Modeling and numerical simulation of crack ... - Wiley Online Library

GAMM-Mitt. 39, No. 1, 55 – 77 (2016) / DOI 10.1002/gamm.201610004

Modeling and numerical simulation of crack growth and damage with a phase field approach Kerstin Weinberg†,∗ , Tim Dally†, , Stefan Schuß†, , Marek Werner†, , and Carola Bilgen†, †

Lehrstuhl für Festkörpermechanik, Department Maschinenbau, Universität Siegen, Paul-Bonatz-Straße 9-11, 57068 Siegen, Germany

Received 14 April 2016 Published online 31 May 2016 Key words phase field, brittle fracture, NURBS, crack propagation, thermomigration, void growth, spallation Phase field methods allow for convenient and efficient moving interface simulations. In this paper phase field approaches of different order are presented, and applied to simulate damage in solids of temperature dependent and non-linear elastic materials. The numerical framework provides a NURBS based finite element method which minimizes the numerical and computational effort without impairing the smoothness required by the problem. In order to demonstrate the possibilities of such general phase field approaches a series of different models from material science and fracture mechanics is investigated. Specifically, a priori unknown crack propagation in different fracture modes is studied, simulations of thermomigration in a technical alloy and of void growth are presented and an inverse analysis of a dynamic fracture experiment is performed. The examples show the versatility of the presented low-order and high-order phase field approaches. c 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1

Introduction

One of the main challenges in computational mechanics is to predict damage, cracks and fragmentation patterns. Besides the high demands on the modeling side, the complicated structure and non-regular behavior of cracks turn numerical simulations of such problems into a difficult task. A promising tool to overcome such difficulties are phase field methods. The main idea behind it is to mark the material’s different states or phases by continuous order parameter field s(x, t), and to let them evolve in space and time. Since the physical properties within the phases are all in all given, the evolving structure is fully described by the position and motion of the phase interfaces. However, an order parameter -or phase field- is by definition a continuous field and thus, the moving boundaries are ’smeared’ over a small but finite length. Therefore, phase field models constitute so-called diffuse-interface formulations. Originally derived for diffusion problems, phase field models are meanwhile used for a variety of interface problems like decomposition, phase transformations or aging of a microstructure. The core of every model is a Landau free energy functional. For two phases it ∗ Corresponding author +49 271 740 2241

E-mail: [email protected], Phone:

+49 271 740 4641, Fax:

c 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

56

K. Weinberg et al.: Phase field approach to damage and crack growth

states the potential energy of a body B ⊂ R3 E= (Ψcon (s) + Ψsur (∇s)) dV,

(1)

B

where Ψcon (s) denotes a configurational energy density which controls the decomposition of the phases -it often has a double-well shape with minima at s = 0 and s = 1- and Ψsur (∇s) is the interfacial or surface free energy density, cf. Fig. 1. Additional fields may contribute to the phase evolution but are omitted here for brevity. Phase field models of fracture have gained attention only recently, see, e.g., [5, 12, 16, 19, 29]. The main idea is here to regularize the local discontinuities of brittle cracks and to approximate them by diffuse crack zones. The phase field indicates the state of the material which may be solid (s = 1) or, if cracked, empty (s = 0). The field s(x, t) is controlled by an additional differential equation which results in a coupled field problem but completely avoids the resolution of discontinuities. For purpose of illustration let us consider a deforming solid with domain B ⊂ R3 and boundary ∂B ≡ Γ ⊂ R2 . Crack growth corresponds to the creation of new boundaries Γ(t). Hence the total potential energy of a homogenous but cracking solid is composed of its bulk energy with free Helmholtz energy density Ψbulk and of surface energy contributions from growing crack boundaries. E= Ψbulk dV + Gc dS (2) B

Γ(t)

The fracture-energy density Gc quantifies the material’s resistance to cracking, for brittle fracture it corresponds to Griffith’s critical energy release rate. However, the energy functional (2) cannot be optimized in general and even an incremental approach [20] is challenging because of the moving boundaries Γ(t). Highly sophisticated discretization techniques have been developed to solve such problems, e.g. cohesive zone models [21, 24, 32], the extended finite element method [18, 27], eroded finite elements or recently developed eigenfracture strategies [23, 26]. In a phase field approach to fracture the set of evolving crack boundaries is instead replaced by a surface-density functional γ(t) = γ(s(x, t)) and an approximation of the form dS ≈ γ(t) dV , (3) Γ(t)

B

which allows to re-write the total potential energy of a cracking solid and to formulate the optimization problem locally. bulk (4) Ψ + Gc γ dV → optimum E= B

In potential (4) the material’s energy is again composed of two terms, a bulk energy density Ψbulk and a surface energy contribution Gc γ. By definition γ is only different from zero along cracks. Optimization of the potentials (1) or (4) leads to evolution equations for the phase field s(x, t). For a simple ordering type of phase field, the variation of energy leads the wanted

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driving force, s˙ = −c δs E. Herein and below c, c1 , c2 , . . . , c4 ∈ R+ denote unspecified constants. The corresponding Euler-Lagrange equation is typically named Allen-Cahn equation and has the form of a simple reaction-diffusion equation. s˙ = c1 φ(s) + c2 s

(5)

In this formulation φ(s) denotes the reaction term, e.g. φ(s) = 2s3 − s. If the phase field variable is a conserved quantity like mass concentration or volume fraction, its evolution has additionally to account for the continuity equation which leads to an evolution equation of Cahn-Hilliard type. s˙ = c3 φ(s) − c4 s

(6)

Obviously, there is no general phase field evolution equation but instead the specific formulation has to map the physics of the underlying problem. In this paper we apply different phase field models to simulate damage in solids of temperature dependent and non-linear elastic materials. The necessary fundamental equations are stated in Section 2 and the NURBS based finite element framework used for numerical simulation is shortly outlined in Section 3. The use of NURBS minimizes the numerical and computational effort without impairing the smoothness required by the problem. Example problems in Section 4 demonstrate the versatility of the phase field approach for crack growth and damage simulations.

2

Governing equations of phase-field models

2

Ψ

1.5

1

0.5

0

0

0.5

1

s Fig. 1 (online colour at: www.gamm-mitteilungen.org) Energy density as a function of phase field s: classical double well functional of the configurational energy density with two minima and simplified quadratic energy functional employed for phase field fracture.

We start by defining the basic fields. A material point of a solid in its reference configuration B0 is labeled by X = (X1 , X2 , X3 )T and deforms during a time interval [0, t¯] with a mapping χ(X, t) : B0 × [0, t¯] → Rn .


(7)


58


We denote x = χ(X, t) with the gradient of the deformation F : B0 × [0, t¯] → Rn×n . F = ∇X χ =

∂χ ∂X

(8)

The temperature field T (X, t) : B0 × [0, t¯] → R

(9)

effects the free energy potential of the solid. Additionally, the material state is characterized by a phase field function s(X, t) : B0 × [0, t¯] → R.

(10)

All fields are assumed to be sufficiently smooth. For the phase field we define s ∈ [0, 1]. 2.1 Continuum mechanics The elastic boundary value problem follows from the balance of linear momentum, ¯ = ρ0 v˙ ∇X · P + B

in B0 × [0, t¯],

(11)

¯ T¯ the where P is the first Piola-Kirchhoff stress tensor; ρ0 denotes the mass density, B, prescribed body force and traction, and v the material velocity. The boundary of the solid is subdivided into displacement and traction boundaries ∂B0u , ∂B0σ with ∂B0 = ∂B0u ∪ ∂B0σ , ∂B0u ∩ ∂B0σ = ∅ and ¯ on ∂B0u × [0, t¯] x=x

and

P N = T¯

on ∂B0σ .

(12)

The initial conditions are x(X, 0) = x0

and

v(X, 0) = v 0

in B0 .

(13)

Let Ψbulk (F , T, s, . . . ) be the local energy density of the bulk material. There may be additional dependencies of Ψbulk , e.g. on other phase fields or on internal variables, but we will restrict ourselves here to an isotropic non-linear elastic material with one or two phases. This material will develop damage and/or cracks. From physics we know that fracture requires a local state of tension whereas the compressive part of the deformation does not contribute to crack growth. Therefore, we decompose the deformation gradient (8) into elastic and inelastic components, F = F e F i . Moreover it is expressed in principal stretches λa , a = 1, 2, 3, − + − which will be decomposed into tensile λ+ a and compressive components λa via λa = λa λa and with λ± a =

1 [(λa − 1) ± |λa − 1|] + 1 . 2

(14)

Then, the fracture insensitive part of the deformation gradient F i reads F = i

3

(1−s) − (λ+ λa na ⊗ N a , a)

(15)

a=1



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where na and N a denote the principal directions of the left and right stretch tensors, respec(1−s) − tively. In the following we abbreviate λia = (λ+ λa and write, in a slight misuse of noa) tation, the Helmholtz free energy density equivalently as Ψ := Ψ(F , T, s) ≡ Ψ(λa , T, s) ≡ Ψ(λia , T ). The first Piola-Kirchhoff stress tensor then follows as 3

P =

3

∂Ψ ∂Ψ ∂λia = na ⊗ N a = Pa na ⊗ N a , i ∂F ∂λa ∂λa a=1 a=1

(16)

whereas the second Piola-Kirchhoff stress tensor S = F −1 P reads 3

S=2

1 ∂Ψ = Pa N a ⊗ N a ∂C λ a=1 a

(17)

and the principal Cauchy stresses are calculated with J = det F as σa = J −1 F SF T . 2.2 Heat conduction and thermal diffusion The heat equation reads in its classical form cp ρT˙ = ∇ · (k∇T ) + q

in B0 × [0, t¯],

(18)

where q denotes the heat generation rate per unit volume. The corresponding boundary conditions are T = T¯min on ∂B0l ,

T = T¯max on ∂B0h

and

∇T · n = 0 on ∂B0n ,

(19)

where ∂B0l and ∂B0h denote the low and high temperature boundary of the body B0 and it holds ∂B0 = ∂B0l ∪ ∂B0h ∪ ∂B0n and ∂B0l ∩ ∂B0h ∩ ∂B0n = ∅. We assume the material to be composed of two components A and B and choose the mole fraction as an order parameter. Then, the specific heat capacity cp , the mass density ρ and the thermal conductivity k are A A B B B set as a convex combination of the parameter cA p , ρ , k and cp , ρ , k of the individual components. B B cp := sA cA p + s cp ,

ρ := sA ρA + sB ρB ,

k := sA k A + sB k B

(20)

Clearly, the mole fraction is a conservative order parameter with sB = 1 − sA . We define s := sB and for the evolution it holds the continuity equation, s˙ = −∇ · j s . The driving force for the flux j s is the gradient of the chemical potential μ which follows from the variation of energy potential (1), μ = δs E = δs (Ψcon + Ψsur ) = ∂s Ψcon (s, T ) − λΔs . Here the interfacial free energy density is assumed to be of the form Ψsur = The material parameter λ = γ¯lc


(21) 1 2 2 λ(T ) ∇s .

(22)


60


is determined by surface energy density γ¯ . Length lc is the ’interface width’, i.e. the width of the transition region between the two phases. Now, with a chemical mobility Mc and j s = −Mc ∇μ follows the phase field evolution equation s˙ = ∇ · (Mc ∇μ)

in B0 × [0, ¯t],

(23)

which is clearly of Cahn-Hilliard type (6), with source term Mc ∂s Ψcon and c4 = Mc λ for constant mobility. If the phase evolution is additionally driven by a temperature field, the evolution equation (23) needs to be extended. A detailed, thermodynamically consistent derivation of such a thermal diffusion model is given in [3]. The resulting equation reads s˙ = ∇ · (Mc ∇μ − Mq ∇T )

in B0 × [0, t¯]

(24)

with the initial field s(x, 0) = s0 (x), ∀x ∈ B0 and boundary conditions ∇s · n = 0,

(Mc ∇μ − Mq ∇T ) · n = 0 on ∂B0 .

(25)

Here n denotes the outward unit normal on ∂B0 , and Mq is the mobility of thermotransport which is related to the thermodiffusion coefficient DT as Mq = −s(1 − s)DT . 2.3 Phase field fracture The evolution equations for both, conservative and non-conservative phase fields may be restated in a general form s˙ = −M Y (x, s),

(26)

where M denotes a kinematic mobility [1/sec] and Y (x, s) summarizes all (dimensionless) driving forces which typically represent a competition of bulk and surface forces. In phase field fracture such driving force results from a release of stored elastic energy of the body into the formation of free surfaces. Typically it is derived from an energy potential and we reformulate potential (4) as Gc ¯ e ¯ e = lc Ψ . E= (Ψ + lc γ ) dV with Ψ (27) Gc B lc ¯ Ψ

¯ which summarizes elastic and fracture For normalization we introduce here a potential Ψ ¯ =Ψ ¯ e + lc γ, and a characteristic length lc which corresponds to half energy contributions, Ψ of the diffuse ’crack width’, i.e. the transition zone between intact and broken material. The surface-density functional γ may be understood as a wavenumber of the moving disturbance; it characterizes the shape of the diffuse zone. By definition, function γ has a small support and is symmetric to the ’real’ crack path. In the following it is defined as a function of the phase field parameter s solely. γ=

1 f (s) lc

(28)

Then, a general ansatz of the form f (s) = c0 |1 − s|2 + c1 lc2 |∇s|2 + c2 lc4 |s|2 + c3 lc6 |∇3 s|2 + . . .


(29)


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phase field s

1

2nd order 4th order

0

−2lc

0

2lc

x

Fig. 2 (online colour at: www.gamm-mitteilungen.org) Uniaxial model with a crack at x = 0 and with a continuous phase field s ∈ [0, 1]; phase field approximation for a second order and a fourth order approximation of γ.

can be made. Inserting it into (3) and minimizing the corresponding potential (27) analytically, leads for the simplest uniaxial case to an exponential solution of the form s = 1 − exp(−|x|/lc ), see Fig. 2. Now we determine the constants c0 , c1 , c2 , c3 , . . . in such a way, that this disturbance is approximated properly. In consequence we obtain for the surfacedensity function a second order approximation of the form γ=

1 (1 − s)2 + lc2 (∇s)2 . 2lc

(30)

The consideration of higher order terms gives an overall continuous analytical solution. Approximating the corresponding disturbance s = 1 − exp(−|x|/lc ) · (1 + |x|/lc ), the fourth order crack-density functional reads γ=

1 (1 − s)2 + 2lc2 (∇s)2 + lc4 (s)2 . 4lc

(31)

Note that an approximation (29) with the first term only describes a sharp transition and would result in the typical difficulties of moving discontinuities. The gradient term (∇s)2 regularizes the crack zone and renders the method non-local. The additional Laplacian in (31) affects the curvature of the diffuse interface approximation and smoothes the transition. We would like to emphasize that gradient terms are known from continuum damage mechanics. However, in opposite to a damage variable here the material’s state is well defined only for phase field parameter s = 1 (intact) and s = 0 (broken). The transition zone is a consequence of the regularized model and an intermediate value 0 < s < 1 state has no physical meaning. 2.3.1 Variational fracture criterium In a variational approach the driving force of equation (26) is derived from the potential energy of the cracked body (4) or its normalized energy density in (27), i.e., e ¯ = δs Ψ ¯ + l c γ = Y e + l c δs γ (32) Y = δs Ψ where Y e summarizes the normalized crack driving force and lc δs γ represents a kinematic fracture resistance. It evolves for the second order crack-density approximation (30) and for



62


the fourth order crack-density approximation (31) to lc δs γ = −(1 − s) − lc2 s

1 1 and lc δs γ = − (1 − s) − lc2 s − lc4 s , (33) 2 2

¯ e (F , T, s) on the phase field s can respectively. The dependence of the elastic energy term Ψ ¯e = 1 : be modeled in different ways. In the simplest case, a linear-elastic bulk energy, i.e. Ψ 0 2 ¯ with normalized elasticity tensor C ¯ = lc /Gc C(T ), is multiplied with a weight function, C ¯ e = g(s)Ψ ¯e . Ψ 0

(34)

This function g(s) is such that in regions where the material is broken (s = 0), the contribution to the elastic energy is zero, while in the intact regions the elastic energy contribution recovers the one prescribed by the material’s energy density. Preferably it is stationary at both limits. This leads to a weight function of the general form g(s) = (a + 1)sa − asa+1

a ∈ N.

(35)

Often a = 3 is set, i.e. g(s) = s3 (4 − 3s), and together with a double well term in the dimensionless free energy (see red curve in Fig. 1), this approach has been used in the early works of phase field fracturing, e.g. by Karma et al. [16] and Henry et al. [8,12]. Its major drawback is the third order term in the variation (32) which impedes finite element approximations. In order to simplify the numerical solution, a quadratic weight can be employed which, however, cannot result in stationarity at both limits, s = 0 and s = 1. Typically, instead of (35) with a = 1 the function g(s) = s2 is chosen, see blue curve in Fig. 1. The corresponding variational functional of brittle fracture was first introduced by Francfort, Marigo [11] and Bourdin [6] and with slight modifications this ansatz has become very popular since, see, e.g., Kuhn & Müller [17], Miehe et al. [19] and Abdollahi & Arias [1], Borden et al. [5]. Here the local energy function is of the form ¯ Y = δs Ψ

¯ = s2 Ψ ¯ e0 + 1 (1 − s)2 + lc |∇s|2 , with Ψ 2lc 2

(36)

which corresponds to (32) with a second order crack-density approximation (30). Please note that for a finite deformation approach to fracture the elastic energy density always needs to be consistently coupled via Ψe (λia ) as outlined above, [14]. 2.3.2 General damage and fracture criteria A derivation of the driving force from an energy potential, however, is not necessary for a phase field approach to fracture. Likewise the driving force can directly be modeled by a classical failure criterion. It is a common hypothesis, that in brittle materials fracture occurs when the normal stress exceeds the ultimate tensile strength σc of the material. Thus, we may define a Rankine-stress threshold of the form σmax > σc , where σmax is the maximum principal Cauchy stress, σmax = maxa σa (x, s), with a = I, II, III. This leads to a driving force

σmax Y = −1 + lc δs γ(s) (37) σc +



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where, by taking the positiv part of · only, we additionally account for irreversibility of cracking, i.e., we do not allow the crack to close. In a more general way we may define a Mohr-Coulomb failure criterion that describes a maximum principal stress criterion with an isotropic critical failure stress. Since such a definition allows an easy distinction between tensile and compression regions, the decomposition of the deformation gradient (15) is not necessary here, cf. [4]. 2.3.3 Phase field fracture problem Finally we summarize the phase field evolution equation as s˙ = −M (Y e + lc δγ)

in B0 × [0, ¯t]

(38)

with boundary conditions ∇s · n = 0

on ∂B0

(39)

for the second order approach (30). For the fourth order ansatz (31) the boundary conditions are ∇s · n = 0,

s = 0

and

∇(lc3 s − 2lc s) = 0

on ∂B0 .

(40)

The initial conditions are given with s(X, 0) = 1 on B0 , describing an unbroken state.

3

Numerical approximation

The set of equations (11), (18), (24) and (38) forms -with corresponding boundary and initial conditions- the coupled phase field boundary value problem. For finite element approximation we now deduce the weak forms by taking the variations with respect to x, T and s. For the mechanical problem (11-12) we obtain ¯ · δx dV + T¯ · δx dA, ∀δx ∈ V x . (41) ρ0 v˙ · δx dV + S : F T δF dV = B B0

B0

B0

∂B0σ

The functional space of admissible variations is V x = {δx ∈ H1 (B0 )|δx = 0 on ∂B0u }, where H1 denotes the Sobolev functional space of square integrable functions with square integrable weak first derivatives. The terms on the right hand side of eq. (41) represent the external contributions of body force and traction. The weak forms of the thermal diffusion problem (18-25) follows as ρcp T˙ δT dV + k∇T · ∇δT dV − qδT dV = 0, ∀δT ∈ V T , (42)

B0

B0

(Mc ∇∂s Ψcon − Mq ∇T + λΔs∇Mc ) · ∇δs dV +

sδs ˙ dV + B0

B0

B0

(43)

λMc ΔsΔδs dV = 0,

∀δs ∈ V0s .

B0



64


The functional space of temperature variations is V T = {δT ∈ H1 (B0 )|δT = 0 on ∂B0l and δT = 0 on ∂B0h }. Clearly, the variation of the phase field s has to fulfill higher continuity requirements here, V0s = {δs ∈ H2 (B0 )|∇δs · n = 0 on ∂B0 }. The Sobolev space H2 comprises functions with square integrable weak first and second derivatives. ¯ e = lc Ψe /Gc and Finally we restate the fracture evolution (38-40) in its weak form. With Ψ variation of (30) we get ¯ − (1 − s) − lc2 ∇(δs) · ∇(s) dV = 0 δs ∂s Ψ ∀δs ∈ V1s , (44) B0

where the functional space of admissible test functions of the second-order phase field is V1s = {δs ∈ H1 (B0 )|δs = 0 on Γ(t)}. For a fourth order approximation (31) the weak form reads

1 l4 ¯ ∀δs ∈ V1s , (45) δs ∂s Ψ − (1 − s) −lc2 ∇(δs)·∇(s)− c (δs)(s) dV = 0 2 2 B0

where the functional space is V2s = {δs ∈ H2 (B0 )|δs = 0, ∇δs · n = 0 on Γ(t)}.

0

1

2

3

⊗

0

2

1

3

Fig. 3 (online colour at: www.gamm-mitteilungen.org) Two-dimensional B-spline of order p = 2.

Clearly the spaces V0s and V2s require at least C 1 -continuous approximation functions, whereas for V x , V T and V1s the C 0 -continuity of classical finite element basis functions is sufficient. In order to meet these continuity requirements within one finite element framework we employ NURBS (Non-uniform rational B-Splines) as finite element basis, Fig. 3. A multivariate B-spline basis of degree p = [p1 , . . . , pd ] and dimension d ∈ N is defined by the tensor product Θ1 ⊗ ... ⊗ Θd of knot vectors, built by a sequence of knots Θl = [ξ1l ≤ ξ2l ≤ . . . ≤ ξnl l +pl +1 ], l ∈ {1, . . . , d}. In the absence of repeated knots, the partition [ξi11 , ξi11 +1 ] × . . . × [ξidd , ξidd +1 ] forms an element of the mesh in the parametric domain. A single multivariate B-spline B A , A ∈ [1, . . . , n], n := n1 ...nd , is then defined by i (ξ) = B i (ξ 1 , ..., ξ d ) = B A = Bp p

d

Nil ,pl (ξ l ),

(46)

l=1

with multi-index i = [i1 , . . . , id ] and supp(B A ) = [ξi11 , ξi11 +p1 +1 ] × . . . × [ξidd , ξidd +pd +1 ]. It provides the necessary support for the required continuity. The recursive definition of a univariate B-spline is given as follows Nil ,pl (ξ) =

ξill +pl +1 − ξ ξ − ξill N (ξ) + Ni +1,pl −1 (ξ), i ,p −1 ξill +pl − ξill l l ξill +pl +1 − ξill +1 l


(47)


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starting with piecewise constant functions 1 if ξill ≤ ξ < ξill +1 . Nil ,0 (ξ) = 0 otherwise

(48)

The extension to NURBS with weights wi is given by i (ξ) = RA = Rp

d

Nil ,pl (ξ l )wi

d

l=1

ˆ l=1

i

Nîl ,pl (ξ l )wˆ . i

(49)

Linear independence as a fundamental property of finite element basis as well as local support are provided by a NURBS basis. Smoothness is related to knot multiplicity, i.e., the number of repetitions in Θ at node i. Unfortunately, the tensor product structure in (46) impedes standard local refinement strategies which motivated us to introduce a specific hierarchical refinement strategy in [13]. For the time integration we divide the considered time interval [0, ¯t] into nt pairwise disjoint equidistant subintervals In = [tn , tn+1 ] with time step Δt := tn+1 − tn and employ an implicit Crank-Nicolson scheme, known to be second-order accurate.

4

Numerical simulations

4.1 Crack growth in different fracture modes In order to illustrate the versatility of the phase field approach for the computation of crack growth and damage we begin with sample simulations of phase field fracture. Here, the threedimensional blocks are loaded in tension and shear. The size of the blocks is 1 × 0.2 × 1 mm; the finite element mesh consists of 20 × 4 × 20 hexahedral elements with p = 2 before further refinement. The material is assumed to be uniform, non-linear elastic and temperature independent. Its energy density (2) is of Neo-Hookean type, Ψ=

μ λ (ln(J))2 − μ ln(J) + (F : F − 3) 2 2

(50)

with Lamé coefficients typical for steel, μ = 80769 N/mm2 and λ = 121154 N/mm2 . The critical energy release rate is chosen to be Gc = 2.7 N/mm. At first we consider a symmetric tension test with the specific boundary conditions shown in Fig. 4. The load is applied as prescribed displacement at the upper boundary using constant displacement increments of u = 5 · 10−5 mm. The mesh is a priori refined locally around the expected crack path and consists now of 46288 elements which corresponds to a total of 305856 degrees of freedom for both, the mechanical and the phase field. In the refined domain the elements have a size of hmin = 0.0063 mm which allows to choose a small critical length parameter, lc = 0.01575 mm. The growing crack is computed with a fourth-order phase field model (31). In the diagram of Fig. 4 the load-displacement curves are shown, now for plane-strain simulations of a second-order and a fourth-order phase field model, cf. [30]. All parameters, i.e., Lamé coefficients, Griffith’s energy release rate, critical length lc , applied displacement steps as well as the mesh using quadratic NURBS shape functions are exactly identical for both calculations. Obviously, a higher continuity in the phase field leads in the



66


8

4th order PF 2nd order PF

load [kN]

6

4

2

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

displacement ×10−2 [mm]

0.8

Fig. 4 (online colour at: www.gamm-mitteilungen.org) Boundary conditions and computed phase field for mode I fracture and load-deflection curves for the second-order and the fourth-order phase field model.

critical range to a more compliant response. This has been confirmed by further tests. The effect, however, is not necessarily contributed to the order of the phase field only. Additionally, the choice of critical length parameter lc and the compliance of the discretization (quadratic NURBS vs. linear finite elements) play a role. At next we simulate an in-plane shear test with boundary conditions shown in Fig. 5. Analogous to the tension test the load is applied in constant displacement increments, here with u = 1 · 10−3 mm. The refined mesh is composed of 21396 elements that leads to 112192 degrees of freedom in this case. We get an element size of hmin = 0.0125 mm and choose the length-scale parameter as lc = 0.03125 mm. The snapshot of the phase field is presented in Fig. 5 (left). Very similar, with a mesh of 21480 elements, a total of 113632 degrees of freedom, and same element size and length-scale parameter lc we performed the out-of-plane shear test

Fig. 5 (online colour at: www.gamm-mitteilungen.org) Boundary conditions and computed phase field for mode II and mode III fracture



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Fig. 6 (online colour at: www.gamm-mitteilungen.org) Stress distributions in the current configuration for the three fracture modes I,II and III

simulation. The displacement increments of u = 1 · 10−4 mm are imposed in both outof-plane directions. The boundary conditions and simulated phase field pattern of mode III are shown in Fig. 5 (right). The corresponding von Mises-stress distributions in the current configuration are displayed in Fig. 6. The phase field and stress distribution for the secondorder phase field ansatz (30) look basically the same like for the displayed fourth-order ansatz (31). However, our results indicate a significant faster convergence and a better robustness of the numerical simulation by making use of the fourth-order ansatz.

4.2 Material degradation by thermomigration in a Sn-Pb solder bump Solder joints, made of fusible metal alloys with a low melting point, are used to connect metallic surfaces. They cover a broad range of technical applications. In particular in electronics soldering plays an important role, where solder balls provide mechanical as well as electrical connections between different components and, consequently, the reliability of the joints determines the life expectation of the whole electronic device. Here we study eutectic Sn-Pb solder balls subjected to an inhomogeneous temperature field, see Fig. 7. When such initially homogeneous binary alloy is annealed, its microstructure becomes inhomogeneous, e.g. [7,15]. Under a temperature gradient heat conduction, spinodal decomposition and mass diffusion couple and cause an irreversible thermomigration. In the corresponding system of equations (18-25) we set the mole fraction of tin, as our phase field variable s := cSn = 1 − cPb but omit the subscript Sn. In order to obtain a temperature and mole fraction dependent free energy function, we constructed energy functions for a discrete set of temperatures in a first step. Then we used the least square method and the pure metal’s free enthalpy coefficients g˜ik to determine function gk (T ) = Ak + Bk T + Ck T ln(T ) + Dk T 2 + Ek T 3 + g5 (T ) = A5 + B5 T


Fk , T

k = 1, ..., 4,

3

in J/m .


68


Fig. 7 (online colour at: www.gamm-mitteilungen.org) Solder bump of Sn-Pb with an eutectic mole fraction of 0.63 ; distribution of the initial mole fraction field (left), and initial temperature profile (right).

As a result the system’s configurational free energy density can be described by a smooth function in temperature and mole fraction, Ψcon (c, T ) = g1 c+g2 (1−c)+g3 c ln(c)+g4 (1−c) ln(1−c)+g5 c(1−c),

in J/m3 . (51)

The fitted values are listed in table 1, see [2] for details of the method. Table 1 Fit values for the coefficients gi , i = 1, ..., 5, of the free energy function Ψcon . The discrete set of temperatures is Tk = 250 + hk ∈ (250 K, 505 K), h := 85/8334, k = 1, ...25000.

i=1 i=2 i=3 i=4 i=5

MJ Ai m 3 −355.77 −405.42 275.05 −85.964 1224.9

Bi mMJ 3K 3.9628 5.5101 1.338 1.3153 −0.49161

Ci mMJ 3K −0.96695 −1.3286 −0.18453 −0.13691 -

Di m3JK2 −1146 −191.5847 389.8139 244.7001 -

Ei m3JK3 0.1898 −0.0153 −0.1331 −0.0888 -

Fi GJK m3 −3.7692 0.027891 1.9559 1.274 -

For realistic values of the chemical mobility Mc we adapted a work of Ubachs et al. [28] and determined the mobility for a Sn-Pb alloy by comparisons between a two-dimensional simulation and an ageing experiment. Since they derived the mobility only at a homogeneous temperature of 423.12 K and, to the best knowledge of the authors, comparable results at different temperatures are not available in the literature, we follow Dreyer & Müller [10] in order to relate the mobility to the diffusion coefficients. To this end they considered diffusion of the Fickean type and stated the relations ∗ (T ) = MSn/Pb


−1 ∂ 2 Ψcon cβ/α (T ), T γ DSn/Pb (T ) in m5 /(Js), ∂c2

(52)


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∗ ∗ where cα/β are the equilibrium mole fractions and MSn /DSn , MPb /DPb are the mobility/diffusivity of lead in tin and tin in lead, respectively, given by an Arrhenius relationship

DSn (T ) = 4.1 · 10

−5

94400 exp − , RT

DPb (T ) = 3.533 · 10

−6

61370 exp − in m2 /s, RT (53)

with the universal gas constant R = 8.31451 Jmol−1K−1 . The parameter γ will be used later ∗ on to fit MSn/Pb such that the magnitude of MC at T = 423.15 K corresponds to the mobility employed in [28]; for now we set γ = 1. In order to model the equilibrium mole fractions as functions in temperature, we now apply the common tangent construction to the configurational free energy density Ψcon (c, Tk ) at temperatures Tk = 250 + hk ∈ (250 K, 505 K), h := 85/8334, k = 1, ...25000 and obtain a corresponding set cα/β (Tk ) of equilibrium mole fractions and subsequently polynomial functions cα (T ) = 1.1125 · 10−9 T 3 − 5.5931 · 10−7 T 2 + 2.4223 · 10−4 T − 1.2076 · 10−2 , (54) cβ (T ) = −1.8897 · 10−9 T 3 + 1.4722 · 10−6 T 2 − 3.867 · 10−4 T + 1.0342

(55)

by minimising a least square problem. The chemical mobility Mc of equation (24) is given by Mc = Vm s(1 − s) [sMA + (1 − s)MB ]

in m5 /(Js),

(56)

whereas the heat of transport term Mq , the mobility of thermotransport, is defined as Mq = s(1 − s) [MA Q∗A − MB Q∗B ] T −1

in m2 /(sK).

(57)

In this context, Vm is the molar volume, Mi the atomic mobility and Q∗i the molar heat ∗ of transport of element i ∈ {A, B}. Comparison of the units yields the relation MSn/Pb = ∗ −1 Vm MSn/Pb ⇔ MSn/Pb = MSn/Pb Vm and consequently the temperature and mole fraction dependent mobilities for the Sn-Pb alloy. ∗ ∗ (T ) + (1 − c)MSn (T )] Mc (c, T ) = c(1 − c) [cMPb

Mq (c, T ) = c(1 −

∗ c) [MPb (T )Q∗Pb

−

in m5 /(Js),

∗ MSn (T )Q∗Sn ] T −1 Vm−1

(58) 2

in m /(sK)

(59)

By setting γ = 0.0156 we finally obtain mobilities which fit at temperature T = 423.12 K the mobility reported in [28]. In general also the heat of transport terms Q∗Sn/Pb are functions in temperature. However, in lack of experimental data we use here for Pb 25.3 kJ/mol and for Sn -1.36 kJ/mol, according to [22] and [7]. For the surface energy density parameter λ we follow again an idea presented by Dreyer & Müller in [10] to calculate λ from the lattice structure of the equilibrium phases. According to [10], λ is given in [J/m] by λ(T ) =

1125 2

1010 (cα (T ) − cβ (T ))

.

(60)

The remaining material parameter for a Sn-Pb alloy are listed in table 2.



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Table 2 Miscellaneous material data (in SI-units) used in the simulations.

λ 1.364 · 10−7

Vm 17.275 · 10−6

ρSn 7.287 · 103

ρPb 11.342 · 103

cSn p 228

cPb p 131

k Sn 67

k Pb 35

Fig. 8 (online colour at: www.gamm-mitteilungen.org) Microstructural evolution of the Sn-Pb solder bump subjected to a temperature gradient.



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For numerical simulation a Sn-Pb solder bump with a diameter of 0.5 μm and a height of 0.2692 μm is subjected to a temperature gradient with Tmin = 423.1212 K and Tmax = 423.1788 K. The dimensions and the temperature difference between the lower and the upper surface of the bump are scaled such that they correspond to the data from the experiments performed by Hsiao & Chen [15]. Fig. 8. Referring to physical inhomogeneities in alloys the initial setting for the simulation is arranged in such a manner that the alloy has a constant mass fraction of c0 = 0.63 with randomly generated perturbations as shown in Fig. 7. Here and in Fig. 8 the reddish areas denote the tin-rich phase whereas the blue domains represent the lead-rich phase. Similar to the experimental observations, the typical spinodal decomposition of phases occurring in the isothermal case, [2], does not take place here. Instead, a migration of the lead rich phase to the cold end of the bump and one of the tin rich phase to the hot end of the bump can be observed. The solder degradation is illustrated for different aging times in Fig. 8. 4.3 Damage by vacancy condensation and void growth The process of solder joint aging is further complicated by the fact that complex thermodynamic and kinetic phenomena induce the formation of microvoids and facilitate the formation of pores and cracks. The physics behind this mechanism may be sketched as follows: in the solder we observe phase decomposition and diffusion induced aging. Neighboring phases change in a way that the volume of one region grows and the volume of the another phase reduces. In case of Sn-based solders such regions are connections with copper plates e.g., at the microelectronic chip. Specifically, the diffusion of copper from connecting pads into the solder’s boundary layer components is much slower than the diffusion of copper within the solder, it also cannot be “corrected” by the very slow inverse diffusion of tin through the copper-tin interface. Because of this unbalanced Cu-Sn diffusion vacancies are left which condense to form Kirkendall voids, cf. [31]. Additional vacancies and defects in the crystal lattices are generated by plastic deformation of the solder material and assist in the process of void growth and material degradation. In this example we simulate vacancy growth by means of a combined phase field model. Please note that we do not aim at a detailed model of phases in solder but instead we choose a continuum body to model void formation here. The material is assumed to be homogeneous and its deformation is neglected. Thus, we assume an initial vacancy volume fraction c0 which will control the void growth. The volume fraction or concentration c(x, t) serves as a conservative phase field here. Its driving force is a Landau energy potential (1) which accounts for the configurational energy by a simple double well potential Ψcon (c) = 12 (c2 − 1)2 . However, another phase field parameter s marks the state of the material and ensures -similar to phase field fracture- that the emerging voids do not carry any bulk energy. This two-phase field approach is similar but not identical to the one in [25]. Here the s-term works like a weight function -see eq. (34)- for the configurational energy. Thus, the combined Landau free-energy potential has the form

2 1 1 1 2 2 2 2 E= (61) c −1 − (1 − s) − + αc |∇c| − αs |∇s| dV 2 2 B 2 where the surface energy coefficients αc and αs work as penalty terms for emerging voids. The shape of the Landau free-energy potential is shown in Fig. 9 and we clearly see here



72


minima for c = 0, s = 1 (no vacancies, no void damage) and c = 1, s = 0 (empty void full of vacancies). The states at c = 1, s = 1 and c = 0, s = 0 which describe physically unlikely states have a high energy potential. Ψcon

α = const. α = α(c)

0

s 1

1

0

c

Δt

Fig. 9 (online colour at: www.gamm-mitteilungen.org) Void growth in a homogeneous continuum: Landau free-energy potential for a combined phase field model (left), two-dimensional simulations of voids growing by vacancy concentration within 200 seconds (middle), and for different surface coefficients αc in potential (61) void volume vs. time (right).

First results of void growth are also shown in Fig. 9. Because we do not consider vacancy generation at this point, the voids grow slowly. In this computation we use a mesh of 128×128 NURBS based finite elements over a domain of 128 nm2 and an initial concentration of 10−6 . In our computation we assume the penalty parameter αc to be constant. This, however, is not realistic because it favors unlimited void growth - until all vacancies are absorbed. In practise we typically have an equilibrium concentration of vacancies in every material. Therefore, we express the coefficient αc as a positive function of concentration c with an ansatz α(c) = A exp(k(c)) .

(62)

The exponential form allows a simple derivation ∂c α = α ∂c k. For the equilibrium state the chemical potential assumed to be constant. Then, using a Legendre transformation of the (squared) concentration gradient (∂c/∂x)2 = c2 , and the consequence, that the square of the concentration gradient divided by its second derivative is proportional to the concentration, we obtain an equation for the exponent in (62).

∂c Ψcon k(c) = ln (63) A(2c − c2 ∂c k) After a lengthy calculation the simplified solution leads to an approximation αc = 1 − c which, in turn, renders the variation of the surface energy term in (61) non-linear. The effect of a concentration dependent αc is visualized in the diagram of Fig. 9. We observe a saturation of growth at a specific vacancy concentration here. 4.4 Dynamic fracture by wave reflection In this example we consider wave propagation and reflection in a spallation experiment. Such tests are performed in our lab with a Hopkinson-Bar setup where a cylindrical specimen is positioned next to a steel bar (incident bar). The strike of a projectile generates a compressive stress pulse traveling through the incident bar into the adjacent specimen. At the free end of the specimen the stress pulse is reflected with a phase reversal. If the tensile stress of the superposition of both pulses is beyond the dynamic tensile strength σc of the investigated www.gamm-mitteilungen.org


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Im

Fig. 10 (online colour at: www.gamm-mitteilungen.org) Hopkinson-Bar spallation experiment of UHPC, the incident bar is left of the specimen. Only the first crack in the middle of the specimen is considered in the simulations, the right crack happens later.

Fig. 11 (online colour at: www.gamm-mitteilungen.org) Influence of critical energy-release rate Gc on the crack evolution: phase field parameter s(x, t = 120μs) for Gc = 40 N/m (top), the experimentally determined value of G = 90 N/m (middle), and G = 360 N/m (bottom). All specimen are loaded with a stress pulse of σmax = 17.3 MPa.

material, fracture occurs. We used this experimental setup in order to determine the fracture properties of an Ultra-High Performance Concrete (UHPC), see Fig. 10. For finite element analysis we work with an axisymmetric model of the spallation experiment. The mechanical problem is described with the balance of momentum (11) in domain B0 = [0, rs ] × [0, l], where rs = 10 mm and l = 200 mm denote radius and length of the UHPC specimen. Stress boundary conditions are applied on B0σ = {(r, z) ∈ B0 : z = l}, i.e. at the left side of the cylindrical specimen. The incoming stress pulse has a rectangular shape with maximum σmax . Its length and value are adapted to the experimentally measured data. For the specimen we use the material parameters of UHPC, E = 59 GPa, ν = 0.2, ρ = 2370 kg/m3 . The specimen has the typical properties of a brittle ceramic and thus, we employ a small strain approach. The linear-elastic material is presumed to follow Hooke’s law with elastic free energy density Ψ (u) =

1 ε(u) : C : ε(u), 2


(64)


74


where strain ε(u(x)) denotes the symmetric gradient of displacement u(x) and C is the Hookean tensor. Also, the split into tensile and compressive components (14) is linearized, ε = [ε+ ] + [ε− ], where [ε+ ] denote the positive and tensile components of the principal strains, and [ε− ] the remaining compressive parts. This leads to an elastic energy density function which only accounts for tension: Ψe (u) =

1 [ε+ (u)] : C∗ : [ε+ (u)] 2

(65)

Here the tensor C∗ = g(s)C is multiplied with a weight function (34), specifically we set g(s) = s2 + η. The choice of η 1 but η = 0 avoids numerical problems and results in a substitute-material approach for the damaged zones. In dynamical computations η can be set to zero. The resolution of the diffuse crack zone is determined by the critical length lc which depends on the finite element mesh size. We used here a uniform mesh with 5760 triangular elements and lc = 2.5 mm. A finer mesh would lead to improved results in the sense of a narrower crack zone, see Fig. 4. However, also with such a rather coarse mesh a quantitative agreement to cohesive element simulations was obtained, cf. [9]. The mobility parameter in equation (38) is set to M = 106 /s. It needs to be chosen in such a way that the phase field is able to decrease locally to zero within a time range of few microseconds, once the energy-release rate exceeds the specific fracture energy. We performed our numerical simulations with the aim to validate experimental measurements of Griffith’s critical energy release rate Gc in a dynamic regime. Out of the experimental data Gc is calculated from an energy balance,

1 1 1 1 Gcexp = mall (v(t1 ))2 − m1 (v1 (t2 ))2 − m2 (v2 (t2 ))2 , (66) Ac 2 2 2

Gc [N/m]

as the energy decreases between t1 immediately before crack initiation and t2 immediately after fragmentation of the specimen; v(t1 ) denotes the specimen’s velocity, v1 (t2 ), v2 (t2 ) the velocities of the fragments and mall , m1 , m2 the corresponding masses. The specific fracture energy then follows by division with the fractured area Ac . This value is now compared to the input value Gc of the phase-field model. In Fig. 11 we illustrate qualitatively in the crack formation for different values of Gc which -by definition200 180 160 140 120 100 80 60 40 20 0 0

20

40

60

80

100

120

140

160

180

200

input-Gc [N/m]

Fig. 12 Comparison between the input Gc -value of the phase-field model and the Gc -value calculated by means of (66) with the variables v, v1 , v2 , m1 , m2 observed in the finite element simulation.



GAMM-Mitt. 39, No. 1 (2016)

v

75

m s

8 7 6 5 4 3 2 1 0

0

0.2

0.4

0.6

0.8

1

time [10−1 ms]

Fig. 13 (online colour at: www.gamm-mitteilungen.org) Computed velocity of the specimen before crack initiation and velocity of the two fragments after crack initiation with Gc = 90N/m, lc = 20μm, M = 106 /s, σmax = 17.3MPa. The experimental velocity-time diagram of the spallation experiment is shown on the right. The red dots represent the velocity vs,1 of the first fragment (average of 31 control points), the blue squares represent the velocity vs,2 of the second fragment (average of 8 control points) and the green diamonds represent the extrapolated initial center of mass velocity vi .

have a significant effect on the crack evolution. If Gc is set to a high value, Griffith’s criterion cannot be fulfilled anymore and the phase field parameter s remains at the constant value 1 during the whole simulation. A too small value leads to ’distributed’ crack, i.e., a wide range of s < 1. Also, Gc influences the crack position of the damaged zone, a reduction of Gc moves the crack position closer to the free end of the specimen because the driving force ∂Ψe /∂s exceeds the fracture resistance at earlier time. Additionally the specific fracture energy of UHPC is determined in the sense of an inverse analysis. In Fig. 12 the values of Gc , with eq. (66) inversely calculated out of the fragments velocities, are plotted over the input values of second-order phase field fracture calculations. They show a good agreement, all points are located near the bisector that declares the ’exact’ values. Finally, Fig. 13 illustrates the velocity of the specimen and of the two fragments. We assume an initial velocity of v0 = 5 m/s for the specimen due to the acceleration of the incident bar and a specific fracture energy of Gc = 90 N/m. A permanent crack is defined if the phase field parameter s is below a value of 0.3 over the whole width of the specimen. As can be seen from Fig. 13, the phase field fracture simulation agrees well with the fragments’ velocity measured in the experiments. Immediately after the cracking this difference is approximately 1 m/s.

5

Summary

In our paper we address the problem of computing material failure within a finite element framework. We chose phase fields of different type and approximation order to resolve and regularize the discontinuities typically arising between intact and damaged material. By means of a phase field ansatz such sharp boundaries are replaced by diffuse interfaces which



76


avoid a difficult explicit tracking of the boundaries. Here we discuss the theoretical framework of phase diffusion and phase field fracture methods and suggest a numerical solution of the resulting multi-field problems with spline-based finite elements. These NURBS functions are an efficient tool to provide the continuity required for different phase field approaches. Computations of several multi-field problems such as quasi-static crack propagation in a hyperelastic material, thermomigration and void growth in solder alloys, and crack initiation by wave reflection have been modeled and solved numerically.

Acknowledgment The authors gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG) under the grant WE2525/4-1, WE2525/5-1, HE4593/3-2 and WE2525/8-1.

References [1] A. Abdollahi and I. Arias. Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions. Journal of the Mechanics and Physics of Solids, 60(12):2100 – 2126, 2012. [2] D. Anders, C. Hesch and K. Weinberg. Computational modeling of phase separation and coarsening in solder alloys. Int. J. Solids Structures 49, 1557–1572, 2012 [3] D. Anders and K. Weinberg. Thermophoresis in binary blends. Mechanics of Materials, 47:33–50, 2012. [4] C. Bilgen, T. Reppel and K. Weinberg. Determining fracture properties of polyurea by an inverse analysis using phase field fracture. subm. to Experimental Mechanics, 2016. [5] M.J. Borden, C.V. Verhoosel, M.A. Scott, T.J.R. Hughes, and C.M. Landis. A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Engrg., 217–220:77–95, 2012. [6] B. Bourdin. The variational formulation of brittle fracture: Numerical implementation and extensions., Volume 5 of IUTAM Symposium on Discretization Methods for Evolving Discontinuities, IUTAM Bookseries, Chapter 22, pages 381–393. Springer Netherlands, 2007. [7] C. Chen, H.-Y. Hsiao, Y.-W. Chang, F. Ouyang, K.N. Tu. Thermomigration in solder joints. Materials Science and Engineering R 73, 85–100, 2012 [8] F. Corson, M. Adda-Bedia, H. Henry, and E. Katzav. Thermal fracture as a framework for quasistatic propagation. International Journal of Fracture, 158:1–14, 2009. [9] T. Dally and K. Weinberg. The phase-field approach as a tool for experimental validations in fracture mechanics. Continuum Mech. Thermodyn., 10.1007/s00161-015-0443-4, 2015 [10] W. Dreyer and W.H. Müller. Modeling diffusional coarsening in eutectic tin/lead solders: a quantitative approach. Int. J. Solids Structures 38, 1433–1458, 2001 [11] G.A. Francfort and J.-J. Marigo. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46:1319–1342, 1998. [12] H. Henry and H. Levine. Dynamic instabilities of fracture under biaxial strain using a phase field model. Physics Review Letters, 93:105505, 2004. [13] C. Hesch, S. Schuß, M. Dittmann, M. Franke and K. Weinberg. Isogeometric analysis and hierarchical refinement for higher-order phase-field models. Compt. Methods Appl. Mech. Engrg., 2016 [14] C. Hesch and K. Weinberg. Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture. Int. J. Numer. Meth. Engng. 99, 906–924, 2014 [15] H.-Y. Hsiao and C. Chen. Thermomigration in flip-chip SnPb solder joints under alternating current stressing. Applied Physics Letters 90, 152105, 2007 [16] A. Karma, D.A. Kessler, and H. Levine. Phase-field model of mode iii dynamic fracture. Physical Review Letter, 81:045501, 2001.



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[17] C. Kuhn and R. Müller. A continuum phase field model for fracture. Engineering Fracture Mechanics, 77:3625–3634, 2010. [18] S. Mariani and U. Perego. Extended finite element method for quasi-brittle fracture. Int. J. Numer. Meth. Engng. 58, 103–126, 2003 [19] C. Miehe, M. Hofacker, and F. Welschinger. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Engrg., 199:2765–2778, 2010. [20] M. Ortiz and L. Stainier. The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering, 171(3-4):419–444, 1999. [21] M. Ortiz and A. Pandolfi. Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int. J. Numer. Meth. Engng. 44, 1267–1282, 1999 [22] F.Y. Ouyang, K.N. Tu, Y.S. Lai and M. Gusak. Effect of entropy production on microstructure change in eutectic snpb flip chip solder joints by thermomigration Applied Physics Letters 89, 2006 [23] A. Pandolfi and M. Ortiz. An eigenerosion approach to brittle fracture. Int. J. Numer. Meth. Engng., 92:694–714, 2012. [24] K.L. Roe and T. Siegmund. An irreversible cohesive zone model for interface fatigue crack growth simulation. Eng. Fract. Mech. 70, 209–232, 2003 [25] S. Rokkam, Anter El-Azab, P. Millett and D. Wolf. Phase field modeling of void nucleation and growth in irradiated metals. Modelling and Simulation in Materials Science and Engineering, 17 064002, 2009. [26] B. Schmidt, F. Fraternali and M. Ortiz. Eigenfracture: An eigendeformation approach to variational fracture. Multisclae Model. Simul. 7,1237–1266, 2009 [27] N. Sukumar, D.J. Srolovitz, T.J. Baker and J.-H. Prevost. Brittle fracture in polycrystalline microstructures with the extended finite element method. Int. J. Numer. Meth. Engng. 56,2015–2037, 2003 [28] R.L.J.M Ubachs, P.J.G Scheurs, and M.G.D Geers. A nonlocal diffuse interface model for microstructural evolution of tin-lead solder. J. Mech. Phys. Solids 52, 1763–1792, 2004 [29] C.V. Verhoosel and R. de Borst. A phase-field model for cohesive fracture. Int. J. Numer. Methods Eng., page in press, 2013. [30] K. Weinberg and C. Hesch. A high-order finite-deformation phase-field approach to fracture. Continuum Mech. Thermodyn., DOI 10.1007/s00161-015-0440-7, 2015 [31] K. Weinberg, T. Böhme, and W. H. Müller. Kirkendall voids in the intermetallic layers of solder joints in MEMS. Computational Materials Science, 45(3):827–831, 2009. [32] X.P. Xu and A. Needleman. Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42,1397–1434, 1994

