Implementation aspects of a phase-field approach for brittle fracture

doi:10.1007/s11709-018-0477-3

Frontiers of Structural and Civil Engineering

2019, Vol. 13

Issue (2): 417-428 https://doi.org/10.1007/s11709-018-0477-3

本期目录

Implementation aspects of a phase-field approach for brittle fracture

G. D. HUYNH¹(

), X. ZHUANG¹(

), H. NGUYEN-XUAN²

¹. Institute of Continuum Mechanics, Leibniz-Universität Hannover, 30167 Hannover, Germany
². Center for Interdisciplinary Research in Technology, Ho Chi Minh City, University of Technology (HUTEH), Ho Chi Minh City, Vietnam

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Abstract：

This paper provides a comprehensive overview of a phase-field model of fracture in solid mechanics setting. We start reviewing the potential energy governing the whole process of fracture including crack initiation, branching or merging. Then, a discretization of system of equation is derived, in which the key aspect is that for the correctness of fracture phenomena, a split into tensile and compressive terms of the strain energy is performed, which allows crack to occur in tension, not in compression. For numerical analysis, standard finite element shape functions are used for both primary fields including displacements and phase field. A staggered scheme which solves the two fields of the problem separately is utilized for solution step and illustrated with a segment of Python code.

Key words： phase-field modeling FEM staggered scheme fracture

收稿日期: 2017-08-25 出版日期: 2019-03-12

Corresponding Author(s): G. D. HUYNH,X. ZHUANG

引用本文:

. [J]. Frontiers of Structural and Civil Engineering, 2019, 13(2): 417-428.
G. D. HUYNH, X. ZHUANG, H. NGUYEN-XUAN. Implementation aspects of a phase-field approach for brittle fracture. Front. Struct. Civ. Eng., 2019, 13(2): 417-428.

链接本文:

https://academic.hep.com.cn/fsce/CN/10.1007/s11709-018-0477-3
https://academic.hep.com.cn/fsce/CN/Y2019/V13/I2/417

Fig.1

Fig.2

Fig.3

parameter	unit	description	value
$μ$	N/mm²	shear modulus	80.77 × 10³
$λ$	N/mm²	bulk modulus	121.15 × 10³
V	–	Poisson’s ratio	0.3
E	N/mm²	Young’s modulus	2.09 × 10⁵
g_c	N/mm	energy release rate	2.7
l_c	mm	length scale	0.015

Tab.1

Fig.4

Fig.5

Fig.6

Fig.7

parameter	unit	name	value
$μ$	N/mm²	shear modulus	8.0 × 10³
$λ$	N/mm²	bulk modulus	12.00 × 10³
V	–	Poisson’s ratio	0.3
E	N/mm²	Young’s modulus	2.08 × 10⁴
g_c	N/mm	energy release rate	0.5
l_c	mm	length scale	0.03

Tab.2

Fig.8

Fig.9

Fig.10

parameter	unit	name	value
$μ$	N/mm²	shear modulus	8.0 × 10³
$λ$	N/mm²	bulk modulus	12.00 × 10³
v	–	Poisson’s ratio	0.3
E	N/mm²	Young’s modulus	2.08 × 10⁴
g_c	N/mm	energy release rate	1
l_c	mm	length scale	0.01

Tab.3

Fig.11

Fig.12

Fig.13

Fig.14

parameter	unit	name	value
$μ$	N/mm²	shear modulus	1.94 × 10³
$λ$	N/mm²	bulk modulus	2.45 × 10³
v	–	Poisson’s ratio	0.22
E	N/mm²	Young’s modulus	2.09 × 10⁵
g_c	N/mm	energy release rate	2.28
l_c	mm	length scale	0.1

Tab.4

Fig.15

Fig.16

def SolveStaggeredOneStep(self):
	# phase?field dofs are frozen
	free_node_list= []
	for not node.IsFixed(PHASE FIELD):
			free_node_list .append(node)
			node . Fix (PHASE FIELD)
			# displacements are solved and updated
			self.model part.ProcessInfo[FRACTIONAL STEP] = 0
			converged= self.SolveOneStep()
			for node in self.model_part.Nodes:
		if node.IsFixed(DISPLACEMENT_X) and node.SolutionStepsDataHas(PRESCRIBED_DELTA _DISPLACEMENT
		ux= node . GetSolutionStepValue (DISPLACEMENT_X)
		dux= node . GetSolutionStepValue (PRESCRIBED_DELTA _DISPLACEMENT_X) node.SetSolutionStepValue(DISPLACEMENT_X, ux+ dux)
		if node.IsFixed(DISPLACEMENT_Y) and node.SolutionStepsDataHas(PRESCRIBED_DELTA _DISPLACEMENT
		uy= node . GetSolutionStepValue (DISPLACEMENT_Y)
		duy= node . GetSolutionStepValue (PRESCRIBED_DELTA DISPLACEMENT_Y) node.SetSolutionStepValue(DISPLACEMENT_Y, uy+ duy)
		if node.IsFixed(DISPLACEMENT Z) and node.SolutionStepsDataHas(PRESCRIBED_DELTA DISPLACEMENT
		uz= node . GetSolutionStepValue (DISPLACEMENT Z)
		duz= node . GetSolutionStepValue (PRESCRIBED_DELTA _DISPLACEMENT_Z) node.SetSolutionStepValue(DISPLACEMENT_Z, uz+ duz)
# phase?field dofs are unfrozen
for node in free_node_list :
node . Free (PHASE FIELD)
	# displacement dofs are frozen
	free_node_list_x= []
	free_node_list_y= []
	free_node_list_z= []
	for node in self.model part.Nodes:
		if not node.IsFixed(DISPLACEMENT_X):
		free_node_list_x. append (node)
		node. Fix (DISPLACEMENT_X)
		if not node.IsFixed(DISPLACEMENT_Y):
		free_node_list_y .append(node)
		node. Fix (DISPLACEMENT_Y)
		if not node.IsFixed (DISPLACEMENT_Z):
		free_node_list_z. append (node)
		node. Fix (DISPLACEMENT_Z)
	# phase field are solved and updated
	self.model_part.ProcessInfo[FRACTIONAL_STEP] = 1
	converged= self.SolveOneStep()
	print(”Solve for phase field completed”)
	# displacement dofs are unfrozen
	for node in free_node_list_x :
	node. Free (DISPLACEMENT_X)
	for node in free_node_list_y :
	node. Free (DISPLACEMENT_Y)
	for node in free_node_list_z :
	node. Free (DISPLACEMENT_Z)

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