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Analytical Fracture Mechanics
By David J. Unger Dover Publications, Inc.
Copyright © 1995 David J. Unger
All rights reserved.
ISBN: 978-0-486-14385-9
CHAPTER 1
On the Continuance of an Analytical Solution across the Elastic–Plastic Boundary of a Mode I Fracture Mechanics Problem
Analytical elastoplastic solutions involving linear elastic and finite-dimensional plastic regions have been found only for mode III fracture mechanics problems. Under conditions of small-scale yielding, the elastic–plastic boundary for the mode III problem can be determined by substituting the stresses from linear elastic fracture mechanics into the Mises/Tresca yield condition. In contrast, numerical studies for mode I problems involving the Mises yield condition [Tub 66, LMOR 71, KPPC 70a, KPPC 70b, PM 78, Cher 79] have shown boundaries that differ in shape from loci determined from linear elastic fracture mechanics in general (see Sections 1.10 and 4.2). Here, we will analytically investigate a mode I elastoplastic problem for the Tresca yield criterion under plane stress loading conditions.
In this study, the mode I small-scale yielding stresses are substituted into the Tresca yield condition to obtain a prescribed elastoplastic boundary. A plastic stress function for a perfectly plastic material is sought—a function that satisfies both equilibrium and the yield condition across the elastic–plastic interface. An intermediate integral is found for the governing partial differential equation, which is a nonlinear, second-order equation of the Monge–Ampere class. From a complete solution of the intermediate integral, the initial value problem is solved analytically as in [Ung 90a].
The plastic stress field that is determined from this function exhibits a discontinuity in the trailing portion of the plastic zone. Physical implications of this solution together with possible applications are discussed in the Preface and Section 1.5.
In the course of investigating this mode I problem, we develop a method of solving elastoplastic problems based on concepts from differential geometry. The solution technique finds an integral plastic surface that circumscribes a known elastic surface. This method of solution can also be applied to mode III problems, as well as other elastoplastic problems where a complete solution can be found to the governing partial differential equation of the plastic stress function.
We first demonstrate the solution process for a mode III problem that has a simple solution. This problem has been solved previously by other investigators using different techniques. Once insight is gained from working the mode III problem, we can apply a modified version of the solution scheme to the mode I problem, which has a higher-order partial differential equation.
1.1 ELASTOPLASTIC STRESS ANALYSES FOR MODES I AND III
Mode III
The stress function φE of small-scale yielding and its relationship to the elastic antiplane shear stresses τExz and τEyz for the mode III fracture mechanics problem with a stress intensity factor KIII follow: mode III:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-1)
(1.1-2)
(1.1-3)
where the commas in (1.1-2) and (1.1-3) denote partial differentiation. The relationships between the polar coordinates (r, θ) and the rectangular Cartesian coordinates (x, y) employed in (1.1-1)–(1.1-3) are
x = r cos θ = r]1 - 2 sin2(θ/2)], y = r sin θ = 2r sin(θ/2)cos(θ/2),
(1.1-4)
where the trigonometric identities involving θ/2 in (1.1-4) are cited for later use.
The potential φE is a harmonic function and the stresses (1.1-2)–(1.1-3) derived from it satisfy equilibrium equation (2.1-3).
The notation for the first partial derivatives of the plastic stress function φ (x, y) with respect to x and y is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-5)
In the mode III problem all stresses are zero except for the antiplane stresses. Consequently, the Tresca or Mises yield condition for a perfectly plastic material takes the following form [Hut 79]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-6)
where k is the yield stress in pure shear.
If the elastic mode III stresses (1.1-2)–(1.1-3) are substituted into yield condition (1.1-6), the locus of points that satisfy this relationship is a circle of radius R where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-7)
This circle, which is shown in Fig. 1.1-1, is the assumed elastic–plastic interface [partial derivative]Ω.
Let us now parametrize the elastic stress function ([??]), its first partial derivatives ([??],[??]), and the Cartesian coordinates ([??],[??]) on the elastoplastic interface in terms of the parameter a; i.e.,
[partial derivative]Ω: a = sin(θ/2) -> cos(θ/2) = (1 - a2)1/2.
(1.1-8)
When (1.1-7) and (1.1-8) are substituted in (1.1-1)–(1.1-4), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-9)
(1.1-10)
(1.1-11)
Interior to [partial derivative]Ω, the definitions of the functions [??], [??], [??], [??], and [??] in terms of the parameter a remain the same as in (1.1-9)–(1.1-11), but the relationship between a and the coordinates differs from that in (1.1-8).
Methods for finding complete solutions of nonlinear, first-order equations are detailed in [Sne 57]. It is easily verified that the following relationship for φ is a solution to partial differential equation (1.1-6):
φ = -k(1 - a2)1/2x - kay + b
(1.1-12)
provided a and b are constants. Equation (1.1-12) is a complete solution to (1.1-6) as it involves two arbitrary parameters a and b.
Sneddon [Sne 57] describes a method of finding an integral surface of a first-order, nonlinear partial differential equation that circumscribes a given surface. This procedure finds the envelope of a one-parameter subsystem of a complete solution. An imposed condition is [partial derivative]Ω:
p/[??] = q/[??] = 1.
(1.1-13)
In our analysis the prescribed surface is related to the elastic stress function φE. The governing equation is (1.1-6), and the surface that circumscribes φE is related to the plastic stress function φ. The imposed condition (1.1-13) suits our purpose as it ensures that the antiplane stresses will be continuous across the elastic–plastic boundary. This fulfills an equilibrium requirement.
The technique described in [Sne 57] requires the elimination of the parameters a, b from relationships corresponding to (1.1-9)–(1.1-13). The first step is to reduce the two-parameter complete solution to a one-parameter subsystem such that φ equals φE on the elastic–plastic boundary. This is accomplished by setting b = b(a) where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-14)
Thus (1.1-12) assumes form (1.1-15) after the substitution of (1.1-10) and (1.114), i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-15)
On the elastic–plastic boundary φ = [??] as x = [??] and y = [??].
In (1.1-12) a and b were assumed to be constants; however, in (1.1-15) the parameter a must be a function of (x, y) as it varies with θ on the boundary by (1.1-8). Consequently [??], [??], [??], x, and [??], which are functions of a, must also be functions of the coordinates. This implies that (1.1-15) cannot satisfy (1.1-6) with the parameter a as a function of the coordinates without some additional condition. This restriction may be found by taking the derivative of (1.1-15) with respect to the parameter a while treating x, y, and φ as constants; i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-16)
Equation (1.1-16) may be interpreted as the family of characteristic lines of (1.1-6) and (1.1-15).
We now prove that (1.1-15) is a solution of (1.1-6) provided that (1.1-16) is satisfied. Taking the partial derivative of (1.1-15) with respect to x and using a chain rule for differentiations of [??], [??], q, [??], and [??], which are functions of the parameter a, we find
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-17)
By substituting (1.1-16) into (1.1-17), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-18)
Similarly, we can prove that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-19)
Thus any exclusive relationship that exists between partial derivatives of φ on the boundary is also satisfied in the region interior to [partial derivative]Ω; i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1.20)
The function f is determined by eliminating a between equations (1.1-10). This procedure yields
[??]2 + [??]2 = k2 ->p2 + q2 = k2,
(1.1-21)
which proves our original assertion about (1.1-6), (1.1-15), and (1.1-16).
We now seek the explicit form of our solution φ(x, y). Substituting the relationships for [??], [??], q, [??], and [??] from (1.1-9)–(1.1-11) into (1.1-16) and taking derivatives with respect to a as indicated, we find
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1-22)
Equation (1.1-22) provides the parameter a as a function of (x, y) in the region interior to [partial derivative]Ω, i.e., the plastic region. It will naturally reduce to (1.1-8) on the boundary [partial derivative]Ω. By substituting a(x, y) from (1.1-22) into (1.1-9)–(1.1-11), we infer from (1.1-15) that
φ(x, y) = -k[(x + R)2 + y2]1/2
(1.1-23)
Solution (1.1-23) is the required envelope of the one-parameter subsystem (1.115) of the complete solution (1.1-12). It represents geometrically a surface that circumscribes the elastic surface defined by (1.1-1).
(Continues...)
Excerpted from Analytical Fracture Mechanics by David J. Unger. Copyright © 1995 David J. Unger. Excerpted by permission of Dover Publications, Inc..
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