Z.M. Huang / Computers and Structures 80(2002)1177-1199 both thermal and in-plane mechanical fatigue loads. The ( composite was made from ceramic SCS-6 fibers and TIMETAL 21S matrix, with a fiber volume fraction of 0.35[ 8]. Both the constituent materials are isotropic where (delg is obtained from Eq.(2. 1). Calculate in- Whereas the SCs-6 fibers are linearly elastic until rup- equations. (14. 1) and(14.2) of Ref. [i. and update ture. the titanium TIMetAL 2IS matrix exhibits ar elastic-visco-plastic behavior [8,9], for which the Bod Io=oh+ido]k and omh=omh+ido Check if there is any ply ko such that one of the updated er-Partom constitutive model as summarized in Sec tion 6.2 of Ref [1] should be employed stresses,o h, and omIk, has attained its ultimate value Suppose that the composite laminate cools down If yes, ko=ko+1. If all the plies in the laminate have failed, stop calculation. from a stress-free temperature, Tzero to a reference tem- (7) Calculate the plastic strain rate in the matrix of perature, Trer Let the cooling rate be denoted by Tool. If the kth lamina according to Tiero# Tref thermal residual stresses will be generated in the composite. For a general purpose, let the composite 1=(-s=)(ah ycling variation in temperature from Tini to Ifin. If any 2oM(S( je (12) pair of them coincides such as Tini Tin the thermal load and the mechanical load are independently applied. We where [se mlk is the elastic component of the instanta- Tin simultaneously. According to the theories presented equivalent plastic strain rate of us trix material.The can thus suppose that the composite reaches alfin and neous compliance matrix of the in Ref. [I] and in Section 2, a simulation procedure for determined from the formula the in-plane thermal-mechanical fatigue response of the titanium mMc laminate is summarized below (2))=V2(em)(em)/3 (13) (1)LetE,v,r,Pm,四,m,喟,聞,o,,,, f=n, and 0) be given. Calculate all coordinate Note that Eaml should also be included in Eq(13),be- transformation matrixes, ((Te) cause it is not always zero according to a three-dimen- according to equations(44)and(45)of Ref [l] with O sional constitutive description. Suppose that from Eq instead of s. Determine the thermal residual stresses in (12), we hav Sudu+Sdn+s23 d21. Then from the fiber and matrix phases, of) and omIR, of each k- dim)=0 it follows that Es(m)=s2dI+s23d21.Hence, ply of the laminate. Apply to the composite the initial load increment from (0) to (o) increment from Tref to Tini and calculate the initial in ternal stress increments in the fiber and matrix, olk (8)Update T=T+AT and t=t+Af. Define fom. Let oh=o +oh and oml. uniaxial stress-strain curve of the matrix material in the kth lamina using the Bodner-Partom model and the (2)Take 1=0 and M>0. Define dT=(Tin-Tni ) current plastic strain rate. (e)gm). Define(Em)g,(vm L, dt= 1/2oM, and do)(@=(o] -io)ini)/ where o is the cycling frequency(Hz). Define the stress (如)(),(碑)k,(),and(m) Go to step(3) The above process should be repeated until all the and temperature rates as o=2o(alfin-ialini) laminae in the laminate have failed, or, until the maxi- and T= 2o(Tfin-Tini) Calculate the laminate stiffness mum incremental step, M, is attained. In the latter case, elements, @l, Oll, and oll, from Eq (6). Set ko= null. a reversed process(with a changed sign of the tempe 3)According to the current matrix sti Home, ature and load increments) begins. It must be realised the given material parameters, calculate sm], (Ak, that a complete cycling consists of a forward and a Bebe,(aJk, and IBIg(see Ref [ID respectively. reversed processes. If the composite is subjected to a Set (b)=-Vm(bm)/V. Evaluate [(CH)=([Tle) tensile fatigue, the resulting matrix will be generally ()-([r2)4 subjected to a loading condition only in one directional (4)Calculate the load incremental vectors, IdN process, either in the forward or in the reversed process dNrY, dNxy, dMxx, dMn, dMxr. Update the laminate depending on the thermal residual stress level and the stifness and equivalent thermal load elements using Eqs. laminate lay-up arrangement. While in the other direc- (9)and(10), if necessary tional process, the matrix will be subjected to an un- ()Calculate the laminate in-plane strain and cur- loading. In the case of the unloading, only the elastic ature increments, daxx, dery, and dexy and dxxx, dry, component of the material instantaneous compliance and dx Dy, using Eq. (5) matrix will retain. In other words the matrix constitu- (6) For each un-failed kth lamina, calculate its load tive equations must be described using Hooke's law share in the local coordinate system according to during the unloadingboth thermal and in-plane mechanical fatigue loads. The composite was made from ceramic SCS-6 fibers and TIMETAL 21S matrix, with a fiber volume fraction of 0.35 [8]. Both the constituent materials are isotropic. Whereas the SCS-6 fibers are linearly elastic until rup￾ture, the titanium TIMETAL 21S matrix exhibits an elastic-visco-plastic behavior [8,9], for which the Bod￾ner–Partom constitutive model as summarized in Sec￾tion 6.2 of Ref. [1] should be employed. Suppose that the composite laminate cools down from a stress-free temperature, Tzero to a reference tem￾perature, Tref Let the cooling rate be denoted by T_ cool. If Tzero 6¼ Tref thermal residual stresses will be generated in the composite. For a general purpose, let the composite be subjected to a cycling load from frgini to frgfin and a cycling variation in temperature from Tini to Tfin. If any pair of them coincides such as Tini ¼ Tfin the thermal load and the mechanical load are independently applied. We can thus suppose that the composite reaches frgfin and Tfin simultaneously. According to the theories presented in Ref. [1] and in Section 2, a simulation procedure for the in-plane thermal–mechanical fatigue response of the titanium MMC laminate is summarized below. (1) Let Ef , mf , af ; Em, mm, am, rm Y, Em T , rf u, rf u;c; rm u , rm u;c, fzkgðNÞ k¼0, and fhkgðNÞ k¼1 be given. Calculate all coordinate transformation matrixes, ð½T  cÞk  ðNÞ k¼1 and ð½T  sÞk  ðNÞ k¼1, according to equations (44) and (45) of Ref. [1] with hk instead of n. Determine the thermal residual stresses in the fiber and matrix phases, frf gR k and frmgR k , of each k￾ply of the laminate. Apply to the composite the initial load increment from {0} to frgini and the temperature increment from Tref to Tini and calculate the initial in￾ternal stress increments in the fiber and matrix, frf g 0 k and frmg0 k . Let frf gk ¼ frf g R k þ frf gk and frmgk ¼ frmgR k þ frmg0 k . (2) Take t ¼ 0 and M > 0. Define dT ¼ ðTfin  TiniÞ/ M, dt ¼ 1/2xM, and fdrgðGÞ ¼ ðfrgfin  frginiÞ=M, where x is the cycling frequency (Hz). Define the stress and temperature rates as fr_gðGÞ ¼ 2x frgfin  frgini ð Þ and T_ ¼ 2xð Þ Tfin  Tini . Calculate the laminate stiffness elements, QI ij; QII ij , and QIII ij , from Eq. (6). Set k0 ¼ null. (3) According to the current matrix stresses, frmgk , and the given material parameters, calculate ½Sm k ; ½A k , ½B k , fbmgk , fagk , and fbgG k (see Ref. [1]) respectively. Set fbf gk ¼ Vmfbmgk=Vf . Evaluate ½ðCG ij Þk  ¼ ð½T  cÞk  ð½S k Þ 1 ð½T  T c Þk . (4) Calculate the load incremental vectors, fdNXX ; dNYY ; dNXY ; dMXX ; dMYY ; dMXY gT. Update the laminate stiffness and equivalent thermal load elements using Eqs. (9) and (10), if necessary. (5) Calculate the laminate in-plane strain and cur￾vature increments, de0 XX , de0 YY , and de0 XY and dj0 XX , dj0 YY , and dj0 XY , using Eq. (5). (6) For each un-failed kth lamina, calculate its load share in the local coordinate system according to fdrgk ¼ ð½T  T s Þk ½ðCG ij Þk fdeg G k   fbgG k dT  ; where fdegG k is obtained from Eq. (2.1). Calculate in￾ternal stress increments, fdrf gk and fdrmgk , from equations. (14.1) and (14.2) of Ref. [1], and update frf gk ¼ frf gk þ fdrf gk and frmgk ¼ frmgk þ fdrmgk . Check if there is any ply k0 such that one of the updated stresses, frf gk and frmgk , has attained its ultimate value. If yes, k0 ¼ k0 þ 1. If all the plies in the laminate have failed, stop calculation. (7) Calculate the plastic strain rate in the matrix of the kth lamina according to e_ I n oðmÞ k ¼ ð½Sm k  ½Se;m k Þfr_gk ¼ 2xMð½Sm k  ½Se;m k Þfdrgk ; ð12Þ where ½Se;m k is the elastic component of the instanta￾neous compliance matrix of the matrix material. The equivalent plastic strain rate of the matrix, ðe_ I Þ ðmÞ k , is determined from the formula, ðe_ I Þ ðmÞ k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðe_ I;ðmÞ ij Þk ðe_ I;ðmÞ ij Þk=3 q : ð13Þ Note that e_ I;ðmÞ 33 should also be included in Eq. (13), be￾cause it is not always zero according to a three-dimen￾sional constitutive description. Suppose that from Eq. (12), we have e_ I;ðmÞ 22 ¼ 121r_ 11 þ 122r_ 22 þ 123r_ 21. Then from r_ I;ðmÞ 33 ¼ 0 it follows that e_ I;ðmÞ 33 ¼ 121r_ 11 þ 123r_ 21. Hence, ðe_ I Þ ðmÞ k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½ðe_ I;ðmÞ 11 Þ 2 k þ ðe_ I;ðmÞ 22 Þ 2 k þ ðe_ I;ðmÞ 33 Þ 2 k þ 2ðe_ I;ðmÞ 12 Þ 2 k =3 q : (8) Update T ¼ T þ DT and t ¼ t þ Dt. Define the uniaxial stress–strain curve of the matrix material in the kth lamina using the Bodner–Partom model and the current plastic strain rate, ðe_ I Þ ðmÞ k . Define ðEmÞk , ðmmÞk , ðamÞk , ðrm YÞk , ðEm T Þk , ðrm u Þk , and ðrm u;cÞk . Go to step (3). The above process should be repeated until all the laminae in the laminate have failed, or, until the maxi￾mum incremental step, M, is attained. In the latter case, a reversed process (with a changed sign of the temper￾ature and load increments) begins. It must be realised that a complete cycling consists of a forward and a reversed processes. If the composite is subjected to a tensile fatigue, the resulting matrix will be generally subjected to a loading condition only in one directional process, either in the forward or in the reversed process depending on the thermal residual stress level and the laminate lay-up arrangement. While in the other direc￾tional process, the matrix will be subjected to an un￾loading. In the case of the unloading, only the elastic component of the material instantaneous compliance matrix will retain. In other words, the matrix constitu￾tive equations must be described using Hooke’s law during the unloading. 1186 Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199