Yield Functions and Flow Rules for Porous Pressure-Dependent Strain-Hardening Polymeric Materials

Abstract

To characterize the response of progressively damaged glassy polymers due to the presence and evolution of voids, yield functions and flow rules were developed systematically for a pressure-dependent matrix following the modified von Mises criterion. A rigid-perfectly plastic material was first assumed. The upper bound method was used with a velocity field which has volume preserving and shape changing portions. Macroscopic yield criterion in analytical closed form was first obtained for spherical voids which is valid for all possible macroscopic strain rate fields. Macroscopic yield criteria in analytical closed form were then obtained for cylindrical voids for the special cases of axisymmetric and plane-strain modes of deformation. The upper-bound solutions were subsequently improved to better match analytical solutions for pure hydrostatic loading. Characteristics of the yield function as a function of pressure dependency and void fraction were studied in detail. Generalization of the model for spherical voids to include elasticity as well as strain hardening of the matrix was then obtained. An example for the uniaxial response of a progressively damaged material was then used to illustrate one possible application of the full set of constitutive equations. [S0021-8936(00)02902-0]

A new macroscopic criterion of porous materials with a Mises-Schleicher compressible matrix

“…A good agreement is found not only for a Mises-Schleicher type matrix with weak asymmetry between tensile and compressive yield stresses, but also for a strongly asymmetric matrix with low and high porosities. The new macroscopic criterion improves the existing ones proposed by (Lee and Oung, 2000) and (Durban et al, 2010). …”

“…To consider this effect, some studies have been devoted to porous materials with a Drucker-Prager type compressible matrix (see for instance (Jeong, 2002;Guo et al, 2008;Trillat and Pastor, 2005)); a Green type matrix (an elliptic isotropic criterion (Green, 1972)) has been recently studied by (Shen et al, 2012;Pastor et al, 2013a;Fritzen et al, 2013;Shen et al, 2013) to consider the compressibility in the matrix; some works focus on the double porous material with two populations of voids at different scales (Vincent et al, 2008(Vincent et al, , 2009Shen et al, 2012;Shen et al, 2014). Concerning a pressure-sensitive matrix obeying the Mises-Schleicher criterion (Schleicher, 1926) (parabolic type), many works (such as (Raghava et al, 1973), (Zhang et al, 2008), (Kovrizhnykh, 2004), (Aubertin and Li, 2004)) have been done: (Lee and Oung, 2000) and (Durban et al, 2010) have proposed macroscopic criteria for porous materials with a MisesSchleicher type matrix to consider the effects of porosity and the compressibility of the matrix on the macroscopic behavior; recently, (Monchiet and Kondo, 2012) established the closed-form expressions of the velocity and of the stress field for the case of hydrostatic tension and compression; (Pastor et al, 2013b) proposed numerical bounds (lower and upper bounds) for this class of pressure sensitive porous materials by means of 3D finite element formulations of the static and kinematic methods of Limit Analysis. Comparing with the numerical results, these criteria proposed by (Lee and Oung, 2000) and (Durban et al, 2010) can not be validated for high compressible matrix (strong asymmetry between tension and compression).…”

“…Concerning a pressure-sensitive matrix obeying the Mises-Schleicher criterion (Schleicher, 1926) (parabolic type), many works (such as (Raghava et al, 1973), (Zhang et al, 2008), (Kovrizhnykh, 2004), (Aubertin and Li, 2004)) have been done: (Lee and Oung, 2000) and (Durban et al, 2010) have proposed macroscopic criteria for porous materials with a MisesSchleicher type matrix to consider the effects of porosity and the compressibility of the matrix on the macroscopic behavior; recently, (Monchiet and Kondo, 2012) established the closed-form expressions of the velocity and of the stress field for the case of hydrostatic tension and compression; (Pastor et al, 2013b) proposed numerical bounds (lower and upper bounds) for this class of pressure sensitive porous materials by means of 3D finite element formulations of the static and kinematic methods of Limit Analysis. Comparing with the numerical results, these criteria proposed by (Lee and Oung, 2000) and (Durban et al, 2010) can not be validated for high compressible matrix (strong asymmetry between tension and compression).…”

Plastic flow and failure in single point incremental forming of PVC sheets

“…Because PVC is nearly incompressible [11], the normality rule typical of associated plasticity does not hold in case of pressure dependent yield surfaces (Equation (2)) since it leads to volume variations that are at least one order of magnitude larger than observed experimental values [12]. Instead, a non-associated flow rule where the plastic potential is taken as J 2 of the von-Mises plasticity should be employed [11,13]. Under these circumstances, and assuming isotropic plastic deformation under plane stress conditions " 3 = 0, the constitutive equations relating the inplane strain increments with the applied stresses are given by Equation (3) …”

Numerical limit analysis and plasticity criterion of a porous Coulomb material with elliptic cylindrical voids

“…Recent studies were devoted to porous materials with a matrix exhibiting a pressure-sensitive behavior ( [2][3][4], etc.). Other extensions of the Gurson model accounting for void-shape effects have been also proposed: see among others the references [5][6][7][8][9][10] for spheroidal voids, and [11,12] for ellipsoidal cavities.…”