Compliant Core Research Articles

Volcanotectonic interactions between inclined sheets, dykes, and faults at the Santorini Volcano, Greece

Dykes and inclined sheets are known occasionally to exploit faults as parts of their paths, but the conditions that allow this to happen are still not fully understood. In this paper, we report field observations from a swarm composed of 91 segments of dykes and inclined sheets, the swarm being particularly well-exposed in the mechanically layered caldera walls of the Santorini volcano, Greece. Here the focus is on dykes and sheets in the swarm that are seen deflected into faults and the mechanical conditions that encourage such deflections. In particular, we present new analytical and numerical models to explain the mechanical principles of dyke/sheet deflections into faults. The numerical models are applied to a normal-fault dipping 65° with a damage zone composed of parallel layers or zones of progressively stiffer rocks with increasing distance from the fault rupture plane. We model a sheet-intrusion, dipping from 0° to 90° and with an overpressure of alternatively 1 MPa and 5 MPa, approaching the fault. We further tested the effects of changing (1) the thickness of the sheet-intrusion, (2) the fault-zone thickness, (3) the fault-zone dip-dimension (height), and (4) the loading by, alternatively, regional tension and compression. We find that the stiffness of the fault core, where a compliant core characterises recently active fault zones, has pronounced effects on the orientation and magnitudes of the local stresses and, thereby, on the likelihood of dyke/sheet deflection into the fault zone. Similarly, the analytical models, focusing on the fault-zone tensile strength and energy conditions for dyke/sheet deflection, indicate that dykes/sheets are most likely to be deflected into and use steeply dipping recently active (zero tensile-strength) normal faults as parts of their paths.

Geometrical nonlinear thermal response of sandwich panels with temperature dependent mechanical properties—Extended high-order approach

The thermal and the thermo-mechanical responses of a sandwich panel with a compliant core are investigated within the framework of the extended high-order approach where the core properties are temperature dependent or independent. Loads schemes include thermal field within temperature working range simultaneous with in-plane compressive loads applied to the core only and to the face sheets and core in the form of the uniform end—shortening of edge of panel. The mathematical formulations use the extended high-order sandwich panel theory approach that takes into account the in-plane rigidity of the core and uses the deformation patterns of the high-order sandwich panel theory. The linear and nonlinear field equations along with the appropriate boundary conditions are presented. A numerical study is conducted, and it investigates the thermal response with temperature independent and temperature dependent mechanical properties of the core as well as the thermo-mechanical response due to in-plane compressive loads. The results include displacements, stress resultants, and stress at critical locations along the panel as well as equilibria curves. They reveal that, in general, the panel with temperature independent properties response remains almost linear while with temperature dependent ones it takes a general nonlinear response. The addition of an external mechanical load changes the response from a linear/nonlinear one that may be allowable stress controlled to a case where loss of stability occurs.

Time Domain Spectral Element method to Study Response of a Sandwich Beam with Compliant Core Subjected to Under Water Explosion

A spectral element formulation is developed for a sandwich beam with stiff face sheets and a compliant core. Higher order displacement functions are taken for the core in order to incorporate the core compression effects. Highly efficient spectral element method is used for the study. Accuracy and the computational efficiency of this method are compared with the conventional finite element method in solving transient blast loading case. Using this spectral element method, under water blast loading response of a sandwich beam with composite face sheets and core is studied. A comparison is made between the response of stiff core and compliant core sandwich beam models. The study presents a design approach for an optimal configuration for under water blast resistant structures.

Non-linear response of curved sandwich panels – extended high-order approach

The geometrical non-linear behavior a curved sandwich panel with a stiff or compliant core when subjected to a pressure load using the Extended High-Order Sandwich Panel theory (EHSAPT), is presented. The formulation follows the EHSAPT procedure where the in-plane. i.e circumferential rigidity of the core is considered and the distribution of the displacements through the depth of the core are presumed. These displacement distributions are the closed-form solutions of the 2D governing equations of the curved core without circumferential rigidity that appear in the HSAPT curved sandwich panel model. The mathematical formulation includes the field equations along with the appropriate boundary and continuity conditions that take into account the high-order stress resultants in the core due to the presumed distributions. Finally a numerical study is conducted for a panel loaded by a distributed pressure at the upper face sheet. It reveals that the post-buckling response of a curved sandwich panels is associated with shallow to deep wrinkling deformations of the upper face sheet in the case of a simply-supported panel or a general non-linear pattern without wrinkles in the case of pinned supports with a short span. In both cases a stable post-buckling response is observed similar to that of a plate one.

On the use of the first order shear deformation plate theory for the analysis of three‐layer plates with thin soft core layer

Three‐layer laminates with thin soft core layer can be found in many engineering applications. Examples include laminated glasses and photovoltaic panels. For such structures high contrast in the mechanical properties of faces and core requires the use of advanced methods to determine effective material properties of the laminate. In this paper we address the application of the first order shear deformation plate theory to the analysis of laminates with thin and soft core layer. In particular, transverse shear stiffness parameters for three‐layered plates with different symmetric configurations are analyzed. For classical sandwiches with thick core layer the result coincides with the Reissner's formula. For the case of thin and compliant core layer the new expression for the effective shear stiffness is derived.

Geometrically nonlinear creep behavior of debonded sandwich panels with a compliant core

The paper investigates the effects of creep of the core material on the geometrically nonlinear behavior of sandwich panels with debond at one of the core–face interfaces within the framework of the high-order sandwich panel theory (HSAPT). The theoretical model accounts for the viscoelasticity of the core via an incremental exponential law that corresponds to the rheological generalized Maxwell model obtained from the expansion of the relaxation moduli into Prony series. The HSAPT structural modeling accounts for the deformability of the core in shear and through its thickness, as well as for the large displacements of the facesheets. An incremental time-stepping analysis is conducted, which accounts for the time variations of the stresses, deformations, and the geometrically nonlinear effects. The incremental governing equations and the stresses and deformations fields through the thickness of the core are derived, and the capabilities of the model are studied numerically. The results show that creep of the core leads to significant variations in the deformations and stresses over time, which when combined with the geometrically nonlinear effects of the debonded facesheet, can have an important influence on the performance of sandwich panels under sustained loads. It is shown that stress relaxation due to creep and/or contact may restrain the buckling of the debonded facesheet in some cases.

Brazier effect of single- and double-walled elastic tubes under pure bending

The cross sections of hollow cylindrical tubes ovalise under a pure bending condition, and this reduces their flexural stiffness as their curvatures increase. It is important to accurately evaluate this phenomenon, known as the 'Brazier effect', to understand the bending behaviour of the systems considered. However, if the tubes are supported by an elastic medium or foundation, the ovalisation displacements of their cross sections may decrease. From this point of view, the purpose of this research is to analytically investigate the bending characteristics of single- and double-walled elastic tubes contacted by an elastic material by considering the Brazier effect. The Brazier moment, which is the maximum moment-carrying capacity of the ovalised cross section, can be calculated by introducing the strain energy per unit length of the tube in terms of the degree of ovalisation for the tube and the curvature. The total strain energy of the double-walled system is the sum of the strain energies of the outer and inner tubes and that of the compliant core. Results are comparatively presented to show the variation in the degree of ovalisation and the Brazier moment for single- and double-walled tubes.

The variation in Poisson’s ratio caused by interactions between core and wrap in helical composite auxetic yarns

Abstract Materials with a negative Poisson’s ratio are referred to as auxetic. One recently invented example of this is the helical auxetic yarn (HAY). This has been proved to successfully exhibit auxetic behaviour both as a yarn and when incorporated into fabric. The HAY is based on a double-helix geometry where a relatively stiffer ‘wrap’ is helically wound around a compliant core fibre. This paper studies the effect of the interaction between the core and the wrap fibre on the auxetic behaviour of the HAY, including the effect of their relative moduli. Assessment of the Poisson’s ratio of the HAYs has revealed that an elevated difference in component moduli causes the wrap fibre embedding itself into the core fibre, thus decreasing the auxetic effect. Careful determination of an optimum core–wrap moduli ratio where the ratio is high enough to yield an auxetic effect and low enough to prevent the core-indentation effect can lead to the fabrication of a yarn with largest negative Poisson’s ratio.

A practical, nondestructive method to determine the shear relaxation modulus behavior of polymeric interlayers for laminated glass

A practical, nondestructive method to determine the shear relaxation modulus behavior of viscoelastic polymer interlayers used in laminated glass is demonstrated. The method is based on a direct measurement of the laminate effective thickness using a four point bend test geometry or uniform pressure loading of laminate plates. The effective thickness expressions that incorporate the polymer shear modulus explicitly in the formulation are then used to calculate the polymer shear modulus in an inverse fashion. Comparison of this approach with dynamic mechanical analysis shows the new approach to be accurate over a wide range of test conditions. The approach described has the advantage of being able to characterize the polymer interlayer in situ. We have focused on laminated glass in this study, but the approach is applicable to any laminated combination of different material types consisting of a compliant core sandwiched between stiff skins.

High-Order Modeling of Circular Cylindrical Composite Sandwich Shells with a Transversely Compliant Core Subjected to Low Velocity Impact

In this study, a high-order model for the analysis of circular cylindrical composite sandwich shells subjected to low-velocity impact loads is presented. The sandwich shell is composed of two composite face sheets and a transversely compliant core. The impact behavior of the cylindrical composite sandwich shells is described by a high-order sandwich shell theory. The interaction between the impactor body and the sandwich shell is approximated using a spring mass model. The present analysis is based on an iteration procedure, and yields analytic functions describing the contact force history. The contact force is considered to be distributed uniformly over a contact patch, the size of which depends on the magnitude of the impact load as well as the elastic properties and geometry of the impactor. Finally, the obtained results have been compared with the available experimental results, and a good correlation has been found.

Non-linear behavior of a face-sheet debonded sandwich panel – Thermal effects

Geometrical non-linear analyses of a sandwich panel with a prescribed debond within one of the face sheets where the core is either incompressible (stiff) or compressible (compliant) are presented. A loading scheme of thermal or mechanical loads independently or combined thermo-mechanical load is considered. The debond, in one of the face sheets, consists of a through-the-width crack that divides it into two layers, one with a free surface (upper or lower) and the other attached to the core. The interfaces of the debonded crack are free of shear stresses but can only accommodate vertical normal compressive stresses. In the case of a compliant core the high-order sandwich panel (HSAPT) model is adopted and in case of an incompressible core the ordinary beam theory (OBT) and the first-order shear deformable theory (FOSDT) (equivalent single layer, ESL) are used. The mathematical formulation outlines the field equations along with the stress and displacements fields for the cases where the core (stiff or compliant) properties are either temperature independent (TI) or dependent (TD). The governing equations along with the appropriate boundary and continuity conditions are derived using variational principles following the principles of the HSAPT or ESL models. The non-linear analysis includes geometrical non-linearities in the face sheets caused by rotation of the face cross sections and high-order effects that are the result of the compliant core. The core stress and displacements fields with temperature-dependent (TD) mechanical properties are determined in a closed form using properties that are vertically coordinate dependent. The numerical study looks into the effect of the thickness (thin, moderate and very thick) of the debonded layer that is not attached to the core in the presence of mechanical and thermal loads on the non-linear response. An inner delamination is considered, around mid-span, for specific lengths with TI or TD mechanical properties. The results reveal that the ESL models cannot be used to predict the non-linear response of a sandwich panel with a compliant core. In addition, the existence of a debond crack nearby the face–core interface (very thick layer) may be associated with local buckling of the thin layer that is bonded to the core.

Improved contact load-bearing capacity of ultra-fine grained titanium: Multilayering and grading

The contact load-bearing response and surface damage resistance of multilayered hierarchical structured (MHSed) titanium were determined and compared to monolithic nanostructured titanium. The MHS structure was formed by combining cryorolling with a subsequent Surface Mechanical Attrition Treatment (SMAT) producing a surface structure consisted of an outer amorphous layer containing nanocrystals, an inner nanostructured layer and finally an ultra-fine grained core. The combination of a hard outer layer, a gradual transition layer and a compliant core results in reduced indentation depth, but a deeper and more diffuse sub-surface plastic deformation zone, compared to the monolithic nanostructured Ti. The redistribution of surface loading between the successive layers in the MHS Ti resulted in the suppression of cracking, whereas the monolithic nanograined (NG) Ti exhibited sub-surface cracks at the boundary of the plastic strain field. Finite element models with discrete layers and mechanically graded layers representing the MHS system confirmed the absence of cracking and revealed a 38% decrease in shear stress in the sub-surface plastic strain field, compared to the monolithic NG Ti. Further, the mechanical gradation achieves a more gradual stress distribution which mitigates the interface failure and increases the interfacial toughness, thus providing strong resistance to loading damage.

Analysis of Sandwich Beams With a Compliant Core and With In-Plane Rigidity—Extended High-Order Sandwich Panel Theory Versus Elasticity

A new one-dimensional high-order theory for orthotropic elastic sandwich beams is formulated. This new theory is an extension of the high-order sandwich panel theory (HSAPT) and includes the in-plane rigidity of the core. In this theory, in which the compressibility of the soft core in the transverse direction is also considered, the displacement field of the core has the same functional structure as in the high-order sandwich panel theory. Hence, the transverse displacement in the core is of second order in the transverse coordinate and the in-plane displacements are of third order in the transverse coordinate. The novelty of this theory is that it allows for three generalized coordinates in the core (the axial and transverse displacements at the centroid of the core and the rotation at the centroid of the core) instead of just one (midpoint transverse displacement) commonly adopted in other available theories. It is proven, by comparison to the elasticity solution, that this approach results in superior accuracy, especially for the cases of stiffer cores, for which cases the other available sandwich computational models cannot predict correctly the stress fields involved. Thus, this theory, referred to as the “extended high-order sandwich panel theory” (EHSAPT), can be used with any combinations of core and face sheets and not only the very “soft” cores that the other theories demand. The theory is derived so that all core/face sheet displacement continuity conditions are fulfilled. The governing equations as well as the boundary conditions are derived via a variational principle. The solution procedure is outlined and numerical results for the simply supported case of transverse distributed loading are produced for several typical sandwich configurations. These results are compared with the corresponding ones from the elasticity solution. Furthermore, the results using the classical sandwich model without shear, the first-order shear, and the earlier HSAPT are also presented for completeness. The comparison among these numerical results shows that the solution from the current theory is very close to that of the elasticity in terms of both the displacements and stress or strains, especially the shear stress distributions in the core for a wide range of cores. Finally, it should be noted that the theory is formulated for sandwich panels with a generally asymmetric geometric layout.

A high-order theory for the analysis of circular cylindrical composite sandwich shells with transversely compliant core subjected to external loads

A new model based on the high order sandwich panel theory is proposed to study the effect of external loads on the free vibration of circular cylindrical composite sandwich shells with transversely compliant core, including also the calculation of the buckling loads. In the present model, in contrast to most of the available sandwich plate and shell theories, no prior assumptions are made with respect to the displacement field in the core. Herein the displacement and the stress fields of the core material are determined through a 3D elasticity solution. The performance of the present theory is compared with that of other sandwich theories by the presentation of comparative results obtained for several examples encompassing different material properties and geometric parameters. It is shown that the present model produce results of very high accuracy, and it is suggested that the present model, which is based on a 3D elasticity solution for the core material, can be used as a benchmark in future studies of the free vibration and buckling of circular cylindrical composite sandwich shells with a transversely compliant core.

Geometrically nonlinear thermomechanical response of circular sandwich plates with a compliant core

The geometrically nonlinear response of a circular sandwich plate that consists of two face sheets and a compliant (“soft”) core with mechanical properties that may be either independent or dependent of temperature and subjected to both mechanical loads and thermal induced deformations, but remain elastic linear throughout the loading process, is presented. The mathematical formulation follows the principles of the high-order sandwich panel theory (HSAPT) and includes the vertical flexibility of the core in addition to the temperature dependency of the mechanical properties of the core material. The mathematical formulation outlines the set of governing partial differential equations as well the appropriate boundary conditions for a general sandwich layout. The particular case of an axisymmetric circular sandwich plate subjected to axisymmetric mechanical and thermal loads, and with axisymmetric boundary conditions is studied analytically and numerically. The numerical study includes an interaction of mechanical and thermal loads which is presented through results within the plate for various load levels of various structural quantities as well as equilibrium curves of temperatures versus these structural quantities. The results reveal that the combination of mechanical and thermal loads along with a compliant core material with mechanical properties that degrade with increasing temperatures shifts the behavior from a linear and stable (strength controlled) response into a strongly nonlinear response with limit point behavior and associated loss of stability, when large displacements and large rotations (geometrical nonlinearity) are included in the modeling.

On wrinkling of a sandwich panel with a compliant core and self-equilibrated loads

The nonlinear response of a unidirectional sandwich panel that is associated with wrinkling of the face sheets, due to a self-equilibrated loading scheme is presented. This loading scheme may be a results of a manufacturing process where the core is tensioned first and the sandwich panel is formed through bonding of the face sheets to the tensioned core while wrinkling occurs as a result of the release of the tensile force of the core, i.e. similar to the manufacturing of electroactive polymers (EAPs) [Wantanaba et al. (2002)] or due to prestressing of the core which is associated with the pre-strain the edge of the core only. These self-equilibrated loads yield compression in the face sheets as well as in the core which may be associated with loss of stability as a result of overall buckling of the entire panel or wrinkling of the face sheets. Thus, for such loading scheme the longitudinal rigidity of the core must be considered although it is small as compared with that of the face sheets. The governing equations along with the appropriate boundary conditions are derived through the introduction of longitudinal normal stresses in the core along with high-order distribution functions for the displacements through its depth. The mathematical formulation is based on variational principles along with moderate type of deformations for the kinematic relations. The results of the various structural quantities in the form of curves along the panel, equilibrium curves and deformed shapes for a particular sandwich panel are presented. The study discusses the effects of the transfer of the compressive load from the core to the face sheets, either directly through an edge beam or without it. Conclusions are drawn and presented.

Non-linear thermo-mechanical behaviour of delaminated curved sandwich panels with a compliant core

A non-linear analysis of a delaminated curved sandwich panel with a compliant core, and a delamination (debond) at one of the face–core interfaces, and subjected to a thermal field and a mechanical loading or combined is presented. The mathematical formulation outlines the governing equations along with the stress and displacements fields for the cases where the core properties are either temperature independent ( TI) or temperature dependent ( TD). A variational formulation is used following the principles of the high-order sandwich panel theory (HSAPT) to derive the field equations along with the appropriate continuity conditions. The non-linear analysis includes geometrical non-linearities in the face sheets caused by rotation of the face cross sections, and high-order effects that are the result of the radially (transversely) flexible (or compliant) core. The core stress and displacements fields with temperature-dependent ( TD) mechanical properties are determined in closed form using an equivalent polynomial description of the varying properties. The numerical study describes the non-linear response of delaminated curved sandwich panels subjected to mechanical concentrated loads, thermally induced deformations and simultaneous thermal and mechanical loads. In the combined loading case the mechanical loads are below the limit point load level of the mechanical response, and the imposed temperature field is varied. The results are displayed in terms of plots of various structural quantities along the sandwich panel length (circumference), equilibrium curves and strain energy release rate curves. It is shown that the combined thermo-mechanical response shifts the linear or non-linear responses,observed for the separate cases of either temperature induced deformations or mechanical loading, into a strongly non-linear response with limit point behaviour and large stresses in vicinity of supports, loads and tips of delaminated zone.

Nonlinear behavior of thermally loaded curved sandwich panels with a transvesely flexible core

The nonlinear analysis of a curved sandwich panel with a compliant core subjected to a thermal field and mechanical load is presented. The mathematical formulation is developed first, along with the solution of the stress and displacement fields for the case of a sandwich core with mechanical properties that are independent of the temperature. The nonlinear analysis includes geometrical nonlinearities in the face sheets caused by rotation of the face cross sections and high-order effects due to the transversely flexible core. The mathematical formulation uses the variational principle of minimum energy along with HSAPT (high-order sandwich panel theory) to derive the nonlinear field equations and the boundary conditions. The full displacement and stress fields of the core with uniform temperature-independent mechanical properties and the appropriate governing equations of the sandwich panel are given. This is followed by the general solution of the core stress and displacement fields when the mechanical core properties are dependent on the radial (through-the-thickness) coordinate. The displacement fields of a core with temperature-dependent mechanical properties are determined explicitly using an equivalent polynomial description of the varying properties. A numerical study then describes the nonlinear response of curved sandwich panels subjected to concentrated and distributed mechanical loads, thermally induced deformations, and simultaneous thermal and mechanical loads where the mechanical load is below the limit load level of the mechanical response and the imposed temperature field is made to vary. The results reveal that the thermomechanical response is linear when the sandwich panel is heated, but becomes nonlinear with limit point behavior when the panel is cooled down.

Wave propagation in a sandwich structure

The propagation of elastic waves in a sandwich structure with two thin stiff face-plates and a thick compliant core is considered in this paper. A complete description of the dispersion relation with no restrictions on frequency and wavelength is provided. This is accomplished by transforming the wave equation to a Hamiltonian system and then using a transfer matrix approach for solving the Hamiltonian system. To provide insight, particular regimes of the frequency–wavelength plane are then considered. First, an explicit formula is derived for all natural frequencies at the long wavelength limit. It is shown that all waves with finite limiting frequency have zero group velocity, while those with vanishing limiting frequency correspond to longitudinal, shear and flexural waves. The displacement of the flexural waves are reminiscent of Mindlin plates, and an asymptotic procedure to find the shear correction factor is presented. Second, the lowest branch of the dispersion relation is studied in detail and mode shapes are used to motivate explicit but accurate description of this lowest branch. This approximate model is anticipated to be useful in simulations of large structures with sandwich structures.

Elastica of sandwich panels with a transversely flexible core—A high-order theory approach

The elastica behavior of an extensional sandwich panel with a “soft” core when subjected to in-plane compressive loads is presented and it is compared with the response of its extensional equivalent single layer (ESL) with shear deformations model. The field equations along with the appropriate boundary conditions for the sandwich and the ESL panels have been derived through a variational approach following the High-order SAndwich Panel Theory (HSAPT) approach that takes into account the vertical flexibility of the core. The governing equations include the effects of the extension of the mid-surfaces of the face sheets of the sandwich panel or the mid-plane of the ESL model which the classical elastica approach misses. The results of the elastica response of a clamped-simply-supported sandwich panel and its ESL counterpart are presented and compared. They include the response along the panel, deformed shapes and equilibrium curves of in-plane loads versus structural quantities such as displacements and internal stress resultants and stresses. These results reveal that the predicted buckling load of the ESL panel is larger than that of the sandwich panel and that deep in the non-linear range the upper face sheet wrinkles with increasing overall and edge displacements and a release of the load. Hence, the use of an equivalent single layer panel especially when a sandwich panel with a compliant core is considered may lead to unsafe and unreliable predictions when large displacements and large rotations are considered.