Mechanical and numerical modeling of a porous elastic–viscoplastic material with tensile failure

The objective of this paper is to develop simple but comprehensive constitutive equations that model a number of physical phenomena exhibited by dry porous geological materials and metals. For geological materials the equations model: porous compaction; porous dilation due to distortional deformation and tensile failure; shear enhanced compaction; pressure hardening of the yield strength; damage of the yield strength due to distortional deformation and porosity changes; and dependence of the yield strength on the Lode angle. For metals the equations model: hardening of the yield strength due to plastic deformation; pressure and temperature dependence of the yield strength, and damage due to nucleation of porosity during tensile failure. The equations are valid for large deformations and the elastic response is hyperelastic in the sense that the stress is related to a derivative of the Helmholtz free energy. Also, the equations are viscoplastic with rate dependence occurring in both the evolution equations of porosity and elastic distortional deformations. Moreover, formulas are presented for robust numerical integration of the evolution equations at the element level that can be easily implemented into standard computer programs for dynamic response of materials.

[1]  D. Steinberg,et al.  A constitutive model for metals applicable at high-strain rate , 1980 .

[2]  D. Grady,et al.  Simulation of spall fracture of aluminum and magnesium over a wide range of load duration and temperature , 1997 .

[3]  P. Flory,et al.  Thermodynamic relations for high elastic materials , 1961 .

[4]  M. B. Rubin,et al.  On the treatment of elastic deformation in finite elastic-viscoplastic theory , 1996 .

[5]  R. Hill.  The mathematical theory of plasticity , 1950 .

[6]  M. B. Rubin,et al.  CALCULATION OF HYPERELASTIC RESPONSE OF FINITELY DEFORMED ELASTIC‐VISCOPLASTIC MATERIALS , 1996 .

[7]  M. Rubin.  An Elastic-Viscoplastic Model for Metals Subjected to High Compression , 1987 .

[8]  M. Rubin.  An elastic-viscoplastic model exhibiting continuity of solid and fluid states , 1987 .

[9]  P. M. Naghdi,et al.  On thermodynamics and the nature of the second law , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[10]  Carl Eckart,et al.  The Thermodynamics of Irreversible Processes. IV. The Theory of Elasticity and Anelasticity , 1948 .

[11]  Y. Gupta,et al.  Shock Waves in Condensed Matter , 1986 .

[12]  Alan K. Miller,et al.  Unified constitutive equations for creep and plasticity , 1987 .

[13]  M. Wilkins.  Calculation of Elastic-Plastic Flow , 1963 .

[14]  J. N. Johnson,et al.  Tensile plasticity and ductile fracture , 1988 .

[15]  M. B. Rubin,et al.  A time integration procedure for plastic deformation in elastic-viscoplastic metals , 1989 .

[16]  S. Bodner,et al.  A Large Deformation Elastic-Viscoplastic Analysis of a Thick-Walled Spherical Shell , 1972 .

[17]  M. Rubin.  Plasticity theory formulated in terms of physically based microstructural variables—Part I. Theory , 1994 .

[18]  George Y. Baladi,et al.  Soil Plasticity: Theory and Implementation , 1985 .

[19]  Lynn Seaman,et al.  Dynamic failure of solids , 1987 .

[20]  J. F. Besseling.  A Thermodynamic Approach to Rheology , 1968 .

[21]  M. Rubin.  An elastic-viscoplastic model for large deformation , 1986 .

[22]  Albert C. Holt,et al.  Suggested Modification of the P‐α Model for Porous Materials , 1972 .

[23]  M. Rubin,et al.  On the relationship between phenomenological models for elastic-viscoplastic metals and polymeric liquids , 1993 .

[24]  A. Gurson.  Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I—Yield Criteria and Flow Rules for Porous Ductile Media , 1977 .

[25]  P. M. Naghdi,et al.  The second law of thermo-dynamics and cyclic processes , 1978 .

[26]  A. I. Leonov.  Nonequilibrium thermodynamics and rheology of viscoelastic polymer media , 1976 .

[27]  S. Bodner.  Review of a Unified Elastic—Viscoplastic Theory , 1987 .

[28]  J. Gillis,et al.  Methods in Computational Physics , 1964 .

[29]  M. Rubin,et al.  Modeling added compressibility of porosity and the thermomechanical response of wet porous rock with application to Mt. Helen Tuff , 1996 .

[30]  J. W. Swegle,et al.  Shock viscosity and the prediction of shock wave rise times , 1985 .

[31]  H. C. Heard,et al.  High-pressure mechanical properties of Mt. Helen, Nevada, tuff , 1973 .