مقاله ی زیر روبه صورت تخصصی برام ترجمه کن 1
Background on Continuum Damage Mechanics
This chapter gives the reader the necessary and sufficient information to deal
with practical applications of Continuum Damage Mechanics (the term was
first introduced by J. Hult in 1972) to predict the crack initiation in structures
subjected to heavy loadings. This ranges from the early definition of a
scalar damage variable by L.M. Kachanov in 1958 to the anisotropic tensorial
damage variable, from classical thermodynamics aspects to the effective
stress that take into account the closure of microdefects in compressionlike
loadings, from simple measurement of local damage to fields of damage,
from a simple damage law to induced anisotropy and a generalization
to quasi-brittle materials, from elasto-(visco-)plasticity coupled with damage
to the phenomenon of localization, the ultimate stage before crack
initiation.
The basic concepts are written in a concise and sufficiently general way in
order to be applied to any material: metals and alloys, polymers, elastomers,
composites, concretes, ceramics . . . For more basic details please refer to
the book A Course on Damage Mechanics written by Jean Lemaitre and
published by Springer-Verlag in 1992 and 1996. Other (good!) references are
given at the end of the book.
1.1 Physics and Damage Variables
Damage, in its mechanical sense in solid materials is the creation and growth
of microvoids or microcracks which are discontinuities in a medium considered
as continuous at a larger scale. In engineering, the mechanics of continuous
media introduces a Representative Volume Element (RVE) on which
all properties are represented by homogenized variables. To give an order of
magnitude, its size can vary from about (0.1 mm)3 for metals and ceramics to
about (100 mm)3 for concrete. The damage discontinuities are “small” with
respect to the size of the RVE but of course large compared to the atomic
spacing (Fig. 1.1).
2 1 Background on Continuum Damage Mechanics
From a physical point of view, damage is always related to plastic or
irreversible strains and more generally to a strain dissipation either on the
mesoscale, the scale of the RVE, or on the microscale, the scale of the discontinuities:
Fig. 1.1. Examples of damage in a metal (left, micro-cavities in copper from L. Engel
and H. Klingele), in a composite (middle, microcracks in carbon-fiber/epoxy
resin laminate from O. Allix), and in a concrete (right, crack pattern in a ASTM I
cement, 4.8 and 9.5 mm aggregates, petrography from S.P. Shah)
• In the first case (mesolevel), the damage is called ductile damage if it is
nucleation and growth of cavities in a mesofield of plastic strains under
static loadings; it is called creep damage when it occurs at elevated temperature
and is represented by intergranular decohesions in metals; it is
called low cycle fatigue damage when it occurs under repeated high level
loadings, inducing mesoplasticity.
• In the second case (microlevel), it is called brittle failure, or quasi-brittle
damage, when the loading is monotonic; it is called high cycle fatigue
damage when the loading is a large number of repeated cycles. Ceramics,
concrete, and metals under repeated loads at low level below the yield
stress are subjected to quasi-brittle damage.
In all cases, these are volume defects such as microcavities, or surface defects
such as microcracks. This is the reason to have several definitions of a damage
variable:
• If only ductile damage is considered, it may be defined as the volume
density of microvoids,
Dv = δVvoids
δVRVE
= fv . (1.1)
This is the starting point of the Gurson model described in Sect. 3.4.5.
1.1 Physics and Damage Variables 3
• For a larger generality where microcavities and microcracks may exist, the
damage variable is physically defined by the surface density of microcracks
and intersections of microvoids lying on a plane cutting the RVE of cross
section δS (Fig. 1.2). For the plane with normal n where this density is
maximum, we have
D(n) = δSD
δS
. (1.2)
Fig. 1.2. Physical damage and mathematical continuous damage
1.1.1 Definition of a Scalar Damage Variable
If the damage is isotropic, the scalar variable D(n) does not depend on the
normal. The intrinsic variable is a scalar (L.M. Kachanov 1958):
D = δSD
δS
(1.3)
It can be used as such for one-dimensional problems. It can also be used as
an easy evaluation of the approximate damage in three-dimensional problems
(particularly in proportional loading).
1.1.2 Definition of Several Scalar Damage Variables
If several mechanisms of damage occur, simultaneously or not, each of them
may be represented by a scalar variable (Dk) with the same physical meaning
as above. This is the case in classical composites where delamination of fibers
and matrix cracking may occur. Two or three independent variables are used
(P. Ladev`eze 1983), for instance:
• DF for quasi-brittle fibers breakage,
• DT for transverse matrix cracking, and
• DS acting on shear for splitting.
4 1 Background on Continuum Damage Mechanics
1.1.3 Definition of a Tensorial Damage Variable
Damage is often non-isotropic due to microcracking more or less perpendicular
to the largest positive principal stress. Then, the surface density of
microdefects in a plane with normal n acts through an operator which transforms
the surface δS and n of Fig. 1.2 into a smaller but continuous area,
δ ˜ S = δS − δSD, and into another normal ˜n (S. Murakami 1981).
To keep the same physical meaning as above the damage acts through the
operator (1 −D) and
(δij − Dij) njδS = ˜niδ ˜ S , (1.4)
where δij is the Kronecker delta andD is a second order tensor. As there is no
distorsion of the surface δS, it induces the property of orthotropy consistent
with the fact that the damage is governed by the plastic strain represented
by the second order tensor p.
Remark – In fact, the largest generality for a damage variable is a representation
by a fourth order tensor, as it can be shown in several ways (J.L. Chaboche 1978,
D. Krajcinovic 1981, F.A. Leckie and E.T. Onat 1981, C.L. Chow 1987). Such
a tensor is difficult to use and is not necessary for damage-induced by meso- or
microplasticity.
• As in the previous section, consider a damage plane area, δS, with normal
n and a reference vector ν such that the tensor νinjδS defines the geometrical
reference configuration. Continuum Damage Mechanics defines the effective
continuous configuration by a modified area δ ˜ S and a modified normal ˜n, as
shown in Fig. 1.3.
The damage D is the operator which transforms the second order tensor νinjδS
of the reference configuration into the tensor of the effective configuration
νi˜njδ ˜ S. This is a fourth order tensor, where
(Iijkl − Dijkl)νknlδS = νi˜njδ ˜S , (1.5)
with the following symmetries: Dijkl = Dijlk = Djikl = Dklij .
Fig. 1.3. Reference and effective configurations
1.1 Physics and Damage Variables 5
• Another way to define a fourth order tensor D is to consider the operator which
changes the initial elasticity tensor Eijkl into the actual elasticity tensor ˜E ijkl
softened by damage:
(Iijrs − Dijrs)Erskl = ˜Eijkl . (1.6)
From a purely theoretical point of view, this definition does not yield a real state
variable because it requires the knowledge of a particular behavior (elasticity
here).
1.1.4 Effective Stress Concept
The constitutive equations for strain and damage characterize the plain material
itself without the volume or the surface discontinuities. An interesting
concept to introduce next is the stress acting on the resisting area,
δ ˜ S = δS − δSD, as shown in Fig. 1.2 – the effective stress.
• In the uniaxial case of isotropic damage without the closure effect of
microcracks in compression, this mean value of the microstresses is simply
given by the force equilibrium (Y.N. Rabotnov 1968):
˜σδ ˜ S = σδS with D = δSD
δS
= δS − δ ˜ S
δS
(1.7)
or
˜σ = σ
1 − D
. (1.8)
• In the multiaxial case of isotropic damage, all the stress components act
on the same effective area. The effective stress tensor is simply
˜σij = σij
1 − D
. (1.9)
• The case of anisotropic damage is much more complicated to ensure
a good representation of the physics as well as compatibility with
thermodynamics. In fact, the effective stress with a second order damage
tensor representation is an approximation of the exact effective stress deduced
from the general representation of the damage by the fourth order
tensor D, as defined in the remark in the previous section. This effective
stress is defined as previously, but, here, by the projection of the stress
vector on the reference vector ν in Fig. 1.3:
νi˜σij ˜njδ ˜ S = νiσijnjδS (1.10)
or
˜σij (Iijkl − Dijkl)νknlδS = σklνknlδS (1.11)
using the preceeding equation (1.5).
6 1 Background on Continuum Damage Mechanics
With the symmetry properties of the damage tensor Dijkl given in
Sect. 1.1.3, the effective stress is the symmetric tensor,
˜σij = σkl (I −D)−1
klij , (1.12)
which may also be written as
˜σij = Mijklσkl , with Mijkl = (I −D)−1
klij . (1.13)
Back to a second order damage tensor, there are several possibilities to
write a restriction of the fourth order formulation but only one fulfills the
following conditions:
– Symmetry of the effective stress (note that ˜σij = σik(1 −D)−1
kj is not
symmetric!).
– An effective stress independent of the strain behavior and, in particular,
independent from the Poisson ratio (˜σij = Eijkl ˜E
−1
klrsσrs may
depend on the Poisson ratio).
– Compatibility with the thermodynamics framework: existence of strain
potentials and principle of strain equivalence. The symmetrization
˜σij = 1
2

σik(1 −D)−1
kj + (1 −D)−1
ik σkj

is not derived from a potential
(J.P. Cordebois and F. Sidoroff 1979).
– Different effect of the damage on the hydrostatic behavior represented
by the stress (σH) and on the deviatoric part (σD
ij) by means of a hydrostatic
sensitivity parameter (η) that is easy to identify.
This restriction is represented by
˜σij = (HikσD
klHlj)D + σH
1 − ηDH
δij , with Hij = (1 −D)−1/2
ij , (1.14)
where DH = 1
3Dkk is the hydrostatic damage and H is the effective
damage tensor. The corresponding fourth order tensor Mijkl is
Mijkl = HikHlj − 1
3

H2
klδij + H2
ijδkl

+
1
9H2
ppδijδkl +
1
3(1 − ηDH) δijδkl .
(1.15)
For the uniaxial case, the effective von Mises stress (˜σeq) differs from the
effective tensile stress (˜σ1) (assumed here in direction 1) as follows:
˜σeq =
2
3
σ
1 − D1
+
1
3
σ
1 − D2
˜σ1 =
4
9
σ
1 − D1
+
2
9
σ
1 − D2
+
1
3
σ
1 − ηDH
,
(1.16)
where D3 = D2. The coefficient η does not depend very much on the
materials as most often η ≈ 3. The isotropic damage case corresponds to
Dij = Dδij and η = 1.
The effective stress taking into account the quasi-unilateral conditions of
microdefects closure is formulated in Sect. 1.2.4