Belytschko T. - Introduction (779635), страница 5

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 5)

Therefore only those equations which arerelated to finite element implementations will be given in Voigt notation.Voigt notation usually refers to the procedure for writing a symmetrictensor in column matrix form. However, we will use the term for all conversionsof higher order tensor expressions to lower order matrices.The Voigt conversion for symmetric tensors depends on whether a tensoris a kinetic quantity, such as stress, or a kinematic quantity, such as strain. Wefirst consider Voigt notation for stresses.

In Voigt notation for kinetic tensors, thesecond order, symmetric tensor σ is written as a column matrix:tensor→ Voigtin two dimensions:σ 11  σ 1 σ 11 σ 12    σ≡→ σ 22  = σ 2  ≡ {σ}σ 21 σ 22    σ 12  σ 3 (A.1.1)in three dimensions:σ 11 σ 12σ ≡ σ 21 σ 22σ 31 σ 32σ 11  σ 1 σ  σ 222σ 13     σσ 33   3 σ 23  →   =   ≡ {σ}σ4σ 23σ 33     σ 13  σ 5   σ 12  σ 6 (A.1.2)We will call the correspondence between the square matrix form of the tensor andthe column matrix form the Voigt rule.

For stresses the Voigt rule resides in therelationship between the indices of the second order tensor and the column matrix.The order of the terms in the column matrix in the Voigt rule is given by the linewhich first passes down the main diagonal of the tensor, then up the last column,and back across the row (if there are any elements left). As indicated in thebottom row, the square matrix form of the tensor is indicated by boldface,whereas brackets are used to distinguish the Voigt form. The correspondence isalso given in Table 1.TABLE 1Two-Dimensional Voigt Rule1-16T.

Belytschko, Introduction, December 16, 1998σ iji123σa¨a123j123Three-Dimensional Voigt Ruleσ ijij112233231312σaa123456When the tensors are written in indicial notation, the difference betweenthe Voigt and tensor form of second order tensors is indicated by the number ofsubscripts and the letter used. We use subscripts beginning with letters i to q fortensors, and subscripts a to g for Voigt matrix indices. Thusσ ij is replaced by σ ain going from tensor to Voigt notation. The correspondence between thesubscripts (i,j) and the Voigt subscript a is given in Table 1 for two and threedimensions.For a second order, symmetric kinematic tensor such as the strain ε ij , therule is almost identical: the correspondence between the tensor indices and therow numbers are identical, but the shear strains, i.e. those with indices that are notequal, are multiplied by 2.

Thus the Voigt rule for the strains istensor→ Voigttwo dimensions ε11   ε1 ε11 ε12    ε≡→  ε 22  = ε 2  ≡ {ε}ε 21 ε 22    2ε12  ε 3 in three dimensions1-17(A.1.3)T. Belytschko, Introduction, December 16, 1998 ε11 ε12ε 22ε ≡ sym ε11 ε 22 ε13  ε 33 ε 23  →  ≡ {ε}2ε 23 ε 33 2ε13 2ε12 (A.1.4)The Voigt rule requires a factor of two in the shear strains, which can beremembered by observing that the strains in Voigt notation are the engineeringshear strains.The inclusion of the factor of 2 in the Voigt rule for strains and strain-like tensorsis motivated by the requirement that the expressions for the energy be equivalentin matrix and indicial notation. It is easy to verify that an increment in energy isgiven byρdwint = dε ijσ ij = dε : σ = {dε}T {σ}(A.1.5)For these expressions to be equivalent, a factor of 2 is needed on the shear termsin the Voigt form; the factor of 2 can be added to either the stresses or the strains(or a coefficient of 2 on both the stresses and strains), but the preferredconvention is to use this factor on the strains because the shear strains are thenequivalent to the engineering strains.The Voigt rule is particularly useful for converting fourth order tensors, which areawkward to implement in a computer program, to second order matrices.

Thusthe general linear elastic law in indicial notation involves the fourth order tensorCijkl :σ ij = Cijkl ε klor in tensor notationσ = Cε(A.1.6)The Voigt or matrix form of this law is{σ} = [C]{ε}(A.1.7)or writing the matrix expression in indicial form: σ a = Cabε b(A.1.8)and as indicated on the right, when writing the Voigt expression in matrix indicialform, indices at the beginning of the alphabet are used. The Voigt matrix form ofthe elastic constitutive matrix isC11 C12[C] = C21 C22C31 C32C13  C1111 C1122C23  = C2211 C2222C33  C1211 C12221-18C1112 C2212 C1212 (A.1.9)T.

Belytschko, Introduction, December 16, 1998The first matrix refers to the elastic coefficients in in tensor notation, the secondto Voigt notation; note that the number of subscripts specifies whether the matrixis expressed in Voigt or tensor notation. The above translation is completelyconsistent.

For example, the expression for σ 12 from (A.1.6) isσ 12 = C1211ε11 + C1212ε12 + C1221ε 21 + C1222ε 22(A.1.10)The above translates to the following expression in terms of the Voigt notationσ 3 = C31ε1 + C33ε 3 + C32ε 2(A.1.11)which can be shown to be equivalent to (A.1.10) if we use ε 3 = ε12 + ε 21 = 2ε12and the minor symmetry of C : C1212 = C1221 .It is convenient to reduce the order of the matrices in the indicial expressionswhen applying them in finite element methods.

We will denote nodal vectors bydouble subscripts, such as uiI , where i is the component index and I is the nodenumber index. The component index is always lower case, the node numberindex is always upper case; sometimes their order is interchanged. The followingrule is used for converting matrices involving node numbers and components tocolumn matrices:matrix uiI is transformed to a column matrix {d} by(A.1.12a)elements of {d} are ua where a = ( I − 1)nSD + i(A.1.12b)(The symbol for the column matrix associated with displacements is changedbecause u is used for the components, i.e.

u = ux , uy , uz .) This rule is combinedwith the Voigt rule whenever a pair of indices on a term pertain to a second ordersymmetric tensor. For example in the higher order matrix BijKk is often used torelated strains to nodal displacements byε ij = BijKk ukK(A.1.13)ui ( x ) = N I ( x )uiI ,(A.1.14) ∂u ∂u j  1  ∂N I∂Nε ij = 12  i +δ ik + I δ jk  ukI ≡ BijIk ukI= 2∂xi ∂x j ∂xi  ∂x j(A.1.15)whereTo translate this to a matrix expression in terms of column matrices for ε ij and arectangular matrix for Bija , the kinematic Voigt rule is used for ε ij and the firsttwo indices of BijKk and the nodal component rule is used for the second pair ofindices of BijKk and the indices of ukK .

Thus1-19T. Belytschko, Introduction, December 16, 1998elements of [B] are Bab where (i , j ) → a by the Voigt rule,(A.1.16a)b = ( K − 1)nSD + k(A.1.16b)The expression corresponding to (??) can then be written asε a = Babub or{ε} = [B]d(A.1.17)The correspondence of the indices depends on the dimensionally of the problem.In two dimensional problems, the matrix counterpart of BijKk is then written as B11xKB K =  B22 xK2 B12 xKB11yK B22 yK 2 B12 xK (A.1.18)The full matrix for a 3-node triangle is Bxx1x[B] =  Byy1x2 Bxy1xBxx1yByy1y2 Bxy1yBxx 2 xByy 2 x2 Bxy 2 xBxx 2 yByy 2 y2 Bxy 2 yBxx 3 xByy3 x2 Bxy3 xBxx 3 y Byy3 y 2 Bxy3 y (A.1.19)where the the first two indices have been replaced by the corresponding letters.The Voigt rule is particularly useful in the implementation of stiffness matrices.In indicial notation, the stiffness matrix is written asK IJrs = ∫ BijIr Cijkl BklJs dΩ(A.1.20)ΩThe above stiffness is a fourth order matrix and maultiplying it with a matrix ofnodal displacements is awkward.

The indices in the above matrices can beconverted by the Voigt rule, which givesTKab = ∫ BaeCef B fg dΩ → [K] = ∫ [B] [C][B]dΩΩ(A.1.21)Ωwhere the indices "Ir" and "Js" have been converted to "a" and "b" , respectively,by the column matrix rule and the indices "ij" and "kl" have been converted to "e"and "f" respectively by the Voigt rule.

Another useful form of the stiffness matrixis obtained by transforming only the indices "ij" and "kl", which gives[K]IJ = ∫ [B]TI [C][B]J dΩ(A.1.22)Ωwhere [B]I is given in Eq. (A.1.18).1-20T. Belytschko, Introduction, December 16, 19984.5.3. Indicial to Voigt Notation. The above equations in Voigt notationcan also be derived from the indicial equations if we start with the followingexpressions for the displacement and velocity fieldsui = Nib dbvi = Nib d˙ b(4.5.20)The key difference in (4.5.20) as compared to (4.4.5) is that the component indexis ascribed to the shape function.

In this notation, different displacement andvelocity components can be approximated by different shape functions. Thisadded generality is seldom used because it generally destroys the capability of theelement to represent rigid body rotation. The gradient of the velocity field in thisnotation is obtained by differentiating (4.5.20):∂vi ∂Nib ˙=db∂x j ∂x j ∂v ∂v j  1  ∂Nib ∂N jb Dij = 12  i += 2+d˙ b = Bijb d˙ b (4.5.21)∂xi  ∂x j ∂xi  ∂x jwhere ∂N∂N jb ∂NBijb = 12  ib += sym ib∂xi ∂x j ∂x j(4.5.22)so the Bijb extracts the symmetric part of the gradient of the shape functions. Therectangular B matrix is defined byBab = Bijb where (i , j ) → a by the symmetric kinematic Voigt rule, seeAppendix Band Eqs.(4.5.18) and (4.5.19) hold as before.

Характеристики

Список файлов книги