T-stress extraction

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The asymptotic expansion of the stress field near a sharp crack in a linear elastic body with respect to r, the distance from the crack tip, is

\begin{matrix} σ_{i j} = & \frac{K_{I}}{\sqrt{2 π r}} f_{i j}^{I} (θ) + \frac{K_{I I}}{\sqrt{2 π r}} f_{i j}^{I I} (θ) + \frac{K_{I I I}}{\sqrt{2 π r}} f_{i j}^{I I I} (θ) + T δ_{1 i} δ_{1 j} + t_{1}^{o} δ_{1 i} δ_{2 j} + t_{2}^{o} δ_{2 i} δ_{2 j} + t_{3}^{o} δ_{2 i} δ_{3 j} + \\ + σ_{13}^{o} δ_{1 i} δ_{3 j} + (ν T + E ε_{33}) δ_{3 i} δ_{3 j} + O (r^{1 / 2}) \end{matrix}

(Williams, 1957), where r and $θ$ are the in-plane polar coordinates centered at the crack tip. The local axes are defined so that the 1-axis lies in the plane of the crack at the point of interest on the crack front and is perpendicular to the crack front at this point; the 2-axis is normal to the plane of the crack (and thus is perpendicular to the crack front); and the 3-axis lies tangential to the crack front. $t^{o}$ is the surface traction on the crack surfaces at the crack tip, and $σ_{13}^{o}$ is a constant stress term for $σ_{13}$ . $ε_{33}$ is the extensional strain along the crack front. In plane strain $ε_{33} = 0$ ; in plane stress the term $(ν T + E ε_{33}) δ_{3 i} δ_{3 j}$ vanishes.

The T-stress usually arises in the discussions of crack stability and kinking for linear elastic materials. For small amounts of crack growth under Mode I loading, a straight crack path has been shown to be stable when $T < 0$ , whereas the path will be unstable and, therefore, will deviate from being straight when $T > 0$ (Cotterell and Rice, 1980). A similar trend has been found in three-dimensional crack propagation studies by Xu, Bower, and Ortiz (1994). Hutchinson and Suo (1992) also showed how the advancing crack path is influenced by the T-stress once cracking initiates under mixed-mode loading. (The direction of crack initiation can be otherwise predicted using the criteria discussed in Prediction of the direction of crack propagation.)

The T-stress also plays an important role in elastic-plastic fracture analysis, even though the T-stress is calculated from the linear elastic material properties of the same solid containing the crack. The early study of Larsson and Carlsson (1973) demonstrated that the T-stress can have a significant effect on the plastic zone size and shape and that the small plastic zones in actual specimens can be predicted adequately by including the T-stress as a second crack-tip parameter. Some recent investigations (Bilby et al., 1986; Al-Ani and Hancock, 1991; Betegón and Hancock, 1991; Du and Hancock, 1991; Parks, 1992; and Wang, 1991) further indicate that the T-stress can correlate well with the tensile stress triaxiality of elastic-plastic crack-tip fields. The important feature observed in these works is that a negative T-stress can reduce the magnitude of the tensile stress triaxiality (also called the hydrostatic tensile stress) ahead of a crack tip; the more negative the T-stress becomes, the greater the reduction of tensile stress triaxiality. In contrast, a positive T-stress results only in modest elevation of the stress triaxiality. It was found that when the tensile stress triaxiality is high, which is indicated by a positive T-stress, the crack-tip field can be described adequately by the HRR solution (Hutchinson, 1968; Rice and Rosengren, 1968), scaled by a single parameter: the J-integral; that is, J-dominance will exist. When the tensile stress triaxiality is reduced (indicated by the T-stress becoming more negative), the crack-tip fields will quickly deviate from the HRR solution, and J-dominance will be lost (the asymptotic fields around the crack tip cannot be well characterized by the HRR fields). Thus, using the T-stress (calculated based on the load level and linear elastic material properties) to characterize the triaxiality of the crack-tip stress state and using the J-integral (calculated based on the actual elastic-plastic deformation field) to measure the scale of the crack-tip deformation provides a two-parameter fracture mechanics theory to describe the Mode I elastic-plastic crack-tip stresses and deformation in plane strain or three dimensions accurately over a wide range of crack configurations and loadings.

To extract the T-stress, we use an auxiliary solution of a line load, with magnitude f, applied in the plane of crack propagation and along the crack line:

σ_{11}^{L} = \frac{f}{π r} \cos^{3} θ, σ_{22}^{L} = \frac{f}{π r} \cos θ \sin^{2} θ, σ_{12}^{L} = \frac{f}{π r} \sin θ \cos^{2} θ,

σ_{33}^{L} = \frac{f}{π r} ν \cos θ, σ_{13}^{L} = σ_{23}^{L} = 0.

The term $σ_{33}^{L} = 0$ for plane stress.

The interaction integral used is exactly the same as that for extracting the stress intensity factors:

I_{i n t} = lim_{Γ \to 0} \int_{Γ} n \cdot M \cdot q d Γ,

with $M$ as

M = σ : ε_{a u x}^{L} I - σ \cdot {(\frac{\partial u}{\partial x})}_{a u x}^{L} - σ_{a u x}^{L} \cdot \frac{\partial u}{\partial x} .

In the limit as $r \to 0$ , using the local asymptotic fields,

T = \bar{E} [- \frac{I_{i n t} (s)}{f} + ν ε_{33} (s) - ξ α Δ Θ] + t_{2}^{o},

where $\bar{E} = E$ and $ξ = 1$ for plane stress; $\bar{E} = E / (1 - ν^{2})$ and $ξ = 1 + ν$ for plane strain, axisymmetry, and three dimensions; $ε_{33}$ is zero for plane strain and plane stress; $α$ is the thermal expansion coefficient; and $Δ Θ$ is the temperature difference.

$I_{i n t} (s)$ can be calculated by means of the same domain integral method used for J-integral calculation and the stress intensity factor extraction, which has been described in J-integral evaluation, and Stress intensity factor extraction. $I_{i n t} (s)$ is doubled if only half the structure is modeled.