力学专业英语_11.pdf

Lesson 12 Elastic constitutive If the equations of equilibrium and compatibility are examined,it is found that the equations which involve stresses involve only stresses,body forces,and accelerations,but do not involve strains or displacements.Furthermore,those equations which involve strains and displacements do not contain stresses.Since the application of forces to a body produces both stress and deformation,it is expected that the stresses on an element can be related to the deformation which these stresses produce.Such a set of relationships will then complete the description of the action of loads on the body.Experience has demonstrated that these relationships will depend on the material in question.For most engineering materials,elastic strains are always sufficiently small that the relationship between the stress and strains is linear.The most general set of linear relations which could be written is.The Cs are proportionality constants,where i and j can have any value from 1to 3.The number of constants C required is therefore 33=81.From equilibrium it is known that ij=ji and by definition ij=ji.From energy arguments it can be shown that.These conditions combined imply that the number of independent constants which must be specified in the most general linear elastic formulation is 21.Since materials generally have a great deal of symmetry,arguments based on the equivalence of various directions in a material can be used to further reduce the number of constants required.For an isotropic material,one in which all directions are equivalent,it can be shown that the material has only two independent elastic constants and a great many zeros in the general formulation above.Hence,it is convenient to rewrite the general equation above in the more familiar form of Hookes law.As was stated above,only two of the three constants represented by the symbols E,v,and G can be independent.It can be shown that.When the constants E and G are specified,the six equations of Eqs.(2a,b)form the link between the stress and the strain relations of continuum mechanics.Another formulation which will prove useful in relating elastic behavior to other types of material behavior separates the volume change portion of the strain from the distortional portion.For any state of strain,the change in volume per unit volume is the sum of the three normal strain components,that is.The symbol I is used to denote the first strain invariant.So we have,where is a 3 3 unit matrix.The first matrix on the right-hand side represents a state of strain for which the volume change is zero,while the second matrix represents a pure change in volume with no distortion.The state of strain represented by the second matrix will be called the dilatation strain.The terms of the first matrix on the right-hand side are called deviator strain components and they are denoted by the symbol defined by where the symbol is called the Kronecker delta.A state of stress may be similarly decomposed into deviator and hydrostatic components.The hydrostatic or mean normal stress is defined as where J is the first invariant of the stress tensor.A state of stress can then be decomposed into the deviator stress and the hudrostatic stress.The second matrix on the right-hand side is called the hudrostatic stress,which is the same in all directions.The first matrix on the right-hand side is the deviator stress,which is a state of stress with zero hydrostatic component.The deviator stress components are therefore defined as.In an isotropic elastic body,a state of stress with a zero hydrostatic component only distortion and no volume change in the body.A purely hudrostatic state of stress produces only a volume change with no distortion.Hookes law,expressed in terms of the decomposed stresses and strains,is where B is the bulk modulus,i.e.Notice that in the form of the deviator stress-deviator strain relationship is the same for all components,and that the dilatation strain is a function of only the hydrostatic stress.The elastic constitutive relations,the equilibrium equations,and the compatibility conditions form a complete description of the behavior of linearly elastic materials under the action of loads.Since the number of those equations equals the number of unknown quantities,it is possible,in principle,to determine the distribution of stress in a body when the distribution of tractions on the surface is given.In practice,however,the solution of a boundary value problem in three dimensions is a nearly impossible task.The methods which involve various approximate mathematical techniques have been shown to be most successful in generating solutions of many engineering problems.