Rheology, Physical and Mechanical Behavior of Materials 3 (eBook)

Rigidity and Resistance of Materials, Sizings, Pieces and Structures

Maurice Leroy (Autor)

eBook Download: EPUB

2024
568 Seiten
Wiley-Iste (Verlag)
978-1-394-34067-5 (ISBN)

Infos

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.

Maurice Leroy is a lecturer and professor at the University of Nantes, France, as well as director of the Composite and Metallic Formations research laboratory at the IUT. He was instrumental in the creation of France's first Materials Science and Engineering department.

Textauszug

1
Elasticity, 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 [σij]:

  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 [εij] in relation to the strain tensor [eij], 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, Sijkl = Smn;
• 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

ε1 = S11 σ1 + S12 σ2 + S13σ3

ε2 = S12 σ1 + S11 σ2 + S12σ3

ε3 = S12 σ1 + S12 σ2 + S11σ3

ε4 = 2 (S11 − S12) σ4

ε5 = 2 (S11 − S12) σ5

ε6 = 2 (S11 − S12) σ6

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

σ1 = (2μ + λ) ε1 + λ ε2 + λ ε3

σ1 = C11 ε1 + C12 ε2 + C12 ε3

σ2 = λ ε1 + (2μ + λ) ε2 + λ ε3

σ2 = C12 ε1 + C11 ε2 + C12 ε3

σ3 = λ ε1 + λ ε2 + (2 μ + λ) ε3

σ3 = C12 ε1 + C12 ε2 + C11 ε3

σ4 = μ ε4

σ5 = μ ε5

σ6 = μ ε6

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 = σm, with σm, the average stress;
• tr [ε] = Θ dilatation with Θ/3 = εm, 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:

We will establish that .

We obtain the Lamé coefficients in the stresses:

[1.1]

and thus:

[1.2]

and thus:

[1.3]

The same is attained by permutation for SII and SIII, hence:

[1.4]

1.1.4.1.1. Relationship between stress and strain deviators

The relationship between the stress and strain deviators is independent of the reference area in use. Thus, for any reference area x, y, z, we obtain:

and thus, for example:

1. but according to [1.2]:

   and two other relationships via permutation:

2. and thus:

   [1.5]

• From [1.5]: shear strain for i ≠ j equal to:

  with G, the elastic modulus for shearing (or transverse) (or μ) in Pascals.

• From [1.5]: the elongation strain is equal to:

EXAMPLE 1.1.–

We have that i = j:

but [1.2] gives:

and thus:

or:

With:

• Values that can be measured by tests:
  [1.6]
• Values that can be measured by permutation:
  [1.7]

REMARK.–

[1.8]

1.1.4.1.2. A few values: λ, E, G in daN/mm2

Table 1.4. Elasticity coefficients of materials

	λ 10−3	E 10−3	G 10−3	ν

Steel

9–13

20–22

7.9–8.4

0.27–0.31

Brass

8.5

11

4.1

0.33

Copper

9–13

13

4.8

0.33–0.38

Lead

3.5

1.6

0.56

0.43

Glass

2.7–3

6

2.38

0.26

The following relationships are obtained in the main reference area:

and thus the shift to the stress and deformation states for elastic isotropic material in Figure 1.1.

Figure 1.1. Representation of Mohr domains from stress and strain states

Table 1.5. Young’s moduli values (Ashby and Jones 1980)

Material	E in GPs	Material	E in GPs

Diamond

1000

Niobium and alloys

80–110

Tungsten carbide WC

450–650

Silicon

107

Osmium

551

Zirconium and alloys

96

Cobalt/tungsten carbide cements

400–530

Silica glass and SiO2 (quartz)

94

Ti, Zr, and Hf borides

500

Zinc and alloys

43–96

Silicon carbide siC

450

Gold

82

Boron

441

Calcite...

Produkt-Details

• Erscheint lt. Verlag: 4.12.2024
• Reihe/Serie: ISTE Invoiced
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Chemie
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: composite materials • Displacement Method • elastic energy • Force Method • Maxwell's influence coefficients • mechanical behavior • Metallic materials • Mohr's integrals • rheology • Shear Forces • Stiffness • Strain • Strength • Stress
• ISBN-10: 1-394-34067-2 / 1394340672
• ISBN-13: 978-1-394-34067-5 / 9781394340675