CHARACTERIZATION OF MATERIALS IN AN ELASTO-PLASTIC THERMO ORTHOTROPIC SHELL Pankaj Thakur1and Monika Sethi21,2Department of mathematics, ICFAI University Himachal Pradesh, IndiaKeywords: spherical shells, temperature, analytical modeling, orthotropic material.Abstract: The purpose of this paper is to present the study of characterization of materials in an orthotropic shell subjected to temperature gradient by using Seth’s transition theory. It has been observed that the spherical shell made of orthotropic material required higher pressure to yield at the inner surface as compared to a shell made of isotropic material at the initial yielding but reverse results are formed for the fully-plastic state.
Thermal effect decreased value of hoop stress at the internal surface of the spherical shell made of isotropic material as well as an orthotropic material. The value of hoop stress for an isotropic spherical shell has a maximum at the inner surface as compared to a spherical shell made of orthotropic materials and is on the safe side of the design.
1. Introduction Material science is constantly evolving; new materials with complex properties appear. Many classic constructional materials exhibit structural anisotropic properties and different resistant. Different resistant is a material whose deformational and structural properties depend on the type of stress state. Orthotropic materials have material properties that differ along three mutually-orthogonal two-fold axes of rotational symmetry. Orthotropic materials are the subset of anisotropic materials because their properties change when measured from different directions. Shells formed of metal or of other solid materials are expected to find numerous technical and industrial applications e.g. metal shells might be used as inertial confinement fusion targets, or be employed as containers for phase-change heat-storage media or DOD large antennas and mirrors. Spherical shells are extensively used in many structural engineering applications such as storage of gas, liquid, aerospace vehicles, ballistic missile bulkhead and submarines, adaptive smart membranes and active shells, laminated transducers and sensors and soon. Guz  and Savin  et al. analyzed stress state in a sloping spherical shell with a curved hole by using the method of boundary form disturbance with the mechanical loading. Shnerenko /6/ discuss stresses in a spherical shell with a perfectly rigid, curved ring. Podstrigach et al.  investigated thermal stress state of sloping shells around circular holes in. Gupta et al.  discussed the problem of elastic-plastic transition in an orthotropic shell under internal pressure by using Seth’s transition theory. Peter et al.  discuss the problem of the layered wooden shell using orthotropic elastoplastic model for multi-axial loading of clear spruce wood. Nie et al.  investigated asymptotic solution for nonlinear buckling of orthotropic Shells on Elastic Foundation. Thakur  successfully analyzed creep transition stresses of an orthotropic Thick-walled cylinder under combined axial load under internal pressure. Maksimyuk et al.  stress-strain state of thin spherical, conical, and ellipsoidal shells made of nonlinear elastic orthotropic composites by using the finite-difference method. Alexandr et al.  investigated the influence of temperature differences for the analysis of thin orthotropic cylindrical shell. In this research paper, we investigated characterization of materials in an elasto-plastic thermo orthotropic shell by using Seth transition theory. This theory  does not acquire any assumptions like incompressibility condition, yield condition and thus possess and solves a more general problem from which cases pertaining to the above assumptions can be worked out. According to this theory, we use the concept of generalized strain measure and asymptotic solution at the turning points of the differential equations defining the deformed field and successfully applied to a large number of problems Seth [3, 7, 8]; Gupta ; Thakur et al. [14, 15, 17, 18. 20].2. Governing Equations416623518351500Let us consider a spherical shell with the central bore of radius a and external radius b respectively, subjected to uniform pressure on the inner surface of the shell say p. The thickness of the spherical shell assumed to be constant. The temperature at the central bore of the spherical shell is at r = a, shown as shown in Figure 1. Displacement coordinates: The components of displacement in spherical coordinatesare taken as: ; v = 0; (1)3521710364812Figure 1 Geometry of orthotropic spherical shell00Figure 1 Geometry of orthotropic spherical shellwhere u, v, w (displacement components); is position function depending on Generalized Strain Components: The generalized components of strain are given by [3, 7] as:,, (2)where r, , be spherical coordinates; n be measure and Stress-strain relation: The thermo-elastic constitutive equations for the material are given  as: (3) where and are stress and strain tensor; elastic constants and being the coefficient of thermal expansion, and is the temperature. The temperature satisfying Laplace eq. with boundary condition:at r = a, at r = b (4)where is constant, is given by : (5)Substituting from Eq. (2) into Eq. (3), we get: ; ; (6)Equation of equilibrium: The equations of equilibrium are : ;; (7)Substituting Eq. (5) in Eq. (7), we see that the equations of equilibrium are all satisfied except: and (8) From Eq. (8), the only case of interest is , which gives from Eq. (6) if and ; then Eq. (8) becomes: (9) Critical points: By substituting Eq. (6) into Eq. (9) and using Eq. (5), we obtain a non-linear differential equation with respect to: (10) where and P is function of and is function of r only. Transition points: The transition points of in Eq. (10) are and Boundary condition: The boundary conditions of the problem are given by: r = a , , r = b, (11)where p is pressure applied at the inner surface of the spherical shell.3. Problem SolutionFor finding the plastic stresses distribution, the transition function is taken through the principal stresses (see [ 3, 7, 8, 11, 14, 15, 17, 18, 20]) at the transition point, we define the transition function as: (12) where and be the transition function unction of r only. Taking the logarithmic differentiation of Eq. (12), with respect to r and using Eq. (10), we get (13) Taking the asymptotic value of Eq. (13) as and integrating, we get: (14) where is a constant of integration and . From Eq. (12) and Eq. (14), we have (15)The relation between yielding stress in tension and material constants at the transition range are is given by : ;where Y is the yielding stress. Substituting the value of yielding stress in tension Y in Eq. (15), we get: (16) Using boundary condition from Eq. (11) in Eq. (16), we get . Now substituting the value of constant A in Eq. (16), we get: (17)Investigation Pressure for spherical shell: Using boundary condition Eq. (11) in Eq. (17), we have (18) Substituting Eq. (17) in Eq. (9), we get: (19) Initial Yielding: From Eq. (19), it has been seen that is maximum at the inner surface (that is at r = a), therefore yielding of the spherical shell will take place at the inner surface and Eq. (19) can be written as: (20) where is the initial yielding stress. Substituting the value of Y in term of Y* from Eq. (20) into Eqs. (17) ” (19), we get: And (21) Eq. (21) gives orthotropic transitional stresses in a shell. Non-dimensional components: We introduce the following non-dimensional components:,,,, and . Eq. (21) in non-dimensional form become: ;and (22)where Fully-Plastic state: For fully-plastic case  has investigated following relation relationship between the constants , and. Eq. (22) become for fully-plastic state:;and (23)where Isotropic case: For isotropic material the material constants reduces to two only ;, and , , , and . Eq. (15) for isotropic case becomes:;and (24)where and E is the Young’s modulus. Fully plastic state for isotropic case: For fully plastic state (i.e. ), Eq. (24) becomes:;and (25) where and E is the Young’s modulus. Table 1. Influence Temperature distribution.Radii ratio Ro (degree F) Material c11 c12 c21 c22 Temperature distribution(T=T0/Y*) initial yielding Temperature distribution(= T0/Y*) fully-plastic state0.51000 Orthotropic(barite) 0.8941 0.4614 0.4614 0.7842 0.02419752 0.006146Isotropic(steel) 5.326 3.688 3.688 5.31 0.01537739 0.000150.5 10000 Orthotropic(barite) 0.8941 0.4614 0.4614 0.7842 0.241975171 0.061458Isotropic(steel) 5.326 3.688 3.688 5.31 0.153773940.0015024. Numerical Discussion To see the effect of temperature gradient in an orthotropic shell following values have been considered: = 1,0000 F, 10,0000 F; = degree F-1for Methyl Methacrylate /9 /. As a numerical example elastic constant cij (unit of 1011 N/m2) have been given for isotropic material (say steel); c11 = 5.326 ; c12 = 3.688; c13 = 3.688 ; c21= 3.688 ; c22 = 5.3 ; c23 = 3.688 with density 7.849 and orthotropic material (say barite); c11 = 0.8941; c12 = 0.4614; c13 = 0.2691; c21= 0.4614; c22 = 0.7842; c23 = 0.2676 with density 4.4 respectively. From Table 1, we calculated the value influence of temperature gradient in spherical shell made of an orthotropic and isotropic materials at the initial yielding and fully-plastic state along the radii ratio Ro = a/b. Curves are produced, between influence of temperature gradient along the radii ratio Ro = a/b (see Figure 2(a)-(b)) of the spherical shell made of orthotropic as well as isotropic materials with different values of . It is also observed from (Figure 2(a)-(b)) that spherical shell made of orthotropic material required large value of influence of temperature as compared to the spherical shell made of an isotropic material at the initial yielding stage. Curves are drawn between pressure along the radii ratio Ro = a/b (see Figure 3(a)-(b)) for spherical shell made of orthotropic and isotropic material at different influence of temperature gradient T = 0.241975171; 0.02419752; 0.061458; 0.006146 (i.e. orthotropic case) and 0.15377394; 0.01537739 0.001502; 0.001502 (i.e. isotropic case) at the initial and fully-plastic state. It has been observed from (Figure 3(a)-(b)) that the spherical shell made of orthotropic material required higher pressure to yield at the inner surface as compared to a shell made of isotropic material at the initial yielding but reverse results are formed for the fully-plastic state. Curves are produced between stresses distribution along the radii ratio R = r/b (see Figure 4(a)-(b)) for spherical shell made of orthotropic and isotropic material at the different influence of temperature gradient. It is also observed from (Figure 4(a)-(b)) that the hoop stress has a maximum value at the internal surface of the spherical shell made of isotropic material as compared to the shell made of an orthotropic material at the initial and fully-plastic state. The authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest in the subject matter or materials discussed in this manuscript.REFERENCESTimoshenko, S.P., et al., Theory of elasticity , McGraw-Hill, Book Company, Third Edition, 1951.Sokolinikoff, I.S., Mathematical theory of Elasticity, Second edition, McGraw-Hill Book Co., New York, 1956.Seth, B.R. (1962), Transition theory of elastic-plastic deformation, creep and relaxation, Nature, 195(1962), pp. 896-897, doi: 10.1038/195896a0.Guz O. M., An Approximate method of determining the stress concentration near curved holes in shells, Prikl. Mekh. , 8(1962), 6, pp. 605″612.Savin G. N., et al., Stress state around curved holes in shells, Izv. Akad. Nauk SSSR, Mekh. Mashinostr, 8(1962), 6, pp. 96″105 .Shnerenko K. I., Stress distribution in a spherical shell around reinforced curved holes, Prikl. Mekh.,. 11(1966), pp. 55″62.Seth, B.R., Measure concept in mechanics, International Journal of Non-linear Mechanics, 1 (1966),1, pp. 35-40.Seth, B.R., Transition condition: the yield condition, International Journal of Non-linear mechanics, 5(1970), 2, pp. 279-285.Levitsky, M., et al., Residual thermal stresses in a solid sphere from a thermosetting material, Jr. of Appl. Mech., Trans of ASME, 43(1975), 3, pp. 651-655.Podstrigach Ya. S., et al., Thermo-elasticity of Thin Shells [in Russian], Naukova Dumka, Kiev, 1978.Gupta S.K. , et al., Elastic-plastic transition in an orthotropic shell under internal pressure, Indian J. of pure appl. Math., 23(1991), 3, pp. 243-250.Peter M. H., et al., Analysis of layered wooden shell using orthotropic elasto-plastic model for multi-axial loading of clear spruce wood, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 21-24, pp. 2661-2685.G. H. Nie, et al., Asymptotic Solution for Nonlinear Buckling of Orthotropic Shells on Elastic Foundation, AIAA Journal,. 47 (2009), 7, pp. 1772-1783..Thakur, P., Elastic-plastic transition stresses in rotating cylinder by finite deformation under steady- state temperature, Thermal Science, 15(2011), 2, pp. 537-543.Thakur, P., Creep transitional stresses of Orthotropic Thick-Walled cylinder under combined axial Load under internal Pressure, Facta Universities Series: Mechanical engineering, 11(2013), 1, pp. 13-18.Maksimyuk V. A. , et al., Nonlinear deformation of thin isotropic and orthotropic shells of revolution with reinforced holes and rigid inclusions, International Applied Mechanics, 49(2013), 6, pp. 685″692.Thakur, P., et al., Thermal creep stresses and strain rates in a circular disc with shaft having variable density, Engineering Computation, 33(2016), 3, pp. 698-712.Thakur, P., et al., Creep strain rates analysis in cylinder under temperature gradient materials by using Seth’s theory, Engineering Computation, 34(2017), 3, pp. 1020-1030.Alexandr A.T. , et al. , The influence of temperature differences for the analysis of thin orthotropic cylindrical shell, Scientific Review ” Engineering and Environmental Sciences, 26 (2017), 4, pp. 528″537.Thakur, P., et al., Thermal creep stress and strain analysis in a non-homogeneous spherical shell, J Theor. and Appl. Mech., Warsaw, 55(2017), 4, pp. 1155-1165. Figures (b) Figure 2 Influence of Temperature gradient in spherical shell made of orthotropic and isotropic material at the initial yielding (a) and fully plastic state (b) along the radii ratio Ro = a/b. (b)Figure 3 Pressure required for initial yielding (a) and fully plastic state (b) along the radii ratio R0 = r/b. (a)(b)Figure 4 Stresses distribution along the radii ratio R = r/b at the initial yielding (a) and fully-plastic state (b).