Elsevier

Thin-Walled Structures

Volume 91, June 2015, Pages 38-49
Thin-Walled Structures

Effect of thermal environment on free vibration of cracked rectangular plate: An analytical approach

https://doi.org/10.1016/j.tws.2015.02.004Get rights and content

Highlights

  • Free vibration of thin cracked rectangular plate in thermal environment is studied.

  • A relation is devised for buckling temperature of thin cracked rectangular plate.

  • The effect of crack location and length on buckling temperature is analyzed.

  • Variation of buckling temperature with plate aspect ratio is established.

  • The influence on crack length, location and temp. rise on frequency is studied.

  • Natural frequency is least when the crack is internal and symmetric about mid plane.

Abstract

An analytical model is proposed for free vibration and geometrically linear thermal buckling phenomenon of a thin rectangular isotropic plate containing a continuous line surface or internal crack located at the centre of the plate using classical plate theory. The crack terms are formulated using Line Spring Model. The in-plane forces are due to the uniform heating of the cracked plate. Applying the Galerkin׳s method, the equation is transformed into Duffing equation which shows the effect of uniform rise in temperature on natural frequencies. A classical relation for buckling temperature for a cracked plate is also determined.

Introduction

Plate as a basic structural element is widely used in many fields such as shipbuilding, aerospace, mechanical and civil installations. Quite often plates are exposed to uniform or non-uniform heating during their service life. This exposure affects the stiffness, thus changing its dynamic characteristics. As an example, the rotor blades or wings of an aircraft suffer aerodynamic heating inducing thermal stresses which may induce buckling or dynamic instability. Also the presence of singularities in the form of Crack(s) affects the vibration characteristics. In order to improve the design reliability of plated structures, knowledge of vibration characteristics of cracked plate along with their response to thermal environment becomes important. Murphy et al. [1] considered clamped rectangular plate in their theoretical and experimental approach in order to assess the effect of uniform rise in temperature on natural frequencies. The authors considered initial geometric imperfections and concluded that the first mode natural frequency becomes zero at critical buckling temperature. The results for thermal buckling load for clamped rectangular plates were obtained by Murphy and Ferreira [2] based on energy formulation. Yang and Shen [3] studied the dynamics of initially stressed functionally graded plates in thermal environment. The authors presented results for the effect of volume fraction index, temperature rise, initial stresses and boundary conditions on free and forced vibration characteristics. Kim [4] used Rayleigh–Ritz procedure to formulate the equation of frequency for initially stressed functionally graded rectangular plates using third order shear deformation theory. They considered a prescribed temperature distribution through the thickness of the plate and demonstrated the effect of temperature field on the vibration characteristics. Jeyaraj et al. [5] presented vibration analysis and acoustic response of an isotropic plate subjected to uniform temperature rise using ANSYS and SYSNOISE. It shows that the increase in temperature of the plate decreases the frequency and increases the sound power level for all boundary conditions. Jeyaraj et al. [6] analyzed vibration response and sound radiation for a composite plate with different fiber orientation in thermal environment considering inherent material damping using classical laminate plate theory. The free vibration characteristics of functionally graded plates were studied by Li et al. [7] using three dimensional theory of elasticity. The authors presented parametric results for the effect of boundary conditions, uniform-non uniform temperature rise and volume fraction indices on vibration. Raki et al. [8] derived equations of equilibrium and stability of functionally graded plates (FGP) using higher order shear deformation theory and concluded that the critical buckling temperatures for FGP׳s are lower than the corresponding values for isotropic plates. As opposed to isotropic plates, the critical buckling temperature of FGP׳s increases with increase in plate aspect ratio. Baferani and Saidi [9] investigated stability analysis of thick annular plates subjected to thermal and mechanical loads using third order shear deformation theory. They presented results for nine different boundary conditions and concluded that for boundary conditions involving simply supported and clamped edges, the classical plate theory over-predicts and the first order shear deformation theory under-predicts the thermal and mechanical buckling load. Paik and Kim [10] considered both the axial loading and lateral pressure to develop the ultimate strength expressions for stiffened panels. Considering the initial deflection and residual stresses, they calculated the ultimate strengths for all potential collapse modes. Brighenti [11] studied the effect of through crack length and orientation on buckling load of rectangular plates considering different boundary conditions and Poisson׳s ratio in their finite element formulation. They considered compression for buckling, tension for fracture and concluded that the presence of crack affects the buckling load. The work of Rigo et al. [12] shows the application of nonlinear finite element analysis to calculate the ultimate compressive strength of unstiffened and stiffened plates with through thickness cracks. They presented results for in-plane compressive strength as affected by the length and location of cracks. Rahbar-Ranji and Zarookian [13] considered uniaxial compression to study the effect of crack on ultimate strength of stiffened steel plates. They presented results as affected by crack length, position, plate thickness and size of stiffener from their nonlinear finite element model. Vescovini and Bisagni [14] represented the composite panel as an assembly of plates and presented a semi-analytical method for buckling under the action of biaxial load and shear. They obtained the buckling equations based on minimum potential energy principle. Alinia et al. [15] studied the influence of central cracks on buckling and post-buckling of shear panels. The authors presented parametric results involving crack length, orientation of crack, aspect ratio and concluded that the buckling characteristics degrade due to presence of cracks. Recently, Rad and Panahandeh-Shahraki [16] investigated the tensile buckling of functionally graded cracked plates through their finite element formulation based on classical plate theory. The authors considered uniaxial and biaxial loads for results affected by gradient index, plate dimensions and material properties. The work of Seifi and Khoda-yari [17] shows the results for buckling of cracked thin-plates under partial and full edge loading. The authors verified their numerical results with experiments. Natarajan et al. [18] presented a parametric study on buckling of functionally graded plates with discontinuities like and cracks and cutouts using the extended finite element method. They concluded that the buckling load decreases with increase in through crack length, radius of cutout and gradient index. Rice and Levy [19] formulated the Line Spring Model (LSM) using classical plate theory for finding stress intensity factors, wherein the crack is represented by continuously distributed line spring with stretching and bending compliances. The authors matched the crack compliance coefficients with those for an edge cracked strip under plane strain. Israr et al. [20] developed an approximate analytical model for vibrations of cracked plate using the LSM wherein, the surface crack is parallel to one of the edges of the plate and is located at the centre. They considered three different boundary conditions of support and employed a perturbation technique called method of multiple scales to obtain amplitude response. Using classical plate theory and LSM, they formulated relation between tensile-bending force effects at far sides of the plate and at crack location. It can be concluded from their work that the natural frequencies decrease as the surface crack length increases. Recently Ismail and Cartmell [21] developed an analytical model for vibration analysis of a cracked isotropic rectangular plate considering various angular orientation of a surface crack. More recently Bose and Mohanty [22] also considered arbitrary position and orientation of part through crack in a thin isotropic plate for vibration analysis and deduced that the orientation of crack affects the vibration characteristics of the plate. Huang and Chan [23] applied moving least squares interpolation functions in their application of the Ritz method for vibration of cracked plate. Bachene et al. [24] used extended finite element method for an approximate solution of cracked plates. Viola et al. [25] applied differential quadrature finite element method to investigate dynamics of thick composite plates containing crack and extended it to plates of arbitrary shapes. Although, the modeling of plates with defects is one of the currently developing areas of research, the literature survey shows that not much work has been done in the area of vibration response of cracked plate in thermal environment. The work in Refs. [20], [21], [22], [23], [24], [25], [26], [27], [28], [29] shows that the presence of crack either partial or through, affects the vibration characteristics and dynamic stability of plates. Literature research (Refs. [30], [31]) shows that the presence of hole also affects mechanical and thermal buckling load. Sahin et al. [32] considered anti-symmetric composite square plates with angle crack for studying thermal buckling behavior. The authors formulated finite element scheme and concluded that the presence of a through crack decreases the critical buckling temperature. Also the crack angle affects the critical buckling temperature more than the horizontal crack. Natarajan et al. [33] studied the free flexural vibrations of a functionally graded plate containing a through crack located at the centre of the plate using 8-noded shear flexible element in their finite element formulation. The authors concluded that the natural frequency decreases by increasing the temperature and crack length.

Section snippets

Problem description

The present work proposes an analytical model for vibration analysis of a cracked rectangular plate in thermal environment. New configuration of the crack (internal crack, located along the thickness of the plate) is considered, thus extending the currently developing field of analytical modeling of cracked plates. Starting with the dynamic equilibrium principle of the plate based on the classical plate theory, the effect of crack is considered in the form of moment and membrane force

Governing equation

The classical form of the equation for thermal bending and buckling is rigorously treated in Refs. [34], [35], [36]. Consider a thin rectangular isotropic plate subjected to a known temperature distribution which is linear through the thickness of the plate. The continuous line crack at the centre has length 2a and is parallel to x axis as shown in Fig. 1(b). The governing equation of motion is derived using equilibrium principle based on the classical plate theory. The assumptions involved are

In-plane forces due to temperature

With the in-plane displacements in both x and y direction being restricted, the uniform or non uniform heating of the cracked plate causes membrane forces. The essence of LSM is the relation between nominal bending and tensile stresses at the crack location as a function of bending and tensile stresses at the far edges of the plate by using crack compliance coefficients. These relations were converted into force effects which reduce the stiffness of the plate by Israr et al. [20]. Using

Crack compliance coefficients

The crack compliance coefficients appearing in the crack terms of Eq. (8) are functions of the thickness of the plate, crack depth and can be found by curve fitting the shape functions defining the stress intensity factor as deduced in the LSM. These coefficients are different for various values of offset distance ‘d’. Aksel and Erdogan [37] analyzed the surface crack problem in a flat plate for stress intensity factors using LSM. The authors found stress intensity factors for the case of

Solution of governing equation

The presence of thermal environment in the governing equation of cracked plate (Eq. (12)) is in the form of thermal bending and in-plane forces. This gives rise to three cases of interest for its solution. (1) The plate is subjected to uniform rise in temperature throughout its volume (MT=0) with in-plane deflections restricted. (2) No thermal in-plane forces (no in-plane restrictions), only thermal moment (MT). (3) Both in-plane forces and moments due to temperature gradient along the

Thermal buckling analysis

Thermal buckling implies the existence of an out of plane deflection that satisfies the equations of equilibrium and boundary conditions and it is governed by the in-plane thermal forces and stiffness. In-plane restrained plates without imperfections when uniformly heated, develop thermal stresses and buckle at a specific temperature [34]. The aim of this section is to derive classical relation for critical buckling temperature for cracked plate, shown in Fig. 2(b), using equilibrium principle.

Results and discussion

Not much work has been done on the vibrations of internally cracked plate in thermal environment and hence the validation of the present model is carried out for a surface crack (d/h=0.2) in absence of thermal environment. Table 1 shows the first mode natural frequencies for such a validation for SSSS and CCSS boundary conditions. It is thus seen that the present model reduces to the model proposed in Ref. [20], for a surface crack in absence of thermal environment. The present relation for

Conclusion

An analytical model is proposed for vibration analysis of cracked rectangular plate subjected to uniform temperature rise. It is shown that the first mode natural frequency is most affected when the crack is internal and symmetric about the mid-plane of the plate and is least affected when the crack is at the surface. This frequency further decreases by rise in temperature for all crack locations. The decrease in natural frequency with increase in crack length is less for internal crack as

Acknowledgment

The authors are thankful for the constructive comments from the reviewers. This work is not funded by any organization.

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