Kapiel 2 Ohoopic Maeials 2. Elasic Sain maix Elasic sains ae elaed o sesses by Hooke's law, as saed below. The sesssain elaionship is in each maeial poin fomulaed in he local caesian coodinae sysem. ε e = Cσ () σ is he sess veco, which uses he same convenion as he sain veco. The sess veco is given below. σ = [σ l σ σ τ l τ l τ ] T (2) C is he maeial compliance maix. The compliance maix is given as C = ν l ν l E ν E ν E G l 0 0 0 G l 0 0 0 G ν l ν l (3) E's ae moduli of elasiciy, G's ae shea moduli and ν's ae Poissons aios. The compliance maix should be symmeic, which gives ise o he following esic- 4
ions: ν l = ν l ; ν l = ν l E ; ν = ν E (4) The invese of he compliance maix is he maeial siness maix, D, which give he sesses poduced by an elasic sain sae. D = ( ν ν )/k (ν ν l + ν l )/k (ν l ν + ν l )/k (ν l ν + ν l )/k ( ν l ν l )/k (ν l ν l + ν )/k E (ν l ν + ν l )/k E (ν l ν l + ν )/k E ( ν l ν l )/k G l 0 0 0 G l 0 0 0 G (5) whee k = ν ν ν l ν l ν l ν l ν l ν ν l ν l ν ν l. The siness maix is also symmeic, which follows fom he esicions given in (4). Fuhemoe D should be posiive deni i.e. ε T Dε > 0 (The maeial esiss defomaions), which leads o he following esicions: ν ν ν l ν l ν l ν l ν l ν ν l ν l ν ν l > 0 ν ν > 0 ν l ν l > 0 ν l ν l > 0 (6) The maeial siness maix is used when he equilibium equaions ae solved, using he Finie elemen mehod. 2.2 Coodinae sysems Thee dieen caesian coodinae sysems ae used in he spaial disceizaion, one global and wo dieen local associaed wih each elemen. The local elemen coodinae sysems ae called elemen coodinae sysem and maeial coodinae sysem, especively. The hee dieen coodinae sysems ae shown in gue 2.2. Global coodinae sysem The global coodinaes ae emed (X, Y, Z). The oigin is locaed in global node numbe. The Global coodinae sysem is used o dene he geomey of he 5
x y x X Z x y y Y z Figu : Local and global coodinae sysems imbe beam, and he bounday condiions. 2.2.2 Elemen coodinae sysem The s of he local coodinae sysems is called he elemen coodinae sysem. The coodinae axes ae emed (x, y, z). The oigin is locaed in he cene of he elemen. The posiive diecion fo he x-axis is fom node 2 owads node. The posiive diecion fo he y-axis is fom node 4 owads node and nally he posiive diecion fo he z-axis is fom node 3 owads node. The elemen coodinae sysem is used fo he inepolaion beween he elemen nodes and he elemen inegaion. 2.2.3 Maeial coodinae sysem The second coodinae sysem is he elemen maeial coodinae sysem, wih axes (l,, ). The oigin of he maeial coodinae sysem is also locaed in he cene of he elemen, and he maeial diecions ae assumed o be consan in he elemen, which make i necessay wih a lage numbe of elemens in plane pependicula o he gain diecion. 6
2.2.4 Tansfomaion beween coodinae sysems The ansfomaion of popeies descibed in he maeial coodinae sysem o he elemen coodinae sysem is descibed in his secion. Popeies can be geomeic poins, maeial popeies eg. maeial siness o physical quaniies eg. sess o hea ux. The ansfomaion of geomeical poins beween he wo coodinae sysems ae: x y z = A T l (7) Whee A is given by (8) A = l x l y l z x y z x y z (8) The s ow in A is a uni veco in he l-diecion, descibed in he xyz-coodinae sysem, he second is a uni veco in he -diecion and he las ow is a uni veco in he -diecion. The elemens in A ae denoed diecion cosines. The deeminaion of diecion cosines is given i secion 2.2.5. A is ofen called a oaion maix, and is an ohogonal maix; i.e. A = A T. Hence he invese of he ansfomaion in (7) is given below l = A x y z (9) The ansfomaion of sesses and sains, fom one coodinae sysem o anohe is as given in (0) ε l = Gε xyz σ xyz = G T σ l (0) σ l = ( G T) σxyz whee σ l = [σ l σ σ τ l τ l τ ] T is he sess veco, descibed in he maeial coodinae sysem and σ xyz is he sess veco descibed in he elemen coodinae 7
sysem. The ansfomaion maix G can be deduced fom he elemens in (8), and is as given hee: G = lx 2 ly 2 lz 2 l x l y l x l z l y l z x 2 y 2 z 2 x y x z y z 2 x 2 y 2 z x y x z y z 2l x x 2l y y 2l z z l x y + l y x l z x + l x z l y z + l z y 2l x x 2l y y 2l z z x l y + y l x z l x + x l z y l z + z l y 2 x x 2 y y 2 z z x y + y x z x + x z y z + z y () The ( G T) maix is given in (2), when he G maix is subdivided ino fou 3 3 maices, emed G, G 2, G 2 and G 22. G = [ ] G G 2 G 2 G 22 [, G G 2G = 2 G 2 2 G 22 ] T (2) The ansfomaion of siness o exibiliy popeies, fom he maeial o he elemen coodinae sysem, is pefomed by a enso-like ansfomaion, as saed below. D xyz = G T D l G (3) whee D l is he maeial siness maix, fomulaed in he maeial coodinae sysem, given by (5) and D xyz is he maeial siness maix fomulaed in he elemen coodinae sysem. The G maix is he ansfomaion maix given in () The ansfomaion of hemal conduciviies and moisue anspo popeies is done in a simila enso-like way, as saed below λ xyz = A T λ l A (4) whee λ l is he maeial conduciviy maix fomulaed in he maeial coodinae sysem, which is a 3 by 3 diagonal maix conaining he hemal conduciviies in he pincipal diecions. λ xyz is he maeial conduciviy maix fomulaed in he elemen coodinae sysem. A is he ansfomaion maix given in (8). 2.2.5 Diecion Cosines The diecion cosines ae deemined fom he dieen elemens locaion elaive o he pih, see gue, which ae emed A 0. A 0 has he same sucue as he maix given in (8). Fuhe he diecions cosines ae depended on wo gowh phenomena, being conical angle and spial gowh, as given below. A = A 0 A c A s (5) 8
Whee A c and A s ae given by (6). cos φ sin φ 0 A c = sin φ cos φ 0 A s = 0 0 cos ϖ 0 sin ϖ 0 0 sin ϖ 0 cos ϖ (6) φ is he conical angle and ϖ is he spial gain angle. Boh angles ae illusaed in gue 2 φ Figu 2: conical angle and spial gowh 9