{"loaderData":{"0-0-57":{"chapterData":{"_id":"68e626fe9caaa78da88af2e8","book_id":"68b7d9c95419de3bf4488686","chapter_name":"Linear Stress-Strain Relations","public_visibility":true,"allow_part_see":false,"book_slug":"solid-mechanics","slug":"linear-stress-strain-relations","regular_authors":"","express_authors":"","regular_content":"

In previous chapters, we developed definitions and equations for stress, strain, and displacement. The next step is discuss the relations between stress and strain. In this chapter, we consider only linear elastic behavior.

","express_content":"","book_name":"Solid Mechanics","permissions":{"book":"solid-mechanics","chapter":"linear-stress-strain-relations","header1":"","header2":"","header3":"","bookType":"main","queryOn":"chapter","rolesData":[{"type":"PUBLIC","read":true,"write":false,"edit":false}]},"authors":[{"_id":"630da3e25d2b6424645ccb69","full_name":"Kamyar M. Davoudi","image":"/user-images/KamyarProfilejpg-1729801219447.jpg","email":"kaamyaar@gmail.com"}]},"bookData":{"_id":"68b7d9c95419de3bf4488686","category_id":"66cecf01c3d3f3f05eac01b6","author_id":["630da3e25d2b6424645ccb69"],"book_name":"Solid Mechanics","book_image":"/books/6571e32ec18d4af4a1bb64146b997900png-1756879305198.png","book_description":"

This book is an introduction to solid mechanics, designed for students who have already completed an introductory course in mechanics of materials or strength of materials. It covers the fundamental concepts of the subject, including stress and strain, elastic behavior of materials, two-dimensional elasticity problems, plastic behavior, time-dependent material response, and an introduction to finite element modeling.

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What Is Solid Mechanics?

Solid mechanics is a branch of continuum mechanics, also known as mechanics of a continuous medium. The word “continuum” reflects the assumption that matter is considered completely continuous without any gaps or spaces between its its constituents, disregarding its atomic structure. It is clear that this assumption fails if we zoom in too closely. The level of magnification at which this assumption breaks down depends on the problem; but the is often on the scale of a few tens of nanometers. Still, sometimes it is helpful or even necessary to think about the atomic structure.

The other major branch of continuum mechanics is fluid mechanics. A solid is distinguished from a fluid by its ability to support substantial shearing forces over a specific time scale. While all materials, both fluids and solids, can support normal forces to some extent, it is this resistance to shear forces that characterizes substances like rock, steel, and rubber as solids. However, this distinction is not a clear-cut and sometimes depends on the timeframe of interest. For example, Earth’s mantle acts like a solid when seismic waves move through it quickly, but over millions of years, it flows like a fluid. Silly Putty is another example: it spreads out like a liquid if left on a table (see the following figure), but bounces like a solid when thrown at a wall (see the following video). This shows that whether a material is a solid can depend on how long the force is applied.¹

Silly Putty behaves like a solid when bounced. Video from Wikipedia.

Mechanics is the field of study concerning the response of bodies to forces. Therefore, the central question in solid mechanics is: how does a solid deform when a load is applied? The load can be of any type, but we mainly focus on mechanical and to some extent thermal loads. We want to determine how much (and in which direction) every particle² moves when a load is applied. The vector function that assigns to each point of a body the displacement it undergoes when the body moves or deforms under applied loads is called the displacement field³. The gradient, or spatial derivative, of the displacement field is called strain.

In mechanics, the displacement field is a vector function \$\\mathbf{u}(\\mathbf{X})=\\mathbf{x}(\\mathbf{X})-\\mathbf{X}\$ which gives, for each material point initially at position \$\\mathbf{X}\$ the displacement vector to its new position \$\\mathbf{x}\$ after deformation.

As the application of loads changes the distance between atoms, they move from their equilibrium positions, and internal forces develop. As we explained before, continuum mechanics adopts a macroscopic perspective; instead of analyzing individual atomic forces, we consider their collective effect. We speak of an average force intensity—known as stress—acting over a small volume that may contain millions of atoms. Therefore, the three fields we often work with are displacement, strain, and stress.

If, upon removing the forces, the displacement of particles returns to zero—meaning particles move back to their original places—the material is called elastic. If the strain is linearly proportional to the stress (equivalently, displacement is linearly proportional to forces), the material is considered linearly elastic. Not all elastic materials are linearly elastic. Rubber is an example of a nonlinear elastic material. If some displacement remains after the forces are removed, the remaining deformation is known as plastic deformation.

The study of the relationship between internal forces and displacement (or equivalently stress and strain) is another central topic in solid mechanics. The relationship varies from a material to another, and it is often a function of temperature and in some cases the loading rate. The study of this relationship requires some knowledge in materials science and atomic structures.

The response of a material can be time-dependent. If the load remains constant, the displacement may very gradually increase. This phenomenon is called creep. For example, a new bookshelf may be perfectly straight but can be seen to sag after bearing the weight of books for a long time. Generally creep is more severe at higher temperatures.

In some materials, the stress may decrease over time even if the displacement is held constant. This phenomenon is called stress relaxation.

A material may break into two or more separate pieces as a result of applied stress. This is what a branch of solid mechanics called fracture mechanics.

Solid mechanics is not a field that can be studied in a vacuum. A true understanding often requires knowledge of thermodynamics, materials science, or even quantum mechanics. For example, the classical theories of solid mechanics reach their limits when a metal’s internal microstructure begins to dominate its behavior. Examples of such as case include when a metal’s structure continuously evolves at high temperatures, or when it undergoes a transition from ductile to brittle. In these moments, we see a clear intersection where the principles of mechanics and materials science must work together.

Beyond the Basics: How This Book is Different

Elementary solid mechanics, also known as mechanics of materials or mechanics of deformable bodies, is a subject students typically take in their undergraduate studies. Although it introduces the concepts of stress and strain, the material is almost always assumed to be linearly elastic. In addition, only very simple cases, such as the bending of a beam or the torsion of a circular bar (shaft), are considered. This leaves open important questions, such as how to calculate the stress and strain fields when arbitrary forces are applied to a body of an arbitrary shape. Another simplification often made in elementary mechanics of materials is the assumption that the material is isotropic, meaning its properties are the same in all directions. A common example of an anisotropic material is wood, which exhibits different strengths along its fibers compared to perpendicular to them. Finally we will also study the rate-dependent behavior of materials.

In this book, we start with the analysis of stress and strain, learn about the stress-strain relationship. We derive the equations of elasticity and learn how we can solve some problems in elasticity analytically and more generally numerically. We learn about yield criteria and plastic deformation. At the end, we study elements of fracture mechanics and time-dependent behavior.

Further Studies

James R. Rice. “Mechanics of Solids.” Encyclopædia Britannica. Accessed September 2, 2025. https://www.britannica.com/science/mechanics-of-solids.
Dieter, George E. Mechanical Metallurgy. 1st ed. New York: McGraw-Hill, 1961.
Poirier, Jean-Paul. Creep of Crystals. Cambridge University Press, 1985.

Reference https://www.britannica.com/science/mechanics-of-solids by James Rice.↩︎
In continuum mechanics, whenever we talk about a particle, we mean a mathematical point in space.↩︎
In physics and mathematics, a field is a physical quantity that has a value for each point in space and time. Fields are classified based on the type of quantity they represent at each point. A scalar field assigns a single numerical value, or scalar (such as temperature), to every point in space. A vector field assigns a vector—a quantity with both magnitude and direction (such as velocity or displacement)—to every point. More complex quantities called tensors can also be represented by fields. A tensor field associates a tensor with every point in space and time. Stress and strain are two common examples of tensor fields that are fundamental to solid mechanics.↩︎

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In this chapter, we talk about one of the three central fields in continuum mechanics: stress.

","chapter_express_content":"","public_visibility":true,"allow_part_see":false,"book_slug":"solid-mechanics","slug":"stress","regular_authors":"","express_authors":"","created_at":"2025-08-29T05:02:47.681Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"body-and-surface-forces","header_name":"Body and Surface Forces","type":"header-1","header_data":[],"_id":"68be7d9f5419de3bf4488ded"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"traction-or-stress-vector","header_name":"Traction or Stress Vector","type":"header-1","header_data":[],"_id":"68be7de65419de3bf4488e03"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"stress-components","header_name":"Stress Components","type":"header-1","header_data":[],"_id":"68be7e195419de3bf4488e19"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"traction-vector-and-stress-tensor-relationship","header_name":"Traction Vector and Stress Tensor Relationship","type":"header-1","header_data":[],"_id":"68be7e6e5419de3bf4488e2f"},{"sequence":5,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"resultant-force-and-moment-and-stress-relations","header_name":"Resultant Force and Moment and Stress Relations","type":"header-1","header_data":[],"_id":"68c134635419de3bf448933c"},{"sequence":6,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"symmetry-of-the-stress-tensor","header_name":"Symmetry of the Stress Tensor","type":"header-1","header_data":[],"_id":"68c128c05419de3bf448919c"},{"sequence":7,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"variation-of-stress-within-a-body","header_name":"Variation of Stress Within a Body","type":"header-1","header_data":[],"_id":"68c134055419de3bf44892ff"},{"sequence":8,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"planar-stress-stress-transformation-in-2d","header_name":"Planar Stress Stress. Transformation in 2D","type":"header-1","header_data":[],"_id":"68c7cc315419de3bf448a1d7"},{"sequence":9,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"stress-transformation-in-3d","header_name":"Stress Transformation in 3D","type":"header-1","header_data":[],"_id":"68c7cc745419de3bf448a1ed"},{"sequence":10,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"stress","slug":"principal-stresses","header_name":"Principal Stresses","type":"header-1","header_data":[],"_id":"68ca6b795419de3bf448aef9"}]},{"_id":"68da50939caaa78da88ae1a4","book_id":"68b7d9c95419de3bf4488686","chapter_name":"Displacement and Strain ","sequence":3,"chapter_content":"

In this chapter, we explore two other key concepts in solid mechanics: displacement and strain.

","chapter_express_content":"","public_visibility":true,"allow_part_see":false,"book_slug":"solid-mechanics","slug":"displacement-and-strain","regular_authors":"","express_authors":"","created_at":"2025-09-28T15:55:16.885Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"displacement-and-strain","slug":"displacement-and-strain","header_name":"Displacement and Strain","type":"header-1","header_data":[],"_id":"68da50bd9caaa78da88ae1bb"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"displacement-and-strain","slug":"transformation-of-strain","header_name":"Transformation of Strain","type":"header-1","header_data":[],"_id":"68dcd8f49caaa78da88ae3ef"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"displacement-and-strain","slug":"measurement-of-surface-strain","header_name":"Measurement of Surface Strain","type":"header-1","header_data":[],"_id":"68dd3ffb9caaa78da88ae7a1"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"displacement-and-strain","slug":"compatibility-equations","header_name":"Compatibility Equations","type":"header-1","header_data":[],"_id":"68e394999caaa78da88af006"}]},{"_id":"68e37fb69caaa78da88aee95","book_id":"68b7d9c95419de3bf4488686","chapter_name":"Introduction to Crystallography","sequence":4,"chapter_content":"

A crystal is formed by the periodic repetition of a group of atoms in all directions. That group of atoms is called the basis.

The basis of a crystal can be one or more atoms.
An ideal crystal consists of infinite repetitions of the basis.

Lattice

The infinite array of mathematical points, which describes how the basis is repeated, forms a periodic spatial arrangement is called the lattice.

The lattice looks identical from whichever points you view the array.
Note that lattice is not a crystal.

In three dimensions, the lattice can be identified by three independent vectors \$\\mathbf{a}_1\$, \$\\mathbf{a}_2\$ and \$\\mathbf{a}_3\$.
The position of each point \$\\mathbf{R}\$ can be written as a linear combination of these vectors, \\[\\mathbf{R}=n_1 \\mathbf{a}_1+n_2 \\mathbf{a}_2+n_3 \\mathbf{a}_3 \\tag{1}\\] where \$n_1\$, \$n_2\$, and \$n_3\$ are integers.
The vectors that can generate or span the lattice are not unique. For example, see the following figure.

\"\" — Different choices for vectors spanning the lattice

Recall that the volume of a parallelepiped with axes \$\\mathbf{a}_1\$ , \$\\mathbf{a}_2\$ and \$\\mathbf{a}_3\$ is given by \$|\\mathbf{a}_1\\cdot(\\mathbf{a}_2\\times \\mathbf{a}_3)|\$ .

Primitive Cells vs. Unit Cells

Primitive Cell

Of the vectors satisfying Eq. (1), those that form a parallelepiped with the smallest volume, are called primitive translation vectors, and the parallelepiped they form is known as the primitive cell.

Unit Cell

To better demonstrate the symmetry of the entire lattice, sometimes non-primitive translation vectors are used to specify the lattice. In this case, the parallelepiped is called the unit cell.

The unit cell may or may not be identical to the primitive cell.

Unit cell of a face-centered cubic structure

Primitive unit cell of a face-centered cubic structure. As drawn, the primitive translation vectors are \\[\\mathbf{a}_1=\\frac{a}{2}(\\hat{\\mathbf{x}}+\\hat{\\mathbf{y}})\\quad \\mathbf{a}_2=\\frac{a}{2}(\\hat{\\mathbf{y}}+\\hat{\\mathbf{z}})\\quad \\mathbf{a}_3=\\frac{a}{2}(\\hat{\\mathbf{x}}+\\hat{\\mathbf{z}})\\]

Lattice Parameters and Packing Fraction

Lattice Parameters

Lattice Parameters

The lengths of axes of the unit cell (called lattice constants \$a\$, \$b\$, and \$c\$) and the angles between the axes (\$\\alpha\$, \$\\beta\$ and \$\\gamma\$) specify the unit cell. The lattice constants and the three angles between them are termed lattice parameters. See the following figure.

Packing Fraction

We try to pack \$N\$ hard spheres (cannot deform)
The total volume of the spheres is \\[V_s=N \\frac{4}{3}\\pi R^3\\]
The volume these spheres occupy \$V>V_S\$ (there are spacing)\\[{\\rm Packing\\ Fraction}=\\frac{N \\frac{4}{3}\\pi R^3}{V}\\]

Bravais Lattices

Auguste Bravais, a French physicist, identified 14 distinct lattices in three dimensions.
There are 5 Bravais lattices in two dimensions (shown below)
In crystallography, all lattices are traditionally called Bravais lattices or translation lattices.

Five Bravais Lattices in Two Dimensions

Seven Crystal Systems

Isometric (or Cubic)

$a=b=c$

$\\alpha=\\beta=\\gamma=90^\\circ$

Tetragonal

$a=b\\ne c$

$\\alpha = \\beta = \\gamma = 90^\\circ$

Orthorhombic

$a\\ne b\\ne c$

$\\alpha = \\beta = \\gamma = 90^\\circ$

Hexagonal

$a=b\\ne c$

$\\alpha = \\beta = 90^\\circ, \\gamma = 120^\\circ$

Triclinic

$a\\ne b\\ne c$

$\\alpha \\ne \\beta \\ne \\gamma$

Monoclinic

$a\\ne b\\ne c$

$\\alpha = \\beta = 90^\\circ \\ne \\gamma$

Rhombohedral (or Trigonal)

$a=b=c$

$\\alpha = \\beta = \\gamma < 120^\\circ, \\ne 90^\\circ$

Cubic Lattices


Primitive	Body-Centered	Face-Centered

Simple Cubic (SC) Crystal

Location of all lattice points \\[\\mathbf{R}=a(u\\, \\hat{\\mathbf{x}}+v\\, \\hat{\\mathbf{y}}+w\\, \\hat{\\mathbf{z}})=a[u\\ v\\ w]\\]
Polonium (Po) is the only metal that forms a simple cubic unit cell.
Each lattice point is shared by 8 neighboring units
average volume occupied by each lattice point\$=\$
average volume occupied by each atom in SC\$=\\Omega_{SC}=\\frac{a^3}{8\\times\\frac{1}{8}}=a^3\$

Body-Centered Cubic (BCC) Crystal

Location of all lattice points \\[\\begin{aligned} \\mathbf{R}&=a(u\\, \\hat{\\mathbf{x}}+v\\, \\hat{\\mathbf{y}}+w\\, \\hat{\\mathbf{z}})=a[u\\ v\\ w]\\\\ \\mathbf{R}&=a((u+0.5)\\, \\hat{\\mathbf{x}}+(v+0.5)\\, \\hat{\\mathbf{y}}+(w+0.5)\\, \\hat{\\mathbf{z}})=a[u+0.5\\ v+0.5\\ w+0.5]\\end{aligned}\\]
average volume occupied by each lattice point\$=\$
average volume occupied by each atom in BCC\$=\\Omega_{BCC}=\\frac{a^3}{8\\times\\frac{1}{8}+1}=\\frac{a^3}{2}\$
Nearest neighbor distance = \$d=2r=a\\frac{\\sqrt{3}}{2}\$
\\[\\text{Packing fraction}=\\frac{2\\times\\frac{4\\pi}{3}\\times(\\frac{a\\sqrt{3}}{4})^3}{a^3}=\\frac{\\pi\\sqrt{3}}{8}\\approx 0.68\\]

The primitive translation vectors are: \\[\\mathbf{a}_1=\\frac{a}{2}(\\hat{\\mathbf{x}}+\\hat{\\mathbf{y}}-\\hat{\\mathbf{z}})\\] \\[\\mathbf{a}_2=\\frac{a}{2}(-\\hat{\\mathbf{x}}+\\hat{\\mathbf{y}}+\\hat{\\mathbf{z}})\\] \\[\\mathbf{a}_3=\\frac{a}{2}(\\hat{\\mathbf{x}}-\\hat{\\mathbf{y}}+\\hat{\\mathbf{z}})\\]

The primitive unit cell is a rhombohedron of edge \$a\\frac{\\sqrt{3}}{2}\$
The angle between adjacent edges is \$109^\\circ 28'\$

Number of nearest neighbors = 8
Number of second nearest neighbors = 6
Second nearest neighbor distance = \$a\$

Some Elements with BCC structure

Barium	Ba	Chromium	Cr
Cesium	Cs	\$\\alpha-\$Iron	\$\\alpha-\$Fe
Potassium	K	Lithium	Li
Molybdenum	Mo	Sodium	Na
Niobium	Nb	Rubidium	Rb
Tantalum	Ta	Titanium	Ti
Vanadium	V	Tungsten	W

Face-Centered Cubic (FCC) Crystal

Location of all lattice points \\[\\begin{aligned} \\mathbf{R}&=a[u\\ v\\ w]\\\\ \\mathbf{R}&=a[u+0.5\\ v\\ w]\\\\ \\mathbf{R}&=a[u\\ v+0.5\\ w]\\\\ \\mathbf{R}&=a[u\\ v\\ w+0.5]\\end{aligned}\\]
average volume occupied by each lattice point\$=\$
average volume occupied by each atom in FCC\$=\\Omega_{FCC}=\\frac{a^3}{8\\times\\frac{1}{8}+6\\times\\frac{1}{2}}=\\frac{a^3}{4}\$
Nearest neighbor distance = \$d=2r=\\frac{a}{\\sqrt{2}}\$
\\[\\text{Packing fraction}=\\frac{4\\times\\frac{4\\pi}{3}\\times(\\frac{a\\sqrt{2}}{4})^3}{a^3}=\\frac{\\pi\\sqrt{2}}{6}\\approx 0.74\\]

The primitive cell vs the unit cell

The primitive translation vectors are: \\[\\mathbf{a}_1=\\frac{a}{2}(\\hat{\\mathbf{x}}+\\hat{\\mathbf{y}})\\] \\[\\mathbf{a}_2=\\frac{a}{2}(\\hat{\\mathbf{y}}+\\hat{\\mathbf{z}})\\] \\[\\mathbf{a}_3=\\frac{a}{2}(\\hat{\\mathbf{x}}+\\hat{\\mathbf{z}})\\]

The angle between adjacent edges is \$60^\\circ\$

The reference atom is red. Blue, green, and orange points are the 12 nearest neighbors of the reference atoms, and the purple points are its 6 next nearest neighbors atoms.

Packing fraction \$\\frac{1}{6}\\pi\\sqrt{2}\\approx 0.740\$
Some elements with fcc structure: Ar, Ag, Al, Au, Ca, Ce, \$\\beta-\$Co, Cu, Ir, Kr, La, Ne, Ni, Pb, Pd, Pr, Pt, \$\\delta-\$Pu, Rh, Sc, Sr, Th, Xe, Yb

Characteristics of Cubic structure

	simple cubic	b.c.c.	f.c.c.
volume of conventional cell	\$a^3\$	\$a^3\$	\$a^3\$
no. of lattice points per cell	1	2	4
no. of nearest neighbors (coordination number)	6	8	12
no. of 2nd nearest neighbors	12	6	6
nearest neighbor distance	\$a\$	\$\\frac{\\sqrt{3}}{2}a\\approx 0.866 a\$	\$\\frac{a}{\\sqrt{2}}\\approx 0.707a\$
2nd nearest neighbor distance	\$a\\sqrt{2}\$	\$a\$	\$a\$
packing fraction	\$\\pi/6\\approx 0.52\$	\$\\pi\\sqrt{3}/8\\approx 0.68\$	\$\\pi\\sqrt{2}/6\\approx 0.74\$

[111], [101], and [110] describe directions \$m\$, \$t\$, and \$n\$, respectively.

Crystal Structure May Change with Temperature

If the temperature changes, some materials might undergo a phase change.
For example, at atmospheric pressure (10⁵ Pa) and below 912 °C, pure iron (Fe) has a BCC structure, known as ⍺-iron. If we heat iron above 912 °C, its structure changes to an FCC structure, known as 𝛾-iron. Above 1394 °C, the structure changes back to BCC, known as 𝛿-iron. You may see the phase diagram of pure iron in the following figure.

Hexagonal Close-Packed (HCP) Structure

30 elements crystallize in hcp form.
HCP crystal has a hexagonal lattice and a multi-atom basis.
It can be viewed as two nested simple hexagonal Bravais lattice shifted by \$\\mathbf{a}_1/3+ \\mathbf{a}_2/3+\\mathbf{a}_3/2\$.
where

\\[\\mathbf{a}_1=a\\hat{\\mathbf{x}},\\ \\mathbf{a}_2=\\frac{a}{2}\\hat{\\mathbf{x}}+\\frac{a\\sqrt{3}}{2}\\hat{\\mathbf{y}},\\quad \\mathbf{a}_3=c\\,\\hat{\\mathbf{z}}\\]

- average vol. occupied by each lattice point \\[\\Omega_{hex\\ lattice}=\\frac{\\sqrt{3}}{2}a^2 c\\]
- average vol. occupied by each atom \\[\\Omega_{HCP}=\\frac{\\Omega_{hex\\ lattice}}{2}\\]

Comparing Close-Packed Structures

\"\" — The left is the hcp structure, and the right is the fcc structure.


hcp	fcc

In this figure, the left structure is hcp, and the right is fcc
volume fraction = 0.74
number of nearest neighbors (coordination number) is 12 for both hcp and fcc structures
Although a hexagonal close-packing of equal atoms is only obtained if \$c/a=\\sqrt{8/3}\\approx 1.63\$, the term hcp is used for any structure described previously.

Elements with hcp structures

Element	$\"\\large$	Element	$\"\\large$
Ideal	1.63
Be	1.56	Cd	1.89
Ce	1.63	\\(\\alpha\\)-Co	1.62
Dy	1.57	Er	1.57
Gd	1.59	He (2K)	1.63
Hf	1.58	Ho	1.57
La	1.62	Lu	1.59
Mg	1.62	Nd	1.61
Os	1.58	Pr	1.61
Re	1.62	Ru	1.59
Tb	1.58	Ti	1.59
Tl	1.60	Tm	1.57
Y	1.57	Zn	1.59

[adapted from Ashcroft, Mermin, Solid State Physics]

Diamond Crystal

It is the structure of carbon in a diamond crystal
It can be viewed as two interpenetrating fcc lattices displaced by \$\\frac{a}{4}(\\hat{\\mathbf{x}}+\\hat{\\mathbf{y}}+\\hat{\\mathbf{z}})\$
Or it can be imagined as the fcc lattice with two point basis \$\\mathbf{0}\$ and \$\\frac{a}{4}(\\hat{\\mathbf{x}}+\\hat{\\mathbf{y}}+\\hat{\\mathbf{z}})\$.
Coordination number is 4
Packing fraction is \$\\frac{\\sqrt{3}}{16}\\pi\\approx 0.34\$

3D Diamond Structure

Elements with diamond structure: C (diamond), Si, Ge, \$\\alpha\$-Sn (gray)
average volume occupied by each atom: \$\\Omega_{DC}=\\frac{\\Omega_{FCC}}{2}=\\frac{a^3}{8}\$

Sodium Chloride Structure

Na\$^{+}\$ and Cl\$^-\$ ions are placed on alternate points of a simple cubic structure
The lattice is fcc; the basis consists of Na\$^{+}\$ and Cl\$^-\$

NaCl structure, Blue atoms represent Na atoms and Green ones represent Cl atoms.

Some compounds with sodium chloride structure

LiF	LiCl	LiBr	LiI
NaF	NaCl	NaBr	NaI
RbF	RbCl	RbBr	RbI
CsF
AgF	AgCl	AgBr
MgO	MgS	MgSe
CaO	CaS	CaSe	CaTe
SrO	SrS	SrSe	SrTe
BaO	BaS	BaSe	BaTe

[Adapted from Ashcroft, Mermin, Solid State Physics]

Cesium Chloride Structure

Cs\$^{+}\$ and Cl\$^-\$ ions are placed at \$\\mathbf{0}\$ and body center position \$\\frac{a}{2}(\\hat{\\mathbf{x}}+\\hat{\\mathbf{y}}+\\hat{\\mathbf{y}})\$, respectively.
The lattice is simple cubic; the basis consists of Cs\$^{+}\$ and Cl\$^-\$

CsCl structure. Blue spheres represent Cl atoms and the big red sphere represents the Cs atom which is much larger than the Cl atoms.

Some compounds with the cesium chloride structure:

CsCl	CsBr	CsI
TlCl	TlBr	TlI

Miller Indices

Miller Indices for Directions in Cubic Structure

It is often required to specify certain directions and planes in crystals. To this end, we use Miller indices.
\$[hkl]\$ represents the direction vector \$h\\hat{\\mathbf{x}}+k\\hat{\\mathbf{y}}+l\\hat{\\mathbf{z}}\$ of a line passing through the origin.
These integers \$h\$, \$k\$ and \$l\$ must be the smallest numbers that will give the desired direction. i.e. we write \$[111]\$ not \$[222]\$.
If a component is negative, it is conventionally specified by placing a bar over the corresponding index. For example, we write \$[1\\bar{1}1]\$ instead of \$[1 -1 1]\$.
Coordinates in angle brackets such as \$\\langle123\\rangle\$ denote a family of directions that are equivalent due to symmetry operations, such as [123], [132], [321], [\$\\bar{1}23\$], [\$\\bar{1}\\bar{2}3\$], etc.

[111], [101], and [110] describe directions \$m\$, \$t\$, and \$n\$, respectively.

Miller Indices for Planes in Cubic Structure

A crystallographic plane is denoted by the Miller indices of the direction normal to the plane, but instead of brackets we use parenthesis, i.e. (\$hkl\$)


$(100)$	$(110)$	$(111)$

“Coordinates in curly brackets or braces such as \$\\{100\\}\$ denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.”
For a cubic system, a family \$\\{hkl\\}\$ consists of all the planes given by the permutations of the number \$h\$, \$k\$, \$l\$ and their negatives.
If the symmetry of the system is lower than that of cubic, not all the planes given by the permutations necessarily belong to a family. For example, in a rhombohedral system, we have \$\\{100\\}=\\{(100),(\\bar{1}00),(010),(0\\bar{1}0),(001),(00\\bar{1})\\}\$, but in an orthorhombic system \$\\{100\\}\$ family has only two members \$(100)\$ and \$(\\bar{1}00)\$.

Miller Indices for hcp

Miller indices contain 4 digits instead of 3 digits
\$[hkil]\$ means: \\[\\{\\alpha (h \\mathbf{a}_1+k\\mathbf{a}_2+i\\mathbf{a}_3+l \\mathbf{c}): \\alpha\\in\\mathbb{R}\\}\\] and \\[h+k+i=0\\]
Directions along axes \$\\mathbf{a}_1\$, \$\\mathbf{a}_2\$ and \$\\mathbf{a}_3\$ are of type \$\\langle \\bar{1} 2\\bar{1} 0\\rangle\$.
\$(hkil)\$ is a plane the normal direction of which is \$[hkil]\$.

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","chapter_express_content":"","public_visibility":true,"allow_part_see":false,"book_slug":"solid-mechanics","slug":"linear-stress-strain-relations","regular_authors":"","express_authors":"","created_at":"2025-09-28T15:55:16.885Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"linear-stress-strain-relations","slug":"strain-energy","header_name":"Strain Energy","type":"header-1","header_data":[],"_id":"68e627e79caaa78da88af2ff"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"linear-stress-strain-relations","slug":"generalized-hookes-law","header_name":"Generalized Hookes Law","type":"header-1","header_data":[],"_id":"68e629039caaa78da88af358"}]},{"_id":"68ef540c9caaa78da88b0377","book_id":"68b7d9c95419de3bf4488686","chapter_name":"Governing Equations of Linear Elasticity and Boundary Conditions","sequence":6,"chapter_content":"

The previous chapter introduced the key concepts of elasticity by developing separate equations for stress, strain, and their constitutive relationship. Now, in this chapter, we will put all those pieces together to form the complete set of governing equations. We will then turn our attention to the practical side of solving problems by discussing boundary conditions. Finally, we will cover Saint Venant's principle, an important concept that helps us simplify these conditions in real-world applications.

","chapter_express_content":"","public_visibility":true,"allow_part_see":false,"book_slug":"solid-mechanics","slug":"governing-equations-of-linear-elasticity-and-boundary-conditions","regular_authors":"","express_authors":"","created_at":"2025-09-28T15:55:16.885Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"governing-equations-of-linear-elasticity-and-boundary-conditions","slug":"the-governing-equations-of-elasticity","header_name":"The Governing Equations of Elasticity","type":"header-1","header_data":[],"_id":"68ef54409caaa78da88b038e"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"governing-equations-of-linear-elasticity-and-boundary-conditions","slug":"boundary-conditions","header_name":"Boundary Conditions","type":"header-1","header_data":[],"_id":"68ef54799caaa78da88b03a5"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"governing-equations-of-linear-elasticity-and-boundary-conditions","slug":"the-navier-equation","header_name":"The Navier Equation","type":"header-1","header_data":[],"_id":"68ef54b99caaa78da88b03bc"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"governing-equations-of-linear-elasticity-and-boundary-conditions","slug":"saint-venant-s-principle","header_name":"Saint Venant’s Principle","type":"header-1","header_data":[],"_id":"68ef55189caaa78da88b03d3"}]},{"_id":"68ef56eb9caaa78da88b03ea","book_id":"68b7d9c95419de3bf4488686","chapter_name":"Plane Elasticity","sequence":7,"chapter_content":"

Solving the full three-dimensional equations of elasticity can be incredibly complex for many practical applications. Fortunately, a large class of important engineering problems, such as those involving thin plates or very long uniform bodies, can be reasonably simplified to a two-dimensional analysis. This chapter is dedicated to the theory of plane elasticity, which provides the framework for these simplifications. We will explore the two primary idealizations used for this purpose: plane stress, which is suitable for thin bodies, and plane strain, which applies to long prismatic members. A significant portion of this chapter will be dedicated to a powerful mathematical tool known as the Airy stress function, which cleverly reduces the problem to solving a single governing equation. Finally, we will apply these concepts to solve several practical examples, demonstrating how this 2D framework is used to analyze real-world engineering structures.

","chapter_express_content":"","public_visibility":true,"allow_part_see":false,"book_slug":"solid-mechanics","slug":"plane-elasticity","regular_authors":"","express_authors":"","created_at":"2025-09-28T15:55:16.885Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"plane-elasticity","slug":"plane-stress","header_name":"Plane Stress","type":"header-1","header_data":[],"_id":"68ef572d9caaa78da88b0401"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"plane-elasticity","slug":"plane-strain","header_name":"Plane Strain","type":"header-1","header_data":[],"_id":"68ef57749caaa78da88b0418"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"plane-elasticity","slug":"the-stress-function-method","header_name":"The Stress Function Method","type":"header-1","header_data":[],"_id":"68f604c9feb3469c1121132b"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"plane-elasticity","slug":"uniaxial-tension","header_name":"Uniaxial Tension","type":"header-1","header_data":[],"_id":"68f60526feb3469c11211342"},{"sequence":5,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"plane-elasticity","slug":"pure-bending","header_name":"Pure Bending","type":"header-1","header_data":[],"_id":"68f6055dfeb3469c11211359"},{"sequence":6,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"plane-elasticity","slug":"a-cantilever-beam-with-an-end-load","header_name":"A Cantilever Beam with an End Load","type":"header-1","header_data":[],"_id":"68f605b5feb3469c11211370"},{"sequence":7,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"plane-elasticity","slug":"bending-of-a-beam-by-uniform-transverse-loading","header_name":"Bending of a Beam by Uniform Transverse Loading","type":"header-1","header_data":[],"_id":"68f8bd0afeb3469c112116e2"}]},{"_id":"6901dcc89b1d7200dbfd9b68","book_id":"68b7d9c95419de3bf4488686","chapter_name":"Finite Element Method","sequence":8,"chapter_content":"

Finite Element Method (FEM) or Finite Element Analysis (FEA) is a powerful tool to numerically solve partial differential equations (PDEs). Although it was originally developed for structural analysis, it now has various applications in various majors including civil and mechanical engineering, electrical engineering, physics, and applied mathematics. There are various ways to develop the formulation, in this chapter, we apply the principle virtual work to obtain the stiffness matrix and solve some 1D and 2D problems.

","chapter_express_content":"","public_visibility":true,"allow_part_see":false,"book_slug":"solid-mechanics","slug":"finite-element-method","regular_authors":"","express_authors":"","created_at":"2025-10-29T08:15:22.148Z","__v":0,"header_data":[{"sequence":1,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"the-principle-of-virtual-work-in-the-finite-element-method","header_name":"The Principle of Virtual Work in the Finite Element Method","type":"header-1","header_data":[],"_id":"6901df819b1d7200dbfd9bbd"},{"sequence":2,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"truss-element","header_name":"Truss Element","type":"header-1","header_data":[],"_id":"6901dfc69b1d7200dbfd9c00"},{"sequence":3,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"the-assembly-process--the-method-of-direct-superposition","header_name":"The Assembly Process: The Method of Direct Superposition","type":"header-1","header_data":[],"_id":"690b3733a3c639fb1eb4ca96"},{"sequence":4,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"stiffness-matrix-of-an-arbitrarily-oriented-truss-element","header_name":" Stiffness Matrix of an Arbitrarily Oriented Truss Element","type":"header-1","header_data":[],"_id":"690b377ea3c639fb1eb4caad"},{"sequence":6,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"beam-elements-1762769578","header_name":"Beam Elements","type":"header-1","header_data":[],"_id":"6911baaa9e7236b8c20607d3"},{"sequence":7,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"equivalent-nodal-forces","header_name":"Equivalent Nodal Forces","type":"header-1","header_data":[],"_id":"6911bb139e7236b8c206083b"},{"sequence":8,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"the-two-dimensional-triangular-element","header_name":"The Two-Dimensional Triangular Element","type":"header-1","header_data":[],"_id":"6911bb569e7236b8c2060852"},{"sequence":9,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"rectangular-elements","header_name":"Rectangular Elements","type":"header-1","header_data":[],"_id":"6911bbc09e7236b8c206086a"},{"sequence":10,"public_visibility":true,"header_type":"TEXT","book_slug":"solid-mechanics","chapter_slug":"finite-element-method","slug":"distorted-quadrilateral-element--natural-coordinate-system","header_name":"Distorted Quadrilateral Element: Natural Coordinate System","type":"header-1","header_data":[],"_id":"6911bc6c9e7236b8c2060881"}]},{"_id":"69631b07867864be40b381fc","book_id":"68b7d9c95419de3bf4488686","chapter_name":"Elements of Theory of Plasticity","sequence":9,"chapter_content":"

If a material is deformed beyond its elastic limit, it will retain some deformation even after all external forces are removed. When this happens, the material is described as plastic. Many substances, such as clay, putty, and ductile metals, become plastic when stress exceeds a certain threshold. Conversely, the concept of plastic deformation generally does not apply to brittle materials, which fracture rather than stretch when forces reach a limit. Notably, while fracture is induced by normal stresses, plastic deformation is primarily driven by shear stresses.

Modeling and understanding plastic deformation is significantly more complex than linear elasticity for several reasons:

Nonlinearity: Plastic deformation is inherently nonlinear. In mathematics and physics, nonlinear systems are generally more difficult to solve than linear ones, much like how nonlinear differential equations are far harder to solve than linear differential equations. While there are established methods for solving systems of linear equations, no such universal method exists for nonlinear equations.
Path Dependence: In linear elasticity, stress is uniquely determined by the current strain. However, in plasticity, a single strain state can be associated with various stress values depending on the loading history. Because the material behavior is path-dependent, we cannot simply relate total stress to total strain; instead, we must relate increments of stress to increments of strain.
Constitutive Variability: Plastic behavior varies fundamentally from one material to another. Also various parameters, such as temperature, loading rate, and grain size, can fundamentally alter a material's plastic behavior. In contrast, linear elasticity is uniform across materials, differing only by their compliance constants (the slope of the stress-strain curve).
Irreversibility and Energy Dissipation: Elastic deformation is a conservative process; the energy stored in the material is essentially a potential energy that is fully recoverable upon unloading. Plastic deformation, however, is dissipative. Energy is permanently lost (mostly converted to heat) and cannot be recovered. This means plastic modeling must satisfy strict thermodynamic laws to ensure that the model is physically valid, adding a layer of thermodynamic complexity that is not present in elasticity.
\n Geometric Nonlinearity and Large Deformations: In linear elasticity, deformations are assumed to be infinitesimal. This allows for major simplifications: equilibrium is calculated on the original shape, and all stress/strain definitions essentially coincide. In plasticity, these assumptions break down due to large stretches and rotations, introducing three layers of complexity:\n
- Distinction of Configurations: Because the geometry changes significantly, we cannot assume a fixed shape. We must strictly distinguish between the reference (undeformed) configuration and the current (deformed) configuration, and enforce equilibrium on a continuously moving and deforming body.\n
- Multiplicity of Stress and Strain Measures:\n While a single definition suffices for small strains, large deformations require specific measures to ensure physical accuracy and energy balance (work conjugacy):\n
  - Cauchy Stress: Defined on the current configuration (force per current area).
  - Piola-Kirchhoff Stresses: Mapped back to the reference configuration. The 1st Piola-Kirchhoff relates current force to original area, while the 2nd Piola-Kirchhoff transforms the force vector itself to account for material rotation.
  - Conjugacy:Each stress measure must be paired with a mathematically corresponding strain rate to ensure the calculation of internal work is correct.
  \n
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$\"\"$	$\"\"$	$\"\"$
Oblique	Rectangular	Centerd Rectangular

$\"\"$	$\"\"$
Square	Hexgonal (Rhombic)

$\"\"$	$\"\"$
Determination of indices for a digonal axis if Type I - $[2\\overline{11}0]$	Determination of indices for a digonal axis if Type II - $[10\\overline{1}0]$

	simple cubic	b.c.c.	f.c.c.
volume of conventional cell	\\(a^3\\)	\\(a^3\\)	\\(a^3\\)
no. of lattice points per cell	1	2	4
no. of nearest neighbors (coordination number)	6	8	12
no. of 2nd nearest neighbors	12	6	6
nearest neighbor distance	\\(a\\)	\\(\\frac{\\sqrt{3}}{2}a\\approx 0.866 a\\)	\\(\\frac{a}{\\sqrt{2}}\\approx 0.707a\\)
2nd nearest neighbor distance	\\(a\\sqrt{2}\\)	\\(a\\)	\\(a\\)
packing fraction	\\(\\pi/6\\approx 0.52\\)	\\(\\pi\\sqrt{3}/8\\approx 0.68\\)	\\(\\pi\\sqrt{2}/6\\approx 0.74\\)

	$\"\"$
(a) Tetrahedral bond in a diamond structure	(b) Diamond structure projected on a cube face. Fractions denote the height above the base in units of \\(a\\)

What Is Solid Mechanics?

Beyond the Basics: How This Book is Different

Further Studies

Primitive Cells vs. Unit Cells

Lattice Parameters and Packing Fraction

Lattice Parameters

Packing Fraction

Bravais Lattices

Five Bravais Lattices in Two Dimensions

Seven Crystal Systems

Cubic Lattices

Simple Cubic (SC) Crystal

Body-Centered Cubic (BCC) Crystal

Some Elements with BCC structure

Face-Centered Cubic (FCC) Crystal

Characteristics of Cubic structure

Crystal Structure May Change with Temperature

Hexagonal Close-Packed (HCP) Structure

Comparing Close-Packed Structures

Elements with hcp structures

Diamond Crystal

Sodium Chloride Structure

Cesium Chloride Structure

Miller Indices

Miller Indices for Directions in Cubic Structure

Miller Indices for Planes in Cubic Structure

Miller Indices for hcp

Further Reading

Linear Stress-Strain Relations

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