On a Nonlocal Finite Element Model for Mode-III Brittle Fracture with Surface-Tension Excess Property S. M. Mallikarjunaiah Department of Mathematics Texas A&M University College Station, TX S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 1 / 25
Overview Notation and Preliminaries 1 Mode-III Fracture Model with Surface-Tension Excess Property 2 Reformulation of Jump Momentum Balance Boundary Condition 3 Nonlocal Finite Element Method 4 Numerical Results 5 References 6 S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 2 / 25
Notation and preliminaries Let X be any arbitrary point in the reference configuration and x denotes the corresponding point in the deformed configuration. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 3 / 25
Notation and preliminaries Let X be any arbitrary point in the reference configuration and x denotes the corresponding point in the deformed configuration. Then the mapping x = f ( X ) represents the motion of the body. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 3 / 25
Notation and preliminaries Let X be any arbitrary point in the reference configuration and x denotes the corresponding point in the deformed configuration. Then the mapping x = f ( X ) represents the motion of the body. u := x − X S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 3 / 25
Notation and preliminaries Let X be any arbitrary point in the reference configuration and x denotes the corresponding point in the deformed configuration. Then the mapping x = f ( X ) represents the motion of the body. u := x − X F := ∂ f ∂ X = I + ∇ u S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 3 / 25
Notation and preliminaries Let X be any arbitrary point in the reference configuration and x denotes the corresponding point in the deformed configuration. Then the mapping x = f ( X ) represents the motion of the body. u := x − X F := ∂ f ∂ X = I + ∇ u B := FF T , C := F T F and S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 3 / 25
Notation and preliminaries Let X be any arbitrary point in the reference configuration and x denotes the corresponding point in the deformed configuration. Then the mapping x = f ( X ) represents the motion of the body. u := x − X F := ∂ f ∂ X = I + ∇ u B := FF T , C := F T F and E = 1 2 ( C − I ) S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 3 / 25
Notation and preliminaries Let X be any arbitrary point in the reference configuration and x denotes the corresponding point in the deformed configuration. Then the mapping x = f ( X ) represents the motion of the body. u := x − X F := ∂ f ∂ X = I + ∇ u B := FF T , C := F T F and E = 1 2 ( C − I ) � ∇ u + ∇ u T � ǫ = 1 2 S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 3 / 25
Notation and preliminaries Let X be any arbitrary point in the reference configuration and x denotes the corresponding point in the deformed configuration. Then the mapping x = f ( X ) represents the motion of the body. u := x − X F := ∂ f ∂ X = I + ∇ u B := FF T , C := F T F and E = 1 2 ( C − I ) � ∇ u + ∇ u T � ǫ = 1 2 If we linearize using the assumption of displacement gradients are small, we can approximate E and ǫ . Then there is no distinction between reference and deformed configuration. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 3 / 25
Mode-III Fracture The Mode-III fracture (or anti-plane shear fracture): The fracture surfaces slide relative to each other skew-symmetrically with shear stress acting as shown in the figure. Displacement: u 1 = 0 and u 2 = 0 u 3 = u 3 ( x 1 , x 2 ) S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 4 / 25
Classical LEFM model The classical Linearized Elastic Fracture Mechanics (LEFM) model has two well known inconsistencies: S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 5 / 25
Classical LEFM model The classical Linearized Elastic Fracture Mechanics (LEFM) model has two well known inconsistencies: It predict singular crack-tip strains and stresses. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 5 / 25
Classical LEFM model The classical Linearized Elastic Fracture Mechanics (LEFM) model has two well known inconsistencies: It predict singular crack-tip strains and stresses. Also it predicts an elliptical crack-surface opening displacement with a blunt crack-tip. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 5 / 25
Classical LEFM model The classical Linearized Elastic Fracture Mechanics (LEFM) model has two well known inconsistencies: It predict singular crack-tip strains and stresses. Also it predicts an elliptical crack-surface opening displacement with a blunt crack-tip. Thus, several remedies have been attempted: appealing to a non-linear theory of elasticity, the introduction of a cohesive zone around the crack tip and non-local theories. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 5 / 25
Mode-III Brittle Fracture Problem Formulation σ ∞ 23 ⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗ The problem studied here is the Ω straight, static, anti-plane shear crack, lying on | x 1 | < a , x 2 = 0 in X 2 an infinite, isotropic, linear elastic body subjected to uniform far-field X 1 anti-plane shear loading ( σ ∞ 23 ). The a − a Σ stress-strain relations are: τ 23 = µ ∂ u 3 τ 13 = µ ∂ u 3 and , ∂ x 2 ∂ x 1 where τ 23 and τ 13 are the relevant ⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙ stress components, and u 3 denotes σ ∞ 23 the z − displacement. Figure: Physical description of the problem. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 6 / 25
Mode-III fracture problem formulation To derive the governing equations for this problem, we follow the study of Sendova and Walton 1 . 1 T. Sendova and J. R. Walton, A New Approach to the Modeling & Analysis of Fracture through an Extension of Continuum Mechanics to the Nanoscale. Math. Mech. Solids, 15(3), 368-413, 2010. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 7 / 25
Mode-III fracture problem formulation To derive the governing equations for this problem, we follow the study of Sendova and Walton 1 . The equilibrium equation, without the body force term, is the Laplace equa- tion for u 3 − ∆ u 3 = 0 . 1 T. Sendova and J. R. Walton, A New Approach to the Modeling & Analysis of Fracture through an Extension of Continuum Mechanics to the Nanoscale. Math. Mech. Solids, 15(3), 368-413, 2010. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 7 / 25
Mode-III fracture problem formulation To derive the governing equations for this problem, we follow the study of Sendova and Walton 1 . The equilibrium equation, without the body force term, is the Laplace equa- tion for u 3 − ∆ u 3 = 0 . Then we consider a surface tension model which depend on (linearized) curvature of the out-of-plane displacement by: γ = γ 0 + γ 1 u 3 , 11 ( x 1 , 0) , where γ 0 and γ 1 are surface tension parameters. 1 T. Sendova and J. R. Walton, A New Approach to the Modeling & Analysis of Fracture through an Extension of Continuum Mechanics to the Nanoscale. Math. Mech. Solids, 15(3), 368-413, 2010. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 7 / 25
Mode-III fracture problem formulation To derive the governing equations for this problem, we follow the study of Sendova and Walton 1 . The equilibrium equation, without the body force term, is the Laplace equa- tion for u 3 − ∆ u 3 = 0 . Then we consider a surface tension model which depend on (linearized) curvature of the out-of-plane displacement by: γ = γ 0 + γ 1 u 3 , 11 ( x 1 , 0) , where γ 0 and γ 1 are surface tension parameters. Then the resulting bound- ary condition on the upper crack-surface is give by: u 3 , 2 ( x 1 , 0) = − γ 1 u 3 , 111 ( x 1 , 0) 1 T. Sendova and J. R. Walton, A New Approach to the Modeling & Analysis of Fracture through an Extension of Continuum Mechanics to the Nanoscale. Math. Mech. Solids, 15(3), 368-413, 2010. S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 7 / 25
Mode-III fracture BVP x 2 Γ 3 ( b , b ) − ∆ u 3 ( x 1 , x 2 ) = 0 , in Q Boundary conditions: Γ 4 Q Γ 2 on Γ 0 u 3 , 2 ( x 1 , 0) = − σ ∞ 23 − γ 1 u 3 , 111 , on Γ 1 u 3 = 0 , 1 Γ 0 (0 , 0) on Γ 2 � n · ∇ u 3 = 0 , x 1 Γ 1 - σ ∞ on Γ 3 n · ∇ u 3 = 0 , � 23 on Γ 4 � n · ∇ u 3 = 0 . Figure: Finite computational domain Q . S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 8 / 25
Weak Formulation and Numerical Strategy Appealing to the BVP, the weak formulation for the problem on hand is found by integrating the PDE against a test function v over Ω. This yields � � ∇ v · ∇ u 3 dQ − v ( � n · ∇ u 3 ) d ∂ Q = 0 . Q ∂ Q S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 9 / 25
Weak Formulation and Numerical Strategy Appealing to the BVP, the weak formulation for the problem on hand is found by integrating the PDE against a test function v over Ω. This yields � � ∇ v · ∇ u 3 dQ − v ( � n · ∇ u 3 ) d ∂ Q = 0 . Q ∂ Q There is no contribution from the second term on the left-hand side of the above equation except over the crack-surface Γ 0 . S. M. Mallikarjunaiah (TAMU) Deal.II Workshop 2015 9 / 25
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