System Of Exact Kinematic Conditions Research Articles

Higher-order semi-layerwise models for doubly curved delaminated composite shells

This work deals w ith the development and extension of higher-order models for delaminated doubly curved composite shells with constant radii of curvatures. The mechanical model is based on the method of four equivalent single layers and the system of exact kinematic conditions. A remarkable addition of this work compared to some previous ones, is a modified and improved continuity condition between the delaminated and undelaminated parts of the shell. Using the principle of virtual work, the equilibrium equations of the shell systems are brought to the stage and solved by using the classical Lévy plate formulation under simply supported conditions. Four different scenarios of elliptic and hyperbolic delaminated shells are investigated providing the solutions for the mechanical fields as well as for the J-integral. The analytical results are compared to 3D finite element calculations, and excellent agreement was obtained for the displacement components and normal stresses. On the contrary, it was found that the transverse shear stresses are captured quite differently by the proposed method and the finite element models. Although the role of shear stresses should not be underrated, they seem to be marginal because the distributions of the J-integral components are in very good agreement with the numerically determined energy release rates.

An analytical solution for buckling and vibration of delaminated composite spherical shells

Abstract The estimation of the critical buckling loads and eigenfrequencies are among the most common problems of mechanical engineering. These parameters are very important measures to avoid the loss of stability of the designed structures. In this work a progressive analytical model of doubly curved shells will be presented and applied to a delaminated spherical shell. The equations are derived using an improved version of the Sanders shell theory and the System of Exact Kinematic Conditions (SEKC). The solution method is based on the Levy formulation. With this method the governing partial differential equation (PDE) can be reduced to an ordinary differential equation (ODE) with the use of Fourier-series. The resulting set of equations are solved using a variant of the state-space method which is able to solve systems with non-constant system matrix.

Fracture and mode mixity analysis of shear deformable composite beams

To analyse delaminated composite beams with high accuracy under mixed-mode I/II fracture conditions first-, second-, third- and Reddy’s third-order shear deformable theories are discussed in this paper. The developed models are based on the concept of two equivalent single layers and the system of exact kinematic conditions. To deduce the equilibrium equations of the linearly elastic system, the principle of virtual work is utilised. As an example, a built-in configuration with different delamination position and external loads are investigated. The mechanical fields at the delamination tip are provided and compared to finite element results. To carry out the fracture mechanical investigation, the J-integral with zero-area path is introduced. Moreover, by taking the advantage of the J-integral, a partitioning method is proposed to determine the ratio of mode-I and mode-II in-plane fracture modes. Finally, in terms of the mode mixity, the results of the presented evaluation techniques are compared to numerical solutions and previously published models in the literature.

The role of transverse stretching in the delamination fracture of softcore sandwich plates

• The first and second-order plate theories are complemented with transverse stretching. • The thickness locking was eliminated by applying a simplified stiffness matrix. • The transverse stretching effect involves a significant improvement of the first-order theory. • The comparison of analytical and 3D finite element results was carried out. • The energy release rates are calculated by the 3D J-integral. This paper presents the semi-layerwise analysis of structural sandwich plates with through-width delamination. The mechanical model of rectangular plates is based on the method of four equivalent single layers and the system of exact kinematic conditions. An important improvement compared to a previous formulation is the consideration of linear and quadratic stretching term in the transverse displacement component. Three different delamination scenarios are investigated: core-core failure, face-core delamination and the face-face failure. By applying the first- and second-order laminated plate theories and the principle of virtual work the governing equations are derived. The equilibrium equations are solved under Lévy type boundary conditions using the state-space approach. Solutions for the mechanical fields are provided and compared to 3D finite element results. The energy release rate distributions along the delamination front are also determined using the J-integral. Although the stress resultants by transverse stretching do not influence directly the J-integral, the results indicate that this effect improves the accuracy of the model in general, and substantially influences the results of the first-order plate theory in the case of the face-face delamination.

Analytical solution of some delamination scenarios in thick structural sandwich plates

The first-, second- and third-order shear deformation plate theories are applied in this work to model thick rectangular sandwich plates with through-width delamination. The models are based on the concept of the four equivalent single layers and the system of exact kinematic conditions. Three different scenarios are considered: the failure of the core, the delamination between the top facesheet and the core, and finally, the case when the delamination takes place in the local midplane of the top facesheet. A general model is derived and applied to sandwich plates with Lévy type boundary conditions. The governing equations are summarized and the state-space model of the system is created. The mechanical fields are calculated and compared to finite element results. The comparison shows that the first-order sandwich plate model is inaccurate, on the other hand, the second- and third-order theories capture very well the mechanical fields compared to finite element results. The J-integral distribution is also calculated along the delamination front and it is concluded that the third- and second-order models give very good approximations of the results by finite element analysis and the virtual crack closure technique.

Antiplane–inplane shear mode delamination between two second-order shear deformable composite plates

The second-order laminated plate theory is utilized in this work to analyze orthotropic composite plates with asymmetric delamination. First, a displacement field satisfying the system of exact kinematic conditions is presented by developing a double-plate system in the uncracked plate portion. The basic equations of linear elasticity and Hamilton’s principle are utilized to derive the system of equilibrium and governing equations. As an example, a delaminated simply supported plate is analyzed using Lévy plate formulation and the state-space model by varying the position of the delamination along the plate thickness. The displacements, strains, stresses and the J-integral are calculated by the plate theory solution and compared with those by linear finite-element calculations. The comparison of the numerical and analytical results shows that the second-order plate theory captures very well the mechanical fields. However, if the delamination is separated by only a relatively thin layer from the plate boundary surface, then the second-order plate theory approximates badly the stress resultants and so the mode-II and mode-III J-integrals and thus leads to erroneous results.

Natural vibration-induced parametric excitation in delaminated Kirchhoff plates

This paper revisits the problem of free vibration of delaminated composite plates with Lévy type boundary conditions. The governing equations are derived for laminated Kirchhoff plates including through-width delamination. The plate is divided into two subplates in the plane of the delamination. The kinematic continuity of the undelaminated part is established by using the system of exact kinematic conditions. The free vibration analysis of orthotropic simply supported Lévy plates reveals that the delaminated parts are subjected to periodic normal and in-plane shear forces. This effect induces parametric excitation leading to the susceptibility of the plates to dynamic delamination buckling during the vibration. An important aspect is that depending on the vibration mode the internal forces have a two-dimensional distribution in the plane of the delamination. To solve the dynamic stability problem the finite element matrices of the delaminated parts are developed. The distribution of the internal forces in the direction of the delamination front was considered. The mode shapes including a half-wave along the width of the plate accompanied by delamination buckling are shown based on the subsequent superimposition of the buckling eigenshapes. The analysis reveals that the vibration phenomenon is amplitude dependent. Also, the phase plane portraits are created for some chosen cases showing some special trajectories.

Semi-layerwise analysis of laminated plates with nonsingular delamination—The theorem of autocontinuity

The proposed semi-layerwise approach captures the mechanical behavior of delaminated composite plates using four equivalent single layers independently of the lay-up. Two equivalent single layers are applied for both the top and bottom parts of a delaminated plate. The updated version of the system of exact kinematic conditions formulates the continuity of the in-plane displacements between the neighboring layers, the location of the global reference plane of the plate and – as important additions compared to previous papers – the continuity of shear strains, their derivatives and curvatures, respectively. The method is demonstrated using the first-, second- and third-order plate theories. As examples, simply supported delaminated plates are considered.The continuity between the delaminated and undelaminated plate regions is established through the theorem of autocontinuity. The J-integral is calculated along the straight delamination front and compared to the results of the virtual crack closure technique. The results indicate that the first- and third-order plate theories provide the best solutions, and give good approximation even in those cases when the previous models failed, i.e., when the delamination is asymmetrically placed between two layers and it is close to the free surface of the plate.

Nonsingular crack modelling in orthotropic plates by four equivalent single layers

In this work the second- and third-order laminated plate theories are applied to model delaminated composite plates with material orthotropy. The method of four equivalent single layers is proposed and a general third-order displacement field is utilized in each layer. The kinematic continuity between the layers is established by the system of exact kinematic conditions. Apart from the continuity of the in-plane displacements between the interfaces of the layers even the continuity of shear strains, their derivatives and curvatures is imposed. As a novelty a so-called shear strain control condition is introduced, which means that the shear strains at two or more points located along the thickness are imposed to be the same. Using the proposed conditions the equilibrium equations are derived for the delaminated and undelaminated regions of the plate. Plates with different boundary conditions are solved as examples and the theorem of autocontinuity is introduced, which is essentially related to the continuity conditions between the delaminated and undelaminated parts. The stress and displacement fields as well as the J-integral are determined in the examples and compared to finite element calculations. The results indicate that the control condition works very well in the case of the second-order plate theory, in contrast it is rather a disadvantage in the case of the third-order approximation.

A special case of parametrically excited systems: Free vibration of delaminated composite beams

The problem of free vibration of delaminated composite beams is revisited in this paper by showing the existence of parametric excitation, bifurcation and stability. Why and when the delaminated part buckles during the vibration? To answer this question, first a finite element model is developed for the uncracked beam portion using the theory of layered beams and the so-called system of exact kinematic conditions. Second, a transition element is developed to establish the kinematic connection and continuity among the delaminated and uncracked portions. Thirdly, the delaminated beam portions are modeled by simple layered beam elements. The coupling between the flexural and longitudinal vibration is taken into consideration in the delaminated part and therefore it is shown that this part is loaded by periodic normal forces during the free vibration of the system. The time dependent stiffness because of the normal forces does not influence the frequencies globally at all, however, locally delamination buckling can take place, as well. The bifurcation and stability of the delaminated parts are analyzed using the harmonic balance method and the critical forces in delaminated layered beams under dynamic stability are determined. The free vibration mode shapes in the state of instability are predicted based on the critical dynamic force and the condition of constant arc length of the delaminated parts. Finally the phase plane portraits of the motion of the midpoint in the delaminated region are presented giving a global view on the behavior of composite beams under harmonic vibration.

Bending solution of third-order orthotropic Reddy plates with asymmetric interfacial crack

In this paper Reddy’s third-order shear deformable plate theory is applied to asymmetrically delaminated orthotropic composite plates under antiplane–inplane shear fracture mode. A double-plate system is utilized to capture the mechanical behavior of the uncracked plate portion. An assumed displacement field is used and modified in order to satisfy the traction-free conditions at the top and bottom plate boundaries. Moreover, the system of exact kinematic conditions was also implemented into the novel plate model. An important improvement of this work compared to previous papers is the continuity condition of the shear strains at the interface of the double-plate system. Applying these conditions it is shown that the nineteen parameters of the third-order displacement field can be reduced to nine. Using the simplified displacement field the governing equations are derived, as well. The solution of a simply-supported delaminated plate is presented using the state-space model and the displacement, strain and stress fields are determined, respectively. The energy release rate and mode mixity distributions are calculated using the 3D J-integral. The analytical results are compared to those by finite element computations and it is concluded that the present model is the most accurate one among the previous plate theory-based approaches.

The system of exact kinematic conditions and application to delaminated first-order shear deformable composite plates

In this work the analysis of unsymmetrically delaminated composite plates with straight crack front is presented. It is shown that in the undelaminated region of laminated plates the problem can be captured by applying a double-plate system. The displacement continuity between the top and bottom plates requires five conditions, three of them are known in the literature, however further two are necessary to define the reference plane of the undelaminated region. This so-called system of exact kinematic conditions can be applied to any plate theory to satisfy the requirements against the displacement field. The first-order shear deformable plate theory is applied to demonstrate the applicability of the system of exact kinematic conditions. The key step of the formulation is that the in-plane displacement and the rotation parameters are not independent of each other, therefore the size of the problem can be significantly reduced. Laminated simply supported plates are analyzed by varying the position of the delamination in the thickness direction. The energy release rates and the mode mixity are calculated by using the J-integral. It is shown based on the comparison of the analytical results to those by finite element solution that if the delamination is close to the midplane of the plate, then good agreement can be obtained. In contrast, if the delamination is close to the top or bottom plate surfaces, then the first-order plate theory is not appropriate to estimate the fracture mechanics parameters of the plates.