Lecture 25 Bodies With Holes / Stress Perturbations Around Cracks Geol 542 Textbook reading: p.313-318; 354-357.

Slides:



Advertisements
Similar presentations
Common Variable Types in Elasticity
Advertisements

Common Variable Types in Elasticity
Stress in any direction
Griffith Cracks Flaws Make the World Beautiful! Were it not for the flaws, rocks and mountains would have been perfectly boring.
CHAPTER 4: FRACTURE The separation or fragmentation of a solid body into two or more parts, under the action of stresses, is called fracture. Fracture.
Fracture Mechanics Brittle fracture Fracture mechanics is used to formulate quantitatively The degree of Safety of a structure against brittle fracture.
Fracture and Failure Theory. Defining Failure Failure can be defined in a variety of ways: Unable to perform the to a given criteria Fracture Yielding.
1 Thin Walled Pressure Vessels. 2 Consider a cylindrical vessel section of: L = Length D = Internal diameter t = Wall thickness p = fluid pressure inside.
Mohr Circle for stress In 2D space (e.g., on the s1s2 , s1s3, or s2s3 plane), the normal stress (sn) and the shear stress (ss), could be given by equations.
Distribution of Microcracks in Rocks Uniform As in igneous rocks where microcrack density is not related to local structures but rather to a pervasive.
STRESS CONCENTRATION AT NOTCHES One of the fundamental issues of designing a resistant structure (specially in “design against fracture” and “design against.
Chapter Outline Shigley’s Mechanical Engineering Design.
PLANE STRESS TRANSFORMATION
CHAPTER OBJECTIVES Apply the stress transformation methods derived in Chapter 9 to similarly transform strain Discuss various ways of measuring strain.
Principle and Maximum Shearing Stresses ( )
Analysis of Stress and Strain
Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros.
An Introduction to Stress and Strain
Application Solutions of Plane Elasticity
Joints and Shear Fractures
Stress Transformation
Chapter 4: Solutions of Electrostatic Problems
Mechanics of Materials(ME-294)
Stress II Cauchy formula Consider a small cubic element of rock extracted from the earth, and imagine a plane boundary with an outward normal, n, and an.
Thermal Strains and Element of the Theory of Plasticity
7.2 Shear and Moment Equations and Diagrams
CHAPTER OBJECTIVES Analyze the stress developed in thin-walled pressure vessels Review the stress analysis developed in previous chapters regarding axial.
Stress Fields and Energies of Dislocation. Stress Field Around Dislocations Dislocations are defects; hence, they introduce stresses and strains in the.
Slip-line field theory
6. Elastic-Plastic Fracture Mechanics
Content Stress Transformation A Mini Quiz Strain Transformation
Load and Stress Analysis
CHAP 1 STRESS-STRAIN ANALYSIS
Stress II. Stress as a Vector - Traction Force has variable magnitudes in different directions (i.e., it’s a vector) Area has constant magnitude with.
Transformations of Stress and Strain
APPLICATIONS/ MOHR’S CIRCLE
10.7 Moments of Inertia for an Area about Inclined Axes
CHAPTER OBJECTIVES Analyze the stress developed in thin-walled pressure vessels Review the stress analysis developed in previous chapters regarding axial.
If A and B are on the same side of the origin (i. e
Chapter 9 DISTRIBUTED FORCES: MOMENTS OF INERTIA x y y dx x The rectangular moments of inertia I x and I y of an area are defined as I x = y 2 dA I y =
1 Principal stresses/Invariants. 2 In many real situations, some of the components of the stress tensor (Eqn. 4-1) are zero. E.g., Tensile test For most.
Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros.
Introduction to Seismology
Introduction Stress: When some external system of forces act on a body, the internal forces are set up at various sections of the body, which resist the.
Elasticity I Ali K. Abdel-Fattah. Elasticity In physics, elasticity is a physical property of materials which return to their original shape after they.
Chapter 4: Solutions of Electrostatic Problems 4-1 Introduction 4-2 Poisson’s and Laplace’s Equations 4-3 Uniqueness of Electrostatic Solutions 4-4 Methods.
Ch 4 Fluids in Motion.
Chapter 3 Force and Stress. In geology, the force and stress have very specific meaning. Force (F): the mass times acceleration (ma) (Newton’s second.
Mechanics of Materials(ME-294) Lecture 12: YIELD and Failure CRITERIA.
CHAPTER OBJECTIVES To show how to transform the stress components that are associated with a particular coordinate system into components associated with.
COMBINED LOADING.  Analyze the stress developed in thin-walled pressure vessels  Review the stress analysis developed in previous chapters regarding.
Stress and Strain ( , 3.14) MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical & Aerospace Engineering Stress.
Triaxial State of Stress at any Critical Point in a Loaded Body
Transformations of Stress and Strain
Failure I. Measuring the Strength of Rocks A cored, fresh cylinder of rock (with no surface irregularities) is axially compressed in a triaxial rig.
1 INTRODUCTION The state of stress on any plane in a strained body is said to be ‘Compound Stress’, if, both Normal and Shear stresses are acting on.
1 Structural Geology Force and Stress - Mohr Diagrams, Mean and Deviatoric Stress, and the Stress Tensor Lecture 6 – Spring 2016.
Force and Stress – Normal and Shear Stress Lecture 5 – Spring 2016
MOHR'S CIRCLE The formulas developed in the preceding article may be used for any case of plane stress. A visual interpretation of them, devised by the.
Fracture Mechanics Brittle fracture
1. PLANE–STRESS TRANSFORMATION
Fracture of Solids Theoretical tensile strength of a solid U(r) a r
DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
If A and B are on the same side of the origin (i. e
Thin Walled Pressure Vessels
BDA30303 Solid Mechanics II.
326MAE (Stress and Dynamic Analysis) 340MAE (Extended Stress and Dynamic Analysis)
Fracture of Solids Theoretical tensile strength of a solid U(r) a r
LINEAR ELASTIC FRACTURE MECHANICS
Yielding And Fracture Under Combine Stresses
Presentation transcript:

Lecture 25 Bodies With Holes / Stress Perturbations Around Cracks Geol 542 Textbook reading: p ;

We can also take the relations from the Airy stress function: 2D Elastic Solutions in Polar Coordinates and relate these to the polar coordinate system to derive general relationships (see handout):

If the principal stresses act along the coordinate axes, we have:  xx r =  1 r  yy r =  2 r  xy r = 0 Circular Hole in a Biaxial Stress Field The remote boundary conditions can be expressed in polar coordinates as: Note, these are simply the Mohr equations used at r = ∞, where  rr   ii (normal stress on a cubical element) and  r    ij (i≠j). Along the hole boundary, we have local boundary conditions for a shear stress free surface:  rr =  r  = 0(note,   ≠ 0)

The stress function for a circular hole is given by: Circular Hole in a Biaxial Stress Field where A, B, C, E, and F are constants dictated by the boundary conditions. Using the Airy equations in polar coordinates, we get: These are the general solutions for stress around a circular hole (for any loading condition).

Then solve for A, B, C, E, and F using the boundary conditions at r = a and r = ∞ to get: Circular Hole in a Biaxial Stress Field These are the specific solutions for a circular hole with biaxial loading.

We can use the specific solutions to solve for a circular hole for various remote boundary conditions and for any spatial location (r,  ). First we consider the case of a uniform remote compression of magnitude – S (i.e.,  xx r =  yy r = – S;  xy r =0) and zero stress on the hole boundary (i.e., p f = 0). Circular Hole in an Isotropic Stress Field Substituting into the specific solution stress equations in polar coordinates, we get: So the stress intensity factor is 1+(a/r) 2.

Circular Hole in an Isotropic Stress Field At the hole boundary (r = a),  rr =  r  = 0, and   = -2S everywhere (i.e., circumferential compression). So there is a stress concentration factor of 2, independent of hole size. This becomes important if 2S is greater than the uniaxial compressive strength of the rock.

Circular Hole in an Isotropic Stress Field Around the hole, principal stresses form radial and concentric stress trajectories. The mean stress (  rr +   )/2 is constant everywhere and equal to –S. The maximum shear stress (  rr –   )/2 is equal to S(a/r) 2. So the contours (isochromatics) are concentric around the hole. Note that despite the isotropic loading, the hole perturbation creates shear stress. As r  ∞, (a/r)  0, so  s(max)  0.

Next, we consider the case of a uniform remote compression of magnitude – S (i.e.,  xx r =  yy r = – S;  xy r =0) and an internal fluid pressure acting on the hole boundary (i.e., p f = – P = – S). This condition reflects a pressurized borehole, an oil well, or magma pressure in a cylindrical conduit. Tension is positive. Circular Hole With an Internal Fluid Pressure Substituting into the specific solution stress equations in polar coordinates, we get: The result is a homogeneous, isotropic state of stress. It’s as if the hole isn’t even there.

We now consider the case of zero remote stress (i.e.,  xx r =  yy r = 0 ;  xy r =0) and an internal fluid pressure acting on the hole boundary (i.e., p f = – P). Circular Hole With an Internal Fluid Pressure The stress components for this problem are: The result is a tension all around the hole equal in magnitude to the fluid pressure inside the hole (i.e., a stress concentration factor of -1). If this tension exceeds the tensile strength of the rock, hydrofracturing may occur.

Spanish Peaks Dikes Muller and Pollard, 1977

We now consider the case of isotropic remote stress (i.e.,  xx r =  yy r = – S;  xy r =0) and an internal fluid pressure acting on the hole boundary (i.e., p f = – P), where P ≠ S. Remote Stress Plus Internal Fluid Pressure The stress components for this problem are: If P = S, this result reduces to the equations derived previously. ≠

The specific solutions for a circular hole can also be used for the boundary conditions of biaxial loading. Biaxial Loading For example, the circumferential stress component can be solved at r = a to show: Hence, at  = 0,  :   = 3S h – S H. At  =  /2, 3  /2:   = 3S H – S h. (i.e., as described previously)

Circular Hole in a Biaxial Stress Field Around the hole, principal stresses are perturbed.

A similar approach can be applied to the problem of stresses around elliptical holes. e.g., dikes, sills, veins, joints, Griffith flaws Stress Around Elliptical Holes If the hole is oriented with long axes parallel to the x and y coordinate axes, respectively, the hole boundary is defined by: (x/a) 2 + (y/b) 2 = 1

Just as it was more useful to use a polar coordinate system for circular holes, it is appropriate to use an elliptical curvilinear system for elliptical holes, with components  (xi) and  (eta). Stress Around Elliptical Holes The transformation equations are: x = c cosh  cos  y = c sinh  sin  where 2c is the focal separation (see figure).

The stress components are:       which act on any particular element in this coordinate system. Stress Around Elliptical Holes It is the   component that tells us of the circumferential stress acting along the hole boundary, and always acts along lines of constant .

As with the circular hole, solutions are found by specifying the boundary conditions both at infinity (remote) and on the hole boundary, designated at  =  o. Stress Around Elliptical Holes The semi-major and semi-minor axes are given by: a = c cosh  o and b = c sinh  o As  o  0, a  c, and b  0. This produces a pair of straight lines connecting the foci and is the special case of a crack (cf. Griffith’s approximation).

Boundary conditions: Uniform remote tension of magnitude S (i.e.,  xx r =  yy r = S;  xy r =0) and zero stress on the hole boundary (i.e., p f =   =   = 0). The solution to the circumferential stress on the hole boundary is given by: The maximum values occur at the crack tips where  = 0, , so cos 2  = 1. This can be solved to show: Elliptical Hole in an Isotropic Tension The stress concentration factor is thus 2a/b (i.e., hole shape is important). e.g., if a = 5b, then   (max) = 10S.

The minimum values occur along the crack edges where  =  /2, 3  /2, so cos 2  = -1. This can be solved to show: Elliptical Hole in an Isotropic Tension The stress diminution factor is thus 2b/a. So if a = 5b, then   (min) = (2/5)S. We can reduce our solution to two special cases: (1)Circular hole: a = b    (max) =   (min) = 2S (2)Crack:b  0    (max) = ∞ Infinite stresses are predicted at the crack tip. This is referred to as a stress singularity in linear elastic fracture mechanics.

Pressurized Elliptical Hole with Zero Remote Stress Boundary conditions: Zero remote stress (i.e.,  xx r =  yy r =  xy r =0) and an internal pressure on the hole boundary (i.e., p f =   = – P;   = 0). In this case we get: For the special case of a crack-like hole, a>>b, so the stress concentration factor becomes ~2a/b. Also: If a>2b, this is a compressive stress, and in the limit a>>b, the stress approaches –P. In other words, a pressure acting on a flat surface induces a compressive stress of the same magnitude parallel to the surface.  = 0,   =  /2, 3  /2

Elliptical Hole with Orthogonal Uniaxial Tension Boundary conditions: Uniaxial remote tension parallel to minor axis b (i.e.,  xx r = 0;  yy r = S;  xy r =0) and an zero pressure on the hole boundary (i.e., p f =   =   = 0). The general solution is: We thus get the same result determine by Inglis: So   (min) is independent of hole shape. If a = 5b,   (max) = 11S and   (min) = -S. If a = b,   (max) = 3S and   (min) = -S (as we determined earlier in polar coords). For a crack, a>>b, so   (max) = 2Sa/b (  ∞ ) and   (min) = -S.  = 0,  and  =  /2, 3  /2

Elliptical Hole with Various Loadings Plots of tangential stress around two elliptical holes (a/b = 2 and 4) with three loading configurations:

Elliptical Hole with Stress at an Angle to Crack Boundary conditions: S 2 at  to x-axis S 1 at  +  /2 to x-axis (S 1 >S 2 OR S 2 >S 1 ) The general solution is: This is the equation that Griffith solved with respect to  to develop his compressive stress failure criterion. So we’ve already examined an application of this. x y S2S2 S2S2 S1S1 S1S1 

Elliptical Hole with Stress at an Angle to Crack Jaeger and Cook, 1969

Solutions for Holes with Other Shapes Analytical methods for determining solutions for holes with other shapes were introduced by Greenspan (1944) and are reviewed in the book “Rock mechanics and the design of structures in rock” by Obert & Duvall (1967). One of the most important considerations when addressing holes in rock is the effect of sharp corners on stress concentration. The sharper a corner, the greater the concentration of stress. We can explain this by re-examining the elliptical hole problem. For a uniaxial tension T applied orthogonal to the long axis of an elliptical hole, the circumferential stress at the tip is: So the stress scales as 2a/b. From the geometry of an ellipse, the radius of curvature at the end of the ellipse  = b 2 /a. Substituting b = √  a into the above equation, we get: where  is the remote stress acting perpendicular to the crack. So as  is decreased,   gets bigger = bad!

Solutions for Holes with Other Shapes Obert and Duvall, 1967 Note: even for rounded corners, the stress concentrations are greatest at the corners.

Solutions for Holes with Other Shapes Obert and Duvall, 1967

A reminder of why it matters… THE END!