Results and Discussion

Let us first examine if the use of deformed configuration in the analysis has any influence on stresses. Consider the basic assumptions. Both linear elastic fracture mechanics and the classical infinitesimal strain theory assume small deformation. Therefore, displacement z-Z is considered small and hence z = Z in lieu of deformed position z on the basis that this choice is expected to introduce negligible error. In view of this argument, the two theories use a/ = 0 for the crack line, and assume the crack tip at a = в = 0. Or, in terms of cylindrical polar coordinates with origin at the crack tip, the two theories use в = n for the crack surface, and assume the crack-tip at r = 0. The two stress fields (5)3 and (5)4 both display singular behavior in the limit as a, в ^ 0 or as the crack-tip is approached. Both linear elastic fracture mechanics and HRR solutions exhibit such singularity (Barenblatt, 1962; Hutchinson, 1968; Irwin, 1957).

We adopt another view point. Let us assume small deformation and consider the displacement z-Z to be small such that z = Z. In that case, we propose to use the deformed position z of the particle in place of its un-deformed position Z in solving a boundary value problem. This particular choice can be justified on the basis of the argument employed in the classical theory that replacing one by the other is expected to introduce negligible error. There is yet another reason in favor of the present choice. If the boundary value problem is formulated in terms of true stress and true traction, and if the analysis is to be consistent with the formulation, solution must depend on the deformed geometry. Note that true stress and true traction are defined in terms of load over unit current or deformed area. The use of undeformed geometry in boundary value problems formulated in terms of true stress is only a convenient approximation. It should not be considered a requirement that must be imposed.

Suppose the crack opens under and the crack opening parameter af has non-zero value, implying crack tip blunting. The tip is initially at (a = 0, в = 0) and is displaced under deformation to (a = af, в = 0). The stress field in (5) is no longer singular even though the stresses near the tip are still high. Stress distribution depends on af, which can be used as a parameter that controls stresses. Under sufficiently high stresses, the crack tip and the area around it yield and undergo plastic deformation. The extent of plasticity along the crack surface can be determined from (5)3.

The problem involving elastic behavior was solved by Singh et al. (1994). They obtained the value

E + S E – 3S

for the crack opening parameter. With af = ae in Equations (5)3 and (5)4, stresses can be obtained from (6).

Equation (5)i with C = 0 and u = c cosh(ae + ів) – a0 cos в yields the location

of points on the deformed crack surface.

For a bilinear solid, the crack tip opening and consequent blunting is governed by the modulus at the tip. The parameter

En + S

En – 3S controls stresses in the plastic domain and determines the deformed shape of the crack opening at the tip. Since the tangential stress along the crack surface must remain continuous across elastic – plastic boundary, the parameter a must have a common value ab at that point. The parameter ae that determines the shape of the elastic part of the crack is no longer linked to Young’s modulus E. It is therefore possible to assume af = ae = an = ab. In that case, the parameter an controls the stress field. The choice also ensures the continuity of stresses across the elastic-plastic boundary.

Since ayy = En at the crack tip, a solid with lower modulus entails larger crack opening para­meter, more blunting and lower stress at the tip. However, this solution is not applicable if the yield stress dominates stress field of the plastic domain.

In the case of ideal plastic behavior, there is no strain hardening and therefore En = E. It is inconceivable that Young’s modulus will control stress field in the plastic domain. It is more likely
to be controlled by the yield stress aY which is expected to play an important role in stress analysis in the plastic zone.

Let us assume that the plate is in a state of plane stress and obeys Mises yield criterion. Consider Mises effective stress ae — aY and rewrite it in the form

To satisfy the above equation, choose Oyy + Oxx — 2^y sin t,

Note that we use linearized stress-strain curve in the case of work hardening material. The curve is linear even for non-hardening material. Since the sum ayy + axx known to be harmonic for linear solids, choose an analytic function fp (z) in the plastic domain such that

ayy + axx = (fp + f p )•

Therefore,

Re f’p — aY sin t,

where Re f’p is the real part of df p/dz.

Since the crack surface is traction free, the only non-zero stress on it is tangential. It can be identified with the effective stress. In other words, ayy + axx — aY or t — n/6, in the plastic domain of the crack surface.

On the surface y — 0, the shear stress must vanish because of symmetry. Therefore, x – and y-surfaces on the extended crack line are principal planes on which the principal stresses are

2 = cry (sin f + cosf/л/З),

err = ay (sin f – cosf/л/З).

Some general conclusions regarding the stress distribution for x > a on the line y — 0 can be drawn even if the exact form of the function fp is unknown. For example, at the crack tip, a1 — 0, a2 — aY, hence Re fp — aY/2. Thereafter, both principal stresses increase with Re fp until at Re fp = х/Ъ/А aY, eri = ау/л/З and аг attains a maximum value of 2ау/л/3. Subsequently, аг decreases while a1 continues to increase until the elastic plastic boundary is reached.

In view of the piece-wise linear stress-strain behavior assumed in the analysis, the analytic func­tion f, or rather its derivative df/dz of the elastic domain can be used in the plastic domain as well. However, the analytic function g has no role in the plastic domain in which the yield criterion must be used for evaluating deviatoric stresses. It means that an additional term must be added to df/dz of the elastic domain to obtain the corresponding function of the plastic domain. Accordingly, choose

fp — A1 + A2 coth f + A3/sinh %•

This choice immediately leads to

A yield criterion must be used to obtain deviatoric stress components. For a Mises solid in plane stress for example, the yield criterion can be rearranged to obtain

(ayy axx + 2iaxy) (Oyy axx 2i(7Xy) 1 I 2 (ayy + axx

2

To find the values of the constants A1, A2 and A3, it is necessary to impose the condition that the stresses must be continuous across the elastic-plastic interface. Moreover, for a non-hardening material, stress along the crack surface in the plastic domain must remain at the yield value aY. The second condition can be satisfied easily by choosing A1 = ay/8 and

A2 cosh af + A3 cos в = 0.

Both af and в must therefore vary in the plastic domain. At the elastic-plastic interface, af = ae and в = вУ and hence

cosh ae

A3 =————- /Аг-

cos в

The value a = at at the crack tip can be obtained from

A3 cosh ae

cosher =——– =————— .

A2 cos/Sy

To find the value of A2, use the expression for stresses on the line в = 0 on which

A2cosh a + A3 cosh a — cosh at

Oyy + axx = ay + 8———– ———— = oy + 8A2-

For continuity across the elastic-plastic boundary a = ay, the above equation must yield a value equal to that obtained from the elastic solution. Therefore,

cosh ay — cosh at 2a 2a

ay + 8A2————————- = S(-e2 e + (e2 e + l)cotho! y).

sinh ay

The above equation can be solved for

Hence on в = 0,

The difference in stresses is obtained with the help of the yield criterion and it can be expressed in the form

14aY — (ayy + axx)2

3

The above two equations can be used to find stresses axx and ayy between the crack tip and the yield point on в = 0.

For a work hardening solid in plastic deformation, a > ay and hence the above equation can be rearranged to yield

un и 1

En E + 2EPn

Suppose the crack surface open in the form used in LEFM solution or as predicted by the solution of Singh et al. (1994).

Suppose a crack a = 0 opens under external load into a surface a = af. The presence of crack is likely to induce non-homogeneity in the stress field. Hence, to solve for stresses, choose

f = cA cos f + cB sin f + C = c(A + B) cosh Ц — cBe— + C,

where A, B, C and c are constants. Since the crack surface remains free of traction,

I + g + C = A constant on a = af.

To satisfy the above condition, choose

In view of these choices,

cosh 2a — cosh2af

2cA{coshf — cosh (2a/ — f)} + cB—————– =——- J – + C.

sinh f

To evaluate the constants A and B, observe that the above stress field must be consistent with the applied far-field stress at large distances from the origin. In other words, they must satisfy the condition that both Y,^ and J]d ^ S as a ^ro. These conditions lead to

A — —Se2af /8, B — S(e2af + 1)/8.

When these constants are substituted in (2), (3) and (4), they lead to cS

f = —{-e2af coshf + (e2af + 1) sinhf} + C,

8

™ (-2^/(cosh{ – cosh(2a/ – I» + <«*>■ + 1>C°Sh2a-COSh2“/|,

dz 8 [ sinh f J

y – ^ ayy – axx – 2iaxy = 2af sinh(2a/ – f) 2a/ cosh 2a f cosh f – cosh g 4^ S 6 sinh I 1 6 (sinh I)3 ‘

To find stress components, it is necessary to evaluate the real and imaginary parts, Re d and Im d of d. Subsequently, stresses can be obtained from

The last two equations in (5) suggest that for a given applied load S, stresses in the plate depend only on the crack opening parameter af. It is therefore reasonable to assume that the parameter controls the stress field around the crack. To find its value, consider the displacement of points on the crack surface.

References

Barenblatt, G. I., 1962, “The mathematical theory of equilibrium cracks in brittle fracture”, Adv. AppliedMech., 7, 55-129.

Hutchinson, J. W., 1968, “Singular behavior at the end of a tensile crack in a hardening material”, Journal of the Mechanics and Physics of Solids, 16, 13-31.

Irwin, G. R., 1957, “Analysis of stresses and strains near the end of a crack traversing a plate”, Journal of Applied Mechanic, 24, 361-364.

Rice, J. R. and Rosenberg, G. F., 1968, “Plane strain deformation near a crack tip in a power-law hardening material”, Journal of the Mechanics and Physics of Solids, 16, 1-12.

Singh, M. N.K., Dubey, R. N. and Glinka, G., 1994, “Notch and Crack analysis as a moving boundary problem”, Engineering Fracture Mechanics, 135,479-492.