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DTD Handbook

Handbook for Damage Tolerant Design

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    • Sections
      • 1. Introduction
      • 2. Fundamentals of Damage Tolerance
        • 0. Fundamentals of Damage Tolerance
        • 1. Introduction to Damage Concepts and Behavior
        • 2. Fracture Mechanics Fundamentals
          • 0. Fracture Mechanics Fundamentals
          • 1. Stress Intensity Factor – What It Is
          • 2. Application to Fracture
          • 3. Fracture Toughness - A Material Property
          • 4. Crack Tip Plastic Zone Size
          • 5. Application to Sub-critical Crack Growth
          • 6. Alternate Fracture Mechanics Analysis Methods
            • 0. Alternate Fracture Mechanics Analysis Methods
            • 1. Strain Energy Release Rate
            • 2. The J-Integral
              • 0. The J-Integral
              • 1. J-Integral Calculations
              • 2. Engineering Estimates of J
            • 3. Crack Opening Displacement
        • 3. Residual Strength Methodology
        • 4. Life Prediction Methodology
        • 5. Deterministic Versus Proabilistic Approaches
        • 6. Computer Codes
        • 7. Achieving Confidence in Life Prediction Methodology
        • 8. References
      • 3. Damage Size Characterizations
      • 4. Residual Strength
      • 5. Analysis Of Damage Growth
      • 6. Examples of Damage Tolerant Analyses
      • 7. Damage Tolerance Testing
      • 8. Force Management and Sustainment Engineering
      • 9. Structural Repairs
      • 10. Guidelines for Damage Tolerance Design and Fracture Control Planning
      • 11. Summary of Stress Intensity Factor Information
    • Examples

Section 2.2.6.2.2. Engineering Estimates of J

While the J-Integral was developed for nonlinear elastic material behavior, it has been extensively studies for its direct application to describing elastic-plastic material behavior [Begley & Landes, 1972; Verette & Wilhem, et al., 1973; Landes, et al., 1979; Paris, 1980; Roberts, 1981].  Its nonlinear elastic foundation has provided engineers with some techniques which allow them to focus on the combination of linear-elastic and plasticain hardening behavior and then to separate these two components for further study of the plastic behavior.  The J-Integral for an elastic-plastic material is taken as the sum of two components parts: the linear elastic part (Jel) and the plasticain hardening part (Jpl), i.e.,

(2.2.43)

which when used in conjunction with Equation 2.2.36 becomes

(2.2.44)

Engineering estimates of J then focus on the development of the plasticain hardening part Jpl.  Recently, Shih and coworkers have published a series of reports and technical papers [Shih & Kumar, 1979; Kumar, et al., 1980; Shih, 1976; Kumar, et al., 1981] detailing how the Jpl term can be calculated from a series of finite element models that consider changes in material properties for the same structural geometry.  The following briefly describes the Shih and coworkers method for estimating Jpl.

First, the material is assumed to behave according to a power hardening constitutive (s - e) law of the form

(2.2.45)

where a is a dimensionless constant, so = Eeo, and n is the hardening exponent.  For n = 1, the material behaves as a linearly elastic material; as n approaches infinity, the material behaves ore and more like a perfectly plastic material.  For a generalization of Equation 2.2.45 to multiaxial states via the J2 deformation theory of plasticity, Ilyushin [1946] showed that the stress at each point in the body varies linearly with a single load such as s, the remotely applied stress, under certain conditions.

Ilyushin’s analysis allowed Shih and Hutchison [1976] to use the relationship for crack tip stresses under contained plasticity, i.e. to use [Hutchinson, 1968; Rice & Rosengren, 1968]

(2.2.46)

and similar equations for syy, sxy, etc., to relate the crack tip parameters uniquely to the remotely applied load.  Note that Jpl term in Equation 2.2.46 acts as a (plastic) stress field magnification factor similar to that of the stress-intensity factor in the elastic case.  The form of the relationship that Shih and Hutchinson postulated is given by

(2.2.47)

where  is a function only of relative width (a/b) and n.  An alternate form of Equation 2.2.47 that has been previously used in computer codes [Kumar, et al., 1980; Kumar, et al., 1981; Weerasooriya & Gallagher, 1981] is

(2.2.48)

where P is the applied load (per unit thickness), PTo is the theoretical limit load (per unit thickness), f1 is a function only of geometry and crack length, while h1 depends on geometry, crack length, and the strain hardening exponent n. Shih and coworkers [Kumar, et al., 1980; Kumar, et al., 1981] have tabulated the functions for a number of geometries for conditions of plant stress and plane strain.  From the reference tabulated data [also see Weerasooriya & Gallagher, 1981], these functions can be obtained by interpolation for any value within the a/b and n limits given; thus, the plastic (strain hardening) component of Equation 1.3.44 can be computed for any given applied load P from Equation 2.2.48.

 

 

 

EXAMPLE 2.2.1         J Estimated for Center Crack Panel

Figure 2.2.10 describes the geometry for this example wherein the width W is set equal to 2b and the load P is expressed per unit thickness.  Using Equation 2.2.44 to describe the relationship between the elastic and plastic components, we have

             

From elastic analysis, the stress-intensity factor is known to be (see section 11):

           

For the strain hardening analysis, Equation 2.2.48 is employed, i.e., we use

           

For a center crack panel, the function f1 is given by [Kumar, et al., 1980; Kumar, et al., 1981]

           

and the limit load (per unit thickness) is given by either

           

for plane strain or by

           

for plane stress.  The supporting data for calculating the function h1 is supplied by the following tables for plane strain conditions and plane stress conditions.  The other functions (h2 and h3) contained in these tables support displacement calculations.  As indicated above, data are available for estimating the J-integral according to this approach for a number of additional (simple geometries).  See Kumar, et al. [1981] and Weerasooriya & Gallagher [1981] for further examples.


 

Table of Values of h1, h2, and h3 for the Plane Strain CCP in Tension
[Shih, 1979; Kumar, et al., 1980; Weerasooriya & Gallagher, 1981]

a/b

 

n = 1

n = 2

n = 3

n = 5

n = 7

n = 10

n = 13

n = 16

n = 20

 

1/4

h1

2.535

3.009

3.212

3.289

3.181

2.915

2.625

2.340

2.028

h2

2.680

2.989

3.014

2.847

2.610

2.618

1.971

1.712

1.450

h3

0.536

0.911

1.217

1.639

1.844

1.554

1.802

1.637

1.426

 

3/8

h1

2.344

2.616

2.648

2.507

2.281

1.969

1.709

1.457

1.193

h2

2.347

2.391

2.230

1.876

1.580

1.276

1.065

0.890

0.715

h3

0.699

1.059

1.275

1.440

1.396

1.227

1.050

0.888

0.719

 

1/2

h1

2.206

2.291

2.204

1.968

1.759

1.522

1.323

1.155

0.978

h2

2.028

1.856

1.600

1.230

1.002

0.799

0.664

0.564

0.466

h3

0.803

1.067

1.155

1.101

0.968

0.796

0.665

0.565

0.469

 

5/8

h1

2.115

1.960

1.763

1.616

1.169

0.863

0.628

0.458

0.300

h2

1.705

1.322

1.035

0.696

0.524

0.358

0.250

0.178

0.114

h3

0.844

0.937

0.879

0.691

0.522

0.361

0.251

0.178

0.115

 

3/4

h1

2.072

1.732

1.471

1.108

0.895

0.642

0.461

0.337

0.216

h2

1.345

0.857

0.596

0.361

0.254

0.167

0.114

0.081

0.051

h3

0.805

0.700

0.555

0.359

0.254

0.168

0.114

0.081

0.052

 

Table of Values of h1, h2, and h3 for the Plane Stress CCP in Tension
[Shih, 1979; Kumar, et al., 1980; Weerasooriya & Gallagher, 1981]

a/b

 

n = 1

n = 2

n = 3

n = 5

n = 7

n = 10

n = 13

n = 16

n = 20

 

1/4

h1

2.544

2.972

3.140

3.195

3.106

2.896

2.647

2.467

2.196

h2

3.116

3.286

3.304

3.151

2.926

2.595

2.288

2.081

1.814

h3

0.611

1.010

1.352

1.830

2.083

2.191

2.122

2.009

1.792

 

3/8

h1

2.344

2.533

2.515

2.346

2.173

1.953

1.766

1.608

1.431

h2

2.710

2.621

2.414

2.032

1.753

1.473

1.279

1.134

0.988

h3

0.807

1.195

1.427

1.594

1.570

1.425

1.267

1.133

0.994

 

1/2

h1

2.206

2.195

2.057

1.809

1.632

1.433

1.300

1.174

1.000

h2

2.342

2.014

1.703

1.299

1.071

0.871

0.757

0.666

0.557

h3

0.927

1.186

1.256

1.178

1.040

0.867

0.758

0.668

0.560

 

5/8

h1

2.115

1.912

1.690

1.407

1.221

1.012

0.853

0.712

0.573

h2

1.968

1.458

1.126

0.785

0.617

0.474

0.383

0.313

0.256

h3

0.975

1.053

0.970

0.763

0.620

0.478

0.386

0.318

0.273

 

3/4

h1

2.073

1.708

1.458

1.208

1.082

0.956

0.745

0.646

0.532

h2

1.611

0.970

0.685

0.452

0.361

0.292

0.216

0.183

0.148

h3

0.933

0.802

0.642

0.450

0.361

0.292

0.216

0.183

0.149

 


In the application of Equation 2.2.44 to structural material problems, it has been found [Bucci, et al., 1972] that better correlation with experimental results is obtained if one uses the plasticity enhanced, effective crack length (ae) in plane of the physical crack length (a) in the elastic component expressions.  The effective crack length utilized by Bucci, et al. [1972] was based on the Irwin plastic zone size correction, i.e. the effective crack length was given by

(2.2.52)

where

(2.2.53)

with x = 2 for plane stress and x = 6 for plane strain.  K represents the stress-intensity factor.