The Elastic-plastic Constitute Relation Connected With Texture Coefficients And Its Ultrasonic Measurement

The sheet metal is an orthorhombic aggregate of cubic crystallites(or hexagonal close packed crystallites).The orthorhombic sheet contains 9 independent elastic constants which is listed as follows,C11,C12,C13,C22,C23,C33,C44,C55,C66.Hill proposed a famous yield function which includes 6 plastic parameters.This yield function is described as (?)-1=0.The common method to measure mechanic properties is uniaxial tensile test while this cannot completely determine the whole elastic constants or plastic parameters.Elastic and plastic properties are in connection with the symmetrical characteristic of a single crystal and the orientation distribution of polycrystalline materials.Taking this into account,only 5 independent elastic constants and 4 independent plastic parameters are left.Because macroscopic elastic and plastic properties of polycrystalline materials are all related with texture coefficients,there is a corresponding relationship between elastic constants and plastic parameters.Finding this corresponding relationship makes it possible for uniaxial tensile test and ultrasonic test to determine all the elastic constants and plastic parameters.Based on others' research,there is no large works had been done on this corresponding relationship.This paper's first important work has been listed as follows:1)Use the elastic constitutive and yield function which include the symmetrical characteristic of a single crystal and the orientation distribution,give the relationship between 9 elastic constants and the orientation distribution function,2)Determine the relationship between the plastic parameters in Hill's yield function and the texture coefficients.3)Finding the expression between elastic constants and plastic parameters by texture coefficients.4)The simulation of finite element analysis which has been done to verify this expression can prove the rationality of this correlation expression.Roe and Bunge introduced orientation distribution function w(R)to describe the probability density of R orientation.Expand w(R)based on Wigner D-function and the coefficients in the expansion are texture coefficients.Through the measurement of texture coefficients,we can get the elastic constants and the plastic parameters of an orthorhombic sheet.Ultrasonic measurement of texture is an attractive alternative because it is cheap,fast and non-destructive and it can provide a bulk measurement.In 1980s,Sayers and Morris have already derived expressions between the ultrasonic velocity and texture coefficients c204,c004 while there is an large effect with the elastic constants ?,? on the expression of c004,which makes the result cannot be exact.Meanwhile,Sayers and Morris did not give the explicit expression between ultrasonic velocity and c404.In conclusion,the second important work in this paper is listed as follows:1)Introduce the texture coefficients in the Christoffel equation and deduce explicit expressions between c204?c004?c404 and ultrasonic velocity by ignoring the higher order terms in the derivation.Compared with Sayers and Morris's explicit expressions,our explicit expressions have less impact by elastic constants,which makes our result more reliable.2)By using the instrument of RETIC-SNAP-5000,we measured C004,4204,c404 of a set of different thickness steel sheets.Compared them with the results got by tensile test,both of the result are nearly identical.Lord Rayleigh first found in 1887 that there is a surface Rayleigh wave whose speed is slower than velocity of shear wave in the free surface along the elastic half space.He Lingfeng derived the surface wave velocity of the isotropic material in her book?acoustic elastic technology?,and for anisotropic materials,surface wave velocity is associated with the texture of materials.Therefore,the third focus of this paper is introducing texture of materials and deducing the relationship between surface wave velocity and the texture coefficient.Compared with stroh's formula,the result of derived expressions is consistent,which proves the rationality of the expression of this theory.According to the relationship between surface wave velocity and texture coefficients,we can get texture coefficient by measuring the surface wave velocity.Using transverse wave and vertical wave velocity expression of texture coefficient and surface wave velocity and texture coefficient's expression,measurements have been carried out on an orthogonal sheet.The two methods to calculate the texture coefficient get the similar results,which indirectly proves the rationality and applicability of this expression on the theory of two groups.