J.Am. Ceran.Sor,881879-1885(2005 DO:l0.lll1551-2916.2005.0037x urna Ceramic Composites with Three-Dimensional Architectures designed to Produce a Threshold Strength--Il Mechanical Observations Geoff E. Fair,*, T-f Ming Y. He, Robert M. McMeeking, and F. F. Lange* Materials Department, University of California, Santa Barbara. California 93106 Finite element modeling and linear elasti mechanics strated for laminate composites. The stress intensity function, are used to model the residual stresses and ramic composites consisting of polyhedral rounded by thin alumina/mullite layers in residual compression K=DaVia+ocia (3) his type of composite architecture is expected to exhibit iso- tropic threshold strength behavior, in which the strength of the was developed to explain the growth of a slit crack in a thick composite for a particular assumed fiaw will be constant and ndependent of the orientation of tensile loading. The results of layer through two thin, bounding compressive layers by an ap- plied tensile stress, Oa, and where 2a is the crack length, IA and he modeling indicate that the strengths of such architectures will be higher than those of laminates of similar architectural layers, respectively, and o is the biaxial cor in the compressive layers. Since the second term in Eq. ( 3) trength behavior for a particular fiaw type Flexural testing duces the stress intensity at the crack tip, the applied stress must of the polyhedral architectures reveals that failure is dominated by processing defects found at junctions between the polyhedra constantly increased in order for the crack to grow through he compressive layer in a stable manner. Namely, Eq (3)pre- Fractography revealed the interaction of these defects with the dicts that crack growth through the bounding compressive residual stresses in the compressive layers that separate the layers is stable, i.e., the crack exhibits an R-curve behavior; ex- perimental results have confirmed this prediction.2 Although the R-curve behavior predicted by Eq (3)and ob- L. Introducti rved for a number of laminar composites is interesting by it- self, of greater interest is the fact that the limits of Eq (3)can be WHsi laye st res ad enteri i thie the de together uses ive haoer s hat lashihai thresphosi s rent hing pei inc lumber of phenomena including differential thermal contrac- lure due to a particular type of defect(assumed or real) does tion during cooling from the fabrication temperature, a phase ot occur until a well-defined stress, o,hr, hereafter referred to as change during cooling, or a molar volume change due to a re- the threshold strength, is reached. Thus, as shown by Eq(3), action that forms one of the materials within the laminate. Using with increasing applied stress, the slit crack that spans the thick the example where the stresses arise due to differential therm layer will extend across the much thinner compressive layers ontraction, a biaxial compressive strain e develops in one layer til it reaches the next thick layer. At this applied stress, the because of its smaller thermal expansion coefficient. As reviewed crack tip is no longer shielded and the crack propagates across by Ho et al, the biaxial compressive stress within the thin layers the remaining layers to produce catastrophic failure. The ap- (material A)is given by plied stress needed to extend the slit crack across the compress- ive stresses can be determined by substituting 2a=/B+2IA and K= Ke into Eq. (3)and rearranging. The result is as follows: I+(IAEA/IBEB here E= El(1-v), E is the Youngs modulus, v is the Pois- thr sons ratio of the material, and ia and ib are the thicknesses of (1+) the thin and thick layers, respectively. The stress within the thick layers(material B)is given by IA\2 1-(1+ (4) IA Equation(4)shows that the threshold strength of the laminate is d on the ude of the residual compress It is clear that as (A/tg approaches zero, i.e., for the case of very ive stress in the thin layers, the fracture toughness of the thin compressive layers, the stress in the thick layers disappears. layers, and the thickness of both the thick and thin layer thin he concept of using compressive regions within brittle ma- Threshold strength behavior has been experimentally ob- terials to stop cracks and to provide an increasing resistance to rved for symmetric, periodic laminates consisting of nearly crack extension with increasing applied stress has been demon- stress-free thick layers(200-650 um)of alumina and thin layers (20-75 um)of mixtures of alumina and mullite(0.1-0.85 volume R. K. Bordiacontributing editor relative to the thick alumina layers. To demonstrate the Manuscript No. 10791. Received January 9, 200-4 approved December 31, 2004. threshold strength behavior of the laminates, indentation pre- I Research. under cracks of varying size (10-225 um) were placed in the center thick layer of the specimens. When the laminates were tested in 'Current address: Materials and Manufacturing Directorate, AFRL/MLLN, Air Force four-point bending with traverse loading with respect to the arch Laboratory, Wright-Paterson Air Force Base, OH 45433 layers, the strengths were observed to be independent of fil 1879Ceramic Composites with Three-Dimensional Architectures Designed to Produce a Threshold Strength—II. Mechanical Observations Geoff E. Fair,* ,w,z Ming Y. He, Robert M. McMeeking, and F. F. Lange* Materials Department, University of California, Santa Barbara, California 93106 Finite element modeling and linear elastic fracture mechanics are used to model the residual stresses and failure stress of ce￾ramic composites consisting of polyhedral alumina cores sur￾rounded by thin alumina/mullite layers in residual compression. This type of composite architecture is expected to exhibit iso￾tropic threshold strength behavior, in which the strength of the composite for a particular assumed flaw will be constant and independent of the orientation of tensile loading. The results of the modeling indicate that the strengths of such architectures will be higher than those of laminates of similar architectural dimensions that were previously found to exhibit threshold strength behavior for a particular flaw type. Flexural testing of the polyhedral architectures reveals that failure is dominated by processing defects found at junctions between the polyhedra. Fractography revealed the interaction of these defects with the residual stresses in the compressive layers that separate the polyhedra. I. Introduction WHEN layers of dissimilar materials are bonded together, residual stresses may develop within the layers by a number of phenomena including differential thermal contrac￾tion during cooling from the fabrication temperature, a phase change during cooling, or a molar volume change due to a re￾action that forms one of the materials within the laminate. Using the example where the stresses arise due to differential thermal contraction, a biaxial compressive strain e develops in one layer because of its smaller thermal expansion coefficient. As reviewed by Ho et al.,1 the biaxial compressive stress within the thin layers (material A) is given by sA ¼ eE0 A 1 þ ðtAE0 A=tBE0 BÞ (1) where E0 i 5 Ei/(1
ni), E is the Young’s modulus, n is the Pois￾son’s ratio of the material, and tA and tB are the thicknesses of the thin and thick layers, respectively. The stress within the thick layers (material B) is given by sB ¼
sA tA tB (2) It is clear that as tA/tB approaches zero, i.e., for the case of very thin compressive layers, the stress in the thick layers disappears. The concept of using compressive regions within brittle ma￾terials to stop cracks and to provide an increasing resistance to crack extension with increasing applied stress has been demon￾strated for laminate composites.2,3 The stress intensity function, K ¼ sa ffiffiffiffiffi pa p þ sc ffiffiffiffiffi pa p 1 þ tA tB 2 p sin
1 tB 2a  
1   (3) was developed to explain the growth of a slit crack in a thick layer through two thin, bounding compressive layers by an ap￾plied tensile stress, sa, and where 2a is the crack length, tA and tB are the thickness of the thin compressive and thicker tensile layers, respectively, and sc is the biaxial compressive stress with￾in the compressive layers.3 Since the second term in Eq. (3) re￾duces the stress intensity at the crack tip, the applied stress must be constantly increased in order for the crack to grow through the compressive layer in a stable manner. Namely, Eq. (3) pre￾dicts that crack growth through the bounding compressive layers is stable, i.e., the crack exhibits an R-curve behavior; ex￾perimental results have confirmed this prediction.2 Although the R-curve behavior predicted by Eq. (3) and ob￾served for a number of laminar composites is interesting by it￾self, of greater interest is the fact that the limits of Eq. (3) can be used to show that laminar composites containing periodic com￾pressive layers can exhibit threshold strength behavior in which failure due to a particular type of defect (assumed or real) does not occur until a well-defined stress, sthr, hereafter referred to as the threshold strength, is reached. Thus, as shown by Eq. (3), with increasing applied stress, the slit crack that spans the thick layer will extend across the much thinner compressive layers until it reaches the next thick layer. At this applied stress, the crack tip is no longer shielded and the crack propagates across the remaining layers to produce catastrophic failure. The ap￾plied stress needed to extend the slit crack across the compress￾ive stresses can be determined by substituting 2a 5 tB12tA and K 5 Kc into Eq. (3) and rearranging. The result is as follows: sthr ¼ Kc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p tB 2 1 þ 2tA tB r   þ sc 1
1 þ tA tB 2 p sin
1 1 1 þ 2tA tB " # ! (4) Equation (4) shows that the threshold strength of the laminate is expected to depend on the magnitude of the residual compress￾ive stress in the thin layers, the fracture toughness of the thin layers, and the thickness of both the thick and thin layers. Threshold strength behavior has been experimentally ob￾served for symmetric, periodic laminates consisting of nearly stress-free thick layers (200–650 mm) of alumina and thin layers (20–75 mm) of mixtures of alumina and mullite (0.1–0.85 volume fraction), which is placed in residual compression during cooling from the densification temperature due to a lower average CTE relative to the thick alumina layers.2,3 To demonstrate the threshold strength behavior of the laminates, indentation pre￾cracks of varying size (10–225 mm) were placed in the center thick layer of the specimens. When the laminates were tested in four-point bending with traverse loading with respect to the layers, the strengths were observed to be independent of flaw 1879 Journal J. Am. Ceram. Soc., 88 [7] 1879–1885 (2005) DOI: 10.1111/j.1551-2916.2005.00377.x R. K. Bordia—contributing editor Supported by the Office of Naval Research, under contract N00014-03-1-0305. *Member, American Ceramic Society. w Author to whom correspondence should be addressed. e-mail: geoff.fair@wpafb.af.mil z Current address: Materials and Manufacturing Directorate, AFRL/MLLN, Air Force Research Laboratory, Wright-Paterson Air Force Base, OH 45433. Manuscript No. 10791. Received January 9, 2004; approved December 31, 2004.