Crack propagation and residual stress in laminated Si₃N₄/BN composite structures

Abstract

The level of residual stress and crack propagation in a new generation of laminates, based on silicon nitride (Si₃N₄) layer and a mixture of boron nitride (BN) and alumina (Al₂O₃) interlayer, was presented. The structure consists of alternated concentric rings of Si₃N₄ separated by the weak BN interlayer possessing no planes of easy crack propagation and fracture resistance much larger than that of any classical planar laminates. The results on direction of crack propagation and residual stress in relation to inter-layer composition, the number of layers, and their thickness are investigated and reported. The effect of residual stress on crack propagation was studied by using Vicksrs intentation. The highest compressive residual stress of ∼170 MPa was found in samples with five layers possessing an average layer thickness of ∼310 × 10⁻⁶ m.

Introduction

A new generation of laminated structures has been designed and fabricated using a modified slip casting method followed by pressureless sintering.1, 2, 3 Instead of layers being deposited in the planar form, as is the case with planar laminates, in the new design they are arranged in the form of concentric rings along the length of the structure thus avoiding the creation of plane for easy crack propagation.²

Two different laminate compositions (in the terms of interlayer phase composition) were designed. The first one, consists of Si₃N₄ layers, and interlayers containing 10 wt.% of Si₃N₄ and 90 wt.% of BN marked as SN − (SN + BN); and the second one, consists of Si₃N₄ layers, and interlayers containing 50 wt.% of Al₂O₃ and 50 wt.% of BN marked as SN − (BN + Al₂O₃). In these types of structures the objective is to create interfacial layer sufficiently weak to allow crack deflection along the interface. On the other hand, the strength of the Si₃N₄ layer must be sufficiently high to prevent an easy crack initiation at the surface of the next layer. The residual stress is known to strongly affect mechanical behaviour of the laminated composite by controlling the crack propagation at the interface between the two dissimilar materials.4, 5 According to He et al., there are three possibilities for the crack propagation: the crack may arrest at the interface under compressive stress, it may penetrate the interface, or it may deflect from the interface. The structural model which will provide this to take place is proposed in Fig. 1.

During the fabrication of composites with two dissimilar materials (Si₃N₄ layers and BN − Al₂O₃ interlayer) residual stresses will arise due to the mismatch in thermal expansion and differences in elastic properties.6, 7, 8 In the case of laminates with different ceramics, the differences in shrinkage during sintering will also contribute to the residual stress.⁹ Perhaps, the most important factor which contributes to the rise in residual stress is the difference in the linear coefficient of thermal expansion between the two materials used to fabricate laminates. The residual strain, responsible for the creation of residual stress, which develops in a composite laminate consisting of materials with different thermal expansion coefficients, as proposed by the Oechsner et al.¹⁰ can be expressed by equation:

$${\epsilon }_M={\int }_T^{T_0}({\alpha }_2-{\alpha }_1)dT$$

where α₁ and α₂ are the thermal expansion coefficients of the two materials and dT is the temperature difference T₁ − T₀.

In the Si₃N₄ laminates the residual bi-axial compressive stress in the Si₃N₄ is given by Blugan et al.¹¹:

$${\sigma }_C=-\frac{{\epsilon }_M\cdot {E^{\prime }}_1}{1+(t_1\cdot {E^{\prime }}_1/t_2\cdot {E^{\prime }}_2)}$$

where E′ = E/(1 − ν), E₁ and E₂ are Young's modulus of materials with thermal expansion coefficients of α₁ and α₂, respectively, ν is the Poisson's ratio and t₁ and t₂ are the thickness of the layer 1 and layer 2, respectively. In the interlayer (Al₂O₃ + BN) which has larger thermal expansion coefficient compared to silicon nitride, the bi-axial tensile stress is given by¹¹:

$${\sigma }_T=-{\sigma }_C\frac{t_1}{t_2}$$

If t₁/t₂ approaches 0 then the bi-axial stress in material with lower thermal expansion coefficient (i.e. Si₃N₄) reduces to:

$${\sigma }_C=-\frac{{\epsilon }_M\cdot E_1}{1-\nu }$$

and the stress in the material with larger thermal expansion coefficient vanishes (σ_T approaches 0).

The effect of residual stress on crack propagation was studied using Vickers indentation by Hsueh and Evans⁶ who developed a method of measuring the residual stress in a ceramic matrix by comparing the growth of crack from indentations made in unstressed and stressed materials. The same method was used by Marshal and Lawn¹² to evaluate the residual stress in tempered glass. In 1994 Zeng and Rowcliffe¹³ developed a method of mapping the residual stress in glass. First, they introduced a residual stress in the glass by applying high external loads. Then, using low loads, small indentations were made at various angles and distances from the cracks extending from the large indentations. It was than possible to map the residual stress field created on the materials surface by the indentation.

By using indentation fracture mechanics theory it has been possible to determine the fracture toughness, K_IC, of the material by measuring the load and crack length C₀, via equation of the form¹⁴:

$$K_{IC}=\alpha \cdot {\left(\frac{E}{H}\right)}^{1/2}\cdot \left(\frac{P}{{C_0}^{3/2}}\right)$$

where P is the indentation loads, α is an empirical geometric constant, and E and H are Young's modulus and hardness, respectively. Following Zeng and Rowcliffe¹³ it has been shown that, under the influence of a prevailing residual stress, σ_a, the crack will assume a new equilibrium length, C, when the surface of the specimen is loaded at the same indentation load, P, as in the unstressed case. At equilibrium, the crack will then experience composite stress intensity, described by the fracture toughness, K_IC, and is given by the expression:

$$K_{IC}=\alpha \cdot {\left(\frac{E}{H}\right)}^{1/2}\cdot \left(\frac{P}{{C_0}^{3/2}}\right)\pm \psi {\sigma }_a{C}^{1/2}$$

The first term in Eq. (6) represents the stress intensity due to the indentation load, P, and the second term corresponds to the contribution of a prevailing residual stress. The sign ± refers to the tensile or compressive stress. Under the tensile stress, the second term is added to the first term, and under the compressive stress the second term is subtracted from the first term. In Eq. (6), ψ is a crack geometry factor and describes the nature of the surface-to-depth ratio of the crack dimensions. The commonly used value for ψ for Vickers indenter is $\sqrt{\pi }$ (Lawn).¹⁵ Combining Eqs. (5), (6) and noting that the same peak load is involved in both expressions, the unknown residual stress is obtained through the expressions:

$${\sigma }_T=K_{IC}\cdot \left[\frac{1-{(C_0/C)}^{3/2}}{\psi \cdot \sqrt{C}}\right]$$

$${\sigma }_C=-K_{IC}\cdot \left[\frac{1-{(C_0/C)}^{3/2}}{\psi \cdot \sqrt{C}}\right]$$

where σ_T and σ_C represent residual tensile and compressive stresses, respectively. A tensile residual stress will act to extend the stress-free crack so that C > C₀, while a compressive residual stress will shorten it so C < C₀.

The objective of this paper is to investigate the effect of number of layers, their thickness and thermal properties on the level of residual stress developed in the layers of the laminates. Also, an additional objective of this paper was to study the role of the residual stress in toughening of the laminated composites.