H. Ismar, F. Streicher Computational Materials Science 16(1999)17-24 a(o1+a2+)+a1(+2+3+2 and compressive loading. If the stress vector in a matrix element reaches the fracture surface. a +2G3+206)+2a(0102+0203 crack is initiated. The modelling is carried out with +0301-02-G3-03)=1 (2) a crack model that does not change the topology of the finite element structure. but varies the ele The remaining constants al, au and aj can be determined from the values of strength under ment specific elastic constants [5]. The orientation of the crack is determined by using the energy re- uniaxial tension (ut), uniaxial compression (uc) lease rate [6]. The difference in the potential energy between the state before and the state after crack strength values is taken into account by statistics. initiation in relation to the crack surface has to be They can be considered to be Weibull distributed [3]. The Weibull distribution describes the proba- maximum. The energy release rate is given by bility of failure of a specimen at a given stress o as a function of its volume v. with△A=A-A· denoting the change in the crack P(a)=1-exp-vx urface, and AU=U-U the change in the po- tential energy. The parameters without an asterisk The Weibull-shape parameter m characterizes the represent the state before crack initiation; the pa- dispersion of the failure stress. The smaller the rameters with an asterisk. the situation afterwards value of m the greater dispersion. The constant After the crack orientation is determined the crack Go can be interpreted as the stress up to which 63% is modelled by reducing elastic constants in of the specimens have failed. Vo is the reference in the direction of the normal of the crack surface volume used to determine go. Ou is the stress below x, the Poissons ratios vry, nd the shear which no failure occurs. In the case of ceramic moduli Gv, Grg within the elasticity matrix are materials one cannot exclude failure even in the reduced. Afterwards the tensor of elastic constants case of very small stresses because of inhomoge- is transformed back to the original system(x, y, z) neities that are always present. Therefore, u is assumed to be zero [4]. Thus, Eq(3)can be re- By introducing this first crack the behavior of the written a matrix element changes from isotropic to trans- versalisotropic. It is possible to insert three or- (4) thogonal cracks into each element. If, in the case of load reversal. a crack is loaded under com- pression, the original stiffness is restored, because the crack closes and can transfer load again damage under compression occurs, the material G (5) collapses completely and all stiffnesses of the ele- ment are reduced Failure is checked independently for each finite element. The average volume of a matrix element 2. 2. Fiber conducts V= 4x10-8 m3. The values used re ferring to this volume are out= 500 MPa The fiber also shows isotropic behavior, so the 00= 1490 MPa, 00=675 MPa and mm=9.7. criterion defined in Eq. (2)can be used to describe For these strength values the surface of fracture is fiber failure. The elastic and thermal constants represented by a spheroid in the principle stress E= 200 000 MPa, v=0.25,G=80 000 MPa and space. The axis of the spheroid coincides with the a=3x 10-K- are taken from the literature pace diagonal of the principle stress state. Its [7, 8]. If a fiber of constant diameter is regarded center is moved towards the seventh octant be the fracture probability for a fiber with a length L cause of the differing strength values for tension is defined [9] by Eq (4)witha1…r1 ‡ r2 ‡ r3† ‡ a11 r2 1 ÿ ‡ r2 2 ‡ r2 3 ‡ 2r2 4 ‡ 2r2 5 ‡ 2r2 6
‡ 2a12…r1r2 ‡ r2r3 ‡r3r1 ÿ r2 4 ÿ r2 5 ÿ r2 6† ˆ 1: …2† The remaining constants a1, a11 and a12 can be determined from the values of strength under uniaxial tension (ut), uniaxial compression (uc) and pure shear (s). The strong scattering of the strength values is taken into account by statistics. They can be considered to be Weibull distributed [3]. The Weibull distribution describes the proba￾bility of failure of a specimen at a given stress r as a function of its volume V: Pf…r† ˆ 1 ÿ exp  ÿ V V0 r ÿ ru r0  m : …3† The Weibull-shape parameter m characterizes the dispersion of the failure stress. The smaller the value of m the greater the dispersion. The constant r0 can be interpreted as the stress up to which 63% of the specimens have failed. V0 is the reference volume used to determine r0. ru is the stress below which no failure occurs. In the case of ceramic materials one cannot exclude failure even in the case of very small stresses because of inhomoge￾neities that are always present. Therefore, ru is assumed to be zero [4]. Thus, Eq. (3) can be re￾written as Pf…r† ˆ 1 ÿ exp  ÿ r r 0  m …4† with r 0 ˆ V0 V  1=m r0: …5† Failure is checked independently for each ®nite element. The average volume of a matrix element conducts V ˆ 4 10ÿ18 m3: The values used re￾ferring to this volume are r 0;ut ˆ 500 MPa, r 0;uc ˆ 1490 MPa, r 0;s ˆ 675 MPa and mm ˆ 9:7. For these strength values the surface of fracture is represented by a spheroid in the principle stress space. The axis of the spheroid coincides with the space diagonal of the principle stress state. Its center is moved towards the seventh octant be￾cause of the di€ering strength values for tension and compressive loading. If the stress vector in a matrix element reaches the fracture surface, a crack is initiated. The modelling is carried out with a crack model that does not change the topology of the ®nite element structure, but varies the ele￾ment speci®c elastic constants [5]. The orientation of the crack is determined by using the energy re￾lease rate [6]. The di€erence in the potential energy between the state before and the state after crack initiation in relation to the crack surface has to be maximum. The energy release rate is given by G ˆ DU DA …6† with DA ˆ A ÿ A denoting the change in the crack surface, and DU ˆ U ÿ U the change in the po￾tential energy. The parameters without an asterisk represent the state before crack initiation; the pa￾rameters with an asterisk, the situation afterwards. After the crack orientation is determined the crack is modelled by reducing elastic constants in the crack system (xr ; yr ;zr ). Young's modulus Exr in the direction of the normal of the crack surface xr , the Poisson's ratios mxryr , mxrzr and the shear moduli Gxryr , Gxrzr within the elasticity matrix are reduced. Afterwards the tensor of elastic constants is transformed back to the original system (x; y;z). By introducing this ®rst crack the behavior of the matrix element changes from isotropic to trans￾versalisotropic. It is possible to insert three or￾thogonal cracks into each element. If, in the case of load reversal, a crack is loaded under com￾pression, the original sti€ness is restored, because the crack closes and can transfer load again. If damage under compression occurs, the material collapses completely and all sti€nesses of the ele￾ment are reduced. 2.2. Fiber The ®ber also shows isotropic behavior, so the criterion de®ned in Eq. (2) can be used to describe ®ber failure. The elastic and thermal constants E ˆ 200 000 MPa, m ˆ 0:25, G ˆ 80 000 MPa and a ˆ 3 10ÿ6 Kÿ1 are taken from the literature [7,8]. If a ®ber of constant diameter is regarded, the fracture probability for a ®ber with a length L is de®ned [9] by Eq. (4) with H. Ismar, F. Streicher / Computational Materials Science 16 (1999) 17±24 19