Fracture Mechanics and Crack Growth E-Book

Naman Recho

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 491

Veröffentlichungsjahr: 2012

Beschreibung

This book presents recent advances related to the following two topics: * how mechanical fields close to material or geometrical singularities such as cracks can be determined; * how failure criteria can be established according to the singularity degrees related to these discontinuities. Concerning the determination of mechanical fields close to a crack tip, the first part of the book presents most of the traditional methods in order to classify them into two major categories. The first is based on the stress field, such as the Airy function, and the second resolves the problem from functions related to displacement fields. Following this, a new method based on the Hamiltonian system is presented in great detail. Local and energetic approaches to fracture are used in order to determine the fracture parameters such as stress intensity factor and energy release rate. The second part of the book describes methodologies to establish the critical fracture loads and the crack growth criteria. Singular fields for homogeneous and non-homogeneous problems near crack tips, v-notches, interfaces, etc. associated with the crack initiation and propagation laws in elastic and elastic-plastic media, allow us to determine the basis of failure criteria. Each phenomenon studied is dealt with according to its conceptual and theoretical modeling, to its use in the criteria of fracture resistance; and finally to its implementation in terms of feasibility and numerical application. Contents 1. Introduction. Part 1: Stress Field Analysis Close to the Crack Tip 2. Review of Continuum Mechanics and the Behavior Laws. 3. Overview of Fracture Mechanics. 4. Fracture Mechanics. 5. Introduction to the Finite Element Analysis of Cracked Structures. Part 2: Crack Growth Criteria 6. Crack Propagation. 7. Crack Growth Prediction in Elements of Steel Structures Submitted to Fatigue. 8. Potential Use of Crack Propagation Laws in Fatigue Life Design.

Leseprobe

Table of Contents

Preamble

Preface

Notations

Chapter 1

PART 1: STRESS FIELD ANALYSIS CLOSE TO THE CRACK TIP

Chapter 2. Review of Continuum Mechanicsand the Behavior Laws

2.1. Kinematic equations

2.2. Equilibrium equations in a volume element

2.3. Behavior laws

2.4. Energy formalism

2.5. Solution of systems of equations of continuum mechanicsand constitutive behavior law

2.6. Review of the finite element solution

Chapter 3. Overview of Fracture Mechanics

3.1. Fracture process

3.2. Basic modes of fracture

Chapter 4. Fracture Mechanics

4.1. Determination of stress, strain and displacement fields arounda crack in a homogeneous, isotropic and linearly elastic medium

4.2. Plastic analysis around a crack in an isotropichomogeneous medium

4.3. Case of a heterogeneous medium: elastic multimaterials

4.4. New modeling approach to singular fracture fields

Chapter 5. Introduction to the Finite Element Analysisof Cracked Structures

5.1. Modeling of a singular field close to the crack tip

5.2. Energetic methods

5.3. Nonlinear behavior

5.4. Specific finite elements for the calculation of cracked structures

5.5. Study of a finite elements program in a 2D linearelastic medium

5.6. Application to the calculation of the J-integral in mixed mode

5.7. Different meshing fracture monitoring techniques by finite elements

PART 2: CRACK GROWTH CRITERIA

Chapter 6. Crack Propagation

6.1. Brittle fracture

6.2. Crack extension

6.3. Crack extension criterion in an elastic plastic medium

6.4. Crack-extension criterion from V-notches

6.5. Fracture following crack growth under high-cyclenumber fatigue

6.6. Crack propagation laws

6.7. Approaches used for the calculation of fatigue lifetime

6.8. Case of the variable amplitude loading

6.9. Crack retardation effect due to overloading

6.10. “Reliability–failure” in the presence of random variables

Chapter 7. Crack Growth Prediction in Elements of SteelStructures Submitted to Fatigue

7.1. Significance and analysis by calculation of stressesaround the local effect

7.2. Crack initiation under fatigue

7.3. Localization and sensitivity to rupture of cracks

7.4. Extension of the initiated crack under fatigue

Chapter 8. Potential Use of Crack Propagation Laws in FatigueLife Design

8.1. Calculation of the crack propagation fatigue lifeof a welded-joint

8.2. Study of the influence of different parameters on fatigue life

8.3. Statistical characterization of the initial crack size accordingto the welding procedure

8.4. Initiation/propagation coupled models: two phase models

8.5. Development of a damage model taking into accountthe crack growth phenomenon

8.6. Taking into account the presence of residual weldingstresses on crack propagation

8.7. Consideration of initial crack length under variableamplitude loading

8.8. Propagation of short cracks in the presence of a stress gradient

8.9. Probabilistic approach to crack propagation fatigue life:reliability–failure

Conclusion

Bibliography

Index

First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Naman Recho to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Recho, Naman.   Fracture mechanics and crack growth / Naman Recho.       p. cm.   Includes bibliographical references and index.   ISBN 978-1-84821-306-7   1. Fracture mechanics. 2. Materials--Fatigue. I. Title.     TA409.R44 2012     620.1′126--dc23

2011051809

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library

PreambleHow to Comprehend This Work?

The aims of fracture mechanics are twofold: on one hand they concern the description of mechanical fields in the neighborhood of the tip of the crack and the energies that are associated therewith; and on the other hand, they deal with the evaluation of the harm of a crack in terms of its propagation.

Two fields of study constitute the structure of this work. The first one is relative to the modeling of the singularity induced by the crack tip that is described in Part I entitled Stress Field Analysis Close to the Crack tip. It deals with the modeling of mechanical fields at the crack or singularity tip. The second part, entitled Crack Growth Criteria, deals with crack initiation and propagation under monotonic and cyclic loadings.

In Part I, an introduction to continuum mechanics is given. Then, an approach which consists of studying the way which enables us to see how to introduce a singularity in a continuum is developed. To that end, two methods are detailed:

– a local one, which describes the stress (or even strain) functions as being continuous everywhere except at the crack tip and which introduces the free boundary conditions relative to the lips of the crack; this forms the asymptotic analysis. In a two-dimensional linear elastic medium, the asymptotic analysis leads to the following description of the stress field:

with KI and KII being two load and geometry functions, describing the amplitude of the stress fields, called stress intensity factors. fij (θ) and gij (θ), which are two exclusive functions of θ. r and θ, respectively, represent the distance to the crack tip of the volume element studied to the crack tip and its orientation with respect to the axis of the crack.

– For the second method, we can evaluate G, which is the release rate of potential energy, Wpot, subsequent to an infinitesimal increase Δa in the crack length:

In a linear elastic plane medium we can, easily link G to the stress intensity factor in crack-opening mode KI:

, in plane stress

, in plane strain

where E is Young’s modulus and v is Poisson’s ratio

Part II, entitled Crack Growth Criteria deals with the propagation and extension criteria of a crack in elastic and elastic-plastic media under constant and dynamic loads (fatigue fracture).

The analytical solutions obtained cannot be used in the structures with variable geometry and boundary conditions, so we need to use methods of numerical analysis, and in particular the finite element method. Two chapters deal with these numerical applications: one, in Chapter 5 relative to the introduction to the finite element calculations of cracked structures and the other, in Chapter 7 dealing with the forecast of the failure by crack growth of elements of steel structures subjected to fatigue.

Each phenomenon studied is dealt with according to:

– its conceptual and theoretical modeling;

– its use in the criteria of fracture resistance; and

– its implementation in terms of feasibility and numerical application.

The reader is warned that the bulk of the developments in this book concern metal materials. The extension of the conclusions to composite, elastomeric or plastic materials is unreliable.

Preface

“The fact is that there is no opposition between constraint and liberty,and that, on the contrary, they support each other – since all libertyattempts to overturn or overcome a constraint and every constraint hascracks or points of least resistance that invite liberty to pass through.”

The View from Afar, Claude Lévi-Strauss(p. 17, Plon, Paris, 1983)

What are the reasons that have led to the writing up of the present work? When re-reading the first French edition of September 19951, I noticed that as a conclusion and a final paragraph of the conclusion, I had written:

“Presently, such a complete analysis (of crack growth in structures, developed in the book) still faces two kinds of difficulties; technical for the determination of certain variables, and economical because of the relatively high cost of numerical analyses and experimental measurements.”

In the 16 years since, numerical methods have evolved in capacity and are now accessible to a larger number of engineers and researchers. Equally, the feasibility of certain experimental measurements has increased with the availability of analysis techniques (fractographic aspects, strain fields, etc.) allowing us to monitor and record the crack growth history of a structure and in particular to measure the mechanical fields in the neighbourhood near the tip of a crack, or even a singularity.

In addition, original works carried out by PhD students have led to significant advances in various aspects covered in the earlier edition, namely:

– the analysis of the reliability of welded components [AP.1.5, AP.1.7];

– the analysis of the effect of overload on the elements of cracked structures [AP.1.2, AP.1.5];

– the follow up of crack growth by numerical methods such as crack bow techniques [AP.1.2];

– the development of specific crack propagation and crack extension criteria [AP.1.1, AP.1.3, AP.1.4];

– the establishment of a new approach to fracture mechanics to find new solutions to problems such as the presence of a crack in an anisotropic elastic material [AP.1.4];

– the analysis of geometric singularities, such as the V-shaped notch [AP.1.1];

– the establishment of new models of coupling – “initiation, propagation” – seen in their local aspects in terms of fracture mechanics ([AP.1.6], and in terms of damage analysis [AP.2.1]), in their global aspects in terms of the S-N curve [AP.1.5, AP.1.9, AP.2.2] or in their numerical aspects in the finite element analysis of an industrial structure [AP.1.2];

– the study of the influence of local effects at the weld toes on the fatigue design of welded joints [AP.1.2, AP.1.10, AP.1.11].

All of these advances have enabled the establishment of a comprehensive global approach, applicable to real structures and not just test specimens, in that the flexibility of a structure, which is affected by the size of the crack, is considered in the calculation.

Finally, the lessons in “fracture mechanics, damage analysis and fatigue design” given at the ETH (engineering schools, Switzerland), the CHEC (Centre des Hautes Etudes de la Construction, Paris) and Research Masters and Professional training atthe Université Blaise Pascal in Clermont-Ferrand, have provided some subjects for students and teachers who are interested in research.

This edition originated during the establishment of specific scientific seminars given in China, Italy and France ([AP.3.1] to [AP.3.7]) targeted at researchers and engineers. It incorporates the findings of the work done in collaboration with doctoral students since 1995 and published in journals and presented at international and national conferences. This first book treats the theoretical, conceptual and numerical aspects of fracture mechanics and divided into two parts:

Part I: Stress Field Analysis Close to the Crack tip,

Part II: Crack Growth Criteria.

Given the amount of material involved, there may be another book dealing with industrial applications and exercises published in the near future.

Naman RECHOJanuary 2012

1 It corresponds to a book known as: Rupture par fissuration des structures, collection Traité des nouvelles technologies, série Matériaux, Hermès, Paris, September 1995.

Notations

(• ) Partial derivatives relative to the base coordinate

H (p, q, z)= Hamiltonian

λ

ET

Mode I, mode II, mode III are the elementary fracture modes, opening mode in plane shear and out of plane shear modes, respectively

Chapter 1Introduction

The rupture of a mechanical specimen can be interpreted primarily as an interruption in the continuity of the specimen (this is in fact a particular definition of “failure”). In this case, the application of continuum mechanics faces a singularity due to the presence of cracks in the specimen.

Fracture mechanics is simply the application of continuum mechanics and the behavior laws of a material to a body whose boundary conditions are introduced in the presence of crack geometry.

Rupture can occur after a large deformation, usually after a plastic instability resulting from the presence of two opposite effects: one reducing the section; and the other consolidation of the material by hardening. It can, however, occur without significant prior deformation under generalized stresses that is often the case in the elastic domain. We are then in the presence of a brittle fracture.

The analysis of stresses and strains near the crack tip is a basis for understanding the behavior of cracks. Although a plastic or damaged zone is present at the tip of the crack, the linear elastic analysis provides us with an accurate enough mapping of reality for materials such as steel. In the case of ductile materials or extreme loads, however, we need to take into account the elastic-plastic behavior laws.

Fracture mechanics assumes the existence of an initial crack in the structure being studied. This introduces geometric discontinuity singularity to the stress and strain fields and deformations at the crack tip.

The phase that explains the behavior of the structure of intact state where the structure contains a macroscopic crack is called the initiation phase of the crack. Priming of a crack is usually in the vicinity of defects in the design of the structure (e.g. geometric discontinuities) due to poor execution or welding, etc. These defects create local high stresses that promote the initiation of cracks without generalized stresses that exceed the yield strength of the material.

When the cracks are initiated, their propagation can be sudden or gradual. This may result in brittle fracture or crack growth by fatigue. When the propagation of these cracks is accompanied by plastic deformations it is the plastic fracture mechanics, if not a mechanical linear elastic fracture, that will be responsible.

Table 1.1 shows the different types of failure mentioned. Indeed, each type of rupture is a set of assumptions, definitions and analysis.

Table 1.1.Types of failure according to the behavior laws

We will mainly study the two types of failure — I and III — in the context of this book.

The basic problem in linear fracture mechanics can be seen as the analysis of a stress field in plane linear elastic cracked media. This is for theoretical reasons (since the elastic plane is the means by which we find analytical solutions), and for technical and practical reasons (there are structures that are cracked, in which generalized constraints are below the elastic limit).

The definition of a failure criterion (or security) is a specific preoccupation, and is of major importance. This definition comes from a collection of reflections the engineer has on the basis of disparate elements, such as the behavior of the structure, industry, socioeconomics, etc. The safety criterion is given in Figure 1.1 and determines how it can be structured.

Figure 1.1.The safety criterion

Three essential elements exist for any judgments on the safety behavior of a structure:

– global and local geometry;

– the boundary conditions (by forces and displacements); and

– the safety criterion (or failure).

It is the comparison between the solution obtained from the first two elements and the safety criterion that is essential. In the triangle created by the geometry, the failure criterion and boundary conditions (see Figure 1.2), the intervention of one or more peaks can resize the structure.

Figure 1.2.The interaction of geometry, boundary conditions and the failure criterion

To study failure, it is essential to analyze the stress, strain and displacement fields in the cracked structure, especially near the tips of existing cracks. This is encompassed in the study of fracture mechanics. This theoretical study should interpret the phenomenological aspects of the rupture, yet these aspects cannot be addressed without experimentally observing the fracture surface, the rate of crack growth, etc. A presentation of the results of continuum mechanics and the behavior laws, however, appears to be necessary to determine the mechanical fields (displacements, strains and stresses) near the tip of a crack (or a singularity). A review of experimental observations, in light of the calculated mechanical fields gives us a better understanding of failure criteria under quasi-static loading and fatigue. Practical applications for the propagation of cracks in welded joints are detailed in Chapter 7 to illustrate the analytical process.

PART I Stress Field Analysis Close to the Crack Tip

Chapter 2Review of Continuum Mechanics and theBehavior Laws

Suppose a given structure, with known geometry and constituent materials, is subjected to boundary conditions in force (load). When there are sufficient1 boundary conditions in the displacements (tie), a displacement field is generated in the structure that determines for each point P(x, y, z) belonging to the structure, the position P’ (x′, y′, z′) after loading where u, v, and w are the displacement of P to P’ based on x, y, z, or:

[2.1]

The displacement field generates a stress field in the structure. The stress from a point O of the structure is defined in terms of force acting on an infinitesimal area plane through O.

The orientation of this area can be described by the unit normal vector. The force is also a vector. It is appropriate to describe the stress in the form of the components of these two vectors in a coordinate system that has been defined previously. Each area and force vector has three components in three dimensions, so that it is expected to describe the stress in nine terms. The nine components (terms) of the stress are plotted on a three-dimensional element (dx, dy, dz) in Cartesian coordinates, see Figure 2.1.

Figure 2.1.Representation of the components of stress in Cartesian coordinates

Each component of the stress is defined by two indices. The first indicates the side on which the stress is applied (1 for x, 2 for y and 3 for z). The second index indicates the direction of the force-generating component of the stress.

We can take the example: , which is a normal stress perpendicular to side A1 (perpendicular to x) in the direction of x. The stress state at the point O (the stress tensor) is called σij . σij represents nine components where i and j independently take the values 1, 2, and 3.

Figure 2.2.Representation of the components of stresses usingcylindrical coordinates

The displacement field also generates a strain field, where the strain is defined as the relative displacement of points belonging to a structure to each other. The strain is closely related to stress by a behavior law and is written in the form of the εij tensor, which consists of nine components:

2.1. Kinematic equations

Suppose that point P belonging to a deformed body, with coordinates P (x, y, z ), is associated with point Q at a distance, ds, from P. Its coordinates are: Q (x + dx, y + dy, z + dz ). Thus, we have: . Then a load is applied on this body. Segment PQ will move to P′Q′ with the following coordinates and .

Thus: (see Figure 2.3).

Figure 2.3.A solid deformation

[2.2]

By definition:

[2.3]

Replacing [2.3] in [2.2], we have:

[2.4]

Considering that the environment (deformable body) that belongs to PQ is continuous, we can write:

[2.5]

Replacing [2.5] in [2.4]:

[2.6]

with:

[2.7]

we notice that the left-hand side of equation [2.6] may be written as:

Naming lengthening of segment PQ, we have:

Equation [2.6] can then be written as:

[2.8]

Neglecting (∆ ℓ )2, equation [2.8] gives a physical significance to each term in the strain tensors εij of equation [2.7].

Equation [2.7] is written with the index notation in the form of:

[2.9]

where k takes the values of 1, 2 and 3 for every pair of values given to i and j . We consider that u1 ≡ u, u2 ≡ v, u3 ≡ w and x1≡x, x2≡ y , x3 ≡ z.

For example:

Equations [2.7] and [2.9] are called kinematic equations.

In the case of small strains (small displacements), by neglecting the second-order derivatives equation [2.9] can be written:

[2.10]

or:

[2.11]

From equations [2.10] or [2.11], in the case of small displacements we can write the displacement field as:

[2.12]

or using index notation:

[2.13]

or for the (i) given, (j) takes the values 1, 2 and 3.

Considering a two-dimensional application, suppose that point P(x,y ) moves after loading to P’(x’,y’ ), where:

with:

Figure 2.4 shows an element as a plane (dxdy) described in a Cartesian coordinate system (element PACB). After loading, this element becomes (P’A’C’B’), with a left warping and a shift in translation.

Figure 2.4.Deformations of a volume element in a two-dimensional medium with Cartesian coordinates

The displacement of point A to A’ occurs via a translation ν , and an increase in ν due to a shift of P’ to A’ over x, or , etc. The rotation of segment P’B’ relative to P’A’is therefore equal to .

Similar to stress fields, strain fields can also be written in symmetrical tensor form, where εijji , or:

In the case of polar coordinates, where:

[2.14]

where ur and uθ are the displacement of point A (r, θ ) (which becomes A’ (r’,θ ) after the strain) based on the axes :

The strains εrr and εθθ are the relative addition of sides AB and AD:

[2.15]

Figure 2.5.Deformations of a volume element in a two dimensional medium shown bypolar coordinates

These relations can be deduced from the formulas of the transformation of the strain tensor from Cartesian coordinates to polar coordinates:

[2.16]

ε11, ε22 and ε12 are given by equation [2.11], where u and v are linked to ur and uθ by the following relations:

[2.17]

Thus equations [2.14] and [2.15] can be obtained.

In the case of cylindrical coordinates, where we have:

the kinematic equations are written as follows:

[2.18]

These equations are equivalent to equations [2.11] in Cartesian coordinates, and therefore only analyze small strain cases.

2.2. Equilibrium equations in a volume element

If we consider a volume element (dx dy dz) belonging to a deformable body, there are six facets of this element on which there are nine pairs of stress components. There are nine stress components on three facets formed by three planes — xoy, xoz and yoz — and nine other stress components on the three facets opposite. Figure 2.6 shows three pairs of components in the direction of x.

Figure 2.6.Equilibrium in a volume element

When loading is applied to the structure, it is assumed that it is in equilibrium. In other words, any volume element belonging to this structure (body deformation) is in equilibrium. If we write the balance of forces from Figure 2.6 along x, we obtain:

Applying the six equilibrium equations, we obtain:

[2.19]

Equations [2.19] are known as the Cartesian equilibrium equations in a volume element. In this case, we have ignored the volume forces acting on the element, and we are in a quasi-static state.

Equations [2.19] are written in index form as follows:

[2.20]

and in vectorial form as follows:

[2.21]

When volume forces are considered, we have:

[2.22]

where:

In the planar case, considering volume forces equations [2.19] are written as follows:

[2.23]

Figure 2.7 shows the stress components on a planar volume element (dr rdθ) of uniform thickness in the polar coordinate case.

By projecting along the normal , by considering equilibrium we obtain:

(dθ) being infinitesimal, we have: cos (dθ) ≅ 1, sin (dθ) ≅ dθ.

Figure 2.7.Equilibrium of a two-dimensional volume element in polar coordinates

Neglecting the infinitesimal third-order terms, we obtain:

[2.24]

In the context of cylindrical coordinates, neglecting volume forces the equilibrium equations of the volume elementare as follows:

[2.25]

2.3. Behavior laws

A behavior law is a relationship between the components of stress and components of strain. This relationship depends on the variables intrinsic to the material. In fact, it was experimentally observed in the tensile test specimens of a simple one-dimensional ε11 that the strain varies with the stress, σ11.

The shape of the (σ11 ~ε11) curve obtained is closely related to the quality of the material in the specimen. Hooke observed that with a simple loading generating a small value of σ11, the strain ε11 is linearly related to σ11 with:

where E is Young’s modulus (which is intrinsic to the material). Hooke also noted that when the load generating σ11 is removed, ε11 becomes zero, resulting in a behavior that is termed reversible. Note that when you exceed a threshold σ11, known as σy, the relation between σ11 and ε11 becomes nonlinear, and when the load is removed a permanent strain εp remains, which is called the plastic residual strain (see Figure 2.8).

Figure 2.8.Schematic of a one-dimensional behavior law

When reloading, the elastic range is exceeded with a higher value of σ11 placed on the loading monotonic curve. In other words, the value of σy varies during cyclic loading in a phenomenon known as “hardening”. During hardening, a material has an increase in yield strength and its plastic range is restricted.

Figure 2.9.Different types of one-dimensional behavior

Several types of behavior may occur at point M of the behavior law, σ11 ~ε11 (see Figure 2.9):

– unloading: where we are back in the elastic region;

– continued monotonic loading, with the continuity of the work hardening phenomenon;

– maintenance of the stress level and where the evolution of ε11 is observed as a function of time. This is the creep phenomenon that is often observed in the thermomechanical beyond 400°C in the case of steel; and

– maintenance of the strain level where the evolution of σ11 is observed as a function of time. This refers to the relaxation phenomenon:

[2.26]

Figure 2.10.Modeling of a one-dimensional behavior law

2.3.1. Modeling the linear elastic constitutive law

The linear elastic behavior law linearly connects the stress field to the strain field. It is written as follows for a one-dimensional element:

[2.27]

Factor E is the elasticity modulus (Young’s modulus) and factor (1/E ) is known as the elastic compliance modulus. The value of E is in the region of 20,000 MPa/mm2 for most steels:

Figure 2.11.Schematic of Poisson’s ratio

Based on this one-dimensional case, the Poisson ratio, ν, is defined as being the ratio between lateral contraction, δt, and longitudinal extension δl:

[2.28] (see Figure 2.11):

The value of ν is equal to 0.28 to 0.33 for most metals.

A problem with modeling the elastic behavior law occurs in the real three-dimensional case where we have six independent stress components and six independent strain components, and a linear relationship between εij and σij . In this case, 36 constants are essential:

[2.29]

After considering the linearity assumption, two other assumptions can be made. The medium is homogeneous and isotropic; implying that the axes of the stress and strain components are similar in the three axes, x, y and z, considered (homogeneity).

Equations [2.29], considering the two previous assumptions, become:

[2.30]

where μ and λ are known as the Lamé coefficients:

These equations can then be written as follows:

[2.31]

where:

– δij is known as Kroneker coefficient.

By inverting equation [2.31], we obtain:

[2.32]

with:

[2.33]

or vice versa:

2.3.2. Definitions

Principal stresses and strains

In three-dimensions, the stress is applied to the faces of an orthonormal system, with an arbitrary orientation passing through point O (see Figure 2.1). This stress is expressed with six components, σij . It is possible to determine three specific orientations X, Y and Z perpendicular to six faces of a three-dimensional element on which no shear stress is present. The three faces perpendicular to X, Y and Z are known as the “principal faces”. The three orthonormal vectors and are known as the “principal directions”. Finally, the three normal stresses σI , σII and σIII on the principal faces are known as “principal stresses”. Conventionally, σI > σII > σIII . σI is the highest stress component in the structure. The values σI, σII and σIII are obtained by diagonalizing the stress tensor in order to obtain a stress tensor having only values σI , σII and σIII on the diagonal (not shear). Thus σI , σII and σIII willbe the eigenvalues of the tensor {σij} and and will be the eigenvectors.

2.3.2.1. Calculation of eigenvalues

[2.34]

leading to:

[2.35]

with:

[2.36]

2.3.2.2. Calculation of eigenvectors

If we consider ℓ, m and n to be the direction cosines of each eigenvector, for each value of λ (that is σI , σII and σIII ) we have the following equations:

[2.37]

with:

[2.38]

The maximum shear stresses act on facets that are equal to 45° with the principal facets. The value of the maximum shear stress is given by:

[2.39]

Analogically with the stresses, it is possible to determine the axial system defining the faces in which there is no shear strain. For isotropic solids, we can show that the principal axes X, Y and Z of the principal stresses are identical to the principal strains.

2.3.2.3. Two-dimensional applications

and:

Figure 2.12. Representation of the principal stresses in a plane medium

By dividing the two equations by (dy), and knowing that:

we obtain:

[2.40]

where:

[2.41]

By analogy:

From equations [2.41], we determine σI , σII and θ as functions of σ11, σ22 and σ12:

[2.42]

Equations [2.42] can be represented in a graphical form that is commonly known as Mohr’s circle (see Figure 2.13).

Point M represents the stress state (σ11, σ22 and σ12). M M’’ and M’ M’’ are the facets where the principal stresses σI and σII act with the values on the normal stress axes. Equations [2.41] and [2.42] can also be obtained from equations [2.35] and [2.37] in a two-dimension medium.

Figure 2.13.Representation of a stress state in Mohr’s circle

2.3.2.4. Equations of compatibility

One of the principles of continuum mechanics is that the strains must be continuous; this is known as the “compatibility condition”. The equation of the compatibility condition can be more clearly illustrated in a two-dimensional medium. In the case of small deformations, the equations of kinematics are expressed, from equation [2.10], by differentiating ε11 twice with respect to y , ε22, twice with respect to x , and ε12 with respect to x and y :

[2.43]

This equation is known as the “compatibility equation for a two-dimensional medium”.

If we consider the kinematic equations expressed for a two-dimensional medium, and the polar coordinates given in equation [2.18], the compatibility equation is written as:

[2.44]

In the context of small strains, equations [2.43] and [2.44] are viable irrespective of the behavior law.

The continuity of the medium in question is provided by the satisfaction of this equation. Equation [2.43] can be expressed using the law of material behavior. In the case of a linear elastic behavior law (equation [2.32]), we obtain:

[2.45]

In a three-dimensional environment, the condition of compatibility is written in the form of six equations developed from the equations of kinematics in the case of small strains:

[2.46]

2.3.2.5. Boundary conditions

If we consider a solid in stable equilibrium, the boundary conditions applied to the solid are of two natures:

– boundary conditions for displacements applied on the surface Su of the structure (solid) where the displacements are given and the forces (reactions) are unknown; and

– boundary conditions for forces applied on surface SF of the solid where the forces are given and the displacements are unknown.

The combination of these displacements and forces (Su and SF) represent the entire surface of the structure.

The loads applied to a structure are always on its surface. We can, however, list some forces that are applied in the volume, such as: inertial forces, thermal forces and volume forces.

Displacements applied to a structure may represent a recess (where rotations and displacements are completely blocked), a joint or movement imposed by actuators, spring, etc. In any case, we consider that any force applied to the solid three-dimensional component is:

All surface displacement under force also has three components u, v and w.

Figure 2.14.Boundary conditions under loading

Let us consider a solid, D, in a three-dimensional environment composed of infinite volume elements. On the surface of the structure, the load is applied to an oblique surface where the normal is .

By taking force and the internal forces generated by the stress components σij at equilibrium, we have:

[2.47]

where A is the surface abc, and A.n1, A.n2 and A.n3 are the projections of this surface on planes yoz, xoz and xoy, respectively.

We name: the force vector spread over the abc surface boundary. We therefore obtain:

[2.48]

This equation represents the boundary conditions of force applied to the surface SF of the structure.

NOTE 2.2.– In order to apply the boundary conditions, we use the Saint-Venant assumption, thus eliminating the local effects.

2.3.2.6. The position of the problem of computing a structure

The data for the calculation of a structure are of three types:

– the geometry of the structure: this refers to the overall geometry and local geometry of this structure;

– the intrinsic mechanical characteristics of the material(s) constituting the structure; and

– the boundary conditions, see Table 2.1, which are:

– the load applied to the structure on the SF boundary (force boundary conditions), and

– the fasteners in the structure applied on the Su boundary (displacement boundary conditions).

From these data, before finding a solution, a definition of the solution must be explained. A solution to a structural analysis is defined as the knowledge of the stress tensor σij , strain tensor εij , and displacement tensor u, v, w at any point in the structure. To achieve this, we go through a “black box”, known as “analysis”. This black box comprises three systems of equations allowing the attainment of a solution from the data.

The three systems of equations are:

– the equilibrium equations in a volume element (see equations [2.19] to [2.25]);

– the kinematic equations (see equations [2.11] or [2.18]); and

– the behavior law (see equations [2.31] or [2.32] on elasticity and section 2.3.3 on elastoplasticity).

The first two systems use “continuum mechanics”, in which the nature of the material is not mentioned. The third system takes into account the nature of the material.

In the three-dimensional case, the three systems of equations represent 15 equations, namely:

– three equilibrium equations in a volume element,

– six kinematic equations; and

– six equations for the behavior law.

These contain 15 unknowns (three displacements u, v and w; six εij strain components; and six σij stress components) for each volume element. This is therefore a well-posed problem. Solving these equations is quite difficult, however, because we have partial differential equations. Indeed, the integration of these equations has led to integration constants and their determination is based on the boundary conditions of forces and displacements. Similarly, to ensure the continuity of displacement and strain fields, we should check the compatibility equations.

In a two-dimensional case, the three systems’ equations are reduced to eight equations with eight unknown functions of (x, y) which are: (u, v, ε11, ε22, ε12, σ11, σ22 and σ12). The two-dimensional problems may be solved analytically in some cases. In the case of a one-dimensional element, the three systems’ equations are reduced to three equations.

The solution of the problem is thus obvious if the boundary conditions are known.

2.3.2.7. Mechanical properties of displacement and stress fields

There are a number of mechanical properties to be considered:

– The admissible kinematic (CA) displacement. For a displacement field to be CA, it must:

– be continuous throughout the object’s volume;

– be continually differentiable for piecewise volume; and

– verify the boundary conditions (boundary in displacements on the Su part of the surface), see Table 2.1.

– The statistically admissible (SA) stress field. For a stress field to be SA, it must:

– be continuous throughout the volume;

– be continually differentiable for piecewise volumes;

– verify the boundary conditions (boundary with forces on the SF part of the surface), see Table 2.1;

– verify the equilibrium equations in the volume element (see equation [2.20]).

– The solution field:

– must satisfy the equilibrium equations in the volume element if a KA field is to be the solution field;

– must satisfy the kinematic equations as well if a SA field is to be the solution field;

– can be a simultaneous KA and SA field.

– The plastically admissible stress field refers to the continuous and differentiable field that does not violate the plasticity criteria (plastic limit, for example: Von Mises or Tresca criteria).

Table 2.1.Schematic representation of a structural analysis: data, analysis, solutions

2.3.3. Modeling of the elastic-plastic constitutive law

Figure 2.15.Representation of an elastic-plastic law in a one-dimensional medium

We notice in both cases that during unloading-reloading, σy varies as a function of actual loading, the previous plastic strain and n (or ET, depending on the law being considered).

If the function f is defined as:

[2.49]

when:

f < 0, the medium is elastic;

f > 0, the medium is plastic.

In the case of residual plastic deformations, the previous three possibilities for stress state exist when the threshold is plastic:

(II) F=K , no_additional_load

Function F (σij ) may be represented by a surface in a six-dimensional space (σij ).

(I) implies that the stress varies outside this surface where plastic deformation is obtained is dF > 0.

(III) implies that the stress state varies on the inside of this surface, dF < 0 (see Figure 2.16).

Figure 2.16.Schematic of a plastic boundary

We set:

[2.50]

Function f has two properties:

– it represents a surface with six convex dimensions. This implies that every line in the stress space meets this surface at a maximum of two points; and

[2.51]

where (dλ ) is a scalar of dλ >0.

The above assumptions form the basis of the plasticity theory.

2.3.3.1. Modeling the yield stress

It was found, experimentally, that there is crystal slip during plasticity. The slip lines correspond to the dimensions of the volume element where the maximum shear occurs.

We can express the stress components as follows:

[2.52]

where sij represents the components of the deviatoric stress and σkk is the summation of the stresses σ11 + σ22 + σ33 that are known as the hydrostatic stresses:

[2.53]

2.3.3.2. Von Mises criterion

This is a criterion that allows the determination of the plastic limit. According to this criterion, it is the deviatoric energy (constructed from the deviatoric stress and strain) that produces the plasticity dependant on the maximum shear planes.

[2.54]

with:

[2.55]

When the deviatoric energy (Wd ) reaches an intrinsic value (B ), the medium is at the plastic limit.

The value of B is determined from the analysis of a one-dimensional element subjected to tension where the stress tensor is written as:

[2.56]

We deduce from equation [2.52] that the deviatoric tensor sij is:

[2.57]

[2.58]

[2.59]

[2.60]

The comparison between equations [2.55] and [2.60] leads to the Von Mises criterion, which is written in the following form:

[2.61]

Replacing [2.53] in [2.59] and [2.59] in [2.61], we have the Von Mises Criterion written as a function of σij:

[2.62]

2.3.3.3. Tresca criterion

[2.63]

[2.64]

From equations [2.63] and [2.64] the Tresca criterion is written as:

[2.65]

Comparing the two plastic boundaries — Von Mises (equation [2.61]) and Tresca (equation [2.65]) — with equation [2.50], we deduce that:

– for the Von Mises criterion:

[2.66]

– for the Tresca criterion:

[2.67]

2.3.3.4. Application on a plane stress case

The stress and strain tensors are written as:

In this case, the Von Mises criterion, considering the principal stresses σI and σII , is written as follows:

[2.68]

where σI, σII and θ are given in equation [2.42].

Here θ represents the angle of rotation between the planes yoz and xoz on one hand, and principal stresses’ planes on the other. The Tresca criterion is written as follows:

[2.69]

Figure 2.17 shows the two plastic boundaries given in equations [2.68] and [2.69].

Figure 2.17.Schematic of the Von Mises and Tresca criteria on a plane medium

2.3.3.5. Modeling elastic-plasticity with the isotropic hardening law

We consider the Von Mises plastic boundary. Based on equation [2.51], we can determine the plastic strain as follows:

[2.70]

where: (Levy-Mises conditions)

On isotropic hardening, the successive loading and unloading always replaces σy on the initial elastic-plastic curve (see Figure 2.18).

Figure 2.18.Evolution of the plastic boundary during loading and unloading in a one-dimensional medium

For each reloading, the plastic boundary increases by the same proportion as all stress components (see Figure 2.19).

Figure 2.19.Evolution of the plastic boundary during loading-unloading in a two-dimensional medium

To determine the plastic strains from equation [2.70], the scalar (dλ ) must be calculated from writing equation [2.70] in one-dimension:

[2.71]

in the one-dimensional case: (see equation [2.57]).

[2.72]

In the three-dimensional case, from the Von Mises criterion we have:

Replacing in [2.72], we have:

[2.73]

Replacing [2.73] in [2.70], we obtain the plastic strain in three-dimensions:

[2.74]

Equation [2.74] is known as the Prandlt-Reuss elastic-plastic law. For this law, the assumptions are that there is isotropic hardening and Von Mises criterion is taken as a plastic boundary.

2.3.3.6. Case of simple loading

This is the usual case for structural analysis where we assume that the stress vector { σij } does not change when the load is increased. The following can thus be written:

where α is a time-dependent variable corresponding to the increase in load.

Thus:

Integrating equation [2.74] relative to time, we have:

Finally, we obtain:

[2.75]

Returning to the one-dimensional case, we can see the effects in Figure 2.20.

Figure 2.20.Stress and strain in a one-dimensional hardened plastic medium

where in three-dimensions:

Equation [2.75] can thus be written:

[2.76]

Equation [2.75] is known as the Hencky-Mises equation. It is to be noted that an elastic-plastic law can be modeled with kinematic hardening. In this case, during loading and unloading, the initial monotonic curve is not plastic, and the Von Mises plastic boundary is written as:

[2.77]

with:

2.3.4. Modeling the law of perfect plastic behavior in plane stress medium

Here, we consider the material to be isotropic, homogeneous, rigid, perfectly plastic and incompressible and the plastic boundary to be described by the Von Mises criterion.

Three principal characteristics are rigid , perfectly plastic and incompressible :

– σy is independent of εp . There is only one limit (see Figure 2.21);

Figure 2.21.Behavior law in a perfectly plastic one-dimensional medium

Figure 2.22 shows the conventions used. σI and σII are the principal stresses in a two-dimensional medium. X and Y indicate the faces on which these stresses are applied. α and β are the faces on which the maximum shear components are applied. The vectors and are the tangential vectors to faces α and β.

Figure 2.22.The orientation of stresses

If we consider the case of plane strain where:

the Von Mises criterion is written as follows (see Equation [2.62]):

[2.78]

The equilibrium equations in a volume element are written as follows:

[2.79]

We make the following change in the variables:

[2.80]

If equation [2.80] is replaced in [2.78], we have:

[2.81]

whose value is constant as σy is constant:

Replacing [2.80] in [2.79]:

[2.82]

p and θ being two continuous and differentiable functions (in continuum), we write:

[2.83]

Equations [2.82] and [2.83] are written in the following matrix form:

[2.84]

There is only ever a solution to this equation when the determinant of [A] is zero:

[2.85]

In other words:

[2.86]

This equation defines two characteristic equations known as and for which a solution must be found. For this solution to exist, the following must be true:

[2.87]

In other words:

Replacing with equation [2.86], we have:

[2.88]

Referring to Figure 2.22, this differential equation is written as:

[2.89]

These two relations are known as Hencky ’s relations.

Going from the boundary conditions, the two equations in [2.89] allow us to know the value of (p ) for every given θ along the lines β and α. This problem is thus solved. Figure 2.23 shows the solution for a uniform pressure on frictionless surface, A’A. AB and A’B’ are two free surfaces. The material has infinite plane strain and no hardening occurs (σy is constant). From the boundary conditions of surfaces A’A, AB and A’B’, the orientation of lines α and β can be determined, along with the values of p . This allows the calculation of force F that produces plastic flow along lines α and β where the maximum shear stress is located:

Figure 2.23.Plastic flow in semi-infinite perfectly plastic medium under uniform pressure

2.4. Energy formalism

We now move from the simple idea that in order to study the forces of friction on a body, it is slid on a surface, and to study gravitational forces it is elevated, etc. In other words, to study any force, a movement is required. Consider a solid made of n material points with two neighboring points Mp and Mq (see Figure 2.24).

Figure 2.24.A solid with two neighboring points

It is said that:

In each point, we can write: In other words:

[2.90]

with:

Consider λp to be a scalar (that may be a displacement of point Mp) by multiplying equation [2.90] by λp and summing:

[2.91]

This energy equilibrium is the origin of the principles that we will now develop.

2.4.1. Principle of virtual power

This principle links two separate and distinct systems. The first is a set offorces in equilibrium (with F and σ being the external forces and internal stresses, respectively). The second system is a set of consistent strains and displacements (with ∆ and ε being displacement and strain, respectively). The general rule is as follows: for any system in equilibrium that is quasi-static, the virtual power of forces (or displacements) must be equal outside, in absolute terms, with the virtual power offorces (stresses) balance:

[2.92]

Note that in practice, one of the two systems is virtual while the other is real, and vice versa. It is therefore possible to present the virtual power principle in two forms:

– the case where the displacement-strain system is real, and coupled with the virtual system offorces and stresses:

(virtual stress).(real strain) d(volume)

– the dual case where the displacement-strain system is virtual and coupled with a real system of forces and stresses:

(real stresses).(virtual strains) d(volume)

The second form will be developed, as follows.

Consider volume V with an exterior surface S . The virtual power principle is written as follows (see Figure 2.25):

[2.93]

with:

[2.94]

[2.95]

Figure 2.25.Volume subjected to external surface loads

where:

In quasi-static state and neglecting volume forces, equation [2.93] is written as:

[2.96]

This is comparable to equation [2.92].

Assuming small perturbations (small strains), and replacing εij with equation [2.10], we obtain:

[2.97]

[2.98]

Using integration by parts, we have:

replacing that in equation [2.98], we have:

[2.99]

Using the Ostrogradsky theorem to transform the volume integral into the surface integral, we obtain:

where nj is the normal to surface S. Equation [2.99] is therefore written as:

[2.100]

When ui is considered a virtual displacement that is not zero, by identifying the surface and volume integrals we obtain:

and

Note that in this development continuum mechanics is used, and not the material behavior law.

2.4.2. Potential energy and complementary energy

For a supposed displacement field, KA, the potential energy is defined as follows:

[2.101]

where W (ε ) is the strain energy from the displacement field, CA.

[2.102]

NOTE 2.4.– the difference between the two integrals of equation [2.102] is that the first is performed on the geometric volume and the second is mechanical, carried out on ε .

Consider the one-dimensional case:

hatched surface in Figure 2.26

(linear elasticity)

Figure 2.26.Strain energy density in a one-dimensional medium

Wext (Td ) is the work of given external forces Td on the surface SF of the solid V (see Figure 2.25):

[2.103]

In the case where the one-dimensional structure is a section bar Ω that is blocked at A and loaded at B by force F , the potential energy is written in linear elasticity as such:

[2.104]

where:

– UB is the displacement at B ;

– SF is identified by point B ;

– (σ11)B and UA are the two stress components on SF and the displacement at SU ;

– is the force boundary condition on SF ; and

– σ11 and ε11 are independent of x.

Figure 2.27.Bar subjected to tension

For a supposed stress field, KA, the complementary energy is defined as follows:

[2.105]

Where W* (σ ) is the stress energy from the stress field, KA.

[2.106]

with:

NOTE 2.5.– the difference between the two integrals contained in equation [2.106] is noticeable.

Consider the one-dimensional case:

(linear eleasticity)

Figure 2.28.Stress energy density in a one-dimensional medium

W*ext (ud ) is the work of given external displacement ud on the surface Su from surface S of the solid V (see Figure 2.25):

[2.107]

It can be shown from a one-dimensional medium that:

[2.108]

Figure 2.29.Representation of strain and stress energy

Figure 2.29 shows there is only equality when ε11 and σ11 are linked by a behavior law. (This is only true for the behavior laws where εij grows with σij .)

Generalizing equation [2.108] in a three-dimensional case and integrating it using volume, we obtain:

[2.109]

From the virtual power principle (see equation [2.98]), we obtain:

[2.110]

Equation [2.109] thus becomes:

[2.111]

In other words: W*co + Wpot ≥ 0.

When [εij ] is linked to [σij