力學(Mechanics)

It is observed from frequent experimentation that drawing identical bars from steel and rubber by same axial force and consequently the same axial stress, results in different elongations from the two bars. In mechanics, the differences of the two bars is represented by the relationship between the components of stress and strain. The writing of these like a matrix of column, i.e
(1)
(2)

we have
(3)
where X is a 5 x 5 matrix that characterizes the material of the body, and is generally called the matrix of elasticity and its components are called elasticity’s or constants of elasticity for the material of the body. Note that shear strains have been multiplied by 2 in eqn. (2); the definitions (1) and (2) make sure that . The equation (3), in other words the relation between the stresses and strains, are called the relation constitutive for the material of the body. It is assumed in Eq. (3) that the body is stress free in the configuration of reference from which the strain d is measured.

It is observedfrom frequent experimentation that drawing pulling 1identicalbars from steeland rubber by the same axial force,and consequentlythe sameaxial stress, results of stress indifferent elongations from the two bars.  In mechanics, the differences difference inthe materials 2of the two bars is represented by the relationship betweenthe components of stress and the strain. TheBywriting each ofthese likeasa matrix of column matrix,i.e.,3

(1)

(2)

we have

   (3)

where X is a 5 x 5 matrix that characterizes thematerial of the body, and is generallycalled the matrix of elasticityand itscomponents are called elasticity’s elasticities 4or elastic constantsof elasticity 5for the material of the body.Note that shear strains have been multiplied by 2 in eqn. (2); the definitions (1) and(2) make  ensure6 that.  Theequation (3), in other words the relation between the stresses and strains, areiscalled the relationconstitutive relation for the material of the body.It is assumed in Eq. (3) that the body is stress free in the reference configuration of reference from which the strain d is measured.

It is observed from fFrequent1experimentation has shown that drawing pulling 2the identical barsof from steel and rubber having the same initial lengths by the sameaxial force, and consequently thus thesame axial stress, results ofstress in different elongations from of these two bars.  In mechanics, the differences difference in the materials 3of the two bars is represented by the relationship betweenthe components of stress and the strain.TheBywritingexpressing 4each of these likeasa matrix of column matrix5,i.e.,6

(1)

(2)

we haveobtain

   (3)

whereHere,X is a 57 x× 5 matrix that characterizes thematerial of the body,and is generally. It is generally calledknown asthe matrix of elasticity andmatrix;8 itscomponents are called elasticity’s elasticities 9or elastic constants10of elasticity forthe material of the body. Note thatThe shear strains have been multiplied by 2 in eqn.Eq. (2); the definitions (1)and (2) makeensure11 that.  Thestress-strainequation (3), in other words the relation shown inbetween the stresses and strains, are Eq. (3) is called known as therelation constitutiverelationfor the material of the body. 12It is assumed in Eq. (3) that thebody is stress-13free in the reference configuration of reference fromwhich the strain d is measured.