- Email: [email protected]
Nonlinear evolution of aggregates with inextensible constraints
11:
Nonlinear Evolution Constraints l
of Aggregates
with
X/d=25 Y/d=0
Inextensible
Ming-Xiang CBEN, Wei YANG and Quan-Shui ZBENG ( Department of Engineering &chanics, Tsinghus University, Beijing 100084, China) Email: [email protected] Abstract: Crydlline and aemicrystalline polymers are formed ZM aggregates of grains with evoIving inextensible axes. This inextensibIe constraint leads to texture evolution under large plastic deformation. This paper reveals the nonlinear texture evolution of crystalline polymers under axiqmmetric straining. Key Words: Nonlinear texture evolution, crystalline polymers, inextensible constraint.
Introduction Crystalline and aemicryetalline polymers are formed as aggregates of grains with evolving inextensible axes. Their mechanical behavior is dominated by the phase of crystallinity, ‘Thi8pspC-l!tSraceivedopM~10,1~
CHEN. et d.: Nonlmear Evolution of.. .
9
usually in a configuration of chain folding. Due to the stiff covalent bonding of molecular chain, a crystalline polymer is almost inextensible in the chain direction. Only four independent slip systems may operate within a grain, leading to texture evolution under large plastic deformation. The crystallographic chain axes tend to align with the direction of maximum stretch. The preferential orientation induces anisotropy of the macroscopic mechanical behavior. The present paper describes nonlinear evolution of such aggregates. A continuous orientation distribution function (ODP) is introduced to describe the orientation distribution of the inextensible axes. The ODF is expanded as a convergent series of irreducible tensors, whose co&&&s are termed moment tensors of different orders. Microscopic-macroscopic transition for the deformation accounting for chain inetiensibility ahd the overall material symmetry is proposed using tensor function representation theory. The model is applied to simulate the texture evolution and texture hardening under axizymmetric straining. 1. Constrained
Single
Crystals
The crystal lattice of crystalline polymers such as HDPE is orthorhombic-with the c-axis in the chain diiection. The slip planes are parallel to the c-axis and a slip direction can be either parallel to the c-axis (chain slip) or perpendi&lar ti> it (transverse slip). Let sB and nfl be unit vectors representing the slip direction and the slip plane normal of the b-th slip system, and +fl the shear rate of that slip system. The symmetric part of the velocity gradient is expressed by D = c
‘& -1 ,+(s$
n> + nd 18,d) = &“Rp
9=1*
ill
J=l
Simmilar expression can be drived for the skew-symmetric crystallographic axis c is
part W. The rotating rate of a
k=Wc-[(c@c)D-D(c@c)]c
12)
Each slip system, with slip planes penciled by the chain direction, ailows no stretching in the chain direction. This inextensible constraint can be expressed as C’: R’ = 0
(3)
wher6.C =.c Q c - (l/3) 1 and 1 the second order identity tensor. The constitutive description of a single crystal consists of a relation between the shear rate +@ and the resoIved shear stress 78. Following Parks and Ahzi (1990), we assume that they are related through a power law +a = +;f
I$~‘-’
(4)
where n is the rate exponent, +o the reference shear rate and gp the shear strength associated with the @th slip system. The locai Cauchy stress deviator in the crystal can be decomposed into the direct sum of a stress tensor S’ normal to C’ and a component SC aligned with C’ S = S’ + % (trSC’) C’ = S’ + SCC’
(5)
Relations (4), (5) and (1) lead to a constitutive equation for a constrained single crystal :s’
The reaction stress component SC has to be determined from the equilibrium.
(‘3)
10
Cdcdio~~~
2. Orientation
in Nonlinear
Vol.1. NQJul.
Science& Numerical Simulation
1996)
Distribution
A representative aggregate element contains many individual grains. The orientation of the chain axis of a crystal grain is described by the polar angles (0,4) of the c-axis. An orientation distribution function, @(e, d), is introduced to describe the orientation distribution of the chain axes within an aggregate element. It ia normalized and its evolution satisfies an orientation conservation law. We assume that the microscopic variables depend on the grain orientation and the orientation diitribution of the aggregate, and neglect the influences by the grain shape, grain size and grain boundaries. Under prescribed velocity boundary condition, one has the following self-consistent conditions for the deformation within the aggregate (I))
= b,
(W)
= w
where cusped parentheses denote the average over orientations ODF can be expanded in term of irreducible tensors
(7)
weighted by @(e,#).
@(c)z;ga;,.c@2n
The
(8)
n=l
where cg2” denotes the Zn-th tensor power of c. a;, is the deviatoric part of the moment tensor az,,, defined as az,, = (cB2”). Micro-macro Transition under Transverse Isotropy Undeformed melt-crystallized semi-crystalline polymers show a spherulitic structure. The aggregate initially behaves isotropically. Under axi-symmetric straining, the deformed spherulites exhibit a global transverse isotropy. Let k be a unit vector along the principal loading axis. The structure tensor that characterizes the transverse isotropy can be formed as M = k@k. For incompressible materials under ax&symmetric straining, the macroscopic deformation is fi = (g/2) (3M - 1) and %r = 0, where 2 ia the straining rate along the symmetry axis k. By the ieotropicization theorem, the ODF is an isotropic function of C’ and M. Denote at, w,. . . as the coefficients of a series of even order moment tensors. Two assumptions about micro-macro transition are adopted. (A) The local deformation rate D is a quadratic function of the current orientation C’; and (B) the local spin W linearly depends on the macroscopic deformation b and the current orientation C’, and only the second-order moment a2 is accounted for its orientation distribution effect. It was shown (Chen et al., 1996) that the microscopic deformation, D and W, can be expressed a~ w = C’ (c’b
- dc’)
where I denotes the fourth-order identity tensor. The first expression for D confirms and further details the form proposed by Parka and Ahzi (1990) using an intermediate traceless deformation rate. It is straightforward to verify that the consistent condition (7) is satisfied by (9). The coefficient [’ relates to the second and fourth order moments a2 and a4 by
and C” depends only on the coefficient a2. The &f-consistent condition for global equilibrium within the aggregate requires that the orientation average of the local stress be equal to the macroscopic stress s . lgquating the macroscopic and microscopic power, Chen et al. (1996) showed that 3 relates to the local stress S’ by s = 4’ (S’). The local stress S’ can be determined in terms of the macroscopic deformation rate b by combining the constitutive equations for single crystals (6) and the micro-macro transition (9).
CHEN. et al.. Nodoncar
Texture
3.
Evolution
Evolution
of
Under Axisymmetric
11
Straining
Under axisymmetric straining, the orientation angles can be solved analytically. From an initially isotropic distribution. the ODF under axisymmetric straining is given by 1 @((e.cp) = 4l;e \ (e’xcos~~ + e-2rsgqFurthermore,
+
the second and the fourth moment coeflkients are given by a2
= ; (b2+ b’) (1 - barctan;> - i
~4
= -&
[
(28b”
+ 24) a2 + 77b* +631
(13)
where b=
J
(t’ + C’kdt
(14)
We now prescribe the material parameters suitable for HDPE. The strain-rate sensitivity exponent n is ass’igned the same value 9 as given by Lee et RI. (1993). The strain hardening of the slip systems is neglected. From the normalized resistance. # iso, of the slip system is chosen the same value as given in Table 1 of Lee et nl (1993). where TOis the initial shear strength on the easiest slip system The remaining parameter <‘ iz to be identified by fitting the experiment data of stress response. First consider the caSe of u&axial stretching. By fitting the experiment data of the stress strain curve. we take <’ + <” to be a constant of 1.1. As the material sample elongates. the ODF is focused to the value of 0 = 9 and 8 = K . Fig. 1 shows the curve of . versus the macroscopic equivalent strain. macroscopic equivalent stress. ?q = c’s = j dmcft. In the drawing, 5’s is normalized by ~~ . and Ps by +o/&-,. The n calculakd stress&rain curve exhibits a strong textural hardening, in agreement with the experiment data obtained by G’Sell and Jonas(1979). The strong textural hardening is obviously due to the rotation of inextensible chain axes towards the stretching direction. Next consider the case of globally unkxial compression. The same set of material constants are employed. The texture evolves to align the e -axes of various grams in the plane normal to the compression direction, namely 8 = r/2. The calculated equivalent stress-strain curve is included in Fig. 1. Contrasted with the case of stretching, the texture hardening in compression is rather mild. The microscopic deformation mode for uniaxial compression is featured by tranzverse slip. resulting in slow texture formation and mild texture hardening. The experimental results by Bartczak et aL (1992) under uniaxial compression exhibits more hardening than ou~.prediction, possibly due to the end-friction effect and the strain hardening in single crystal constitutive law.
References [l]
Bartczak. Z., Cohen. R. E. and Argon, A. S. (1992). Evolution of the crystalline texture of high-density polyethylene during uniaxial compression. Mcrcro-molecules. 25, 4692. [2] C&en, M. X., Yang, W. and Zheng Q.-S. (1996). Texture evolution of crystalline polymers under axi-symmetric straining. Submitted to Ini. J. Plasticity. [3] G’Sell. C. and Jonas. J. J. (1979). Determination of the plastic behaviour of solid polymers at constant true stain rate. J. Muter. Sci. 14 . 583-591.
12
Communications
in Nonlinear
Science
k Nuxnuicd
Vol.1.
Simulation
No.3(Jul.
1996)
[4] Lee, B. J.. Parks, D. M. and Ahzi. S. (1993). Micromechanical modelling of large plastic deformation and texture evolution in semi-crystalline polymers. J. Mech. Phys. Solids, 41. 1651-1687. [5] Parks, D. M. and Ahzi, S. (1990) Polycrystalline plastic deformation and texture evolution for crystals lacking independent slip systems. J. Mech. Phys. Solids. 38. 701-724.
0.0
0.5 Non-naked
Fig. 1
Ferromagnetism
Stress-strain
1.0 Equivalent
Strain
1.5
-
P(s
/ y,,)
responses under axisymmetric
in an Itinerant
D. F. WANG (Institut de Physique Th6orique Ecole Polytechnique PHH-Ecublens, CH-1015 Lausanne, Switaerland) Emaik [email protected]
Electron
2.0
straining.
System
1
F&l&ale de Lausanue,
Abstract: In this work, the ground states of the Hubbard model on complete graph are studied, for a 6nite lattice size L aud arl$trary on-site energy IT. We construct explicitly the ground states of the system when the number of the electrons N, 2 L + 1. In particular, for N. = L + 1, the ground state is ferromagnetic with total spin sI = (N, - 2)/2. Key Words: Ferromaguetism, Hubbard model, Kondo-lattice model Hubbard model hae been of considerable interest since the discovery of the high temperature superconductivityfll In one dimension, the Hubbard model is solvable with Betheauaatdq. The system exhibits an interesting SO(d) symmetry[~. One particular~feature is that at halMill ing, any small positive onsite interaction would make the sydem a Mottinsulato~*~. At Lessthan half filling, the low lying excitations of the system are characterized 'ThcPaPcrrr4sreceiTcdonMsS.23,1996
Nonlinear Evolution Constraints l
of Aggregates
with
X/d=25 Y/d=0
Inextensible
Ming-Xiang CBEN, Wei YANG and Quan-Shui ZBENG ( Department of Engineering &chanics, Tsinghus University, Beijing 100084, China) Email: [email protected] Abstract: Crydlline and aemicrystalline polymers are formed ZM aggregates of grains with evoIving inextensible axes. This inextensibIe constraint leads to texture evolution under large plastic deformation. This paper reveals the nonlinear texture evolution of crystalline polymers under axiqmmetric straining. Key Words: Nonlinear texture evolution, crystalline polymers, inextensible constraint.
Introduction Crystalline and aemicryetalline polymers are formed as aggregates of grains with evolving inextensible axes. Their mechanical behavior is dominated by the phase of crystallinity, ‘Thi8pspC-l!tSraceivedopM~10,1~
CHEN. et d.: Nonlmear Evolution of.. .
9
usually in a configuration of chain folding. Due to the stiff covalent bonding of molecular chain, a crystalline polymer is almost inextensible in the chain direction. Only four independent slip systems may operate within a grain, leading to texture evolution under large plastic deformation. The crystallographic chain axes tend to align with the direction of maximum stretch. The preferential orientation induces anisotropy of the macroscopic mechanical behavior. The present paper describes nonlinear evolution of such aggregates. A continuous orientation distribution function (ODP) is introduced to describe the orientation distribution of the inextensible axes. The ODF is expanded as a convergent series of irreducible tensors, whose co&&&s are termed moment tensors of different orders. Microscopic-macroscopic transition for the deformation accounting for chain inetiensibility ahd the overall material symmetry is proposed using tensor function representation theory. The model is applied to simulate the texture evolution and texture hardening under axizymmetric straining. 1. Constrained
Single
Crystals
The crystal lattice of crystalline polymers such as HDPE is orthorhombic-with the c-axis in the chain diiection. The slip planes are parallel to the c-axis and a slip direction can be either parallel to the c-axis (chain slip) or perpendi&lar ti> it (transverse slip). Let sB and nfl be unit vectors representing the slip direction and the slip plane normal of the b-th slip system, and +fl the shear rate of that slip system. The symmetric part of the velocity gradient is expressed by D = c
‘& -1 ,+(s$
n> + nd 18,d) = &“Rp
9=1*
ill
J=l
Simmilar expression can be drived for the skew-symmetric crystallographic axis c is
part W. The rotating rate of a
k=Wc-[(c@c)D-D(c@c)]c
12)
Each slip system, with slip planes penciled by the chain direction, ailows no stretching in the chain direction. This inextensible constraint can be expressed as C’: R’ = 0
(3)
wher6.C =.c Q c - (l/3) 1 and 1 the second order identity tensor. The constitutive description of a single crystal consists of a relation between the shear rate +@ and the resoIved shear stress 78. Following Parks and Ahzi (1990), we assume that they are related through a power law +a = +;f
I$~‘-’
(4)
where n is the rate exponent, +o the reference shear rate and gp the shear strength associated with the @th slip system. The locai Cauchy stress deviator in the crystal can be decomposed into the direct sum of a stress tensor S’ normal to C’ and a component SC aligned with C’ S = S’ + % (trSC’) C’ = S’ + SCC’
(5)
Relations (4), (5) and (1) lead to a constitutive equation for a constrained single crystal :s’
The reaction stress component SC has to be determined from the equilibrium.
(‘3)
10
Cdcdio~~~
2. Orientation
in Nonlinear
Vol.1. NQJul.
Science& Numerical Simulation
1996)
Distribution
A representative aggregate element contains many individual grains. The orientation of the chain axis of a crystal grain is described by the polar angles (0,4) of the c-axis. An orientation distribution function, @(e, d), is introduced to describe the orientation distribution of the chain axes within an aggregate element. It ia normalized and its evolution satisfies an orientation conservation law. We assume that the microscopic variables depend on the grain orientation and the orientation diitribution of the aggregate, and neglect the influences by the grain shape, grain size and grain boundaries. Under prescribed velocity boundary condition, one has the following self-consistent conditions for the deformation within the aggregate (I))
= b,
(W)
= w
where cusped parentheses denote the average over orientations ODF can be expanded in term of irreducible tensors
(7)
weighted by @(e,#).
@(c)z;ga;,.c@2n
The
(8)
n=l
where cg2” denotes the Zn-th tensor power of c. a;, is the deviatoric part of the moment tensor az,,, defined as az,, = (cB2”). Micro-macro Transition under Transverse Isotropy Undeformed melt-crystallized semi-crystalline polymers show a spherulitic structure. The aggregate initially behaves isotropically. Under axi-symmetric straining, the deformed spherulites exhibit a global transverse isotropy. Let k be a unit vector along the principal loading axis. The structure tensor that characterizes the transverse isotropy can be formed as M = k@k. For incompressible materials under ax&symmetric straining, the macroscopic deformation is fi = (g/2) (3M - 1) and %r = 0, where 2 ia the straining rate along the symmetry axis k. By the ieotropicization theorem, the ODF is an isotropic function of C’ and M. Denote at, w,. . . as the coefficients of a series of even order moment tensors. Two assumptions about micro-macro transition are adopted. (A) The local deformation rate D is a quadratic function of the current orientation C’; and (B) the local spin W linearly depends on the macroscopic deformation b and the current orientation C’, and only the second-order moment a2 is accounted for its orientation distribution effect. It was shown (Chen et al., 1996) that the microscopic deformation, D and W, can be expressed a~ w = C’ (c’b
- dc’)
where I denotes the fourth-order identity tensor. The first expression for D confirms and further details the form proposed by Parka and Ahzi (1990) using an intermediate traceless deformation rate. It is straightforward to verify that the consistent condition (7) is satisfied by (9). The coefficient [’ relates to the second and fourth order moments a2 and a4 by
and C” depends only on the coefficient a2. The &f-consistent condition for global equilibrium within the aggregate requires that the orientation average of the local stress be equal to the macroscopic stress s . lgquating the macroscopic and microscopic power, Chen et al. (1996) showed that 3 relates to the local stress S’ by s = 4’ (S’). The local stress S’ can be determined in terms of the macroscopic deformation rate b by combining the constitutive equations for single crystals (6) and the micro-macro transition (9).
CHEN. et al.. Nodoncar
Texture
3.
Evolution
Evolution
of
Under Axisymmetric
11
Straining
Under axisymmetric straining, the orientation angles can be solved analytically. From an initially isotropic distribution. the ODF under axisymmetric straining is given by 1 @((e.cp) = 4l;e \ (e’xcos~~ + e-2rsgqFurthermore,
+
the second and the fourth moment coeflkients are given by a2
= ; (b2+ b’) (1 - barctan;> - i
~4
= -&
[
(28b”
+ 24) a2 + 77b* +631
(13)
where b=
J
(t’ + C’kdt
(14)
We now prescribe the material parameters suitable for HDPE. The strain-rate sensitivity exponent n is ass’igned the same value 9 as given by Lee et RI. (1993). The strain hardening of the slip systems is neglected. From the normalized resistance. # iso, of the slip system is chosen the same value as given in Table 1 of Lee et nl (1993). where TOis the initial shear strength on the easiest slip system The remaining parameter <‘ iz to be identified by fitting the experiment data of stress response. First consider the caSe of u&axial stretching. By fitting the experiment data of the stress strain curve. we take <’ + <” to be a constant of 1.1. As the material sample elongates. the ODF is focused to the value of 0 = 9 and 8 = K . Fig. 1 shows the curve of . versus the macroscopic equivalent strain. macroscopic equivalent stress. ?q = c’s = j dmcft. In the drawing, 5’s is normalized by ~~ . and Ps by +o/&-,. The n calculakd stress&rain curve exhibits a strong textural hardening, in agreement with the experiment data obtained by G’Sell and Jonas(1979). The strong textural hardening is obviously due to the rotation of inextensible chain axes towards the stretching direction. Next consider the case of globally unkxial compression. The same set of material constants are employed. The texture evolves to align the e -axes of various grams in the plane normal to the compression direction, namely 8 = r/2. The calculated equivalent stress-strain curve is included in Fig. 1. Contrasted with the case of stretching, the texture hardening in compression is rather mild. The microscopic deformation mode for uniaxial compression is featured by tranzverse slip. resulting in slow texture formation and mild texture hardening. The experimental results by Bartczak et aL (1992) under uniaxial compression exhibits more hardening than ou~.prediction, possibly due to the end-friction effect and the strain hardening in single crystal constitutive law.
References [l]
Bartczak. Z., Cohen. R. E. and Argon, A. S. (1992). Evolution of the crystalline texture of high-density polyethylene during uniaxial compression. Mcrcro-molecules. 25, 4692. [2] C&en, M. X., Yang, W. and Zheng Q.-S. (1996). Texture evolution of crystalline polymers under axi-symmetric straining. Submitted to Ini. J. Plasticity. [3] G’Sell. C. and Jonas. J. J. (1979). Determination of the plastic behaviour of solid polymers at constant true stain rate. J. Muter. Sci. 14 . 583-591.
12
Communications
in Nonlinear
Science
k Nuxnuicd
Vol.1.
Simulation
No.3(Jul.
1996)
[4] Lee, B. J.. Parks, D. M. and Ahzi. S. (1993). Micromechanical modelling of large plastic deformation and texture evolution in semi-crystalline polymers. J. Mech. Phys. Solids, 41. 1651-1687. [5] Parks, D. M. and Ahzi, S. (1990) Polycrystalline plastic deformation and texture evolution for crystals lacking independent slip systems. J. Mech. Phys. Solids. 38. 701-724.
0.0
0.5 Non-naked
Fig. 1
Ferromagnetism
Stress-strain
1.0 Equivalent
Strain
1.5
-
P(s
/ y,,)
responses under axisymmetric
in an Itinerant
D. F. WANG (Institut de Physique Th6orique Ecole Polytechnique PHH-Ecublens, CH-1015 Lausanne, Switaerland) Emaik [email protected]
Electron
2.0
straining.
System
1
F&l&ale de Lausanue,
Abstract: In this work, the ground states of the Hubbard model on complete graph are studied, for a 6nite lattice size L aud arl$trary on-site energy IT. We construct explicitly the ground states of the system when the number of the electrons N, 2 L + 1. In particular, for N. = L + 1, the ground state is ferromagnetic with total spin sI = (N, - 2)/2. Key Words: Ferromaguetism, Hubbard model, Kondo-lattice model Hubbard model hae been of considerable interest since the discovery of the high temperature superconductivityfll In one dimension, the Hubbard model is solvable with Betheauaatdq. The system exhibits an interesting SO(d) symmetry[~. One particular~feature is that at halMill ing, any small positive onsite interaction would make the sydem a Mottinsulato~*~. At Lessthan half filling, the low lying excitations of the system are characterized 'ThcPaPcrrr4sreceiTcdonMsS.23,1996