Section 6: Material modelling in solid mechanics - GAMM 2012

Section 6: Material modelling in solid mechanics 19
A New Continuum Approach to the Coupling of Shear Yielding and Crazing with
Fracture in Glassy Polymers
Lisa Schänzel, Christian Miehe (Universität Stuttgart)
Over the past decades, considerable effort was made to develop constitutive models that account
for finite viscoplasticity and failure in glassy polymers. Recently, we developed a new model
of ductile thermoviscoplasticity of glassy polymers in the logarithmic strain space [1]. However,
depending on thermal and loading rate conditions to which the material is subjected, the response
might change from ductile to brittle. This brittle response is characterized by inelastically deformed
zones, so-called crazes, having the thickness of micrometers and spanning at some fractions of
a millimeter [2]. The crazing is associated with considerable dilatational plasticity, containing a
dense array of fibrils interspersed with elongated voids. The shear yielding and crazing are not
completely independent excluding each other. In this lecture, we outline an extension of the ductile
plasticity model [1] towards the description of (i) volumetric directional plasticity effect due to
crazing and (ii) the modeling of the local failure due to fracture. To this end, the first extension
accounts for a (i) dilatational plastic deformation mechanism in the direction of the maximum
principle tensile stress. The ultimate amount of this volumetric plastic craze strain is bounded
by a limiting value, where failure occurs. Then, in a second step, the (ii) modeling of subsequent
failure mechanisms is realized by the introduction of a fracture phase field, characterizing via an
auxiliary variable the crack topology. Here, we adopt structures of a recently developed continuum
phase field model of fracture in brittle solids [3], and modify it for a fracture driving term related
to the volumetric plastic deformation of the crazes. We demonstrate the performance of proposed
formulation by means of representative boundary value problems.
[1] Miehe, C.; Mendez, J.; Göktepe, S.; Schänzel, L. [2011]: Coupled thermoviscoplasticity of
glassy polymers in the logarithmic strain space based on the free volume theory. International
Journal of Solids and Structures, 48: 1799–1817.
[2] Kramer, E. J.[1983]:Microscopic and Molecular Fundamentals of Crazing. Advances in Polymer
Science, 52/53: 1–56.
[3] Miehe, C; Hofacker, M.; Welschinger, F. [2010]: A phase field model for rate-independent
crack propagation: Robust algorithmic implementation based on operator splits. Computer
Methods in Applied Mechanics and Engineering, 199: 2765-2778.
Anisotropic finite strain hyperelasticity based on the multiplicative decomposition of
the deformation gradient
Raad Al-Kinani, Kaveh Talebam, Stefan Hartmann (TU Clausthal)
Frequently, the case of finite strain anisotropy, particularly, the case of transversal isotropy, is
applied to biological applications or to model fiber-reinforced composite materials. In this article
the multiplicative decomposition of the deformation gradient into one part constrained in the direction
of the axis of anisotropy and one part describing the remaining deformation is proposed.
Accordingly, a form of additively decomposed strain-energy function is proposed. This leads to a
clear assignment of deformation and stress states in the direction of anisotropy and the remaining
part. The decomposition of the case of transversal isotropy is explained. The behavior of the
model is investigated at different simple analytical examples, such as uniaxial tension along and
perpendicular to the axis of anisotropy, and simple shear. In addition, the model is also studied for
the case of a thick-walled tube under internal pressure, where a second order ordinary differential