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Plasticity of Porous Materials
Plasticity of Porous Materials N. Cristescu DepU of Mechanics, Univ. of Bucharest, Bucharest, ROMANIA
ABSTRACT The irreversible volumetric deformation of several porous materials is described and the appropriate constitutive equations are formulated. Both irreversible dilatancy and/or compressibility are taken into account. Several examples are given. KEYWORDS Pores, microcracks, dilatancy, compressibility, rock, sand. VOLUMETRIC PLASTICITY Typical for the mechanical behaviour of porous materials is the irreversible deformation of the volume, either an irreversible decrease or an irreversible increase. This typical behaviour is due to the presence of a great number of voids, either pores or/and microcracks. These properties are studied in hydrostatic compression creep tests and in triaxial deviatoric creep tests. As an example in Fig.l is shown the uniaxial compression stress-strain curves for sandstone (from Secu-Ranchina) obtained in short-term creep test, i.e. at each stress level shown the stress is kept constant for 15 minutes before being increased again. From this figure follows that at relatively small stresses the volume is highly compressible, but dilatant at higher stresses. The yield stress is practically zero. The "elastic" slopes are shown by two segments of straight lines. Such behaviour is typical for rocks of small to medium porosity (less than 5%, say) due mainly to the presence of microcracks and not to pores. Similar properties are exhibited by other porous materials as sand, cement concrete, wood, broken rock, etc. CONSTITUTIVE EQUATION In order to describe the volumetric viscoplastic compressibi1iy and/or di-
ll
12
(%) Fig.1
Stress-strain curves for sandstone in uniaxial short-term creep test.
latancy a constitutive equation of the form (Cristescu, 1987, 1989) was developed -
°
+
(1
MAI * u/
W'(Q\3F
(1)
where G and K(=A the positive part of the expression shown. For instance for sandstone shown in Fig.1 we have Η(σ,σ):-
(σ·
+ a.
+ a,
♦W
(«- f)J- ♦
+ b„
σ
bl
l)(f)2+cf
(2)
where σ is the equivalent stress and all the other parameters involved are material constants. According to (1) the irreversible volumetric behaviour is governed by
·<
w'(t)'
3σ
(3)
if F(a,a) depends on the two invariants shown alone. In Fiq.2a is shown the constitutive domain for sandstone, with dash-dot line the compressibility/ dilatancy boundary of equation 3F/9o - 0, and several surfacesH ■ const, as dotted lines. The associative law (i.e. H=F) seems reasonable as shown in Fig.2b where experimental data (full lines) and theoretical predictions(dotted lines) match quite well. For dense sand using the experimental data by Goldscheider (1982) we have determined a constitutive equation of the form (1) but with
13
OjR(MPa)
60
+60
40
w
·/«<
20
20
^2=5MPa
//
COMPRESSIBLE :
(a)
20
Fig.2
Η(σ,σ):
σ(ΜΡα)
40
^
•0.1
'I
(b) 0.1
C o n s t i t u t i v e domain ( a ) and s t r e s s - s t r a i n in t r i a x i a l t e s t s (b) f o r sandstone. a ( e - ~*G)—
+ a,
VK
/O \2
curves
2— 1 b (σ- ^ σ ) — + b, o 3 σ* 1
(JSL)
+ c, (™)
0.2
(%)
(4)
+
CL-oj(kPa)
800 k i ,R
.x
600
x
^
xd
x·
■L
€i
400 • xL
i ; J-x.
• · EXPERIMENTAL * * ASSOCIATIVE MODEL 1 1 N0NASS0CIATIVE MODEL
xl d
^ = 203kPa
-4 Fiq.3
AP-B
-2
-1
Comparison o f e x p e r i m e n t a l c o n s t i t u t i v e equations.
1
(%)
data w i t h p r e d i c t r o n
of
14 For sand a nonassociative constitutive equation is expected (Lade QÄ. at. t 1987) and that is why function F(s,a) was chosen of the same form as (k) but other constitutive constants. The comparison of exnerimental data with either associative or nonassociative constitutive eauation are shown in Fip.3. The models are only tentative pending additional comnlete set of experimental data. The constitutive domain for sand is shown in Fig.** with the com-
.. ^F=const H=const^.
er Σ lb
Ν
,
H= constCOMPRESSIBLE
0.1
Fig.4
0.2
0.3
0.4
σ(ΜΡα)
Q5
Constitutive domain for sand.
pressibi 1 i ty/di latancy boundary 9F/90 = 3Η/3σ = 0 shown as dash-dot line. The surface F = const, passing by point A and shown as interrupted line is slightly above the surface H « const, passinp by the same point (dotted line). The same relative positions of these two surfaces were found experimentally by Lade dt al. (I987). Similar constitutive equations have been determined for a variety of other materials as: various kinds of rocks, cement concrete, various woods, a stratum of broken rock, etc. The form of the constitutive equations depends on the kind of voids existing in the porous material considered. REFERENCES C r i s t e s c u , N. (1987). Int.3.Rock Mach ΛΙίη. Sei. & Gaomack. kbbtx. 2^, 271-282. C r i s t e s c u , N. (1989). Rock Rhaolocny. Kluwer Academic P u b l . , Dordrecht. Goldscheider, M. (1982) I n : Con6titutiva Relation* &OK Soih (G. Gudehus at aJL.t E d . ) , Dp. 11-5*». Lade, P.V., R.B. Nelson and Y.M. I to (1987). J.Ewg.Mecfi. J J £ , 1302-1318. S u l i c i u , I . (1989). Int.3.Plasticity (in press).
ABSTRACT The irreversible volumetric deformation of several porous materials is described and the appropriate constitutive equations are formulated. Both irreversible dilatancy and/or compressibility are taken into account. Several examples are given. KEYWORDS Pores, microcracks, dilatancy, compressibility, rock, sand. VOLUMETRIC PLASTICITY Typical for the mechanical behaviour of porous materials is the irreversible deformation of the volume, either an irreversible decrease or an irreversible increase. This typical behaviour is due to the presence of a great number of voids, either pores or/and microcracks. These properties are studied in hydrostatic compression creep tests and in triaxial deviatoric creep tests. As an example in Fig.l is shown the uniaxial compression stress-strain curves for sandstone (from Secu-Ranchina) obtained in short-term creep test, i.e. at each stress level shown the stress is kept constant for 15 minutes before being increased again. From this figure follows that at relatively small stresses the volume is highly compressible, but dilatant at higher stresses. The yield stress is practically zero. The "elastic" slopes are shown by two segments of straight lines. Such behaviour is typical for rocks of small to medium porosity (less than 5%, say) due mainly to the presence of microcracks and not to pores. Similar properties are exhibited by other porous materials as sand, cement concrete, wood, broken rock, etc. CONSTITUTIVE EQUATION In order to describe the volumetric viscoplastic compressibi1iy and/or di-
ll
12
(%) Fig.1
Stress-strain curves for sandstone in uniaxial short-term creep test.
latancy a constitutive equation of the form (Cristescu, 1987, 1989) was developed -
°
+
(1
MAI * u/
W'(Q\3F
(1)
where G and K(
(σ·
+ a.
+ a,
♦W
(«- f)J- ♦
+ b„
σ
bl
l)(f)2+cf
(2)
where σ is the equivalent stress and all the other parameters involved are material constants. According to (1) the irreversible volumetric behaviour is governed by
·<
w'(t)'
3σ
(3)
if F(a,a) depends on the two invariants shown alone. In Fiq.2a is shown the constitutive domain for sandstone, with dash-dot line the compressibility/ dilatancy boundary of equation 3F/9o - 0, and several surfacesH ■ const, as dotted lines. The associative law (i.e. H=F) seems reasonable as shown in Fig.2b where experimental data (full lines) and theoretical predictions(dotted lines) match quite well. For dense sand using the experimental data by Goldscheider (1982) we have determined a constitutive equation of the form (1) but with
13
OjR(MPa)
60
+60
40
w
·/«<
20
20
^2=5MPa
//
COMPRESSIBLE :
(a)
20
Fig.2
Η(σ,σ):
σ(ΜΡα)
40
^
•0.1
'I
(b) 0.1
C o n s t i t u t i v e domain ( a ) and s t r e s s - s t r a i n in t r i a x i a l t e s t s (b) f o r sandstone. a ( e - ~*G)—
+ a,
VK
/O \2
curves
2— 1 b (σ- ^ σ ) — + b, o 3 σ* 1
(JSL)
+ c, (™)
0.2
(%)
(4)
+
CL-oj(kPa)
800 k i ,R
.x
600
x
^
xd
x·
■L
€i
400 • xL
i ; J-x.
• · EXPERIMENTAL * * ASSOCIATIVE MODEL 1 1 N0NASS0CIATIVE MODEL
xl d
^ = 203kPa
-4 Fiq.3
AP-B
-2
-1
Comparison o f e x p e r i m e n t a l c o n s t i t u t i v e equations.
1
(%)
data w i t h p r e d i c t r o n
of
14 For sand a nonassociative constitutive equation is expected (Lade QÄ. at. t 1987) and that is why function F(s,a) was chosen of the same form as (k) but other constitutive constants. The comparison of exnerimental data with either associative or nonassociative constitutive eauation are shown in Fip.3. The models are only tentative pending additional comnlete set of experimental data. The constitutive domain for sand is shown in Fig.** with the com-
.. ^F=const H=const^.
er Σ lb
Ν
,
H= constCOMPRESSIBLE
0.1
Fig.4
0.2
0.3
0.4
σ(ΜΡα)
Q5
Constitutive domain for sand.
pressibi 1 i ty/di latancy boundary 9F/90 = 3Η/3σ = 0 shown as dash-dot line. The surface F = const, passing by point A and shown as interrupted line is slightly above the surface H « const, passinp by the same point (dotted line). The same relative positions of these two surfaces were found experimentally by Lade dt al. (I987). Similar constitutive equations have been determined for a variety of other materials as: various kinds of rocks, cement concrete, various woods, a stratum of broken rock, etc. The form of the constitutive equations depends on the kind of voids existing in the porous material considered. REFERENCES C r i s t e s c u , N. (1987). Int.3.Rock Mach ΛΙίη. Sei. & Gaomack. kbbtx. 2^, 271-282. C r i s t e s c u , N. (1989). Rock Rhaolocny. Kluwer Academic P u b l . , Dordrecht. Goldscheider, M. (1982) I n : Con6titutiva Relation* &OK Soih (G. Gudehus at aJL.t E d . ) , Dp. 11-5*». Lade, P.V., R.B. Nelson and Y.M. I to (1987). J.Ewg.Mecfi. J J £ , 1302-1318. S u l i c i u , I . (1989). Int.3.Plasticity (in press).