This paper presents a new approach based on deformation
reinforcement theory [ 22 ]toevaluatecracksgrowthinthree-
dimensional structure. Unbalanced force is proposed as a
set of equivalent nodal forces of overstress beyond the
yield surface. Unbalanced force drives time-dependent defor-
mation, as well as damage evolution. The distribution of
unbalanced force indicates crack initiation area, while its
direction predicts potential crack propagation path. Uniaxial
compression test of precrack specimen is performed, which is
in a good agreement with the numerical results. The method
isalsoappliedinfractureanalysisofXiaowanhigharchdam.
2. Fracture Analysis Method Based on
Unbalanced Force
2.1. Deformation Reinforcement Theory.Most geotechnical
structures are under complicated configurations and working
conditions. Stability analysis of these engineering structure
couldbesummarizedasaboundaryvalueproblemwithsome
basic equations, including kinematic admissibility, equilib-
rium condition, and constitutive equations. The classical
elastoplastic theory aims at solving the displacement and
stress fields that simultaneously satisfy all the aforementioned
equations. However, the existence of such solution requires
that the structure is stable. Structural instability occurs when
action is greater than resistance. The difference between
action and resistance is overstress, which is the key concept of
the Deformation Reinforcement Theory (DRT) and defined
as the unbalanced force.
For perfect elasto-plastic materials with associative flow
rule, the constitutive equations can be stated as
휀̇=휀̇푒+휀̇푝, 휀̇푒=C:휎̇;퐷푝(휎,휀̇푝)≥퐷푝(휎yc,휀̇푝), ∀푓(휎yc)≤0,(1)
where퐷푝isdefinedastheplasticdissipation,퐷푝(휎yc,휀̇푝):=
휎yc:휀̇푝.휀̇푒and휀̇푝are elastic and plastic strain rates, respec-
tively.Cis the flexibility tensor.푓is the yield function and
휎ycis an arbitrary stress state satisfying the yield criterion.
Inequality ( 1 ) is known as principle of maximum plastic
dissipation [ 23 ], which covers associative flow rule and the
Kuhn-Tucker condition.
Assuming that푡is a pseudotime, the stress and plastic
strain of material are휎 0 and휀푝 0 at time푡 0. With a strain incre-
ment in time intervalΔ푡,anewstate휎and휀푝is achieved.
Thusthelinearizedplasticstrainrateis
휀̇푝=Δ휀푝
Δ푡
=
Δ휀−C:(휎−휎 0 )
Δ푡
=
C:(휎eq−휎)
Δ푡, (2)
where휎eqis termed the trial elastic stress while휀푝̇ =0.휎eqis
also an equilibrium stress field under given loads;
휎eq=휎 0 +C−1:Δ휀. (3)
Applying ( 2 )intoInequality( 1 ), the process of solving
real stress field is turned into the following minimization
problem:
min퐸(휎yc), ∀푓(휎yc)≤0,퐸(휎yc)=1
2
(휎eq−휎yc):C:(휎eq−휎yc).(4)
Equation ( 4 ) is known as the closest-point projection
method (CPPM) [ 24 ], as shown inFigure 1.휎ycis an arbitrary
stress field on the yielding surface, which represents the
material resistance after previous minimization process.휎eqis
an equilibrium stress field, which could be regarded as certain
stress under external actions.
The minimization objective퐸represents the difference
between the plastic dissipations of the external action and the
material resistance. It is defined as the volume density of the
plastic complementary energy (PCE);퐸=퐷푝(휎eq,Δ휀푝)−퐷푝(휎yc,Δ휀푝). (5)Thus, instability of a material point is equal to the
statement that the external action is greater than the material
resistance:퐸>0,forall푓(휎yc)≤0.Asindicatedbythe
minimization problem, if휎eq > 휎yc, the real stress state
minimizes the resistance deficiency in the sense of PCE. In
other words, material resistance capacity is fully developed.
Drucker-Prager yield criterion is adopted in the following
nonlinear calculation:푓(휎)=훼퐼 1 +√퐽 2 −퐻, (6)where퐼 1 = tr휎,퐽 2 = 휎 : 휎/2 + 퐼 1 /6,and훼and퐻are
material parameters. With the real stress field휎solved by ( 4 ),
the following characteristic can be proved [ 22 ]:휕푓
휕휎儨儨
儨儨儨
儨儨儨휎=휎=
휕푓
휕휎
儨儨
儨儨儨
儨儨儨휎=휎eq. (7)Eventuallythefollowinganalyticalsolutionof휎can be
derived:휎=(1−푛)휎eq+푝I,푛=
푤퐺
√퐽 2
,푝=−푚푤+3푛퐼 1 ,
푚=훼(3휆 + 2퐺),푤=
푓
(3훼푚 + 퐺)
,
(8)
where퐼 1 and퐽 2 are invariants of휎eqand휆and퐺are the Lame ́
constants.2.2. Demonstration in FEM.PCE and unbalanced force are
implemented in FEM analysis. Since the process is restricted
to displacement method, the kinematic admissibility is natu-
rally satisfied.
In order to evaluate the global stability of structure under
certain actions, two stress field sets,푆eqand푆yc,aredefinedfor
elastoperfectplasticproblem.Thetwostresssets,respectively,
satisfy the equilibrium condition and the yield criterion as푆eq={휎eq|F=∫
푉B푇휎eq푑푉},푆yc={휎yc|푓(휎yc)≤0},(9)
whereBis the displacement gradient matrix,Fis equivalent
nodal force vector of external loads, and푉is structure