This document provides a summary of key concepts relating to stress, strain, and Hooke's law: 1) It defines stress and strain, describing normal stress, shear stress, normal strain, and shear strain. 2) It explains that stress and strain are second-order tensors with six independent components that define their state at a point, and how their values depend on the coordinate system used. 3) It describes plane stress and plane strain conditions, as well as transformations between coordinate systems.

1. Review of the Concepts of Stress,
Strain, and Hooke's Law
by:
Prof. Mark E. Tuttle
Dept. Mechanical Engineering
Box 352600
University of Washington
Seattle, WA 98195
1

2. Prof. M. E. Tuttle
University of Washington
Force Vectors
• A force, F , is a vector (also called a "1st-order tensor")
• The description of any vector (or any tensor) depends on
the coordinate system used to describe the vector
100 lbf 100 lbf 100 lbf
+y
+x"
+x'
+y'
45 degs +y" 60 deg
+x
F = 70.7 lbf (x'-direction) F = 86.6 lbf (x"-direction)
F = 100 lbf
+ +
(y-direction)
70.7 lbf (y'-direction) 50 lbf (y"-direction) 2

3. Prof. M. E. Tuttle
University of Washington
Normal and Shear Forces
• A "normal" force acts perpindicular to a surface
• A "shear" force acts tangent to a surface
P = Normal Force
V = Shear Force
3

4. Prof. M. E. Tuttle
University of Washington
Forces Inclined to a Plane
• Since forces are vectors, a force inclined to a plane can always
be described as a combination of normal and shear forces
Inclined Force
P = Normal Force
V = Shear Force
4

5. Prof. M. E. Tuttle
University of Washington
Moments
• A moment (also called a "torque" or a "couple") is a force
which tends to cause rotation of a rigid body
• A moment is also vectoral quantity (i.e., a 1st-order tensor)...
M
M
5

6. Prof. M. E. Tuttle
University of Washington
Static Equilibrium
• A rigid solid body is in "static equilibrium" if it is either
- at rest, or
- moves with a constant velocity
• Static equilibrium exists if: Σ F = 0 and Σ M = 0
50 lbf 50 lbf
+y
(40 lbf) (40 lbf)
(30 lbf) +x (30 lbf)
60 lbf
(30 lbf) (BALL ACCELERATES) (30 lbf) (NO ACCELERATION)
(40 lbf) (40 lbf)
50 lbf 50 lbf 6

7. Prof. M. E. Tuttle
University of Washington
Free Body Diagrams
and Internal Forces
• An imaginary "cut" is made at plane of interest
• Apply Σ F = 0 and Σ M = 0 to either half to determine
internal forces
F F
R (= F)
"cut"
(or)
R (= F)
F F 7

8. Prof. M. E. Tuttle
University of Washington
Free Body Diagrams
and Internal Forces
• The imaginary cut can be made along an arbitrary plane
• Internal force R can be decomposed to determine the normal
and shear forces acting on the arbitrary plane
F
R (= F) P
"cut" V
F F F 8

9. Prof. M. E. Tuttle
University of Washington
Stress:
Fundamental Definitions
• Two "types" of stress:
normal stress = σ = P/A
shear stress = τ = V/A
where P and V must be uniformly distributed over A
P = Normal Force σ = P/A
V = Shear Force
τ = V/A
9
A = Cross-Sectional Area

10. Prof. M. E. Tuttle
University of Washington
Distribution of Internal Forces
• Forces are distributed over the internal plane...they may
or may not be uniformly distributed
F M
σ = F/A σ= ?
"cut" "cut"
F F M M 10

11. Prof. M. E. Tuttle
University of Washington
Infinitesimal Elements
• A free-body diagram of an "infinitesimal element" is used to
define "stress at a point"
• Forces can be considered "uniform" over the infinitesimally
small elemental surfaces
+x
+z
+y
dx
dy
dz 11

12. Prof. M. E. Tuttle
University of Washington
Labeling Stress Components
• Two subscripts are used to identify a stress component, e.g.,
"σxx" or "τxy" (note: for convenience, we sometimes write
σx = σxx, or σxy = τxy)
• 1st subscript: identifies element face
• 2nd subscript: identifies "direction" of stress
τ
xy
+y
σ xx +x
σ
xx
12

13. Prof. M. E. Tuttle
University of Washington
Admissable Shear
Stress States
τyx
+y +y +y
+x
τxy τxy +x
τxy τxy +x
τxy
τyx
ΣF = 0 ΣF = 0 If: τyx = τxy
ΣM = 0 ΣM = 0 ΣF = 0
(inadmissable) (inadmissable) ΣM = 0
(admissable)
13

14. Stress Sign Conventions
• The "algebraic sign" of a cube face is positive if the outward
unit normal of the face "points" in a positive coordinate
direction
• A stress component is positive if:
• stress component acts on a positive face and "points" in
a positive coordinate direction, or
• stress component acts on a negative face and "points"
in a negative coordinate direction.
14

15. Stress Sign Conventions
+y +y
σyy σyy
τxy τxy
σxx σxx
σxx σxx
+x +x
σyy σyy
σxx and σ yy Positive
All Stresses Positive
τxy Negative
15

16. Prof. M. E. Tuttle
University of Washington
3-Dimensional Stress States
• In the most general case, six independent components of
stress exist "at a point"
+x
F1 +z σxx
F2 +y
M1 τxz
τzy
τxy
σzz
F3 σyy
F4
M2 16

17. Prof. M. E. Tuttle
University of Washington
Plane Stress
• If all non-zero stress components exist in a single plane, the
(3-D) state of stress is called "plane stress"
+y
+y
σyy σyy
+x τxy
+z
τxy
σxx
σxx
σxx σxx
+x
σyy
σyy 17

18. Prof. M. E. Tuttle
University of Washington
Uniaxial Stress
• If only one normal stress exists, the (3-D) state of stress is
called a "uniaxial stress"
+y
+y
+x
σyy σyy
+z
+x
σyy
σyy
16

19. Prof. M. E. Tuttle
University of Washington
Free Body Diagram Defines
the Coordinate System
F
+y +y
F σyy
"cut"
+x +x
F F F
17

20. Prof. M. E. Tuttle
University of Washington
Free Body Diagram Defines
the Coordinate System
F
y' y'
σy'y'
P V x' τx'y' x'
"cut"
F F F
18

21. Prof. M. E. Tuttle
University of Washington
Free Body Diagram Defines
the Coordinate System
F
"cut" y" y"
(a
plane) Py"y" σy"y"
x" x"
Vy"x" τy"x"
Vy"z" τy"z"
z" z"
F F F
21

22. Prof. M. E. Tuttle
University of Washington
Stress Transformations
Within a Plane
• Given stress components in the x-y coordinate system (σxx,
σyy, τxy), what are the corresponding stress components in
the x'-y' coordinate system?
σyy
+y σy'y' τx'y'
τxy
+y' σx'x'
? +x'
σxx
θ
+x
20

23. Prof. M. E. Tuttle
University of Washington
Stress Transformations
• Stress components in the x'-y' coordinate system may be
related to stresses in the x-y coordinate system using a free
body diagram and enforcing Σ F = 0
Σ Fx' = 0
+y σyy +y' +x'
τxy τx'y' θ
σxx σxx σx'x'
+x τxy
"cut" σyy
23

24. Prof. M. E. Tuttle
University of Washington
Stress Transformation
Equations
• By enforcing ΣFx' = 0, ΣFy' = 0, it can be shown:
σ xx + σ yy σ xx − σ yy
σ x' x' = + cos 2θ + τ xy sin2θ
2 2
σ xx + σ yy σ xx − σ yy
σ y' y' = − cos2θ − τ xy sin2θ
2 2
σ xx − σ yy
τ x'y' = − sin 2θ + τ xy cos 2θ
2
22

25. Prof. M. E. Tuttle
University of Washington
Use of Mohr's Circle of
Stress
σyy
+y
τxy
σxx τ
(σyy,τ xy) (σx'x' ,−τx'y' )
+x
2θ 2θ
(Equivalent)
σ
σy'y' τx'y' (σxx,−τxy)
(σy'y' ,τ x'y' )
+y' σx'x'
+x'
θ
25

26. Prof. M. E. Tuttle
University of Washington
Principal Stresses
• In the principal stress coordinate system the shear stress is
zero...
σyy
+y
τxy
τ
σxx
(σyy,τ xy)
+x
2θ'
(Equivalent) σ22 2θ' σ11 σ
σ22
+2
σ11 (σxx,−τxy)
+1
θ'
26

27. Prof. M. E. Tuttle
University of Washington
"Stress":
Summary of Key Points
• Normal and shear stresses are both defined as a (force/area)
• Six components of stress must be known to specify the state
of stress at a point (stress is a "2nd-order tensor")
• Since stress is a tensoral quantity, numerical values of
individual stress components depend on the coordinate
system used to describe the state of stress
• Stress is defined strictly on the basis of static equilibrium;
definition is independent of:
material properties
27
strain

28. Prof. M. E. Tuttle
University of Washington
Strain:
Fundamental Definitions
• "Strain" is a measure of the deformation of a solid body
• There are two "types" of strain; normal strain (ε) and shear
strain (γ)
(change in length )
ε= ; units = in / in, m / m, etc
(original length)
γ = (change in angle ) ; units = radians
28

29. Prof. M. E. Tuttle
University of Washington
3-Dimensional Strain States
• In the most general case, six components of strain exist "at
a point":
εxx, εyy, εzz, γxy, γxz, γyz
• Strain is a 2nd-order tensor; the numerical values of the
individual strain components which define the "state of
strain" depend on the coordinate system used
• This presentation will primarily involve strains which exist
within a single plane
29

30. Prof. M. E. Tuttle
University of Washington
Strain Within a Plane
• We are often interested in the strains induced within a
single plane; specifically, we are interested in two distinct
conditions:
• "Plane stress", in which all non-zero stresses lie within
a plane. The plane stress condition induces four
strain components: εxx, εyy, εzz, and γxy
• "Plane strain", in which all non-zero strain
components lies within a plane. By definition then,
the plane strain condition involves three strain
components (only): εxx, εyy, and γxy 30

31. Prof. M. E. Tuttle
University of Washington
Strain Within a Plane
∠ abc = π/2 radians ∠ abc < π/2 radians
+y +y
a
a ly + ∆ly
ly c
c
b b
+x +x
lx lx + ∆lx
Original Shape Deformed Shape
εxx = ∆lx/lx
εyy = ∆ly/ly
γxy = ∆(∠ abc)
29

32. Strain Sign Convention
• A positive (tensile) normal strain is associated with an
increase in length
• A negative (compressive) normal strain is associated with a
decrease in length
• A shear strain is positive if the angle between two positive
faces (or two negative faces) decreases
+y +y
+x +x
All Strains Positive εxx Positive
εyy and γ xy Negative 32

33. Prof. M. E. Tuttle
University of Washington
Visualization of Strain
F
+y (Loading)
εyy > 0
+x εxx < 0
γxy = 0
F 33

34. Prof. M. E. Tuttle
University of Washington
Visualization of Strain
F
+y' +x' (Loading)
εx'x' = εy'y' > 0
γx'y' > 0
45 deg
F 34

35. Prof. M. E. Tuttle
University of Washington
Strain Transformations
• Given strain components in the x-y coordinate system (εxx, εyy,
γxy), what are the corresponding strain components in the x'-y'
coordinate system?
+y +y'
+x'
?
ly + ∆ly θ
+x
lx + ∆lx εx'x' = ?
εxx = ∆lx/lx εy'y' = ?
εyy = ∆ly/ly γx'y' = ?
γxy = ∆(∠ abc) 35

36. Prof. M. E. Tuttle
University of Washington
Strain Transformation
Equations
• Based strictly on geometry, it can be shown:
ε xx + ε yy ε xx − ε yy γ xy
ε x' x' = + cos 2θ + sin2θ
2 2 2
ε xx + ε yy ε xx − ε yy γ xy
ε y' y' = − cos2θ − sin2θ
2 2 2
γ x'y' ε xx − ε yy γ xy
=− sin 2θ + cos 2θ
2 2 2
36

37. Prof. M. E. Tuttle
University of Washington
Well, I'll Be Darned!!!
• The stress transformation equations are based strictly on
the equations of static equilibrium
• The strain transformation equations are based strictly on
geometry
• Nevertheless, the stress and strain transformation
equations are nearly identical!!
37

38. Prof. M. E. Tuttle
University of Washington
"Strain":
Summary of Key Points
• ε = (∆ length)/(original length) γ = (∆ angle)
• Six components of strain specify the "state of strain"
• Strain is a 2nd-order tensor; numerical values of individual
strain components depend on the coordinate system used
• Strain is defined strictly on the basis of a change in shape;
definition is independent of:
material properties
stress
• "Suprisingly," the stress and strain transformation equations
are nearly identical
38

39. Prof. M. E. Tuttle
University of Washington
"Constitutive Models"
• The structural engineer is typically interested in the state of
stress induced in a structure during service
• The state of stress cannot be measured directly...
• The state of strain can be measured directly...
• Hence, we must develop a relation between stress and
strain...this relationship is called a "constitutive model," and
the most common is "Hooke's Law"
39

40. Prof. M. E. Tuttle
University of Washington
Material Properties:
The Uniaxial Tensile Test
• Specimen is subjected to an axial tensile force, inducing a
uniform tensile stress in the "gage" region of the specimen
• Stress is increased until fracture occurs; strains are measured
throughout the test
• The ASTM has developed "standard" specimen
styles/procedures for most materials
F F
"Gage
Region" 40

41. Prof. M. E. Tuttle
University of Washington
The Tensile
Stress-Strain Curve
σ
Tensile Strength
Fracture Stress
x
0.2% Offset
Yield Strength
ε 41
0.2% Strain to Fracture

42. Prof. M. E. Tuttle
University of Washington
Load-Unload Cycles
σ σ σ
Tensile Strength Tensile Strength Tensile Strength
0.2% Offset 0.2% Offset 0.2% Offset
Yield Strength Yield Strength Yield Strength
ε ε ε
Elastic Strain Elastic Strain Inelastic Strain Elastic Strain
Inelastic
Strain
42

43. Prof. M. E. Tuttle
University of Washington
Material Property:
Young's Modulus
σ
• At stress levels below the
yield stress the response is
called "linear elastic"
0.2% Offset
Yield Strength
• The slope of the linear
In Linear Region:
region is called "Young's
E σ = Eε
modulus" or the "modulus of
elasticity", E
1
• In the linear region (only):
ε
σ = Eε (or) ε = σ/E

44. Prof. M. E. Tuttle
University of Washington
Material Property:
Poisson's Ratio
• A uniaxial tensile stress causes both a tensile axial strain (εa)
as well as a compressive transverse strain (εt)
• Poisson's ratio is defined as: ν = -(εt/εa)
F F
εa
εt 44

45. Prof. M. E. Tuttle
University of Washington
Material Properties:
The Torsion Test
• Cylindrical specimen is subjected to a torque, inducing a
uniform shear stress τxy in the gage region of the
specimen
• Shear stress increased until fracture occurs; shear strain
measured throughout test
+y
τ xy
+x
T
T
45

46. Prof. M. E. Tuttle
University of Washington
Material Property:
Shear Modulus
• At linear levels, the
slope of the shear τ
stress vs shear strain
curve is called the
"shear modulus," G
• In the linear region G
(only):
τ = Gγ (or) γ = τ/G
γ
46

47. Prof. M. E. Tuttle
University of Washington
"Isotropic" vs
"Anisotropic" Materials
• A material which exhibits the same properties in all
directions is called "isotropic"
2 2
1 1 2 1 1 2
E1 ≅ E2 E1 > E2
Isotropic Materials Anisotropic Materials
47

48. Prof. M. E. Tuttle
University of Washington
Material Properties
Summary
• Three material properties are commonly used to
describe isotropic materials in the linear-elastic region:
E, ν, and G
• It can be shown that (for isotropic materials) only two
of these three properties are independent. For
example:
E
G=
2(1 + ν)
48

49. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
3-Dimensional Stress States
• Assumptions:
• Linear-elastic behavior; i.e., yielding does not occur, so
the principle of superposition applies
• Isotropic material behavior
• Constant environmental conditions (e.g., constant
temperature)
49

50. Strains Induced by Individual Stress Components
(That is, Strain Caused During Six Different Tests)
=
εx xσ σxx /E
xx γx y τ= τx y /G = 2(1+ν)τx y /E
= -νεx x
εyy Induces:= -νσx x /E xy
εx x = εyy = εzz = γxz = γyz = 0
Induces:
εzz = -νεx x = -νσx x /E
γx y = γxz = γyz = 0
εyyσ= σyy /E
yy
= -νεyy
εx x Induces:= -νσyy /E γxz τ= τxz /G = 2(1+ν)τxz /E
xz
εzz = -νεyy = -νσyy /E εx x = εyy = εzz = γx y = γyz = 0
Induces:
γx y = γxz = γyz = 0
=
εzzσ σzz/E
zz
εx x Induces:= -νσzz/E
= -νεzz γyz τ= τyz /G = 2(1+ν)τyz /E
yz
εyy = -νεzz = -νσzz/E εx x = εyy = εzz = γx y = γxz = 0
Induces:
γx y = γxz = γyz = 0

51. Prof. M. E. Tuttle
University of Washington
3-D Form of Hooke's Law:
Apply Principle of Superposition
• The total strain εxx caused by all stresses applied
simultaneously is determined by "adding up" the strain εxx
caused by each individual stress component
 σ xx   −νσ yy   −νσ zz 
ε xx =
 E   +  E  +  E  + [0 ] + [0 ] + [0 ]
  
or
ε xx
1
[
= σ xx − νσ yy − νσ zz
E
]
51

52. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
3-Dimensional Stress States
• Following this procedure for all six strain components:
2(1+ ν)τ yz
1
[
ε xx = σ xx − νσ yy − νσ zz
E
] γ yz =
E
2(1+ ν)τ zx
1
[
ε yy = −νσ xx + σ yy − νσ zz
E
] γ zx =
E
2(1+ ν)τ xy
1
[
ε zz = −νσ xx − νσ yy + σ zz
E
] γ xy =
E
52

53. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
3-Dimensional Stress States
 ε xx  1 −ν −ν 0 0 0  σ xx 
    σ 
 ε yy  −ν 1 −ν 0 0 0   yy 
ε 
 zz  1 −ν −ν 1 0 0 0   σ zz 
 
 =   
 γ yz  E  0 0 0 2(1+ ν) 0 0   τ yz 
γ    τ 
 zx  0 0 0 0 2(1+ ν) 0   zx 
 γ xy 
  0
 0 0 0 0 2(1+ ν)  τ xy 
 
53

54. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
3-Dimensional Stress States
[ ]
E Eγ yz
σ xx = (1− v)ε xx + νε yy + νεzz τ yz =
(1 + ν)(1− 2 ν) 2(1 + ν)
σ yy =
E
(1+ ν)(1 − 2ν)
[
vε xx + (1 − v)ε yy + νε zz ] τ zx =
Eγ zx
2(1 + ν)
[ ]
E Eγ xy
σ zz = vε xx + νε yy + (1 − v)ε zz τ xy =
(1+ ν)(1− 2ν) 2(1 + ν)
54

55. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
3-Dimensional Stress States
(1− ν) ν ν 0 0 0 
σ xx    ε 
   ν (1− ν) ν 0 0 0  xx 
σ yy   ν ε
ν (1− ν) 0 0 0  yy 
σ    ε 
 zz  0  zz 
E  0 (1− 2ν)
 = 0 0 0  
 τ yz  (1+ ν)(1− 2ν)  2  γ yz 
τ   (1− 2ν)  
 0 0 0 0 0  γ zx
 zx 
2  
 τ xy    γ 
(1− 2ν)  xy 
   0 0 0 0 0 
 2 
55

56. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
3-Dimensional Stress States
• "Shorthand" notation:
{σ } = [ D]{ε }
where:
(1− ν) ν ν 0 0 0 
 
 ν (1− ν) ν 0 0 0 
 ν ν (1− ν) 0 0 0 
 (1− 2ν) 
[ D] =
E  0 0 0 0 0 
(1 + ν)(1− 2 ν)  2 
 (1− 2ν) 
 0 0 0 0
2
0 
 (1 − 2ν) 
 0 0 0 0 0 
 2  56

57. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
• For thin components (e.g., web or flange of an I-beam,
automobile door panel, airplane "skin", etc) the "in-plane"
stresses are much higher than "out-of-plane" stresses:
(σxx,σyy,τxy) >> (σzz,τxz,τyz)
• It is convenient to assume the out-of-plane stresses equal
zero...this is called a state of "plane stress"
+y
+x
+z
57

58. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
1
ε xx = (σ xx − νσ yy )
E
1
σ yy ε yy = (σ yy − νσ xx )
τ xy E
−ν
σ xx ε zz = (σ xx + σ yy )
E
τ xy 2(1+ ν)τ xy
γ xy = =
G E
γ xz = γ yz = 0 58

59. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
E
σ xx = 2 (ε xx + νε yy )
(1- ν )
σ yy E
τ xy σ yy = 2 (ε yy + νε xx )
(1- ν )
σ xx E
τ xy = Gγ xy = γ xy
2(1+ ν)
σ zz = τ xz = τ yz = 0 59

60. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
σ xx    
1 ν 0  ε xx
  E   
σ yy  = 2 ν 1 0   ε yy 
  (1− ν ) 1 − ν  
 τ xy  0

0   γ xy 
2 
60

61. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Stress
• "Shorthand" notation:
{σ } = [ D]{ε }
where
 
1 ν 0 
E 
[ D] = 2 ν 1 0 
(1 − ν ) 1− ν
0 0 
 2 
61

62. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Strain
• For thick or very long components (e.g., thick-walled pressure
vessels, buried pipe, etc) the "in-plane" strains are much
higher than "out-of-plane" strains: (εxx,εyy,γxy) >> (εzz,γxz,γyz)
• It is convenient to assume the out-of-plane strains equal
zero...this is called a state of "plane strain"
+y
+z +x
62

63. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Strain
σ xx =
E
(1 + ν)(1− 2ν)
[ (1− ν)ε xx + νε yy ]
σ yy
τ xy σ yy =
E
(1+ ν)(1− 2ν)
[ νε xx + (1− ν)ε yy ]
νE
σ xx σ zz =
(1+ ν)(1− 2ν)
[ ] [
ε xx + ε yy = ν σ xx + σ yy ]
Eγ xy
τ xy = Gγ xy =
2(1+ ν)
τ xz = τ yz = 0 63

64. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Strain
  
σ xx  (1− ν) ν 0  ε xx
  E  ν ε 
σ yy  = (1− ν) 0   yy 
 
 (1+ ν)(1− 2ν)  (1 − 2v)   γ 
 τ xy  
0 0
  xy 
2
64

65. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Plane Strain
• "Shorthand" notation:
{σ } = [ D]{ε }
where
 
(1− ν) ν 0 
[ D] =
E  ν (1− ν) 0 
(1 + ν)(1− 2ν)  
(1− 2v) 
 0 0
 2 
65

66. Hooke's Law
Plane Strain
1 − ν2   ν  
ε xx = σ xx −   σ yy 
E   1− ν  
σ yy 1− ν 2   ν  
τ xy ε yy = σ yy −   σ xx 
E   1− ν  
2(1+ ν)
σ xx γ xy = τ xy
E
ε zz = γ xz = γ yz = 0
[
....and: σ zz = ν σ xx + σ yy ] 66

67. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Uniaxial Stress
• Truss members are designed to carry axial loads only (i.e.,
uniaxial stress)
• In this case we are interested in the axial strain only (even
though transverse strains are also induced)
67

68. Prof. M. E. Tuttle
University of Washington
Hooke's Law:
Uniaxial Stress
• For: σyy = σzz = τyz = τxz = τzy = 0,
"Hooke's Law" becomes:
σ xx = Eε xx
• "Shorthand" notation:
{σ } = [ D]{ε } where [ D] = [ E ]
68

69. Prof. M. E. Tuttle
University of Washington
Hooke's Law
Summary of Key Points
• Hooke's Law is valid under linear-elastic conditions only
• The mathematical form of Hooke's Law depends on whether
the problem involves:
• Structures for which 3-D stresses are induced
• Structures for which (due to loading and/or geometry)
plane stress conditions can be assumed
• Structures for which (due to loading and/or geometry)
plane strain conditions can be assumed
• Structures for which (due to loading and/or geometry)
uniaxial stress conditions can be assumed
69