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FIGURE 10.4 Stress distribution in nickel-alumina joint with (solid line) and without (dashed line) graded interlayer. Joint has experienced temperature drop of 100° C from stress-free temperature. (Figure reprinted from Suresh and Mortensen [6].)

FIGURE 10.4 Stress distribution in nickel-alumina joint with (solid line) and without (dashed line) graded interlayer. Joint has experienced temperature drop of 100° C from stress-free temperature. (Figure reprinted from Suresh and Mortensen [6].)

FIGURE 10.5 Possible graded architectures for joint between two materials. Volume fraction of material 2, f2, as function of distance across graded joint normalized to graded region thickness, x/t. Linear gradient is defined as p = 1, parabolic as p > 1, and square root as p < 1. See Fig. 10.2 for definition of x and t.

FIGURE 10.5 Possible graded architectures for joint between two materials. Volume fraction of material 2, f2, as function of distance across graded joint normalized to graded region thickness, x/t. Linear gradient is defined as p = 1, parabolic as p > 1, and square root as p < 1. See Fig. 10.2 for definition of x and t.

h a joints [6, 11, 12]. However, stress reduction in one region may enhance stresses in another; furthermore, stress relaxation through plastic strain may not be desired for some applications [11]. Whether or not the presence of a graded interlayer is desired to reduce effects of thermal residual stress depends on the combination of component geometry and size and material properties.

TABLE 10.1 The Type of Geometry Alters the Stress Distribution"'*

Relative Level of Residual Stress in Material with Lowest Type of Geometry Thermal Coefficient of Expansion (see Figure 10.3) (TCE)C

Plate High

Cylinder Low

Coating Medium aThree generic geometrical configurations are listed: 1. a plate; 2. a cylinder; and 3. a coating.

b Quantitative information concerning locations and magnitudes of stresses may be found in references [7, 8] cIn most metal/ceramic joints, the material with the lowest TCE is the most brittle.

TABLE 10.2 The Type of Gradation Alters the Stress Distribution"'*

Relative Level of Residual Type of Gradation Stress in Material with

(see Figure 10.2 and Lowest Thermal Coefficient of Figure 10.5) Expansion (TCE)C

Parabolic (V = bx2) High

Square-root (V = bx1'2) Low aThree general types of gradients are listed: 1. the volume fraction of material 1, V, varies linearly with distance, x; 2. V varies parabolically with x; and 3. V exhibits a square root dependence on x.

b Quantitative information locations and magnitudes of stresses may be found in references [7-9]

cIn most metal/ceramic joints, the material with the lowest TCE is the most brittle.

### 10.2.2 Mechanical Behavior 10.2.2.1 Elastic

If the constituent properties of materials comprising an FGM are known, then the overall elastic response of the FGM component can be predicted, either by analytic means in the simplest cases or by numerical methods. When the gradation is accomplished by discrete layering, then a prediction of the elastic response is usually a relatively simple mechanics problem. If each layer is considered to be an isotropic composite, then the elastic modulii of that layer should not depend on the fact that the layer is surrounded by other layers. The few reported cases where the modulii of a layer within an FGM are different than those of the same composition composite processed separately are due to slight processing variations that probably result in microstructural differences. For discrete layers where the microstructural features within the layer are much smaller than the layer thickness, classical plate and beam theories of continuum mechanics apply [6]. For example, to predict the bending of an inhomogeneous beam, a homogeneous beam of equivalent cross section should be constructed, and standard mechanics methods should be applied [13]. FGMs frequently experience thermal loading, and, in this case, a system of linear equations that contain the thermal strain and plate curvature should be developed and solved to predict elastic response [6]. The solutions to the thermal residual stress distribution in Fig. 10.4 was solved using these types of classical theories. Detailed application of these techniques to FGMs may be found in [6]. When gradation is continuous, elastic response prediction may be more challenging, and numerical procedures such as the finite-element method are likely the most efficient approaches.

### 10.2.2.2 Plastic Deformation and Fracture

Fracture mechanics concepts can be applied to FGMs in a way that is similar to monocomposition materials, though there may be some deviations. The linear elastic fracture mechanics approach of stress intensity factors can be applied to FGMs if the elastic constants are continuous and differentiable [14]. However, for layered materials in which a crack propagates between the layers, interface fracture mechanics may have to be invoked [15]. The two main differences between a crack that propagates in a graded material perpendicular to the direction of the gradient and one that propagates in a homogeneous material are the amount of shear loading on the crack tip and the precise level of the stress intensity factor [16]. For a given geometry and load application, a crack in a graded material will experience more shear stress and have a lower driving force for propagation than the same crack in a homogeneous material or at a bimaterial interface. The end result is that an FGM is slightly more resistant to crack propagation than a homogeneous material, all else except elastic properties being equal. The magnitudes of the differences in crack propagation driving force depend on the specific geometries, the material properties, and type of gradient but, in general, are in fact quite small. Little is currently known about crack propagation perpendicular to the direction of the gradient when plasticity is relatively large, though it appears that the effect of elastic mismatch is much stronger than plastic mismatch [17].

Numerical simulation studies have shown that as a crack propagates in a linear elastic graded material parallel to the direction of the gradient from a more compliant to a less compliant material, the driving force for its continued propagation (i.e., the mode I stress intensity factor) typically increases [18]. In other words, if the fracture toughness were the same everywhere, the FGM would "appear" less resistant to fracture than a homogeneous material with the same fracture toughness. When plasticity is considered, it has been shown that when a crack grows toward a material with a higher yield stress, the crack's driving force decreases [6]. Since the driving force must be compared with a fracture criterion to predict crack propagation, it is necessary to superimpose the elastic and plastic effects and to compare the resultant driving forces with values of the fracture resistance. More recent mechanics studies on ceramic/metal graded materials in which crack bridging occurs have shown that the fracture resistance increases as the crack propagates toward the metal [19]. In this case, the fracture resistance increases because the crack bridging forces reduce the driving force of the crack tip. Thus, the elastic, plastic, and fracture properties together dictate when the crack driving force will exceed the fracture resistance of the graded material. This complex interplay of loading, component geometry, and distribution of material properties makes it difficult to arrive at a set of general guidelines for design purposes.

The wear of material is a complex process that may occur by a variety of mechanisms that depend on several properties such as the fracture toughness and the hardness, as well as the environmental conditions. Many applications, for example, cutting tools, gears, and prosthetic implants, utilize concepts of wear to engineer a component that exhibits a hard surface and a tough interior. Since for most materials there is an inverse correlation between hardness and toughness, a gradient frequently produces the optimum structure. A compositionally graded approach has been applied to at least three different materials systems: Co-WC, TiC-NiMo, and diamond-SiC. These have been shown to exhibit superior wear resistance compared to monocomposition tools [3]. In the case of TiC-NiMo, optimum graded composites were fabricated in which the compositions ranged from 95 wt % TiC at the surface to 86 wt % at the transition site to plane steel [20]. In the case of diamond-SiC, a graded layer between the diamond chip and the SiC shank is formed through a powder metallurgy-reaction sintering approach.

Elastic modulus gradients are believed to alter wear resistance of materials [6, 21]. While it is difficult to separate the specific effects of elastic modulus gradient on wear, model studies conducted on alumina and glass suggest that wear resistance is improved when the surface is more compliant than the interior [22].

### 10.2.3 Thermomechanical Behavior

Thermomechanical behavior in graded materials refers to nonuniform deformation during heating or cooling induced by differential thermal expansion coefficients. Many of the thermomechanical descriptions for graded materials are only applicable for small deformations in the context of beam and plate theories of classical continuum mechanics [6]. Fortunately, these descriptions are not only simpler than large deformation models but are also applicable to a wider range of applications, since in many applications, large deformations are not acceptable. On the other hand, they do not include dynamic effects, thermal gradients, stress relaxation due to plasticity, and edge and singular effects. User-friendly computer programs have been developed for predicting this kind of deformation [23]. After a component has been exposed to a particular temperature excursion, the resulting deformation and residual stress (e.g., Fig. 10.4) are uniquely determined by the component geometry and material constitutive properties. Specific situations have been analyzed by several authors [24, 25].

The presence of plastic deformation alters the stress distribution from the elastic case above. The von Mises stress criterion is most commonly used in describing plasticity in graded materials. One of the challenges in obtaining accurate predictions is to specify this criterion as a function of position within the graded structure. Because of microstructural changes along the compositional gradation, the effective plastic flow stress may change as well (e.g., the degree of constraint around a plastic phase is a function of composition). The current models do not consider these microstructural variations.

### 10.2.4 Electrical and Optical Characteristics

In describing the electrical conduction of FGMs, it is necessary to consider whether the potential is applied parallel or perpendicular to the gradient. When considering a potential applied perpendicular to the gradient, the electrical conductivity through a particular compositional layer depends on the relative contiguity of the two or more phases within that layer. Applying percolation theory, the electrical conductivity of the two-phase composite A is proportional to the volume fraction of phase 1, V1, and the percolation threshold Vc, which is the spatial location where phase 1 becomes continuous [26, 27]:

where n is an exponent with a value between 1.6 and 2.0 [26, 27]. The percolation threshold Vc depends on the relative particle sizes of the two phases and is approximately 0.16 when the particles are spheres of identical radius. The percolation threshold for the second phase may be different from Vc. In general, the spatial location of the percolation threshold for the two phases are not equivalent, implying that there exists a band of a three-dimensional interpenetrating composite between the two percolation thresholds. Thus, the electrical characteristics of FGMs may be dramatically altered by changes in the microstructure.

In the case where the electrical potential E is applied parallel to the gradient, the current through a graded structure with thickness t is [3]

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