MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS

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MECHANICS OF MATERIALS. EQUATIONS AND
THEOREMS
Version 2011-01-14
Stress tensor
Definition of traction vector
(1)
Cauchy theorem
(2)
Equilibrium
(3)
Invariants
(4)
(5)
(6)
or, written in terms of principal stresses,
(7)
(8)
(9)
Coordinate transformation
(10)
or, inverted
(11)
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Stress deviator
(12)
Stress deviator invariants
(13)
Displacement and strain
Definition of displacement
(14)
Definition of infinitesimal strain
(15)
‘Engineering’ shear
;
(16)
Voigt notation of stress and strain
Elastic anisotropy
A material is symmetric with respect to the transformation
→
if
(17)
or, using Voigt notation
(18)
3
where L is the transformation matrix for the transformation
→
in Voigt notation:
(19)
Stiffness (C) and ‘engineering’ compliance (S) matrices for
some important classes of materials
Linearly orthotropic material
(20)
and
(21)
In Eq. (21), the actual number of independent material constants is reduced to 9 by the relations
(22)
Transversely isotropic material (symmetry axis
)
(23)
and
4
(24)
In Eq. (23), the actual number of independent material constants is reduced to 5 by the relation
(25)
Plasticity. Yield criteria
Flow function and equivalent stress: von Mises
(26)
Flow function and equivalent stress: Tresca
(27)
Plasticity. Flow rules
General
(28)
(29)
(30)
Perfect plasticity
(31)
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Perfect plasticity, von Mises
(32)
where the equivalent plastic strain increment d
is defined as
(33)
Isotropic hardening, von Mises
(34)
or, in most cases,
(35)
where
(36)
(cf Eq. (31)). Flow rule:
(37)
or
(38)
Linear isotropic hardening, von Mises
(39)
Eqs. (37) and (38) can now be simplified into
(40)
and
(41)
(since
during plastic flow).
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In the uniaxial tensile test, during plastic flow
Kinematic hardening, von Mises
The hardening is described by a backstress
(42)
The flow rule is
(43)
Kinematic hardening, Prager/von Mises
Prager’s linear hypothesis:
(44)
This leads to a simplified expression for the flow rule:
(45)
In the uniaxial tensile test, during plastic flow
(Note the factor 3/2 in the denominator, which is a difference against the corresponding isotropic
uniaxial test!)
Plasticity. Computational aspects
Continuum tangent stiffness matrix
(46)
where
is the continuum tangent matrix.
has the following principal structure:
(47)
One common way of writing it in detail is
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(48)
Viscoplasticity
Additive decomposition:
(49)
Norton uniaxial creep law for stationary creep
(50)
Multiaxial creep laws
(51)
with
Stationary creep (von Mises/Norton/Odqvist)
(52)
Multiplicative isotropic hardening
(53)
Perzyna overstress model
(54)
Viscoelasticity
Maxwell material
(55)
(56)
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(57)
Kelvin material
(58)
(59)
(60)
Standard linear solid
(61)
(62)
(63)
Relaxation modulus
The Laplace transform
of the relaxation modulus can be computed from the Laplace transform
of the creep compliance as
(64)
Hereditary integrals
For a given stress history
, the strain response
can be computed as the hereditary integral
(65)
or, in the Laplace transform space,
(66)
For a given strain history
, the stress response
can be computed as the hereditary integral
9
(67)
or, in the Laplace transform space,
(68)
Multiaxial hereditary integral
(69)
or, split into a deviator (
and a bulk (
part
(70)
(71)
In analogy with the previous uniaxial hereditary integrals, these can also be written as Laplacetransformed equations [cf Eq. (68)]:
(72)
(73)
(74)
(be careful with the notations here:
is the full 4th order relaxation modulus tensor, while
shear modulus;
is the stress deviator, while is the Laplace space variable)
is the
These equations can be used together with ‘the viscoelastic correspondence principle’ for solving
multiaxial problems.
Damage
Isotropic damage postulate
(75)
which replaces
in the constitive laws. For instance, in linear elasticity:
(76)
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or, if
(77)
(78)
Elastic damage: evolution law
(79)
(
is the maximum experienced value of the largest principal strain during the elastic history,
a fracture strain, and
is a threshold strain.)
is
Plastic damage: evolution lkaw
(80)
where
is the critical damage
Creep damage: Kachanov damage evolution law:
(81)
in which
(82)
where
is the largest principal stress and
is von Mises equivalent stress.

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