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IN THIS ISSUE

### Guest Editorial

J. Appl. Mech. 2013;80(5):050301-050301-1. doi:10.1115/1.4023474.

The Twelfth Pan American Congress of Applied Mechanics (PACAM XII) was held at The University of the West Indies, St. Augustine, Port of Spain in Trinidad & Tobago in Jan. 2012. The congress proceedings, comprising 153 short papers, were published at http://www.aamech.org/PACAM12/Home.html. During, and shortly after the congress, based on the four-page versions and the oral presentations, the best papers were selected by the session chairs and the scientific committee of the congress. The authors of these papers were invited to submit full-length papers to be published in this special issue of Journal of Applied Mechanics, although some authors already had other plans. As a result, six papers were finalized, all of which were peer reviewed.

Commentary by Dr. Valentin Fuster

### Research Papers

J. Appl. Mech. 2013;80(5):050901-050901-7. doi:10.1115/1.4023472.

The fate of malignant glioma (MG) is governed by a multifaceted and dynamic circuit that involves the surrounding cellular and molecular tumor microenvironment. Despite extensive experimental studies, a complete understanding of the complex interactions among the constituents of this microenvironment remains elusive. To clarify this, we introduce a biologically based mathematical model that examines the dynamic modulation of glioma cancer stem cells (GSC) by different immune cell types and intracellular signaling pathways. It simulates the proliferation of glioma stem cells due to macrophage-induced inflammation, particularly involving two microglia phenotypes. The model can be used to regulate therapies by monitoring the GSC self-renewal rates that determine tumor progression. We observe that the GSC population is most sensitive to its own proliferation rate and the relative levels of the activating natural killer (NK) cell stimulatory receptors (NKG2D) versus killer inhibitory receptors (KIR) on NK cells that influence the proliferation or demise of the GSC population. Thus, the two most important factors involved in tumorigenesis or tumor regression are (1) GSC proliferation and (2) the functional status of NK cells. Therefore, strategies aimed at blocking proliferation and enhancing NKG2D and KIR signals should have a potentially beneficial impact for treating malignant gliomas.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):050902-050902-9. doi:10.1115/1.4023495.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):050903-050903-6. doi:10.1115/1.4023473.

Parametric excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. Evolutionary dynamics refers to a mathematical model of natural selection (the “replicator” equation) which involves a combination of game theory and differential equations. In this paper we apply perturbation theory to investigate parametric resonance in a replicator equation having periodic coefficients. In particular, we study evolution in the Rock-Paper-Scissors game, which has biological and social applications. Here periodic coefficients could represent seasonal variation. We show that 2:1 subharmonic resonance can destabilize the usual “Rock-Paper-Scissors” equilibrium for parameters located in a resonant tongue in parameter space. However, we also show that the tongue may be absent or very small if the forcing parameters are chosen appropriately.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):050904-050904-8. doi:10.1115/1.4023496.

The steady, axisymmetric base flow and instabilities in a rotating lid-driven cylinder are investigated experimentally via ultrasonic Doppler velocimetry and verified with computations. The flow is governed by two parameters: the Reynolds number (based on the angular velocity of the top lid, the cylinder radius, and kinematic viscosity) and the aspect ratio (cylinder height/radius). Base states and instabilities are explored using ultrasonic Doppler velocimetry in two mixtures of glycerol and water. Velocity profiles in the cylinder are constructed for aspect ratio 2.5 and Reynolds numbers between 1000 and 3000. The results are compared to computational spectral element simulations, as well as previously published findings. The base flow velocity profiles measured by ultrasonic Doppler velocimetry are in good agreement with the numerical results below the critical Reynolds number. The same is true for time-averaged results above the critical Reynolds number. Prediction of the first axisymmetric instability is demonstrated, although not always at the expected critical Reynolds number. Advantages and limitations of ultrasonic Doppler velocimetry are discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):050905-050905-11. doi:10.1115/1.4023475.

The asymptotic homogenization method (AHM) yields a two-scale procedure to obtain the effective properties of a composite material containing a periodic distribution of unidirectional circular cylindrical holes in a linear transversely isotropic piezoelectric matrix. The matrix material belongs to the symmetry crystal class 622. The holes are centered in a periodic array of cells of square cross sections and the periodicity is the same in two perpendicular directions. The composite state is antiplane shear piezoelectric, that is, a coupled state of out-of-plane shear deformation and in-plane electric field. Local problems that arise from the two-scale analysis using the AHM are solved by means of a complex variable method. For this, the solutions are expanded in power series of Weierstrass elliptic functions, which contain coefficients that are determined from the solutions of infinite systems of linear algebraic equations. Truncating the infinite systems up to a finite, but otherwise arbitrary, order of approximation, we obtain analytical formulas for effective elastic, piezoelectric, and dielectric properties, which depend on both the volume fraction of the holes and an electromechanical coupling factor of the matrix. Numerical results obtained from these formulas are compared with results obtained by the Mori–Tanaka approach. The results could be useful in bone mechanics.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):050906-050906-10. doi:10.1115/1.4023497.

This paper proposes an effective numerical method to generate approximate solutions for the overall nonlinear elastic response of isotropic filled elastomers subjected to arbitrarily large deformations. The basic idea is first to idealize the random microstructure of isotropic filled elastomers as an assemblage of composite spheres and then to generate statically admissible numerical solutions, via finite elements, for these material systems directly in terms of the response of a single composite sphere subjected to affine stress boundary conditions. The key theoretical strengths of the method are discussed, and its accuracy and numerical efficiency assessed by comparisons with corresponding 3D full-field simulations. The paper concludes with a discussion of straightforward extensions of the proposed method to account for general classes of anisotropic microstructures and filler-elastomer interphasial phenomena, features of key importance in emerging advanced applications.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051001-051001-9. doi:10.1115/1.4023476.

In this paper the problem of transformation toughening in anisotropic solids is addressed in the framework of Stroh formalism. The fundamental solutions for a transformed strain nucleus located in an infinite anisotropic elastic plane are derived first. Furthermore, the solution for the interaction of a crack tip with a residual strain nucleus is obtained. On the basis of these expressions, fundamental formulations are presented for the toughening arising from transformations using the Green's function method. Finally, a representative example is studied to demonstrate the relevance of the fundamental formulation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051002-051002-6. doi:10.1115/1.4023535.

When a buckle is initiated in a pipe subjected to external pressure, it will propagate along the longitudinal direction of the pipe if the external pressure is greater than its buckle propagation pressure. For a steady state condition, the propagation is simply considered as the translation of the buckle along the pipeline. This paper presents a unique approach to determine the length of the transition zone in a buckle propagating pipe by analyzing the mechanism of postbuckling of the pipe subjected to the external pressure. Buckling is considered to occur locally in the shell, spreading over a certain length along the longitudinal axis of the shell. The governing equations are derived from the postbuckling theory. Approximate solutions are obtained from the Ritz method, using a plausible function of the flexural displacement created based on Timoshenko's ring solution of the transverse collapse mode. The postbuckling equilibrium path shows that the pipeline experiences unstable collapse until the two opposite points on the inner surface contact each other. The length of the transition zone is found to be proportional to the ratio of (radius)3/2/(thickness)1/2 and is hardly affected by the material properties. The analysis is performed by comparing the obtained results with well-established predictions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051003-051003-8. doi:10.1115/1.4023477.

This paper presents a study of fracture in nickel using multiscale modeling. A comparison of six concurrent multiscale methods was performed in their application to a common problem using a common framework in order to evaluate each method relative to each other. Each method was compared in both a quasi-static case of crack tip deformation as well as a dynamic case in the study of crack growth. Each method was compared to the fully atomistic model with similarities and differences between the methods noted and reasons for these provided. The results showed a distinct difference between direct and handshake coupling methods. In general, for the quasi-static case, the direct coupling methods took longer to run compared to the handshake coupling methods but had less error with respect to displacement and energy. In the dynamic case, the handshake methods took longer to run, but had reduced error most notably when wave dissipation at the atomistic/continuum region was an issue. Comparing each method under common conditions showed that many similarities exist between each method that may be hidden by their original formulation. The comparison also showed the dependency on the application as well as the simulation techniques used in determining the performance of each method.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051004-051004-11. doi:10.1115/1.4023623.

This paper presents the derivation of a new boundary element formulation for plate bending problems. The Reissner's plate bending theory is employed. Unlike the conventional direct or indirect formulations, the proposed integral equation is based on minimizing the relevant energy functional. In doing so, variational methods are used. A collocation based series, similar to the one used in the indirect discrete boundary element method (BEM), is used to remove domain integrals. Hence, a fully boundary integral equation is formulated. The main advantage of the proposed formulation is production of a symmetric stiffness matrix similar to that obtained in the finite element method. Numerical examples are presented to demonstrate the accuracy and the validity of the proposed formulation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051005-051005-6. doi:10.1115/1.4023536.

A refined molecular life prediction scheme for single-walled carbon nanotubes (SWCNTs), taking into consideration C–C bond rotation and preexisting strain under mechanical loads, is proposed. The time-dependent fracture behavior of 12 different cases of zigzag (18,0) SWCNT, each embedded with either a single Stone–Wales (SW) defect of different types or two interacting or noninteracting defects, is studied under axially applied tensile load. It is shown that the patterns of atomistic crack propagation and fatigue lives of SWCNTs are influenced by the type and orientation of the SW defect(s), inter-defect distance, as well as the magnitude of externally applied stress. For SWCNTs with two SW defects, if the inter-defect distance is within the so called indifference length, defect-defect interaction does exist, and it has pronounced effects on diminishing the lives of the nanotubes. Also, the defect-defect interaction is stronger at shorter inter-defect distance, resulting in shorter fatigue lives.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051006-051006-6. doi:10.1115/1.4023478.

A semi-analytic solution for plastic collapse of a thin annular disk subject to thermomechanical loading is presented. It is assumed that the yield criterion depends on the hydrostatic stress. A distinguished feature of the boundary value problem considered is that there are two loading parameters. One of these parameters is temperature and the other is pressure over the inner radius of the disk. The general qualitative structure of the solution at plastic collapse is discussed in detail. It is shown that two different plastic collapse mechanisms are possible. One of these mechanisms is characterized by strain localization at the inner radius of the disk. The entire disk becomes plastic according to the other plastic collapse mechanism. In addition, two special regimes of plastic collapse are identified. According to one of these regimes, plastic collapse occurs when the entire disk is elastic, except its inner radius. According to the other regime, the entire disk becomes plastic at the same values of the loading parameters at which plastic yielding starts to develop.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051007-051007-10. doi:10.1115/1.4023537.

The deformation mechanics of dry networks of large-aspect-ratio fibers with random orientation controls the processing of long-fiber thermoplastics (LFTs) and greatly affects the mechanical properties of the final composites. Here, we generate initial geometries of fiber networks in a cubic unit cell with a fiber aspect ratio of l/d = 100 and fully periodic boundary conditions for later numerical simulation. The irreversible random sequential adsorption (RSA) process is first used to generate a quasi-random structure due to the excluded-volume requirements. In order to investigate the nonequilibrium character of the RSA, a second method, which is similar to the mechanical contraction method (MCM) (Williams and Philipse, 2003, “Random Packings of Spheres and Spherocylinders Simulated by Mechanical Contraction,” Phys. Rev. E, 67, pp. 1–9) and based on a simplified Metropolis Monte Carlo (MC) simulation is then developed to produce quasi-equilibrium fiber geometries. The RSA packing results (ϕ ≈ 4.423% when using a fiber aspect ratio of 100) are in good agreement with the maximum unforced random packing limits (Evans and Gibson, 1986, “Prediction of the Maximum Packing Fraction Achievable in Randomly Oriented Short-Fibre Composites,” Compos. Sci. Technol., 25, pp. 149–162). The fiber structures were characterized by several distribution functions, including pair-spatial and pair-orientation distributions, based on either the center-to-center distance or the shortest distance between the particles. The results show that the structures generated by the RSA have an easily-detectable long-range spatial correlation but very little orientational correlation. In contrast, the quasi-equilibrium structures have reduced spatial correlation but increased short-range orientational correlation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051008-051008-7. doi:10.1115/1.4023538.

One-dimensional wave propagation in granular flow has been investigated using a three-dimensional discrete element model (DEM). Cohesionless, dry, smooth, elastic, hard spheres are randomly distributed in a cylinder-piston system with initial granular temperature and solid fraction. Upon a sudden motion of the piston, subsequent wave propagation in granular materials between two ends of the cylinder is numerically simulated. The simulation results of wave speed normalized by the square root of granular temperature are found to be well correlated as a function of solid fraction. Comparison with several analytical works in the literature shows that the simulated wave speed is in good agreement with the wave speed calculated at the isentropic condition but is higher than that at the constant granular temperature condition. Finally the simulation result is employed to describe shock waves observed in the literature. It has been found that, when particles rapidly flow through an orifice, a shock is formed very near the location of the maximum granular temperature. It has also been observed that a shock can be formed even when the flow does not appear to be choked due to its low density upstream of the orifice.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051009-051009-10. doi:10.1115/1.4023519.

In this paper, the stochastic stability of the three elastically connected Euler beams on elastic foundation is studied. The model is given as three coupled oscillators. Stochastic stability conditions are expressed by the Lyapunov exponent and moment Lyapunov exponents. It is determined that the new set of transformation for getting $Ito∧$ differential equations can be applied for any system of three coupled oscillators. The method of regular perturbation is used to determine the asymptotic expressions for these exponents in the presence of small intensity noises. Analytical results are presented for the almost sure and moment stability of a stochastic dynamical system. The results are applied to study the moment stability of the complex structure with influence of the white noise excitation due to the axial compressive stochastic load.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051010-051010-15. doi:10.1115/1.4023617.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051011-051011-11. doi:10.1115/1.4023639.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051012-051012-9. doi:10.1115/1.4023640.

In this paper, the four integral identities satisfied by the fundamental solution for elastostatic problems are reviewed and slightly different forms of the third and fourth identities are presented. Two new identities, namely the fifth and sixth identities, are derived. These integral identities can be used to develop weakly singular and nonsingular forms of the boundary integral equations (BIEs) for elastostatic problems. They can also be employed to show the nonuniqueness of the solution of the traction (hypersingular) BIE for an elastic body on a multiconnected domain. This nonuniqueness is shown in a general setting in this paper. It is shown that the displacement (singular) BIE does not allow any rigid-body displacement terms, while the traction BIE can have arbitrary rigid-body translation and rotation terms, in the BIE solutions on the edge of a hole or surface of a void. Therefore, the displacement solution from the traction BIE is not unique. A remedy to this nonuniqueness solution problem with the traction BIE is proposed by adopting a dual BIE formulation for problems with multiconnected domains. A few numerical examples using the 2D elastostatic boundary element method for domains with holes are presented to demonstrate the uniqueness properties of the displacement, traction and the dual BIE solutions for multiconnected domain problems.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051013-051013-5. doi:10.1115/1.4023680.

It is intriguing how the mechanics of molecular motors is regulated to perform the mechanical work in living systems. In sharp contrast to the conventional wisdom, recent experiments indicated that motor force maintains ∼6 pN upon a wide range of filament loads during skeletal muscle contraction at the steady state. Here we find that this rather precise regulation which takes place in an essentially chaotic system, can be due to that a “working” motor is arrested in a transitional state when the motor force is ∼6 pN. Our analysis suggests that the motor force can be self-regulated through chemomechanical coupling, and motor force homeostasis is a built-in feature at the level of a single motor, which provides insights to understanding the coordinated function of multiple molecular motors existing in various physiological processes. With a coupled stochastic-elastic numerical framework, the kinetic model for a Actin-myosin-ATP cycle constructed in this work might pave the way to decently investigate the transient behaviors of the skeletal muscle or other actomyosin complex structures.

Topics: Engines , Muscle , Cycles
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051014-051014-6. doi:10.1115/1.4023642.

Irwin's model on plastic zone at the crack tip is discussed in many fracture mechanics textbooks. However, we found in Irwin's model that the internal resultant force on the crack plane and the one applied in remote field are not strictly balanced. This imbalance leads to the error in the scenario of small scale yielding, and an improper finite plastic zone size (PZS) is predicted when the remote stress approaches the yielding strength. In this paper, an improved model is developed through surrendering some main assumptions used in Irwin's model and an infinite PZS is then predicted as the remote stress goes up close to yielding strength, which implies that this estimation can be applied to situations with large scale yielding. In small scale yielding cases, the new estimation of PZS agrees well with finite element simulation results. In addition, a more accurate quantitative relation between the PZS and the effective stress intensity factor is derived, which might help characterize fracture behaviors in engineering applications.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051015-051015-9. doi:10.1115/1.4023684.

Coarse-grained molecular dynamics simulations have been performed to investigate the tensile behavior of CNT films. It is found that CNT entanglements greatly degrade the tensile load-bearing capability of CNT films. The effect of twisting on the tensile behavior of CNT fibers spun from CNT films has also been investigated. Results indicate that twisting can make either positive or negative contributions to the mechanical properties of the film, depending on the microstructure. The structural and energy evolution of CNT films and fibers, as well as the stress distributions of CNTs which cannot be easily determined experimentally, have been illustrated. This study provides an effective means of revealing the structure/property relationships of CNT films/fibers, which are essential in designing high performance CNT fibers.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051016-051016-6. doi:10.1115/1.4023625.

Micro and nanomechanics are growing fields in the semiconductor and related industries. Consequently obstacles, such as particles trapped between layers, are becoming more important and warrant further attention. In this paper a numerical solution to the von Kármán equations for moderately large deflection is used to model a plate deformed due to a trapped particle lying between it and a rigid substrate. Due to the small scales involved, the effect of adhesion is included. The recently developed moment-discontinuity method is used to relate the work of adhesion to the contact radius without the explicit need to calculate the total potential energy. Three different boundary conditions are considered—the full clamp, the partial clamp, and the compliant clamp. Curve-fit equations are found for the numerical solution to the nondimensional coupled nonlinear differential equations for moderately large deflection of an axisymmetric plate. These results are found to match the small deflection theory when the deflection is less than the plate thickness. When the maximum deflection is much greater than the plate thickness, these results represent the membrane theory for which an approximate analytic solution exists.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051017-051017-12. doi:10.1115/1.4023626.

A horizontally multilayered Green elastic transversely isotropic half-space is considered as the domain of the boundary value problem involved in this paper, such that the axes of material symmetry of different layers are parallel to the axis of material symmetry of the lowest half-space, which is depthwise. The domain is assumed to be affected by an arbitrary time-harmonic forced vibration due to a rigid rectangular surface foundation. With the use of a potential function method and the Hankel integral transforms, the displacements and stresses Green's functions are determined in each layer. The unknown functions due to integrations in each layer are transformed to the unknown functions of the surface layer with the use of the concept of propagator matrix and the continuity conditions. The mixed boundary conditions at the surface of the whole domain are numerically satisfied with the assumption of piecewise constant distribution of tractions in the contact area. It is numerically shown that the surface displacement and stress boundary conditions are satisfied very well. The vertical and horizontal impedance functions of the rectangular foundation are determined, which may be used as lumped parameters in time-harmonic soil-structure interaction with transversely isotropic horizontally layered domain as the soil. It is shown that the impedance functions determined in this paper coincide with the same functions for the simpler case of isotropic homogeneous half-space as degenerations of this study.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051018-051018-14. doi:10.1115/1.4023627.

Unique, explicit, and exact expressions for the first- and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997,“The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids,” Q. J. Mech. Appl. Math. 50, pp. 407–426) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues $pα$ ($α=1,2,3$) on the oblique plane with normal coincident with the position vector; second, it remains well-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):051019-051019-5. doi:10.1115/1.4023635.

The Green's function for the general anisotropic solid has been the subject of several studies. Here a variation of a standard integral transform approach allows the transient Green's function to be expressed in a somewhat different form. This alternative form is less compact, but features explicit integrals of functions in terms of polar and azimuthal angles defined with respect to the principal basis coordinates. Dimensionless expressions for the three anisotropic wave speeds are also given in terms of these angles, and sample calculations presented that show wave speed dependence on propagation direction. Some standard formalisms of anisotropic elasticity are not invoked, but similar terms are identified in the course of the analysis, and help define the solution expressions.

Commentary by Dr. Valentin Fuster

### Technical Briefs

J. Appl. Mech. 2013;80(5):054501-054501-4. doi:10.1115/1.4023624.

Creasing in thin shells admits large deformation by concentrating curvatures while relieving stretching strains over the bulk of the shell: after unloading, the creases remain as narrow ridges and the rest of the shell is flat or simply curved. We present a helically creased unloaded shell that is doubly curved everywhere, which is formed by cylindrically wrapping a flat sheet with embedded fold-lines not axially aligned. The finished shell is in a state of uniform self-stress and this is responsible for maintaining the Gaussian curvature outside of the creases in a controllable and persistent manner. We describe the overall shape of the shell using the familiar geometrical concept of a Mohr's circle applied to each of its constituent features—the creases, the regions between the creases, and the overall cylindrical form. These Mohr's circles can be combined in view of geometrical compatibility, which enables the observed shape to be accurately and completely described in terms of the helical pitch angle alone.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):054502-054502-9. doi:10.1115/1.4023618.

The present paper deals with the boundary layer flow and heat transfer of a non-Newtonian fluid at an exponentially stretching permeable surface. The Casson fluid model is used to characterize the non-Newtonian fluid behavior, due to its various practical applications. With the help of similarity transformations the governing partial differential equations corresponding to the continuity, momentum, and energy equations are converted into nonlinear ordinary differential equations, and numerical solutions to these equations are obtained. Furthermore, in some specific parameter regimes, analytical solutions are found. It is observed that the effect of increasing values of the Casson parameter is to decrease the velocity field while enhancing the temperature field. Furthermore, it is observed that the effect of the increasing values of the suction parameter is to increase the skin-friction coefficient.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(5):054503-054503-4. doi:10.1115/1.4023643.

The size effect in the failure of a hybrid adhesive joint of a metal with a fiber-polymer composite, which has been experimentally demonstrated and analytically formulated in preceding two papers, is here investigated numerically. Cohesive finite elements with a mixed-mode fracture criterion are adopted to model the adhesive layer in the metal-composite interface. A linear traction-separation softening law is assumed to describe the damage evolution at debonding in the adhesive layer. The results of simulations agree with the previously measured load-displacement curves of geometrically similar hybrid joints of various sizes, with the size ratio of 1:4:12. The effective size of the fracture process zone is identified from the numerically simulated cohesive stress profile at the peak load. The fracture energy previously identified analytically by fitting the experimentally observed size effect curves agrees well with the fracture energy of the cohesive crack model obtained numerically by optimal fitting of the test data.

Commentary by Dr. Valentin Fuster

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