Implementation aspects of a phase-field approach for brittle fracture

doi:10.1007/s11709-018-0477-3

Front. Struct. Civ. Eng.

2019, Vol. 13

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

RESEARCH ARTICLE

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.

Keywords phase-field modeling FEM staggered scheme fracture

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

Just Accepted Date: 10 April 2018 Online First Date: 28 May 2018 Issue Date: 12 March 2019

Cite this article:

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

URL:

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

Fig.1 Representation of the solid body with the boundary conditions

Fig.2 Problem setup and finite element mesh for the single edge notched tension test. (a) Geometry setup; (b) finite element mesh

Fig.3 Single edge notched tension test. Crack pattern at a displacement of (a) u =5.2 × 10^-³ mm, (b) u =5.5 × 10^-³ mm, (c) u =5.7 × 10^-³ mm for an undeformed configuration, and (d) u =5.7 × 10^-³ mm for a deformed configuration with a scale factor 12

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 Material parameters for single edge notched tension test

Fig.4 Single edge notched tension test. Load-deﬂection curves of the obtained result and reference one [²⁰]

Fig.5 Problem setup and finite element mesh for the single edge notched shear test. (a) Geometry setup; (b) finite element mesh

Fig.6 Load-deﬂection curve of the single edge notched pure shear test

Fig.7 Single edge notched shear test. Crack pattern at a displacement of (a) u =9.0 × 10^-³ mm, (b) u = 11.0 × 10^-³ mm, (c) u = 14.8 × 10^-³ mm for an undeformed configuration, and (d) u =5.7 × 10^-³ mm for a deformed configuration with a scale factor 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	0.5
l_c	mm	length scale	0.03

Tab.2 Material parameters for three-point bending test

Fig.8 Problem setup and finite element mesh for the three-point bending test. (a) Problem setup; (b) finite element mesh

Fig.9 Symmetric three-point bending test. Crack pattern at a displacement of (a) u = 3.8 × 10^-² mm, (b) u =4.4 × 10^-² mm, (c) u =6 × 10^-² mm for an undeformed configuration, and (d) u =6 × 10^-² mm for a deformed configuration with a scale factor 6

Fig.10 Load-deﬂection curve of three-point bending test

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 Material parameters for asymmetric three-point bending test

Fig.11 Problem setup and finite element mesh for the asymmetric three-point bending test. (a) Problem setup; (b) finite element mesh

Fig.12 Load-deﬂection curve of asymmetric three-point bending test

Fig.13 Asymmetric three-point bending test. Crack pattern at a displacement of (a) u = 0.19 mm, (b) u =0.21 mm, and (c) u =0.229 mm

Fig.14 Problem setup and finite element mesh for the notched plate test. (a) Problem setup; (b) finite element mesh; (c) experimental crack path for notched plate with hole [³⁵]

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 Material parameters for notched plate test

Fig.15 Notched plate with hole. Crack pattern at a displacement of (a) u =0.3 mm, (b) u =0.4 mm, (c) u =0.7 mm for an undeformed configuration, and (d) u =2.23 mm

Fig.16 Load-deﬂection curve of notched plate with hole

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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