SM 1 10 Continuum Mechanics Constitutive equations CONTINUUM

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SM 1 -10: Continuum Mechanics: Constitutive equations CONTINUUM MECHANICS (CONSTITUTIVE EQUATIONS - HOOKE LAW)

SM 1 -10: Continuum Mechanics: Constitutive equations CONTINUUM MECHANICS (CONSTITUTIVE EQUATIONS - HOOKE LAW) M. Chrzanowski: Strength of Materials /16

SM 1 -10: Continuum Mechanics: Constitutive equations Summary of stress and strain state equations

SM 1 -10: Continuum Mechanics: Constitutive equations Summary of stress and strain state equations 3 internal equilibrium equations (Navier eq. ) 6 unknown functions (stress matrix components) Boundary conditions (statics) 6 kinematics equations (Cauchy eq. ), 9 unknown functions (6 strain matrix components, 3 displacements) 9 equations 15 unknown functions (6 stresses, 6 strains, 3 displacements) M. Chrzanowski: Strength of Materials Boundary conditions (kinematics) From the formal point of view (mathematics) we are lacking 6 equations From the point of view of physics – there are no material properties involved 2/16

SM 1 -10: Continuum Mechanics: Constitutive equations An obvious solution is to exploit already

SM 1 -10: Continuum Mechanics: Constitutive equations An obvious solution is to exploit already noticed interrelation between strains and stresses General property of majority of solids is elasticity (instantaneous shape memory) Material Deformation Strain deformability versusinternal stress forces properties P This observation was made already in 1676 by Robert Hooke: CEIIINOSSSTTUV UT TENSIO SIC VIS which reads: „as much the extension as the force is” u Linear elasticity M. Chrzanowski: Strength of Materials where k is a constant dependent on a material and body shape 3/16

SM 1 -10: Continuum Mechanics: Constitutive equations To make Hooke’s law independent of a

SM 1 -10: Continuum Mechanics: Constitutive equations To make Hooke’s law independent of a body shape one has to use state variables characterizing internal forces and deformations in a material point i. e. stress and strain. Dynamics Kinematics Physical quantities (measurable) Hooke, 1678 Mathematical quantities (non-measurable) Navier, 1822 f- M. Chrzanowski: Strength of Materials linear function of all strain matrix components defining all stress component matrix 4/16

SM 1 -10: Continuum Mechanics: Constitutive equations As Navier equation is a set of

SM 1 -10: Continuum Mechanics: Constitutive equations As Navier equation is a set of 9 linear algebraic equations then the number of coefficient in this set is 81 and can be represented as a matrix of 34=81 components: The coefficients of this equation do depend only on the material considered, but not on the body shape. Summation over kl indices reflects linear character of this constitutive equation. M. Chrzanowski: Strength of Materials 5/16

SM 1 -10: Continuum Mechanics: Constitutive equations Universality of linear elasticity follows observation, that

SM 1 -10: Continuum Mechanics: Constitutive equations Universality of linear elasticity follows observation, that for loading below a certain limit (elasticity limit) most of materials exhibit this property. Nevertheless, the number of assumptions allows for the reduction of the coefficients number: two of them are already inscribed in the formula: 1. For zero valued deformations all stresses vanish: the body in a natural state is free of initial stresses. 2. Coefficients Cijkl do not depend on position in a body – material properties are uniform (homogeneous). M. Chrzanowski: Strength of Materials 6/16

SM 1 -10: Continuum Mechanics: Constitutive equations 3. Assumption of the existence of elastic

SM 1 -10: Continuum Mechanics: Constitutive equations 3. Assumption of the existence of elastic potential yields symmetry of group of indices ij - kl thus reducing the number of independent coefficients to 36 [=(81 -9)/2]. 4. Symmetry of material inner structure allows for further reductions. In a general case of lacking any symmetry (anisotropy) the number of independent coefficients is 21 [=(36 -6)/2+6]. 5. In the simplest and the most frequent case of structural materials (except the composite materials) – the number of coefficients is 2 (isotropy): or in an inverse form: The pairs of coefficients G, i E, ν are interdependent so there are really only two material independent constants. M. Chrzanowski: Strength of Materials 7/16

SM 1 -10: Continuum Mechanics: Constitutive equations 6 9 General anisotropy: 15+6=21 constants 81

SM 1 -10: Continuum Mechanics: Constitutive equations 6 9 General anisotropy: 15+6=21 constants 81 components of Cijkl Symmetrical components Identical components Components dependent on other components M. Chrzanowski: Strength of Materials 0 0 0 0 0 0 Isotropy: 2 constants 8/16

SM 1 -10: Continuum Mechanics: Constitutive equations Lamé constants [Pa] Summation obeys ! Kronecker’s

SM 1 -10: Continuum Mechanics: Constitutive equations Lamé constants [Pa] Summation obeys ! Kronecker’s delta This equation consists of two groups: Normal stress and normal strain dependences Shear stress and shear strain dependences M. Chrzanowski: Strength of Materials 9/16

SM 1 -10: Continuum Mechanics: Constitutive equations Young modulus [Pa] Poisson modulus [0] Summation

SM 1 -10: Continuum Mechanics: Constitutive equations Young modulus [Pa] Poisson modulus [0] Summation obeys ! Kronecker’s delta Normal stress and normal strain dependences Shear stress and shear strain dependences M. Chrzanowski: Strength of Materials 10/16

zmianyequations objętości SM 1 -10: Continuum Mechanics: Prawo Constitutive =3 Mean stress Mean strain

zmianyequations objętości SM 1 -10: Continuum Mechanics: Prawo Constitutive =3 Mean stress Mean strain Volume change law M. Chrzanowski: Strength of Materials 11/16

SM 1 -10: Continuum Mechanics: Constitutive equations Decomposition of symmetric matrix (tensor) into deviator

SM 1 -10: Continuum Mechanics: Constitutive equations Decomposition of symmetric matrix (tensor) into deviator and volumetric part (axiator) deviator = M. Chrzanowski: Strength of Materials axiator + 12/16

SM 1 -10: Continuum Mechanics: Constitutive equations Volume change law Distortion law M. Chrzanowski:

SM 1 -10: Continuum Mechanics: Constitutive equations Volume change law Distortion law M. Chrzanowski: Strength of Materials 13/16

SM 1 -10: Continuum Mechanics: Constitutive equations 0 -1 ν 1/2 G E/3 E/9

SM 1 -10: Continuum Mechanics: Constitutive equations 0 -1 ν 1/2 G E/3 E/9 K - No shape change: stiff material M. Chrzanowski: Strength of Materials 0 Constants cross-relations: No volume change: incompressible material 14/16

SM 1 -10: Continuum Mechanics: Constitutive equations x 3 + x 3 x 2

SM 1 -10: Continuum Mechanics: Constitutive equations x 3 + x 3 x 2 x 1 Volume change x 2 Shape change x 1 Composed (full) deformation M. Chrzanowski: Strength of Materials 15/16

SM 1 -10: Continuum Mechanics: Constitutive equations x 3 FAST + x 3 x

SM 1 -10: Continuum Mechanics: Constitutive equations x 3 FAST + x 3 x 2 x 1 Volume change x 2 Shape change x 1 Composed (full) deformation M. Chrzanowski: Strength of Materials 16/16

SM 1 -10: Continuum Mechanics: Constitutive equations stop M. Chrzanowski: Strength of Materials 17/16

SM 1 -10: Continuum Mechanics: Constitutive equations stop M. Chrzanowski: Strength of Materials 17/16