GANJI NAGESWARA RAO (IN)
NAIK NIRANJAN KRISHNA (IN)
GANJI NAGESWARA RAO (IN)
US4729705A | 1988-03-08 | |||
GB876106A | 1961-08-30 | |||
US5536344A | 1996-07-16 | |||
US4265688A | 1981-05-05 | |||
US6055790A | 2000-05-02 | |||
DE2459608A1 | 1975-08-21 |
1. | Insert assembly of high specific strength for sandwich structures as claimed in claimi wherein insert and potting material extend throughout the thickness of the sandwich structure to form a throughthe thickness configuration. |
2. | Insert assembly of high specific strength for sandwich structures as claimed in claimi wherein insert extends partially from the upper faceplate while the potting material extends throughout the thickness of the sandwich structure to form a fully potted configuration. 3. Insert assembly of high specific strength for sandwich structures as claimed in claimi wherein insert and potting material in an insert assembly extend partially from the upper faceplate through the thickness of the sandwich structure to form a partially potted configuration. 4. Insert assembly of high specific strength for sandwich structures as claimed in claims1,2 with throughthethickness configuration wherein insert materials are selected from • 2D composites • 3D thermoelastic isotropic woven composites • 3D woven composites • 3D woven composites with multiple inserts • 3D functionally gradient woven composites as inserts and their combination. 6. Insert assembly of high specific strength for sandwich structures as claimed in claimsi ,3 with fully potted configuration wherein insert materials are selected from • 2D composites • 3D thermoelastic isotropic woven composites • 3D woven composites • 3D woven composites with multiple inserts • 3D functionally gradient woven composites as inserts and their combination. 7. Insert assembly of high specific strength for sandwich structures as claimed in claimsi ,4 with partially potted configuration wherein insert materials are selected from • 2D composites • 3D thermoelastic isotropic woven composites • 3D woven composites • 3D woven composites with multiple inserts • 3D functionally gradient woven composites as inserts and their combination. 8. Insert assembly of high specific strength for sandwich structures as claimed in claims'! 7 wherein insert is cylindrical in shape with integral/assembled flanges. 9. Insert assembly of high specific strength for sandwich structures as claimed in claims17 wherein potting material is selected from resins such as epoxy, polyester, polyimide and their like. 10. Insert assembly of high specific strength for sandwich structures as claimed in claims17 wherein upper faceplate is selected from material with density of 1700 7800 Kg / m3, Young's modulus of 5 200 GPa, Shear modulus of 2 77 GPa and Poisson's ratio of 0.108 0.35. 11. Insert assembly of high specific strength for sandwich structures as claimed in claims17 wherein lower faceplate is selected from material with density of 1700 7800 Kg / m3, Young's modulus of 5 200 GPa, Shear modulus of 2 77 GPa and Poisson's ratio of 0.108 0.35. 1. |
3. | Insert assembly of high specific strength for sandwich structures as claimed in claims17 wherein the profile of the upper faceplate is flat and curved. 1. |
4. | Insert assembly of high specific strength for sandwich structures as claimed in claims17 wherein the profile of lower faceplate is flat and curved. 1. |
5. | Insert assembly of high specific strength for sandwich structures as claimed in claimi wherein specific strength of the insert enhanced by about 200 % with respect to aluminium and about 500 % with respect to other metals. 1. |
6. | Insert assembly of high specific strength for sandwich structures as claimed in claims17 wherein upper faceplate is selected from materials such as aluminum, aluminum alloys and fiber reinforced plastic composites. 1. |
7. | Insert assembly of high specific strength for sandwich structures as claimed in claims17 wherein lower faceplate is selected from materials such as aluminum, aluminum alloys and fiber reinforced plastic composites. 1. |
8. | Insert assembly of high specific strength for sandwich structures as claimed in claimi wherein the geometrical configurations and materials may be obtained by mapping stress distribution and failure initiation in steps comprising • Establishing frame of reference for sandwich structures with inserts • Applying equilibrium equations, continuity conditions and constitutive relations of the core and the faceplates to obtain 24th order partial differential equation consisting of fundamental variables such as Midplane radial displacement of upper and lower faceplates in radial direction, Midplane circumferential displacement of upper and lower faceplates in circumferential direction, Transverse displacement of upper and lower faceplates, Derivative of transverse displacement with respect to radius of insert assembly of upper and lower faceplates, Derivative of transverse displacement with respect to circumferential direction of upper and lower faceplates and divided by radius of insert assembly, Bending moment resultant of upper and lower faceplates in radial direction, Twisting moment resultant in the plane of radial and circumferential coordinates of upper and lower faceplates, Inplane normal stress resultant in radial direction of upper and lower faceplates, Inplane normal stress resultant in the plane of radial and circumferential coordinates of upper and lower faceplates, Inplane shear stress resultant in the plane of radial and circumferential coordinates of upper and lower faceplates, Transverse shear stress component of core in the plane of radial and circumferential coordinates, Shear stress in circumferential direction on a plane perpendicular to throughthe thickness direction in the core, Derivative of shear stress in circumferential direction on a plane perpendicular to throughthethickness direction in the core with respect to radius of the insert assembly, Derivative of transverse shear stress component of core in the plane of radial and circumferential coordinates with respect to radius of the insert assembly, and further to obtain 24 first order coupled exact differential equations • Applying boundary conditions to the 24 first order coupled exact differential equations and solving two point boundary value problem to obtain stresses, displacements and failure initiation wherein, frame of reference is established based on i. The attachment is infinitely rigid ii. Insert and potting material are an integral part of the core for mathematical formulation iii. Inplane shear stress and inplane normal stresses are negligible in the core material iv. Core material is flexible in nature v. Effective shear modulus of the honeycomb core is considered in modeling vi. Insert assembly is circular in shape vii. Interaction between two adjacent inserts is negligible viii. Interaction between the insert and the honeycomb core along the circumference of the insert assembly is negligible ix. Faceplates are homogeneous and isotropic / quasiisotropic x. Classical plate theory is applicable for the analysis of the faceplates and wherein, the steps to obtain 24 first order coupled exact differential equations involve a. Representing the behavior of the sandwich structure with an insert assembly using a set of plurality of equations based on equilibrium equations, constitutive relations and continuity conditions b. Combining the core and faceplate equations to obtain 24th order governing partial differential equation with 24 unknown fundamental variables c. Rearranging the governing partial differential equation to 24 first order coupled partial differential equations in terms of 24 fundamental variables, their derivatives with respect to circumferential angle and radius using plurality of equations d. Eliminating the dependency of derivatives of circumferential angle in the 24 first order coupled partial differential equations using Fourier expansions to convert them into 24 first order coupled exact differential equations and wherein, stresses, displacements and failure initiation are obtained by I. Specifying 24 boundary conditions with respect to 24 first order coupled exact differential equations with 12 boundary conditions at the interface of attachment and insert and remaining at simply supported outer edge of the insert assembly II. Constituting a two point boundary value problem comprising 24 first order coupled exact differential equations and boundary conditions III. Converting two point boundary value problem into a series of initial value problems by dividing the sandwich structure into a number of segments along radial direction IV. Solving the series of initial value problems numerically using multisegment method of direct integration for 24 fundamental variables at each segment using continuity conditions between two adjacent segments to determine the stresses and displacements throughout the insert assembly for given loading conditions V. Obtaining the failure initiation within the insert assembly using quadratic failure criterion and the corresponding failure initiation load VI. Obtaining transverse, radial and circumferential displacements, throughthethickness normal and shear stress components in the core, induced normal stress resultants, induced shear stress resultants and induced bending moment resultants in the face¬ plates and specific strength of the insert assembly based on geometrical, mechanical and physical properties of the insert assembly and loading conditions. |
and w2), transverse shear stress (rrz), mid-plane radial displacement. of top face-plate (MQ1 ).
mid-plane circumferential displacement of top face-plate (v^). The novel method to reliably map stress distribution is described in the form of f low tliagrams in Figures 4 and 5. The input parameters are: geometry of the insert assembly, elastic properties of different materials used for making the insert assembly and the loading condition. This novel method enables to map the following parameters reliably: * transverse and radial displacements of the insert assembly ■ normal and shear stress components throughout the insert assembly The final governing equations are: The transverse displacement of the core material,
Through-the-thickness normal stress in the core material,
Radial displacement in the core material,
Circumferential displacement in the core material,
here,
All the Ci1 are stiffness constants and are calculated using elastic properties of the material. The normal and shear stress distribution within the entire insert assembly can be obtained using this novel method. The induced stress state can lead to initiation of failure within insert assembly. Initiation of failure is characterized using the following through-the-thickness quadratic interaction failure criterion. ,
Here, σ. - Through-the-thickness normal stress
Tn, τθz - Transverse shear stresses Zt - Through-the-thickness normal strength Sn, S^ - Transverse shear strengths I - Failure function Failure function, I = 1 indicates initiation of failure. In the above method the stress distribution is mapped and failure initiation is obtained in steps comprising ■ Establishing frame of reference for sandwich structures with inserts ■ Applying equilibrium equations, continuity conditions and constitutive relations of the core and the face-plates to obtain 24th order partial differential equation consisting of fundamental variables such as
- Mid-plane radial displacement of upper and lower face-plates in radial direction,
- Mid-plane circumferential displacement of upper and lower face-plates in circumferential direction,
- Transverse displacement of upper and lower face-plates,
- Derivative of transverse displacement with respect to radius of insert assembly of upper and lower face-plates,
- Derivative of transverse displacement with respect to circumferential direction of upper and lower face-plates and divided by radius of insert assembly, - Bending moment resultant of upper and lower face-plates in radial direction,
- Twisting moment resultant in the plane of radial and circumferential coordinates of upper and lower face-plates,
- In-plane normal stress resultant in radial direction of upper and lower face-plates,
- In-plane normal stress resultant in the plane of radial and circumferential coordinates of upper and lower face-plates,
- In-plane shear stress resultant in the plane of radial and circumferential coordinates of upper and lower face-plates,
- Transverse shear stress component of core in the plane of radial and circumferential coordinates,
- Shear stress in circumferential direction on a plane perpendicular to through-the- thickness direction in the core,
- Derivative of shear stress in circumferential direction on a plane perpendicular to through-the-thickness direction in the core with respect to radius of the insert assembly,
- Derivative of transverse shear stress component of core in the plane of radial and circumferential coordinates with respect to radius of the insert assembly, and further to obtain 24 first order coupled exact differential equations ■ Applying boundary conditions to the 24 first order coupled exact differential equations and solving two point boundary value problem to obtain stresses (equation 2), displacements (equations 1 ,3,4) and failure initiation ( equation 5) wherein, frame of reference is established based on i. The attachment is infinitely rigid ii. Insert and potting material are an integral part of the core for mathematical formulation iii. In-plane shear stress and in-plane normal stresses are neglected in the core material iv. Core material is flexible in nature v. Effective shear modulus of the honeycomb core is considered in modeling vi. Insert assembly is circular in shape vii. Interaction between two adjacent inserts is negligible viii. Interaction between the insert and the honeycomb core along the circumference of the insert assembly is negligible ix. Face-plates are homogeneous and isotropic / quasi-isotropic x. Classical plate theory is applicable for the analysis of the face-plates and wherein, the steps to obtain 24 first order coupled exact differential equations involve a. Representing the behavior of the sandwich structure with an insert assembly using a set of plurality of equations based on equilibrium equations, constitutive relations and continuity conditions b. Combining the core and face-plate equations to obtain 24th order governing partial differential equation with 24 unknown fundamental variables c. Rearranging the governing partial differential equation to 24 first order coupled partial differential equations in terms of 24 fundamental variables, their derivatives with respect to circumferential angle and radius using plurality of equations d. Eliminating the dependency of derivatives of circumferential angle in the 24 first order coupled partial differential equations using Fourier expansions to convert them into 24 first order coupled exact differential equations and wherein, stresses, displacements and failure initiation are obtained by i. Specifying 24 boundary conditions with respect to 24 first order coupled exact differential equations with 12 boundary conditions at the interface of attachment and insert and remaining at simply supported outer edge of the insert assembly, ii. Constituting a two point boundary value problem comprising 24 first order coupled exact differential equations and boundary conditions iii. Converting two point boundary value problem into a series of initial value problems by dividing the sandwich structure into a number of segments along radial direction iv. Solving the series of initial value problems numerically using multi-segment method of direct integration for 24 fundamental variables at each segment using continuity conditions between two adjacent segments to determine the stresses and displacements throughout the insert assembly for given loading conditions v. Obtaining the failure initiation within the insert assembly using quadratic failure criterion and the corresponding failure initiation load vi. Obtaining transverse, radial and circumferential displacements (equations 1 ,3,4), through- the-thickness normal (equation 2) and shear stress components in the core, induced normal stress resultants, induced shear stress resultants and induced bending moment resultants in the face-plates and specific strength of the insert assembly based on geometrical, mechanical and physical properties of the insert assembly and loading conditions. The method described above was used to obtain geometrical configuration of the inserts and the displacement and failure initiation were predicted and experimentally verified. The present work establishes the superiority of the inserts of the present invention over inserts of prior art. Example 1 Experimental studies ♦ Fabrication of Through-the-Thickness Insert Assembly The insert assembly comprises of six constituents. They are: insert, potting material, foam core, lower face-plate, upper face-plate and the attachment. Lower face-plate and the upper face¬ plate are made of woven fabric E glass and epoxy resin using matched-die molding technique. The core is made of polyurethane foam. The attachment is made of mild steel. The material used for composite inserts is glass. The potting material is epoxy resin. Three insert assemblies were constructed using the above with aluminum, 2D woven composite and 3D woven composite as insert materials. ♦ Measurement of load, displacements and failure initiation ■ The insert assembly was placed on a support ring and then located on Hounsfield Test Equipment - 450 KS, 50 KN UTM. ■ Compressive load was applied through the attachment on to the insert assembly. ■ The displacement of the attachment, lower face-plate, upper face-plate and the corresponding load were measured at loading rate of 0.25 mm/min. ■ Failure initiation of the insert assembly is obtained from sudden change in the load- displacement plot.
♦ Experimental Results
Transverse displacement as a function of compressive load for through-the-thickness inserts is presented in Figures 6-8. For the same geometrical configurations and material properties (Tables 1 and 2), analytically obtained transverse displacement plots, compressive load at failure initiation and specific strength of inserts are presented in Figures 6-8 and Tables 3 and 4. The compressive loading was applied until the failure initiation took place.
Failure functions (equation 5) are plotted as a function of compressive load in Figure 9. It is observed from Tables 3 and 4 that the compressive load at failure is higher for the case of 3D woven composite compared to the aluminum as insert material. The specific strength of insert is significantly higher for the case of 3D woven composite compared to the aluminum insert case. For the case of 2D woven composite insert, it is in between aluminum and 3D woven composite.
Table 1 : Geometrical configuration of the insert assembly for the experimental studies.
Table 2: Material properties of foam core sandwich structure with inserts: used for experimental studies.
Table 3: Specific strength of through-the-thickness inserts with different materials: experimental studies. Volume of insert, V=1.22x 10"6 m3 Density of aluminum = 2800 Kg /m3 Density of 2D woven composite = 1700 Kg/m3 Density of 3D woven composite = 1700 Kg/m3
Table 4: Specific strength of through-the-thickness inserts with different materials: analytical predictions for experimental configurations. Volume of insert, V=1.22x 10'6 m3 3D woven composite insert: Zt = 45 MPa, Srz = 36 MPa 2D woven composite insert : Zt = 27 MPa, Srz = 36 MPa Aluminum insert : Zt = 150 MPa, Srz = 30 Mpa
Example 2
Comparison of through-the-thickness inserts of present invention with inserts of prior art
Using the method of the present invention, compressive load at failure initiation, failure function and specific strength of inserts are. mapped for the prior art disclosed in US Patent 50532285 (corrugated aluminum insert) and for the insert (3D woven composite insert) of the present invention with the same geometry as used for the prior art (R = 30 mm). The results are given in Figure 10 and Table 5. Further, the geometry of the insert of the present invention was modified (R = 10 mm) to reduce the insert assembly weight. With such a modified configuration, 10 compressive load at failure initiation, failure function and specific strength of inserts are mapped. From Figure 10 and Table 5, it is established that the inserts of the present invention are having higher specific strength compared to the insert of prior art. Table 5: Specific strength of through-the-thickness inserts with corrugated aluminum and 3D woven composite: analytical studies. 15 Density of Corrugated aluminum = 459 Kg / m3 Density of 3D woven composites = 1700 Kg / m3
Example 3 Analytical studies with different insert materials Using the experimentally validated method for mapping of stresses, displacements and failure initiation, the results for aluminum, 2D woven composite, 3D thermoelastic isotropic woven composite, 3D woven composite, 3D woven composite with multiple inserts and 3D functionally gradient woven composite are obtained for the geometrical configuration of the insert assembly as given in Table 6. Material properties of the insert assembly are presented in Tables 7 and 8. Maximum displacement, maximum normal stress and maximum shear stress corresponding to maximum compressive load at failure initiation are presented in Table 9. Specific strength of insert for different materials is also presented in Table 9. Failure function as a function of compressive load for different insert materials is presented in Figure 11. Percentage increase in specific strength of insert and percentage decrease in mass of insert for the composite inserts compared to the aluminum insert is presented in Table 9. It is observed that there is significant increase in specific strength and decrease in mass of insert for the composite inserts. The maximum gain is for the case of 3D functionally gradient woven composite inserts. Table 6: Geometrical configuration of the insert assembly for the analytical studies.
Table7: Material properties of the insert assembly. Volume of insert, V= 4.006x 10'6 m3 3D woven composite insert : Zt = 60 MPa1 Srz = 36 MPa, 2D woven composite insert : Zt = 27 MPa, Srz = 36 MPa Aluminum insert : Zt = 150 MPa, Srz = 30 MPa, 3D thermoelastic isotropic woven composite insert : Zt = 55 MPa, Srz = 36 MPa Epoxy: Zt = 38 MPa, Zc = 85 MPa, S = 42 MPa 3D woven composite with multiple inserts: Zt = 60 -> 38 MPa, Srz = 36 MPa 3D functionally gradient woven composite insert: Zt = 55 -> 38 MPa, Srz = 36 MPa * in-plane properties
Tables 2, 7 and 8 are based on the following references. 3D functionally gradient woven composite inserts are analyzed for the range of properties given in Table 7.
References:
1. Naik, N. K. and E. Sridevi. 2002. An analytical method for thermoelastic analysis of 3D orthogonal interlock woven composites, Journal of Reinforced Plastics and Composites, Vol. 21 , pp. 1149-1191. 2. Naik N. K. et al. 2001. Stress and failure analysis of 3D orthogonal interlock woven composites, Journal of Reinforced Plastics and Composites, VoI: 20, pp. 1485-1523. 3. Naik, N. K. and V. K. Ganesh. 1996. Failure behavior of plain weave fabric laminates under on-axis uniaxial tensile loading: Ii - analytical predictions, Journal of Composite Materials, Vol. 30, pp. 1779-1822. 4. Shembekar, P. S. and N. K. Naik. 1992. Elastic behavior of woven fabric composites: Il - laminate analysis, Journal of Composite Materials, Vol. 26, pp. 2226-2246. 5. Engineered Materials Handbook, Vol. 1 , Composites, 1989, ASM International, Materials Park, OH. Table 8: Elastic properties of orthotopic composite inserts (at θ - 0).
Table 9: Specific strength of through-the-thickness inserts with different materials: analytical studies. Volume of insert, V= 4.006x 10"6 m3 Density of Aluminum = 2800 Kg / m3 Density of 2D and 3D woven composites = 1700 Kg / m3 Density of 3D functionally gradient woven composite = 1700 -> 1100 Kg / m3
It is evident from the examples that the novel method reliably maps the stresses, displacements and failure initiation and enables the judicious selection of the novel insert material and geometry to achieve higher specific strength.