Constitutive Relations in Solids Elasticity H. Garmestani, Professor School of Materials Science and Engineering Georgia Institute of Technology Outline: Materials Behavior Tensile behavior …

The Elastic Solid and Elastic Boundary Value Problems Constitutive equation is the relation between kinetics (stress, stress-rate) quantities and kinematics (strain, strain-rate) quantities for a specific material. It is a mathematical description of the actual behavior of a material. The same material may exhibit different behavior at different temperatures, rates of loading and duration of loading time.). Though researchers always attempt to widen the range of temperature, strain rate and time, every model has a given range of applicability. Constitutive equations distinguish between solids and liquids; and between different solids. In solids, we have: Metals, polymers, wood, ceramics, composites, concrete, soils… In fluids we have: Water, oil air, reactive and inert gases

The Elastic Solid and Elastic Boundary Value Problems (cont.) Load-displacement response

Examples of Materials Behavior Uniaxial loading-unloading stress-strain curves for (a) linear elastic; (b) nonlinear elastic; and (c) inelastic behavior.

Constitutive Equations: Elastic Elastic behavior is characterized by the following two conditions: (1) where the stress in a material () is a unique function of the strain (), (2) where the material has the property for complete recovery to a “natural” shape upon removal of the applied forces Elastic behavior may be Linear or non-linear

Constitutive Equation The constitutive equation for elastic behavior in its most general form as where C is a symmetric tensor-valued function and e is a strain tensor we introduced earlier. Linear elastic  = Ce Nonlinear-elastic  = C(e) e

Equations of Infinitesimal Theory of Elasticity  Boundary Value Problems  we assume that the strain is small and there is no rigid body rotation. Further we assume that the material is governed by linear elastic isotropic material model.   Field Equations (1) (2) Stress Strain Relations (3)Cauchy Traction Conditions (Cauchy Formula) (4)

Equations of the Infinitesimal Theory of Elasticity (Cont'd) In general, We know that For small displacement Thus

Equations of the Infinitesimal Theory of Elasticity (Cont'd) Assume v << 1, then For small displacement, Thus for small displacement/rotation problem

Equations of the Infinitesimal Theory of Elasticity (Cont'd) Consider a Hookean elastic solid, then Thus, equation of equilibrium becomes

Equations of the Infinitesimal Theory of Elasticity (Cont'd) For static Equilibrium Then The above equations are called Navier's equations of motion. In terms of displacement components

Plane Elasticity In a number of engineering applications, the geometry of the body and loading allow us to model the problem using 2-D approximation. Such a study is called ''Plane elasticity''.  There are two categories of plane elasticity, plane stress and plane strain. After these, we will study two special case: simple extension and torsion of a circular cylinder.

Plane Strain &Plane Stress For plane stress, (a)    Thus equilibrium equation reduces to (b)    Strain-displacement relations are (c)    With the compatibility conditions,

Plane Strain &Plane Stress (d)     Constitutive law becomes, Inverting the left relations, Thus the equations in the matrix form become: (e)    In terms of displacements (Navier's equation)

Plane Strain (b) (Cont'd) (b)     Inverting the relations,               can be written as: (c)      Navier's equation for displacement can be written as:

The Elastic Solid and Elastic Boundary Value Problems  Relationship between kinetics (stress, stress rate) and kinematics (strain, strain-rate) determines constitutive properties of materials. Internal constitution describes the material's response to external thermo-mechanical conditions. This is what distinguishes between fluids and solids, and between solids wood from platinum and plastics from ceramics.   Elastic solid Uniaxial test: The test often used to get the mechanical properties

Linear Elastic Solid If      is Cauchy tensor and       is small strain tensor, then in general, where         is a fourth order tensor, since T and E are second order tensors. is called elasticity tensor. The values of these components with respect to the primed basis ei’ and the unprimed basis ei are related by the transformation law However, we know that                   and                     then We have          symmetric matrix with 36 constants, If elasticity is a unique scalar function of stress and strain, strain energy is given by

Linear Elastic Solid Show that if for a linearly elastic solid, then Solution: Since for linearly elastic solid , therefore Thus from , we have Now, since Therefore,

Linear Elastic Solid (cont.) Now consider that there is one plane of symmetry (monoclinic) material, then     One plane of symmetry =>     13 If there are 3 planes of symmetry, it is called an ORTHOTROPIC material, then     orthortropy =>  3 planes of symmetry =>   9 Where there is isotropy in a single plane, then     Planar isotropy   =>      5 When the material is completely isotropic (no dependence on orientation)     Isotropic  =>      2

Linear Elastic Solid (cont.) Crystal structure Rotational symmetry Number of independent elastic constants Triclinic Monoclinic Orthorhombic Tetragonal Hexagonal Cubic Isotropic None 1 twofold rotation 2 perpendicular twofold rotations 1 fourfold rotation 1 six fold rotation 4 threefold rotations   21 13 9 6 5 3 2

Linear Isotropic Solid A material is isotropic if its mechanical properties are independent of direction Isotropy means Note that the isotropy of a tensor is equivalent to the isotropy of a material defined by the tensor. Most general form of (Fourth order) is a function

Linear Isotropic Solid Thus for isotropic material   and are called Lame's constants. is also the shear modulus of the material (sometimes designated as G).

Relationship between Youngs Modulus EY, Poisson's Ratio g, Shear modulus m=G and Bulk Modulus k We know that So we have Also, we have

Relationship between EY, g, m=G and k (Cont'd) Note: Lame’s constants, the Young’s modulus, the shear modulus, the Poisson’s ratio and the bulk modulus are all interrelated. Only two of them are independent for a linear, elastic isotropic materials,