Constitutive Relations in Solids Elasticity

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Constitutive Relations in SolidsElasticity

• H. Garmestani, Professor
• School of Materials Science and Engineering
• Georgia Institute of Technology
• Outline
• Materials Behavior
• Tensile behavior

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The Elastic Solid and Elastic Boundary Value
Problems

• Constitutive equation is the relation between
  kinetics (stress, stress-rate) quantities and
  kinematics (strain, strain-rate) quantities for a
  specific material. It is a mathematical
  description of the actual behavior of a material.
  The same material may exhibit different behavior
  at different temperatures, rates of loading and
  duration of loading time.). Though researchers
  always attempt to widen the range of temperature,
  strain rate and time, every model has a given
  range of applicability.
• Constitutive equations distinguish between solids
  and liquids and between different solids.
• In solids, we have Metals, polymers, wood,
  ceramics, composites, concrete, soils
• In fluids we have Water, oil air, reactive and
  inert gases

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The Elastic Solid and Elastic Boundary Value
Problems (cont.)
Load-displacement response
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Examples of Materials Behavior
Uniaxial loading-unloading stress-strain curves
for (a) linear elastic (b) nonlinear elastic
and (c) inelastic behavior.
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Constitutive Equations Elastic

• Elastic behavior is characterized by the
  following two conditions
• (1) where the stress in a material (?) is a
  unique function of the strain (?),
• (2) where the material has the property for
  complete recovery to a natural shape upon
  removal of the applied forces
• Elastic behavior may be Linear or non-linear

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Constitutive Equation

• The constitutive equation for elastic behavior in
  its most general form as

where C is a symmetric tensor-valued function and
e is a strain tensor we introduced earlier.
Linear elastic ? Ce Nonlinear-elastic
? C(e) e
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Equations of Infinitesimal Theory of Elasticity
 Boundary Value Problems  we assume that the
strain is small and there is no rigid body
rotation. Further we assume that the material is
governed by linear elastic isotropic material
model.   Field Equations (1) (2) Stress
Strain Relations (3)Cauchy Traction Conditions
(Cauchy Formula) (4)
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Equations of the Infinitesimal Theory of
Elasticity (Cont'd)
In general, We know that For small
displacement Thus
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Equations of the Infinitesimal Theory of
Elasticity (Cont'd)
Assume v ltlt 1, then For small
displacement, Thus for small
displacement/rotation problem
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Equations of the Infinitesimal Theory of
Elasticity (Cont'd)
Consider a Hookean elastic solid, then Thus,
equation of equilibrium becomes
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Equations of the Infinitesimal Theory of
Elasticity (Cont'd)
For static Equilibrium
Then The above equations are called
Navier's equations of motion. In terms of
displacement components
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Plane Elasticity
In a number of engineering applications, the
geometry of the body and loading allow us to
model the problem using 2-D approximation. Such a
study is called ''Plane elasticity''.  There are
two categories of plane elasticity, plane stress
and plane strain. After these, we will study two
special case simple extension and torsion of a
circular cylinder.
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Plane Strain Plane Stress
For plane stress, (a)    Thus equilibrium
equation reduces to (b)   
Strain-displacement relations are (c)   
With the compatibility conditions,
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Plane Strain Plane Stress
(d)     Constitutive law becomes, Inverting the
left relations, Thus the equations in
the matrix form become (e)    In terms of
displacements (Navier's equation)
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Plane Strain (b) (Cont'd)
(b)     Inverting the relations,              
can be written as
(c)      Navier's equation for displacement can
be written as
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The Elastic Solid and Elastic Boundary Value
Problems
 Relationship between kinetics (stress, stress
rate) and kinematics (strain, strain-rate)
determines constitutive properties of
materials. Internal constitution describes the
material's response to external thermo-mechanical
conditions. This is what distinguishes between
fluids and solids, and between solids wood from
platinum and plastics from ceramics.   Elastic
solid Uniaxial test The test often used to
get the mechanical properties
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Linear Elastic Solid
If      is Cauchy tensor and       is small
strain tensor, then in general,
where         is a fourth order tensor, since
T and E are second order tensors. is
called elasticity tensor. The values of these
components with respect to the primed basis ei
and the unprimed basis ei are related by the
transformation law
However, we know that                   and    
                 then
We have          symmetric matrix with 36
constants, If elasticity is a unique scalar
function of stress and strain, strain energy is
given by
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Linear Elastic Solid

• Show that if for a linearly
  elastic solid, then
• Solution
• Since for linearly elastic solid
  , therefore
• Thus from , we have
• Now, since
• Therefore,

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Linear Elastic Solid (cont.)
Now consider that there is one plane of symmetry
(monoclinic) material, then     One plane of
symmetry gt     13 If there are 3 planes
of symmetry, it is called an ORTHOTROPIC
material, then     orthortropy gt  3 planes of
symmetry gt   9 Where there is isotropy in a
single plane, then     Planar isotropy   gt
     5 When the material is completely
isotropic (no dependence on orientation)    
Isotropic  gt      2
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Linear Elastic Solid (cont.)
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Linear Isotropic Solid

• A material is isotropic if its mechanical
  properties are independent of direction
• Isotropy means
• Note that the isotropy of a tensor is equivalent
  to the isotropy of a material defined by the
  tensor.
• Most general form of (Fourth order)
  is a function

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Linear Isotropic Solid

• Thus for isotropic material
•  
• and are called Lame's constants.
• is also the shear modulus of the material
  (sometimes designated as G).

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Relationship between Youngs Modulus EY, Poisson's
Ratio g, Shear modulus mG and Bulk Modulus k
We know that So we have Also, we have
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Relationship between  EY, g, mG and k  (Cont'd)
Note Lames constants, the Youngs modulus, the
shear modulus, the Poissons ratio and the bulk
modulus are all interrelated. Only two of them
are independent for a linear, elastic isotropic
materials,