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A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads



K. Jayakumar, D. Yadav and B. Nageswara Rao
 
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ABSTRACT

Utilizing the stress formulation approach a closed-form solution is obtained for a long multi-layer cylindrical shell subjected to electro-thermo-mechanical loads. The present analytical solution holds good to examine the elastic behaviour of laminated composite shells with anisotropic piezoelectric layers acting as sensor and actuator under electro-thermo-mechanical loads. Standard finite elements are not adequate for modelling of the incompressible nature of the solid propellant grains in rocket motors. Utilizing MSC/MARC© software, finite element analysis has been carried out on a three-layered (solid propellant grain/insulation/metallic casing) cylindrical shell subjected to thermal and internal pressure loads. The structure is idealized using the eight node quadrilateral isoparametric Hermann element for incompressible materials like propellant grain and insulation and regular elements for casing material. The finite element analysis results are found to be in good agreement with the present closed-form solution. This analytical solution can serve as benchmark to finite element solutions. Stress analysis has also been carried out on a three-layer cylindrical shell to examine the piezoelectric effects of one of the layers as composed of piezoelectric layers.

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  How to cite this article:

K. Jayakumar, D. Yadav and B. Nageswara Rao , 2006. A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads. Trends in Applied Sciences Research, 1: 386-401.

DOI: 10.3923/tasr.2006.386.401

URL: https://scialert.net/abstract/?doi=tasr.2006.386.401

Introduction

Piezoelectric materials have many applications in various fields such as: smart structures, electric resonators, filters, actuators, sensors and more. The electro-mechanical response of piezoelectric materials is complex as it involves a mechanical response, an electrical response and a mutual coupling between the mechanical and electrical domains. Shell type smart structures have constituted the subject of study to fairly good number of researchers in the aerospace industry (Tzou et al., 1989, 1991, 1992, 1994a, b). Using the first-order shear deformation theory, Miller and Abramovich (1995) have made analytical studies on thick shell having distributed self-sensing piezoelectric actuators. Cheng and Shen (1996a, 1997), Cheng et al. (1996b) Tarn (2002) and Jayakumar et al. (2006) have examined piezoelectric circular cylindrical shell problems. Smart/intelligent materials and structures refer to structures with surface-mounted or embedded sensors and actuators. Figure 1 shows a typical smart laminated cylindrical shell. In most MEMS (micro electro-mechanical systems) applications, multi-layer piezoelectric structures are being considered as thin structures and edge effects are neglected. Paul (1966) and Paul and Raju (1982) computed frequencies for solid cylinders, whereas Paul and Venkatesan (1987) have extended the analysis including hollow cylinders. Balamurugan and Narayan (2001) have carried out the coupled analysis of piezo-laminated plate and piezo-laminated curvilinear shell structures and their vibration control performance considering the random response to random input loads. Adelman and Stavsky (1973) and Adelman (1975) have considered the coupling between the constitutive relation and the charge equations of electrostatics for the analysis of laminated composite cylinders. Mitchell and Reddy (1995) have obtained a power series solution for an axisymmetric composite cylindrical truss type structure, which are radially polarised. Chen and Shen (1997) have studied linear buckling of piezoelectric circular cylindrical shells of infinite length subjected to axisymmetric external pressure and electrical field. In the solution of Bhaskar and Varadan (1993) for an n-layer cylindrical shell, the 6n boundary and interface conditions yield a system of 6n algebraic equations to determine the 6n unknown coefficients. The method becomes complex for a cylindrical with very large number of layers.

Motivated by the work of the above researchers, an attempt is made to obtain a closed-form solution for a multi-layer cylindrical shell subjected to electro-thermo-mechanical loads utilizing the stress formulation approach. In the present formulation, each layer is idealised as a single piezoelectric circular cylinder under internal and external pressure. The interface pressures in the multi-layer cylindrical shell are evaluated assuming the continuity of the displacements at the interface. From the known interface pressure values, each layer is analysed utilizing the derived closed-form solution for a single cylindrical shell. Finite element analysis results on a three-layer cylindrical shell subjected to internal pressure are found to be in good agreement with the closed-form solution. The present analytical solution can serve as benchmark to finite element solutions. The present formulation can be applied to cylindrically anisotropic materials (electro-thermo-elastic coupling). Piezoelectric effects are also examined through the stress analysis considering one of the layers as composed of piezoelectric material.

Analytical Solution for a Single-layer Piezoelectric Cylindrical Shell
An elasto-electro-thermo analysis of generalised plane-strain of a piezoelectric circular cylindrical shell (having a and b as inner and outer radii) under thermal and pressure load is presented here (Fig. 1). This corresponds to the case of rotational symmetric loading of a piezoelectric right circular cylinder of sufficiently high ratio of length to diameter to justify the assumption of plane-strain (i.e., εz = constant). Referred to the cylindrical (r, θ, z) coordinates, the piezoelectric material is cylindrically anisotropic of the most general kind (Nye, 1957). For mathematical simplicity, the strain-displacement relations, the equations of equilibrium, the equations of electrostatics are expressed in cylindrical coordinate system. In this system, u, v and w are the displacement components along the radial (r), circumferential (θ) and axial (z) directions of the cylindrical shell (Fig. 2). σr, σθ, σz, are the normal stress components, whereas τθz, τ, τrz, are the transverse (shear) stress components. εr, εθ, εz, are the normal strain components and γθz, γ, γrz are shear strain components. The details of stress analysis for a piezoelectric circular cylindrical shell under thermal and pressure loads are presented below.

The constitutive equations are written in the form

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(1)

Where, the strain and stress components are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

The electric displacements and electric field components are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
Fig. 1: A typical smart laminated cylindrical shell

The coefficients of thermal expansion measured at constant electric field and the pyroelectric coefficients measured at constant stress are {α} = {α1, α2, α3, α4, α5, α6}T and {pσ} = {pσ1, pσ2, pσ3}T. ΔT is the temperature change, [S] is a 6x6 elastic compliance matrix, whose elements sij are measured at a constant electric field and constant temperature. Elements Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads in the 3x6 matrix of [dp] correspond to the coefficients of converse piezoelectric effect measured at constant temperature. Elements κij in the 3x3 matrix of [κ] correspond to the permittivity constants measured at constant stress and constant temperature.

For axial symmetry with v ≠ 0, the displacement components are independent of θ and hence, the strain-displacement relations for the cylindrical shell are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(2)

The equations of equilibrium for the generalized plane-strain conditions in the absence of body forces are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(3)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
Fig. 2: Cylindrical coordinate system and six stress components on an infinitesimal volume of the axisymmetric continua
Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(4)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(5)

The equations of electrostatics without free charges are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(6)

The electric field components for this problem are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(7)

Here, φ is the electric potential.

The solutions of Eq. (4) to (5) are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(8)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(9)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(10)

Here, ci’s (I = 1, 2, 3) are arbitrary constants, which are to be determined from the boundary conditions.

The end conditions require that the stress resultants over the cross-section reduce to an axial force (Pz) and a torque (Mt) such that (Tarn, 2002):

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(11)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(12)

When the cylindrical shell is subjected to electromechanical loadings that do not vary axially, the resultant shear forces and moments over a cross-section, are zero identically. When the cylindrical shell is subjected to in-plane and anti-plane shears as well as uniform electric charges or voltages on the inner and outer surfaces, the boundary conditions are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(13)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(14)

Where, the prescribed loads must satisfy the conditions: a2 τa = b2 τb and aSa = bSb for static equilibrium. Existence of an electrostatic solution requires that aDa = bDb. Hence, the arbitrary constants in Eq. 8 to 10 become:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(15)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(16)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(17)

From the radial component of the electric displacement (Dr) in equation (10) and the circumferential and the axial components of the electric field (Eθ and Ez) in Eq. (7), the radial component of the electric field (Er) can be expressed in terms of stress components and the radial electric displacement (Dr) utilizing the constitutive relations (1). It is also possible to express the strain components as well as the circumferential and axial electric displacement components (Dθ and Dz) in terms of the stress components.

The stress-strain relations (1) is modified in the following form:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(18)

Here, the elements in the 6x6 matrix of [Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads] are: Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads. The elements in the 6x1 matrix {α*} are: α*I = αi – βip1σ. The elements in the 6x1 vector of {β} are: Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

For simplicity Eq. 11 is rewritten in the form:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(19)

Here,

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Proper care is taken while arranging the elements of the 6x6 matrix of [Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads] to define the elements aij, bij, cij and dij in the 3x3 matrices of A, B, C and D. Since, the stress components and for the problem under consideration is known from Eq. 8 and 9, the stress-strain relation (19) is further modified to:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(20)

Where Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads,

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

For the present case of generalized plane strain condition,

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(21)

From Eq. 10, one can write

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(22)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(23)

Here, Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads, Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads, Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads and Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads are the elements of the matrices Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads, Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads, Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads and Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads, respectively.

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Using Eq. 3 for σθ and Eq. 2 for εr and εθ in Eq. 12 and 23, one can obtain coupled first-order differential equations for σr and u. After eliminating u from these differential equations, a second-order differential equation in σr is obtained as:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(24)

The associated boundary conditions for the differential Eq. 24 are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(25)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(26)

Here, pa and pb are the applied internal and external pressures, respectively. a and b, are inner and outer radii of the cylindrical shell. Other constants in Eq. 24 are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

The solution of Eq. 24 to 26 can be written in the form:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(27)

Where,

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

m1 and m2 are the roots of the characteristic equation: m2 + L1m + L2 = 0. If the roots of the characteristic equation are not equal to unity, then Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads and η3 = 0. Otherwise η1 = 0 and Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads.

After determining the radial stress component (σr), σθ is obtained directly from the equilibrium Eq. 3. The circumferential strain (εθ) can be obtained from Eq. 22. Using the strain-displacement relation (2) for εθ, the radial displacement u is determined. All other stress and strain components can be obtained directly from the modified stress-strain relation (20). From the known stresses and strains, the radial component of the electric field (Er) can be evaluated for the specified constant c3(≡ aDa = bDb) to the radial electric displacement (Dr) in Eq. 10. The circumferential and the axial electric displacement components (Dθ, Dz) can be evaluated directly from the constitutive relation (1). Since, the radial component of the electric field (Er) is a function of stress components and the radial electric displacement (Dr), the potential difference Δφ(≡φa – φb), can be worked out by integrating Er with respect to r from r = a to r = b. Here φa and φb are the electrical potentials at inner (r = a) and outer (r = b) surfaces of cylindrical shell.

The stress and strain components are independent of z in generalized plane-strain and torsion. From the strain-displacement relation (2), the displacement components are found to be dependent on z, which can be expressed in the form:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(28)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(29)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(30)

Here, fu, fv, fw, are functions of r alone. Θ, is the twisting angle per unit length of the cylindrical shell. ω, is the rigid-body rotation about the z-axis. The constant, εz0 corresponds to uniform extension and w0 is the rigid-body displacement. In the present study, evaluation of radial displacement (u) is essential to understand the deformation pattern of a piezoelectric cylindrical shell under thermal and pressure loads.

Evaluation of Interface Pressures in a Multi-layered Cylindrical Shell
The formulation and solution developed in the preceding section for a single-layer circular cylindrical shell can be easily extended to the shells composed of multiple layers after evaluating the interface pressures assuming the radial displacement continuity at each interface.

The multi-layer cylindrical shell is composed of n co-axial cylindrical shells. For ith cylinder the inner and outer radii are denoted by Ri and Ri+1, respectively. The interface pressures Pi and Pi+1 acting on this cylinder are specified as the inner and outer surface pressures. The radial displacement for the ith cylinder at the outer surface ui(at r = Ri+1) is expressed as a function of Ri, Ri+1, Pi, Pi+1 and the respective material properties as well as the temperature change and applied charge. Similarly, the radial displacement for the (I + 1)th cylinder at the inner surface ui+1 (at r = Ri+1) is expressed as a function of Ri+1, Ri+2, Pi+1, Pi+2 and the respective material properties, temperature change and applied electric charge.

At the interface of the cylinders (at r = Ri+1), this condition provides the relation for unknown interface pressures as:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(31)

Where, ai+1,i, ai+1, I+1, ai+1,i+2 and bi+1 are known constants in terms of the geometric and material properties of the ith and (I + 1)th cylinders. By substituting I = 1, 2, 3…, (n – 1) in Eq. (31) one can get interface relations for unknown interface pressures at r = R2, R3,…, Rn. It should be noted that the applied internal pressure (P1 = Pint ernal) is at the inner surface (r = R1) of the multi-layer cylindrical shell and the applied external pressure (Pn+1 = Pexternal) is at the outer surface (r = Rn+1) of the multi-layer cylindrical shell.

The interface pressures P2, P3,…, Pn are evaluated by solving the matrix equation:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(32)

Where, the size of the tri-diagonal square matrix ⌊aij⌋ is (n+1) x (n+1). Size of column matrices {Pj} and {bj} is (n+1)x1.

After determining Pj’s (I = 1, 2,…, (n+1)) from Eq. 32 for the applied internal and external pressures and also for the uniform temperature change (ΔT) and applied surface charge, the stress, strains and the radial displacements can be determined from the derived solution in section-2 for a single piezoelectric cylindrical shell.

Results and Discussion

The solution of the problem is examined by considering the test data on rocket motors, whose casings are made of metallic/composite materials. It is further verified by the finite element analysis of the cylindrical shell portion of a solid rocket motor composed of solid propellant grain followed by insulation and metallic casing (Fig. 3). Finally, the effect of piezoelectric response is examined for a multi-layer cylindrical shell subjected to electro-thermo-mechanical loads.

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
Fig. 3: Cross-section of a typical rocket motor. A) Chamber; B) Head end dome; C) Nozzle; D) Igniter; E) Nozzle convergent portion; F) Nozzle divergent portion; G) Port; H) Inhibitor; I) Nozzle throat insert; J) Lining; K) Insulation; L) Propellant; M) Nozzle exit plane; N) SITVC system; O) Segment joint

Table 1: Experimental stress analysis and comparison of analytical results on the hydro-burst pressure tested AFNOR 15CDV6 steel rocket motor case
Inner radius, Ri = 103.3 mm; Outer radius, RO = 105.9 mm
Young’s modulus, E = 203.558 GPa; Poisson’s ratio, v = 0.3
Ultimate tensile strength, σULT = 1030 MPa; 0.2% of proof stress, σYS = 915 MPa.
The stress-strain curve is represented by:
Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads where εo = 0.005 and n = 3.2825.
Actual burst pressure of two rocket motors (Beena et al., 1995): 28.86 MPa, 29.59 MPa.
(a) Experimental stress analysis
Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

(b) Comparison of analytical and experimental results
Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
+Results in parenthesis corresponds to test results

A comparative study is made considering test data on rocket motors (Beena et al., 1995), whose casing is made of AFNOR 15CDV6 steel. The pressure chambers of these motors, having diameter about 200 mm were fabricated and hydro-burst pressure tested. Since, the cylindrical portion of the motor is known to be stressed maximum under internal pressure strain gauges are mounted mostly in that portion. Using the stress-strain data from the tensile tests, stresses induced in the motor case at various pressure levels are computed for the recorded strains. Stress analysis has been carried out using the properties of the material given in Table 1. Comparison of analytical and experimental results is also given in Table 1. The analytical results of hoop stress (σθ) and hoop strain (εθ) are found to reasonably in good agreement with test results. The discrepancy in the results of meridional (σz) is mainly due to the assumption on the meridional strain (εz) assumed as zero in the present analytical solution, which is valid for cylindrical vessels having large length to diameter ratio. The measured longitudinal strains (εz) in the cylindrical portion of the rocket motor case presented in Table 1 were non-zero values. To bridge the gap between the present analytical solution and the measured values, a value of E εz is superimposed to the meridional stress (σz). With this the meridional stress (σz) values are reasonably in good agreement with test results. The failure pressure estimates of the rocket motors using the Faupel’s formula:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
(33)

This is close to the actual burst pressure values (28.86 and 29.59 MPa) of the two rocket motors. The rocket motors after the burst test are shown in Fig. 4.

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
Fig. 4: AFNOR 15CDV6 steel rocket motors after the hydro-burst test (Beena et al., 1995)

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
Fig. 5: A typical composite motor case

To have low mass in upper stage systems for enhancement of payload capacity of launch vehicles, upper stage rocket motors having casing made of composite materials have been designed. Strains and displacements on the outer surface of the casing were measured during proof-pressure tests. From the overall orthotropic properties in the cylindrical portion of a typical composite motor case (Fig. 5), elastic compliances are determined for stress analysis to have comparison with the proof pressure test results. Appendix gives the overall orthotropic properties and the corresponding elastic compliances considered in the stress analysis of the cylindrical shell portion of the composite motor case. The inner radius of the casing is about 980 mm and the average thickness of the cylindrical shell is around 13 mm. The measured hoop strains under proof pressure of 4.905 MPa, in the cylindrical portion of composite motor casing is 12000 (μ) whereas the present analysis gives the results as 12282 (μ). The measured radial displacement is 11.5 mm, whereas the analysis result is 11.917 mm. The analytical results are found to be in good agreement with the measured values of the proof pressure test.

The solution of the multi-layer cylindrical shell problem is examined considering the cylindrical shell portion of a solid propellant rocket motor. Stress analysis has been carried out on a three-layered (solid propellant grain/insulation/metallic casing) cylindrical shell subjected to thermal loads and pressure loads. This structure is also idealized using the eight node quadrilateral, isoparametric Hermann element of MSC/MARC® for elements of incompressible materials viz., propellant grain, insulation and regular elements for the casing material. Both axisymmetric and plane strain models were used in the comparative study. Table 2 gives the material properties for the three-layers and also provides comparison of the results for the three-layered cylindrical shell under internal pressure (4.905 MPa) and thermal load (-38°C). The finite element analysis results are in good agreement with the present analysis results.

Stress analysis is performed on a three-layered cylindrical shell that has a central piezoelectric (PZT5A) layer. Properties of the materials considered in the analysis are given in the appendix. The inner and outer layers are composed of carbon-epoxy.

Table 2: Comparison of analytical and finite element analysis results of a three-layer cylindrical shell subjected to thermal and pressure loads
Inner radius of the propellant grain, R1 = 500 mm
Inner radius of the insulation, R2 = 1389 mm
Inner radius of the casing, R3 = 1394 mm
Outer radius of the casing, R4 = 1401.8 mm
Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Table 3: Stress analysis results for a multi-layer piezoelectric cylindrical shell subjected to internal pressure of 1MPa using the present formulation
Pressure vessel composed of Carbon epoxy-PZT5A-Carbon epoxy.
- Inner radius = 1000 mm
- Thickness of first layer of Carbon epoxy = 10 mm
- Thickness of second layer of PZT5A = 5 mm
- Thickness of third layer of Carbon epoxy = 10 mm
Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads
The electrical potential difference across the piezoelectric layer is 582.24 Volts

Table 4: Statistical analysis results on voltage output for a multi-layer piezoelectric cylindrical shell subjected to internal pressure of 1 MPa using the present formulation
Pressure vessel composed of Carbon epoxy-PZT5A-Carbon epoxy.
- Inner radius =1000 mm
- Thickness of first layer of Carbon epoxy = 10 mm
- Thickness of second layer of PZT5A = 5 mm
- Thickness of third layer of Carbon epoxy = 10 mm
Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

The stress analysis results, when the cylinder is subjected to internal pressure of 1 MPa are given in Table 3. The electrical potential difference across the thickness of the piezoelectric layer is 582.24 Volts.

The use of probabilistic concepts to assess the reliability of structures, require the knowledge about the statistical scatter of parameters used in the models. In the design process it is rarely the case that only one variable parameter is of interest. It is with the combinations of many variable parameters that we must deal. Statistical analysis has been carried out considering a three-layer cylindrical shell that has central piezoelectric (PZT5A) layer. Material properties for carbon/epoxy and PZT5A, given in the Appendix are assumed as mean values in the present statistical analysis. Table 4 gives the dimensions of the three-layer cylindrical shell.

For the present problem, the electric potential difference Δφ(≡ φa – φb) is functionally related to the material properties Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads as well as the specified change in temperature (ΔT), electric displacement (Da) and internal pressure (pa) and external pressure (pb). The temperature change, electric displacements, internal pressure and external pressure specified from Table 3 while evaluating Δφ are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Appendix

Material Properties
Composite motor casing
The overall orthotropic properties in the cylindrical shell region of the composite motor casing are:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Compliances:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Poisson’s ratio and coefficient of thermal expansion:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Carbon-Epoxy
Compliances:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Coefficient of thermal expansion:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Piezoelectric material PZT5A
Compliances:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Piezoelectric strain coefficients:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

Electrical permittivity:

Image for - A Multi-layer Cylindrical Shell Under Electro-thermo-mechanical Loads

The inner and outer radii are: a = 1.01 m and b = 1.015 m.

We perform the random analysis for the piezoelectric layer, as output voltage is our prime variable of interest. The output voltage statistics is dependent on the interfacial pressure (pa = 0.5074 Mpa and pb = 0.4721 MPa). The interfacial pressures developed are linearly dependent on the applied internal pressure of 1 MPa for the multi-layer cylindrical shell. For the statistical analysis, variations in material properties are only considered.

Following the multivariate concept (Haugen, 1968; Bowker and Lieberman, 1972; Jeyakumar et al., 2005), the expected values of the output potential difference and its coefficient of variation are presented in Table 4. The coefficient of variation in the material properties varies from 1 to 10%. From the statistical analysis it was also seen that potential difference is sensitive to the material properties S33, d12 and κ11.

Conclusions

Analytical solution is obtained for a generalized plain-strain of a multi-layer cylindrical shell subjected to electro-thermo-mechanical loads. The solution of the problem is verified by considering the test data on the rocket motors, finite element analysis results of a three-layer solid propellant rocket motor and a three-layer cylindrical shell having a central piezoelectric layer, the other layers being composed of carbon-epoxy. The electrical potential difference across the thickness of the piezoelectric layer is computed from the stress analysis results. The present analytical solution can serve as benchmark to finite element solutions.

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