Z.M. Huang /Computers and Structures 80(2002)1159-1176 and the matrix, thermal stresses will be generated in the The overall thermal expansion coefficients of the constituent materials during the temperature variation, composite are thus determined from dT=T-To. The general constitutive equations of the fiber, matrix, and the composite are thus modified to {x}=H{x}+{}+(S]-S]){b}.(17 {d}={s]{dd}+{x}dr, (131) If there is no mechanical load applied to the lamina namely, Ida)=10f, the pure thermal stress increments de"=s"do"1+(a")dT. (13.2) in the constituents are simply expressed as idom]=bm)dT and do])=-bmdT Ide)=[sfda)+iadr where a, a, and ai, respectively, are the thermal ex pansion coefficients of the fiber, matrix, and the com- From Eq (18), the thermal residual stresses in the con- posite at the initial temperature To. In the case of a plane stituents of the lamina can be estimated.Furthermore,if problem, we have of=of=0. It should be noted that the initial interval of the temperature variation, [To, Til. the constituent compliance matrices in Egs (13.1) and is large, a sub-division is required and the final thermal (13.2),S] and Im], are also defined at the initial tem- stresses, o o and om, are obtained by the sum- perature To. For example, sm may not be merely the mation of those from each sub-interval contribution On the other hand the internal stress increments can be related to the overall applied stress incre idah 5. Failure criteria and the temperature variation, dT, through The total stresses in the constituent materials are Ido)=([n+Iml)do)+b)dT obtained by summing up the thermal-mechanical stress =[]{da}+{b}d7 (14.1) wIth residual stress components. Namely, the current total stresses in the en by {d}=团(+m4)-{da}+{b=}dr {d}+{o =[B]{da}+{b"}dT (14.2) where [B]([B) and [B] are called the stress concen tration matrices of the fiber and matrix materials. and where (b) and (bm) are thermal stress concentration factors of the fibers and the matrix, respectively. It is evident o)+=fo0M-)+ do)m-n that the latter two factors satisfy K=0,1,…,.with{o}={0} (19.3) H{b}+n{b}={(0} Therefore, only one concentration factor needs to be om) M-n, K+=fom) M-n,K+ dom)(M-n, determined. This factor is uniquely expressible in terms of the concentration matrices. Choosing bmI as inde 1,…,with{o"n0={0} (194) pendent, Benveniste and Dvorak [12] derived a rigorous The superscript"R refers to residual stresses. If no relation as follows other but thermal residual stresses are involved, they are {b}=(-{=)(]-S)-({2}-{x} calculated using Eq (18). The superscript"M-T"indi- cates that the quantity involved is resulted from the By means of the bridging matrix, the last equation coupled thermo-mechanical effect. The stress incre- calculated from }=(-4(+=)-)s Eqs. (14. 1)and(14.2), rather than from(2)and (3), with constituent properties specified at the current temper S])-({x2}-{x}). (16.1) ture. T. When the stress state in either the fiber or the matrix has attained its ultimate value, the corresponding The fiber thermal stress concentration factor is then composite is considered to have failed. In this way, the given by constituent materials (16.2) In detecting the constituent failure, the maximum
and the matrix, thermal stresses will be generated in the constituent materials during the temperature variation, dT ¼ T1 T0. The general constitutive equations of the fiber, matrix, and the composite are thus modified to fde f g¼½Sf fdrf gþfaf gdT ; ð13:1Þ fde mg¼½SmfdrmgþfamgdT ; ð13:2Þ and fdeg¼½SfdrgþfagdT ð13:3Þ where af i , am i , and ai, respectively, are the thermal expansion coefficients of the fiber, matrix, and the composite at the initial temperature T0. In the case of a plane problem, we have af 3 ¼ am 3 ¼ 0. It should be noted that the constituent compliance matrices in Eqs. (13.1) and (13.2), ½Sf and ½Sm, are also defined at the initial temperature T0. For example, ½Sm may not be merely the elastic component. On the other hand, the internal stress increments can be related to the overall applied stress increments, fdrg, and the temperature variation, dT , through fdrf g¼ðVf ½I þ Vm½AÞ 1 fdrgþfbf gdT ¼ ½Bf fdrgþfbf gdT ; ð14:1Þ fdrmg¼½AðVf ½I þ Vm½AÞ 1 fdrgþfbmgdT ¼ ½BmfdrgþfbmgdT ; ð14:2Þ where ½Bf ( ½B) and ½Bm are called the stress concentration matrices of the fiber and matrix materials, and fbf g and fbmg are thermal stress concentration factors of the fibers and the matrix, respectively. It is evident that the latter two factors satisfy Vffbf g þ Vmfbmg¼f0g: ð15Þ Therefore, only one concentration factor needs to be determined. This factor is uniquely expressible in terms of the concentration matrices. Choosing fbmg as independent, Benveniste and Dvorak [12] derived a rigorous relation as follows fbmg ¼ ð½I ½BmÞð½Sf ½SmÞ 1 ðfamg faf gÞ: By means of the bridging matrix, the last equation becomes fbmg¼ ½I ½AðVf ½I þ Vm½AÞ 1 ð½Sf ½SmÞ 1 ðfamg faf gÞ: ð16:1Þ The fiber thermal stress concentration factor is then given by fbf g¼ Vm Vf fbmg: ð16:2Þ The overall thermal expansion coefficients of the composite are thus determined from fag ¼ Vffaf g þ Vmfamg þ Vmð½Sm ½Sf Þfbmg: ð17Þ If there is no mechanical load applied to the lamina, namely, fdrg¼f0g, the pure thermal stress increments in the constituents are simply expressed as fdrmg ðTÞ ¼ fbmgdT and fdrf g ðTÞ ¼ Vm Vf fbmgdT ð18Þ From Eq. (18), the thermal residual stresses in the constituents of the lamina can be estimated. Furthermore, if the initial interval of the temperature variation, [T0; T1], is large, a sub-division is required and the final thermal stresses, frf gðTÞ and frmgðTÞ , are obtained by the summation of those from each sub-interval contribution. 5. Failure criteria The total stresses in the constituent materials are obtained by summing up the thermal–mechanical stress with residual stress components. Namely, the current total stresses in the constituents are given by rf Kþ1 ¼ rf ðRÞ þ rf ðM–TÞ;Kþ1 ; ð19:1Þ rm f gKþ1 ¼ rm f gðRÞ þ rm f gðM–TÞ;Kþ1 ; ð19:2Þ where rf ðM–TÞ;Kþ1 ¼ rf ðM–TÞ;K þ drf ðM–TÞ ; K ¼ 0; 1; ... ; with rf ðM–TÞ;0 ¼ f0g; ð19:3Þ rm f gðM–TÞ;Kþ1 ¼ rm f gðM–TÞ;K þ drm f gðM–TÞ ; K ¼ 0; 1; ... ; with rm f gðM–TÞ;0 ¼ f0g: ð19:4Þ The superscript ‘‘R’’ refers to residual stresses. If no other but thermal residual stresses are involved, they are calculated using Eq. (18). The superscript ‘‘M–T’’ indicates that the quantity involved is resulted from the coupled thermo-mechanical effect. The stress increments, fdrf gðM–TÞ and fdrmg ðM–TÞ are calculated from Eqs. (14.1) and (14.2), rather than from (2) and (3), with constituent properties specified at the current temperature, T. When the stress state in either the fiber or the matrix has attained its ultimate value, the corresponding composite is considered to have failed. In this way, the composite strength is defined in terms of those of its constituent materials. In detecting the constituent failure, the maximum normal stress criterion is among the best applicable. This 1164 Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176
Z-M. Huang / Computers and Structures 80(2002)1159-1176 criterion is certainly applicable to isotropic matrix ma instantaneous compliance matrix of the matrix material terial. For transversely isotropic fiber, however, the. is given below application is still reasonable. Due to its small cross- ectional dimension. the failure of the fiber material in Sm] when≤哩 he composite is most probably resulted from an exces- [sm+[sm, when om >oy, ive stress in its longitudinal direction, which corre- sponds to the maximum normal stress. Hence, the where composite failure is assumed if either of the following 0 conditions is satisfied 0 d≤d,≤(-d),山≤,≤(-m)(20 where af and af are the first and the third principal stresses in the fiber, of and o are the fiber dndn dadu 20nd tensile and compressive strengths, am and 4M(o) and the third principal stresses in the matrix, and om and symmetry 4012d1 ous are the matrix ultimate tensile and compressive strengths, respectively It has been recognised that the maximum normal EEm stress criterion may not be very accurate if two or three (22.3) principal stresses are close to each other. In such a case, a generalised maximum normal stress criterion [13] may be pertinent. Details are referred to Ref. [13] d 0,ifi≠j i,j=1, 6. Constitutive description for constituent materials lI.if i=i' The three-dimensional Prandth-Reuss theory formu- It is seen from Eq(4)that the overall compliance lae are summarized in Appendix B matrix of the composite relies upon the instantaneous compliance matrices of the fiber and matrix materials. In practice, the fiber material used is generally at most 6.2. Bodner and Partom theory transversely isotropic. Furthermore, most fibers such Some materials such as titanium alloys display sig- glass, carbon/graphite,Kevlar, boron, alumina, etc can nificant temperature dependent thermoelastic/viscoplas be regarded as linearly elastic until rupture. The com- tic behavior. Their constitutive relationships can be best pliance matrix of such a fiber is simply defined using represented using the unified Bodner and Partom model Hooke's law, and keeps unchanged up to failure On the other hand, most matrix materials especially with directional hardening [I5. This unified theory as- metal and polymer materials possess ability of undergo sumes that the total strain of the material is the sum of ing significant inelastic deformation before failure. These the elastic, thermal, and viscoplastic strains. The in taneous compliance matrices of them can be described 6. strains are controlled by the following flow rule g a number of well-developed constitutive theories 1(z+ For an explanation purpose, only two such theories D summarized in this section. One is the prandtReuss theory and another is the Bodner-Partom theory. The where former is applicable to an elasto-plastic material, whereas the latter to an elastic-visco-plastic material. If the ma- J2=l0 d, d=di-doHdjit trix used is a rubber or elastomer material. its instanta- cous compliance matrix can be defined using a model given in Ref. [6 z=mW(z1-2)-A1Z1 6.1. Prandtl-Reuss theory Z\aZ he Prandtl-Reuss plastic flow theory is well known in the literature [4, 14], and only related formulae are Wp=dj 4p z(0)=z0 summarized herein. According to this theory, the planar
criterion is certainly applicable to isotropic matrix material. For transversely isotropic fiber, however, the application is still reasonable. Due to its small crosssectional dimension, the failure of the fiber material in the composite is most probably resulted from an excessive stress in its longitudinal direction, which corresponds to the maximum normal stress. Hence, the composite failure is assumed if either of the following conditions is satisfied: r1 f 6 rf u; r3 f 6 ð rf u;cÞ; r1 m 6 rm u ; r3 m 6 ð rm u;cÞ ð20Þ where r1 f and r3 f are the first and the third principal stresses in the fiber, rf u and rf u;c are the fiber longitudinal tensile and compressive strengths, r1 m and r3 m are the first and the third principal stresses in the matrix, and rm u and rm u;c are the matrix ultimate tensile and compressive strengths, respectively. It has been recognised that the maximum normal stress criterion may not be very accurate if two or three principal stresses are close to each other. In such a case, a generalised maximum normal stress criterion [13] may be pertinent. Details are referred to Ref. [13]. 6. Constitutive description for constituent materials It is seen from Eq. (4) that the overall compliance matrix of the composite relies upon the instantaneous compliance matrices of the fiber and matrix materials. In practice, the fiber material used is generally at most transversely isotropic. Furthermore, most fibers such as glass, carbon/graphite, Kevlar, boron, alumina, etc. can be regarded as linearly elastic until rupture. The compliance matrix of such a fiber is simply defined using Hooke’s law, and keeps unchanged up to failure. On the other hand, most matrix materials especially metal and polymer materials possess ability of undergoing significant inelastic deformation before failure. These materials, however, are generally isotropic. The instantaneous compliance matrices of them can be described using a number of well-developed constitutive theories. For an explanation purpose, only two such theories are summarized in this section. One is the Prandtl–Reuss theory and another is the Bodner–Partom theory. The former is applicable to an elasto-plastic material, whereas the latter to an elastic–visco-plastic material. If the matrix used is a rubber or elastomer material, its instantaneous compliance matrix can be defined using a model given in Ref. [6]. 6.1. Prandtl–Reuss theory The Prandtl–Reuss plastic flow theory is well known in the literature [4,14], and only related formulae are summarized herein. According to this theory, the planar instantaneous compliance matrix of the matrix material is given below. Sm ½ ¼ Sm ½ e ; when rm e 6 rm Y Sm ½ e þ Sm ½ p ; when rm e > rm Y; ð21Þ where ½Sm e ¼ 1 Em mm Em 0 1 Em 0 symmetric 1 Gm 2 6 4 3 7 5; ð22:1Þ Sm ½ p ¼ 9 4Mm T ðrm e Þ 2 r0 11r0 11 r0 22r0 11 2r0 12r0 11 r0 22r0 22 2r0 12r0 22 symmetry 4r0 12r0 12 2 6 4 3 7 5 rij¼rm ij ; ð22:2Þ Mm T ¼ EmEm T Em Em T ; ð22:3Þ r0 ij ¼ rij 1 3 ðr11 þ r22Þdij; dij ¼ 0; if i 6¼ j 1; if i ¼ j ; i;j ¼ 1; 2: ð22:4Þ The three-dimensional Prandtl–Reuss theory formulae are summarized in Appendix B. 6.2. Bodner and Partom theory Some materials such as titanium alloys display significant temperature dependent thermoelastic/viscoplastic behavior. Their constitutive relationships can be best represented using the unified Bodner and Partom model with directional hardening [15]. This unified theory assumes that the total strain of the material is the sum of the elastic, thermal, and viscoplastic strains. The inelastic strains are controlled by the following flow rule [16,17] e_ I ij ¼ D0 exp " 1 2 ðZI þ ZDÞ 2 3J2 !n# r0 ij ffiffiffiffi J2 p ; ð23Þ where J2 ¼ 1 2 r0 ijr0 ij; r0 ij ¼ rij 1 3 rkkdij; Z_ I ¼ m1W_ pðZ1 ZI Þ A1Z1 ZI Z2 Z1 r1 þ T_ ZI Z2 Z1 Z2 oZ1 oT þ Z1 ZI Z1 Z2 oZ2 oT ; W_ p ¼ rije_ I ij; ZD ¼ bijuij; uij ¼ rij ffiffiffiffiffiffiffiffiffiffiffi rklrkl p ; ZI ð0Þ ¼ Z0; Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176 1165