Engineering Plasticity
Description
* Provides a solid grounding in plasticity fundamentals and material properties
* Features models, theorems and analysis of processes and relationships related to plasticity, supported by extensive experimental data
* Offers a detailed discussion of recent advances and applications in metal forming
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Persons
Z. R. Wang, Harbin Institute of Technology, China.
W. L. HU, Troy Design and Manufacturing Co., USA.
S. J. Yuan, Harbin Institute of Technology, China.
X. S. Wang, Harbin Institute of Technology, China.
Content
Preface xiii
1 Fundamentals of Classical Plasticity 1
1.1 Stress 1
1.1.1 The Concept of Stress Components 1
1.1.2 Description of the Stress State 2
1.1.2.1 Stresses on an Arbitrary Inclined Plane 2
1.1.2.2 Stress Components on an Oblique Plane 4
1.1.2.3 Special Stresses 6
1.1.2.4 Common Stress States 7
1.1.3 Stress Tensors and Deviatoric Stress Tensors 7
1.1.4 Mohr Stress Circles 9
1.1.4.1 Mohr Circles for a Two-Dimensional Stress System 9
1.1.4.2 Mohr Circles for a Three-Dimensional Stress System 12
1.1.5 Equations of Force Equilibrium 13
1.2 Strain 15
1.2.1 Nominal Strain and True Strain 15
1.2.2 Strain Components as Functions of Infinitesimal Displacements 17
1.2.3 The Maximum Shear Strains and the Octahedral Strains 20
1.2.4 Strain Rates and Strain Rate Tensors 21
1.2.5 Incompressibility and Chief Deformation Types 23
1.3 Yield Criteria 25
1.3.1 The Concept of Yield Criterion 25
1.3.2 Tresca Yield Criterion 26
1.3.3 Mises Yield Criterion 26
1.3.4 Twin Shear Stress Yield Criterion 27
1.3.5 Yield Locus and Physical Concepts of Tresca, Mises, and Twin Shear Stress Yield Criteria 27
1.3.5.1 Interpretation of Tresca Yield Criterion 29
1.3.5.2 Interpretation of Twin Shear Stress Yield Criterion 30
1.3.5.3 Interpretation of Mises Yield Criterion 31
1.4 A General Yield Criterion 33
1.4.1 Representation of General Yield Criterion 33
1.4.2 Yield Surface and Physical Interpretation 34
1.4.3 Simplified Yield Criterion 34
1.5 Classical Theory about Plastic Stress-Strain Relation 35
1.5.1 Early Perception of Plastic Stress Strain Relations 36
1.5.2 Concept of the Gradient-Based Plasticity and Its Relation with Mises Yield Criterion 37
1.5.2.1 Concept of the Plastic Potential 37
1.5.2.2 Physical Interpretation of the Plastic Potential 38
1.5.2.3 Physical Interpretation of Mises Yield Function (Plastic Potential) 39
1.6 Effective Stress, Effective Strain, and Stress Type 42
1.6.1 Effective Stress 42
1.6.2 Effective Strain 42
1.6.3 Stress Type 44
References 44
2 Experimental Research on Material Mechanical Properties under Uniaxial Tension 47
2.1 Stress-Strain Relationship of Strain-Strengthened Materials under Uniaxial Tensile Stress State 47
2.2 The Stress-Strain Relationship of the Strain-Rate-Hardened Materials in Uniaxial Tensile Tests 48
2.3 Stress-Strain Relationship in Uniaxial Tension during Coexistence of Strain Strengthening and Strain Rate Hardening 50
2.4 Bauschinger Effect 56
2.5 Tensile Tests for Automotive Deep-Drawing Steels and High-Strength Steels 57
2.5.1 Test Material and Experiment Scheme 57
2.5.2 True Stress-Strain Curves in Uniaxial Tension 58
2.5.3 Mechanical Property Parameters of Sheets 58
2.5.3.1 Strain-Hardening Exponent n 59
2.5.3.2 Lankford Parameter R 62
2.5.3.3 Plane Anisotropic Exponent ¿R 62
2.5.3.4 Yield-to-Tensile Ratio s s /s b 62
2.5.3.5 Uniform Elongation d m 62
2.6 Tensile Tests on Mg-Alloys 63
2.7 Tension Tests on Ti-Alloys 63
2.7.1 Mechanical Properties of Ti-3Al-2.5V Ti-Alloy Tubes at High Temperatures 65
2.7.2 Strain Hardening of Ti-3Al-2.5V Ti-Alloy in Deformation at High Temperatures 69
References 71
3 Experimental Research on Mechanical Properties of Materials under Non-Uniaxial Loading Condition 73
3.1 P-p Experimental Results of Thin-Walled Tubes 73
3.1.1 Lode Experiment 73
3.1.2 P-p Experiments on Thin-Walled Tubes Made of Superplastic Materials 78
3.1.2.1 Experiment Materials and Specimens 78
3.1.2.2 Loading Methods 80
3.1.2.3 Experimental Results and Analysis 80
3.1.3 Experiments on Tubes Subjected to Internal Pressure and Axial Compressive Forces 86
3.1.3.1 Experimental Device 86
3.1.3.2 Material Properties 88
3.1.3.3 Experimental Results 89
3.2 Results from P-M Experiments on Thin-Walled Tubes 91
3.2.1 Taylor-Quinney Experiments 91
3.2.2 P-M Experiments on Superplastic Material 94
3.3 Biaxial Tension Experiments on Sheets 95
3.3.1 Equipment for Biaxial Tension of Cruciform Specimens 96
3.3.2 Design of Cruciform Tensile Specimens 96
3.3.3 Application of Cruciform Biaxial Tensile Test 97
3.3.3.1 Forming Limit 97
3.3.3.2 Prediction of Yielding Locus 97
3.3.3.3 Analysis of Composite Materials 99
3.4 Influences of Hydrostatic Stress on Mechanical Properties of Materials 100
3.4.1 Testing Technique in High-Pressure Experiments 101
3.4.2 Influences of Hydrostatic Stresses on Flow Behavior of Materials 103
3.4.3 Influences of Hydrostatic Pressure on Fracture Behavior of Materials 106
3.5 Experimental Researches Other Than Non-Uniaxial Tension 114
3.5.1 Plane Compression Experiments 114
3.5.2 Loading Experiments along Normal and Tangential Directions 118
3.5.3 Other Combined Loading Methods 119
References 119
4 Yield Criteria of Different Materials 123
4.1 Predicting Capability of a Yield Criterion Affected by Multiple Factors 123
4.2 Construction of a Proper Yield Criterion in Consideration of Multifactor-Caused Effects 129
4.2.1 A Proper Frame of Yield Criterion 130
4.2.2 Practical Yield Criterion with Multifactor-Caused Effects 133
4.2.3 Material Yielding Behavior Affected by Different Factors 136
4.2.3.1 Convexity of Yield Locus at Plane Stress State 137
4.2.3.2 Stress-Type-Caused Effects 143
4.2.3.3 Hydrostatic-Stress-Caused Effects 145
4.2.4 Simplified Forms of the Yield Criterion 148
4.2.5 Verification of the Yield Criterion Through Experiments 151
4.3 Anisotropic Materials 156
4.3.1 Experimental Description of Anisotropic Behavior of Rolled Sheet Metals 156
4.3.1.1 Uniaxial Tension 157
4.3.1.2 Biaxial Tension 159
4.3.2 Brief Review of the Anisotropic Yield and Plastic Potential Functions 160
4.3.3 Nonassociated-Flow-Rule-Based Yield Function and Plastic Potential 165
4.3.3.1 Anisotropic Yield Criterion 165
4.3.3.2 Anisotropic Plastic Potential 172
4.3.4 Associated-Flow-Rule-Based Anisotropic Yield Criterion 174
4.3.5 Experimental Verification of Two Kinds of Anisotropic Yield Criteria 178
References 184
5 Plastic Constitutive Relations of Materials 187
5.1 Basic Concepts about Plastic Deformation of Materials and Relevant Plastic Constitutive Relations 187
5.1.1 Effects of Material Strength Property Transformation on Material Plastic Deformation 187
5.1.2 General Description of Subsequent Hardening Increments and Convexity of Yield Function 189
5.1.3 Effects of Flow Rules on Judgment of Condition of Stable Plastic Deformation of Materials 196
5.2 Equivalent Hardening Condition in Material Plastic Deformation 197
5.2.1 Universal Forms of Plastic Potential and Yield Criterion in Constructing Plastic Constitutive Relations 198
5.2.2 Relationship between Yield Function and Plastic Potential in Describing Equivalent Hardening Increments 199
5.2.3 Equivalent Hardening Condition Corresponding to Associated Flow Rule 201
5.2.4 Equivalent Hardening Condition Related to Nonassociated Flow Rule 206
5.3 "Softening" and Strength Property Changes in Plastic Deformation of Materials 209
5.3.1 Mechanical Models Mimicking Plastic Deformation of Sensitive-to-Pressure Materials 210
5.3.2 Dynamic Models to Imitate the Stress-Strain Relation of Anisotropic Material 215
5.3.3 Softening and Material Strength Property Changes in a Stable Plastic Deformation 219
5.4 Influences of Loading Path on Computational Accuracy of Incremental Theory 227
5.4.1 Discontinuous Stress Path 227
5.4.2 Unrealistic Strain Path 229
References 231
6 Description of Material Hardenability with Different Models 233
6.1 Plastic Constitutive Relations of Sensitive-to-Pressure Materials 233
6.1.1 Experimental Characterizations of Yield Function and Corresponding Plastic Potential 234
6.1.2 Theoretical Predictions in Comparison with Experimental Results 237
6.1.2.1 Influences of Hardening Models upon Description of Plastic Deformation of Materials 238
6.1.2.2 Yieldability and Plastic Flowability of Sensitive-to-Pressure Materials 239
6.1.2.3 Prediction of the Volumetric Plastic Strain 240
6.1.2.4 Predictions of Stress-Strain Relations in Uniaxial Tension and Compression 243
6.1.2.5 Stress-Strain Relations in Compression Affected by Superimposed Pressures 247
6.2 Anisotropic Hardening Model of Rolled Sheet Metals Characterized by Multiple Experimental Stress-Strain Relations and Changeable Anisotropic Parameters 248
6.2.1 A Constitutive Model to Describe Anisotropic Hardening and Anisotropic Plastic Flow of Rolled Sheet Metals 249
6.2.2 Transformation from Special 3D Stress State into 2D Stress States 252
6.2.3 Predictions of Anisotropic Hardening and Plastic Flow Behavior 254
6.2.3.1 Subsequent Yield Locus of Anisotropic Materials 254
6.2.3.2 Predictions of All Experimental Stress-Strain Relations in Yield Function 260
6.2.4 Experimental Verification 262
6.2.4.1 Predictions of Stress-Strain Relations in Uniaxial Tensions in Different Directions 262
6.2.4.2 Predictions of Changeable Anisotropic Parameters 267
6.3 Plastic Constitutive Relation with the Bauschinger Effects 271
6.3.1 Basic Concepts of the Bauschinger Effects 271
6.3.2 Consideration of the Bauschinger Effect in Constructing a Constitutive Relation 274
6.3.3 Exotic Anisotropic Behavior of Material Element Induced by Kinematic Hardening Model Based on Associated Flow Rule 276
6.3.3.1 Anisotropic Flowability Borne of Kinematic Yield Model 276
6.3.3.2 Calculations of the Exotic Anisotropy by Means of Yoshida's Modified Kinematic Model 281
6.3.4 A Method to Generate a Kinematic Plastic Potential Function 286
References 293
7 Sequential Correspondence Law between Stress and Strain Components and Its Application in Plastic Deformation Process 295
7.1 Sequential Correspondence Law between Stress and Strain Components and Its Experimental Verification 295
7.1.1 Sequential Correspondence Law between Stress and Strain Components 295
7.1.2 Experimental Verification of the Sequential Correspondence Law between Stress and Strain Components 298
7.1.3 Application of the Sequential Correspondence Law between Stress and Strain Components 300
7.2 Zoning of Mises Yield Ellipse and Typical Plane Stress Forming Processes on It 302
7.3 Stress and Strain Analysis of Plane-Stress Metal-Forming Processes 306
7.3.1 Tube Drawing 306
7.3.2 Deep Drawing 307
7.3.3 Tube Hydroforming 308
7.4 Spreading of Mises Yield Cylinder and Characterization of Three-Dimensional Stresses Therein 309
7.5 Zoning in Three-Dimensional Stress Yield Locus and Positioning Typical Forming Processes Thereon 311
References 316
8 Stress and Strain Analysis and Experimental Research on Typical Axisymmetric Plane Stress-Forming Process 317
8.1 Incremental-Theory-Based Solution to Stress and Strain Distribution of Steady Axisymmetric Plane Stress-Forming Processes 317
8.1.1 Two Expressions of Stress and Strain Distribution 317
8.1.2 Division of Steady Thin-Walled Tube-Forming Processes 319
8.1.3 Basic Formulas and Assumption 320
8.1.4 Stress and Strain Distribution in Steady Frictionless Forming Process 321
8.1.4.1 General Equilibrium Equation 321
8.1.4.2 Stress Distribution s(r) 322
8.1.4.3 Strain Rate de/df 324
8.1.4.4 Strain Distribution e(f) 325
8.1.5 Stress and Strain Distribution in Steady Forming Processes in the Presence of Friction 328
8.1.5.1 General Equilibrium Equation 329
8.1.5.2 Stress and Strain Distribution 331
8.2 Experimental Study on Thickness Distribution in Tube Necking and Tube Drawing 331
8.2.1 Thickness Distribution in Tube-Necking Processes 331
8.2.2 Experimental Research on Thickness Distribution during Tube Drawing [6] 333
8.3 Experiments on Thin-Walled Tube under Action of Biaxial Compressive Stresses 336
8.3.1 Introduction of Experimental Setup 337
8.3.2 Results and Discussion 339
References 341
9 Shell and Tube Hydroforming 343
9.1 Mechanics of Dieless Closed Shell Hydro-Bulging 343
9.1.1 Equilibrium Equation for an Internally Pressurized Closed Shell 343
9.1.2 Yield Equation of an Internally Pressurized Closed Shell 345
9.1.3 Principle of Spheroidization of Plastic Deformation in Shell Hydro-Bulging 345
9.2 Dieless Hydro-Bulging of Spherical Shells 347
9.2.1 Stress Analysis of Dieless Hydro-Bulging of Spherical Shells 347
9.2.2 Manufacture of Spherical Shells 347
9.2.3 Shell Structure before Hydro-Bulging 348
9.2.4 Dieless Hydro-Bulging of Single-Curvature Polyhedral Shells 349
9.3 Dieless Hydro-Bulging of Ellipsoidal Shells 350
9.3.1 Stress Analysis of Internally Pressurized Ellipsoidal Shells 351
9.3.2 Wrinkling of Internally Pressurized Ellipsoidal Shell and Anti-Wrinkling Measures 352
9.4 Dieless Hydro-Bulging of Elbow Shell 355
9.5 Tube Hydroforming 356
9.5.1 Principle of Tube Hydroforming and Its Stress States 356
9.5.2 Yield Criterion for Tube Hydroforming 357
9.5.3 Position of Tube Hydroforming on Yield Ellipse 358
9.5.4 Typical Stress States and Their Distribution on Yield Ellipse 358
9.5.5 Effect of Stress State on the Tube Deformation Characteristics 359
9.5.6 Formation Mechanism of Wrinkles in Thin-Walled Tube Hydroforming 360
References 362
10 Bulk Forming 365
10.1 Load Calculation in Tool Movement Direction 365
10.2 Upsetting of Cylinders and Rings 368
10.2.1 Load Calculation for Cylinder Upsetting 369
10.2.2 Inhomogeneous Deformation in Cylinder Upsetting 373
10.2.3 Metal Flow and Pressure Distribution during Ring Compression 376
10.3 Characteristics of Die Forgings and Calculation of Required Loads 378
10.4 Isothermal Forging 381
10.4.1 Stress Analysis in Isothermal Forging 381
10.4.2 Stress Analysis of a Single Rib Piece in Isothermal Forging 382
10.4.3 Isothermal Forming of Cross-Rib-Born Pieces 384
10.4.3.1 Analysis of Forming Processes 384
10.4.3.2 Stress Analysis 384
10.4.4 Control and Analysis of Flow Defects during Isothermal Forging 386
10.4.4.1 Folds 386
10.4.4.2 Formation and Control of Flow Lines 388
10.5 Calculation of Required Load in Rolling 389
10.5.1 Derivation of Formula for Calculating Unit Pressure Distribution on Rollers' Contact Arc Surface 391
10.5.2 Total Rolling Force and Average Pressure 395
10.5.3 Rolling Torque 396
10.5.4 Energy Consumption in Rolling 397
10.6 Extrusion and Drawing 397
10.6.1 Extrusion 397
10.6.2 Drawing 400
10.7 Rotary Forging 403
10.7.1 Introduction 403
10.7.2 Stress and Strain Analysis in Rotary Forging of Cylinders 403
10.7.3 Stress-Strain Analysis in Rotary Forging of Discs 409
10.8 Strain Distribution Measurement in Bulk Forming 411
10.8.1 Introduction 411
10.8.2 Screw Method 412
10.8.3 Applications of Screw Method in Determining Strain Distribution 414
References 419
11 Sheet Forming 421
11.1 Deep Drawing 421
11.1.1 Basic Principles 421
11.1.2 Strain Analysis in Flange Area 421
11.1.3 Stress Analysis of the Flange Area 424
11.1.3.1 Equilibrium Equation 424
11.1.3.2 Yield Criteria 425
11.2 Sheet Hydroforming Process 426
11.2.1 Basic Principles 426
11.2.2 Characteristics and Application Scope 427
11.2.3 Assessment of Experimental Parameters 428
11.2.3.1 Critical Liquid Pressure p cr 428
11.2.3.2 Drawing Force 429
11.2.3.3 Blank Holder Force (BHF) 429
11.2.4 Influences of Normal Stress on SHP [10] 430
11.2.5 Influences of Pre-Bulging on the Deformation Uniformity in SHP 430
11.3 Hole-Flanging 434
11.3.1 Basic Principles 434
11.3.2 Analysis of Stress and Strain 434
11.3.3 Limiting Flanging Coefficient 436
11.4 Viscous Pressure Forming 438
11.4.1 Mechanism and Features 438
11.4.1.1 Forming Sequence 438
11.4.1.2 Properties of Pressure Medium 439
11.4.1.3 Reverse Pressure 439
11.4.1.4 Surface Quality 439
11.4.2 Constitutive Equations of Viscous Medium 439
11.4.3 Influences of BHP on Forming Process 441
11.5 Multipoint Sandwich Forming 445
11.5.1 Introduction 445
11.5.2 Working Principles of MPSF 446
11.5.3 Advantages of MPSF and Applications 447
11.5.4 FE Model of MPSF 448
11.5.5 Forming of Ellipsoidal Workpiece 451
11.5.6 Saddle-Type Pieces Forming 455
11.6 Formability of Sheet Metals 462
11.6.1 Introduction 462
11.6.2 Forming Limit Diagram 462
11.6.3 Experimental Determination of FLC 464
11.6.3.1 Uniaxial Tensile Test 465
11.6.3.2 Hydro-Bulging Test 465
11.6.3.3 Nakazima Test 465
11.6.4 Advanced Experimental Methods 466
11.6.5 Theoretical Prediction of FLC 469
11.6.6 New Developments in FLCs 475
11.7 Improvements of Panel Stamping Process 478
11.7.1 Designs of Draw-Bars Corresponding to the Wrinkling Types 479
11.7.2 Replacement of Stretching Wall with Local Nondeformable Design 482
References 484
Index 489
Chapter 1
1.1 Stress
1.1.1 The Concept of Stress Components
When a set of directional forces acts on a deformable material element (see Figure 1.1) and remains balanced without causing a displacement and/or rotation, a set of balanced internal stresses must be generated because of the deformation taking place in the material element. Generally, if stress components distribute uniformly on a plane, the stress unit is equal to the force per unit area. Despite an inherent relation that exists between the stress and the acting force, the stress and the force are entirely different in their physical concepts that we could not confuse.
Figure 1.1 Directional forces acting on a unit element.
In analyzing displacement and rotation of a rigid body, all acting forces are vectors and can be converted into a single one. For example, the forces shown in Figure 1.1 can be turned into a single vector P:
1.1Equation (1.1) means that if , this loaded body must move and if P does not pass the body's center, the body must rotate simultaneously.
However, it is incorrect to use the force P resulted from vector addition to analyze the elastic or plastic changes in shape of the material element. Different sets of directional forces will respond to different stress distribution on a plane cut out of this material element even if they have the same vector composition. Figure 1.2 illustrates the case of a simple uniaxial tension.
Figure 1.2 Relationship between forces and stresses on a plane cut out of a loaded body under uniaxial tension.
Stress components on different planes of this loaded material element are different. For example, the stress component on the plane vertical to the axis in Figure 1.2 can be expressed by
1.2where P is the axial force, and S0 is the cross section.
If this material element is a unit body with each edge equal to 1 unit, Equation (1.2) becomes
1.3Equation (1.3) builds up a relationship between the force and the stress, however, it holds true only in the analysis of equilibrium system. Their physical concepts are essentially different. The force is mainly to make the forced material element to move or move with rotation, and the stress deals with the "shape changing" of the stressed material element.
On a cut plane tilting at an angle a against the axis, when the area of the plane increases from S0 to S0/cos a, the stress on it becomes
1.4With the angle a increasing, the cut plane should get more inclined to decrease the stress sa so as to maintain the force in equilibrium. Equation (1.4) indicates that the value of stress sa is completely predicated on the orientation of the cut plane. But variation of the stress value sa does not bring any influences upon the deformation type of the uniaxial tension, which means that the strain state of the element remains unchanged.
1.1.2 Description of the Stress State
It should thus be clear that the stress state is very important-we must understand how the material element responds to the deformation caused by the stress components. On the other hand, in the case of inhomogeneous stress distribution on cut planes, which is most common in reality, it is required to analyze the stress state of a deforming body from one point inside the body to the other. Generally, the stress condition of a point inside a deforming body is often defined by a cubic element. Further understanding stress components in relation to any complex stress state would be essential to fully grasp the stress and stress tensor concepts.
1.1.2.1 Stresses on an Arbitrary Inclined Plane
Let's investigate the necessary condition by analyzing the stress state at a point inside the deformable body. Suppose that the point to be analyzed is O. Usually, three mutually perpendicular planes XOY, YOZ, and ZOX (see Figure 1.3) are set up to analyze its stress state. Stress components on each plane are divided into one normal stress, symbolized by s, vertical to the plane and two shear stresses by t, parallel to the coordinate axes. In order to identify what plane the stress components act on, one subscript is used for the normal stress and two subscripts for each of the shear stresses. The subscript for the normal stress denotes its acting direction.
Figure 1.3 Stress components on three mutually perpendicular planes.
The first of the two subscripts of the shear stress denotes the normal direction of the acting plane of the shear stress, and the second the acting direction of the shear stress (see Figure 1.3). The value of all stress components is not arbitrary but is determined by the equilibrium between stress components on the cut plane and the associated external force (see the example in Figure 1.2). Customarily, positive normal stresses are supposed to be tensile ones and negative normal stresses are compressive. Normally, it doesn't matter whether a shear stress is positive or negative, because shear stresses always exist in pairs. Nevertheless, because some materials show anisotropic yield behavior and/or strength differential in tension and compression, the change in loading direction or stress state might change the yielding behavior in value of the shear stresses. Therefore, for some materials, when the stress direction changes (e.g., from tension to compression or vice versa), we must still define whether the shear stresses are positive or negative, based on the action direction. Namely, when the second subscript of shear stress implies the positive direction of the axis, this shear stress is positive, and vice versa. Thus, stress components on three planes in a xyz coordinate system can be expressed in the matrix form (see Figure 1.3).
From Figure 1.3, we know that the nine stress components on the three mutually perpendicular planes share a common feature. That is, shear stresses exist in pairs with the same value and the subscripts composed of two identical English letters in opposite order. Namely,
1.5Equation (1.5) means that there are only six independent stress components in a symmetric form (Figure 1.3), which can be represented by a matrix as follows:
1.6In the case that, on the three mutually perpendicular planes, there are only three normal stresses, called the principal stresses: s1, s2, and s3, without any shear stress, Equation (1.6) becomes
1.7which represents the principal coordinate system.
Obviously, once the six stress components at the point O are given with respect to the x, y, and z coordinate axis, the stress state at the point O is fixed. Any change in the value of one of the stress components, as long as it is not due to the conversion of coordinate system, would mean a change in the stress state at the point O. In other words, the stress state at a point must be described with six stress components or three principal stresses.
1.1.2.2 Stress Components on an Oblique Plane
It has been proved that given six stress components or three principal stresses at a point, the normal and shear stresses on any oblique plane relative to the x, y, and z coordinate axes can be determined.
On the oblique plane represented by a triangle ABC (see Figure 1.4), the normal N (ON) is denoted by directional cosines (lmn) with three angles ax, ay and az formed between ON and separately OX, OY, and OZ. Let ?A denote the area of the triangle ABC, ?Ax, ?Ay, and ?Az the areas formed by projecting ?A, respectively, on the three coordinate surfaces:
1.8Figure 1.4 Stress components on an oblique plane.
Let the resultant stress on the triangle ABC be denoted by S, which has a direct stress component sN normal to the plane ABC and a shear stress component t on it. Thus, the equilibrium of the forces on the tetrahedron OABC in the direction OX, OY, and OZ, respectively, can be described by
1.9where Sx, Sy, and Sz are the components of the resultant stress S in parallel with OX, OY, and OZ, respectively.
Simplification of Equation (1.9) gives
1.10According to the principle of parallel hexahedron, the resultant stress S on the oblique plane is
1.11The normal stress sN on the oblique plane can be expressed with the components Sx, Sy and Sz as follows:
1.12Substitution of the...
