REFINED STRAIN ENERGY OF THE SHELL

Transcription

1 REFINED STRAIN ENERGY OF THE SHELL Ryszard A. Walentyński Deartment of Building Structures Theory, Silesian University of Technology, Gliwice, PL44-11, Poland ABSTRACT The aer rovides information on evaluation of the formula for shell strain energy with a comuter algebra system. There is roosed a formula for strain energy comatible with a definition of stress tensor resultant. The integral has been evaluated with the ˇ system. It was shown that the strain energy can be resented as a linear combination of stress resultant comonents multilied by resective strain tensors. The resented numerical examle shows the difference of deformation, stress resultants and energy for the katenoidal shell subjected to thermal influences. There was found that strain energy do not derive the formulas for tensors of stress resultants. KEYWORDS Nonlinear, elastic, theory of shells, constitutive relations, strain energy, comuter algebra, ˇ, MathTensor, tensor analysis. 1 INTRODUCTION Most of shell theories ostulate the strain density functional in a form roosed by Novozhilov (1959), which is comatible with his aroximated constitutive relations derived for orthogonal system of coordinates. Further theories, for examle Kraus (1967), only adoted this aroach to tensor notation. Mazurkiewicz (1995) stated that the alied the strain energy density for the arallel layer does not have to be exressed with the Novozhilov aroach. This aer roosed another functional (2) that can be used to evaluate shell energy density. It is comatible with exact definition of resultant stress comonents (1) and (2). The symbolic and numerical comutations were made with the ˇ system of comuter algebra, Wolfram (1996), and its external ackage for tensor analysis MathTensor, Parker & Christensen (1994). The grahics is also roduced with the system.

2 The considered shell is thin so Kirchhoff assumtions hold. The functional can be, of course, comleted with comonents resonsible for transverse forces influence. The most of notations in the aer are comatible with Bielak (1993). 2 RESULTANT STRESS TENSORS The tensors stress resultants, stretching forces and moments, should be comuted from the following integrals, Bielak (1993) h N i j h M i j h h 1 g 2 a j r z b j r Τir dz (1) 1 g 2 a j r z b j r Τir z dz (2) where strain tensor in 3D sace Γ i j and stress tensor Τ i j are comuted from Γ i j Γ i j 2 z Ρ i j z 2 ϑ i j (3) Contravariant comonets of metric tensor can be derived from Present scalars are defined with Τ i j Λ g i j g 2 Μ g i g j Γ i j (4) g i j ai j 1 z 2 2 H a i j b i j 1 H z z Z 2 (5) Z g a 1 2 H z K z2 (6) Λ Μ Ν E 1 Ν 2 (7) E 2 1 Ν (8) where E is a Young modulus and Ν is a Poisson ratio. H and K denote average and Gaussian curvatures, resectively. Tensors a i j and b i j form coefficients of the first and the second differential form of the reference surface. Invariants g and a are determinants of the metric tensor g i j and a i j, corresondingly. The strains in shell are denoted with three symmetrical tensors Γ i j, Ρ i j and ϑ i j. They are measure of the change of the three differential forms of the reference surface, resectively. For examle Γ i j 1 2 a i j å i j (9) It has been found, Walentyński (1999), that stretching forces tensor (1) can be evaluated as a sum of three tensors. N i j N i j ˆN i j Ñ i j O h 5 (1)

3 Each of the tensors in (1) deend on one of reference surface strain tensors. N i j 2 E h 1 h2 K Ν a i j Γ 1 Ν Γi j 1 Ν 2 4 E h3 Γ Ν b H a i j b i j 1 Ν b i 3 1 Ν 2 b j H j (11) ˆN i j 4 E h3 2 H j b j Ν ai Ρ 1 Ν Ρi 3 1 Ν 2 8 E h3 Ν a i j b Ρ 1 Ν b i Ρ j 3 1 Ν 2 (12) Ñ i j 2 E h3 Ν a i j ϑ 1 Ν ϑi j 3 1 Ν 2 If we comute the (1) with recision to the third ower of the shell thickness we receive that it deends on two strain tensors M i j M i j ˆM i j O h 5 (14) where M i j 4 E h3 Ν a i j b Γ 1 Ν b i Γ j 2 E h3 2 H j b j Ν ai Γ 3 1 Ν Ν 2 1 Ν Γi (13) (15) and ˆM i j 4 E h3 Ν a i j Ρ 1 Ν Ρi j 3 1 Ν 2 The refined formulas (1) and (14) satisfy the last euation of euilibrium (17) (16) ε N b r M r (17) Many theories aly the simlified formulas where stretching forces tensor is comuted from (18) and moment tensor from (16), only. N i j 2 E h Ν ai j Γ 1 Ν Γi j (18) 1 Ν 2 3 STRAIN ENERGY Strain energy in volume V in orthogonal Cartesian system of coordinates is comuted from the integral. E V 1 2 Τ Γ dv (19) By analogy to the definitions (1) and (2) the strain energy will be ostulated here as a following integral E A U da A h h 1 g 2 a r z b r Τ r Γ dz da (2)

4 The aroriate notation in the ˇ/MathTensor of the inner integral in this definition, which will be called further shell energy density, is Energyk Tsimlify AbsorbKdelta Normal Series 1 Z Kdeltal3, u5 z bl3, u5 2 tauu4, u3 gammastarl4, l5, z,, k The result of the comutation with recision to the third ower of the shell thickness is Energy2 E h 3 5 h2 K a t a s Γ Γ st 3 1 Ν E h 3 5 h 2 K Ν a a st Γ Γ st 3 1 Ν 2 2 E h 3 H a r a s b t r Γ Γ st 3 1 Ν 2 E h 3 H Ν a a rs b t r Γ Γ st 3 3 Ν 2 4 E h3 H Ν a st b Γ Γ st 3 3 Ν 2 2 E h 3 Ν a rs b t r b Γ Γ st 3 3 Ν 2 2 E h3 a s b t r b r Γ Γ st 3 1 Ν 4 E h 3 H a s b t Γ Γ st 3 1 Ν 4 E h 3 b t b s Γ Γ st 3 1 Ν 2 E h3 a r b t r b s Γ Γ st 3 1 Ν 2 E h3 Ν a b t r b rs Γ Γ st 3 3 Ν 2 4 E h 3 Ν b b st Γ Γ st 3 3 Ν 2 8 E h3 H a t a s Γ st Ρ 3 1 Ν 8 E h 3 H Ν a a st Γ st Ρ 3 3 Ν 2 2 E h3 a r a s b t r Γ st Ρ 3 1 Ν 2 E h 3 Ν a a rs b t r Γ st Ρ 3 3 Ν 2 8 E h3 Ν a st b Γ st Ρ 3 3 Ν 2 16 E h 3 a s b t Γ st Ρ 8 E h3 Ν a b st Γ st Ρ 3 1 Ν 3 3 Ν 2 2 E h 3 a r a s b r t Γ Ρ st 3 1 Ν 4 E h 3 a t a s Ρ Ρ st 3 1 Ν 2 E h 3 a t a s Γ st ϑ 3 1 Ν 2 E h3 Ν a a rs b r t Γ Ρ st 3 3 Ν 2 4 E h3 Ν a a st Ρ Ρ st 3 3 Ν 2 2 E h3 Ν a a st Γ st ϑ 3 3 Ν 2 The received result is a symmetrical, ositive definite form, which is eual to zero if the shell is subjected to the rigid motion (in constant temerature). In this case strain tensors are eual to zero, so differential forms of the reference surface do not change, comare en. (9). According to the well-known theorem of differential geometry differential forms defines the surface with a recision to the osition in sace.

5 The received formula can be used directly in the further symbolic and numerical comutations but can be simlified with MathTensor. After some simlifications of the formula - like absortion of the metric tensor, canonicalization, alication of geometrical rules and simlification - we receive U E h 1 h2 K 1 Ν Γ Γ Ν Γ Γ 1 Ν 2 2 E h3 H 1 Ν b Γ r Γ r 3 1 Ν 2 Ν b Γ Γ r 2 E h3 b b r s 1 Ν Γ r Γ s Ν Γ Γ r 3 1 Ν 2 8 E h3 H 1 Ν Γ Ρ Ν Γ Ρ 3 1 Ν 2 4 E h3 1 Ν Ρ Ρ Ν Ρ Ρ 3 1 Ν 2 2 E h3 2 1 Ν b 2 E h3 1 Ν Γ ϑ 3 1 Ν 2 s r Γ r Ρ r Ν b Γr r Ρ Γ Ρ r r 1 Ν 2 Ν Γ ϑ O h 5 Now we will show that the shell energy density can be resented as a linear combination of the resultant stress tensors multilied by the aroriate strain tensors. To do this we will introduce five scalars, which exress the work of internal forces. The first three reresent the work of stretching forces tensors N (11), ˆN (12) and Ñ (13) on the first strain tensor Γ i j U ΓΓ 1 2 N Γ (21) E h 1 h2 K 1 Ν Γ Γ Ν Γ Γ 1 Ν 2 2 E h3 H 1 Ν b Γ r Γ r 3 1 Ν 2 Ν b Γ Γ r 2 E h3 b b r s 1 Ν Γ r Γ s Ν Γ Γ s r 3 1 Ν 2 r (22) U ΡΓ 1 2 ˆN Γ 4 E h3 H 1 Ν Γ Ρ Ν Γ Ρ 3 1 Ν 2 2 E h3 3 1 Ν b Γ r Ρ r Ν b 2 Γr r Ρ 3 1 Ν 2 Γ U ϑγ Ñ Γ 2 E h3 1 Ν Γ ϑ Ν Γ ϑ 3 1 Ν 2 Ρ r r (23) (24)

6 The last two reresent the work of moment tensor comonents ˆM i j (16) and M i j (15) on the second strain tensor Ρ i j U ΡΡ ˆM Ρ 4 E h3 1 Ν Ρ Ρ Ν Ρ Ρ 3 1 Ν 2 (25) U ΓΡ M Ρ 4 E h3 H 1 Ν Γ Ρ Ν Γ Ρ 3 1 Ν 2 2 E h3 3 1 Ν b Γ r Ρ r Ν b Γr r Ρ 3 1 Ν 2 2 Γ Ρ r r (26) It can be easily verified that the strain energy density (21) can be exressed with U U ΓΓ U ΡΓ U ΘΓ U ΡΡ U ΡΓ O h 5 (27) It roves that stretching forces comonents work on the first strain tensor and moments work on the second strain tensor. Below we resent an extract from the Mathematica notebook which verifies it. Simlify/@TsimlifyAbsorbKdeltaExandE1 Eng Emr Enr Emg Ent Some shell theories aly the aroximate formula U U ΓΓ U ΡΡ (28) where E h 1 Ν Γ U ΓΓ Γ Ν Γ Γ (29) 1 Ν 2 This definition is comatible with simlified constitutive relations. 4 NUMERICAL EXAMPLE Figure 1: Deformed reference surface. Simlified aroach in the background.

7 Figure 2: Physical dislacement vector w 1 (meridian direction) and w 3 (normal direction) comonents and rotation vector d 1 (meridian direction) comonent. w i Μm and d 1 Μm/m Figure 3: Physical stretching force N 11 (meridian direction), N 22 (arallel direction). N i j N Figure 4: Physical bending moment M 12 (meridian direction), M 21 (meridian direction) and transverse force Q 1 comonent. M i j Nm and Q 1 N U ΓΓ U ΓΡ U Γϑ U ΡΓ Figure 5: Energy comonents distribution for refined aroach. There is considered a katenoidal shell which meridian is defined with a function f x a o cosh x a o (3) The shell subjected to thermal influence consisting in difference of temerature on both limit surfaces. The following numerical data were alied: shell thickness 2 h.2 m, radius in the neck a o 5. m,

8 shell height l 1. m, difference of temerature t 2 o C, Young modulus E 324 kpa, Poisson ratio Ν 1/6, thermal exansion coefficient Α t 1 5. The shell boundaries are fixed. The calculations were made with alication of both refined and simlified constitutive relations. The results were resented in figures. Fig. 1 resents deformed reference surface. As we can see reference surface deforms when the shell is solved with alication of refined constitutive relations. The deformation of the reference surface is small, Fig. 2, but enough to roduce stretching forces, Fig. 3, which are absent in simlified aroach and remarkable difference in moments and transverse forces, Fig. 4. The strain energy density is also remarkably different for both aroaches, Fig. 5. In this case the energy density comonent U Ρ Ρ is crucial and other comonents are mutually comarative. 5 CONCLUSIONS Formula (27) shows that moments tensor can be derived from the density of strain energy (21) as a derivative with resect to the second strain tensor. The stretching forces tensor can be derived similarly by differentiation of strain energy density with resect to the first strain tensor, but only if we neglect influence of the third strain tensor. Presented numerical examle shows that nevertheless the strain energy density usually deends mostly on comonents (22) and (25) but stress resultant tensors cannot be comuted from the simlified formulas (15) and (18) that can be derived from (28). It may result in significant errors in comutations. The received formulas can be alied in nonlinear theory of shells and can be easily exanded to multilayered shells and then alied for large strain theories. It will be shown in further contributions. REFERENCES Bielak S. (1993). Theory of Shells, Silesian University of Technology, Gliwice Kraus H. (1967). Thin Elastic Shells, an Introduction to the Foundations and the Analysis of Their Static and Dynamic Behavior. John Wiley and Sons, Inc., New York-London-Sydney Mazurkiewicz Z.E. (1995). Thin Elastic Shells, Linear Theory. Warsaw University of Technology, Warsaw Novozhilov V.V. (1959). The Theory of Thin Shells. P. Noordhoff Ltd., Groningen, The Netherlands Parker L. and Christensen S.M. (1994). Math Tensor: A System for Doing Tensor Analysis by Comuter. Addison-Wesley, Reading, MA, USA Walentyński R.A. (1999). Refined constitutive shell euations with MathTensor. Proceedings of the 3rd International Mathematica Symosium IMS 99. RISC, Hagenberg-Linz, htt://south.rotol.ramk.fi/ keranen/ims99/ims99aers/ims99aers.html Wolfram S. (1996). The Mathematica book. Wolfram Media/Cambridge University Press, Chamaign, IL, New York, NY, USA