Rheology, Physical and Mechanical Behavior of Materials 3 E-Book

Maurice Leroy

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: ISTE Invoiced
• Sprache: Englisch

Seitenzahl: 256

Veröffentlichungsjahr: 2024

Beschreibung

This book studies metallic and composite materials and their mechanical properties in terms of stiffness and strength, illustrated through several case studies and exercises.

Rheology, Physical and Mechanical Behavior of Materials 3 introduces the concepts of stiffness, strength, elastic energy, generalized stress and strain, as well as the main criteria for dimensioning isotropic and anisotropic materials. It covers the elastic mechanics of pieces and structures using various techniques such as the force method, Maxwell's influence coefficients, Castigliano and Menabrea's work, Mohr’s integrals and the displacement method, as well as the design and use of stiffness matrices. It also compares the behavior of static and dynamic impact actions and studies the elastic limits of plastic hinges, their influences and shear forces.

This book is aimed at those studying technical or technological training courses, researchers involved in the mechanics of deformation, and industrial design and manufacturing departments.

Leseprobe

Series EditorNoël Challamel

Rheology, Physical and Mechanical Behavior of Materials 3

Rigidity and Resistance of Materials, Sizings, Pieces and Structures

Maurice Leroy

First published 2024 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:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2024The rights of Maurice Leroy to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.

Library of Congress Control Number: 2024947083

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-972-3

Preface

In the case of relatively low loads, the deformation mechanisms for materials, parts, and structures are reversible, and the elastic deformations are proportional to the stresses (with E, Young’s modulus of elasticity).

In the case of complex loads, Hooke’s law is generalized into a three-dimensional relationship, and the linear nature of this law results in the following superposition principle: the stresses or deformations produced by the sum of several loading states on an elastic solid are equal to the sum of the stresses or deformations generated by each of the load states applied in isolation to the solid.

If the stress exceeds a certain value σe (or Re, σ0, Y), known as the elasticity limit stress, the phenomenon ceases to be reversible and linear, and the theory of elasticity can no longer be applied. This limit may be difficult to demonstrate experimentally. It is conventionally defined as being the stress that generates an irreversible deformation close to 0.2%.

For three-dimensional loads, different sets of criteria for yield strength will define the corresponding domain in the stress space. These include the Tresca and Von Mises criteria, while Hill’s criteria are suitable for composites, and are often used in the calculations to determine the scale of parts and structures.

In the case that the elastic limit is crossed, slips will occur within the materials (dislocation in the crystals) and irreversible and permanent deformations can occur (in the plasticity domain).

The stress on metals at a temperature exceeding about one-third of the absolute melting temperature has the property of deforming even if the stress remains constant: this phenomenon is known as creep (untreated), which translates into a viscoplastic deformation.

In principle, the laws of elastic, plastic and viscoplastic behavior associated with the equations describing the mechanics of continuous mediums make it possible to calculate the stresses and deformations in parts and structures.

In many cases, it is sufficient to use the theory of elasticity, with the dimension criteria used to address safety concerns for the determination of the maximum permissible stress and/or maximum deformation.

The criteria for ruptures make use of other theories.

Structure of this book

Chapter 1 provides a review of the concepts of rigidity, resistance and elastic energy, as well as stress-strain relationships and domains. Chapter 2 covers the use of scaling criteria for isotropic and anisotropic materials. Chapters 3 and 4 address the elastic mechanics of parts and structures. Chapter 5 covers elastic limit deflections and plastic hinges and Chapter 6 covers shearing and shear force.

NOTE.– The full and detailed Appendix to this book provides diagrams of the 100 examples covered throughout the book and the relevant page numbers and is available to download from www.iste.co.uk/leroy/rheology3.zip.

Maurice LEROYOctober 2024

1Elasticity, Rigidity

1.1. Elasticity and rigidity tensors

1.1.1. Hooke’s law

In the domain of low elastic deformations, the deformation is proportional to the stress:

where S is called elasticity. Similarly:

with C = S−1, where C is the rigidity.

Hooke’s law consists of nine equations with nine terms, totaling 81 Sijkl coefficients.

Through taking into account the physical phenomena, we can reduce the number of these coefficients to 36.

σij is a symmetric tensor, thus Sijkl = Sijlk, and εij is a symmetric tensor, thus, Sijkl = Sijlk.

1.1.2. Matrix notation

The tensors [σij] and [εij] are symmetrical. The notation can be simplified by adopting the following equivalences:

For [σ

ij

]:

Table 1.1.Tensor and matrix notations of [σij]

xx

yy

zz

yz, zy

zx, xz

xy, yx

Tensor notation

11

22

33

23.32

31.13

12.21

Matrix notation

1

2

3

4

5

6

In other words, six coefficients are given as:

For [ε

ij

] in relation to the strain tensor [e

ij

], we obtain:

In Sijkl, the first two indices are contained in a single variant from 1 to 6 and the same is done for the last two indices. That is, Sijkl = Smn. The factors 1, 2 and 4 are introduced at the same time as follows:

if m and n have the values of 1, 2, 3, S

ijkl

= S

mn

;

if m or n have the values of 4, 5, 6, ;

if m and n have the values of 4, 5, 6, .

1.1.3. Relationships between stresses and strains for isotropic bodies

In the classic works on elasticity, the relationships between the stresses and the strains are expressed as a function of different quantities of the coefficients Sij or Cij.

These are as follows:

Young’s modulus: . This equation gives the relative elongation of a rod of section S subjected to an axial force F.

Poisson’s ratio ν. At the same time as a cylinder lengthens, it also narrows: .

The rigidity modulus, G: .

The Lamé coefficients, λ and μ: , .

To establish the relationships between the coefficients Sij and Cij and the classic coefficients, we will develop the matrices that we have obtained for the isotropic solids and compare them with the classic equations.

Table 1.2.Classical expressions and notations matrix of deformations

Classic expressions

Matrix notations

ε

1

= S

11

σ

1

+ S

12

σ

2

+ S

13

σ

3

ε

2

= S

12

σ

1

+ S

11

σ

2

+ S

12

σ

3

ε

3

= S

12

σ

1

+ S

12

σ

2

+ S

11

σ

3

ε

4

= 2 (S

11

− S

12

) σ

4

ε

5

= 2 (S

11

− S

12

) σ

5

ε

6

= 2 (S

11

− S

12

) σ

6

The comparison of the coefficients gives:

and:

and thus the equation G = E / [2 (1 + v)].

We express the stresses as a function of the strains (Table 1.3).

Table 1.3.Classical expressions and matrix notations of strains

Classic expressions

Matrix notations

σ

1

= (2μ + λ) ε

1

+ λ ε

2

+ λ ε

3

σ

1

= C

11

ε

1

+ C

12

ε

2

+ C

12

ε

3

σ

2

= λ ε

1

+ (2μ + λ) ε

2

+ λ ε

3

σ

2

= C

12

ε

1

+ C

11

ε

2

+ C

12

ε

3

σ

3

= λ ε

1

+ λ ε

2

+ (2 μ + λ) ε

3

σ

3

= C

12

ε

1

+ C

12

ε

2

+ C

11

ε

3

σ

4

= μ ε

4

σ

5

= μ ε

5

σ

6

= μ ε

6

According to the results of Table 1.3, we obtain:

1.1.4. Tensors [σ] and [ε] and deviators

The values of the traces in stresses and strains are as follows:

tr [σ] = − 3 p or − p = σ

m

, with σ

m

, the average stress;

tr [ε] = Θ dilatation with Θ/3 = ε

m

, the average strain.

In the primary reference area, we have:

or:

or:

1.1.4.1. The case of isotropic materials in elasticity in the main reference area

Thus, we have:

or: