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Some results concerning continuous media described by quasilinear reducible systems
Physics of the Earth and Planetary Interiors, 50(1988) 46—51 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands
46
Some results concerning continuous media described by quasilinear reducible systems Domenico Fusco Dipartimento di Matematica e Applicazioni, Universith di Napoli, Via Mezzocannone 8, 88010 Napoli (Italy) (Received February 28, 1987) Fusco, D., 1988. Some results concerning continuous media described by quasilinear reducible systems. Phys. Earth Planet. Inter., 50:46—51. An approach is given for reducing to certain canonical forms the hodograph equations associated with a quasilinear hyperbolic system of first order. Such a procedure also provides a suitable mathematical vehicle for characterizing nonlinear material laws.
1. Introduction Several theoretical methods have been developed for determining possible sets of exact solutions to the governing systems of equations in various physical contexts. Among others, the reduction to canonical form of the hodograph equations connected with homogeneous quasilinear systems of first order involving two dependent and two independent variables has received a great deal of attention from several authors investigating initial or boundary value problems arising from a wide range of physical phenomena. Furthermore such an approach has proved to be a useful tool for characterizing classes of constitutive laws to nonlinear media. Usually multiparameter forms for the material response functions are obtained. Therefore the available parameters may be used to approximate real material behaviour suggested by experimental results. The afore-mentioned approach has been successfully used to analyse one-dimensional wave propagation in elastic—plastic media which exhibit strain-hardening behaviour and also in such media as saturated soil, dry sand, clay silt and nonlinear dielectrics. An exhaustive list of references on this subject 0031-9201/88/$03.50
© 1988 Elsevier Science Publishers B.V.
is available (Rogers and Clements, 1975; Rogers et al., 1977; Rogers and Shadwick, 1982). In this paper we shall be concerned with first order hyperbolic systems of the form U, + A(U)U~ 0
(1.1)
=
where
u
=
U
,
A(u)
a11
=
a21
V
a12 a22
In (1.1) x, t are, respectively, space and time coordinates. Moreover, here and in what follows, subscripts will denote partial derivatives with respect to the indicated variables. The hyperbolicity of (1.1) requires the matrix A to admit two real distinct eigenvalues A and ~t with corresponding left eigenvectors i~ and i~ and right eigenvectors d~ and d~ spanning the Euclidean space E 2~ For convenience we assume A > ~t. By means of a standard procedure (see, for example, Jeffrey, 1976), for the systems (1.1) the Riemann invariants r and s can be introduced r=
ç (Ä) ,q1I ~dU,s=
iq21~~dU
J
1.2
47
where q1 and q2 are integrating factors satisfying =
(1.1) of the form 3h(U)/8t+ 8f(U)/3x=a(U, x, 1)
(1.7) leads to the integration of the pair of linear equations
.~~—(qi1~)~
=
(1.8)
h,.=Afr, h~=uf5
Thus, via the hodograph transformation x(r, s), t
=
=
The system (1.8) is similar to (1.5). (1.3)
t(r, s)
under the assumption 3 (x, t) 0 (r, s)
(1.4)
the system (1.1) can be reduced to the pair of linear equations Xr
=
itt,., X~=
Xt~
(1.5)
Our main aim is to provide a systematic approach for obtaining general conditions to be satisfied by the coefficients of the basic system (1.1) in order that the corresponding hodograph equations reduce to suitable canonical forms allowing explicit integration. Before proceeding further, it is convenient to make some remarks on the reasons which motivated the present study. Actually the system (1.1) encompasses a number of one-dimensional models of physical interest such as nonlinear elasticity, isentropic fluid dynamics, dielectrics. Furthermore, there are several wave propagation problems arising from different physical contexts to which the analysis developed here can be applied, although the media which are concerned there are not directly described by a governing system of field equations of the form (1.1). In connection with this last topic some typical cases are listed below. (1) Consider the following quasilinear hyperbolic nonhomogeneous system
(2) Within the framework of ‘Geometrical Shock Dynamics’ considered in Whitham (1974), the study of two-dimensional wave propagation on the shock front leads to the quasilinear (reducible) system of equations °a+ (1/~)M~ = 0
9~ (i/M)~ —
(1.6) with U, B (column) vectors of E 2• The model (1.6) is not reducible to the linear hodograph form. However, the Riemann field variables r(U) and s(U) can still be introduced through the relations (1.2). Therefore it is possible to see that the investigation of the existence of conservation laws to t)
(1.9) (1.10)
0
where the family a(x1, x2) a01 (a0 is the undisturbed sound speed), describes the successive shock positions whereas /3(x1, x2) const. represents the rays (orthogonal trajectories to a(x1, x2) const.). Moreover 9 is the ray inclination and ~ ~(M) denotes the ray tube area, where M is the Mach number. In section 2 we show a method of approach to the integration of the hodograph system (1.5). Furthermore in section 3 we consider, within the theoretical framework developed herein, the (onedimensional) model governing nonlinear elasticity. In particular we are able to characterize a class of stress—strain laws which can possess a point of inflexion. =
=
=
=
2. Integration of the hodograph system From the hodograph system (1.5), by eliminating x, the following linear second order equation is obtained !J.s
U,+A(U)U~B(U, X,
=
Ar
t
trs~A_,Y,± A—~
=0
(2.1)
The classical Riemann method of integration provides the solution of eq. 2.1 in terms of a Riemann function and initial data specified along an initial line in the hodograph plane. However, it has been• shown (Donato and Fusco, 1980) that a remarkable simplification in integrating (2.1) is obtained if
48
)
the following condition holds O / Ar Os A —
=
of g(r, s) is usually possible in all the models of
0 ~. Or ~ A)
(2.2)
—
In fact by introducing g(r, s)
J[Ar dr
=
—
~ ds]/(A
—
~
and using the transformation of variables t* exp(g)t =
(2.3)
eq. 2.1 specializes to exp(g) (2 4) Or Os where the differential operator is self-adjoint. We remark that all the models of physical interest belonging to the class (1.1) satisfy (2.2) without any restriction on the functional forms of the constitutive laws which are involved there. Now a solution of (2.4) is sought in the form 02
t7~=t’~ exp(—g)
t*(r, s)
=
(1/2) ~ [a~(r, s)’F~(r)
+ /3~(r,s)’I’~(s)} (2.5) where a,, and /3,, are to be determined, while and ‘I’,, satisfy the recurrence relations dtI~,,/dr ~ d’Iç/ds 24’,,~~ (2.6) ~,,
can impose a further condition on the coefficients of the basic model (1.1) (i.e. on the material response functions), so that the series solution (2.5) can be truncated after n + 1 terms. In the process, that requirement will suggest a wide class of canonical forms to which the hodograph system can be reduced. Although such an analysis has been developed fully by Currô and Fusco (1988), we shall give here some details for the reader’s convenience. Thus, let us assume in (2.5) that a0, a1,..., ct,~# 0 whereas ak 0, k n + 1 The same assumption is made for the coefficients f3,~.By inspection of the recurrence relations (2.7), (2.8) determining a,,, it is possible to show that the present requirement is satisfied if the following condition holds =
P,,=A_O2ln(P~~Pi...P,,i)/OrOs=0
=
Substitution of (2.5) into (2.4) and use of (2.6) then yields the following system of recurrence relations for a,, and /3,, (2.7) Os2a,, Or O 1 Oa,, Or Os + 2-i-— 2$,,_, OOr Os +
quently a general series solution like (2.5) for the governing equation (2.4) can be sought. However, on account of the complexity of the solution obtained above and with a view to the possibility of obtaining explicit solutions which can be of direct use in physical applications, we
=
n=0
=
physical interest belonging to the class (1.1) without any restriction on the functional form of the constitutive laws which are involved. Conse-
—
A(r, s)a,,1
=
0
(2.8)
0
(2.9)
~,
(2.10)
where 21nP P0=A, P1=A—O 0/OrOs, Pj=A_O2ln(P~P~1...P,i)/OrOs with i= 1,2, n. Since for the models of physical interest of the form (1.1) the constitutive laws are involved in
2—~--— A(r, s)/3,, —
1
=
where A(r, s) exp(—g) ~2 exp(g)/Or Os. Once A is specified (according to the constitutive laws), the coefficients a,, and /3,, can easily be obtained by integrating (2.7)—(2.9). We note that (2.5) can be considered a Bergman-like series whose uniform convergence can be studied, in principle, within the theory developed by Bergman (1971). As we have observed already the introduction =
g(r, the material condition (2.10) represents restriction s), to the response functions. aThe simplest cases are: (1) n 0 (i.e. the series terminates after the first term) so that (2.10) specializes to A 0, (2) n 1 (i.e. truncation at the second term of the series) so that (2.10) gives rise to =
=
=
02
ln A/Or Os
=
A
(2.11)
49
On introduction of w Liouville equation
=
ln A. (2.11) reduces to the
wrs=exp(w)
(2.12)
which, after integration, yields A
{
ew exp( X1 (r)
=
I
—
X2 (c))
so that (2.4) reduces to n+1 r*+s**s*)0
Ir*S*+
(2.15)
Equation 2.15 is similar to the one obtained by Courant and Friedrichs (1948) for a polytropic gas with special values of the adiabatic exponent y (y 3, 5/3, 7/5,...). Therefore the solution of
}
r
=
xI(8/2)fexP(Xi(E)) d~ (2.15) can be written in the form I=exp(g)t/(r*+s*)~+l 0~ I L(r~)
r~
+6~J (-x2(E)) dEl so
-2
where X1(r) and X2(s) are arbitrary functions which must be specified further to characterize the constitutive laws for the model of interest and also 6 is a nonzero constant known as a Bäcklund parameter. The condition (2.10) for n> 1 is fulfilled when A
=
(n(n + 1)/2)(exp(X1(r)
[
(6/2)1~ exp( x1 ( ~
+
6’j
PS
—
X2(s))}
~ —2
exp(—X 2(E)) dEl
(2.13)
~
L [
~ i +
~
(r* + 5* M(s *) (r * + * P1 + I
~)
]
with arbitrary functions L and M and any constant r. In closing this section, we remark that it is a simple matter to ascertain that the canonical forms and the sets of explicit solutions to the hodograph system which have been obtained by different authors in several problems of physical interest emerge fromShadwick, our results particular cases (see Rogers and 1982asand references quoted there).
Now let us point out that the requirement (2.10) (or (2.13)) gives rise to some canonical forms for the governing equation (2.4). If A = 0, then (2.4) specializes to the wave equation rs
=
0
3. Physical example: nonlinear elasticity Here the governing model is given by
(2.14)
whose immediate integration yields
u,
t*exp(g)1L(r)+M(s)
v,U~0
where L ( r) and M(s) are arbitrary functions which must be determined from given initial or boundary conditions, If n ? 1 and A is given by (2.13) it is convenient to introduce the following change of variables
where x is a Lagrangian coordinate; U denotes the particle velocity and v represents the strain. Moreover, T= T(v) is the stress with T’ = dT/dv, while p is the constant reference density. The characteristic speeds are 2, ~t = wave [T /p ~ ]I/2 A = [T’/p]” and the Riemann invariants are given by
*
=
(6/2)Jexp(Xi(r)) dr,
=
(1/6)fexp( —X 2(s)) ds,
=
1(r*
+
s*)~~
—
(T’(v)/p)v~ = 0
(3.1) (3.2)
—
2r =
f[TP/p]1/2
dv + U,
2s =
f
dv
[TP/p]v2
—
u
50
Furthermore g
=
ln A~2.As in the present case
ion at v~= v 0 n/(1 + 2n)b. Therefore such a law can be used, for example, to model the response of polycrystalline materials under axial —
g = g( r + s), then we are led to the following form of the functions X1(r) and X2(s) X1(r)
=
ar + /3, X2(s) = as + ~
(3.3)
where a, /3, y are constants. Therefore several possibilities arise for integrating eq. 2.13 which must be satisfied by g(r + s) to reduce the governing system of eq. 3.1 and 3.2 to linear canonical forms. In connection with the simplest choice a = 0 and (6/2) exp( /3) = (1/6) exp( y) it is possible to characterize the following nonlinear stress—strain relation —
T— T0=w2n~±2n[3+2fl
2)(3 + 2n)w 4n + b(1 (1 + 2n)(1 4n2)b2w2}
—
—
—
x ((1 + 2n)(1
—
4n2)(3
+ 2n)}1
(3.4) where w = (v
—
—
v0)]
and b, T0, v0 are constants. The arbitrary parameters appearing in (3.4) can be used for approximating nonlinear material laws derived empirically. Such an analysis can be carned out along the lines suggested by Cekirge and Varley (1973) and Kazakia and Varley (1974) and it will not considered here, However, we describe, in a few words, the wide versatility of the constitutive relation (3.4) for describing real material behaviour. Actually the theoretical model laws which have been derived by Cekirge and Varley (1973) and in Kazakia and Varley (1974) can usually approximate the response of materials over ranges in which the Lagrangian sound speed A has either a monotonically increasing or decreasing dependence on the strain, As far as the constitutive equation (3.4) with n > 0 is concerned, it is easy to ascertain that if <
4. Conclusions and remarks In this paper we have shown a method of approach that allows one to obtain possible sets of exact solutions to a class of quasilinear hyperbolic systems of first order by requiring that the associated hodograph equations reduce to certain canonical forms. These solutions can be relevant
v 0)/(1 + 2n)[1 + b(v
b
course if in (3.3) oneclasses assumes acompression. ~ 0 then, inOf principle, other possible of stress—strain laws can be determined by requiring the condition (2.13) to hold. Further examples of nonlinear model laws generated through the use of the present reduction procedure in a different physical context such as nonlinear dielectric media are considered by Currô and Fusco (1988).
0 and
V0
+
b
-
in problems such as simple wave interactions or wave propagation through layered media (Seymour and Varley, 1982; Rogers and Shadwick, 1982). Furthermore the differential condition which the coefficients of the governing system must satisfy, if the present reduction procedure is to hold, also provides a suitable mathematical motivation for characterizing classes of functional forms for the constitutive relations in nonlinear models of physical interest. In particular, in the case of a nonlinear elastic rod, we deduced a theoretical stress—strain relation which can approximate material laws with a point of inflection. Finally we remark that the governing model must be reducible to the hodograph form to apply the present approach for finding out exact solutions. Hence it cannot be used to analyse materials whose equation of state also depends explicitly on x and/or t. These laws occur, for example, in soil mechanics during unloading (Cristescu, 1967) or in nonlinear elastic rods with a variable cross-section. However, in these cases a powerful method
or, alternatively, if
either for determining exact solutions or for char-
b
acterizing functional the constitutive laws is provided by theforms groupfor theoretical analysis (Ames, 1972; Bluman and Cole, 1974); Ovsianni-
>
0 and v0
—
b~ <.~, <~0
then the stress—strain law admits a point of inflex-
51
kov, 1982). Some recent results of physical interest in such a context are presented by Fusco, (1984) and by Currô and Fusco (1986). Acknowledgements The author expresses his sincere thanks to the Soviet Geophysical Conimittee as well as to Prof. A.V. Nikolaev and to Dr. M.A. Grinfeld for the kind hospitality shown to him during the symposium on Nonlinear Seismology. This research was partially supported by C. N. R. through Gruppo Nazionale per la Fisica Matematica of Italy.
References Ames, W.F., 1972. Nonlinear Partial Differential Equations rn Engineering, Vol. II. Academic Press, New York. Bergman, S., 1971. Integral Operators in the Theory of Linear Partial Differential Equations. Springer, Berlin. Bluman, G.W. and Cole, J.D., 1974. Similarity Methods for Differential Equations. Springer, Berlin. Cekirge, H.M. and Varley, E., 1973. Large amplitude waves in bounded media: I. Reflexion and transmission of large amplitude shockless pulses at an interface. Philos. Trans. R. Soc. London Ser. A, 273: 261—313. Courant, R. and Friedrichs, K.O., 1948. Supersonic Flow and Shock Waves. Wiley, New York.
Cristescu, N., 1967. Dynamic Plasticity. North Holland, Amsterdam. Currô, C. and Fusco, D., 1986, Invariant solutions and constitutive laws for a nonlinear elastic rod of variable crosssection, ZAMP, 37: 244—255. Curio, C. and Fusco, D., 1988. Reduction to linear canonical forms and generation of conservation laws for a class of quasilinear hyperbolic systems. mt. J. Non-Linear Mech., in press. Donato, A. and Fusco, D., 1980, Some applications of the l~.iemannmethol to electromagnetic wave propagation in nonlinear media. ZAMM, 60: 539-542. Fusco, D., 1984, Group analysis and constitutive laws for fluid filled elastic tubes. mt. J. Non-linear Mech., 19: 565—574. Jeffrey, A., 1976. Quasilinear Hyperbolic Systems and Waves. Pitman, London. Kazakia, J.Y. and Varley, E., 1974. Large amplitude waves in bounded media: II. The deformation of an impulsively loaded slab: the first reflexion. Philos. Trans. R. Soc. London Ser. A, 277: 191—237. Ovsiannikov, L.V., 1982. Group Analysis of Differential Equations. Academic Press, New York,. Rogers, C. and Clements, D.L., 1975. On the reduction of the hodograph equations for one-dimensional elastic-plastic wave propagation. AppI. Mat., 37: 469—474. Rogers, C. and Shadwick, W.F., 1982. Bäcklund Transformations and their Applications. Academic Press, New York. Rogers, C., Cekirge, H.M. and Askar, A., 1977. Electromagnetic wave propagation in nonlinear dielectric media. Acta Mechanica, 26: 59—73. Seymour, B.R. and Varley, E., 1982. Exact solutions describing soliton-lilce interactions in a non-dispersive medium. SIAM J. Appl. Math., 42: 804—821. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York.
46
Some results concerning continuous media described by quasilinear reducible systems Domenico Fusco Dipartimento di Matematica e Applicazioni, Universith di Napoli, Via Mezzocannone 8, 88010 Napoli (Italy) (Received February 28, 1987) Fusco, D., 1988. Some results concerning continuous media described by quasilinear reducible systems. Phys. Earth Planet. Inter., 50:46—51. An approach is given for reducing to certain canonical forms the hodograph equations associated with a quasilinear hyperbolic system of first order. Such a procedure also provides a suitable mathematical vehicle for characterizing nonlinear material laws.
1. Introduction Several theoretical methods have been developed for determining possible sets of exact solutions to the governing systems of equations in various physical contexts. Among others, the reduction to canonical form of the hodograph equations connected with homogeneous quasilinear systems of first order involving two dependent and two independent variables has received a great deal of attention from several authors investigating initial or boundary value problems arising from a wide range of physical phenomena. Furthermore such an approach has proved to be a useful tool for characterizing classes of constitutive laws to nonlinear media. Usually multiparameter forms for the material response functions are obtained. Therefore the available parameters may be used to approximate real material behaviour suggested by experimental results. The afore-mentioned approach has been successfully used to analyse one-dimensional wave propagation in elastic—plastic media which exhibit strain-hardening behaviour and also in such media as saturated soil, dry sand, clay silt and nonlinear dielectrics. An exhaustive list of references on this subject 0031-9201/88/$03.50
© 1988 Elsevier Science Publishers B.V.
is available (Rogers and Clements, 1975; Rogers et al., 1977; Rogers and Shadwick, 1982). In this paper we shall be concerned with first order hyperbolic systems of the form U, + A(U)U~ 0
(1.1)
=
where
u
=
U
,
A(u)
a11
=
a21
V
a12 a22
In (1.1) x, t are, respectively, space and time coordinates. Moreover, here and in what follows, subscripts will denote partial derivatives with respect to the indicated variables. The hyperbolicity of (1.1) requires the matrix A to admit two real distinct eigenvalues A and ~t with corresponding left eigenvectors i~ and i~ and right eigenvectors d~ and d~ spanning the Euclidean space E 2~ For convenience we assume A > ~t. By means of a standard procedure (see, for example, Jeffrey, 1976), for the systems (1.1) the Riemann invariants r and s can be introduced r=
ç (Ä) ,q1I ~dU,s=
iq21~~dU
J
1.2
47
where q1 and q2 are integrating factors satisfying =
(1.1) of the form 3h(U)/8t+ 8f(U)/3x=a(U, x, 1)
(1.7) leads to the integration of the pair of linear equations
.~~—(qi1~)~
=
(1.8)
h,.=Afr, h~=uf5
Thus, via the hodograph transformation x(r, s), t
=
=
The system (1.8) is similar to (1.5). (1.3)
t(r, s)
under the assumption 3 (x, t) 0 (r, s)
(1.4)
the system (1.1) can be reduced to the pair of linear equations Xr
=
itt,., X~=
Xt~
(1.5)
Our main aim is to provide a systematic approach for obtaining general conditions to be satisfied by the coefficients of the basic system (1.1) in order that the corresponding hodograph equations reduce to suitable canonical forms allowing explicit integration. Before proceeding further, it is convenient to make some remarks on the reasons which motivated the present study. Actually the system (1.1) encompasses a number of one-dimensional models of physical interest such as nonlinear elasticity, isentropic fluid dynamics, dielectrics. Furthermore, there are several wave propagation problems arising from different physical contexts to which the analysis developed here can be applied, although the media which are concerned there are not directly described by a governing system of field equations of the form (1.1). In connection with this last topic some typical cases are listed below. (1) Consider the following quasilinear hyperbolic nonhomogeneous system
(2) Within the framework of ‘Geometrical Shock Dynamics’ considered in Whitham (1974), the study of two-dimensional wave propagation on the shock front leads to the quasilinear (reducible) system of equations °a+ (1/~)M~ = 0
9~ (i/M)~ —
(1.6) with U, B (column) vectors of E 2• The model (1.6) is not reducible to the linear hodograph form. However, the Riemann field variables r(U) and s(U) can still be introduced through the relations (1.2). Therefore it is possible to see that the investigation of the existence of conservation laws to t)
(1.9) (1.10)
0
where the family a(x1, x2) a01 (a0 is the undisturbed sound speed), describes the successive shock positions whereas /3(x1, x2) const. represents the rays (orthogonal trajectories to a(x1, x2) const.). Moreover 9 is the ray inclination and ~ ~(M) denotes the ray tube area, where M is the Mach number. In section 2 we show a method of approach to the integration of the hodograph system (1.5). Furthermore in section 3 we consider, within the theoretical framework developed herein, the (onedimensional) model governing nonlinear elasticity. In particular we are able to characterize a class of stress—strain laws which can possess a point of inflexion. =
=
=
=
2. Integration of the hodograph system From the hodograph system (1.5), by eliminating x, the following linear second order equation is obtained !J.s
U,+A(U)U~B(U, X,
=
Ar
t
trs~A_,Y,± A—~
=0
(2.1)
The classical Riemann method of integration provides the solution of eq. 2.1 in terms of a Riemann function and initial data specified along an initial line in the hodograph plane. However, it has been• shown (Donato and Fusco, 1980) that a remarkable simplification in integrating (2.1) is obtained if
48
)
the following condition holds O / Ar Os A —
=
of g(r, s) is usually possible in all the models of
0 ~. Or ~ A)
(2.2)
—
In fact by introducing g(r, s)
J[Ar dr
=
—
~ ds]/(A
—
~
and using the transformation of variables t* exp(g)t =
(2.3)
eq. 2.1 specializes to exp(g) (2 4) Or Os where the differential operator is self-adjoint. We remark that all the models of physical interest belonging to the class (1.1) satisfy (2.2) without any restriction on the functional forms of the constitutive laws which are involved there. Now a solution of (2.4) is sought in the form 02
t7~=t’~ exp(—g)
t*(r, s)
=
(1/2) ~ [a~(r, s)’F~(r)
+ /3~(r,s)’I’~(s)} (2.5) where a,, and /3,, are to be determined, while and ‘I’,, satisfy the recurrence relations dtI~,,/dr ~ d’Iç/ds 24’,,~~ (2.6) ~,,
can impose a further condition on the coefficients of the basic model (1.1) (i.e. on the material response functions), so that the series solution (2.5) can be truncated after n + 1 terms. In the process, that requirement will suggest a wide class of canonical forms to which the hodograph system can be reduced. Although such an analysis has been developed fully by Currô and Fusco (1988), we shall give here some details for the reader’s convenience. Thus, let us assume in (2.5) that a0, a1,..., ct,~# 0 whereas ak 0, k n + 1 The same assumption is made for the coefficients f3,~.By inspection of the recurrence relations (2.7), (2.8) determining a,,, it is possible to show that the present requirement is satisfied if the following condition holds =
P,,=A_O2ln(P~~Pi...P,,i)/OrOs=0
=
Substitution of (2.5) into (2.4) and use of (2.6) then yields the following system of recurrence relations for a,, and /3,, (2.7) Os2a,, Or O 1 Oa,, Or Os + 2-i-— 2$,,_, OOr Os +
quently a general series solution like (2.5) for the governing equation (2.4) can be sought. However, on account of the complexity of the solution obtained above and with a view to the possibility of obtaining explicit solutions which can be of direct use in physical applications, we
=
n=0
=
physical interest belonging to the class (1.1) without any restriction on the functional form of the constitutive laws which are involved. Conse-
—
A(r, s)a,,1
=
0
(2.8)
0
(2.9)
~,
(2.10)
where 21nP P0=A, P1=A—O 0/OrOs, Pj=A_O2ln(P~P~1...P,i)/OrOs with i= 1,2, n. Since for the models of physical interest of the form (1.1) the constitutive laws are involved in
2—~--— A(r, s)/3,, —
1
=
where A(r, s) exp(—g) ~2 exp(g)/Or Os. Once A is specified (according to the constitutive laws), the coefficients a,, and /3,, can easily be obtained by integrating (2.7)—(2.9). We note that (2.5) can be considered a Bergman-like series whose uniform convergence can be studied, in principle, within the theory developed by Bergman (1971). As we have observed already the introduction =
g(r, the material condition (2.10) represents restriction s), to the response functions. aThe simplest cases are: (1) n 0 (i.e. the series terminates after the first term) so that (2.10) specializes to A 0, (2) n 1 (i.e. truncation at the second term of the series) so that (2.10) gives rise to =
=
=
02
ln A/Or Os
=
A
(2.11)
49
On introduction of w Liouville equation
=
ln A. (2.11) reduces to the
wrs=exp(w)
(2.12)
which, after integration, yields A
{
ew exp( X1 (r)
=
I
—
X2 (c))
so that (2.4) reduces to n+1 r*+s**s*)0
Ir*S*+
(2.15)
Equation 2.15 is similar to the one obtained by Courant and Friedrichs (1948) for a polytropic gas with special values of the adiabatic exponent y (y 3, 5/3, 7/5,...). Therefore the solution of
}
r
=
xI(8/2)fexP(Xi(E)) d~ (2.15) can be written in the form I=exp(g)t/(r*+s*)~+l 0~ I L(r~)
r~
+6~J (-x2(E)) dEl so
-2
where X1(r) and X2(s) are arbitrary functions which must be specified further to characterize the constitutive laws for the model of interest and also 6 is a nonzero constant known as a Bäcklund parameter. The condition (2.10) for n> 1 is fulfilled when A
=
(n(n + 1)/2)(exp(X1(r)
[
(6/2)1~ exp( x1 ( ~
+
6’j
PS
—
X2(s))}
~ —2
exp(—X 2(E)) dEl
(2.13)
~
L [
~ i +
~
(r* + 5* M(s *) (r * + * P1 + I
~)
]
with arbitrary functions L and M and any constant r. In closing this section, we remark that it is a simple matter to ascertain that the canonical forms and the sets of explicit solutions to the hodograph system which have been obtained by different authors in several problems of physical interest emerge fromShadwick, our results particular cases (see Rogers and 1982asand references quoted there).
Now let us point out that the requirement (2.10) (or (2.13)) gives rise to some canonical forms for the governing equation (2.4). If A = 0, then (2.4) specializes to the wave equation rs
=
0
3. Physical example: nonlinear elasticity Here the governing model is given by
(2.14)
whose immediate integration yields
u,
t*exp(g)1L(r)+M(s)
v,U~0
where L ( r) and M(s) are arbitrary functions which must be determined from given initial or boundary conditions, If n ? 1 and A is given by (2.13) it is convenient to introduce the following change of variables
where x is a Lagrangian coordinate; U denotes the particle velocity and v represents the strain. Moreover, T= T(v) is the stress with T’ = dT/dv, while p is the constant reference density. The characteristic speeds are 2, ~t = wave [T /p ~ ]I/2 A = [T’/p]” and the Riemann invariants are given by
*
=
(6/2)Jexp(Xi(r)) dr,
=
(1/6)fexp( —X 2(s)) ds,
=
1(r*
+
s*)~~
—
(T’(v)/p)v~ = 0
(3.1) (3.2)
—
2r =
f[TP/p]1/2
dv + U,
2s =
f
dv
[TP/p]v2
—
u
50
Furthermore g
=
ln A~2.As in the present case
ion at v~= v 0 n/(1 + 2n)b. Therefore such a law can be used, for example, to model the response of polycrystalline materials under axial —
g = g( r + s), then we are led to the following form of the functions X1(r) and X2(s) X1(r)
=
ar + /3, X2(s) = as + ~
(3.3)
where a, /3, y are constants. Therefore several possibilities arise for integrating eq. 2.13 which must be satisfied by g(r + s) to reduce the governing system of eq. 3.1 and 3.2 to linear canonical forms. In connection with the simplest choice a = 0 and (6/2) exp( /3) = (1/6) exp( y) it is possible to characterize the following nonlinear stress—strain relation —
T— T0=w2n~±2n[3+2fl
2)(3 + 2n)w 4n + b(1 (1 + 2n)(1 4n2)b2w2}
—
—
—
x ((1 + 2n)(1
—
4n2)(3
+ 2n)}1
(3.4) where w = (v
—
—
v0)]
and b, T0, v0 are constants. The arbitrary parameters appearing in (3.4) can be used for approximating nonlinear material laws derived empirically. Such an analysis can be carned out along the lines suggested by Cekirge and Varley (1973) and Kazakia and Varley (1974) and it will not considered here, However, we describe, in a few words, the wide versatility of the constitutive relation (3.4) for describing real material behaviour. Actually the theoretical model laws which have been derived by Cekirge and Varley (1973) and in Kazakia and Varley (1974) can usually approximate the response of materials over ranges in which the Lagrangian sound speed A has either a monotonically increasing or decreasing dependence on the strain, As far as the constitutive equation (3.4) with n > 0 is concerned, it is easy to ascertain that if <
4. Conclusions and remarks In this paper we have shown a method of approach that allows one to obtain possible sets of exact solutions to a class of quasilinear hyperbolic systems of first order by requiring that the associated hodograph equations reduce to certain canonical forms. These solutions can be relevant
v 0)/(1 + 2n)[1 + b(v
b
course if in (3.3) oneclasses assumes acompression. ~ 0 then, inOf principle, other possible of stress—strain laws can be determined by requiring the condition (2.13) to hold. Further examples of nonlinear model laws generated through the use of the present reduction procedure in a different physical context such as nonlinear dielectric media are considered by Currô and Fusco (1988).
0 and
V0
+
b
-
in problems such as simple wave interactions or wave propagation through layered media (Seymour and Varley, 1982; Rogers and Shadwick, 1982). Furthermore the differential condition which the coefficients of the governing system must satisfy, if the present reduction procedure is to hold, also provides a suitable mathematical motivation for characterizing classes of functional forms for the constitutive relations in nonlinear models of physical interest. In particular, in the case of a nonlinear elastic rod, we deduced a theoretical stress—strain relation which can approximate material laws with a point of inflection. Finally we remark that the governing model must be reducible to the hodograph form to apply the present approach for finding out exact solutions. Hence it cannot be used to analyse materials whose equation of state also depends explicitly on x and/or t. These laws occur, for example, in soil mechanics during unloading (Cristescu, 1967) or in nonlinear elastic rods with a variable cross-section. However, in these cases a powerful method
or, alternatively, if
either for determining exact solutions or for char-
b
acterizing functional the constitutive laws is provided by theforms groupfor theoretical analysis (Ames, 1972; Bluman and Cole, 1974); Ovsianni-
>
0 and v0
—
b~ <.~, <~0
then the stress—strain law admits a point of inflex-
51
kov, 1982). Some recent results of physical interest in such a context are presented by Fusco, (1984) and by Currô and Fusco (1986). Acknowledgements The author expresses his sincere thanks to the Soviet Geophysical Conimittee as well as to Prof. A.V. Nikolaev and to Dr. M.A. Grinfeld for the kind hospitality shown to him during the symposium on Nonlinear Seismology. This research was partially supported by C. N. R. through Gruppo Nazionale per la Fisica Matematica of Italy.
References Ames, W.F., 1972. Nonlinear Partial Differential Equations rn Engineering, Vol. II. Academic Press, New York. Bergman, S., 1971. Integral Operators in the Theory of Linear Partial Differential Equations. Springer, Berlin. Bluman, G.W. and Cole, J.D., 1974. Similarity Methods for Differential Equations. Springer, Berlin. Cekirge, H.M. and Varley, E., 1973. Large amplitude waves in bounded media: I. Reflexion and transmission of large amplitude shockless pulses at an interface. Philos. Trans. R. Soc. London Ser. A, 273: 261—313. Courant, R. and Friedrichs, K.O., 1948. Supersonic Flow and Shock Waves. Wiley, New York.
Cristescu, N., 1967. Dynamic Plasticity. North Holland, Amsterdam. Currô, C. and Fusco, D., 1986, Invariant solutions and constitutive laws for a nonlinear elastic rod of variable crosssection, ZAMP, 37: 244—255. Curio, C. and Fusco, D., 1988. Reduction to linear canonical forms and generation of conservation laws for a class of quasilinear hyperbolic systems. mt. J. Non-Linear Mech., in press. Donato, A. and Fusco, D., 1980, Some applications of the l~.iemannmethol to electromagnetic wave propagation in nonlinear media. ZAMM, 60: 539-542. Fusco, D., 1984, Group analysis and constitutive laws for fluid filled elastic tubes. mt. J. Non-linear Mech., 19: 565—574. Jeffrey, A., 1976. Quasilinear Hyperbolic Systems and Waves. Pitman, London. Kazakia, J.Y. and Varley, E., 1974. Large amplitude waves in bounded media: II. The deformation of an impulsively loaded slab: the first reflexion. Philos. Trans. R. Soc. London Ser. A, 277: 191—237. Ovsiannikov, L.V., 1982. Group Analysis of Differential Equations. Academic Press, New York,. Rogers, C. and Clements, D.L., 1975. On the reduction of the hodograph equations for one-dimensional elastic-plastic wave propagation. AppI. Mat., 37: 469—474. Rogers, C. and Shadwick, W.F., 1982. Bäcklund Transformations and their Applications. Academic Press, New York. Rogers, C., Cekirge, H.M. and Askar, A., 1977. Electromagnetic wave propagation in nonlinear dielectric media. Acta Mechanica, 26: 59—73. Seymour, B.R. and Varley, E., 1982. Exact solutions describing soliton-lilce interactions in a non-dispersive medium. SIAM J. Appl. Math., 42: 804—821. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York.