MATERIAL ELASTIC HERRMANN INCOMPRESSIBLE command Synopsis The MATERIAL ELASTIC HERRMANN INCOMPRESSIBLE command is used to specify the parameters associated with an isotropic, linear elastic material idealization valid for incompressible as well as compressible materials The associated constitutive relation was first proposed in 1965 by L R Herrmann [1] The MATERIAL ELASTIC HERRMANN INCOMPRESSIBLE idealization is intended for use with mixed elements for the analysis of incompressible or nearly incompressible solids Consequently, this material idealization should be used in conjunction with mixed elements possessing discontinuous pressure approximations Presently APES contains the following elements of this type: Q4P1d quadrilateral elements Q9P3d quadrilateral elements T6P1d triangular elements T7P3d triangular elements H8P1d hexahedral elements H20P4d hexahedral elements H27P4d hexahedral elements Syntax The following syntax is used to describe a typical isotropic elastic Herrmann material idealization: MATerial ELAstic HERrmann incompressible NUMber # (DEScription string ) (DENsity ##) (LAMbda ## ) (SHEar modulus ##) 1 V N Kaliakin
Explanatory Notes The NUMBER keyword is used to specify the (global) number of the material associated with the incompressible isotropic elastic idealization The default material number is one (1) The default material DENSITY is zero (00) The optional alphanumeric string associated with the DESCRIPTION keyword must be enclosed in double quotes ( ) It is used solely to describe the material being idealized to the analyst The DESCRIPTION string is printed as part of the echo of the material information For mixed (as opposed to mixed/penalty) versions the Q4P1, Q9P3, T6P1 and H8P1 elements the keyword LAMBDA is used to input an appropriate value for the Lamé parameter λ Denoting the elastic modulus by E and Poisson s ratio by ν, λ = Eν (1 + ν)(1 2ν) The default value of LAMBDA is 250e+05 The inverse of λ is used in the element pressure submatrix; viz, ( ) 1 L (e) = λ (e) N(p)T N (p) dω Ω e where N (p) is a vector of pressure interpolation functions used to approximate the pressure in the element domain Ω e For the case of ν = 00 one cannot input λ = GAMMA = 00, for this would cause a divideby-zero runtime error In such cases, the analyst should simply input a small number (eg, 10) Past experience shows that the element will then yield results that are identical to those associated with the corresponding mixed elements with the incompressibility constraint exactly enforced (but with L (e) = 0 and thus a computationally more complex solution) As a material approaches the incompressible limit λ approaches infinity For slightly compressible materials, experience shows that the mixed versions of the Q4P1, Q9P3, T6P1 and H8P1 elements give accurate results If the mixed/penalty version of the Q4P1, Q9P3, T6P1 and H8P1 elements is used, the submatrix L (e) is replaced by (1/λ) I, where λ now represents a large but finite penalty number that approximately enforces the incompressibility constraint and I is an identity matrix of appropriate size In this case, LAMBDA should be taken equal to a value that is approximately one order of magnitude greater than the actual value of λ associated with the material idealization As the incompressible limit is approached, the check on the suitability of the value chosen for LAMBDA is the magnitude of the element volumetric strain, which should be zero (at least to the significant digits displayed in the result) 2 V N Kaliakin
The keyword SHEAR MODULUS is used to specify the elastic shear modulus G = µ, where E G = 2(1 + ν) The default SHEAR MODULUS is 1150e+05 Example of Command Usage Assuming that E = 20 x 10 5 and ν = 04999, and noting that and that λ = Eν (1 + ν)(1 2ν) = (20 x 10 5 )(04999) = 3333 x 109 (1 + 04999)(1 2(04999)) G = E 2(1 + ν) = (20 x 105 ) = 6667 x 105 2(1 + 04999) the following commands are sufficient to specify an incompressible Herrmann isotropic elastic material for these parameter values: mat elas Herrmann_incomp num 1 des "nearly incompressible idealization" & lambda 3333e+09 shear 6667e+05 3 V N Kaliakin
Theoretical Considerations The isotropic, linear elastic material idealization valid for incompressible as well as compressible materials was first proposed in 1965 by L R Herrmann [1] This formulation is based on the observation that the main difficulty in applying standard displacement-based formulations to incompressible or nearly incompressible media lies in the determination of the mean stress or pressure, which is related to the volumetric part of the strain For this reason, it is convenient to separate this pressure from the total stress field and to treat it as a primary dependent variable independent from the displacement Utilyzing a special case of the Hellinger-Reissner mixed variational principle in which displacements and the average hydrostatic pressure were selected as primary dependent variables, Herrmann [1] proposed a mixed formulation consisting of the constitutive relation and the following constraint equation for the additional unknown p: σ ij = pδ ij + 2µε ij (1) u i,i + p λ = 0 (2) In equations (1) and (2) compressive pressure is taken as being negative For the incompressible case, ν = 1/2 λ = Equation (2) thus correctly enforces the kinematic constraint of u i,i = 0 For compressible materials ν < 1/2 and λ is finite Solving equation (2) gives p = λu i,i Substituting this relation into equation (1), and noting that δ ij u k,k = ε 11 + ε 22 + ε 33 = ε kk gives σ ij = λδ ij ε kk + 2µε ij (3) which is the standard constitutive relation for isotropic linear elasticity Remark 1 The pressure p appearing in equations (1) and (2) does not necessarily equal the mean pressure From equation (3) it follows that σ ii = 3λε kk + 2µε ii = (3λ + 2µ) ε kk From the definition of the mean or hydrostatic pressure p, it follows that p = σ ii (λ 3 = + 23 ) µ ε kk = Kε kk Now from equation (2) p = λu i,i = λε kk Since K λ, it follows that for the compressible case, p does not represent the hydrostatic pressure Indeed, p λ = p K For the nearly incompressible case µ << λ, K λ, and thus p p Finally, for the incompressible case, the standard constitutive relations from isotropic elasticity do not apply, thus no 4 V N Kaliakin
meaningful relationship can be made However, for the mixed method, for λ =, equation (2) correctly gives u i,i = ε kk = 0 Thus, from equation (1) σ kk = 3p + 2µε kk = 3p showing that now p = σ kk /3 = p 5 V N Kaliakin
Bibliography [1] Herrmann, L R, Elasticity Equations for Nearly Incompressible Materials by a Variational Theorem, AIAA Journal, 3: 1896-1900 (1965) 6