2. S t r e s s e d S t a t e of N o n l i n e a r l y Elastic B o d i e s w i t h D i s l o c a t i o n s and D i s c l i n a t i o n s Finding the Stressed State of Elastic Body with Given Characteristics of Isolated Defect; Set u p o f the Problem 2.1 Given the Burgers' and Frank's vectors, bk and qk, let us consider the problem of finding displacements and stresses in multiply connected nonlinear elastic body with Volterra dislocations. There are two lines of attack on the problem. On the first way, the displacement vector components are unknown variables satisfying the nonlinear equilibrium equations and force conditions on the boundary surface of multiply connected volume v . We are setting up the boundary value problem in a simply connected domain that is obtained by making cuts in v . On the cut surfaces, the displacements must satisfy the jump conditions (1.3.6). It is clear that we can find the vector/~ instead of the displacement vector u . For unique solvability of the problem in question, we need to determine values of the displacement vector and the rotation tensor at some point of the body. From continuity at the cuts of Cauchy strain measure, A , it follows continuity of Kirchhoff stress tensor, P . Thus by equations (1.1.11), (1.3.2), it follows a relation for the jump on a cut in Piola stress vector, n . D , n.D+ =n.D_.Qk, (2.1.1) where n is the normal vector to the cut surface Tk • On the second way, the boundary value problem is formulated in the multiply connected domain; now the components of Cauchy strain measure, G sk , which are determined by the compatibility conditions (1.2.4) and the equilibrium equations (1.1.12)1, are unknowns. If there are no body forces these equations are easily reduced to a form containing only unknown G sk • Indeed, multiply equation (1.1.12)1 by C , with regard for D = P . C , we obtain [div P . C)]. C -~ -- 0. By equation (1.2.2), we have (2.1.2) 42 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations div P + r k. P . Hk = 0. (2.1.3) Since the tensors Hk are expressed in terms of Gmn and the Kirchhoff stress tensor for any material ( and thus for anisotropic elastic material) is expressed in terms of Cauchy strain measure, we can see that the only unknown variables in equations (2.1.3) are Gm~. Assume that the surface load is of tracking type, i.e. the vector f in (1.1.12)2 is of the form f(r) = fo(r, A(r)) • C(r). (2.1.4) In view of equation (2.1.4), the boundary conditions (1.1.12) are n . P = f0, (2.1.5) so they are also expressed completely in terms of Gmn • Combining the integral relations (1.3.13), (1.3.17), which express the defect parameters, bk, qk , in terms of the tensor field A , with equations (1.2.4), (2.1.3) and the boundary conditions (2.1.5), we get the final set up of the problem of equilibrium of a body with isolated defects in Cauchy strain measure components. Let us consider the case when the Frank's tensor of each defect is zero; all the defects are translational dislocations. Now the strain gradient C is a singlevalued function in a multiply connected domain, and to specify the strained state, one can formulate the boundary value problem with the components of C taken as unknowns. The compatibility equations for the deformation gradient consist of 9 linear equations, rot C -- 0 (2.1.6) Combining these with the equilibrium equations and boundary conditions (1.1.12) and, besides, with the integral relations ~ dr . C = bk, (2.1.7) k we get the set up of the problem on translational dislocations in terms of the deformation gradient. In equations (2.1.7) % is a contour enclosing the line of kth dislocation with Burgers' vector bk . Dependence of intensities of loads k and f on the deformation gradient is arbitrary. In particular, the vectors k and f may be some given functions of Lagrangian coordinates (the dead load). Under dead load, the equilibrium equations (1.1.12)1 is the identity if D = rot ,I, + D*, (2.1.8) where ,I~ is a twice-differentiable tensor of second order, D* is a particular solution to the equations (1.1.12)1. As well as for the given reference configuration, we can consider a case of given deformed configuration with prescribed characteristics of defect and 2.1 Finding the Stressed State 43 compose equations with respect to components of Almansi strain tensor. Such a set up, when Eulerian coordinates are independent variables, is possible only for isotropic material because the Cauchy stress tensor is fully determined by Almansi strain tensor only in isotropic elastic solids, that is tMN= FMN(gps) , tMN= nM. T. R N. (2.1.9) The functions gps(Q,M) must satisfy the compatibility equations, the equilibrium equations, DivT - 0, (2.1.10) the boundary conditions N . T = F and some relations which are similar to the equations (1.3.13), (1.3.17). If equations (2.1.9) can be uniquely inverted then the problem of Volterra dislocations can be stated in stresses. Now the equilibrium equations (2.1.10) may be identically satisfied with the tensor of stress functions, O = O T, Rot O = R M × 0 0 / i ) Q M. T = Rot ( Rot o)T, The tensor O must satisfy the compatibility equations, boundary conditions, and integral relations determining Burgers' and Frank's vectors of given defects. Consider now formulation of the system of equilibrium equations for a nonlinearly elastic body with continuously distributed field of translational dislocations when the density c~ is given in advance. Note that in the constitutive relation (1.1.20) for the Cauchy stress tensor, by (Vakulenko 1991), the deformation gradient must be replaced by the elastic distortion tensor, C (e) • det C (e)) T 0A(e) , In view of equations takes the form , (2.1.11) A(e) - C(e) • (c(e)) T (1.1.19), (1.5.7), and (2.1.11), 0A(e) the Piola stress tensor • On substituting equation (2.1.12) into equation (1.1.12)1, we obtain a vector equation that, being combined with equations (1.5.8), forms the full system for the unknowns C (p) and C (e). 44 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations 2.2 Variational F o r m u l a t i o n of t h e P r o b l e m of Volterra D i s l o c a t i o n s in N o n l i n e a r Elasticity Let external forces be conservative; we apply Lagrange variational principle to an elastic body with Volterra dislocations, that is the elementary work of external forces in equation (1.1.9) is the variation of a functional ~, called the potential of external loads, with respect to the displacement vector, ~ pk . ~Rdv + ~ f . SRda = 5E. (2.2.1) Transform the multiply connected volume v into a simply connected one making a sufficient number of partitions and consider the potential energy functional for the elastic body, A = .~,,YYdv - t~. (2.2.2) The functional is defined on the set of displacement fields, which are twice differentiable in the simply connected domain and satisfy the jump conditions (1.3.6) with given parameters bk, qk on the cuts 7k. Since 5u = 5R we can consider the vector field R , specifying the location of body particles in the deformed configuration, as an argument of the functional ,4 instead of the displacement field, u. The jump conditions for the vector R are equations (1.3.4). With regard for (1.1.20), (2.2.1), we obtain the variation of functional A (f,4= jfvtr (D T. grad~R) d v - ~ p k . ~ R d v - ~ f . S R d a . Using the divergence theorem and the equation (1.3.4) in the stationary condition, 5A = 0 we obtain - ~(div D + pk). 3Rdv + f ( n . D - f ) . ~ R d v (2.2.3) +E/(nk" k D _ . ~R_ - nk. D+. QT. ~R_)da = O. k Here, nk is the unit normal to the partition Tk which directs from the partition side marked by "-" to the "+"side. From the variational equation (2.2.3) it follows the differential equilibrium equations (1.1.12)I, the boundary conditions n. D = f on the body surface, and the dynamic conjugation conditions (2.1.1) on the cuts which transform the domain to be simply connected. Thus, the Euler equations for the potential energy functional are the equilibrium equations in displacements and the natural boundary conditions are the force conditions and dynamic conjugation conditions on the cuts. Now let us turn to the variational formulation of the problem of Volterra dislocations which is based on the concept of complementary energy. The specific 2.2 Variational Formulation 45 complementary energy of elastic body is introduced (Zubov 1970) as a function of Piola stress tensor. This function relates with the function of specific potential energy, )/V(C), by Legendre transformation I;(D) = tr (D. CT(D)) - W(D), C = 01)/0D. (2.2.4) To construct the function ~;(D), we need to invert the relationship for Piola stress tensor D(C) with respect to the deformation gradient, i.e., to express the deformation gradient as a function of Piola stress tensor. For an isotropic elastic body, the problem of representation of the deformation gradient in terms of Piola stress tensor has been fully studied in (Zubov 1976). This paper states in particular that there is a unique inverse of the relation D(C) if the rotation angles of material fibers are not too large. As to anisotropic material, the above problem remains to be unsolved. In this case, one can use a weaker formulation (de Veubeke 1972; Christoffersen 1973) of the complementary energy principle, in which the specific complementary energy is a function of two tensors: the Piola stress tensor and the rotation tensor A. In this case, the constitutive relation for elastic material takes the form S=~1 ( p . U + U . P) = ~1 (D. A T + A . D T) = 0)4;/OV. (2.2.5) So it follows tr (D. C T) - 14; = tr (S. U ) - 14;. This enables us to represent the specific complementary energy V as a function of the tensor S called Jaumann stress tensor: l)(S) = tr (S. U(S)) - )4; [U(S)]. (2.2.6) To construct the function (2.2.6), it is necessary to invert the dependence S(U) linking two symmetric tensors. This problem is much easier than that of defining the dependence C(D). In the weaker complementary energy principle, Jaumann tensor is assumed to be expressed, in accordance with equation (2.2.5), in terms of independent variations, Piola stress tensor, D, and the rotation tensor, A. By varying equation (2.2.6), we obtain (iV = tr (U. 5S) = t r ( U . A . 5D T) + tr [(A T. U . D ) . (A T. 5A)] (2.2.7) = t r ( C . 5D T) + tr [(C T. D ) . (A T. 5A)]. 46 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations Piola stress tensor fields satisfying the equilibrium equations and the force boundary conditions will be called statically admissible. These fields, accordingly to equation (2.1.8), are expressed in terms of the tensor of stress functions, ,~, which should meet the boundary conditions on the surface of multiply connected body n . (rot • + D*) = f. (2.2.8) We have assumed that the load is dead, i.e. the vectors f and k are some given functions of Lagrangian coordinates. The expression (2.2.8) can be transformed into the following V. ( e . ¢ ) = f - n . D * , e = -n × E, V ~ = (E - n n ) . grad ~, V . q =_ r ~ . O~/Oy ~ (2.2.9) (a=l,2), where y~ are the Gaussian coordinates on the surface a; r ~ is the reciprocal vector basis; V is the gradient operator on the surface a; e is the surface discriminant tensor (Zubov 1982). The tensor stress function satisfies equation (2.2.9) and the dynamic conjugation conditions (2.1.1) on the cuts converting the body into simply connected, that is ~7. (ek" O+) = V. (ek-O_). Qk, ek = --nk × E. (2.2.10) The complementary energy functional for elastic body, A1, is defined on the set of differentiable orthogonal tensors and twice differentiable tensors of stress functions undergoing the restrictions (2.2.9) and (2.2.10). This functional is AI[~I,, A ] = f V[D(~I,), A]dv+ E bk "~k V " (ek " cI'+)da" (2.2.11) k We will show that stationary points of .AI satisfies the compatibility equations (2.1.6), written in terms of the rotation tensor and stress function tensor, the symmetry of the tensor C. D T and the jump conditions (1.3.4) for the position vector R. Using equations (2.1.8) and (2.2.7), for the variation of ~41 we obtain: 5A1 = ~ [ t r ( C T. rot~O) + t r ( C T. D . A T. ~A)] dv (2.2.12) + E bk" ~k V . e k . 5"~+da k With regard for the identity tr rot (x. y) = tr (y. rot x) - tr (x T. rot yT) 2.2 Variational Formulation 47 integration by parts in (2.2.12) implies 5A1 = ~ [tr(bO T. rotC) + tr(C T. D- A T. hA)] dv - .~. tr (e. 5O. CT)da +~/ k [tr(ek. ~ + . C T) - tr(ek. ~ _ - c T ) ] da .Irk +~bk" k f V . e k . 5O+da. J~'k Let ~A1 = 0. First take 5~ = 5~+ = 5cI,_ = 0 on a and Tk (this is compatible with the restrictions (2.2.9), (2.2.10)). In view of the fundamental lemma of calculus of variations we derive the continuity equations (2.1.6). These equations mean that in the domain occupied by the body there is a vector field R, whose gradient is C(O,A). In a simply connected volume, I / i s a single-valued function of coordinates, which is defined up to an additive vector constant. Thus the stationary condition for A1 takes the form tr (C T. D. A T. 6 A ) d v - £ , V. (e. ~ . +~ R)da + j f R . ( V . e . 6O)da J~~k[R_. (V. (ek. 5O_)) - R+. (V. (ek. 5¢+))] da k +~bk" ~k V . e k . ~ + d a = 0. k Here, a' is a closed surface constituted by a and both sides of all the cuts By the divergence theorem on a surface (Zubov 1982), the integral over a' vanishes. The integral over a becomes zero because of equation (2.2.9). From orthogonality of the tensor A it follows that A T. (~A is skew-symmetric. Hence the stationary state of the functional A1 implies the tensor C T. D to be symmetric that is equivalent, in view of equation (1.1.19), to symmetry of Cauchy stress tensor and provides the balance of moments of all the forces acting on any part of the body (Lurie 1970). With regard for equation (2.2.10), the variational equation becomes Tk. /(R+ k - R _ . Qk - bk). [V. (ek. ~¢+))]da = 0. JTk Since ~ + on ~'k is arbitrary, we establish the jump condition (1.3.4). On the basis of the two fundamental variational principles for bodies with Volterra dislocations, discussed above, we can construct functionals for other stationary principles similar to those in (Zubov 1971). A limitation of above variational theorems is that the varied functions are defined in a simply connected volume obtained by making cuts but not in the 48 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations initial multiply connected domain. The additional boundary surfaces complicate the set up and solving the equilibrium problem and bring some elements of artificiality. In a specific case of plane or generalized plane deformation without disclinations, there is a variational principle without making the cuts, i.e., the principle in which the functions to be varied are defined and single-valued in multiply connected plane region. Under deformation of the form (1.4.1), for isotropic, orthotropic and some other materials, Piola stress tensor will have the representation 3 3 D = D{f~fl + D2flf2 + D~f2fl + D2f2f2 + Daf f3, (2.2.13) m D~¢ = D~(¢, ¢). Let body forces be absent; in terms of the complex stress components the equilibrium equations are OD~/O-¢ + 0D~/0¢ = O. (2.2.14) On introduction of the complex function of stresses, ~(¢, ~), D{ = 00/0¢, D~ = -0(I)/0¢, (2.2.15) equation (2.2.14) becomes identical. The force conditions on the boundary contour of the plane region, ¢(s), s being the length parameter, can be written as follows dCD{ de D~ = iF1 , d---~ - ~s F 1 = f " af1, (2.2.16) where F 1 is the complex component of the vector of external dead load. To derive equation (2.2.16), we used a formula for the normal-to-contour vector, o- -i :l + i :2 From equations (2.2.15), (2.2.16)we have d~ d--7 = iF1 (2.2.17) on the domain boundary. The boundary of multiply-connected plane domain w consists of the outer contour, 70, and the inner contours, % (k - I, 2, ...m). In what follows we shall suppose, for conciseness, that the boundaries of holes are load-free, i.e., F 1 - 0 along %. Therefore, (I)IT k - - Olk~ 2.2 Variational Formulation 49 where ak are complex constants. By equation (2.2.15), the stress function ¢ is defined by the stress field only up to a constant, so we may arbitrarily choose the outer-contour constant arising while we integrate equation (2.2.17). We choose it such that ¢[7o = 0 when F 1 = 0; hence ~1-~o = i Flds. (2.2.18) If in the body there are only translational dislocations then the rotation angle X (see Sec.l.4) and Piola stress tensor are single-valued functions in multiply connected domain. Now the integral in (1.4.20) is zero for any closed contour. The complex parameters characterizing the Burgers' dislocation vectors are ~k = ~£ eix(U~d( + Uld~). J.yk (2.2.19) Instead of the contour % in equation (2.2.19), we can take any closed contour enclosing k-th hole only. If the relations (2.2.15) are considered to be equations to define the function • when the Piola tensor is given, then they reduce to the problem of finding the stress function with total differential given in advance. Using equation (2.2.16) we can easily show that • is single-valued in the multiply connected domain if and only if the principal vector of external load applied to each hole contour is zero. This condition holds in the case under consideration (of load-free holes). As in (2.2.6), we consider the specific complementary energy of an elastic material under plane deformation, V = S~U~ - IV, S~ = f a . S. f~ (a,/3 = 1,2), to be a function of Jaumann stress tensor, whose complex components are expressed in terms of the rotation angle and the stress function as follows S~ S~ = = Re (e-iXOgP/O~), (2.2.20) S~ = S~ = -eixO~/O(. On the basis of (2.2.20), we have = U.SS~ " = U~5(e -~xo 1o + eiXO-~/O-~)-U~5(e-ixO'blO~)-U~5(eiXO-~lO~). Let us consider the following functional in which the rotation fields and stress functions, both single-valued in the multiply connected domain w, as well as the contour constants, a k , are varied independently: m ( / 3 k ~ k _ ~kak). A2[~, ~:,al,...,a~] = Of Vdw + ~i E k=l (2.2.21) The varied stress function must satisfy equation (2.2.18) and the conditions = ak on the contours 7k. 50 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations It is easy to verify that at the stationary state of the functional (2.2.21) it follows the compatibility equations (1.4.14) expressed in terms of the stress function and the rotation field, the symmetry property of Cauchy stress tensor Im ( V l e - i X O ~ / o q ¢ - U2 e-iX o(~ / (~-~)= 0 and the integral relations (2.2.19). 2.3 The Semi-inverse M e t h o d for Solving Quasi-static Problems of Nonlinear Mechanics of Solids By semi-inverse methods in continuum mechanics one usually means the techniques for constructing specific solutions such that the system of resolving equations reduce to equations in a less number of independent variables. The semiinverse methods are of great importance when exact statements of boundary value problems are so complex that cannot be solved by other methods. It is well known a great role of the semi-inverse method, proposed by St. Venant to solve the problems on torsion and bending of prismatic beams in development of linear elasticity as applied science. By St. Venant, threedimensional problems of elasticity were reduced to two-dimensional ones for the Laplace equation, that enabled us to obtain efficient solutions for a number of problems of practical importance. In theory of finite strains of elasticity, the use of semi-inverse methods is primarily due to Rivlin, Ericksen, Green, Adkins, Lurie (Rivlin 1948; Ericksen 1977; Green, Adkins 1960; Lurie 1980). These authors found types of deformations for which the system of equilibrium equations reduces to one or more ordinary differential equations. For incompressible materials, the method results in a series of exact solutions to large deformation problems of practical importance. It should be noted that using semi-inverse methods it is hard to expect to satisfy boundary conditions exactly. On a part of the boundary we are forced to "weaken" them, i.e. to satisfy these conditions approximately. For example, in the classical St. Venant problem, the boundary conditions on the beam ends are satisfied only in integral sense within the accuracy in forces up to a force system being statically equivalent to zero. Although for our needs it is enough to consider only elastic bodies, we shall formulate the semi-inverse method for simple media with memory with regard for links between mechanical and thermal effects (Zubov 1981). Proposed here is the procedure for constructing such types of quasi-static deformations of bodies with arbitrary memory as those for which equilibrium equations have only one independent spatial variable. Unlike usual semi-inverse approach, which consists in specifying the deformation type based on some semi-intuitive and hardly formulated consideration and subsequent testing of adopted assumptions, we suggest a procedure to construct a class of needed deformations. This procedure is based on the analysis of equilibrium equations 2.3 The Semi-inverse Method 51 for continuum with rheology, which are written in a particular form using expansion of tensors in elements of dual Lagrange-Euler basis. In so doing, the invariance of the equations with respect to certain transformations, conditioned by flame-independence and material symmetry properties (isotropy, orthotropy, etc.) of continuum is essentially used. Consider the equation system describing a quasi-static process of deformation of thermomechanical continuum with memory. It consists of the equilibrium equations, the energy conservation law, and the constitutive equations: DivT + pk = 0, Divh+ tr(T.6)+pq=p~, (2.3.1) 2~: "-- C -1./~k. C -T. P(t) = ¢[Zt(s)], J(t)T(t) = cT(t) • P(t). C(t), e(t) = ¢[Zt(s)], = h(t) = cT(t) -- C = grad R, g = Grad0, if(s) - f ( t - s), • ez[Zt(s)], (2.3.2) 0 J = det C, s > O. Here, T is the symmetric Cauchy stress tensor, P is the symmetric Kirchhoff stress tensor, p is the material density in the actual configuration, k is the body force vector, h is the heat flux vector, q is the mass density of volume heat sources, e, r/are the mass densities of the internal energy and entropy, respectively, Grad is the spatial gradient operator, grad is the gradient operator in the reference configuration metric, R is the radius-vector of a particle in the actual configuration, O is the absolute temperature, g is the temperature gradient, A is the Cauchy deformation measure, e is the rate-of-strain tensor, t is the time, if(s) is the history of f up to time t (this history also includes the actual value of a function). The over-dot denotes the material derivative with respect to time. The relation (2.3.2) is the general form of constitutive equations for simple materials with memory which is consistent with the requirement of frameindifference (Truesdell 1977). The operators ¢, ¢, ~, w, defined on the histories of the deformation gradient, the temperature, and the temperature gradient, must satisfy the restrictions following from Clausius-Duhem inequality that is the most commonly used formulation of the second law of thermodynamics in continuum mechanics (Truesdell 1977), p//>_ Div (O-lh) + O-ipq. (2.3.3) 52 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations On introducing the Piola stress tensor, D = J C - T . T, and vector d J C - T . h, the equations (1.2.1) can be written as div D + pok = 0, div d + tr (D. c T ) + Poq = P0~, (2.3.4) P0 being the density in the reference configuration. Let Yk (k = 1, 2, 3) be orthogonal curvilinear coordinates in the reference configuration of material body (Lagrange's coordinates), Y~ (s = 1,2,3) arbitrary orthogonal curvilinear coordinates in the space (Euler's coordinates). The Lame coefficients of these coordinates are denoted by am(yk) and An(Y~), respectively, and the orthonormal vector bases associated with the above coordinates are im(yk) and ln(Ys), respectively. We assume that the vector bases im and In are of the like orientation (the left or the right simultaneously). We shall use the following decompositions of the above vectors and tensors k = knln, l = grad 0 = C . g = lmim, h = hnln, D = dmni,~ln, C = Cmni~ln, d = dmim, A = Amnimin, P = P,~nimin, (2.a.5) T = Tmnlmln. From equation (2.3.5) it follows that Tn~ = J-1CmnDm~ = J-1CmnP~kCks, J = ICm l, (2.3.6) Amn = CmkCnk, Dm~ = P,~nCn~, hn = J-1Cmndm. Although C ¢ (Cmn)" imIn, the identity t r ( D T. 12) = Dm,(Cmn)" holds. The proof is based on the symmetry of tensor C T. D and orthonormality of bases ik and In. Let functions Y~ = Y~(yk,t) describe motion of continuum. There is the representation Cks = a-klAs(OY~/Oyk) (no summing over s and k!). (2.3.7) = ala2a3. (2.3.8) The energy balance equation takes the form 1 0 (x/"gds ~ + Dkn(Ckn).Zc POq = PO~, x/~ Oys as ,/ ~ A special form of the equilibrium equations, found in (Zubov 1981), appeared to be convenient to construct semi-inverse solutions. These equations 2.3 The Semi-inverse Method 53 are formulated with respect to Piola tensor components in the dual LagrangeEuler basis and have the form 1 0 x/~ Oys (.) Dsm as + AmAk OYk Dsm- ~Dsk OYm ) +pokm = 0. (2.3.9) In (2.3.8), (2.3.9), we use summation over indices s, k, n from 1 to 3 and no summation over m. The easiest way to derive equation (2.3.9) is to use, with regard for (2.3.7), the virtual work principle t" f tr (D T. 3C)dv - / v pok. 3Rdv = 0. (2.3.10) In equation (2.3.10), dR = 0 on the body boundary. We shall consider gyrotropic material media. The fact that material is gyrotropic imposes the following restrictions on the material response operator in equation (2.3.2) ¢[z:(~)] = o . ¢[z'(~)]. o ~, ¢[z:(~)] = ¢[z'(~)], w[Z~(s)] = w[Zt(s)], ~[z,'(~)] = o . ~[z'(~)], (2.3.11) z:(~) = { o . A'(~). 0 ~, 0~(~), O. V(~)}, where O is an arbitrary properly orthogonal tensor, which is constant (i.e., independent of t and s). Write now the component representation of the constitutive relation in orthonormal basis P~. = ¢~.( A t~, 0t ,~). (2.3.12) The property of gyrotropness (2.3.11) means that this component representation does not depend on the choice of orthonormal basis of the like orientation. In other words, the form of operators Cmn, i.e., dependence of components of tensor P on pre-history of the Cauchy strain measure and the temperature gradient, is the same in all orthonormal bases of the like orientation. It follows, in particular, that, in spite of dependence of vector basis im on the coordinates Yk, the relation of components in this bases of P with components of the tensor A t and the vector I t is the same in all points of homogeneous body. A similar statement is true for operators ¢, w, ~. In what follows, the coaxiality theorem will be of major importance. Theorem. Assume that in some motion of gyrotropic medium, for a fixed particle, there exists a unit vector i having properties: a) i and I are coUinear vectors for all t, s; b) i is an eigenvector of tensor At(s) for all t, s. Then at this particle the vector d(t) is coUinear to i, and i is an eigenvector of P(t). To prove this, in (2.3.11) we take a proper orthogonal tensor O of the form 0 = 0 T = 2 i i - E. (2.3.13) 54 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations Under the conditions of theorem, we have O . At(s) • O T = At(s). From equation (2.3.11), it now follows the relations O . P(t) = P ( t ) . O, O - d ( t ) = d(t). (2.3.14) Substituting equation (2.3.13) into these and multiplying the first equality in (2.3.14) by the vector i, we obtain ( i . P . i ) i = i.P, that is i is an eigenvector of tensor P. The second equality in equations (2.3.14) gives us i(i. d) = d. Q.E.D. In the system of equations (2.3.8), (2.3.9), the Lagrangian coordinates, Ys , and the time, t, are independent variables whereas Eulerian coordinates, Yn(Ys, t), and the temperature, O(y~, t), are unknown functions. We will seek the continuum motions for which the number of independent spatial variables reduces to one, i.e., all the quantities in equations (2.3.8), (2.3.9) depend on the only spatial coordinate yl and the time. We may satisfy this condition if the coordinate frames y~, Yn are such that the corresponding Lame coefficients of the systems depend on a single coordinate, am = am (Yl), An = An (]I1), respectively. Using the Lame relations, we may show that the only coordinates which satisfy these requirements are Cartesian and circular cylindrical coordinates and those which differ from them in transformation of the form: y~ = ~(yl), y~ = c2y2, y~ = caYa, c2 , Ca being constants. In so far, it is necessary that the given values of km, q do not depend on y~, Ya. For the Lame coefficients An to be independent of y2, Ya in the Lagrangian representation, it should be taken Y1 = c~(yl, t). By (2.3.7), it follows that C,1 = 0 (# = 2, 3). Let the magnitudes of Cmn and 9 be functions of the coordinate yl and time t only. From gyrotropy of medium, for homogeneous body, it follows that the quantities Pm~, din, e, 77 (and thus all the values defined by equations (2.3.6)) will also depend on the coordinate yl and the time only. From equation (2.3.7) it follows that Cta = 0 and Cmn are independent of y2, ya if and only if the body motion is of the form Y1 = o~(yl, t), Y2 ~- ~(yl, t) + p2(t)y2 + T2(t)y3, (2.3.15) Y3 - "Y(yl, t)+ p3(t)y2 + T3(t)y 3. The functions in (2.3.15) must satisfy the condition J > 0. Under above conditions, both sides of inequality (2.3.3) do not depend on y2, Ya. Thus, the above assumptions do not contradict the Clausius-Duhem inequality. For motions of the form (2.3.15), the system of resolving equations (2.3.8), (2.3.9) reduces to a system of four equations for four functions of a single spatial variable and the time: a, /3, 7, 9. As boundary conditions on two surfaces yl = constant, we can specify, say, the temperature and stress components Tim (m = 1, 2, 3) being functions of the time only. On the rest part of body surface, the boundary conditions may be satisfied in the integral (averaged) sense, by matching the functions pi(t), ri(t). 2.3 The Semi-inverse Method 55 Note that the Lagrangian and Eulerian coordinates may be of different type. For example, we can choose the Cartesian coordinates to be Lagrangian whereas the cylindrical coordinates to be Eulerian, or vice versa. Therefore, the representations (2.3.15), in spite of their simplicity in writing, contain a rather wide set of types of deformation. The semi-inverse method also enables us to find out the continuum strains which satisfy identically some part of the equilibrium equations. For the motions (2.3.15), let il be an eigenvector of A'(s). In view of C,1 = 0 (# = 2, 3), this implies the equalities C12C22 ÷ C13C23 = 0, C12C32 ÷ Ci3C33 = 0, which hold if C12 = C13 = 0. The general representation of deformation satisfying the above conditions has the form of equation (2.3.15), with ~ and ~ depending on the time only. By coaxiality theorem and relations (2.3.6), we have P1, = T1, = D1, = d, = h , = 0 (m = 2, 3). If k2 = k3 = 0 and m = 2, 3, the equilibrium equations (2.3.9) are identical. The system of resolving equations consists of the only equilibrium equation and the heat influx equation (2.3.8). On the surfaces yl = constant, it may be specified the uniform normal pressures, being the time functions, as well as the temperature or the heat flux. Let us list some important cases of deformation described by equations (2.3.15) with ~ = ~ = 0: inflation, torsion and extension of a hollow cylinder; bending of a rectangular block into a sector of a circular cylinder or into a complete circular-cylindrical tube; straightening of a sector of circular cylinder into a rectangular block; cylindrical bending of a hollow-cylindrical sector; shearing of a sector of a circular tube along the axis; eversion of a cylinder; etc. Using expressions (2.3.15) we can describe combinations of all these deformations. In problems for circular cylinder, indefiniteness in deformation of the form (2.3.15) is such that we can satisfy end force conditions in St. Venant sense. In pure mechanical theory of viscoelasticity with no regard for thermal e f fects, the constitutive equations for stresses do not include the temperature, and no consideration is given to the heat influx equation. In this case for isotropic homogeneous bodies, under the condition k2 = 0, it can be proved that the second equilibrium equation (m = 2), is identically fulfilled in the motions of the type (2.3.15) with ~ = P2 = r3 = 0 or ~ = P3 = r2 = 0. If ~ = P2 = r3 = 0 or "~ = P3 = r2 = 0, the third equilibrium equation is satisfied when k3 = 0. These semi-inverse solutions were found by systematic trials, based on search of tensors C such that one of ik is an eigenvector for At(s), using essentially the coaxiality theorem. In a similar way it can be proved that the above semi-inverse solutions with a single independent spatial variable remain valid for orthotropic bodies with predominant directions il, i2, i3 and arbitrary non-homogeneity with respect to coordinate yl. 56 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations 2.4 Combination of Screw Dislocation and Wedge Disclination in Nonlinear Elastic Cylinder Isolated defects such as screw dislocations, wedge disclinations or their combination are rather simple for investigation in the framework of nonlinear elasticity. Let r, ~, z be the cylindrical coordinates in the reference configuration of elastic body; R, ~, Z be the cylindrical coordinates in space. The unit vectors tangential to the coordinate lines in reference configuration and in space will be by denoted er, e,, ez and eR, e¢, ez , respectively. Consider the following deformation of continuum R = R(r), ¢ = ~ + Cz, Z = (27r)-~b~ + az, (2.4.1) where ~, ¢, b, a are constants and a > 0. This deformation relates to the class of continuum motions considered in Sec.2.3. For this class, the system of resolving equations reduces to an ordinary differential equation. When ~ > 1, relations (2.4.1) describe the deformation appearing if one has removed the sector 27r~-1 _< ~ _< 27r from a circular hollow cylinder, rotated the section ~ = 27r~-1 about the cylinder axis up to coincidence with the plane ~ - 0, then made translational displacement of this section by b~ -1 along the cylinder axis, and at last fastened tightly the planes ~ = 0 and = 27r~-1. Moreover, the cylinder is subjected to torsion with angle of twist ¢, axial extension (or compression), and inhomogeneous radial deformation defined by a function R(r). If ~ > 1, we have: 0 _< ~ _ 27r~-1, 0 _< (I) _ 27r. This means that in the reference configuration the body occupies a simply connected domain but in the deformed configuration the domain is doubly connected. The case 0 < ~ < 1 corresponds to addition of a material wedge into the cylinder which was cut by the half-plane ~ - 0. In this case, 0 _ ~ < 27r, 0 _ • _< 2r~, i.e., the body without additional material occupies the doubly connected domain in the reference configuration but in the deformed state the domain is simply connected. Using the formulae of Sec.2.3, let us find the deformation gradient associated with transformation (2.4.1) ~R C = R'ereR + ~ e ~ e ¢ + b e~ez + CReze¢ + C~ezez, det C = RR'( be) r ~ " R' _- dR dr" (2.4.2) (2.4.3) The deformation measures, defined by equation (2.4.2), are as follows 2.4 Combination of Screw Dislocation and Wedge Disclination A = R + ( ab /2 erer + ~¢R2 + n2 R2 b2 / - ~ + 47r2r2 + 57 e~e~ + + r A_1 = R ,2 e R e R + ~2 r2 + R2¢~ e,~e,~ (2.4.4) + (e~ez+eze¢)+ 2rr2+aCR 47r~r2 + a 2 ezez, A = (R')-2eReR + (27ra~ -- b¢)2R ~ - aCR+27rr 2 ( e c e z + e z e ~ ) + c~2 + 47r~r9 ¢2R2+ r2 e~e~ ezez • The invariants of the Cauchy strain measure, A, are R2 b2 I1 -- tr A = R '2 + n 2--~ + ¢2R2 + 47r2r2 + a 2, /2 - ~1 tr ( tr 2A - tr h 2) (2.4.5) = an- be ~ 2 R 2 R '2 a 2 b2 + ¢ 2 R 2 + n 2 ~-~ + + 41r2r2 -~ R'2R2( /3-= d e t A = r2 , be) 2 an-~ . If n > 1 the vectors eR, e,~ are single-valued and continuous in doubly connected domain occupied by the deformed body. Hence from equation (2.4.4) it follows that A -1 keeps continuity when it passes through the surface (I) a-lCZ which is the final position of the plane ~ = 0 after deformation whereas the deformation gradient, C, takes a jump, C+=Q1.C_, C_=limC (~ --, Ca-~z + 0), C+ -- lim C (~ --+ 27r + ~2o/-1Z - 0), (2.4.6) Q1 = Q~-T = (E - ezez)cos 27r(1 - n -1) +ezez - ez x Esin21r(1 - n-l). The relation (2.4.6) is tested by using equations (2.4.1), (2.4.2), and the apparent representations e~ = eR cos(~ -- ~) + e¢ sin(qo -- ~), 58 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations e~ = --eR sin(~ -- ~) + e~ cos(~o -- ~). The formulae (2.4.4), (2.4.6) show that the Volterra dislocation occurs in the cylinder. In this case, by equation (1.3.28), the Frank and Burgers vectors take the form bl "- bn-lez, ql = - 2 t g ~ ( 1 - ~-l)ez. Here, the origin lies on the cylinder axis. If 0 < k < 1, the vectors er, e~ are single-valued and continuous in the doubly connected domain occupied by the body in reference configuration. From equation (2.4.4) it follows that the Cauchy strain measure, A, is continuous and single-valued in this doubly connected domain. Thus, if 0 < n < 1, it is the case of Volterra dislocation characterized by the jump equation (1.3.6) with the parameters b = b~-lez, q = - 2 t g ~ ( 1 - n)ez. In linear theory of isolated defects (de Wit 1977), the Volterra dislocation, whose Frank vector is zero and the Burgers' vector is parallel to the dislocation axis, is called the screw dislocation, and the defect with Burgers vector of zero magnitude and the Frank vector being parallel to the defect axis is called the wedge disclination. By analogy, the isolated defect under consideration can be called the combination of screw dislocation and wedge disclination. Negative values of the parameter n correspond to generation of screw dislocation and wedge disclination in the cylinder which is initially inverted. From the constitutive relations of isotropic or orthotropic material, for deformation of the form (2.4.1) it follows that the Cauchy and Piola stress tensors are T = aR(r)eReR + a¢(r)e~e¢ + T¢z(r)(ecez + eze¢) + az(r)ezez, D = DrR(r)e~en + D~(r)e~e~ + D~z(r)e~ez (2.4.7) +Dz~(r)eze¢ + Dzz(r)ezez. The first two of the equilibrium equations in the form (2.3.9) are satisfied identically, whereas the third equation is a nonlinear ordinary differential equation with respect to the function R(r), namely, dDrR dr DrR- ~D~ r - CDz¢ = O. In terms of physical components of Cauchy stress tensor, the equation (2.4.8) daR aR -- a¢ dR R =0. (2.4.9) 2.4 Combination of Screw Dislocation and Wedge Disclination 59 Boundary conditions for equation (2.4.8) are the load-free conditions on the lateral surfaces of the cylinder, DrR = 0. For a whole cylinder (without a hole), the condition on the inner surface is replaced by the requirement R(0) = 0. It can be shown that stresses in any cross-section of the cylinder (Z = const) are statically equivalent to a longitudinal force applied at the center of crosssection and to a twisting torque. By matching the parameters a and ¢ we can make these force factors to be zero. Thus, the problem of determining stresses caused by an isolated defect, being a combination of screw dislocation and wedge disclination, was reduced to a boundary value problem for an ordinary differential equation. For incompressible material, the Cauchy stress tensor is expressed in terms of strains up to an arbitrary spherical tensor. Now the function R(r) is found implicitly from the incompressibility condition, det C =. 1, and the spherical tensor is defined by equilibrium equations (2.4.8) or (2.4.9) with quadratures. The solution (2.4.1), describing formation of a screw dislocation and wedge disclination in an elastic circular cylinder, relates with medium deformation such that the system of equilibrium equations contains a single independent spatial variable. Let us now study a dislocation nonlinear problem wherein application of the semi-inverse approach reduces to a three-dimensional problem to the twodimensional form, i.e., to a system of differential equations and boundary conditions in two independent variables. Consider an elastic body that is a prismatic rod, i.e. noncircular cylinder, in the reference configuration (undeformed state). Cartesian coordinates xl, x2 refer to the plane of prism cross-section whose center of inertia is the origin, x3 refers to the rod axis. The coordinate basis vectors shall be denoted by ek (k = 1, 2, 3). Xn (n = 1, 2, 3) stand for Cartesian coordinates of points in the deformed body, they refer to the same directions ek and origin. Using the complex coordinates and complex bases introduced in Section 1.4, let us define deformation of the prismatic rod as follows z -- z0(~, ~)e i¢x3, b X3 = c~x3 + ~ arg (~+ w(~, ~), (2.4.10) where ¢, a, and b are constants, z0 is a complex-valued function and w is a real single-valued function. Geometrically, the representations (2.4.10) mean that in the prismatic body there is a screw dislocation whose axis directs along the x3axis and the Burgers vector is equal be~; the bar cross-section, at a distance x~ from the origin, being deformed (plane deformation, given by the function z0) , rotates about the rod axis through the angle ¢x3, translationally moves along the axis for distance (a-1)x3, and warps (the form of warping is described by the function w). The deformation gradient C corresponding to the transformation (2.4.10) is defined by 60 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations c = c~f-/~, C1 C~ --2 c~=~, C1 0Z0 _i¢x3 = C 2 = --~-e -- -C3 ~ -- C~ i¢zoeiCXa, --2 : C 1 : OZo • _ o¢ e ~¢~3, --a Ow = C2 = -~ (2.4.11) ib 4~r(" The constitutive relations for isotropic material with respect to the Piola stress tensor may be written as follows D = a~ (Iq)C + a~(Iq)A. C + o/3(/q)C -T, (2.4.12) 11 = tr A, 1 tr 2A - tr A2), I~ = ~( Ia = det A, where aj (j = 1, 2, 3) are scalar functions of the invariants Iq (q = 1, 2, 3). For complex components of Cauchy strain measure, by equations (2.4.11), we have All = A22 = OZo O-2-d Jr-~2 o¢ o¢ 1 (OzoO~o A12=A21=2 A13 = A23 = A31 = ~oOzo) 0( 0~ { 0( 0-~ 1 A32 - ~i¢ (0-20 +7~, i)Zo'~ Zo--~- - -20 0¢ ,] + a'7, (2.4.13) Aaa = ¢~Zo~o + a 9, Ank = f , . A . f k, Ow "/= O~ ib 4r~ The invariants Iq are as follows 11 -- 4A12 + Aaa, (Ozo O~o 12 = 4[(A~2) 2 + A12A3a- AliA22- A~aA2a], oozo) + 2¢ Im (7-2o-~ ozo- "yZoozo 0( ] " (2.4.14) 0( 0-~ The relations (2.4.13), (2.4.14)indicate that under deformation (2.4.10), the tensor A and its invariants do not depend on the coordinate x3. The tensors A . C and C involved in the constitutive relations (2.4.12) take the form 2.4 Combination of Screw Dislocation and Wedge Disclination 61 X/~3C-T= Ls fm f s, A . C = K~nfm f ~, --~. [OZo (OZo___ 1 2 ) O~o ,,0-2o0Zo K1 = K2 = L o¢ \ o¢ ~ +-~-£(-~ + 2~-~+ -~¢ zo-~o ( zo) + ~ 2 ~ - ~ + i¢~zo - ~¢Zo-~ , =_K1rozo(.ozo o oozo = Loc: ~ - ~ + - -o¢ -- £ e *~, 1¢~zo~o) + 2~+ ( zo ) Lro 0¢ zo( o i,zoOZO ~ + iCzo--~ + lCZo-~- + ~ 27--0-~- + iCazo - ~¢z o--~j e'¢x3, K~ - K3 (2.4.15) OZo ] + 2~V-b-~- + iCzo(¢~zo~o + £ ) ei¢~, K3--3 ( OzoOzo )1 (0-2o OZo) = K~ = ~ 2 o¢ o~ + 4~2 + ~ i¢,:,,, zo--5-( - -~o-y( (azo 0~o 0~oa~o ) O'~o7) /(33 = 4a'y~ + iCzo (0-2o7 \--~ + -0~ 0Zo Ozo_~ -i*~o -~- + -O-~7) + a(¢2ZoZo + a2), = L2 = c~-~ - iCzo"/ e*~ L~= L-~I = ( iCzo~-C~--~je Ozo'~ '¢xa, L~ -2 ( Ozo Ozo • = L3 = 7"~- - V"~') e'¢~3, L~ -3 = r~ = 1. ( 0zo 0-20 ~,¢ _~°~ + zo-b-(], 0zo 0-20 0-200Zo The representations D 1 1 - p.(~, ~)e i~, 2 D.2 -- p.(~', ~)e -i~x3 D~ = p~(¢,-~), (~ = 1,2,3). (2.4.16) 62 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations follow from the equations (2.4.11), (2.4.12), (2.4.15), where quantities p~ are independent of the axial coordinate x3. With regard for (2.4.16), the equilibrium equations are Op~ Op1 i 1 O~ F - - ~ + ~¢P3 = O, Op~ ~- Op~ O~ --~ = O. (2.4.17) With given constants b, a, and ¢, the relations (2.4.17) make up a system of equations with respect to the complex and real functions, z0 and w respectively, in two variables. To these equations it is necessary to associate boundary conditions on the lateral surface of the prism which, for the free load surface, are 0~ O~ ~ = O, ~88/.,2 -- ~88b,1 0~3 0¢ ~3 = O, ~8/j2 -- ~88b,1 (2.4.18) where ~(s) describes the contour of cross-section of the bar, s is the arc-length of the contour. If z0 and w is a solution to the boundary value problem (2.4.17), (2.4.18) it is easy to see that the functions z0eiw and w + a with w and a being real constants also satisfy the conditions (2.4.17), (2.4.18). This ambiguity can be eliminated if we require Re (~Z0 - l ° z ° O~ 1) da = O, f~ wda = O, (2.4.19) a being a cross-section of the undeformed bar. Under the conditions (2.4.19), we can expect uniqueness of solution to the problem (2.4.17), (2.4.18), at least for not too large strains. Assume that the bar cross-section is centrally symmetric, i.e. remains the same under rotation about the rod axis through 180 degrees (an example is a cross-section of the form of the letter "Z"). Cross-sections with two axes of symmetry are obviously in this class too. Make the following changes of independent variables and unknown functions in the system (2.4.17) and boundary conditions (2.4.18): ~' - -~, ~ " = -~, -~' - -~,- -~" = ~, Zo' = -z0, z0" = -zo, w' = w, ~" = - w . (2.4.20) (2.4.21) There is the following Theorem. For homogeneous isotropic elastic body, under transformation (2.~.20), the boundary value problem (2.~.17)-(2.~.19) in a centrally symmetric 2.4 Combination of Screw Dislocation and Wedge Disclination 63 domain is invariant. If the domain a possesses two axes of symmetry coincident with the axes x l and x2 , the boundary value problem is invariant under transformations (2.~.20), (2.~.21). The theorem is proved by substituting equations (2.4.20), (2.4.21) into the boundary value problem. Let z0 = h(¢,~), w = m(~,~) be a solution to the boundary value problem (2.4.17)-(2.4.19). By the theorem, the functions Zo = - h ( - ~ , - ~ ) , w = m ( - ~ , - ~ ) satisfy the same boundary value problem. Uniqueness of the solution implies h(~, ~) = - h ( - ~ , - ~ ) , m(~, ~) = m ( - ~ , - ~ ) . So, if the cross-section is centrally symmetric, the solution possesses the property ~ o ( - ~ , - ~ ) = -zo(¢, ~), ~ ( - 4 , - ~ ) = w(¢, ~). (2.4.22) For bisymmetric cross-section, in addition to (2.4.22), there is zo(¢, ¢) = ~o(¢, ¢), ~(¢, ;) = -w(¢, ¢), (2.4.23) From the equations (2.4.11)-(2.4.15), (2.4.22)it follows p~(-¢,-¢) =p~(¢,¢), _ m p ~ ( - ~ , - ¢ ) = -p~(¢,¢), (2.4.24) p~ (_if, _~) = _pa (~.,~) (o~,/3= 1, 2), p~(-¢, -~) = p](¢, ~). For the bisymmetric cross-section, besides, there holds p~(~, ~) = -p~ (~, ~), -- Ot - - 3 -- 3 m P3(~, ~) = Pa(~, ~). (2.4.25) The principal vector F and principal moment M of forces acting in a rod section, xa = const, are F=~fa'Dda, M=- R = Xmem = zkfk, f fa'Dx(R-Ro)da, (2.4.26) RO = X3(0, 0, xa)f3, R being the radius-vector of a point of deformed body; Ro is the radius-vector of the reference point which is the center of inertia of the deformed bar section. Based on equations (2.4.10), (2.4.16), and (2.4.26), we have 64 2. Stressed State of Nonlinearly Elastic Bodies with Dislocations F = f~/~1,~i¢x3 ~2,a-iCx3 3 re3 ~ f l + ~'3 ~ f2 + Pafa) da, M= if -~ l ( z o p ~ - -z0p1) - zoP3) e -'~3 f2] da, + (w • 1 - z0p ) e fl (2.4.27) = w(~, ~) - w(O, 0). Using equations (2.4.22), (2.4.24) and (2.4.27) we obtain F.fZ=M.f~=0 (/3=1,2). (2.4.28) The relations (2.4.28) mean that under deformation of the form (2.4.10), the stresses in a cross-section of the cylinder can be reduced to a longitudinal force F3 and a torque M3. These quantities remain constant in all prism crosssections; on solving the boundary value problem (2.4.17)-(2.4.19), they become known functions of the parameters ¢, a in accordance with formulae F3 = F . f3 = f~ p~da, M3 = M . f3 = - f~ Im (zop~)da. (2.4.29) Thus, the assumptions (2.4.10) for a cylinder with centrally symmetric crosssection enable us, by solving two dimensional boundary value problem, to satisfy the equilibrium equations and boundary conditions on the lateral surface exactly, whereas the boundary conditions at the bar ends only in the integral sense, matching the constants ~, a.