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Research Articles

# Stress Relaxation Near the Tip of a Stationary Mode I Crack in a Poroelastic Solid

[+] Author and Article Information
Chung-Yuen Hui

Field of Theoretical and Applied Mechanics,
Cornell University,
Ithaca, NY 14853

Rong Long

Department of Mechanical Engineering,
Boulder, CO 80303

Jing Ning

Department of Mechanical and Aerospace Engineering,
Cornell University,
Ithaca, NY 14853

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 5, 2012; final manuscript received June 29, 2012; accepted manuscript posted July 25, 2012; published online January 22, 2013. Assoc. Editor: Chad Landis.

J. Appl. Mech 80(2), 021014 (Jan 22, 2013) (9 pages) Paper No: JAM-12-1093; doi: 10.1115/1.4007228 History: Received March 05, 2012; Revised June 29, 2012; Accepted July 25, 2012

## Abstract

We study the short time transient stress and pore pressure fields near the tip of a stationary crack when a sudden load is applied to a poroelastic solid. These fields are determined using a small scale “yielding” (SSY) analysis where the stress relaxation due to fluid flow is confined to a small region near the crack tip. They are found to exhibit the usual inverse square root singularity characteristic of cracks in linear elastic solids. Analysis shows that these fields are self-similar; the region of stress relaxation that propagates outward from the crack tip is proportional to $Dct$, where $Dc$ is the cooperative diffusion coefficient and t is time. The pore pressure at the crack tip vanishes immediately after loading. The stress intensity factor at the crack tip is found to be reduced by a factor of $1/[2(1-v)]$, where $v$ is the Poisson's ratio of the drained solid. Closed form approximations are found for the pore pressure and the trace of the effective stress. These approximate analytical solutions compare well with finite element results.

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## Figures

Fig. 1

Schematic illustration of the SSY problem. The poro-elastic specimen on the left is loaded in Mode I tension. For short times, flow is confined in the region Ω which is very small compared with typical specimen dimensions. As a result, the crack can be modeled as semi-infinite with the elastic KI field prescribed at r=∞.

Fig. 2

(a) Contour plot of the pore pressure in a small region surrounding the crack tip at t=1.8 × 10-4. To avoid large numbers, the stress intensity factor is set to 1. Maximum pore pressure occurs at x=1.98Dct≈0.02 (Dc=1/1.8), y = 0. (b) Close up view of flow directions near maximum pressure. Arrows indicate flow direction.

Fig. 3

Finite element pore pressure distribution directly ahead of the crack tip (θ = 0) for v=0.4 at two different times

Fig. 4

Normalized pore pressure versus similarity variable r/(2Dct). Symbols are finite element results for v=0.4 at two different times. Solid line is obtained using Eq. (17).

Fig. 5

Normalized trace of effective stress directly ahead of the crack tip (θ = 0) versus similarity variable η/2 for two different Poisson ratios v=0.3,0.4 and at two different times t=10-4,10-5. Symbols are finite element results and the solid line is the analytical result given by Eq. (18).

Fig. 6

Crack opening displacement versus distance behind the crack tip r for two different Poisson's ratios v=0.3,0.2 and at two different times t=10-4,10-5. Symbols are finite element results and the solid line is Eq. (25).

Fig. 7

(a) Angular variations of u¯1 at r=δ for v=0.3. Dashed lines are finite element results, the solid line is obtained using Eq. (26a). (b) Angular variations of u¯2 at r=δ for v=0.3. The solid line is obtained using Eq. (26b) and the dashed line is the FEM result.

Fig. 8

Angular variation of p¯/erf(η/2) versus θ for v=0.3. Solid line is the analytical result Eq. (17).

Fig. 9

Angular variation of σ¯/erfc(η/2) versus θ for v=0.3. The dashed line is given by finite element results and the solid line is obtained using Eq. (18).

Fig. 10

Finite element meshes used in calculation

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