Modern interpretation of Saint-Venant’s principle and semi-inverse method

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1. Introduction The main factor for refusing a researcher in recognition of his works becomes fairly meaningful and reasonable lack of trust for something overly unconventional and provocative. If we go back to the image of sphere of knowledge, then any attempt at leaping too far out of the bounds of the sphere will likely be eventually unsuccessful. These are, for example, numerous attempts at creating a “theory of everything” or alternatively subverting some fundamental theory - theory of relativity or evolution. Such “theories of everything” and “subversions of fundamentals” look too mistrustful to spark an interest to seriously consider them. In this case the apriori lack of trust for something excessively strange and pretentious serves as “population control” for disposing of potentially destructive phenomena. It is common for continuum mechanics and the related field of partial derivatives mathematics to gradually develop fundamental ideas, put forth by the “founding fathers” of the science. Progressive development of views on the principles underlying deformable body mechanics, growth of the mathematical analysis field, which is used for computation and accumulation of rules of thumb, obtained by interpreting mathematical results, lead to the fact that the principles are replaced by the other, more general ones, their number decreases, and this field is brought into an increasingly closer relationship with other branches of physics. Besides, same physical principles, in the end, serve as a basis for them all. Elasticity theory is considered to be founded on two basic Saint-Venant’s ideas: the principle and the semi-inverse method. Saint-Venant’s principle, named after Adhemar Jean Claude Saint-Venant, may be expressed as follows: “…the difference between the effects of two unique, but statically equivalent loads, becomes very small at sufficiently large distances from the load”. This statement was published by Saint-Venant in 1855 [1]. Later, von Mises brought forward an assumption that the principle cannot be applied to the bodies of finite dimensions [2]. In this paper, this assertion is refuted. The force and moment based (stress resultants) solution for thin-walled systems had been determining the direction of researchers’ efforts for a long time. After constructing basic theories for plates, shells and thin-walled systems some contradictions were noted: it is impossible to satisfy all the boundary conditions and to evaluate all the stresses and displacements. Then, great difficulties were encountered in solving problems for anisotropic composite materials. Due to this, attention was given to solving problems in stresses, at least in the constraints area. The problem of applying semi-inverse Saint-Venant’s method to mechanics of composite materials was brought to the forefront. Such problems require reconsideration of the accumulated practice and its generalization in order to obtain new possibilities of expanded application of classic ideas to new problems and materials on the basis of extended and generalized formulations. Friedrichs and Dressler [3] and Goldenveiser and Kolos [4] have independently proven that the classical Kirchhoff’s plate theory is the main term of the asymptotic expansion (by small parameter of thickness) for the linear theory of thin-wall isotropic bodies. On the other hand, based on their approach the internal solution, which has a value only close to an edge, is determined by a series of boundary value problems. These problems are very complicated to solve, almost as complicated as the initial problem. Saint-Venant’s principle may be used for boundary stresses to create boundary conditions in the classic plate theory and also for some external expansions of higher order without any reference to the internal boundary solution. Attempts at obtaining the corresponding boundary conditions for displacements were unsuccessful. Gregory and Wan [5] applied a general method developed by them for obtaining the proper boundary condition series for arbitrarily defined allowed boundary conditions (without an explicit solution of internal or boundary layer) for a number of special cases of general interest, including cases with defined boundary displacements. Their overall results demonstrate that in order to be strictly correct, the Saint-Venant’s principle can be used only to the leading terms of the external solution, i.e. classical plate theory. Horgan et al. [6-10] investigated various aspects of applying the Saint-Venant’s principle to the boundary layer and obtained a number of insightful practical results. The practical conclusions for composites are: effect of the end constraints of samples in mechanical tests, effect of support elements, connections, cuts, etc., in composite structures and limitations of strength of materials formulas when applied to composites. It was established that neglecting the elasticity of the end constraints, generally rationalized by the Saint-Venant’s principle, cannot be applied to problems involving composite materials. Particularly, in fiber reinforced composite material the characteristic attenuation length of the end effects is significant, generally several times greater than in isotropic materials. Even though the answers on many of the questions in Saint-Venant’s principle discussion in its classical interpretation were obtained, the full analysis of extended physical and mathematical issues emerging with the asymptotic solution in composites’ elasticity needs to be carried out. Semi-inverse Saint-Venant’s method in its classical interpretation in the publications is used for obtaining solutions of non-linear problems of elasticity theory, linear problems of porous and graded materials, etc. [10-14]. The wide applicability range of the Saint-Venant’s principle and the semi-inverse method may be also explained by that they make it possible to construct insightful analytical solutions, which may serve as a guide for calculation automation by means of numerical methods. In [15-19] the Saint-Venant’s method acquires an extended iteration-based interpretation, allowing to obtain asymptotic analytical solutions without any hypotheses and to satisfy all the boundary conditions. The solution converges satisfying the Banach fixed-point theorem [20]. Saint-Venant’s principle is qualitative and, being applied to an end-loaded bar problem for the first time, it states that a statically equivalent to zero system of forces, distributed over a small area, creates only local disturbances. The disturbances decay rapidly with increasing distance from that area and become negligible at sufficiently large distances compared to its dimensions. The stress state in a long prismatic bar, loaded only at the end sections, practically does not depend on the way of surface forces distribution and is determined at some distance from the ends by their resulting vector and resulting moment. However, for example, this formulation is insufficient for the extended theories of thin-walled systems from isotropic and composite material, which include bars, plates, shells and thin-walled bars. Stresses in corners, caused by the changing lateral dimension of the long elastic strip, rise a question about the formulation of attenuation conditions stated for displacements [19] given on a small area on frontal and side surfaces of a thin-walled body. Iteration-based interpretation of the semi-inverse method expands its application range to composite materials [20]. 2. Generalized formulation of Saint-Venant’s principle for stresses and displacements Let us consider a problem of establishing stress-strain state localization conditions, analogous to the Saint-Venant’s conditions, in a long prismatic bar, given the end displacements rather than stresses. Let a strip, modelling the state of the bar, be defined by , . Long edges of the strip are free from any loading and constraints. On the short edges the displacements are defined as (1) On edges (2) The solution comes down to finding Airy’s stress function , which satisfies biharmonic equation (3) Stresses in terms of function are calculated as follows: ; ; Deformations are evaluated with the help of elasticity relations ; ; Deformation-displacement formulas allow to determine the displacements ; Let us rewrite the equation (3) as follows: Integrating it twice over and twice over , we get (4) where are arbitrary functions of integration. Elasticity relationships and deformation-displacement formulas, taking into account (4), yield (5) Let us define in the equation (5) (6) where the functions satisfy the equation and the conditions on the ends Prime mark designates differentiation with respect to coordinate. In this case the conditions (2) are satisfied and the solution to equation (2) is sought by the Bubnov - Galerkin method. Functions contain exponential multipliers, which provide the decay of the end effect. Substituting expression (6) into formula (5) and integrating accordingly, the following is obtained: (7) Here and are arbitrary functions of integration, the asterisk indicates that integration is carried out with respect to the corresponding coordinate, i.e. . are the eigenfunctions for the problem of free vibration of a bar fixed at both ends and are fairly well examined. For a symmetrical about axis case they become: (8) where satisfies the transcendental equation (9) For asymmetrical about axis case (10) where satisfies the transcendental equation . Let us now consider the possibility of satisfying the conditions (1) by representing displacements in the form of the expressions (7). On the end at there must be (11) Let us determine the conditions, which must be imposed on the functions and , for the expansions (11) to be valid. If the systems of functions and are full, the arbitrary functions must be zero. Let us check the fullness of these systems. May a certain symmetrical with respect to function on the interval be expanded in a series by functions and antisymmetric function in a series by functions . Thus, ; (12) (13) where . Functions , , satisfy the following orthogonality conditions: For symmetric functions it will be . Let us compose expressions for partial sums of the series (12) substituting coefficients (13) into them: (14) Therefore, the problem of correspondence of left and right parts in the formulas (12) results in evaluating the limits of the respective sums , when . Changing the order of summation and integration in (14), the expressions for partial sums can be written as follows: (15) Let us also consider the same way as in [21] the contour integrals corresponding to the relationships (15): (16) which are evaluated by going in positive direction around a circle of radius , circumscribed from the origin in the complex plane . Radius is chosen such that radiuses of real and imaginary roots of the equation (9) get inside the circle. The contour also does not pass the points and , which are the roots of the equation and . Then, every integral in the formulas (16) is equal to the sum of subtractions by all special points inside the circle . With respect to the stated for the formulas (16) above and taking into account that , we obtain (17) The last term in the first expression (17) is subtraction for the integrand of the first integral from (16). Subtraction in zero for the second integral is equal to zero. The expressions (17) lead to (18) Substituting the expressions (18) into the formulas (15), we obtain (19) Thus, the first series in the expression (12) converges to the original function , if (20) The second series converges to the function unconditionally. Let us now consider the case of expanding the asymmetric with respect to function on the interval in a series by antisymmetric functions and symmetric function in a series by antisymmetric functions from the definitions (10). Operating the same way, we obtain the following expressions for the corresponding contour integrals: (21) In both cases the last terms are subtractions at the point . Proceeding to the limit when we obtain the following for the partial sums: (22) The first series converges to the original function , and the second series to the function , if (23) Let us elaborate the meaning of the expressions (20) and (23). Let us consider a problem of a cantilever bar of unit thickness, length and height equal to two. Then in the expressions (1) as for a fixed end. At the non-zero first two displacements (1) are defined. By removing the constraints defining these displacements and substituting them by normal force , shear force and bending moment , statically equivalent to stresses, which do not decay away from edge, we obtain Let us calculate the work done by the non-decaying stresses on the displacements (1) at the end: (24) Imposing a requirement that the emerging in the strip non-decaying stresses do not perform any work on the defined displacements of the bar end, the following expressions are obtained, owing to the independence of the values : which correspond to the conditions (20), (23), where the functions and match in meaning with and respectively. Apparently, analogous to the famous Saint-Venant’s principle formulated for the cases of stresses on a small area, it is possible to formulate the locality of stress-strain state in elastic body, specified by displacements on a small area. Displacements specified on a small area of elastic body produce only local stress-strain state, which decays faster with increasing distance from that area and becomes negligibly small at sufficiently large distances compared to the area dimensions, if the resulting vector and resulting moment, statically equivalent to the stresses on that area, do not produce work on the specified displacements. 3. Generalized iteration-based formulation of semi-inverse Saint-Venant’s method Let us consider a problem of generalizing the semi-inverse Saint-Venant’s method to iteration-based form without resolving the equations in forces and moments by constructing an asymptotic analytical solution. The issues associated with existence and uniqueness of the solutions are formulated in functional analysis as a question about existence and uniqueness of a fixed point when a certain metric space is mapped onto itself. Among the various criteria of existence and uniqueness of a fixed point the most general one is the contraction mapping principle [21], which rationalises the convergence of simple iterations. May a long rectangular strip be located in orthogonal coordinate system such that . The long edges of the strip carry some arbitrary load, the short edges may be loaded or constrained. The plane elasticity equations describing the stress-strain state of such strip are taken as Let us introduce the dimensionless coordinates , , dimensionless displacements , along the axes respectively, and dimensionless stresses , , (dimensional displacements, stresses and loads are marked by an asterisk). Dimensionless equations with these variables become May the above equations be modified such that it is possible to successively calculate the remaining unknown variables by choosing some arbitrary и as initial approximation values by the method of successive approximations with increasing index with respect to the following iteration scheme: From this point on the lower index in parenthesis indicates the number of approximation. We will be interested in the equations of zero and first approximations when choosing values in accordance with the semi-inverse Saint-Venant method in the form of initial approximation Due to the initial approximation values being independent from , all the other remaining unknowns are calculated as a result of the quadratures by : Lower index 0 indicates the arbitrary functions of integration. Calculation of the next values by the previous ones is accompanied by multiplying by a small parameter in order to write the unknowns in the form of asymptotic series based on the power of . It is clear that on this stage of iterations four arbitrary functions are obtained, which allow to satisfy four boundary conditions on the long edges of the strip. From this point on dashes indicate differentiation over ; zero without parenthesis specifies the arbitrary functions of integration that depend only on . Now it is possible to write the expressions for all the unknowns in the problem, assuming that they describe the produced displacements, deformations and stresses sufficiently accurate: (25) Values and are given in the first approximation, the rest are in the zeroth approximation. Boundary conditions that correspond to the loading conditions must be satisfied on the long edges of the strip . In dimensionless form these conditions are written as при при (26) Dimensionless loads are obtained by dividing the dimensional ones by stiffness . Let us assume that - the loads are slowly changing functions of coordinate . Let the conditions (26) be satisfied by the first approximation values from the relationships (25). As a result the equations with respect to the unknowns , defining the bending problem: (27) and with respect to , defining the axial load problem: (28) are obtained. Equations (27) and (28), assuming small variability of the functions and after removing the values with small multipliers, are brought to the classic form (29) (30) confirming that equations (27) and (28) generalize the classic representations of the semi-inverse Saint-Venant’s method and their solutions depend on . The upper index indicates the association with slowly varying components of the stress-strain state. Subtracting by pair the equations (29) from (27) and equations (30) from (28) and taking into account assumptions , , , it is possible to obtain singularly perturbed equations indicated by index : (31) (32) Their solutions differ by a constant that must be removed as a non-conforming to the condition of large variability. Therefore, both solutions of the equations (31) and (32) match: As they depend from the argument , they can be used for satisfying lost boundary conditions and smoothing out discontinuities in slowly varying classical solutions. The upper solution is true when , and the lower is true when . If taking , then . That means that the equation (31) allows to establish a relationship between the ordinary numerical function and the generalized Dirac function. 4. Conclusion Two Saint-Venant’s methods are analysed and modernized. The first method involves evaluation of stress-strain state components for the purpose of simplifying the problem statement of seeking the solution by apriori removal of rapidly varying and decaying solution components. It was introduced because Saint-Venant took into account the complexity of finding general solutions. Hence, by developing the method of solving the problem he came to inventing the principle that rationalizes the components lost in the process of constructing the solution, in particular due to the transition from stresses to forces and moments (stress resultants). An addition to his classical principle for the case of displacements specified on a small area was formulated on an example of a long elastic strip, which is absent in the literature. However, both the Saint-Venant’s principle and the generalized principle cannot give any recommendations to its constructive use, but are practical for mechanical interpretation of the partial approximate solutions obtained by any method. The second method was named semi-inverse, because Saint-Venant suggested to specify a part of the unknown variables and to resolve the rest. With that Saint-Venant moved from stresses to forces and moments, rationalizing the transition by the principle. It can be stated that all the thin-walled body theories are based on forces and moments, assuming the rationalization of transition to them by the Saint-Venant’s principle. It is shown in this paper that if taking the idea of specifying a part of the unknowns, but not transitioning to forces and moments, the semi-inverse method may be expanded into a constructive one and will be converging independently of the choice of the initial approximation. This possibility is based on the Poincare small parameter method, Picard - Lindelöf iteration method and Banach’s fixed point method. With that a transformation of a complex operator in the problem to a series of simple integratable operators and a methodology of separating rapidly varying and slowly varying components of the general solution, provided the satisfaction of all the boundary conditions of the original problem, are proposed. The calculation process may be interpreted as splitting the complex operator into four consecutive Picard’s operators with respect to lateral coordinate and three - with respect to longitudinal coordinate. The accuracy of the obtained solution is evaluated by the order of the first removed term by for slowly varying values. Then the semi-inverse method becomes independent from the Saint-Venant’s principle.

About the authors

Evgeny M. Zveryaev

Keldysh Institute of Applied Mathematics; Moscow Aviation Institute (National Research University)

Author for correspondence.
Email: zveriaev@mail.ru

Doctor of Technical Sciences, Professor, senior researcher of Keldysh Institute of Applied Mathematics, Professor of Moscow Aviation Institute (National Research University)

4 Miusskaya Sq, Moscow, 125047, Russian Federation; 4 Volokolamskoe Highway, Moscow, 125993, Russian Federation

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