K Hbaieb, R M. McMeeking/ Mechanics of Materials 34(2002)755-772 → compressive layer 2→ tensile layer G r2 △aMTE (simulation) Aa△TE Crack Length 2a/t Fig. 4. Comparison of simulation results with theoretical model results for a homogeneous material. Both tensile and compressive layers have same thickness. Erdogan, 1972)arising when the crack tip is ex- starting with a length 2a just greater than t2 so that actly at the interface and because of path de- the crack tip is just inside a compressive layer. The pendence of J when the domain for its calculation total crack tip stress intensity factor, K, is the sum encompasses the neighboring layers. It is worth of the applied load stress intensity factor Kapplied noting here that the trend of our results in the and the residual stress intensity factor Residual close vicinity of the interface is consistent with However, for the crack to grow, the total stress the results of Cook and Erdogan(1972). A com- intensity factor, K must be equal to the fracture parison between the finite element results and the toughness, Ke, of the compressive layer. This al- theoretical model results is also given. In the lows us to calculate the applied stress intensity theoretical model results, it is assumed that both factor needed to sustain crack growth as tensile and compressive layers have the same Applied=Ke-Kresidual elastic properties, E2 and v2. It can 5 that the theoretical model is somewhat in error Defining a parameter S=S(a/t2, t2/t1, E2/E1as makes it clear that the finite element results are s、K的,m叫 d stress Gcrackgrowth nec when the ceramic layers are heterogeneous. This needed for a proper treatment of the threshold strength problem we can The results of the finite element simulation are essary to sustain crack growth by combining Eqs used as follows to obtain the threshold strength. (4)and(5)to give Values of△a△TE, tI and t2 are chosen so that Kresidual for each crack length is fixed. The crack i K=K Ocrackgrow then considered to grow through the material S√2
Erdogan, 1972) arising when the crack tip is exactlyat the interface and because of path dependence of J when the domain for its calculation encompasses the neighboring layers. It is worth noting here that the trend of our results in the close vicinityof the interface is consistent with the results of Cook and Erdogan (1972). A comparison between the finite element results and the theoretical model results is also given. In the theoretical model results, it is assumed that both tensile and compressive layers have the same elastic properties, E2 and m2. It can be seen in Fig. 5 that the theoretical model is somewhat in error when the ceramic layers are heterogeneous. This makes it clear that the finite element results are needed for a proper treatment of the threshold strength problem. The results of the finite element simulation are used as follows to obtain the threshold strength. Values of DaDTE0 1, t1 and t2 are chosen so that Kresidual for each crack length is fixed. The crack is then considered to grow through the material starting with a length 2a just greater than t2 so that the crack tip is just inside a compressive layer. The total crack tip stress intensityfactor, K, is the sum of the applied load stress intensityfactor Kapplied and the residual stress intensityfactor Kresidual. However, for the crack to grow, the total stress intensityfactor, K must be equal to the fracture toughness, Kc, of the compressive layer. This allows us to calculate the applied stress intensity factor needed to sustain crack growth as: Kapplied ¼ Kc Kresidual ð4Þ Defining a parameter S ¼ Sða=t2; t2=t1; E2=E1Þ as S ¼ Kapplied rapplied ffiffiffiffiffiffi p 2 t2 p ð5Þ we can compute the applied stress rcrackgrowth necessaryto sustain crack growth bycombining Eqs. (4) and (5) to give rcrackgrowth ¼ Kc Kresidual S ffiffiffiffiffiffi p 2 t2 p ð6Þ Fig. 4. Comparison of simulation results with theoretical model results for a homogeneous material. Both tensile and compressive layers have same thickness. 760 K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772
K Hbaieb, R M. MeMeeking Mechanics of Materials 34(2002)755-772 E/E1=1.7,t2/t Compressive Layer l→ compressive layer △aATE"1Jm △aTE12 Fig. 5. Simulation results for the case where the elastic modulus in the tensile layer E2 is 1.7 times higher than the elastic modulus in the compressive layer El. The theoretical model results for homogeneous material is also plotted for comparison. Clearly S is given as a function of a t2 for the case all the way to the interface with the tensile layer. In of t2/+1=l, E2/E1=1.7 by the plot of Kapplied/ this case, unstable crack growth will set in while (applied vit/2)for the finite element simulation the tip is still in the compressive layer. However, in results given in Fig. 5. The stress crackgrowth(nor general malized by△a△TE1), needed to sustain crack growth is plotted against crack length in Fig. 6 for the case t2/1=1,E2/E1=1.7andK/(△△TE1 maxS t VIt/2)=0.123 with Kc assumed to be the same in both the tensile and compressive layers. It can where max[ indicates the maximum value of the be seen that this stress rises with crack length until term inside the brackets for crack lengths lying in the crack tip reaches the interface with the tensile the range t2< 2a< t2+ 2t1 yer, indicating stable crack growth to that extent As noted above, in some cases stable growth If the toughness of the tensile layer is equal to or occurs at least until the crack tip is almost at the less than Kc, the crack will begin to propagate interface between the compressive layer and the unstably under load control once it has reached adjacent tensile layer (i.e, as in Fig. 6). Since we the interface. We take this situation to denote the do not have a suitable model for what happens strength of the layered ceramic and define the when a crack tip grows through the interface stress level at that stage to be oth, the threshold( Cook and Erdogan, 1972)we prefer to define the trength of the material, thereby ignoring the threshold strength in this case as the stress neces- possibility of a tougher tensile layer leading to an sary to drive the crack to a position just short of ength. In this interface. The actual threshold strength reaches a maximum before the crack tip penetrates be higher than this due to at least two possibilities
Clearly S is given as a function of a=t2 for the case of t2=t1 ¼ 1, E2=E1 ¼ 1:7 bythe plot of Kapplied= ðrapplied ffiffiffiffiffiffiffiffiffiffiffi pt2=2 p Þ for the finite element simulation results given in Fig. 5. The stress rcrackgrowth (normalized by DaDTE0 1), needed to sustain crack growth is plotted against crack length in Fig. 6 for the case t2=t1 ¼ 1, E2=E1 ¼ 1:7 and Kc=ðDaDTE0 ffiffiffiffiffiffiffiffiffiffiffi 1 pt1=2 p Þ ¼ 0:123 with Kc assumed to be the same in both the tensile and compressive layers. It can be seen that this stress rises with crack length until the crack tip reaches the interface with the tensile layer, indicating stable crack growth to that extent. If the toughness of the tensile layer is equal to or less than Kc, the crack will begin to propagate unstablyunder load control once it has reached the interface. We take this situation to denote the strength of the layered ceramic and define the stress level at that stage to be rth, the threshold strength of the material, therebyignoring the possibilityof a tougher tensile layer leading to an even higher strength. In some cases, rcrackgrowth reaches a maximum before the crack tip penetrates all the wayto the interface with the tensile layer. In this case, unstable crack growth will set in while the tip is still in the compressive layer. However, in general, rth ¼ max Kc Kresidual S ffiffiffiffiffiffi p 2 t2 p " # ð7Þ where max½ indicates the maximum value of the term inside the brackets for crack lengths lying in the range t2 6 2a 6 t2 þ 2t1. As noted above, in some cases stable growth occurs at least until the crack tip is almost at the interface between the compressive layer and the adjacent tensile layer (i.e., as in Fig. 6). Since we do not have a suitable model for what happens when a crack tip grows through the interface (Cook and Erdogan, 1972) we prefer to define the threshold strength in this case as the stress necessaryto drive the crack to a position just short of this interface. The actual threshold strength may be higher than this due to at least two possibilities. Fig. 5. Simulation results for the case where the elastic modulus in the tensile layer E2 is 1.7 times higher than the elastic modulus in the compressive layer E1. The theoretical model results for homogeneous material is also plotted for comparison. K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772 761