ASP01. Constitutive Eq_Linear Elastic

Continuum MechanicsContinuum Mechanics

Constitutive EquationsLinear Ealsiticity


Introduction• Continuum; mathematics teaches us that between any two

points in space we can always find another point, regardless of how close together we choose the original pair

• The analysis of the kinematic and mechanical behavior of materials modeled on the continuum assumption is what we know as continuum mechanics

• Two main themes into which the topics of continuum mechanics;– the first, emphasis is on the derivation of fundamental equations

which are valid for all continuous media– the second, the focus of attention is on the development of so-called

constitutive equations characterizing the behavior of specific idealized materials,


Introduction• Mathematically, the fundamental equations of

continuum mechanics may be developed in two separate but essentially equivalent formulations;– One, the integral or global form, derives from a

consideration of the basic principles being applied to a finite volume of the material

– The other, a differential or field approach, leads to equations resulting from the basic principles being applied to a very small (infinitesimal) element of volume


Introduction


Introduction• Establishment of constitutive models for engineering

materials, which may classified into three groups:– Algebraic constituive relations, such as linear and nonlinear elastic

models,– Differential constitutive realtions, such as plastic and viscoplastic

models, and– Integral constituive relations, such as hereditary viscoelastic models

• It is understood that the constitutive models are linear for performance analysis under service load condition, and highly nonlinear for limit state studies, when material failure is considered in the form of discontinues fracture process.


Generalized Hooke's Law• If the continuum material is a linear elastic material, we

introduce the generalized Hooke's law in Cartesian coordinates


Generalized Hooke's Law• The Hooke's law is a statement that the stress is proportional to the

gradient of the deformation occurring in the material• These equations assume a linear relationship exists between the

components of the stress tensor and strain tensor and we say stress is a linear function of strain

• Constitutive equations serve to describe the material properties of the medium when it is subjected to external forces


Constitutive Equations• The equations (2.4.1) are constitutive equations which are applicable for

materials exhibiting small deformations when subjected to external forces.• The 81 constants cijkl are called the elastic stiffness of the material. The

above relations can also be expressed in the form

where sijkl are constants called the elastic compliance of the material.

• Due to the symmetry of the stress and strain tensors we find that the elastic stiffness and elastic compliance tensor must satisfy the relations

and consequently only 36 of the 81 constants are actually independent.


Restrictions on Elastic Constants due to Symmetry

• The equations (2.4.1) and (2.4.2) can be replaced by an equivalent set of equations which are easier to analyze. This is accomplished by defining the quantities

where

and



• Then the generalized Hooke's law from the equations (2.4.1) and (2.4.2) can be represented in either of the forms

where cij are constants related to the elastic stiffness and sij are constants related to the elastic compliance.


Restrictions on Elastic Constants due to Symmetry• These constants satisfy the relation

Here

and similarly

• These relations show that the constant scij are related to the elastic stiffness coefficient scpqrs by the relations



• A similar type relation holds for the constants sij and spqrs;• The above relations can be verfied by expanding the

equations (2.4.1) and (2.4.2) and comparing like terms with the expanded form of the equation (2.4.4)



• The generalized Hooke's law can now be expressed in a form where the 36 independent constants can be examined in more detail under special material symmetries

• Alternatively, in the arguments that follow, one can examine the equivalent form


Material Symmetries• If we let the x1-x2 plane be a plane of symmetry then the equations (2.4.6)

must remain invariant under the coordinate transformation

which represents an inversion of the x3 axis. That is, if the x1-x2 plane is a plane of symmetry we should be able to replace x3 by −x3 and the equations (2.4.6) should remain unchanged. This is equivalent to saying that a transformation of the type from equation (2.4.7) changes the Hooke's law to the form i =sijj where the sij remain unaltered because it is the same material.

• Employing the transformation equations

• the stress and strain transformation equations


Material Symmetries• If we expand both of the equations (2.4.9) and substitute in the nonzero

derivatives

we obtain the relations


Material Symmetries


Material Symmetries• The equation (2.4.6) then becomes

• If the stress strain relation for the new orientation of the x3 axis is to have the same form as the old orientation, then the equations (2.4.6) and (2.4.12) must give the same results. Comparison of these equations we find that


Material Symmetries• In summary, from an examination of the equations (2.4.6) and (2.4.12) we

find that for an aelotropic material (crystal), with one plane of symmetry, the 36 constants sij reduce to 20 constants and the generalized Hooke's law (constitutive equation) has the form

• Alternatively, the Hooke's law can be represented in the form


Additional Symmetries

• If the material (crystal) is such that there is an additional plane of symmetry, say the x2-x3 plane, then reversal of the x1 axis should leave the equations (2.4.14) unaltered.

• If there are two planes of symmetry then there will automatically be a third plane of symmetry. Such a material (crystal) is called orthotropic.

• Introducing the additional transformation

which represents the reversal of thex1axes, the expanded form of equations (2.4.9) are used to calculate the efect of such a transformation upon the stress and strain tensor. We find 1; 2; 3; 6;e1;e2;e3;e6 remain unchanged while 4; 5;e4;e5 change sign.

• The equation (2.4.14) then becomes



• Note that if the constitutive equations (2.4.14) and (2.4.15) are to produce the same results upon reversal of the x1 axes, then we require that the following coefficients be equated to zero:



• This then produces the constitutive equation

or its equivalent form

• and the original 36 constants have been reduced to 12 constants. This is the constitutive equation for orthotropic material (crystals)


Axis of Symmetry

• If in addition to three planes of symmetry there is an axis of symmetry then the material (crystal) is termed hexagonal.

• Assume that the x1 axis is an axis of symmetry and consider the effect of the transformation

upon the constitutive equations. It is left as an exercise to verify that the constitutive equations reduce to the form where there are 7 independent constants having either of the forms

or


Axis of Symmetry• Finally, if the material is completely symmetric, the x2 axis is also an axis of

symmetry and we can consider the effect of the transformation

upon the constitutive equations. It can be verified that these transformations reduce the Hooke's law constitutive equation to the form

• Materials (crystals) with atomic arrangements that exhibit the above symmetries are called isotropic materials. An equivalent form of (2.4.17) is the relation


Axis of Symmetry

• Elastic stiffness coefficients for some metals which are cubic. Constants are given in units of 1012dynes/cm2

• Under these conditions the stress strain constitutive relations can be written as


Isotropic Material• Materials (crystals) which are elastically the same in all directions are

called isotropic• We have shown that for a cubic material which exhibits symmetry with

respect to all axes and planes, the constitutive stress-strain relation reduces to the form found in equation (2.4.17). Define the quantities

• where E is the Young's Modulus of elasticity, is the Poisson's ratio, and is the shear or rigidity modulus. For isotropic materials the three constants E;; are not independent


Isotropic Material

• EXAMPLE 2.4-1. (Elastic constants) For an isotropic material, consider a cross section of material in the x1-x2 plane which is subjected to pure shearing so that4=12 is the only nonzero stress as illustrated in the figure.

• For the above conditions, the equation (2.4.17) reduces to the single equation

and so the shear modulus is the ratio of the shear stress to the shear angle. Now rotate the axes through a 45 degree angle to a barred system of coordinates where


Isotropic Material

• Expanding the transformation equations (2.4.9) we find that

• In the barred system, the Hooke's law becomes

• This is an important relation connecting the elastic constants associated with isotropic materials. The above transformation can also be applied to triclinic, aelotropic, orthotropic, and hexagonal materials to find relationships between the elastic constants.


Isotropic Material

• The previous discussion has shown that for an isotropic material the generalized Hooke's law (constitutive equations) have the form

where equation (2.4.19) holds. These equations can be expressed in the indicial notation and have the form

where kk= 11+ 22+ 33 is a stress invariant and ij is the Kronecker delta. We can solve for the stress in terms of the strain by performing a contraction on i and j in equation (2.4.21).


Isotropic Material

• This gives the dilatation

• Note that from the result in equation (2.4.21) we are now able to solve for the stress in terms of the strain. We find

• The tensor equation (2.4.22) represents the six scalar equations


Isotropic Material


Alternative Approach to Constitutive Equations

• The constitutive equation defined by Hooke's generalized law for isotropic materials can be approached from another point of view. Consider the generalized Hooke's law:

• If we transform to a barred system of coordinates, we will have the new Hooke's law:

• For an isotropic material we require that • If constants and are called Lame's constants, we find that the constants

and satisfy the relations

• In addition to the constants E; ; ; , it is sometimes convenient to introduce the constant , called the bulk modulus of elasticity, defined by



• EXAMPLE 2.4-2. (Hooke's law) Let us construct a simple example to test the results we have developed so far. Consider the tension in a cylindrical bar illustrated in the figure

• Assume that

where F is the constant applied force and A is the cross sectional area of the cylinder

• Consequently, the generalized Hooke's law (2.4.21) produces the nonzero strains



• From these equations we obtain:– The first part of Hooke's law

– The second part of Hooke's law

• This example demonstrates that the generalized Hooke's law for homogeneous and isotropic materials reduces to our previous one dimensional result given in (2.3.1) and (2.3.2).


Basic Equations of Elasticity


Constitutive equations: Hooke’s LawVoigt-Kelvin notation


Penjelasan Tambahan

Continuum Mechanics


Linear Ealstic Material Behavior

• Generalization of the scalar format of of Hooke’s Law is proportional to to the triaxial state of strain through the linear transformation

• Considering the symmetry of the stress and strain;


Isotropic Linear Elasticity

• Because of the symmtry of stress and strain, the skewed symmetric contribution is inactive, thus isotrpic linear elasticity the material behavior is fully described by two independent elastic constants;


Matrix Form of Elastic Stiffness;

• Isotropic linear behavior of may be cast in matrix format, using the voight notation of symmetric stress and strain tensors and the engineering definition of shear strain ij = 2ij. The elastic stiffness matrix may be written for isotropic behavior as


Matrix Form of Elastic Compliance; C

• The compliance relation for isotropic elasticity in matrix format

• In the isotropic case the shear response is entirely decoupled from the direct response of the normal components. Thus the compliance matrix expands into the partitioned form


Hooke's law for isotropic materials

• In compliance form

• Hooke’s Law in stiffness form


Canonical Format of Isotropic Elasticity

• Decomposing the stress and strain tensors into spherical and deviatoric components leads to stress deviators

and stress deviators

• The decoupled response reduces the elasticity tensor to the scalar form


Canonical Format of Isotropic Elasticity

• In wich the bulk and shear moduli

define the volumetric and deviatoric material stiffness


IsotropicElasiticity under initial volumetric Strain

• In the case of isotropic material behavior, with no directional properties , the size of a representatitve volume element may change due to thermal effect or shrinkage and swelling, but it will not distort. Consequently, the expansion is purely volumetric, i.e. Identical in all direction.

• Matrix equation for the case;

• Considering the special case of plane stress 33=0, the stress-strain relation reduce in the presence of initial volumetric strains to

where the shear components are not affected by the temperature change in the case of isotropy.


Orthothropic Elasiticity• Some engineering materials are orthotropic• By definition, an orthotropic material has at least 2 orthogonal planes of

symmetry, where material properties are independent of direction within each plane. Such materials require 9 independent variables (i.e. elastic constants) in their constitutive matrices.

• In contrast, a material without any planes of symmetry is fully anisotropic and requires 21 elastic constants, whereas a material with an infinite number of symmetry planes (i.e. every plane is a plane of symmetry) is isotropic, and requires only 2 elastic constants

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke.cfm#GeneralizedHooke

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke_isotropic.cfm


Orthothropic ElasiticityHooke's Law in Compliance Form for Orthothropic,• By convention, the 9 elastic constants in orthotropic constitutive

equations are comprised of 3 Young's modulii Ex, Ey, Ez, the 3 Poisson's ratios yz, zx, xy, and the 3 shear modulii Gyz, Gzx, Gxy

• The compliance matrix takes the form

ASP01. Constitutive Eq_Linear Elastic

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