1. A STUDY OF THE DYNAMIC RESPONSE OF A FRACTURED TUNNEL TO PLANE WAVES Pei-cheng Xu SwRI, SanAntonio,Texas Sept. 7, 2000

3. x z y vacuum host medium tunnel fracture

4. Objectives of Study Develop a computational tool to predict dynamic response of a fractured tunnel to plane waves Investigate the effects of fracture properties on the displacement and stress field, using the slip interface model

5. Method of Approach: The Boundary Integral Equation Method Discretize the boundary only Suitable for an unbounded host medium Additional efficiency when large numbers of sources and detectors are involved

6. The Boundary Integral Equation Solve BIE for boundary displacements Traction free

7. The Boundary Integral * Displacement off boundary: direct integration * Stresses: Hooke’s law / differentiation

8. Green’s Functions and Associated Stresses Uniform, isotropic Layered (fractured) x1x1 x3x3 33 11 r x 

9. w h b a 1 2 345 6 7 8 1 2 3 45 6 7 8 9 10 11 12 Boundary Mesh Basic type of elements

10. n 1 columns n 2 rows detector fracture tunnel Detector Mesh

11. The Slip Model of Fracture n s   Stiffness coefficients

12. The Stiffness Calculation in the E-Model of Fracture K n ( Virtual thickness in BIE Slip line stiffness Combined stiffness across the slip line =  1  K n 

13.  I-model ksks knkn

14.  E-model ksks knkn

15.  Fractured host medium I-model ksks knkn

16.  Uniform host medium E-model ksks knkn

17. Comparison of I-Model and E-Model of Fracture Zero thickness Infinite length Longer computer time Numerically unstable when elements are small No mesh on the fracture Finite thickness Finite length More efficient Numerically unstable when virtual thickness is too small Mesh on the fracture I-ModelE-Model

18. Features of the Current Program Predicts displacements and stresses on the boundary and in the surrounding area Handles multiple sources and a large number of detectors efficiently. Two alternative slip interface models for the fracture: Implicit and Explicit. A user friendly interface for input data. A variety of display means of output data, including deformed meshes, hoop stresses on the boundary, and quiver and contour plots of field stresses.

19. Limitations of the Current Program The Implicit slip model encounters numerical instability when boundary elements are small. The Explicit slip models works but the input value of virtual thickness may depend on the frequency and geometry. Needs a user friendly output interface.

20. =+ 11/2 -1/2 Decomposition based on Symmetry nnnnn/ 2  n/ 2

21. Frequency loop Symmetry loop Incidence loop Boundary mesh Decomposed boundary Matrix Input Decomposed boundary response w.o. tunnel Decomposed boundary response w. tunnel Detector mesh Total boundary response PROGRAM STRUCTURE Detector responseOutput files Solving equation

23. I/O Files

24. Geometry of Cavities or Inclusions

25. Geometry of Cavities or Inclusions

26. Input Parameters (1) Enter maximum number of nodes Is the geometry symmetric? Select type of medium

29. Input Parameters (2)

30. Input Parameters (3)

31. Input Parameters (4)

32. Input Parameters (5)

33. Input Parameters (6)

52. Lahey FORTRAN 95