Damage Tensors and Tensors of Continuity in Stress Analysis J. Betten Technical University Aachen
Z. Naturforsch. 38 a, 1383- 1390 (1983); received October 17, 1983*
Starting from a third order skew-symmetric tensor of continuity to represent area vectors (bivectors) of Cauchy's tetrahedron in a damaged state, a second order damage tensor is found which has the diagonal form with respect to the considered coordinate system. The second part of the paper is concerned with the stresses in a damaged continuum. Introducing a linear operator of rank four a net-stress tensor is formulated. This tensor can be decomposed into a symmetric part and into an antisymmetric one, where only the symmetric part is equal to the net-stress tensor introduced by Rabotnov [7], In view of the formulation of constitutive equations the non-symmetric property of the actual net-stress tensor is a disadvantage. Therefore, a pseudo-net-stress tensor is introduced, which is symmetric.
1. Introduction strain rate. Furthermore, it is assumed that the damage rate co or, alternatively, the rate of change The creep process in its tertiary phase is essen- of the continuity y/ is also governed by the nominal tially influenced by the effect of damage. It is well stress and the current state of continuity, i.e. known that the process of creep of a metal is accom- y/ = -g{o, y/). panied by the formation of microscopic cracks on Deliberations about the forms of the functions / the grain boundaries and that damage-accumulation and g were made in detail by many scientists, for occurs. instance by Chrzanowski [1], Goel [2], Hayhurst and In the phenomenological uni-axial theory due to Leckie [3], Hayhurst et al. [4]. In generalisation of Kachanov (1958) material deterioration is consider- Kachanov's theory constitutive equations and aniso- ed by introducing an additional variable co or, alter- tropic damage growth equations are given by ex- natively, y/ = 1 — co into the constitutive equations pressions like i.e. the strain rate can be expressed in the form £ = /((j, i//), where a is the nominal stress of a uni- Eij = fij (a, co) and cb/f = gu (
Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. tors) of Cauchy's tetrahedron in a damaged state, /.!. In three-dimensional space /. can only be equal we finally find a second order damage tensor which to two or to three. has the diagonal form.
3. Geometry 2. Definitions In three-dimensional space a parallelogram form- In this paper rectangular cartesian components of ed by the vectors Aj and 5, can be represented by the tensors are used througout; results discussed here can be expressed in terms of general curvilinear Sj - £jj Aj B (3-1 a) coordinate systems by the standard techniques of k k tensor analysis. Index notation and Einstein's sum- or in the dual form mation convention are employed. Confining to (3.1b) cartesian tensors we define: Sjj = Ejjk Sk<= Si 2 ^'jk Sjk '
In a v-dimensional space a (cartesian) tensor of where Eijk is the third-order alternating tensor
valence p alternating in all indices is called a p- (Sjjk = 1, or — 1 accordingly as i,j, k are even or odd vector, multivector, polyvector, antisymmatric permutations of 1, 2, 3, respectively, otherwise the
tensor or skew-symmetric tensor, components Ejjk are equal to zero). From (3.1a, b) we immediately find = A (2.1)
where p v. Aj Aj Sii — 2\A\jBn = (3.2) Bj Bj Multivectors of orders 2,3,4,...,p are also called bi-vectors, tri-vectors, quadri-vectors, . . . , //-vec- Because of the decomposition (3.2) as an alternating tors, respectively. A skew-symmetric tensor posses- product of two vectors the bi-vector S is called v\ simple and has the following three nonvanishing sesses exactly non zero-indepen- \Pl p\(v-p)\ essential components dent and distinct (essentially distinct, i.e. differing S|2 — A\ B2-A2Bi S23=A2B3-A3B- not only in sign) components. In the following we restrict to a three-dimensional S31 —A3 B\ — A] B3 (3.3) space (v= 3) and consider bi-vectors and tri-vectors: In rectilinear components in three-dimensional space, we see that the absolute values of the com- Am = (Ay — Aji)/2 , (2.2) ponents (3.3) are the projections of the area of the A[ijk] = (Ajjk + Ajki + Akij — Aikj - Ajik - Akji)/6 . (2.3) parallelogram, considered above, on the coordinate If we consider skew-symmetric tensors with respect planes. Thus Sjj, according to (3.2), represents an to / < // indices, for instance area vector in three-dimensional space and has an orientation fixed by (3.1 a). 1 Tiki...k„ == T [kv..kx]k i+1...ku where /. ^v, (2.4) According to (3.1 b) a surface element dS with an unit normal /?,-, i.e. dSj = n, d.S, is expressed by we have in v-dimensional space vM 2 essential
dSjj = £jjk dSk dS, = 4 £ijk dSj, (3.4a) components. In (2.4) the operation of alternation is indicated by placing square brackets only around and (3.4b) those indices to which it applies, that is, the /. 'hi = £ijk nk n, = -j,ijk njk bracketed indizes k]... k, are permutated in all possible ways, while indices which are excluded The components of the bi-vector n are the direction from the alternation are not bracketed. They keep cosines nx, n2. n3: their positions. Thus we obtain /! terms. The terms corresponding to even permutations are given a plus 0 »3 — n 2 sign, those which correspond to odd permutations a -n3 0 minus sign, and they are then added and divided by , n2 -«1 0 The principal invariants of (3.5), defined as
J\ = nu, -Ji = «/[/]«/[/•], x2 h = «/[/] «/•[/•] /»*[*], (3.6 a, b, c) take the following values:
•/,(!») = 0, -y2(«) = «1 + «2+ «3= 1 ,
•/3(») = 0, (3.7a, b,c) i.e. the only nonvanishing invariant is determined by the length of the vector n,. Imagine that at a point o in a continuous medium a set of rectangular coordinate axes is drawn and a Fig. 1. Cauchy's tetrahedron a) in an undamaged state, b) in a damaged state. differential tetrahedron is bounded by parts of the three coordinate planes through o and a fourth plane not passing through o, as shown in Figure 1 a. (3.10). Then the sum of (3.10) yield the vector Such a tetrahedron can be characterised by a system of bi-vectors, / (x — a) \ (3-11)
2T( = d'S/+ ... + d S, = I (x-ß) n2 dS, d '5/ = - T Eijk (d.Y2)/ (d.V3)A: , \{x-y)n3l 2 d Sj =-j Eijk (d.v3)v (d.v ,)*, (3.8) which is not the zero-vector, unless in the isotropic 2 d Si = - \ £iJk (d.Y Oy (dx2)k, damage case (a = ß = y = x) or in the undamaged 4 case (a = ß= y= x = 1) according to (3.9). d 5( = j euk [(d.v,)/ - (d.Y,),] [(d.Y2), - (dx3)*], 1 Furthermore, because of d'S, + 0, d S2 = d'S3 = 0 where the sum is the zero vector: etc., the damage state of the continuum at a point is characterized by the bivectors d'S, + d2S/ + d3S, + d4S, = O,. (3.9)
/ 0 0 0 1 j1 0 -ß\ In a damaged continuum we define a "net cross- a, //= 0 0 ,0 0 (3.12a, b) section" S = y/ S where y/ ^ 1 describes the "con- aJ 1' ßvj- \ 0 \o — 1 0/ 0 0/ tinuity" of the material, as mentioned in the intro- \ß duction. Then, by analogy of (3.8), a tetrahedron in a damaged continuum (Fig. 1 b) can be character- (3.12c) ized by the following system of bi-vectors:
1 d '5/ = - y ctijk (d.Y2)j (dx3)k = a d S,-, In the following we will examine if the bi-vector
2 2 d S, = "T ßijk (d.Y 3)j (d.v,), = ß d Si, (3-10) 3 3 d S, = - \ yijk (d.Y,)/ (d*2)* = 7 d S,, I 0 4 V//= %\ij + ßüj+ 73//= I -y d S, = \ xijk [(d.Y,)/ - (d.Y3)7] \ ß
' [(d-Y )jt — (d.v ),] = x d45,, 2 3 y/jj = a e,u + ß e2ij + y e3// where iijk = a eijk, ßiJk = ß eiJk, etc. are total skew- could be a suitable "tensor of continuity". Then the symmetric tensors of order three, which have the damage tensor co would be of the form essential components a,23 = a, ß\23 = ß, etc., respec- (3.14a) tively. From Fig. 1 (a,b) we find that only dS, = — n\ dS, I 0 1-y -(l-/?)\ 2 d'S, = a d'S,, d S2 = — n2dS, etc. are non vanish- u>ij = S{k)k£kij- y/ij= I -(1-y) 0 1-a ing components of the bi-vector systems (3.8) and \ 1-A -(1-a) 0 / (no sum on the bracketed index k) or By analogy of (3.1b) or (3.4 a. b) the dual rela- tions to;, = (1 - a) £, u + (1 - ß) e2ij + (1 - y) ey.j. (3.14 b) V,jk = Vi[jk] = £jkr Vir <=> Vir = J £rjk Vijk (3.20) If a tensor is symmetric or antisymmetric, respec- tively. in one Cartesian coordinate system, it is sym- OJjjk = OJj[/k] - £jkr °hr OJlr - J £rJk COjjk (3.21) metric or antisymmetric in all such systems; thus are valid. symmetry and antisymmetry are really tensor prop- Contrary to (3.13) and (3.14) the dual tensor of erties. Therefore, the skew-symmetric tensor (3.13 a) continuity y/,j according to (3.20) and the dual has only three essential components in any Carte- damage tensor t//,, according to (3.21) have the sian system, for instance, a, /?, y in relation to the diagonal forms system .v, or a*, ß*, y* with respect of the system A",*. Vij = diag {a, ß, 7} (3.22) The only nonvanishing invariants of the bi-vectors and (3.13) and (3.14) are determined by their lengths: Wij = diag {(1 — a), (1 - /?), (1 - 7)}, (3.23) = = »//„ yj„ =oi2 + ß2+y2, (3.15) respectively. For the undamaged continuum (y/ijk->•
2 2 2 Ejjk) the dual tensor of continuity i//,y is equal to -J2(co) = (1 - x) + (1 - ß) + (1 - y) . (3.16) Kronecker's tensor d(/, as we can see from (3.20) or In the undamaged state (cc = ß=y=\) we have immediately from (3.22). The relations (3.20) and
-/2(
Vijk = IVi\jk) with- y/2jk = ß2jk = ß e2jk. (3.18) dSjj - y/kij dSk dSj = y if/jjk dSjk (3.25) V3jk — 73jk = y £3jk
This tensor is skew-symmetric only with respect to the two bracketed indizes [jk] and possesses three essential components (a, ß, 7), as illustrated in anisotropic domage (a* ß*y*a) Figure 2. Y In the isotropic damage state (a = ß=y= x) the /!" /A tensor (3.18) is total skew-symmetric, and the un- damaged continuum (y. = ß = 7= 1) is characterized / 0 0 -3 / a 0 by the third-order alternating tensor e . ijk / .000 Supplementary to (3.18) we introduce the "dam- / 0 0 ,0 \ 0 /^TT age-tensor" (3.19) \ ^ CO\jk = (1 — a) £ 1 jk 0J \y -a 0 /Xur ak = £ijk ~ V/jk where— co2jk = (\-ß) £2jk .
0)3jk = (1 ~ 7) «3jk Fig. 2. Third order tensor of continuity and its dual form. P.= P./^LNL Denote, that the bi-vector dor dSjk in (3.25) must have the same indizes with respect to which the tensor (3.18) is skew-symmetric. Combining (3.4a) and (3.25) we have the linear transformations
dSjj = y y/ijpq dSpg, dSi = \j/ir dSr, (3.26 a, b)
where y/ir is the tensor (3.20), (3.22) and y/ijpq is a fourth-order wow-symmetric tensor defined as
Vijpq = Vkij £kPq, (3.27 a) which, by using (3.20), can be expressed through net - stress tensor cf;. b) Vijpq = (dip Öjq ~ $iq fyp) Vrr ~ (Vip fyq ~ Viq &jp) Fig. 3. Stress Tensor regarding a) an undamaged, b) a damaged continuum. - tfip Vjq - Öiq Vjp) • (3.27 b)
This tensor has the antisymmetric properties
Vijpq = - Vjipq = - Vijqp = Vjiqp (3-28) is derived from equilibrium, where p, and w, are the and is symmetric only with respect to the index components of the stress vector p and the unit pairs, i.e. vector normal n, respectively. Vijpq = Vpqij • (3.29) In the same way we get to the corresponding rela- tion for a damaged continuum. More briefly, the properties (3.28) and (3.29) can be indicated by Pi V(n) = Vjk Oki rij (4.2) Vijpq = V([ii\[pq]) • (3-30) where y/jk are the components of the continuity The essential components of the tensor (3.27) are tensor ij/ according to (3.20). given by The surface elements dS and dS in Fig. 3 are sub- jected by the same force vector: a, ß, y, if ij is an even permutation ofp q dPi = pidS=pidS = dPi. (4.3) {- i, —ß, — y, if ij is an odd permutation of p q 0, otherwise (3.31) Thus, considering (4.1) and (4.2), we finally find the actual net-stress tensor A as a transformation from that means Cauchy's tensor:
^2323 = ^3232 = «i V3131 = Vl313 = ß\ Oij = Vir Orj = Oji o Oij = '' °rj * Oy, . (4.4)
Vl2l2 = V2I2I = 7 ( 11 By suitable transvections we find a(/- o ~k = yj,k and (3.31b) OijOjk - y/ik • V3223 = ^2332 = ~ Vl331 = V3113 = ~ As indicated in (4.4), the actual net-stress tensor 6 V2!I2 = Vl22l = ~y- is non-symmetric, unless we have isotropic damage In the isotropic damage state (a = ß = y = x) the expressed by y/ir = y/ Öir. tensor (3.27) is proportional to Kronecker's general- Because of the symmetry cr(/ = (cr,y + ay/)/2 of Cauchy's stress tensor a we find the representations ized delta öjjpq = Ekij £kPq, and is identical to that one in the undamaged continuum characterized by Oij = y (Vip Sjq + Siq Vjp) opq - (Pijpq apq, (4.5 a) x = ß=y = 1. 1) 1) Oij = y (y/jp öjq + Vi~q Sjp) apq = 0Upq Gpq (4.5 b) 4. Stresses in a Damaged Continuum from (4.4). We see, the fourth-order tensors and defined as (4.5a, b) are only symmetric with In the undamage continuum (Fig. 3a) Cauchy's respect to two indices: formula Pi = Oy, rij (4.1) Vijpq
This fact is a disadvantage, and it is awkward to 2 2 Cijk,= diag la . ß\ y , \y.ß,\ßy,\ y a}, (4.19b) use the actual net-stress tensor a in constitutive equations with a symmetric strain rate tensor e. and Therefore, we introduce a transformed net-stress (_1) (_1) (_1) (_l) (_l) (_1 L<-i)_ Hiaa
( In the undamaged (^ -+Ö) and total damaged state h\ —t\3i hi ~ i3 > '33= ^33 • (\j/ ->0) we have The results (4.9) and (4.15) can also be found in fv 0 v U ... . . ijpq ijpq 'ij ij (4.12) the following way. Using the linear transformations
( 0 and t,j = | (dir /, ° + V is Öjr) örs (4.22)
C<~ y ccijpq => tjj -> oo ij (singular), (4.13) and Opq = J (Öpr Vqs + dps Vqr) t , (4.23) respectively, where rS which connect a fictitious symmetric tensor t with Eijpq = { (d ö + ö ö ) (4.14) ip iq iq jp the actual non-symmetric net stress tensor 6. we immediately find (4.9) by inserting (4.5b) into is the zero power tensor of rank four. (4.22) and (4.15) by inserting (4.23) into (4.5a). re- The inverse from of (4.9) is given by spectively. Oij Cijpq tpq , (4. 1 5) Because of the non-symmetric property of the actual net-stress tensor we find from (4.23) the where decomposition Cjjpq = J ( Vip y/jq + Viq Vjp) (4.16) Opq -
O(pq) = (tpr Vrq + Vpr trq)/2 (4.25) which is connected with the tensor (4.10) by the and relation 0[pq] = Cpr Vrq ~ Vpr trq)/2 , (4.26) Cijpq Cpqk! \jpq pqkl Cijkl= Ejjk/. (4.18) respectively. In the special case of isotropic damage
Because of the symmetry properties (4.11) and (if/jj = y/ Su) we have a{pq) = kij) Skpq , (4.34 a) 1 dPj = Gjj yj\r ' dSr (4.28 a) Vijpq - (dip Sjq - djq Öjp) ( 1 - COrr) + (Cüjp djq - COiq öjp) or inserting (4.15) we find the relation + (öjp cojq - Siq a>jp). (4.34 b) dPj=¥iptprdSr, (4.28 b) Furthermore, instead of (4.5a) and (4.32b) we find 1 which can be multiplied by y/ki so that we have 1 Oij = 7 [dip djq + diq djp - (OJjp djq + diq CDjp)] Gp t//[~ ' dPj = tkr dSV, or after changing the indices: (4.35 a) l) ^-k dPk = dPj=tjidSj. (4.29) and
Comparing (4.27) and (4.29) we see, that (4.29) can Oij = t (öjj + Gjj) (4.35 b) be interpreted as Cauchy's formula for the damaged - I ((Dip 5jq + diq OJjp + OJjq djp + djp (Djq) (<7pq+ Gqp) configuration, which is subjected to the pseudo- 11 - J (cOip dJq + diq OJjp-OJjq djp-Öjp COJq) (Gpq~Gqp) . force dPj = dPk instead to the actual force dPj. Because of the non-symmetric properties of the By using the inverse "net-stress-tensor" 6 and the operator , i.e. JM> = V< £rqp Eikl Vpk Vql (4.36) Oij - \ (Oij + Gjj) + 3 (Gjj - Gjj) (4.30) 2 det (ip) and and because of the symmetry Ojj = (Ojj + Oji)/2, we (Pijpq = 7 (fpijpq + (pijqp) + J (Vijpq ~
(4.32b) [( 1 - COrr) Gjj + COir Grj (4.37 b) det (5 Because of (4.6a) the right hand sides in (4.32a,b) + 7 Eik/Espq OJpk COq/ Gsj] , are symmetric with respect to the indices / and j. Furthermore, we see the symmetry with respect to On = {[1-7, (CO)-J2(CO)] Gjj (4.37 c) det (ö - co) the indices p and q. This fact can be seen immedi- { + [1 - J, (to)] 00ir Grj + 00 }) G ately from (4.5 a). In the special case of isotropic damage, i.e. i//,7= y/ö/j or 6pq= oqp, the second term where (4.38 a, b) of the right hand side in (4.32b) vanishes. Then, J, (co) = Öjj COjj , J (co) = j (cOjj COjj - COjj COjj) equation (4.32b) is identical to those formulated by 2 Rabotnov [7], are invariants of the damage tensor co. In a similar way, from (4.5 b) we find the decom- Finally, we consider Cauchy's stress equations of position of the "net-stress-tensor" a into a sym- equilibrium, metric and an antisymmetric part: ^ ^ Ojij= 0, (4.39)
Oij = 4 (®ijpq + ®jipq) Opq + \ (^Upq ~ &Hpq) °pq ' in the absence of body forces. Then, by using the 1 / (-') S , (-D S , (-1) S , (-1) X \ Oij = 4 Wip dJq + yjiq djP + y/jp diq + y/jq dip) apq transformation (4.4), we have the equilibrium equa-
{ ]) ,JM) x tions in the net stresses: + T (V ip bjq+ V)q -ip Vjp bjq J-D ~ Vjq dip) Op (4.33b) Gr j y/jrj + y/jr Grjj = 0 . (4.40) The symmetry of Cauehy's stress tensor (cr,,- = cr,-,) into a symmetric part and an antisymmetric one: resulting from moment equilibrium yields the con- 1 <7// = \ (V/? * V/n Vjq) (Opq + oqp) (4 ^) dition ( + i (V iq '' Vjp - Vpq) (Öpq - Oqp) .
1} H>ip °pj = Vjq Cqi or Ojj = y/~ yjp dpq (4.41 a, b) For the isotropic damage case (y/,-,•= (4-42) is equal to the decomposition a^ = (dy,- + öjj)/2 + which states, that the "net-stress tensor" is non sym- (öjj — öjj)/2, i.e. the "net-stress tensor" is symmetric metric. From (4.41 b) we find the decomposition (djj = öjj) in this special case only.
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