G.P.Nikishkov, University of Aizu

Consider a two-dimensional plane strain elastic-plastic body containing Mode I crack. We will use a polar coordinate system r, q for stresses s _r, s q and s _rq and displacements u_r and uq . The problem is defined by equilibrium equations, constitutive strain-stress equations and strain-displacement equations.

The equilibrium equations in the polar coordinate system have the following form:

where comma denotes a partial derivative in respect to a variable after comma.

We adopt the J₂-deformation theory and the Ramberg-Osgood uniaxial strain-stress curve for the description of elastic-plastic material behavior:

where s ₀ is the yield stress; a is the hardening coefficient; n is the hardening exponent (n > 1); e ₀=s ₀/E; E is Young's modulus. In the multi-axial case the corresponding constitutive equations are:

Here s_ij are deviatoric stresses; s _e is the Mises equivalent stress.

The relations between strains and displacements are written as:

To complete the definition of the problem the boundary conditions should be added:

Suppose that in a small vicinity of the crack tip the stresses can be presented as a three-term asymptotic expansion:

where s < t < u; r and s are dimensionless distance from the crack tip and dimensionless stress. Since we are going to obtain an asymptotic solution for near crack tip fields, it is not important now what characteristic length is used for the normalization of the distance r.

We assume that elastic strains are small enough and can be omitted in three-term expansion (this is confirmed by calculations for all n values of practical importance n>=3). The constitutive equation becomes simpler:

The term (s _e/s ₀)^n-1 can be presented in the form:

Now using binomial theorem, performing multiplication and preserving terms up to the power s(n-2)+2t, we arrive at the following expansion for strains:

If u < 2t-s then the term x _ij⁽¹⁾ is a higher order term in comparison to e _ij⁽²⁾ and can be omitted. The other possibility is u = 2t-s (this choice will be explained later) and the strain and displacement asymptotic expansion becomes:

.

Substitution of expansions into the governing equations produces the following sequence of boundary value problems.

Problem (0):

Problem (1):

Problem (2):

For plane strain conditions in the absence of elastic strains, deviatoric stresses and the Mises equivalent stress are equal to:

,

Each system of differential equations has five unknown angular functions. Since the third equation in every system is an algebraic equation, one unknown function can be eliminated.

The system for the problem (0) is a nonlinear homogeneous system of ordinary differential equations with the theoretically known eigenvalue s = -1/(n+1). The system for the problem (1) is a linear homogeneous system with the unknown eigenvalue t. The system for the problem (2) is a linear nonhomogeneous system in which amplitudes of functions depend on the solutions of the problems (0) and (1). The problem (2) can be formulated exactly in the same way as the problem (1) by omitting x _ij⁽¹⁾. As shown by trial solutions, eigenvalues u for the problem (2) after omitting x _ij⁽¹⁾ appear to be larger than (2t-s) for all values of the strain hardening exponent n that are important for practice (n ³ 3). Since this is impossible, the value of u should be set to (2t-s) and the term x _ij⁽¹⁾ should be preserved.

Let us introduce the following notation:

Appropriate substitutions give three systems of equations for the problems (0), (1) and (2). Each system contains four ordinary differential equations and one algebraic equation. The algebraic equation allows to eliminate one unknown function.

The system for the problem (0) is a nonlinear system with a nonlinear algebraic equation. Therefore it is convenient to eliminate u_r⁽⁰⁾ = y₅ by differentiation of the algebraic equation and combining it with the fifth equation. Then the problem (0) can be formulated as the following system of four ordinary differential equations:

where Y, F and G denote the following expressions:

Systems for problems (1) and (2) are linear systems of differential equations with linear algebraic equations. In this case, it is simpler to eliminate s _r^(k)=y₁^(k) using the algebraic equation for obtaining an explicit equation for y₁^(k). Thus the problem (1) can be represented by the system of ordinary differential equations for unknown angular functions:

After the solution of the system, the function y₁⁽¹⁾ can be determined from the expression:

The system of differential equations for the problem (2) has the following form:

,

Here the following notation was introduced:

The function y₁⁽²⁾ is determined by:

The problems (0), (1) and (2) are boundary value problems with the boundary conditions:

We solve these problems by the direct integration of ordinary differential equations using the fourth order Runge-Kutta method. In order to apply the Runge-Kutta method the problems should be formulated as initial value problems. We use the shooting method in conjunction with the Newton-Raphson method for the definition of unknown initial values or eigenvalues.

In general there are two unknown parameters in every problem: eigenvalue s and for the problem (0), eigenvalue t and y₅(0) for the problem (1) and two unknown function values y₂(0) and y₅(0) for the problem (2). Let us denote two unknown parameters as a and b . The shooting method consists in finding such values of a and b that satisfy the last two boundary conditions. The Newton-Raphson iteration procedure for the shooting method can be written as follows:

where detJ is the determinant of the Jacobi matrix:

The three-term asymptotic solution for the stress field near the tip of the mode I crack in an elastic-plastic can be presented in the following form:

.

Here s _ij are stress components in the polar crack tip coordinate system r, q ; s _ij^(k) are dimensionless angular stress functions obtained from the solution of associated asymptotic problems of zero, first and second orders; s is a theoretically known eigenvalue for the problem of zero order, s = -1/(n+1) ; values of an eigenvalue t are determined numerically ; r is a dimensionless radius:

where J is the J-integral proposed by Cherepanov and Rice. The results of asymptotic problems of the order (0) and (1) are normalized so that

where sij are the deviatoric stress components and 'max' means the maximum for all angles.

The coefficient A₀ is equal to:

In this case, a scaling integral I_n can be presented in the form:

Thus the three-term expansions for the crack tip stress and displacement fields are controlled by two parameters: the J-integral and the amplitude parameter A. A natural way of determining the amplitude A is a best fit of the three-term expansion inside some area near the crack tip to the stresses calculated with the help of a numerical method.

The three-term expansion for the ith point can be presented in the form:

where d _i is the deviation of the asymptotic stress field from the finite element stress s ^FEM at the ith point. For fitting the stress components or some combinations of them can be used. Thus, means any stress component or stress combination used for fitting.

Minimization of the sum of squares of the deviations with weights leads to a cubic equation for A:

Here wi is a weight for the ith point. Weights for the points can be proportional to the area, which is represented by this point.

The solution of the cubic equation gives three roots. If all three roots are real, it is necessary to select the root by computing three sums of squares of the deviation corresponding to three roots. The minimal sum indicates the appropriate root. This root has the same sign as the sign of difference between the HRR-solution (zero term of the three-term stress expansion) and the finite element data for normal stresses. This is dictated by the fact that the first term normal stresses are positive for all angles 0<q <p .

Chao and Ji (ASTM STP 1244, 1994) use the three-term asymptotic expansion with different definition of the amplitude parameter A₂. The following relation between our A and their A₂ can be obtained:

,

where L is a crack characteristic length. It is recommended to select L equal to the crack length or the specimen size. For small scale yielding conditions there is no characteristic length. Thus A₂ is not defined for small scale yielding conditions. For finite bodies under low loads when J is close to zero, the amplitude A₂ tends to infinity.

The Q-stress introduced by O'Dowd and Shih (J. Mech. Phys. Solids, 1991, 39, 989-1015) can be defined as the difference:

r = 2, q = 0.

where s q is the angular stress in the body under consideration, s q ^HRR is a zero order term in the asymptotic expansion. The relation between the Q-stress and the amplitude parameter A is:

, r = 2, q = 0.

For a particular value of the hardening exponent n this relation can be presented in the following form:

Values of coefficients a1 and a2 for n = 3 - 15 are listed below:

G.P.Nikishkov, A fracture concept based on the three-term elastic-plastic asymptotic expansion of the near-crack tip stress field. In: Fracture: A Topical Encyclopedia of Current Knowledge (Ed. G.P.Cherepanov), Chapter 29. Krieger, Malabar, FL, 1998, pp. 557-574 [PDF].

G.P.Nikishkov, An algorithm and a computer program for the three-term asymptotic expansion of elastic-plastic crack tip stress and displacement fields. Engineering. Fracture Mechanics, 1995, 50, pp. 65-83.

G.P.Nikishkov, A. Brueckner -Foit and D.Munz, Calculation of the second fracture parameter for finite cracked bodies using a three-term elastic-plastic asymptotic expansion. Engineering Fracture Mechanics, 1995, 52, pp. 685-701.

G.P.Nikishkov, A. Brueckner -Foit and D.Munz, Application of the three-term elastic-plastic asymptotic expansion for the characterization of stress fields near a front of a semi-elliptical crack. Int. Journal of Fracture, 1995,70, R91-R97.

G.P.Nikishkov, A fracture concept based on the three-term elastic-plastic asymptotic expansion of the near-crack tip stress field. In: Fracture: A Topical Encyclopedia of Current Knowledge (Ed. G.P.Cherepanov). Krieger, Melbourne, 1998.

S.Yang, Y.J.Chao and M.A.Sutton, Higher order asymptotic fields in a power-law hardening material. Eng. Fract. Mech., 45, 1-20 (1993).

S.Yang, Y.J.Chao and M.A.Sutton, Complete theoretical analysis for higher order asymptotic terms and the HRR zone at a crack tip for Mode I and Mode II loading of a hardening material. Acta Mechanica, 98, 79-98 (1993).

Y.J.Chao and W.Ji, Cleavage fracture quantified by J and A2. Constraint Effects in Fracture: Theory and Applications, ASTM STP 1244, M.Kirk and A.Bakker Eds., American Society for Testing and Materials, Philadelphia, 1994.

L.Xia, T.C.Wang and C.F.Shih, Higher-order analysis of crack tip fields in elastic power-law hardening materials. J. Mech. Phys. solids, 41, 665-687 (1993).