Chapter 2 Basics of Materials and suemasu/files/ of Materials 2 2.1 Stress 2.1.1 Definition of...

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Transcript of Chapter 2 Basics of Materials and suemasu/files/ of Materials 2 2.1 Stress 2.1.1 Definition of...

  • Mechanics of Materials 1

    Chapter 2

    Basics of Materials

    and Mechanics

    13:14

  • Mechanics of Materials 2

    2.1 Stress

    2.1.1 Definition of stress

    - P

    S

    SS

    P- P

    - P

    P

    P

    P2

    PP1

    V

    V1

    V2

    hypothetical internal

    surface S Body is in

    equilibrium

    Sum of the internal

    forces on S

    =Applied force P

    Stress Vector

    Quantity to express the

    intensity of internal

    force

    SS

    Pp

    0lim (1)

    13:14

  • Mechanics of Materials 3

    Unit of Stress Pa(=N/m2), kgf/m2, psi (=lb/inch2)etc Stress depends not only the internal force (vector) but

    also the direction of the surface considered.

    dS

    dPp

    dS

    dPp

    2

    1

    sin

    cos

    Normal componentNormal Stress

    Inplane componentShear Stress

    13:14

  • Mechanics of Materials 4

    Positive stresses are defined as shown above. In the other sides

    of the cube where the outward normals are opposite to the

    coordinate axes, the directions of positive stresses are opposite.

    Definition of Stresses in the Cartician

    Coordinate System

    x

    y

    z

    P

    B

    A

    O

    C dx

    dy

    dz xxy

    xz y

    yx

    yz

    z

    zx

    zy

    13:14

  • Mechanics of Materials 5

    Stress in an arbitrary surface

    Equilibrium of the force for infinitesimal tetrahedron

    Area of ABCdS Outward normal of ABC : =(l,m,n)

    l, m, n are directional cosines

    OA=dx, OB=dy, OC=dz

    OBC=dydz/2=dSl,

    OCA=dzdx/2=dSm,

    OAB=dxdy/2=dSn

    n

    x

    y

    z

    x

    y

    z

    O A

    B

    C pdS

    -pxldS

    -pymdS

    -pzndS

    13:14

  • Mechanics of Materials 6

    zzyzxz

    xzyyxy

    xzxyxx

    ZYX

    ,,

    ,,

    ,,

    ,,

    p

    p

    p

    p

    Stress vectors

    Equilibrium of the force when no body force exists.

    0 ndSmdSldSdS zyx pppp

    ndSmdSldSdS

    Z

    Y

    X

    z

    zy

    zx

    yz

    y

    yx

    xz

    xy

    x

    x

    y

    z

    OA

    B

    C pdS

    -pxldS

    -pymdS

    -pzndS

    13:14

  • Mechanics of Materials 7

    n

    m

    l

    Z

    Y

    X

    zyzxz

    zyyxy

    zxyxx

    Then

    Stress vector of an arbitrary surface

    Nine components are necessary to express the stress state.

    Note) From the equilibrium of an infinitesimal cube in moment,

    zy=yz , xz=zx, yx=xy

    Six components are practically independent.

    yxxyxzzxzyyzzyx ,,,,,,,,pp

    13:14

  • Mechanics of Materials 8

    2.1.2 Transformation of Coordinate system

    Let (x, y, z) and (x', y', z') be two Cartesian coordinate systems

    with a common origin. The cosines of the angles between the six

    coordinate axes are represented in the following tabular form:

    x y z

    x a11 a12 a13

    y a21 a22 a23

    z a31 a32 a33

    x

    y

    z

    O

    p

    i

    j

    i

    x y

    z k

    k

    j

    Normal to the

    x axis

    13:14

  • Mechanics of Materials 9

    333231

    232221

    131211

    332313

    322212

    312111

    '''''

    '''''

    '''''

    aaa

    aaa

    aaa

    aaa

    aaa

    aaa

    zyzxz

    zyyxy

    zxyxx

    zzyzx

    yzyyx

    xzxyx

    The stresses referring to (x', y', z') coordinate

    system can be derived from the stresses

    referring to (x, y, z) coordinate system .

    13:14

  • Mechanics of Materials 10

    2.1.3 Principal stress

    The values of stresses at a point depend on the plane where the

    stresses are considered. The direction of the plane where the normal

    stress is extremum is called principal axis and the normal stress is

    called principal stress. The principal directions and principal stresses

    are obtained through the following eigenvalue analysis.

    0

    zyzxz

    zyyxy

    zxyxx

    222

    3

    2222

    1

    xyzzxyyzxzyx

    xyzxyzyxxzzy

    zyx

    J

    J

    J

    The coefficients of the equation are called stress invarient.

    13:14

  • Mechanics of Materials 11

    2.1.4 Maximum Shear Stress

    Shear Stress becomes maximum at the surface whose unit

    normal is 0,2/1,2/1,, nml

    212

    1

    212

    1

    The normal stress at the surface of maximum shear stress is

    where the coordinate system accords to the principal directions. 1 is

    maximum principal stress and 2 is minimum principal stress

    13:14

  • Mechanics of Materials 12

    2.1.5 Equilibrium Equation

    0

    0

    0

    Zzyx

    Yzyx

    Xzyx

    zyzxz

    zyyxy

    zxyxx

    x y

    z

    P

    B

    A

    O

    C dx

    dy

    dz

    dxx

    xx

    dxx

    xyxy

    dxxxz

    xz

    dyy

    yy

    dyy

    yxyx

    dyy

    yzyz

    dzz

    zz

    dz

    zzx

    zx

    dz

    z

    zyzy

    From the equilibrium of the force acting on the

    small rectangular parallelepiped

    13:14

  • Mechanics of Materials 13

    A

    B C

    D

    C

    A B

    dz

    D

    dy

    dx

    x

    y

    z

    2.2 Strain 2.2.1 Definition of strain

    Intensity of Deformation

    Normal strain Shear strain

    x

    xxx

    i

    ir

    AB

    ABBA

    ''

    xy

    2

    13:14

  • Mechanics of Materials 14

    Normal strain represents change of length.

    When the length is increased, it is called

    tensile strain. When the length is decreased,

    it is called compressive strain. Normal strain

    is usually positive if tensile. Strain being

    defined as length/length is non-dimensional

    value.

    Shear strain represents change of shape. It is

    defined as reduction of angle between two

    coordinate axes which were normal each

    other.

    13:14

  • Mechanics of Materials 15

    Relations between the strain components and

    displacements (u,v,w)

    x

    w

    z

    w

    x

    v

    z

    v

    x

    u

    z

    u

    x

    w

    z

    u

    z

    w

    y

    w

    z

    v

    y

    v

    z

    u

    y

    u

    z

    v

    y

    w

    y

    w

    x

    w

    y

    v

    x

    v

    y

    u

    x

    u

    y

    u

    x

    v

    z

    w

    z

    v

    z

    u

    z

    w

    y

    w

    y

    v

    y

    u

    y

    v

    x

    w

    x

    v

    x

    u

    x

    u

    zx

    yz

    xy

    z

    y

    x

    222

    222

    222

    2

    1

    2

    1

    2

    1

    13:14

  • Mechanics of Materials 16

    When u and v is infinitesimal,

    x

    w

    z

    u

    z

    v

    y

    w

    y

    u

    x

    v

    z

    w

    y

    v

    x

    u

    zxyzxy

    zyx

    ,,

    ,,

    13:14

  • Mechanics of Materials 17

    2.2.2 Motion of elastic body

    Displacement of P

    dz

    dy

    dx

    dz

    dy

    dx

    w

    v

    u

    zzyzzx

    yzyxy

    zxxyx

    zxy

    xz

    yz

    drp

    2/2/

    2/2/

    2/2/

    0

    0

    0

    0

    0

    0

    0 uuuu

    y

    u

    x

    v

    x

    w

    z

    u

    z

    v

    y

    wzyx

    2

    1,

    2

    1,

    2

    1

    where

    , : deformation

    : rotation

    O

    /2

    P

    O'

    P'/2

    13:14

  • Mechanics of Materials 18

    2.2.3 Transformation of strain component under change of coordinate system

    Two rectangular Cartesian

    coordinate systems x, y, z

    and x, y, z

    333231

    232221

    131211

    21

    21

    21

    21

    21

    21

    332313

    322212

    312111

    '''21

    ''21

    ''21

    '''21

    ''21

    ''21

    '

    aaa

    aaa

    aaa

    aaa

    aaa

    aaa

    zyzxz

    zyyxy

    zxyxx

    zzyzx

    yzyyx

    xzxyx

    Table of directional cosines

    x y z

    x a11 a12 a13

    y a21 a22 a23

    z a31 a32 a33

    13:14

  • Mechanics of Materials 19

    zyxzyx

    zyxyxz

    zyxxzy

    zxzx

    yzyz

    xyxy

    xyzxyzz

    xyzxyzy

    xyzxyzx

    zxxz

    yzzy

    xyyx

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2

    2.2.4 Compatibility Equation

    13:14

  • Mechanics of Materials 20

    2.3 Generalized Hookes Law and elastic constants

    Hookes Law refers that normal strain is directly

    proportional to the relating normal stress. However, in

    general stress state, the whole strain components are

    dependent on whole stress state as

    xy

    zx

    yz

    z

    y

    x

    xy

    zx

    yz

    z

    y

    x

    CCCCCC

    CCCCCC

    CCCCCC

    CCCCCC

    CCCCCC

    CCCCCC

    666564636261

    565554535251

    464544434241

    363534333231

    262524232221

    161514131211

    13:14

  • Mechanics of Materials 21

    For three dime