① The exact solution includes the separation of variables method and the complex variable function method of elasticity. Many exact solutions in elasticity are obtained by separating variables. The steps are as follows: according to the shape of the object, choose the appropriate curvilinear coordinate system, and write the elastic differential equation and boundary conditions corresponding to this coordinate system. If the variables in the differential equation can be separated, the solution of the problem can usually be obtained. The problems that can be solved accurately by the method of separating variables are: infinite body and semi-infinite body, sphere and spherical shell, ellipsoid cavity, cylinder and disk.
Complex variable function method is an effective tool to solve the problem that can be transformed into plane harmonic function or plane biharmonic function.
(2) Approximate solutions In order to solve some complex problems, many approximate solutions have been developed in elasticity, and the energy method is one of the most widely used methods. It transforms the problem of elasticity into a mathematical variational problem (extreme value and stationary value problem of functional), and then uses Rayleigh-Ritz method to find the approximate solution. Energy method is rich in content and has strong adaptability. Finite element method is a new development of energy method, which has been widely used in engineering. Difference method is also a commonly used approximate solution, and its key point is to replace WeChat service with difference quotient approximation, thus transforming the original differential equation into algebraic equation. In addition, boundary integral equation, boundary element method and weighted residual method are also effective means to solve some problems.
Typical problems of mathematical elasticity are as follows:
(1) In general, it discusses the solution of * * * and the general solution. In general theory, the core part is energy principle (theorem), including virtual work principle (virtual displacement principle and virtual stress principle), reciprocal theorem of work, minimum potential energy principle, minimum complementary energy principle, Herlinger-Resner generalized variational principle of the second kind of variables, and Hu Haichang-Kuzinichiro generalized variational principle of the third kind of variables. The existence, uniqueness, analyticity, mean value theorem and convergence of approximate solutions are also closely related to the energy principle. These general theories are the basis of establishing various approximate solutions and practical theories of engineering structures.
Another important aspect of the general theory is the merging theory of unknown functions, whose main content is to classify elastic mechanics problems into solving a few functions, usually called stress functions and displacement functions.
(2) Cylinder Torsion and Bending A slender cylinder with no external force on its side will be twisted and bent under the action of external forces at both ends. According to Saint-Venant's principle, the stress state in the middle of a cylinder is only related to the resultant force and moment of the load acting on the end face, but has nothing to do with the specific distribution of the load. Therefore, the stress in the middle of the cylinder has the following expression:
Here, the X and Y axes are the two principal axes of the cross section; Z axis is parallel to the generatrix of the cylinder; Is the stress component, and a is the area of the cross section; Ix and Iy are the moments of inertia of the cross section with respect to the X axis and the Y axis (see the geometric properties of the cross section); N, Mx and My are the axial resultant force acting on the section and the bending moment acting on the X axis and Y axis respectively. Bending moments Mx and My are linear functions of coordinate Z, which can be obtained by the method of material mechanics. Equation (1 1) gives the same solution as material mechanics, but the shear stress is more accurate than material mechanics. Finally, the problem can be reduced to solving the boundary value problem of a plane harmonic function.
(3) Plane problem Plane problem is a kind of mature and widely used problem in elasticity. Plane problems can be divided into plane stress problems and plane strain problems. Both of them have different application objects, but they all belong to the same mathematical problem-the boundary value problem of plane biharmonic functions.
The plane stress problem is applicable to thin plates. If there is no external force on the two surfaces of the thin plate, but there is a load evenly distributed along the thickness on the side surface (Figure 1), the displacement and stress in the thin plate have the following characteristics:
And the displacements u and v in the x and y directions are independent of the coordinate z. For isotropic materials, the above five quantities which are not equal to zero can be expressed as a stress function φ(x, y) (Airy stress function):
And the stress function φ is a plane biharmonic function, i.e.
The plane strain problem is applicable to the middle of a long cylinder. If the two ends of the cylinder are fixed, the load acting on the side is independent of the coordinate z, and the resultant force and moment are equal to zero (Figure 2), the stress and displacement in the middle of the cylinder have the following characteristics:
The longitudinal displacement ω=0, and u and v are independent of the coordinate z. For isotropic materials, the above five quantities that are not equal to zero can also be expressed as a formula (13) by a biharmonic function φ, but e and v must be replaced by.
④ When the shaft with variable cross-section is twisted, stress concentration often occurs in the transition zone of cross-section (Figure 3). It is convenient to analyze this kind of problems by taking the cylindrical coordinate system (r, θ, z). The displacement component and stress component in cylindrical coordinate system are recorded as u, v, w and respectively.
The mechanical characteristics of this kind of problems are: u=w=0 and.
V, regardless of the coordinate z, the sum of the above two shear stresses which are not equal to zero can be expressed by a stress function (r, z):
But satisfies the following partial differential equation:
This kind of problem finally comes down to the boundary value problem of equation (15).
⑤ Axisymmetric deformation of a body of revolution An isotropic body of revolution will inevitably produce axisymmetric deformation under the action of axisymmetric load. In the cylindrical coordinate system (r, θ, z), the characteristics of axisymmetric deformation are: v=0, =, u, w, and has nothing to do with the coordinate θ. The above six quantities that are not equal to zero can be expressed by the displacement function (x, y):
Where △ is the axisymmetrical La Silas operator, i.e.
But an axisymmetric biharmonic function, i.e.
⑥ Practical theory of engineering structural members The practical theory of various engineering structural members in a broad sense (such as the practical theory of rods, plates and shells) is a special branch of elasticity and the most practical branch. These practical theories make some reasonable simplified assumptions about displacement distribution, and simplify the generalized Hooke's law according to the shape of structural elements and their stress characteristics. In this way, the mathematical equation can be fully simplified while retaining the main mechanical properties. From the point of view of elasticity, the practical theories of these structural elements are all approximate theories, and their approximation is mostly manifested in the fact that the stress and strain calculated according to these theories can not strictly satisfy Hooke's law.