifurcation and stability analysis of the nonlinear elastic response of composite materials with a reinforced microstructure subjected to macrostrain-driven loading conditions inducing contact and decohesion phenomena at reinforcement/matrix interfaces when micro-buckling mechanisms occur. The developed theoretical framework, based on finite strain continuum mechanics and adopting an enhanced cohesive-contact model, provides a way to study the structure and properties of the associated stability and uniqueness problems and is demonstrated by means of an analytical example showing the main failure characteristics of the examined composite solid in conjunction with the role of the cohesive and contact effects on the accuracy of critical loads determination.
2 An introductory 2-DOF example of cohesive and contact instability: structural mechanics approach
A two degrees-of-freedom structural system is here proposed to highlight the primary stability and bifurcation properties for solids with embedded discontinuity interfaces modeled with cohesive laws and unilateral contact. The nonlinear system is subjected to a compressive dead load inducing unilateral contact along an internal interface, allowing only relative frictionless sliding, and it is equipped with a nonlinear tangential spring to simulate cohesive behavior. At this stage, the system is analyzed from a structural mechanics point of view based on member loads and elongations as stress- and strain-like variables, which provides a simple way to obtain primary bifurcation and instability load levels and related deformation modes, but it is not able to clarify the role of cohesive and contact mechanisms in triggering such nonlinear phenomena. To this end, the example will be reexamined later by adopting an alternative methodology based on the continuum mechanics theoretical approach which will be proposed in the present paper.
The geometric arrangement of the system is shown in Fig. 1 where the internal connection between the two rigid rods AB and CD, each one of length \(L\), allows a relative sliding of amount \(\delta\) and prevents relative rotations between its surfaces; the system is hinged at the right end of AB while rigidly connected at the left end C of CD. The rigid rotations of the two rods are denoted by \(\varphi_{1}\) and \(\varphi_{2}\), respectively. An elastic rotational spring of stiffness \(k\) connects the rigid bar AB with the left surface of the internal connection and it assumes a vertical position in the reference configuration. The shortening at the point of load application is denoted by \(\Delta\). The nonlinear spring undergoing relative sliding and simulating cohesive behavior supplies a restoring force \(t(\updelta )\) which is a nonlinear function of the sliding (and, eventually, of its attained maximum to take into account its irreversible nature under unloading). Under compressive loading (\(\lambda > 0\)) the internal connection is in a state of effective unilateral contact conditions (namely the contact pressure is everywhere not zero) and the cohesive force activates only if the system transits from the fundamental equilibrium path (characterized by \(\varphi_{1} = \varphi_{2} = 0\)) to a buckled equilibrium path (\(\varphi_{1} \ne 0\), \(\varphi_{2} \ne 0\)) at the critical load \(\lambda_{c}\). With reference to the deformed configuration sketched in Fig. 1, after some geometric considerations, the shortening and the sliding can be respectively expressed as:
Two degrees-of freedom nonlinear structural model composed by two rigid rods connected by a cohesive-contact interface. In the upper part a representation, in the reference configuration, of the cohesive-contact interface moving with velocity v, is given; the buckled deformation path is also illustrated in the lower part
The above equations can be obtained by the equilibrium condition of the rod AB in the sliding direction and by its rotational equilibrium condition with respect to A, once the normal contact reaction \(R\) is determined by the equilibrium of the rod AB in the direction normal to the internal connections surfaces as \(R = \lambda \cos \varphi_{2}\).
The fundamental equilibrium solution characterized by \(\varphi_{1} = \varphi_{2} = 0\) is theoretically available for all \(\lambda\). Additional buckled equilibrium states can be available for nonzero values of \(\varphi_{1}\) and \(\varphi_{2}\) and are represented by
where \(\varphi_{1}\) and \(\varphi_{2}\) are interrelated by Eq. (2)2 and which can be also reduced to \(\left( {\varphi_{1} + \varphi_{2} } \right)k - t\left( \delta \right)L\frac{{{\rm{sin}}\varphi_{1} }}{{{\rm{sin}}\varphi_{2} }} = 0\).
Assuming that the irreversible cohesive behavior can be neglected in the bifurcation analysis and adopting the following asymptotic expansion for the cohesive force \(t(\delta ) = k_{1} \delta + k_{2} \delta^{2} + ....\,\left( {k_{1} > 0} \right)\), the critical bifurcation load at which the buckled equilibrium path is connected to the fundamental path is given by:
The critical load can be obtained after linearization of Eq. (2), taking into account Eq. (1)2 and extracting the positive value of \(\lambda\) at which non trivial solutions are admitted.
a result that is admissible only for an internal connection with a bilateral contact constraint, but not acceptable for a unilateral contact constraint. As a matter of fact, the proposed example includes the unusual instability phenomena under tensile dead loading similar to those presented in [55] as a very special case (namely bilateral contact, absence of cohesive mechanisms). The corresponding tensile critical mode is characterized by \(\dot{\varphi }_{20} = - \frac{{2k_{1} L^{2} }}{{ - k - k_{1} L^{2} + \sqrt {k^{2} + 6kk_{1} L^{2} + k_{1}^{2} L^{4} } }}\dot{\varphi }_{10}\). The load–displacement behavior of the examined system is illustrated in Fig. 2 where both the equilibrium paths under compression and under tension (for a bilateral internal connection) are shown together with the relevant critical bifurcation loads \(\lambda_{c}^{ + }\) and \(\lambda_{c}^{ - }\), respectively. It is worth noting that in Fig. 2 the assumption \(k < k_{1} L^{2}\) has been made leading to \(\lambda_{c}^{ + } > \left| {\lambda_{c}^{ - } } \right|\), since when \(k > k_{1} L^{2} ,\,\lambda_{c}^{ + } < \left| {\lambda_{c}^{ - } } \right|\).
Fig. 2
Load displacement equilibrium paths for the 2-DOF system of Fig. 1
Under the same hypotheses on the nature of the cohesive force law envisaged above, the above bifurcation analysis can be carried out by means of a work-based approach which will be adopted in the subsequent continuum mechanics formulation of the stability analysis proposed in Sect. 4. In fact, as will be proved by using the proposed continuum mechanics formulation, primary instability and bifurcation critical load levels coincide since effective contact conditions are satisfied along the principal path and the contact rate loading terms and the rate cohesive model admit a potential.
In this context, the second order approximation of the asymptotic expansion, with respect to the fundamental solution, of the difference between the internal deformation work related to both the cohesive interface and the elastic spring and the work done by the external load (note that the contact reactions at the internal connection do not expend power), can be placed in the following form:
leading to Eq. (4) and showing that the buckled equilibrium path exists only under compressive dead loading, while if the internal connection acts as a bilateral constraint buckling under tensile dead loading occurs at the load level of Eq. (8).
3 Problem formulation
The main equations governing the equilibrium problem of a representative volume element (RVE) of a nonlinear composite material with a periodic reinforced microstructure are here briefly stated with reference to both global and rate forms, in order to provide the basis for the subsequent instability and bifurcation failure analysis and to introduce the notation adopted for the kinematical and static variables in the present work.
As illustrated in Fig. 3, the examined RVE, attached to each point of the corresponding homogenized macrostructure and composed by a generic assembly of unit cells in order to capture possible instability and bifurcation phenomena (see [56]), is assumed to occupy the domain \(V_{(i)}\) (with boundary ∂V(i)) in its initial undeformed configuration and to contain a set of internal interfaces where possible displacement discontinuities representing cohesive, traction-free or self-contacting fractures, may occur; the union of surface domains occupied in the undeformed configuration by these interfaces is denoted by Γ(i). The solid part of V(i) is denoted by B(i), whereas its eventual void part by H(i). The deformation process is described by using a time-like parameter t, monotonically increasing with the evolution of the loading process, with t = 0 at the beginning of the deformation process where the body occupies the initial stress-free undeformed configuration. A total Lagrangean formulation is adopted in the description of failure phenomena occurring during the deformation process with the initial configuration occupied by the body taken as reference to write the governing field equations. The deformation process is represented by the one-parameter family of deformations x(X,t): V(i) → Vt mapping points X of the initial configuration V(i) onto points x of the current configuration Vt.
Fig. 3
One-cell RVE of a composite solid with a periodic reinforced microstructure containing displacement discontinuity interfaces where cohesive fracture and self-contact phenomena occur
Each of the above discontinuity interfaces comprises a pair of physical surfaces of the body (the lower and the upper one, respectively denoted as Γ(i)l and Γ(i)u), initially coincident (t = 0) but separating in the subsequent deformation process (t > 0). Moreover, during the deformation process, some portions of the above surfaces may come into unilateral frictionless contact both when no cohesive tractions can be transmitted (owing to complete decohesion or for a preexisting crack) and when negative normal separation occurs during the decohesion process. In the former case the portions of the discontinuity interface will be referred to as “crack-contact interface”, while in the latter “cohesive-contact interface”.
According to a Lagrangean description of a generic phase of the deformation process, the set of all interfaces where cohesion is active in the current configuration Vt is denoted by Γch(i)(t) in the reference configuration, which is composed of pairs of lower and upper reference cohesive surfaces Γlch(i)(t) and Γuch(i)(t), respectively placed on the negative and positive side of the unit normal nl(i)(t) to the oriented interface Γch(i)(t) and separating into the two distinct cohesive surfaces \(\Gamma_{ch}^{l}\)(t) and \(\Gamma_{ch}^{u}\)(t) after deformation (see Fig. 4). The portions of lower and upper surfaces of Γ(i) where contact occurs during the deformation process at time t are respectively denoted by the reference surfaces Γc(i)l(t) and Γc(i)u(t). Owing to large displacements effects, the above surface Γc(i)l(t) and Γc(i)u(t) are generally disjoint, since pair of points in contact in the current configuration are generally not coincident in the reference one. On the other hand, in the current deformed configuration, the intersection of physical surface pairs undergoing contact is denoted by the single interface Γc(t). In what follows the dependence on t will be omitted except when ambiguity must be avoided.
Fig. 4
Deformation process of a cohesive-contact interface from the initial undeformed configuration at t = 0; both the Lagrangean and the Eulerian descriptions are shown with reference to the two pairs of cohesive points (A,A’) and (B,B’) and to the pair of contact points (A,B’)
3.1 Nonlinear cohesive modeling at finite deformations: definition of constitutive variables and representation of interface contact mechanisms
A cohesive point pair is defined as a pair of initially coincident material points \({\varvec{X}}^{l}\) and \({\boldsymbol{X}}^{u}\) respectively placed on opposite sides \(\Gamma_{ch(i)}^{l}\)(t) and \(\Gamma_{ch(i)}^{u}\)(t) of the reference cohesive interface Γch(i)(t), and denoted as
Consequently, in order to introduce the cohesive model, the displacement jump vector across the cohesive interface at a cohesive point pair (Xl, Xu)ch(i) is introduced according to the following expression:
Note that Eq. (14) also defines the meaning of the double bracket symbol denoting the jump in the enclosed quantity. The deformation mapping transforms the unit normal n(i)l in the unit normal nl(t) to the current deformed cohesive surface \(\Gamma_{ch}^{l} (t)\) according to the Nanson’s formula:
where dSch and dSch(i) respectively are the current and the reference surface elements at the same material point. Taking the current unit normal nl(t) as a reference, Δ can be expressed in terms of its normal and tangential components, respectively defined as Δnnl and Δs, according to the following expressions:
As a prelude to define an interface traction–separation cohesive law, the continuity-like condition of the nominal cohesive traction vector tR ch across the reference cohesive interface is usually imposed in the technical literature (e.g. [38‐43]). Consequently, the following relations respectively are satisfied in terms of the nominal (per unit reference cohesive surface area) and the true (per unit current cohesive surface area) cohesive interface traction vectors acting on the lower and upper cohesive surfaces:
where \(dS_{ch}^{u}\)(t) and \(dS_{ch}^{l}\)(t), respectively, are the current deformed surface elements (respectively at xu and xl) on the upper and lower cohesive surfaces, corresponding through the deformation mapping to two initial coincident undeformed reference surface elements (at Xu = Xl) \(dS_{ch(i)}^{u} = dS_{ch(i)}^{l}\).
When contact is absent, it follows from Eq. (17) that \(\llbracket{\boldsymbol{T}}_{R}\rrbracket_{{\Gamma ch(i)} }{\boldsymbol{n}}^{l}_{(i)} = {{\boldsymbol{0}}}\) where TR is the first Piola–Kirchhoff nominal stress tensor. Note that Eq. (17) does not generally follow from the balance of linear momentum across the cohesive interface in the current configuration, due to the finite deformation kinematics of the cohesive surfaces.
Arguments based on material objectivity requirements and thermodynamic consistency (see for instance [41, 43]) have shown that an appropriate cohesive interface law tR ch(Δ) can be defined in terms of interface kinematical quantities referred to the current deformed cohesive interface. Specifically, the formulation of an interface cohesive law based on the current unit normal to the lower deformed cohesive surface allows to be consistent with objectivity requirements and to account for cohesive interface rotations; moreover, this choice is coherent with the finite deformation contact mechanics modeling of the unilateral constraint avoiding interpenetration at the cohesive interface, as will be shown in the following. Different choices of unit normal definition to determine interface displacement jump components can be done not related to physical surfaces but rather to ideal ones, such as the one referred to the so-called deformed cohesive “mid-surface” (see [40‐43], for instance). The choice of a local normal has a significant influence on the values of the normal and tangential interface displacement jump components and, in turn, leads to different traction vector formulations [57, 58], except when a colinear interface constitutive law is adopted where the traction vector is aligned with the interface displacement jump one. Assuming that, owing to interface connection provided by cohesion, the displacement jumps along the active cohesive portions of the interface are small, leading to negligible differences between formulations based on different orientations of the local unit normal, namely those referred to the upper, lower, or middle deformed cohesive surfaces. Only when the effects of finite deformations can be completely neglected, the cohesive interface traction vector formulation can be referred to the normal and tangent unit vectors to the undeformed cohesive interface Γch(i), as done in several studies carried out in the literature (see, for instance [38, 39] and the references cited therein).
In the above introduced framework, the cohesive law can be viewed as a general nonlinear function of the normal and tangential displacement jump \({\boldsymbol{t}}_{R\;ch} = {\boldsymbol{g}}\left( {{{\varDelta}}_{n} {\boldsymbol{n}}^{l}, {\boldsymbol{\varDelta}}_s,{\boldsymbol{n}}^{l} } \right)\), where the dependence on the geometry of the deformed cohesive surface is assumed to be restricted to the unit normal nl(t), and can be defined by using two separate constitutive relations for the normal and tangential cohesive interface traction vector components, respectively denoted by \(t_{n\;ch} {\boldsymbol{n}}^{l}\) and \({\boldsymbol{t}}_{sch}\), according to the following expressions:
A class of mixed-mode cohesive laws can be obtained based on the concept of the effective separation and assuming its maximum attained during the deformation process as the internal/damage variable (see [38, 59] for instance).
Although the following theoretical developments regarding stability and uniqueness will be developed in full generality, namely independently on the specific interface cohesive law and on its deformed local basis choice, some specific results will be given by generalizing the phenomenological interface traction–separation law introduced in [38] and then slightly modified in [39], in such a way to account for finite interface deformations. To this end, according to Eq. (18), the cohesive interface law is formulated by using the local deformed basis:
where Δnc and Δsc are the critical displacements at which complete decohesion respectively occur under purely normal and tangential interface displacement jump, tmax is the maximum traction in purely normal interface displacement jump, \(\hat{\varDelta }\) is the dimensionless effective separation, \(\hat{\varDelta }_{{\rm{max}} }\) is the maximum attained by the dimensionless effective separation during the deformation history, f(\(\hat{\varDelta }\)) describes how the normal traction varies as a function of the normal interface separation, and α specifies the normal to tangential maximum traction ratio.
The above introduced constitutive cohesive model can be conveniently rearranged in a form resembling the damage mechanics framework:
It is worth noting that the cohesive model specified by Eq. (20) in general does not ensure a potential structure, unless \(\alpha = \varDelta_{n}^{c} /\varDelta_{s}^{c}\) [38]. Moreover, when \(\varDelta_{n}^{c} = \varDelta_{s}^{c} = \varDelta^{c}\) and \(\alpha = 1\), the traction vector becomes colinear with the interface displacement jump (see [38, 42]). Such colinear interface constitutive law can be usefully applied to simulate the interface debonding phenomena induced by fibrillation, fiber bridging, and stitching mechanisms in nonlinear composite materials. The former case is typically observed when delamination between a polymer layer and a steel substrate arises, for instance, in adhesively bonded joints [57]; the second is common in both short and long fiber-reinforced composites, where the presence of fibers may delay crack propagation process and consequently enhances the fracture toughness of the material; while the third refers to the use of stitching as a reinforcing strategy in composite laminates, which improves delamination resistance by physically linking adjacent layers through fiber threads and is commonly employed in aerospace structures and marine composites to prevent interlaminar failure.
In the colinear case, the cohesive law structure (21) shows the following attractive structure:
The above introduced interface constitutive law is assumed to be valid also when contact between opposite cohesive surfaces occurs. As a matter of fact, as will be shown in the sequel, contact is modeled separately from the cohesive constitutive model by means of a full finite deformation treatment of the unilateral self-contact condition between adjacent cohesive surfaces, treated as a global constraint, developed as an extension of the formulation introduced in some previous authors’ works where micro-fracture contact has been studied without cohesive effects (see [31, 37], for instance).
It is worth noting that when tangential interface displacement jumps can be assumed small, as may occur in the case of active cohesive zones or near a crack tip, contact conditions can be approximately modeled by a penalty-like linear cohesive law, and cohesive interfacial interpenetration can be detected when negative normal separation occurs \(\varDelta_{n} < 0\) (see, [38, 39, 42, 59]); in such a circumstance the dimensionless effective separation can be defined including only the tangential contribution as \(\hat{\varDelta } = \left| {\left\| {\boldsymbol{\varDelta}_{s} } \right\|/\varDelta_{s}^{c} } \right|\) and the normal cohesive traction, within the cohesive law (Eq. (20)), can be determined as \(t_{n\,ch} = k_{n} \Delta_{n}\) where the penalty stiffness can be assumed \(k_{n} = (27/4)t_{{\rm{max}} } /\varDelta_{n}^{c}\). In particular, this approximate treatment of unilateral cohesive interfacial contact has been adopted in the literature also with reference to the undeformed geometry of the cohesive surface (see [38, 39], for instance), thus neglecting the rotation of the normal to the deformed cohesive surface, an assumption applicable for small deformations and displacements of the cohesive interface.
Note that under the hypothesis of small cohesive interface tangential displacement jumps, the interface normal displacement jump can be determined indifferently at a cohesive point pair and at a contact point pair since \(\left[\kern-0.15em\left[ {u_{n} } \right]\kern-0.15em\right]_{\Gamma ch(i)} = \left[\kern-0.15em\left[ {u_{n} } \right]\kern-0.15em\right]_{\Gamma c}\).
3.2 Rate interface constitutive law and relative convexity conditions
The derivative with respect to a time-like parameter t (at a fixed X) of the cohesive interface law (Eq. (20)) leads to:
where a dot denotes the derivative with respect to t, the prime means differentiation, \({\boldsymbol{L}} = \nabla_{x} \dot{\boldsymbol{u}}\) is the spatial gradient (with respect to x) of the displacement rate and \(L_{n} = {\boldsymbol{Ln}} \cdot {\boldsymbol{n}}\). Equation (24)1 and Eq. (24)2 include both the case of loading (or increasing damage), occurring for \(\dot{\hat{\varDelta }}_{{\rm{max}} } > 0\) (\(\hat{\varDelta } = \hat{\varDelta }_{{\rm{max}} } {\mkern 1mu} {\mkern 1mu} \,{\rm{and}}\,{\mkern 1mu} {\mkern 1mu} \dot{\hat{\varDelta }} > 0\)), and the case of both elastic behavior and elastic unloading (where the accumulated damage remains unvaried) \(\dot{\hat{\varDelta }}_{{\rm{max}} } = 0\) (\(\hat{\varDelta } < \hat{\varDelta }_{{\rm{max}} } \,{\rm{or}}\,\hat{\varDelta } = \hat{\varDelta }_{{\rm{max}} } \,{\rm{and}}\,\dot{\hat{\varDelta }} \le 0\)). The rate counterpart of the formulation (Eq. (22)) can be obtained by taking its straightforward material time derivative.
Note that in the 2D case the tangential interface displacement jump \({\boldsymbol{\varDelta}}_{s}\) can be described in terms.
of a unit tangent s embedded in the deformed cohesive interface as \({\boldsymbol{\varDelta}}_{s} = \left( {\boldsymbol{\varDelta}}_{s} \cdot {\boldsymbol{s}} \right){\boldsymbol{s}}\); therefore the rate of the unit tangent can be written by using the following expression \({\dot{\boldsymbol{s}}} = {\boldsymbol{Ls}} - \left( {\boldsymbol{s}} \cdot {\boldsymbol{Ds}} \right){\boldsymbol{s}},\,\boldsymbol{D}\) being the symmetric part of L. In the general 3D case, the rate of the unit vector aligned with the tangential interface displacement jump must be determined by using its definition \({\boldsymbol{s}} = {\boldsymbol{\varDelta}}_{s} /\left\| {\boldsymbol{\varDelta}}_{s} \right\|\) and involving the projection operator \({\boldsymbol{P}} = {\boldsymbol{1}} - \left( {{\boldsymbol{n}} \otimes {\boldsymbol{n}}} \right)\). Moreover the rate of the interface normal displacement jump \(\dot{\varDelta }_{n} = \left( {\boldsymbol{\varDelta}} \cdot {\boldsymbol{n}} \right)^{\cdot }\) is generally different from the normal component of the interface displacement rate \({\dot{\boldsymbol{\varDelta}}}\cdot {\boldsymbol{n}}\), the above quantities coinciding only when the scalar product \({\boldsymbol{\varDelta}} \cdot {\dot{\boldsymbol{n}}}\) vanishes. This occurs when \({\boldsymbol{\varDelta}}\) and/or \(\dot{\boldsymbol{n}}\) vanish or when they are orthogonal vectors; the former case occurs when the deformation path does not activate interface displacement jumps and/or cohesive interface rotations, the latter one occurs in the case of small (negligible) interface tangential displacement jumps \({\boldsymbol{\varDelta}}_{s} \to {\boldsymbol{0}}\). As a matter of fact, it follows that \({{\boldsymbol{\varDelta}}} \cdot {\dot{\boldsymbol{n}}} = \left( {\varDelta_{n} {\boldsymbol{n}} + {{\boldsymbol{\varDelta}}}_{s} } \right) \cdot {\dot{\boldsymbol{n}}} = \varDelta_{n} {\boldsymbol{n}} \cdot {\dot{\boldsymbol{n}}} = 0\), since \({\boldsymbol{n}} \cdot {\dot{\boldsymbol{n}}} = 0\). Similar considerations can be done for tangential interface displacement jump vector \({\dot{\boldsymbol{\varDelta}}}_{s} = \left( \left\| {{\boldsymbol{\varDelta}}_{{\boldsymbol{s}}}} \right\|{\boldsymbol{s}} \right)^{ \cdot } = \left\| {\boldsymbol{\varDelta}}_{s} \right\|^{ \cdot } {\boldsymbol{s}} + \left\| {\boldsymbol{\varDelta}}_{s} \right\|{\dot{\boldsymbol{s}}}\) which is equal to the first contribution only when the second one vanishes (\(\left\| {{\boldsymbol{\varDelta}}_{s} } \right\|\) and/or \(\dot{\boldsymbol{s}}\) vanishes) as occurs, for instance, when \({\boldsymbol{\varDelta}}_{s} \to {{\boldsymbol{0}}}\).
The cohesive interface displacement jump rate assumes the following expression:
Assuming that contact is approximately included in the cohesive model by using a penalty formulation, the normal cohesive traction rate component in Eq. (24) assumes the following expression:
In the special case of a colinear cohesive interface constitutive law (\(\varDelta_{n}^{c} = \varDelta_{s}^{c} = \varDelta^{c} ,\alpha = 1\)), the incremental interface constitutive law assumes a symmetric structure with the following expression:
showing that the cohesive interface traction vector rate is a positively homogeneous one-degree function of \({\dot{\boldsymbol{\varDelta}}}\) and thus can be derived from a positively homogeneous of degree two rate potential \(U = {1/2}\,{\dot{\boldsymbol{t}}}_{Rch} \cdot {\dot{\boldsymbol{\varDelta}}}\) due to Euler theorem. Moreover, note that the cohesive constitutive law and its product \({\dot{\boldsymbol{t}}}_{Rch} \cdot {\dot{\boldsymbol{\varDelta}}}\) does not depend on the choice of the local basis adopted to describe normal and tangential interface displacement jump components. A similar situation occurs for vanishing interface displacement jumps (Δ = 0), a situation considered in the examined 2-DOF example, where the incremental cohesive law becomes linear and turns out to admit a quadratic rate potential owing to the symmetry of the tangent constitutive operator also when collinearity is not satisfied. As a matter of fact, in this circumstance, in light of Eq. (21) the constitutive law becomes \({\dot{\boldsymbol{t}}}_{R\,ch} = (1 - D){\boldsymbol{K}}{\dot{\boldsymbol{\varDelta}}} - \left[ {D^{\prime}\dot{\hat{\varDelta }}_{{\rm{max}} } {\boldsymbol{K}} - (1 - D){\boldsymbol{\dot{K}}}} \right]{\boldsymbol{\varDelta}} = {\boldsymbol{K}}{\dot{\boldsymbol{\varDelta}}},\) with K symmetric.
As will be shown in the sequel, due to nonlinearities involved by the cohesive constitutive law, a direct application of the uniqueness condition is not straightforward and the use of a less stiff linear comparison rate cohesive law \(\dot{\boldsymbol{t}}_{R\,ch}^{*} [\dot{\boldsymbol{\varDelta}}]\) can be explored in the same spirit of [60], adopting the damaging constitutive branch of the nonlinear rate law (the constitutive branch associated with an increasing damage behavior) obtained by taking \({\dot{\hat{\varDelta }}}_{{\rm{max}}} = {\dot{\hat{\varDelta }}}\) in Eqs. (23) and (24), leading to the following expressions:
The above condition can be proved only under special circumstances, such as when \({\boldsymbol{\varDelta}} = {\boldsymbol{\varDelta}}_{s} \) (since \({\varDelta}_{n} = 0\) ∀t) as occurs when the whole cohesive interface is everywhere under effective contact with a strictly negative contact pressure, or when grazing contact conditions hold along the whole cohesive interface associated with a zero contact pressure implying \(\varDelta_{n} = 0 \, \) but with \(\dot{\varDelta }_{n} \ge 0\). Specifically, in the former case we have:
since \(\dot{\hat{\varDelta }} = \frac{1}{{\left\| {{{\boldsymbol{\varDelta}}}_{s} } \right\|\varDelta_{s}^{c} }}{{\boldsymbol{\varDelta}}}_{s} \cdot \left( {\dot{\boldsymbol{\varDelta}}}_{s} \right)\) and f’ ≤ 0. On the other hand, when one increment is characterized by β = 0 and the other one by β = 1, i.e. \(\dot{\hat{\varDelta }}^{(1)} \le 0,\dot{\hat{\varDelta }}^{(2)} > 0\), we obtain:
As a matter of fact, since
and
\({\boldsymbol{\varDelta}}_{s} \cdot {\dot{\boldsymbol{\varDelta}}} = {{\boldsymbol{\varDelta}}}_{s} \cdot{\dot{\boldsymbol{\varDelta}}}_{s}\;and\; \dot{\hat{\varDelta }} = \frac{1}{{\left\| {{{\boldsymbol{\varDelta}}}_{s} } \right\|\varDelta_{s}^{c} }}{{\boldsymbol{\varDelta}}}_{s} \cdot {\dot{\boldsymbol{\varDelta}}}_{s}\) along a path where \({\varDelta}_{n} = 0\), following the above illustrated arguments we obtain \( \Delta \left[ \left( {\beta - 1} \right)\alpha \frac{ {\boldsymbol{\varDelta}}_{s} } { \varDelta_{s}^{c}}f^{\prime } \left( {\hat{\varDelta}}_{{\rm{max}} } \right)\dot{\hat{\varDelta}} \right] \cdot {\Delta} \left( {{\dot{\boldsymbol{\varDelta}}}} \right) \ge 0 \).
A third circumstance in which the relative convexity condition is satisfied is that of a colinear cohesive interface constitutive. In fact, it follows that:
where \({\dot{\boldsymbol{t}}}_{Rch}^{\,*} = (1 - D)K{\dot{\boldsymbol{\varDelta}}} - D^{\prime}{\dot{\hat{\varDelta}}}K{\boldsymbol{\varDelta}}\), and it can be demonstrated that:
since D’ ≥ 0. On the other hand, when one increment is characterized by β = 0 and the other one by β = 1, i.e. \(\dot{\hat{\varDelta }}^{(1)} \le 0,\dot{\hat{\varDelta }}^{(2)} > 0\), we obtain:
3.3 Equilibrium equations of the microstructure: contact conditions and balance of linear momentum across the discontinuity interface
The RVE is driven along a quasi-static loading path by imposing boundary conditions depending on the macroscopic deformation gradient \(\overline{\boldsymbol{F}}(t)\) defined in terms of boundary deformations by means of the following kinematical macro-to-micro coupling relation:
where dS(i) denotes the reference surface elements and n(i) is the reference outward normal. Thus the microscopic deformation field is written as the sum of a linear part and a fluctuation one:
where the superscripts + and – denote pairs of opposite RVE boundary points. Equation (41)1 prescribes that fluctuation field \({\boldsymbol{w}}\left( {{\boldsymbol{X}},t} \right)\) must be periodic on the RVE boundary, namely \({\boldsymbol{w}}\left( {{\boldsymbol{X}}^{\boldsymbol{ + }} ,t} \right) = {\boldsymbol{w}}\left( {{\boldsymbol{X}}^{{ - }} ,t} \right){\text{ on }}\partial V_{(i)}\). The coupling between the microscopic and macroscopic RVE problem is then obtained by introducing the macroscopic first Piola–Kirchhoff stress tensor \({\overline{\boldsymbol{T}}}_{R}(t)\) defined as:
The succession of RVE equilibrium boundary value problems associated with the loading path \(\overline{\boldsymbol{F}}(t)\), referred to as “principal solution path” when assumed to be unique, is described by the local equilibrium equations of the microstructure under the periodicity boundary conditions (Eq. (41)), the unilateral self-contact constraint between opposite discontinuity surfaces, the balance of linear momentum across the cohesive-contact interface and the equilibrium conditions on the cohesive-contact surfaces.
Specifically, the unilateral contact conditions, in turn consisting of the impenetrability and the mechanical unilateral conditions, can be written as:
where hl = 0 describes the deformed lower discontinuity surface \(\Gamma^{l}\) and \(\sigma_{R}^{u}\) is the nominal contact pressure acting on the upper discontinuity surface. The Lagrangean form of the balance of linear momentum across the current discontinuity interface must account for both cohesive and interface self-contact effects, as shown in Fig. 6. Therefore, when referred to a generic contact point pair (A’, B) respectively placed on the upper and lower discontinuity surfaces where cohesion is active and coming into contact at a generic stage of the deformation process in the current configuration (thus characterized by the same position vector in xA’=xB at this stage), it can be expressed as:
where \(dS_{ch}^{A^{\prime}}\) and \(dS_{ch}^{B}\) respectively are the reference cohesive upper and lower surface elements centered at A’ and B.
Fig. 6
Illustration of the balance of linear momentum across the cohesive-contact interface: loads acting on surface elements are expressed in a Lagrangean form
where it has been considered that \(dS_{ch}^{A^{\prime}}\) = \(dS_{ch}^{B}\) = \(dS_{ch}\) at the cohesive-contact interface Γc, \({\boldsymbol{n}}^{A^{\prime}}=-{\boldsymbol{n}}^{B}\), and the double bracket with the subscript Γc denotes the jump in the enclosed field for a cohesive-contact interface point pair \(\left( {{\boldsymbol{X}}^{l} ,{\boldsymbol{X}}^{u} } \right)_{c}\) respectively defined as follows:
Note that since the reference surface elements dSA’ch(i) and dSBch(i) are deformed in the same current surface element dSch the following relations hold for the reference and current cohesive surface elements involved in Eq. (44):
On the other hand, the equilibrium of a disk-like volume with a base area \(dS_{ch}\) centered at xA’ = xB and contained in the upper or the lower deformed contact-cohesive surfaces (Fig. 7), respectively gives the following conditions in terms of the upper and lower surface nominal contact reactions rR(A’) and rR(B):
\(\sigma_{R}^{A^{\prime}} (\sigma_{R}^{B} )\) being the normal nominal contact reaction on the upper (lower) crack surface. An appropriate combination of Eq. (48), after the use of Eq. (44) leads to:
where \(\sigma\) is the corresponding true normal contact reaction (the same on both the upper and lower contact surfaces). Consequently, the nominal contact pressure on the upper and lower cohesive contact surfaces can be determined respectively as:
showing that the contact pressure corresponds to the normal component of the traction vector acting on the cohesive surface element only when cohesion is absent, as in the case of a crack surface or of a completely cracked cohesive interface.
Fig. 7
Illustration of the local equilibrium conditions at contact-cohesive surfaces in terms of nominal contact reactions
where fourth-order tensor of nominal moduli CR(X,F) satisfies the major symmetry condition and relates the deformation gradient rate \(\dot{\boldsymbol{F}}\) to the first Piola–Kirchhoff stress rate tensor \({\dot{\boldsymbol{T}}}_{R}\). The above rate constitutive law is able to comprehend hyperelastic materials whose associated nominal stress tensor and nominal moduli tensor can be obtained as \(T_{R\;ij} = \partial W\left( {{\boldsymbol{X}},{\boldsymbol{F}}} \right)/\partial F_{ij}\) and \(C_{ijkl}^{R} = \partial^{2} W\left( {{\boldsymbol{X}},{\boldsymbol{F}}} \right)/\partial F_{ij} \partial F_{kl}\), respectively, where \(W\left( {{\boldsymbol{X}},{\boldsymbol{F}}} \right)\) denotes the strain energy–density function.
The subsequent uniqueness and stability analyses performed along the principal equilibrium path driven by the loading process \({\overline{\boldsymbol{F}}}(t)\), require the derivation of the boundary value problem governing the RVE quasi-static rate response induced by a macroscopic deformation gradient rate \(\dot{\overline{\boldsymbol{F}}}\) superposed at the generic time t. The asymptotic expansion of the conditions governing the RVE equilibrium boundary value problem at a generic time t (see [31] for additional details when cohesion is absent) leads to the rate equilibrium conditions of the microstructure. Specifically conditions (Eq. (43)) lead to the following rate interface contact conditions:
where the subscript n denotes projection along the normal to the deformed lower discontinuity surface nl and, owing to the additive decomposition (Eq. (40)) the normal displacement rate jump can be expressed as a function of the normal fluctuation field rate and the superposed macroscopic deformation gradient rate as follows:
In addition, Eq. (45) and Eq. (48) give the following rate equilibrium conditions at the current cohesive-contact interface \(\Gamma_{c}\) for a generic cohesive-contact interface point pair (A’, B):
with \(\dot{\boldsymbol{r}}_{R}^{(A^{\prime})} = {\dot{\boldsymbol{t}}}_{R}^{\,(A^{\prime})} - {\dot{\boldsymbol{t}}}_{Rch}^{\,(A^{\prime})}\) and \(\dot{\boldsymbol{r}}_{R}^{B} = \dot{\boldsymbol{t}}_{R}^{\,B} - \dot{\boldsymbol{t}}_{Rch}^{\,B}\). The above equations are general in the sense that hold \(\forall \left( {{\boldsymbol{X}}^{l} ,{\boldsymbol{X}}^{u} } \right)_{c}\) including when cohesive traction rates are null as in the case of a simple crack contact interface. Note that the rate of the referential to actual area element ratio (for both the upper and the lower discontinuity surfaces) can be obtained by using Nanson’s formula as:
4 Stability and uniqueness properties of the microstructure equilibrium solution
4.1 Derivation of general stability and non-bifurcation conditions
The stability condition of the current equilibrium configuration Vt at a given \(\overline{\boldsymbol{F}}\left( t \right)\) can be obtained by generalizing the results obtained in [31] when cohesive interfaces are absent and considering a small perturbation of the current equilibrium position for a fixed macroscopic deformation gradient (namely for \(\dot{\overline{\boldsymbol{F}}} = {\boldsymbol{0}}\)) due to perturbation forces applied at the time t. Therefore, the introduction of a time-like parameter τ ≥ 0 describing the evolution of the system (with \(\tau\) = 0 corresponding to the time t) and integration over the interval [0,τ] of the identity arising from the theorem of the power expended (see [61]), lead to the following energy balance at time τ:
where u(X,τ) denotes the displacement field induced by the perturbation forces (compatible with self-contact conditions and periodicity boundary conditions for any τ) and such that u(0)(X,τ = 0) individuates the current equilibrium solution, Lper(0,τ) denotes the work done by the perturbation forces, K(τ) is the kinetic energy of the RVE at time \(\tau\) and the term in the curly bracket is the difference between the internal stress power and power expended by the antiperiodic surface tractions tR, the nominal contact reaction rR and the cohesive traction tRch. Considering small values of τ and taking into account the equilibrium condition in V(t) in conjunction with the antiperiodicity condition for the surface tractions, we obtain the following second order asymptotic expansion of Lper(0,τ):
where it has been noted that \({\dot{\boldsymbol{\varDelta}}}\left[ {\left( {{\boldsymbol{X}}^{l} ,{\boldsymbol{X}}^{u} } \right)_{ch(i)} } \right] = - \left[\kern-0.15em\left[ {{\dot{\boldsymbol{u}}}({\boldsymbol{X}})} \right]\kern-0.15em\right]_{\Gamma ch(i)} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \forall \left( {{\boldsymbol{X}}^{l} ,{\boldsymbol{X}}^{u} } \right)_{ch(i)}\), the expression in the round bracket is the second order approximation of quantity in curly bracket of Eq. (59) during the perturbed motion, and it is intended that all the rate quantities are evaluated at τ = 0.
Equation (60) is strictly valid under the assumption that the integration domain of the power expended by the normal contact reaction remains fixed during the perturbation. On the other hand, when the integration domains move during the perturbation, as may often occur for the reference contact surfaces Γc(i)l/u, the rate at which the power expended is carried out of the integration domain across its boundary should be taken into consideration, according to Reynold’s transport theorem (see [61]), when the derivative of the power expended is computed. Specifically, applying the Reynold’s transport theorem leads to:
where n is the outward unit normal on the boundary of Γc(i)l/u and v is the velocity of the moving domain boundary. It follows that in these circumstances, occurring in the examined 2-DOF example, the second contribution on the r.h.s of Eq. (61) with the minus sign must be added in the round bracket of Eq. (60) leading to
which is also satisfied when the kinematical rate quantities are substituted by their first order expansion in the parameter τ (For instance \(\dot{\boldsymbol{u}}\left( {{\boldsymbol{X}},\tau } \right) = \dot{\boldsymbol{u}}\left( {{\boldsymbol{X}},\tau = 0} \right) + ...).\)
It is worth noting that the normal contact reactions do not expend power since:
and owing to the crack self-contact rate conditions (Eq. (53)2), but this does not imply that the work of the contact reaction rate over the displacement rate, namely that the second contribution in the round bracket of Eq. (60), vanishes. As a matter of fact, this term can be conveniently rearranged in a more explicit form as follows:
by using the algebraic manipulations whose details are given in the sequel, showing that, similarly to the incremental cohesive surface loading, incremental self-contact loading is a kind of conditional and deformation dependent rate loading. As a matter of fact, using the expression of the rate of the nominal contact reaction, \({\dot{\boldsymbol{r}}}_{R}^{j} = {\dot{\sigma }}_{R}^{j} {\boldsymbol{n}}^{j} + \sigma_{R}^{j} \dot{\boldsymbol{n}}^{j},\quad j = u,l \), Eq. (50) and its rate version corresponding to:
after considering that \({\boldsymbol{n}}^{l} = - {\boldsymbol{n}}^{u} \,\,{\rm{and}}\,\,\dot{\boldsymbol{n}}^{l} = - \dot{\boldsymbol{n}}^{u}\,\forall \left( {{\boldsymbol{X}}^{l} ,{\boldsymbol{X}}^{u} } \right)_{c}\), it follows that the l.h.s of Eq. (65) can be manipulated as:
where in the last expression it can been noted that \(\dot{\sigma }_{R}^{l} {\boldsymbol{n}}^{l} \cdot \left[\kern-0.15em\left[ {\dot{\boldsymbol{u}}} \right]\kern-0.15em\right]_{{\Gamma_{c} }} = \dot{\sigma }_{R}^{l} \left[\kern-0.15em\left[ {\dot{u}_{n} } \right]\kern-0.15em\right]_{{\Gamma_{c} }}=0\) and \(\sigma_{R}^{l} {\boldsymbol{n}}^{l} \cdot \dot{\boldsymbol{u}}^{l} = \sigma_{R}^{l} {\boldsymbol{n}}^{l} \cdot \dot{\boldsymbol{u}}^{u},\) since \(\sigma_{R}^{l} \left[\kern-0.15em\left[ {\dot{u}_{n} } \right]\kern-0.15em\right]_{{\Gamma_{c} }}=0\), thus providing the r.h.s of Eq. (65).
From Eq. (60) or Eq. (62), it follows that the positivity of the second order term in round bracket implies that the external environment must provide additional energy in order to perturb the system from the examined equilibrium configuration and thus provides an energy based stability criterion.
After considering that the assumption \(\dot{\overline{\boldsymbol{F}}} = {\boldsymbol{0}}\) leads imposing that the displacement rate corresponds to the rate of a fluctuation field \(\dot{\boldsymbol{u}}\left( {\boldsymbol{x}} \right) = \dot{\boldsymbol{w}}\left( {\boldsymbol{x}} \right)\), the stability functional can be cast in the following form:
and its positivity condition for all \(\dot{\boldsymbol{w}}\left( {\boldsymbol{x}} \right) \ne {\boldsymbol{0}}\) belonging to \(A^{*} \left( {\overline{\boldsymbol{F}}},\dot{\overline{\boldsymbol{F}}} = \boldsymbol{0} \right)\), where the more general set \(A^{*} \left( {\overline{\boldsymbol{F}},\dot{\overline{\boldsymbol{F}}}} \right)\) being the set of admissible fluctuation rates defined as:
\(H^{1} \left( {V_{\# } } \right)\) denoting the usual Hilbert space of order one of vector valued functions periodic over V(i), implies stability of the considered equilibrium configuration. It is worth noting that the last term could be also written in the form of a standard external rate loading work contribution as \(- \int\limits_{{\Gamma^{l}_{ch(i)} }} {\dot{\boldsymbol{t}}_{R\,ch} \cdot \left[\kern-0.15em\left[ {\dot{\boldsymbol{w}}\left( {\boldsymbol{X}} \right)} \right]\kern-0.15em\right]_{\Gamma ch(i)} dS_{ch(i)}^{l} }\). Accordingly, a primary instability occurs when the minimum eigenvalue associated with S, \(\Lambda = \mathop {{\mathbf{min}} }\limits_{{\dot{\boldsymbol{w}}(\dot{\boldsymbol{x}}) \in A^{*} ({{\bar{\boldsymbol{F}}}},{\dot{\overline{\boldsymbol{F}}}} = \boldsymbol{0})}} \left\{ {S\left( {\overline{F},\dot{\boldsymbol{w}}} \right)/\int\limits_{{B{}_{(i)}}} {\nabla \dot{\boldsymbol{w}} \cdot \nabla \dot{\boldsymbol{w}}dV_{(i)} } } \right\}\), first vanishes. The corresponding critical loading parameter is denoted as \(t_{cS}\) and the corresponding deformation mode for which the stability functional also vanishes is referred to as the primary instability mode.
Following an approach similar to that adopted in [31], an incremental uniqueness functional \(R\left( {\dot{\overline{\boldsymbol{F}}},\dot{\boldsymbol{w}}^{(1)} ,\dot{\boldsymbol{w}}^{(2)} } \right)\) defined as:
can be introduced, whose positivity condition for every pair of distinct admissible fields \(\dot{\boldsymbol{w}}^{(1)} {, }\dot{\boldsymbol{w}}^{(2)} \in A^{*} (\overline{\boldsymbol{F}},\dot{\overline{\boldsymbol{F}}})\), excludes angular bifurcations. In Eq. (70) the symbol \(\Delta \left( \cdot \right) = ( \cdot )^{(1)} - ( \cdot )^{(2)}\) denotes the difference operator, in terms of the enclosed quantity (vector or tensor fields), between a pair of admissible fields. Note that in the case of the displacement rate the operator \(\Delta \left( \cdot \right)\) corresponds to the difference in the fluctuation rate \(\Delta \left( {\dot{\boldsymbol{u}}} \right) = \dot{\boldsymbol{w}}^{(1)} - \dot{\boldsymbol{w}}^{(2)} = \Delta \dot{\boldsymbol{w}}\), and similarly for the cohesive interface displacement jump rate where the \(\Delta \left( \cdot \right)\) operator leads to \(\Delta \left( {\dot{\boldsymbol{\varDelta }}} \right) = - \Delta \left[\kern-0.15em\left[ {\dot{\boldsymbol{w}}}(\boldsymbol{X}) \right]\kern-0.15em\right]_{\Gamma ch(i)}\).
The uniqueness functional (Eq. (70)) can be derived by considering that the difference of any pair of distinct solutions satisfies appropriate conditions derived from the governing equations of the rate equilibrium problem. Specifically, introducing the solution difference operator, in terms of the enclosed quantity (vector or tensor fields), between a pair of possible solutions \(\dot{\boldsymbol{u}}^{(i)} = {\dot{\overline{\boldsymbol{F}}}X} + \dot{\boldsymbol{w}}^{(i)}_{{{\dot{\overline{\boldsymbol{F}}}}}}\) (i = 1,2) of the rate problem driven by \({\dot{\overline{\boldsymbol{F}}}}\left( t \right)\), these conditions are here detailed:
by taking into account for Eq. (71), (72)2-4 and (74), after considering that \(\Delta \dot{\boldsymbol{T}}_{R}^{j} {\boldsymbol{n}}_{(i)} = \Delta \dot{\boldsymbol{r}}_{R}^{j} + \Delta \dot{\boldsymbol{t}}_{Rch}^{j}, \quad j = u,l\), the above identity becomes:
Noting that \(\Delta \dot{\sigma }_{R}^{l} {\boldsymbol{n}}^{l} \cdot \left[\kern-0.15em\left[ {\Delta \dot{\boldsymbol{w}}} \right]\kern-0.15em\right]_{{\Gamma_{c} }}\) is never positive, it follows that incremental uniqueness is excluded when:
for every pair of analytically admissible fields \(\dot{\boldsymbol{w}}^{(1)} \ne \dot{\boldsymbol{w}}^{(2)} \in A^{*} (\overline{\boldsymbol{F}},\dot{\overline{\boldsymbol{F}}})\).
The condition \(\Delta \dot{\sigma }_{R}^{l} {\boldsymbol{n}}^{l} \cdot \left[\kern-0.15em\left[ {\Delta \dot{\boldsymbol{w}}} \right]\kern-0.15em\right]_{{\Gamma_{c} }} \le 0\) can be derived by means of the following considerations:
a)
when \(\sigma_{R}^{l} < 0\) then \(\Delta \dot{\sigma }_{R}^{l} {\boldsymbol{n}}^{l} \cdot \left[\kern-0.15em\left[ {\Delta \dot{\boldsymbol{w}}} \right]\kern-0.15em\right]_{{\Gamma_{c} }}\) = 0 since \({\boldsymbol{n}}^{l} \cdot \left[\kern-0.15em\left[ {\Delta \dot{\boldsymbol{w}}} \right]\kern-0.15em\right]_{{\Gamma_{c} }} = 0\) from Eq. (74);
b)
when σRl = 0 then \(\Delta \dot{\sigma }_{R}^{l} {\boldsymbol{n}}^{l} \cdot \left[\kern-0.15em\left[ {\Delta \dot{\boldsymbol{w}}} \right]\kern-0.15em\right]_{{\Gamma_{c} }} \le 0\) since
where \(g={\boldsymbol{n}}^{l} \cdot \left[ {{\dot{\overline{\boldsymbol{F}}}}\left( {{\boldsymbol{X}}^{u} - {\boldsymbol{X}}^{l} } \right)} \right]\); moreover the first two terms vanish due to Eq. (53)4 in conjunction with Eq. (54), and the last two are never positive taking into account that \(\dot{\sigma }_{R}^{l(1)}\), \(\dot{\sigma }_{R}^{l(2)}\), \(\left( {{\boldsymbol{n}}^{l} \cdot \left[\kern-0.15em\left[ {\dot{\boldsymbol{w}}^{(1)} } \right]\kern-0.15em\right] - g} \right)\) and \(\left( {{\boldsymbol{n}}^{l} \cdot \left[\kern-0.15em\left[ {\dot{\boldsymbol{w}}^{(2)} } \right]\kern-0.15em\right] - g} \right)\) are never positive too owing to Eq. (53)1 and Eq. (53)3.
4.2 Relations between stability and uniqueness conditions
Some connections between the above introduced stability and the uniqueness conditions can be determined. When the macroscopic incremental loading is assumed homogeneous (\(\dot{\overline{\boldsymbol{F}}} = {\boldsymbol{0}}\)), the stability condition implies incremental uniqueness; in fact, the positivity condition of the uniqueness functional (Eq. (70)) can be expressed by pairing a generic admissible fluctuation rate field \(\dot{\boldsymbol{w}}^{(1)} = \dot{\boldsymbol{w}}\) with the null field \(\dot{\boldsymbol{w}}^{(2)}\) = 0, thus obtaining the positivity condition of the stability functional (Eq. (68)), namely \(R\left( {\dot{\overline{\boldsymbol{F}}}} = {\boldsymbol{0}},\dot{\boldsymbol{w}}^{(1)} = \dot{\boldsymbol{w}},\dot{\boldsymbol{w}}^{(2)} = {\boldsymbol{0}} \right)=S\left( {\overline{\boldsymbol{F}},\dot{\boldsymbol{w}}} \right).\) Along a macro-deformation path beginning where the stability functional is positive, the first load level where the stability functional becomes positive semidefinite (referred to as \(t_{cS}\)), at which the associated minimum eigenvalue first vanishes and a primary instability appears, in general does not correspond to a primary eigenstate (defined as a nontrivial rate solution to the homogeneous rate problem), owing to the absence of a variational structure of the rate problem. It follows that the primary eigenstate loading level \(t_{cE}\), also corresponding to a primary bifurcation load level \(t_{c}\) since \(\dot{\overline{\boldsymbol{F}}} = {\boldsymbol{0}}\), cannot precede the primary instability one, i.e. \(t_{cE} \ge t_{cS}\). As a matter of fact, the admissible field which makes the stability functional vanish and leads to the associated variational inequality, referred to as the primary instability mode, does not correspond to an eigenmode, which, on the contrary, makes the stability functional vanish but does not lead to the same variational inequality since the deformation-sensitive rate loading terms arising from self-contact and cohesive phenomena are not conservative as they do not admit a potential. Specifically, the following condition, including the case of a moving contact interface in the reference configuration, is not satisfied:
On the other hand, when the contact rate loading terms and the rate cohesive model admit a potential (Eq. (80) is satisfied), a primary eigenstate occurs where the stability functional first fails, i.e. primary instability and eigenstate critical load levels coincide \(t_{cE} = t_{cS}\). This occurs, for instance, for the particular class of self-adjoint contact rate loading terms individuated in [36, 45] when the discontinuity interface remains planar during deformation and with a constant nominal contact pressure, in conjunction with a rate cohesive constitutive model admitting a potential function U(\(\dot{\boldsymbol{\varDelta}}\)) of the interface displacement jump rate, namely \(\dot{\boldsymbol{t}}_{Rch} = \partial U/\partial \dot{\boldsymbol{\varDelta}}\). The latter situation is satisfied, for instance, in the case of colinear cohesive law or when the interface displacement jump vector is null along the deformation path.
Owing to the conditionality nature of self-contact unilateral rate contact terms and of the nonlinearities in the cohesive incremental constitutive law, for the rate problem associated with a non-homogeneous macroscopic incremental loading \(\dot{\overline{\boldsymbol{F}}} \ne {\boldsymbol{0}}\), in general, the above introduced connections between primary instability and primary bifurcation cannot be stated. In spite of this some special classes of rate problems can be envisaged where connections between stability and uniqueness can be proved. Specifically, when the effective contact conditions (\(\sigma < 0 \, \forall {\boldsymbol{x}} \in \Gamma_{C}\)) are everywhere satisfied along the displacement discontinuity interface (the unilateral contact condition reducing to a linear constraint (\(\left[\kern-0.15em\left[ {\dot{u}_{n} } \right]\kern-0.15em\right]_{{\Gamma_{C} }} = 0\) on \(\Gamma_{C}\)), and the cohesive rate model specializes to a linear one (as occurs when Δ = 0 along the fundamental macro-deformation path), the connections already determined for homogeneous macroscopic incremental loading remain still valid. Therefore, a primary instability precedes primary bifurcation (\(t_{cS} \le t_{cE}\)) and a primary bifurcation coincides with a primary eigenstate (\(t_{c} = t_{cE}\)). In addition, when contact rate loading terms and the rate cohesive model admit a potential, a special situation occurring for the 2-DOF example examined in Sect. 2, primary instability and bifurcation critical load levels coincide, i.e. \(t_{cS} = t_{c}\). The connections between stability and uniqueness issues are schematized in the following two boxes respectively containing the conditions for the homogeneous and inhomogeneous macroscopic loading conditions:
(81)
In order to deal with the general case of the non-homogeneous macroscopic incremental loading \(\dot{\overline{\boldsymbol{F}}} \ne {\boldsymbol{0}}\), approaches similar to those formulated in [31] and taking advantage of linear comparison rate problems, can be developed in order to provide upper and lower bounds to primary instability and bifurcation load levels. Although, for the sake of brevity, a detailed analysis of such approaches is outside the scope of the present paper and it will be the object of future developments, it is worth noting that useful conservative estimations of critical load levels could be obtained assuming less stiff kinematical constraints at the contact interface in conjunction with a less stiff rate cohesive constitutive response. To this end, the linear comparison rate interface model introduced in the previous section (Eq. (28)) can be adopted under special circumstances ensuring the relative convexity condition (Eq. (29)). For instance, lower bound predictions for the loading levels at the onset of instability and bifurcation can be obtained by determining the primary instability load level, referred to as \(t_{cS}^{F}\), for a linear incremental comparison problem embedding a cohesive-contact interface model assuming free to penetrate rate conditions where the contact pressure vanishes (namely \(\left[\kern-0.15em\left[ {\dot{w}_{(0)n} } \right]\kern-0.15em\right]_{{\Gamma {}_{C}}}\) arbitrary on Γc where σ = 0) and the rate linear cohesive law (Eq. (28)) adopting the damaging constitutive branch of the relevant nonlinear law, leading to \(t_{cS}^{F} \le t_{cS} , \, t_{cS}^{F} \le t_{c}\). In the special case of effective contact conditions, it is sufficient to adopt the comparison rate cohesive model of Eq. (28) to obtain the above critical load lower bounds.
4.3 Full finite deformation versus simplified cohesive-contact interface modelling
As shown in Sect. 4.1, adopting a full finite deformation framework leads to an expression of the stability functional shown in Eq. (68), containing, in addition to the first term representing the nominal incremental material response, two separate terms arising from the deformation sensitive nature of the self-contact reaction rate and a contribution related to the cohesive traction rate. The first one of the two contact terms is associated with the rotations of cohesive-contact interface elements, while the second one is related to the relative deformations of the opposite surfaces of the cohesive-contact interface.
On the other hand, simplified cohesive-contact models can be introduced leading to approximate stability and uniqueness formulations. Without loss of generality, in what follows it is assumed that the cohesive interface \(\Gamma_{ch(i)} \,(t = 0)\) coincides with the whole displacement discontinuity interface \(\Gamma_{(i)}\).
A first simplified model can be obtained when the terms associated with relative cohesive-contact surface deformations are neglected, according to an interface equilibrium condition written with reference to the undeformed interface configuration and to the assumptions of small displacement jumps. In this case Eq. (45) can be written with reference to initially coincident material point pairs placed on opposite sides of the undeformed cohesive interface \(\left( {{\boldsymbol{X}}^{l} ,{\boldsymbol{X}}^{u} } \right)_{c(i)}\) and owing to Eq. (17) we obtain that Eq. (45) reduces to the following continuity condition for the nominal traction vector across the undeformed cohesive-contact interface:
Moreover Eq. (50) gives \(\sigma_{R}^{u} = \sigma_{R}^{l} { = }\sigma\quad \forall \left( {{\boldsymbol{X}}^{l} ,{\boldsymbol{X}}^{u} } \right)_{c(i)}\) since it can be assumed that \(dS_{ch(i)}^{u} = dS_{ch(i)}^{l} = dS_{ch}\), leading to \({\boldsymbol{r}}_{R}^{u} + {\boldsymbol{r}}_{R}^{l} = {\boldsymbol{0}}\) and \(\dot{\boldsymbol{r}}_{R}^{u} + \dot{\boldsymbol{r}}_{R}^{l} = {\boldsymbol{0}}\) with \({\boldsymbol{r}}_{R}^{i} = {\boldsymbol{t}}_{R}^{i} - {\boldsymbol{t}}_{Rch}^{i}\) and \({\boldsymbol{\sigma}}_{R}^{i} = {\boldsymbol{T}}_{R} \left( {{\boldsymbol{X}}^{i} } \right){\boldsymbol{n}}_{(i)}^{i} \cdot {\boldsymbol{n}}^{i} - {\boldsymbol{t}}_{Rch}^{i} \cdot {\boldsymbol{n}}^{i} ,\quad\;i = u,l\). It follows that along the cohesive-contact interface we have:
where it has been assumed that \(\left[\kern-0.15em\left[ {\dot{u}_{n} } \right]\kern-0.15em\right]_{{\Gamma_{c(i)} }}\) = 0 according to a simplified rate contact constraint expressed in the undeformed interface configuration. We thus obtain the following modified stability functional:
If surface rotations at the cohesive-contact interface are also neglected according to small displacement hypotheses, the above functional further simplifies in:
where the contributions arising from the rates of the normal and tangential unit vectors in the cohesive traction rate vector must be accordingly neglected.
The modified exclusion functionals associated with the two simplified formulations (Eq. (84) and Eq. (87)) can be respectively defined as:
for every pair of distinct admissible fields \(\dot{\boldsymbol{w}}^{(1)} {, }\dot{\boldsymbol{w}}^{(2)} \in A^{\prime}(\overline{\boldsymbol{F}},\dot{\overline{\boldsymbol{F}}})\).
Within the first approximated formulation introduced above, the unilateral self-contact constraint \(\left[\kern-0.15em\left[ {\dot{u}_{n} } \right]\kern-0.15em\right]_{{\Gamma_{c(i)} }} = 0\) can be incorporated in the cohesive constitutive law by means of a penalty-like approach as described in Sect. 3.1, as done in several analyses available in the literature (see [38‐42, 59] for instance, and the reference cited therein). In this case, the set of admissible fluctuation rates is characterized by unconstrained functions along the cohesive-contact interface. In addition, in some works (see [38, 39], for example) the unilateral contact constraint has been applied by referring to the undeformed geometry of the cohesive interface, i.e. the following condition has been considered \(\left[\kern-0.15em\left[ {\dot{u}_{{n_{(i)} }} } \right]\kern-0.15em\right]_{{\Gamma_{c(i)} }} = 0\) thus also neglecting the rotations of the unit normal to the deformed cohesive surfaces according to the above introduced second simplified formulation.
5 2-DOF example of cohesive and contact instability revisited: Continuum mechanics approach
The 2-DOF simple system previously analyzed from a structural mechanics point of view, is now reexamined using the continuum mechanics formulation given in Sect. 4. According to this approach, the state of the system is described by the spatial distributions of stresses, strains and displacements within the two bodies constituting the system and at their internal connections, thus highlighting explicitly the role of cohesive and contact internal mechanisms in the analysis. The stability and bifurcation analyses can be both carried out by using the energetic approach leading to the stability functional (Eq. 68) because primary instability and bifurcation critical load levels coincide. As a matter of fact for the examined example, effective contact conditions are satisfied along the principal path, the rate cohesive model is linear, and the contact rate loading terms and the rate cohesive model admit a rate potential. In particular, considering that the second term in Eq. (65) vanishes owing to the rigidity of the contact surfaces, the contribution from the contact reaction rate in Eq. (65) and the cohesive contribution can be written respectively as:
It follows that Eq. (80) is satisfied with equal sign. Additional details about the derivation of Eq. (90) and Eq. (92) are given in the Appendix.
Thus, in order to obtain the second order expansion for small values of τ of the first term in the r.h.s of Eq. (59), the following developments must be considered for the 2-DOF system here analyzed:
where in the asymptotic expansions all the rate quantities are evaluated at τ = 0, that B(i) includes both the rigid rods AB and BC and that the stress power term includes the contribution from the elastic spring \(k\left( {\varphi_{1} + \varphi_{2} } \right)\left( {\dot{\varphi }_{1} + \dot{\varphi }_{2} } \right)\). Note that in the contribution related to the interface self-contact reaction, by means of the Reynold’s transport theorem, it has been considered that the domain of the integration may vary with τ, as occurs along the surface of the internal connection attached to the point B.
Taking into account the equilibrium condition at τ = 0 Eq. (63), it leads to the following second order approximation of the difference between the internal deformation work and the work done by the antiperiodic surface tractions \({\boldsymbol{t}}_{R}\), by the nominal contact reaction \({\boldsymbol{r}}_{R}\) and by the cohesive interface traction \({\boldsymbol{t}}_{R\,ch}\):
the stability and exclusion functional corresponding to the terms in the round bracket of Eq. (94), for the analyzed 2-DOF system assumes the following expression:
and the same critical load given by Eq. (4) \(\lambda_{c} = \lambda_{cS}\) can be then obtained.
It is worth noting that the term in Eq. (90)1 arising from the contact reaction rate is related to the destabilizing effects related to the work of two equal and opposite incremental forces tangential to the contact surfaces, equal to the compression load times the contact surfaces rotation rate \(\lambda \dot{\varphi }_{20}\), whereas the term arising from the moving contact surface (Eq. (95)2) is due to the destabilizing work done by two equal and opposite incremental normal forces equal to \(\lambda L/h\left( {\dot{\varphi }_{10} - \dot{\varphi }_{20} } \right)\), where h is the rod thickness, applied at the two boundary points of the moving cohesive-contact surface attached to the end B of the rigid rod AB and denoted by \(\Gamma_{c(i)}^{l}\), which, owing to the rigidity of the rod, is equivalent to the work of the rate couple \(\lambda L\left( {\dot{\varphi }_{10} - \dot{\varphi }_{20} } \right)\) through the rate rotation \(\dot{\varphi }_{20}\) of the moving cohesive-contact surface. As a matter of fact, in both cases from (Eq. (5)) it follows that \(\left| {\dot{\varphi }_{20} } \right| < \left| {\dot{\varphi }_{10} } \right|\) with the two rotation rates having the same sign and since λ > 0, we obtain \(\lambda L\dot{\varphi }_{20} \left( {\dot{\varphi }_{10} - \dot{\varphi }_{20} } \right)\) > 0. The above destabilizing mechanisms are illustrated in Fig. 8.
Fig. 8
Illustration of the destabilizing effects of the contact reaction rate (left) and of the moving contact reference surface (right)
An additional destabilizing term arises from the contribution of the compressive stresses (times the square of rigid rotations) in the 2nd order internal deformation work (Eq. (95)1), while the contribution of the internal deformation work of the elastic spring together with the cohesive tractions is stabilizing. This shows how the continuum mechanics approach clarifies the physical mechanisms governing the unstable behavior of the 2-DOF system. Note that for the tensile critical load, only the terms related to contact reactions become destabilizing and give an effective explanation of such unusual system unstable behavior under tensile loading. In this case, in fact, the critical mode shape shows that \(\left| {\dot{\varphi }_{20} } \right| > \left| {\dot{\varphi }_{10} } \right|\) with the two rotation rates having opposite signs and since λ < 0, we obtain that \(\lambda L\dot{\varphi }_{20} \left( {\dot{\varphi }_{10} - \dot{\varphi }_{20} } \right)\) > 0.
In order to better analyze the role of the contributions of cohesive-contact reactions acting on the moving cohesive-contact interface, let us now consider what happens if a full finite deformation formulation is not adopted. When cohesive crack surface deformations (including rigid translations) are neglected at the cohesive-contact interface we obtain the following simplified stability functional:
and the system loses its stability at \(\lambda_{cS}^{\prime + } = - 4k_{1} L + 4\sqrt {k_{1} \left( {k + k_{1} L^{2} } \right)}\) (the instability mode being characterized by \(\dot{\varphi }_{20} = \left( {\frac{{2k_{1} L}}{{\sqrt {k_{1} k + k_{1}^{2} L^{2} } }} - 1} \right)\dot{\varphi }_{10}\)), but it does not buckle at the same level since in this case the contact incremental loading does not admit a potential. When k = k1L2 we obtain \(\lambda_{cS}^{\prime + } = 4k_{1} L\left( {\sqrt 2 - 1} \right)\) and \(\dot{\varphi }_{20} = \left( {\sqrt 2 - 1} \right)\dot{\varphi }_{10}\). When k1 approaches to infinite, the critical load becomes 2 k/L and the instability mode becomes characterized by \(\dot{\varphi }_{20} = \dot{\varphi }_{10}\), since the internal sliding connection becomes inactive and the two rods are connected only by an internal elastic hinge.
On the other hand, the bifurcation load which can be obtained by means of the following variational equation:
is equal to \(\lambda_{c}^{ + } = \frac{{k - k_{1} L^{2} + \sqrt {k^{2} + 14kk_{1} L^{2} + k_{1}^{2} L^{4} } }}{2L}\) which reduces to \(\lambda_{c}^{\prime+ } = 2k_{1} L\) when k = k1 L2 and to \(\lambda_{c}^{\prime + } = 4 k/L\) when k1 approaches infinity. The bifurcation load under tension is equal to \(\lambda_{c}^{ - } = - \frac{{ - k + k_{1} L^{2} + \sqrt {k^{2} + 14kk_{1} L^{2} + k_{1}^{2} L^{4} } }}{2L}\) which reduces to \(\lambda_{c}^{\prime - } = - 2k_{1} L\) when k = k1 L2.
If both crack surface deformation contributions and rotations at the cohesive-contact interface are neglected, as in classical cohesive models incorporating contact effects in a simplified way, the above functional further simplifies in:
and the critical load turns out to be \(\lambda_{c}^{\prime \prime + }\) = min\(\left\{ {2k_{1} L,\frac{2k}{L}} \right\}\), the critical mode being \(\dot{\varphi }_{20} = - \dot{\varphi }_{10}\) or \(\dot{\varphi }_{20} = \dot{\varphi }_{10}\) in the former or in the latter case, respectively. When k = k1 L2 we obtain \(\lambda_{c}^{\prime \prime + } = 2k_{1} L\) and we have two simultaneous modes with \(\dot{\varphi }_{20} \ne 0 \, \) or \(\dot{\varphi }_{10} \ne 0\); on the other hand when k1 approaches infinity, the critical load becomes 2 k/L and the instability mode becomes characterized by \(\dot{\varphi }_{20} = \dot{\varphi }_{10}\). Note that no tensile bifurcation is captured in this approximated formulation.
As a matter of fact, in this latter circumstance instability is induced by the destabilizing geometrical effects related to rigid rotations of the rods in presence of dead loading on their ends since the cohesive contact reactions do not follow the rigid rotations of the internal connection. The internal connection behaves as a classical double pendulum imposing the same horizontal displacement to the rod ends B and C (while allowing relative rotations and vertical displacements) and the two rods remain connected by a relative vertical translational spring and a rotational spring (Figs. 9 and 10).
Fig. 9
Primary instability modes when both crack surface deformation contributions and rotations at the cohesive and crack contact interfaces are neglected
Primary instability modes when both crack surface deformation contributions and rotations at the cohesive and crack contact interfaces are neglected, k = k1 L2
As illustrated in Fig. 11, the above developments show that a large overestimation in the critical load determination is obtained when a classical stability approach under dead loading, neglecting cohesive crack surface deformations and rotations, is adopted to carry out the analysis in presence of cohesive-contact interfaces. On the other hand, if the effects of cohesive crack surface deformations are neglected, while the effects of cohesive contact surface rotations are included in the analysis, a smaller overestimation of critical load levels (which turns out to be different for instability and bifurcation) is obtained with respect to the exact full finite deformation approach and the bifurcation problem loses its variational structure. This is a consequence of an accurate representation of the actual full-finite deformation constraints imposed by the internal connection. Moreover, the above analytical results have pointed out how the approximations induced by simplified approaches which do not account for a full finite deformation formulation, become smaller as the cohesive interface stiffness parameter increases, approaching the exact critical load level when the interface stiffness parameters approach infinity. As a matter of fact, when \(k_{1} \to \infty\) from Eq. (4) and Eq. (5) and on the light of the above analytical developments, we can obtain \(\lambda_{cS}^{ + } = \lambda_{cS}^{\prime + } = \lambda_{cS}^{\prime \prime + } = 2k/L\) and \(\dot{\varphi }_{20} = \dot{\varphi }_{10}\) for both the exact and simplified formulations. This shows that an accurate description of the contact cohesive interface model is compulsory when the interface undergoes large damage levels as well as when complete decohesion takes place.
Fig. 11
Equilibrium paths of the 2-DOF system for k = k1 L2: λ is normalized with respect to its critical compressive value \(\lambda_{c}^{ + } = \sqrt 2 k_{1} L\) and the approximated critical load levels are also individuated \(\lambda_{c}^{\prime + } = \lambda_{c}^{\prime \prime + } = 2k_{1} L\), \(\lambda_{c}^{\prime - } = - 2k_{1} L\)
Microscopic instability and bifurcation failure mechanisms in periodic nonlinear elastic composite materials subjected to large deformations have been investigated in the present paper by means of an original theoretical formulation including decohesion and contact phenomena occurring at internal discontinuity interfaces, typical in reinforced composite microstructures. The analysis is firstly developed by using a simple 2-DOF system provided with a special internal connection simulating a cohesive-contact interface and pointing out the main characteristics of the general theoretical problem. To this aim a structural mechanics approach has been initially adopted to calculate primary bifurcation and instability load levels of the 2-DOF example.
A general theoretical analysis, based on finite strain continuum mechanics, has been then developed able to analyze the structure and properties of the equilibrium problem of the composite microstructure driven along a macro-deformation loading path according to a nonlinear periodic homogenization approach. An enhanced nonlinear cohesive-contact interface model is introduced, capable of including both the effects of the progressive damage due to decohesion at discontinuity interfaces embedded in the microstructure (such as reinforcement/matrix interfaces) and of interface unilateral contact occurring at cohesive interfaces under macroscopic loading conditions involving normal interface compression. To this end, the cohesive-contact model adopts a class of irreversible cohesive traction–separation constitutive laws formulated in a finite deformation context to represent the former effects, while a full finite deformation continuum contacts mechanical formulation to simulate the latter ones. The principal governing equations have been derived for both the finite and rate equilibrium problems with special attention to the mechanical conditions at cohesive-contact interfaces, and novel stability and uniqueness conditions associated with the above equilibrium problems are introduced. The hierarchy of the critical load levels corresponding to loss of uniqueness in the rate equilibrium solution and to the onset of instabilities have been determined by investigating the interrelations between the stability and uniqueness criteria and considering special classes of problems (e.g. conservative contact and cohesive rate loadings, effective contact conditions, linear rate cohesive models). Strategies to circumvent the complications introduced in the stability and uniqueness analyses by nonlinearities arising at the cohesive-contact interface both from the cohesive law and unilateral contact constraints have been discussed. To this end, linear comparison rate problems leading to lower bounds critical load predictions have been individuated by exploring the relative convexity properties of the cohesive law under special circumstances (vanishing interface normal displacement jump, grazing contact, collinearity between cohesive traction and displacement jump vectors, for instance) and adopting less stiff kinematical constraints at the contact interface.
In order to clarify the role of unilateral contact and cohesive effects acting along cohesive-contact interface, the consequences on stability and uniqueness analyses of introducing simplified cohesive-contact models, often adopted in the literature, neglecting full finite deformation effects arising from relative deformations and rotations of opposite surfaces comprised in the cohesive-contact interface, have been also studied.
Finally, the 2-DOF example has been reevaluated by applying the previously developed continuum mechanics approach, thus explaining the role played by contact and cohesive mechanisms and the significance of an appropriate modeling of their deformation sensitivity nature to obtain accurate predictions of critical instability and bifurcation load levels. The continuum mechanics approach, in spite of the initially adopted structural mechanics approach, moreover, is able to reveal the physical mechanisms governing the unstable behavior of the 2-DOF system. To this end, comparisons with simplified cohesive-contact models often adopted in the literature have been performed.
Acknowledgements
Fabrizio Greco gratefully acknowledges financial support from the Italian Ministry of Education, University and Research (MIUR) under the P.R.I.N. 2022 National Grant “Innovative tensegrity lattices and architectured metamaterials (ILAM)” (Project Code 20224LBXMZ; University of Calabria Research Unit, CUP H53D23001180006), funded by European Union – Next Generation EU under the National Recovery and Resilience Plan (NRRP), Mission M4, C2 Component- Investment 1.1.
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A theoretical analysis of instability and bifurcation failure phenomena in periodic microstructured nonlinear composite solids embedding discontinuity interfaces
Authors
Fabrizio Greco
Daniele Gaetano
Raimondo Luciano
Andrea Pranno
Girolamo Sgambitterra
Additional details about the derivation of the terms involved in the stability analysis of the 2-DOF system developed in Sect. 5 are given here. Firstly, the displacement field components of the rigid rods AB and CD and their rates are determined as follows:
, and it has been noted that the displacement component rates of the boundary \(\Gamma_{c(i)}^{l}\), attached to the rod AB, along the fundamental equilibrium path are equal to those of the left end of the rod CD.
Moreover, the second order terms arising from the internal deformation work for the two rigid rods can be put in the following form:
\(W_{21} = \frac{{\partial \dot{u}_{20} }}{{\partial x_{1} }} = - \dot{\varphi }_{10}\) for AB \(W_{21} = \frac{{\partial \dot{u}_{20} }}{{\partial x_{1} }} = \dot{\varphi }_{20}\) for CD, in the r.h.s of the above equations, the reference configuration is assumed to coincide with the current one at any stage of the deformation, \({\boldsymbol{C}}_{0}^{(2)}\) is the fourth order tensor of instantaneous moduli relative to the work conjugate stress–strain measure pair (T(2), E(2)) with E(2) the Green–Lagrange strain measure (see [29, 37], for additional details), T0 is the Cauchy stress tensor in the current configuration, D is the rate of strain (identically zero owing to rigidity of the rods) corresponding to the symmetric part of the displacement rate gradient and L thus coinciding with its antisymmetric part W (usually referred to as the “body spin”).
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and
\({\boldsymbol{\varDelta}}_{s} \cdot {\dot{\boldsymbol{\varDelta}}} = {{\boldsymbol{\varDelta}}}_{s} \cdot{\dot{\boldsymbol{\varDelta}}}_{s}\;and\; \dot{\hat{\varDelta }} = \frac{1}{{\left\| {{{\boldsymbol{\varDelta}}}_{s} } \right\|\varDelta_{s}^{c} }}{{\boldsymbol{\varDelta}}}_{s} \cdot {\dot{\boldsymbol{\varDelta}}}_{s}\) along a path where \({\varDelta}_{n} = 0\), following the above illustrated arguments we obtain \( \Delta \left[ \left( {\beta - 1} \right)\alpha \frac{ {\boldsymbol{\varDelta}}_{s} } { \varDelta_{s}^{c}}f^{\prime } \left( {\hat{\varDelta}}_{{\rm{max}} } \right)\dot{\hat{\varDelta}} \right] \cdot {\Delta} \left( {{\dot{\boldsymbol{\varDelta}}}} \right) \ge 0 \).
