COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
New Constitutive Expression of Thermo-Elasto-Plastic Impact Dynamic
Problems for Large-Scale Slab
J. X. Wang 1, 2, S. Wang 1
1
2
College of Architecture and Civil Eng., North China University of Technology, Beijing 100041, China
Key Lab for the Control of Exp Dis., Beijing Institute of Technology, Beijing, 100081, China
Email: [email protected]
Abstract With regard to three-dimensional special large-scale slab, large-scale slab will be provided with special
property. The control equations of stress-strain constitutive relationship are presented, and the evanescence impact
transient thermal loading or thermo-elasto-plastic is considered. For the sake of efficacious theoretics, the research
outcomes are efficacious. When in specific temperature, it incarnates essentiality for the researching on the thermoelasto-plastic evanescence impact problems for the three-dimensional large-scale slab. Thermodynamic forces driving
field singularities in thermo elasticity (or more complex constitutive descriptions) have been shown to belong to the
above-highlighted class of material forces. The singularity sets of interest (and the only ones in three- dimensional
space) are points, lines and walls (transition zones of physically non-zero thickness but viewed mathematically as
singular surfaces of zero thickness). In the case of brittle fracture (line of singularity viewed as a point, the crack tip, in
a planar problem) and the progress of discontinuity fronts (phase-transition fronts and shock waves) which are singular
surfaces of the first order in Hadamard’s classical classification) one shows exactly that dissipation is strictly related to
the power expanded by such forces in the irreversible progress of the singularity set. In this research, from the
concerned mechanics theoretic of large-scale slab, the constitutive relationship for thermo-elasto-plastic evanescence
impact problems will be established. Through analyzing the ultimateness, it brings forth that the evanescence impact
transient thermal loading has an important effect on the constitutive relationship performance of large-scale slab; it is
provided with ardent dynamical characteristic. The several special simulation connection of large-scale slab had been
found.
Key words: thermo-elasto-plastic; constitutive relationship; large-scale slab
INTRODUCTION
When the competitive particles distribute continually in molecule or atom size in space, they can be considered as
continuous material. Otherwise they should be expanded to large-scale slab. For instance, the numerical value reckons
in earthquake response, to define the stress more accurately, it will begin the discussion by taking a free body from
large-scale slab. In respect that the great superficial of the slab, Each particle of the skeleton in the free body is exerted
by the gravity and the contact, the treatment of pining up to groundsill sink and the project of oscillations dame
structure and other engineering issues. Researches on the mechanical behavior of concrete under high loading rates are
of basic importance for the assessment of engineering structures against impact, explosive or seismic loads. Various
kinds of experiments have been carried out to obtain necessary data for engineering practices related to concrete
architectures. Due to high expenses and technical limitation of the dynamical tests, numerical simulation is the
methodology that has been widely accepted and used to study the concrete responses under dynamic loadings. Thus
lead the importance to establish pertinent constitutive model for reproducing accurate predictions in the civil
engineering practice. As to thermo-elasto-plastic evanescence impact dynamic problems analysis of threedimensional large-scale slab, for the complete free body, the magnitudes and the directions of the contact forces are
distributed randomly in the space. It concernfully embodies the inertia loading, for large-scale slab; it can be classified
into continuous materials and porous materials according to the distribution of the compositive particles. It is a very
important a question for discussion when investigating constitutive connection about thermo-elasto-plastic dynamic of
large-scale slab. Biot’s work indicated that three dimensional frame of reference and deduced catholicity, and aspect of
⎯ 398 ⎯
nonlinear scopes. In previous studies, the updated Ottosen’s four-parameter criterion is used to calculate, the dynamic
behavior of concrete under dynamic loading. In order to get more accurate computational results in the crucial
conditions, the postpeak behavior of concrete should be appropriately described. To meet such needs, an
elasto-viscoplastic constitutive model of concrete is proposed, which is developed from the Ottosen’s four-parameter
criterion and Imvan’s empirical loading surfaces for concrete. The expansion and contraction of the viscoplastic
surfaces are considered separately because the properties of concrete are greatly different between the prepeak and
postpeak periods. In continuum mechanics, the stress at a point is defined as the definite limit of the average traction on
an elementary surface [1], in a static state the particles are of equilibrium under the action of all these forces. But for a
single particle, no matter what the magnitudes and the directions of the contact forces are, the resultant of its surface
forces is equal to that of all the forces acting on the particle. The equilibrium of the whole free body depends on the
gravity and the resultant effect of the contact forces, namely, the surface force on the free body. Ashby’s analysis [2] of
polycrystalline was the first research on the elastic-plastic deformation of crystals at finite strain. According to Hill’s
point of view [3], some investigations point out that the math disciplinarian is homologous no matter what was
considered stuff. But little of reporting was seen about thermo-elasto-plastic dynamic problems analysis for three
dimensional large-scale slab in many literatures, there were two physically different mechanisms for deforming and
reorienting material fibers plastic slip and lattice deformation for the side of large-scale slab, the pervasive concept
about liquid of large-scale slab was set up; engineers and theorists have attached importance to it. Lately, a series of
production have been acquired about pervasive liquid for porous medium in geomechanics and biomechanics, but
specially, in the process of constructing Qinghai-Tibet railway engineering, the mechanism of large-scale slab should
be clarity, some conclusions papers [4, 5] in China show that many production has been achieved in the domain of
flexibility hydro coupling seepage and wreck reflected on uninstall abruptness, constitutive relation is the most
important aspect in the investigation, The research will establish the constitutive relation concerning thermoelasto-plastic for large-scale slab in three dimensional. Wang [6, 7] has presented works on theories and applications of
thermoelasticity. It will be acquired for thermo-elasto-plastic idiosyncrasy for large-scale slab. A momentous
complication concerned with plasticity will be taken into account and applied into large-scale slab.
CONSTITUTIVE RELATION FOR LARGE-SCALE SLAB
First, let us consider slab with thermo-elasto-plastic idiosyncrasy with idiosyncratic complexion in all kinds of
basement large-scale slab, for instance, large-scale slab in constructing engineering, it is restricted with
elastic-plasticity specialty of large-scale slab, the problem will be predigested to that of constructing an appropriate
strain energy density function, which will change along with temperature in large-scale slab; otherwise, the research
problem will be of an inadequacy strain energy density function containing based parameters of large-scale slab. When
investigating constitutive relation based on sublimate model or exceptional large-scale slab, there will be several
unbeknown coefficients which should be confirmed by based frozen soil experimental method which can be found in
many literatures, many experiments can get hold of dissimilar aftermath, whereas, the based verdicts will be semblable.
Popularly, two means are available which have been testified efficacious in classical references, one of them is the
direct method, which is a logical consequence of viewing the control object about large-scale slab as an
one-dimensional continuum, actually, large-scale slab relax medium which would be practicality, this method would
be predigestion and could be taken in order of advantages, nevertheless the conclusion would be uncertainty, when this
method is supplemented by empirical evidence based on fractional experimentation, it will become a natural method
which can be used in effective engineering.
1. Basic relation It is linear hypothesis for the flux and drives force with the elementary hypothesis of constitutive
relation; there is one or two original coefficient deservedly. In continuum mechanics, the stress at a point is defined as
the definite limit of the average traction on an elementary surface’s. However, the point must become a representative
elementary volume (REV) for porous medium or soils. Relph, B. Peck had discussed the definition of the stress for dry
soils and indicated that when we talk about the stress at a point within a soil, we must envision a rather large point. This
large point, namely, representative elementary volume, can be defined in a more accurate way. The REV must include
enough particles and voids, and these particles and voids, however, should be small enough for the entire volume of the
porous medium, in which the mean of the state variable is independent from its magnitude and size, and the mean’s
distribution is continual in space and time. Based on theoretic established by Hill and Rice [8], the elastic law in
large-scale slab can be written out as
σ ∇∗ = L : D ∗ = σ& − Π ∗ ⊗ σ + σ ⊗ Π ∗
(1)
where, ⊕ denoting the ratio of two quantities L and D*, σ ∇∗ is Jaumann rate of Kirchhoff stress formed on axes that
spin for the basic large-scale slab lattice, where σ is the Cauchy stress, with σ& the material rate of Kirchhoff stress, the
shear Kirchhoff stress τ is defined as (ρ o ρ )σ ; L is tensor of elastic moduli with components on coordinates fixed of
⎯ 399 ⎯
large-scale slab, and ρo , ρ are material densities in the reference and current state; D and Π are the symmetric rate of
stretching tensor and the rate of spin tensor, respectively. The relation is D ∗ = D + Dθ − D P − D e ,
Π ∗ = Π + Πθ − Π P − Π e , when be writing the constitutive relation of the flux and thermos dynamics, many new
items should be added to the equation.
2. Weight relation When considering the Jaumann rate of Kirchhoff stress formed on axes that rotate with the
large-scale slab, the similar equation can be written out. The flux and thermos dynamics are the most complicated due
to cross transport phenomena, for example, if taking into account the heat diffuses, temperature grads must be accessed
in direct proportion. Thermal effects on large-scale slab need to be considered for many reasons, this outcome is the
same as three-dimensional large-scale slab theory, of which it will be a special case, and this makes it difficult to
evaluate its limitations or to estimate its errors, especially for not-so-incompact slab. There are many types of control
equations of elastic-plasticity when researching on solid problems, if introducing thermal loading, there will be a
complicated style. Two types of Thermo-Elasto-Plastic relation should be considered.
If the Thermo-Elasto-Plastic is unaffected by slip, the relation can be written as
(σ& )11
∇∗
= min {σ& Δer −1 , σ& Δpr −1} + max {σ& Δer +1 , σ& Δpr +1} + Π1∗ ⊗ σ Δer −1 + σ Δer −1 ⊗ Π1∗ + Π ∗2 ⊗ σ Δpr −1 + σ Δpr −1 ⊗ Π ∗2
+ Π ∗3 ⊗ σ Δt r −1 + σ Δt r −1 ⊗ Π ∗3 + Π ∗3 ⊗ σ Δθ r −1 + σ Δθr −1 ⊗ Π ∗3 + σ Δpr −1 ⊗ Π ∗3 + σ Δθ r −1 ⊗ Π ∗2
(σ& )21
∇∗
= min {σ& Δpr −1 , σ& Δt r +1} + max {σ& Δpr +1 , σ& Δt r +1} + Π1∗ ⊗ σ Δer +1 + σ Δer +1 ⊗ Π1∗ + Π ∗2 ⊗ σ Δpr +1 + σ Δpr +1 ⊗ Π ∗2
+ Π ∗3 ⊗ σ Δt r −1 + σ Δt r −1 ⊗ Π ∗3 + Π ∗3 ⊗ σ Δθ r −1 + σ Δθ r −1 ⊗ Π ∗3 + Π ∗3 ⊗ σ Δpr −1 + σ Δθ r −1 ⊗ Π ∗2
(σ& )31
∇∗
= min {σ& Δt r +1 , σ& Δθr +1} + max {σ& Δt r −1 , σ& Δθr −1} + Π1∗ ⊗ σ Δer −1 + σ Δer −1 ⊗ Π1∗ + Π ∗2 ⊗ σ Δpr −1 + σ Δpr −1 ⊗ Π ∗2
+ Π ∗3 ⊗ σ Δt r +1 + σ Δt r +1 ⊗ Π ∗3 + Π ∗3 ⊗ σ Δθr +1 + σ Δθr +1 ⊗ Π ∗3 + Π ∗2 ⊗ σ Δθr +1 + σ Δθ r +1 ⊗ Π ∗2
(2)
(3)
(4)
If the Thermo-Elasto-Plastic is affected by slip, the relation can be written as
(σ& )21
∇∗
= Π ∗2 ⊗ σ Δpr −1 + σ Δpr −1 ⊗ Π ∗2 + Π ∗3 ⊗ σ Δt r −1 + σ Δt r −1 ⊗ Π ∗3 + Π ∗3 ⊗ σ Δθr −1
(5)
+ σ Δθr −1 ⊗ Π ∗3 + σ Δpr −1 ⊗ Π ∗3 + σ Δθ r −1 ⊗ Π ∗2
(σ& )22
∇∗
= Π ∗2 ⊗ σ Δpr +1 + σ Δpr +1 ⊗ Π ∗2
(6)
+ Π ∗3 ⊗ σ Δt r −1 + σ Δt r −1 ⊗ Π ∗3 + Π ∗3 ⊗ σ Δθ r −1 + σ Δθ r −1 ⊗ Π ∗3 + Π ∗3 ⊗ σ Δpr −1 + σ Δθ r −1 ⊗ Π ∗2
(σ& )32
∇∗
= Π ∗2 ⊗ σ Δpr −1 + σ Δpr −1 ⊗ Π ∗2
(7)
+ Π ∗3 ⊗ σ Δt r +1 + σ Δt r +1 ⊗ Π ∗3 + Π ∗3 ⊗ σ Δθr +1 + σ Δθ r +1 ⊗ Π ∗3 + Π ∗2 ⊗ σ Δθr +1 + σ Δθ r +1 ⊗ Π ∗2
where, Π i (i = 1,2,3) are the rates of spin tensor homologous different strains corresponding to large-scale slab.
3. Basic model The basic model of large-scale slab can be described as this, Different superscripts e, p, t, θ denotes the
dissimilar influence of elasticity, plasticity, temperature and density solidity, respectively; (σ& )i (i = 1,2,3) denotes
the Jaumann rate of Kirchhoff stress in response distinct appearance homologous. To define the stress more accurately,
we begin our discussion by taking a free body from dry soil. Each particle of the skeleton in the free body is exerted by
the gravity and the contact forces between the particles. In a static state the particles are of equilibrium under the action
of all these forces. For the complete free body, the magnitudes and the directions of the contact forces are distributed
randomly in the space. But for a single particle, no matter what the magnitudes and the directions of the contact forces
are, the resultant of its surface forces is equal to that of all the forces acting on the particle. The equilibrium of the
whole free body depends on the gravity and the resultant effect of the contact forces, namely, the surface force on the
free body. When the primary influence in large-scale slab is soil, the Thermo-Elasto-Plastic is unaffected by slip
expression (1) will be primary; when the primary influence in large-scale slab is reinforcing steel bar, expression (2)
will be veracious; when the primary influence in large-scale slab is cement, expression (3) will be veracious. When the
Thermo-Elasto-Plastic is affected by slip, the primary influence in large-scale slab is soil, expression (4) will be
primary; when the primary influence in large-scale slab is reinforcing steel bar, expression (5) will be veracious; when
the primary influence in large-scale slab is cement, and expression (6) will be veracious. From Casimir theory, the
∇∗
⎯ 400 ⎯
connection can be given as followings
m
(
)
(8)
(
)
(9)
(
)
(10)
(
)
(11)
(
)
(12)
(
)
(13)
i
ij
j
ij
j
ij
j
K& 11( ) = ∑ γ&11( ) F1( ) + γ&12( ) F2( ) + γ&13( ) F3( )
j =1
m
i
( ij ) ( j )
( ij ) ( j )
( ij ) ( j )
K& 12( ) = ∑ γ&21
F2 + γ&22
F3 + γ&23
F1
j =1
m
i
( ij ) ( j )
( ij ) ( j )
( ij ) ( j )
K& 13( ) = ∑ γ&31
F3 + γ&32
F2 + γ&33
F1
j =1
m
ij
j
ij
j
ij
j
(i )
K& 21
= ∑ γ&11( ) F2( ) + γ&12( ) F1( ) + γ&13( ) F3( )
j =1
m
(i )
( ij ) ( j )
( ij ) ( j )
( ij ) ( j )
K& 22
= ∑ γ&21
F3 + γ&22
F2 + γ&23
F1
j =1
m
(i )
( ij ) ( j )
( ij ) ( j )
( ij ) ( j )
K& 23
= ∑ γ&31
F1 + γ&32
F2 + γ&33
F3
j =1
where, i, j = 1, 2, ..., m , γ&i(，)j =1,2,3 are the self-governed coefficients, respectively, they have nothing to do with
ij
i
i)
)
K& i(,ij)=1,2,3 , Fi =( i1,2,3
, quantities K& i(, j)=1,2,3 , Fi =( 1,2,3
are the flux and thermal dynamics when taking into account the specialty
of the heat transfer from heat barrier to considered point, respectively, Popularly, the flux and thermal dynamics are
tensors that have arbitrary order, paying attention to that the thermal dynamics are extraneous to classical Newtonian
meaning. When taking into account the flux and thermal dynamics, Jaumann rate of Kirchhoff stress is
(min{σ
,σ
e
})
∇
p
[
]
(14)
[
]
(15)
m
= L ⊕ D e + L ⊕ D p − ∑ L ⊕ K& 1(i ) + L ⊕ K& 2(i ) + κ (i ) η& (i )
i =1
(min{σ
p
})
∇
,σ t
m
= L ⊕ D p + L ⊕ D t − ∑ L ⊕ K& 2(i ) + L ⊕ K& 3(i ) + κ (i ) η& (i )
i =1
(min{σ
})
∇
,σ θ
t
[
m
]
= L ⊕ D t + L ⊕ Dθ − ∑ L ⊕ K& 3(i ) + L ⊕ K& 1(i ) + κ (i ) η& (i )
(16)
i =1
(max{σ
,σ
e
p
})
∇
[
]
(17)
[
]
(18)
[
]
(19)
m
= L ⊕ D e + L ⊕ D p + ∑ L ⊕ K& 1(i ) + L ⊕ K& 2(i ) − κ (i ) η& (i )
i =1
(max{σ
p
,σ t
})
∇
m
= L ⊕ D p + L ⊕ D t + ∑ L ⊕ K& 2(i ) + L ⊕ K& 3(i ) − κ (i ) η& (i )
i =1
(max{σ
t
,σ θ
})
∇
m
= L ⊕ D t + L ⊕ Dθ + ∑ L ⊕ K& 3(i ) + L ⊕ K& 1(i ) − κ (i ) η& (i )
i =1
where, connection in the coefficients η& (i ) (I=1,2,3) should be related to either of the stress rates, the deformation rates,
or the current stress state, every expressions deliver themselves of elasticity, plasticity, temperature and different
medium including large-scale slab, each hypothesis includes proportion of the correctitude rates σ e , σ p , σ t , σ θ ;
peradventure, the maximal quantum and minimal quantum possess the similar exterior form, however, the maximal
quantum take on more items in their expressions, on the surface, the equation incarnates with integral in carnet. In a
steady temperature, the heat in large-scale slab will flow in a same direction in a jerkwater circumscription.
4. Constitutive relationship Naturally, field equation of the Thermo-Elasto-Plastic mimicking the steps followed in
⎯ 401 ⎯
(a)
(b)
Figure 1: Characteristic of the large-scale slab: (a) the perfect transformation complexion
aspects in large-scale slab; (b) the heat flow aspects in large-scale slab
general methods, Solid materials can be classified into continuous materials and porous materials according to the
distribution of the competitive particles. When the competitive particles distribute continually in molecule or atom size
in space, they can be considered as continuous material. Otherwise they would be called porous materials. Almost all
the existing constitutive relation models for porous mediums are formulated on a continuum basis, therefore, it is
difficult to tell what the characters of porous medium material are. In addition to definitions of the actual stress and
actual strain of skeleton for porous medium, this paper also presents a model of ideal porous medium followed up by a
discussion of the stress-strain constitutive relation for the ideal porous medium. According to Biot’s principal, total
stresses capacity is
σ ij = σ ije + σ ijp + σ ijt + σ ijθ
= [ Π1 ⊗ dε + Π 2 ⊗ dϑ + Π 3 ⊗ dt ]δ ij + 3Λ1 ⊗ dε ije + 2Λ 2 ⊗ dε ijp + 3Λ 3 ⊗ dε ijt + 3Λ 3 ⊗ dε ijθ
(20)
Where, ϑ is fluid capacity idle archery corresponding to thermal loading, σ ije , σ ijp , σ ijt and σ ijθ are effective stress
weight, plasticity stress weight, temperature stress weight and large-scale slab stress weight. Fig. 1 can show this, the
part of the armored concrete in an exiguity area will enlarge going with diversification of the temperature; if the
temperature change with acuteness, the aspects of the water will be turbulence; if the temperature is not inconstant, it
will change according to the idiosyncrasy of the studied large-scale slab. All these can be expressed by the production
of multiplication for porosity and pressure on fluid corresponding to large-scale slab; and δ ij is Kronecker Delta
symbol; Λ i (i = 1,2,3) are Biot coefficients, dissimilar strain ε ije , ε ijp , ε ijt , ε ijθ should be satisfied the following
connection
α = ⎡⎣ χ1ε kk δ ij n j + 3 ( χ1ε ije + χ 2ε ijp + χ1ε ijt + χ 2ε ijθ ) n j − β ijeuie +β ijp uip − β ijt uit + β ijθ uiθ ⎤⎦ Et μ ni
i
i
i
i
(21)
Where, μ is Poisson’ collapse, n j is unit arrow weight allowing the boundary in collapse direction; coefficients
χ1 , χ 2 are Lamé constant; β ije , β ijp , β ijt and β ijθ are connected coefficients with large-scale slab, respectively. In a
general way, there will exist four kinds of circumstances, the item β ije u ie that have something to do with flexibility will
rise main function, and β ije u ie ≥ 0 , now, item χ 1ε ije in Eq. (20) needs to be noticed, item β ijp u ip in Eq. (20) needs to be
noticed; when the large-scale slab inside water occupies the main composition, the item that have something to do with
temperature will rise main function, now item β ijt u it in Eq. (20) needs to be noticed; when the large-scale slab inside
gas occupies the main composition, the item that have something to do with physical volume will rise main function,
and β ijθ u iθ ≤ 0 , now item β ijθ u iθ in Eq. (20) needs to be noticed. To obtain a complete thermodynamic theory of
large-scale slab, a law of conservation of energy can be postulated, development stress field equation are given by
follows
σ ij , j + f i + α E μ∂ i t + ∂ j (α Et μδ ij ) ≥ 2Λ1u&&ie + 3Λ1u&&ip + 2Λ 2U&&ie + 3Λ 2U&&ip + 3Λ 2U&&it
(22)
σ ij , j + fi − α E μ∂ i t − ∂ j (α Et μδ ij ) ≤ 2Λ1u&&ip + 3Λ1u&&it + 2Λ 2U&&ip + 3Λ 2U&&it + 3Λ 2U&&ie
(23)
⎯ 402 ⎯
σ ij , j + fi + α E μ∂ i t + ∂ j (α Et μδ ij ) ≥ 2Λ1u&&it − 3Λ1u&&iθ + 2Λ 2U&&it − 3Λ 2U&&iθ + 3Λ 2U&&ie
(24)
If the Thermo-Elasto-Plastic is unaffected by slip, the relation can be written as
σ ij , j + f i + α E μ∂ i t ≥ 2Λ1u&&ie + 3Λ1u&&ip + 2Λ 2U&&ie + 3Λ 2U&&ip + 3Λ 2U&&it
(25)
σ ij , j + f i − α E μ∂ i t ≤ 2Λ1u&&ip + 3Λ1u&&it + 2Λ 2U&&ip + 3Λ 2U&&it + 3Λ 2U&&ie
(26)
σ ij , j + f i + α E μ∂ i t ≥ 2Λ1u&&it − 3Λ1u&&iθ + 2Λ 2U&&it − 3Λ 2U&&iθ + 3Λ 2U&&ie
(27)
where, u ie , u ip , u it , u iθ and U ie ,U ip ,U it ,U iθ denote displacements separated capacity for large-scale slab and heat
transfer flow weight corresponding to thermal elasticity, and plasticity, respectively, top signs “.” and “..” indicate
frank quality corresponding to one grand and two grands, α expresses thermal expand coefficient for large-scale slab,
E is Yang’s elasticity coefficient. On account of complicacy in floor slab, especially the dissimilar characteristic
between baton and reinforcing steel bar, all these lead to the difficult when choosing Yang’s model. Which Yang’s
model should be used? Based on engineering experience, the assorted form can be used. As an important phase, the
boundary of the researched control object must be carved up.
NUMERICAL VALUE FOR LARGE-SCALE SLAB AND CONCLUSIONS
Figure 2: Temperature field, t = −30°C
Figure 3: Temperature field, t = +30°C
Figure 4: Displacement field, t = −30°C
Figure 5: Displacement field, t = +30°C
For verifying the method of investigative and dependable, this part will consider the variety moving of large-scale slab
under different temperatures, the temperature field distribution in deferent circumstance, because of the sensitivity to
the temperature in large-scale slab, this research selects by examinations to have representative t = −30°C and t = +30°C,
the simulation results show in Figs. 2-7. Observing the variety trend of every kind of field, relevant the parameter
selects by examinations for two cases. When lucubrating the characteristic in the process of heat transfer or plasticity
orderliness, the textual theoretic will be direction. Constitutive relation of Thermo-Elasto-Plastic dynamic saturation
⎯ 403 ⎯
frozen soil porous medium is considerable concern in the process of investigating stuff characteristic, this article
established the ordinary constitutive relation law of frozen soil porous medium and inner thermal liquid.
Figure 6: Stress field, t = −30°C
Figure 7: Stress field, t = +30°C
Acknowledgements
The work presented here is part of a general investigation of the behavior of thermo quality transmission problems for
large-scale slab with creep. The research is supported by the grant of the Knowledge Innovation Program of the
Chinese Academy of Sciences (KZCX1-SW-04), the Scientific Research Common Program of Beijing Municipal
Commission of Education (KM200610009010) and the Beijing Natural Science Foundation Program and Scientific
Research Key Program of Beijing Municipal Commission of Education (KZ200610009005).
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⎯ 404 ⎯