CH.7. PLANE LINEAR ELASTICITYContinuum Mechanics Course (MMC) - ETSECCPB - UPC
Overview
Plane Linear Elasticity Theory
Plane Stress Simplifying Hypothesis Strain Field Constitutive Equation Displacement Field The Linear Elastic Problem in Plane Stress Examples
Plane Strain Simplifying Hypothesis Strain Field Constitutive Equation Stress Field The Linear Elastic Problem in Plane Stress Examples
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Overview (cont’d)
The Plane Linear Elastic Problem Governing Equations
Representative Curves Isostatics or stress trajectories
Isoclines
Isobars
Shear lines
Others: isochromatics and isopachs
Photoelasticity
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Ch.7. Plane Linear Elasticity
7.1 Plane Linear Elasticity Theory
A lineal elastic solid is subjected to body forces and prescribed traction:
The Linear Elastic problem is the set of equations that allow obtaining the evolution through time of the corresponding displacements , strains and stresses .
Introduction
0t
,0
,0
b x
t x
,
,
t
t
b x
t x
Initial actions:
Actions through time:
, tu x , tx , tx
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The Linear Elastic Problem is governed by the equations:1. Cauchy’s Equation of Motion.
Linear Momentum Balance Equation.
2. Constitutive Equation. Isotropic Linear Elastic Constitutive Equation.
3. Geometrical Equation. Kinematic Compatibility.
Governing Equations
2
0 0 2
,, ,
tt t
t
u x
x b x
, 2t Tr x 1
1, ,2
St t x u x u u
This is a PDE system of 15 eqns -15 unknowns:
Which must be solved in the space.
, tu x , tx
, tx
3 unknowns
6 unknowns
6 unknowns
3R R
6
For some problems, one of the principal stress directions is known a priori: Due to particular geometries, loading and boundary conditions
involved. The elastic problem can be solved independently for this direction. Setting the known direction as z, the elastic problem analysis is
reduced to the x-y plane
There are two main classes of plane linear elastic problems: Plane stress Plane strain
Plane Linear Elasticity
PLANE ELASTICITY
REMARK The isothermal case will be studied here for the sake of simplicity. Generalization of the results obtained to thermo-elasticity is straight-forward.
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Ch.7. Plane Linear Elasticity
7.2 Plane Stress
Simplifying hypothesis of a plane stress linear elastic problem:
1. Only stresses “contained in the x-y plane” are not null
2. The stress are independent of the z direction.
Hypothesis on the Stress Tensor
00
0 0 0
x xy
xy yxyz
, ,
, ,
, ,
x x
y y
xy xy
x y t
x y t
x y t
REMARK The name “plane stress” arises from the fact that all (not null) stress are contained in the x-y plane.
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These hypothesis are valid when:
The thickness is much smaller than the typical dimension associated to the plane of analysis:
The actions , and are contained in the plane of analysis (in-plane actions) and independent of the third dimension, z.
is only non-zero on the contour of the body’s thickness:
Geometry and Actions in Plane Stress
e L
, tb x * , tu x * , tt x
* , tt x
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The strain field is obtained from the inverse Hooke’s Law:
As
And the strain tensor for plane stress is:
Strain Field in Plane Stress
, ,x x x y t
, ,y y x y t , ,x y t
1 02
1, , 02
0 0
x xy
xy y
z
x y t
with 1z x y
000
1
zxzyz
TrE E
1
1 1 2(1 )2
1 1 021 02
x x y xy xy xy
y y x xz xz
z x y yz yx
E E
E
E
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Operating on the result yields:
Constitutive equation in Plane Stress
planestress C
2
1 01 0
110 0
2
x x
y y
xy xy
E
planestress C
Constitutive equation in plane stress
(Voigt’s notation)
1 2(1 )2
1 2 0
2 0
x x y xy xy xy
y y x xz xz
z x y yz yx
E E
E
E
2
2
1
1
2 1
x x y
y y x
xy xy
E
E
E
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The displacement field is obtained from the geometric equations,. These are split into:
Those which do not affect the displacement :
Those in which appears:
Displacement Field in Plane Stress
, ,St tx u x
zu
zu , , x
xux y tx
, , yy
ux y t
y
, , 2 yxxy xy
uux y ty x
, ,, ,
x x
y y
u u x y tu u x y t
Integration in .
Contradiction !!!
0
0
, , ( , , ) ( , , , )1
( , ), , 2 0
( , )( , ), , 2 0
zz x y z
x z zxz xz
zy z z
yz yz
ux y t x y t u x y z tz
u x y u ux y tz x x
u z tu x y u ux y tz y y
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The problem can be reduced to the two dimensions of the plane of analysis. The unknowns are:
The additional unknowns (with respect to the general problem) are either null, or independently obtained, or irrelevant:
The Lineal Elastic Problem in Plane Stress
1z x y
0z xz xz xz yz
, , ,zu x y z t does not intervene in the problem
, , x
y
ux y t
u
u , ,x
y
xy
x y t
, ,x
y
xy
x y t
REMARK This is an ideal elastic problem because it cannot be exactly reproduced as a particular case of the 3D elastic problem. There is no guarantee that the solution to and
will allow obtaining the solution to for the additional geometric eqns.
, ,yu x y t , ,xu x y t
, , ,zu x y z t
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3D problems which are typically assimilated to a plane stress state are characterized by: One of the body’s dimensions is significantly smaller than the other two. The actions are contained in the plane formed by the two “large” dimensions.
Examples of Plane Stress Analysis
Slab loaded on the mean plane Deep beam
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Ch.7. Plane Linear Elasticity
7.3 Plane Strain
Simplifying hypothesis of a plane strain linear elastic problem:
1. The displacement field is
2. The displacement variables associated to the x-y plane are independent of the z direction.
Hypothesis on the Displacement Field
0
x
y
uu
u
, ,
, ,x x
y y
u u x y t
u u x y t
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These hypothesis are valid when:
The body being studied is generated by moving the plane of analysis along a generational line.
The actions , and are contained in the plane of analysis and independent of the third dimension, z.
In the central section, considered as the “analysis section” the following holds (approximately) true:
Geometry and Actions in Plane Strain
, tb x * , tu x * , tt x
0zu
0xuz
0yuz
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The strain field is obtained from the geometric equations:
And the strain tensor for plane strain is:
Strain Field in Plane Strain
1 02
1, , 02
0 0 0
x xy
xy yx y t
REMARK The name “plane strain” arises from the fact that all strain is contained in the x-y plane.
, ,, , 0
, , , ,, , 0
, , , ,, ,, , 0
x zx z
y x zy xz
y yx zxy yz
u x y t ux y tx z
u x y t u x y t ux y ty z x
u x y t u x y tu x y t ux y ty x z y
0zu
0xuz
0yuz
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Introducing the strain tensor into Hooke’s Law and operating on the result yields:
As
And the stress tensor for plane strain is:
Stress Field in Plane Strain
Tr G 1
2
2 0
( ) 0
x x y x xy xy
y x y y xz xz
z x y x y yz yz
G G
G G
v G
2G x y
2 y xG
, ,x y t
, ,x x x y t
, ,y y x y t
, ,z z x y t
, ,xy xy x y t
0
, , 00 0
x xy
xy y
z
x y t
with z x y
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Introducing the strain tensor into the constitutive equation and operating on the result yields:
Constitutive equation in Plane Strain
planestrain C
1 01
11 0
1 1 2 11 20 0
2 1
x x
y y
xy xy
E
planestrain C
Constitutive equation in plane strain
(Voigt’s notation)
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1 1 2 1x x y x y
EG
2 1xy xy xyEG
12
1 1 2 1y y x y x
EG
2Tr 1
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The problem can be reduced to the two dimensions of the plane of analysis. The unknowns are:
The additional unknowns (with respect to the general problem) are either null or obtained from the unknowns of the problem:
The Lineal Elastic Problem in Plane Strain (summary)
0z xz yz xz yz
0zu
z x y
, , x
y
ux y t
u
u , ,x
y
xy
x y t
, ,x
y
xy
x y t
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3D problems which are typically assimilated to a plane strain state are characterized by: The body is generated by translating a generational section with actions
contained in its plane along a line perpendicular to this plane. The plane stress hypothesis must be justifiable. This typically
occurs when:1. One of the body’s dimensions is significantly larger than the other two.
Any section not close to the extremes can be considered a symmetry plane and satisfies:
2. The displacement in z is blocked at the extreme sections.
Examples of Plane Strain Analysis
0z xz yz
0zu
0xuz
0yuz
0
x
y
uu
u
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3D problems which are typically assimilated to a plane strain state are:
Examples of Plane Strain Analysis
Pressure pipe
Tunnel
Continuous brake shoe
Solid with blocked zdisplacements at the ends
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Ch.7. Plane Linear Elasticity
7.4 The Plane Linear Elastic Problem
A lineal elastic solid is subjected to body forces and prescribed traction and displacement
The Plane Linear Elastic problem is the set of equations that allow obtaining the evolution through time of the corresponding displacements , strains and stresses .
Plane problem
Actions:
, ,x y tu , ,x y t , ,x y t
*
*
, ,
, ,x
y
t x y t
t x y t
*t
*
*
, ,
, ,x
y
u x y t
u x y t
*u
On :
On :u
On :
, ,, ,
x
y
b x y tb x y t
b
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The Plane Linear Elastic Problem is governed by the equations:1. Cauchy’s Equation of Motion.
Linear Momentum Balance Equation.
Governing Equations
2
0 0 2
,, ,
tt t
t
u x
x b x
2D
2
2
2
2
2
2
xyx xz xx
xy y yz yy
yzxz z zz
ubx y z t
ub
x y z tub
x y z t
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The Plane Linear Elastic Problem is governed by the equations:2. Constitutive Equation (Voigt’s notation).
Isotropic Linear Elastic Constitutive Equation.
Governing Equations
, :t x C
2D
C
x
y
xy
x
y
xy
With , and
2
1 01 0
10 0 1 2
E
C
E E
PLANE STRESS
21
1
EE
PLANE STRAIN
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The Plane Linear Elastic Problem is governed by the equations:3. Geometrical Equation.
Kinematic Compatibility.
Governing Equations
1, ,2
St t x u x u u
This is a PDE system of 8 eqns -8 unknowns:
Which must be solved in the space.
, tu x , tx
, tx
2 unknowns
3 unknowns
3 unknowns
2
2D
xx
yy
yxxy
uxuy
uuy x
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Boundary conditions in space Affect the spatial arguments of the unknowns Are applied on the contour of the solid, which is divided into:
Prescribed displacements on :
Prescribed stresses on :
Boundary Conditions
u
* **
* *
, ,
, ,x x
y y
u u x y t
u u x y t
u
* **
* *
, ,
, ,x x
y y
t t x y t
t t x y t
t * t n
x
y
nn
n
x xy
xy y
with
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INTIAL CONDITIONS (boundary conditions in time) Affect the time argument of the unknowns. Generally, they are the known values at :
Initial displacements:
Initial velocity:
Boundary Conditions
0t
, ,0 x
y
ux y
u
u 0
00
v, ,, ,0 ,
vx x
y yt
ux y tx y x y
ut
uu v
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The 8 unknowns to be solved in the problem are:
Once these are obtained, the following are calculated explicitly:
Unknowns
, , x xy
xy y
x y t
12, ,
12
x xy
xy y
x y t
( , , ) x
y
ux y t
u
u
PLANE STRESS
PLANE STRAIN
1z x y
z x y
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Ch.7. Plane Linear Elasticity
7.5 Representative Curves
Traditionally, plane stress states where graphically represented with the aid of the following contour lines: Isostatics or stress trajectories Isoclines Isobars Maximum shear lines Others: isochromatics, isopatchs, etc.
Introduction
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System of curves which are tangent to the principal axes of stress at each material point . They are the envelopes of the principal stress vector fields. There will exist two families of curves at each point:
Isostatics , tangents to the largest principal stress. Isostatics , tangents to the smallest principal stress.
Isostatics or Stress Trajectories
1
2
REMARK The principal stresses are orthogonal to each other, therefore, so will the two families of isostatics orthogonal to each other.
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Singular point: characterized by the stress state
Neutral point: characterized by the stress state
Singular and Neutral Points
Mohr’s Circle of a neutral point
Mohr’s Circle of a singular point
0x y xy
0x y
xy
REMARK In a singular point, all directions are principal directions. Thus, in singular points isostatics tend to loose their regularity and can abruptly change direction.
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Consider the general equation of an isostatic curve:
Solving the 2nd order eq.:
Differential Equation of the Isostatics
y f x
2
2 21
xy
x y
yy
2 1 0x y
xy
y y
2
2 2 tgtg 21 tg
tg
xy
x y
dy yd x
2
' 12 2x y x y
xy xy
y
Differential equation of the isostatics
,x y Known this function, the eq. can be integrated to obtain a family of curves of the type:
y f x C
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Locus of the points along which the principal stresses are in the same direction. The principal stress vectors in all points of an isocline are parallel to
each other, forming a constant angle with the x-axis.
These curves can be directly found using photoelasticity methods.
Isoclines
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To obtain the general equation of an isocline with angle , the principal stress must form an angle with the x-axis:
Equation of the Isoclines
1
2
tg 2 xy
x y
Algebraic equation
of the isoclines
,x y
For each value of , the equation of the family of isoclines parameterized in function of is obtained:
,y f x
REMARK Once the family of isoclines is known, the principal stress directions in any point of the medium can be obtained and, thus, the isostatics calculated.
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Locus of the points along which the principal stress ( or ) is constant. The isobars depend on the value of the principal stress but not on
their direction. There will exist two families of isobars at each point, and .
Isobars
1 2
1 2
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Take the equation used to find the principal stresses and principal stress directions given in a certain set of axes:
Equation of the Isobars
Algebraic equation of the isobars
22
1 1 1
22
2 2 2
,2 2
,2 2
x y x yxy
x y x yxy
x y cnt c
x y cnt c
1 1 1
2 2 2
,
,
y f x c
y f x c
This eq. implicitly defines the family of curves of the isobars:
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Envelopes of the maximum shear stress (in modulus) vector fields. They are the curves on which the shear stress modulus is a maximum. Two planes of maximum shear stress correspond to each material
point, and . These planes are easily determined using Mohr’s Circle.
Maximum shear lines
minmax
REMARK The two planes form a 45º angle with the principal stress directions and, thus, are orthogonal to each other. They form an angle of 45º with the isostatics.
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Consider the general equation of a slip line , the relation and
Then,
Equation of the maximum shear lines
4 1tan 2 tan 2
2 tan 2
2tan 2 xy
x y
y f x
2
1 2 tantan 2tan 2 2 1 tan
tan
x y
xy
notdy yd x
22
2 1x y
xy
yy
2 41 0xy
x y
y y
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Solving the 2nd order eq.:
Equation of the maximum shear lines
Differential equation of the
slip lines
22 2
' 1xy xy
x y x y
y
,x y Known this function, the eq. can be integrated to obtain a family of curves of the type:
y f x C
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Ch.7. Plane Linear Elasticity
Summary
For some problems the elastic problem can be solved independently for one of the directions. The analysis is reduced to a plane.
Two problem types:1. Plane stress
Summary
PLANE ELASTICITY
00
0 0 0
x xy
xy yxyz
, ,
, ,
, ,
x x
y y
xy xy
x y t
x y t
x y t
e L
1 02
1, , 02
0 0
x xy
xy y
z
x y t
with
1z x y
The displacement field is obtained from
, ,St tx u x
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2. Plane strain
Summary (cont’d)
0
x
y
uu
u
, ,
, ,x x
y y
u u x y t
u u x y t
In the central section,
0zu
0xuz
0yuz
1 02
1, , 02
0 0 0
x xy
xy yx y t
0
, , 00 0
x xy
xy y
z
x y t
with z x y
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For both problem types the unknowns are:
The additional unknowns (w.r.t. the general problem) are either null, independently obtained or irrelevant:
Summary (cont’d)
1z x y
0z xz xz xz yz
, , ,zu x y z t does not intervene in the problem
, , x
y
ux y t
u
u , ,x
y
xy
x y t
, ,x
y
xy
x y t
z x y
0z xz yz xz yz
0zu
PLANE STRAINPLANE STRESS
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The PLANE Linear Elastic Problem:
Summary (cont’d)
Actions:
*
*
, ,
, ,x
y
t x y t
t x y t
*t
*
*
, ,
, ,x
y
u x y t
u x y t
*u
On :
On :u
On :
, ,, ,
x
y
b x y tb x y t
b
Responses:
, tu x , tx
, tx
2 unknowns
3 unknowns
3 unknowns
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The PLANE Isotropic Linear Elastic Problem:
Summary (cont’d)
Cauchy’s Equation of Motion
Constitutive Equation
2
2
2
2
xyx xx
xy y yy
ubx y t
ub
x y t
Geometric Equation
C
2
1 01 0
10 0 1 2
E
C
;E E PLANE STRESS: 2 ;
1 1EE
PLANE STRAIN:
with
; ;y yx xx y xy
u uu ux y y x
This is a PDE system of 8 eqns -8 unknowns:
Which must be solved in the space.
, tu x , tx
, tx
2 unknowns
3 unknowns
3 unknowns
2R R
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The PLANE Isotropic Linear Elastic Problem (cont’d):
Summary (cont’d)
Boundary Conditions in Space
Initial Conditions
:u
* **
* *
, ,
, ,x x
y y
u u x y t
u u x y t
u
:
* **
* *
, ,
, ,x x
y y
t t x y t
t t x y t
t* t nwith
, ,0x y u 0
00
, ,, ,0 ,
not
t
x y tx y x y
t
u
u v
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Plane stress states can be graphically represented with the curves:
Isostatics or stress trajectories: curves which are at each material point tangent to the principal axes of stress.
Isoclines: locus of the points along which the principal stresses are in the same direction. (Determined using photoelasticity methods.)
Isobars: locus of the points along which the principal stress is constant.
Maximum shear lines: envelopes of the maximum (in modulus) shear stress vector fields.
Isochromatics: curves along which the maximum shear stress is constant. (Determined using photoelasticity methods.)
Isopachs: curves along which the mean normal stress is constant.
Summary (cont’d)
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