Geomechanics

지반공학을 위한 수치해석, 정말 쉽게 한번 접근해 보자.

한번은 들어봤지만 한번에 알수가 없는 지반탄성체에 대해 요점정리 해본다.

Mathematical Preliminaries: 기본 수학

Tensor (in Euclidian Space)

• ( a,b ) = scalar                   = integer, potential
• ( a,b ) = 1st order tensor    = 3x1 matrix vector :  3 components
• ( A,B ) = 2nd order tensor  = 3x3 matrix            :  9 components
• ( A,B ) = 3rd order tensor  = 3x3x3 matrix        : 27 components
• ( A,B ) = 4rd order tensor  = 3x3x3x3 matrix    : 81 components

Product

• inner product                     = u · v = c     : projection, eg) (1,0,0) · (0,1,0) = 0+0+0
• vector (cross) product       = u x v = w   : area,          eg) (1,0,0) x (0,1,0) = (0,0,1)
• triple scalar (box) product = (u x v) · w  : volume
• tensor product (dyad)       = u ⊗ v = U  or  A ⊗ B = D
• trace                                 = tr(U) = tr(u ⊗ v) = u · v
• dot product                       = A · B = C = Aik Bkj
• double contraction            = A : B = D = Aij Bij

Del: Nabla operator (∇)

• gradient:      ∇ p     = u = ( dp/dx, dp/dy, dp/dz )
• divergence: ∇ · u   = a = du/dx + du/dy + du/dz
• curl:             ∇ x u  = v = [ (du3/dy - du2/dz), (du1/dz - du3/dx), (du2/dx - du1/dy) ]
• Laplacian:   ∇ ·∇ p  = d2p/dx2 + d2p/dy2 + d2p/dz2
• Divergence of Gradient (p), 즉 2차 미분된 scalar 값
• Hessian:     H(p) = J(u) = J(∇ p)
• Jacobian of Gradient (p), 이건 2차 미분된 tensor 값

Divergence theorem (Gauss's theorem): 부피 변화는 전체 면적 flux 변화량 합이다.

Stoke's theorem: Cauchy's fomula 의 근간. 점에서의 값은 그 점 주변 둘러싼 닫힌 곡선 위에서 적분한 것과 같다??

Kinematic

가장 기본은 운동 (혹은 변형) 의 기술방법을 구별해 생각하는 일.

• Lagrangian Description (X) : Material coordinates: 물체 따라 같이 움직이는 기술방법
• Eulerian Description (x) : Spatial coordinates: 고정된 관찰자로서의 기술방법

Deformation gradient tensor (F)

• dx = F dX
• F = R U = Rotation tensor x Stretch tensor (symmetric)
• C = F' F : Right Cauchy-Green = Material deformation tensor
• B = F F' : Left Cauchy-Green = Finger tensor = Spatial deformation tensor
• E = 1/2 (C-I)       : Green (Lagrangian) strain tensor
• e = 1/2 (I - invB) : Almansi (Eulerian) strain tensor
• 하지만 결국 Cauchy's infinitesimal strain tensor = 1/2 * (u_j,i + u_i,j)

이게 왜 필요한가?

구조물 모니터링에서 우리가 얻을 수 있는것은 시간에 따른 좌표값 뿐.
그에 따른 displacement 와 strain 을 갖고 stress state 를 제대로 판단해야 파괴 예측 가능.

Stress

Cauchy stresses

• T(n) = s_ji * n_j = (3x1) = (3x3)*(3x1)
• T(n) : traction vector
• s : Cauchy (true) stress tensor
• angle between T(n) & n_j = inv( cos( T· n / |T||n| ) )
• Moment of equilibrium (about x3-axis) : s12 = s21 --> symmetric

Principal stresses (주응력)

• Mohr's circle
• Eigenvalue : det (s -lamda * d_ij) = 0 <--  어렵게만 배우던 고유값의 실제 사용예
• Invariants (I1, I2, I3) : - s^3 + I1*s^2 - I2*s + I3 = 0

Stress-Deviation tensor

• stress tensor = hydrostatic stress tensor (s_m) + deviator stress tensor (s')
• s'_ij = s_ij - s_m * I
• s_m = 1/3 * trace(s)
• I = identity matrix
• It is very important in describing the plastic behavior.
• 주의 : triaxial test 에서 보통 쓰던 deviator stress = s1 - s3 랑은 또 조금 다름.
• 주의 : s' 의 Invariants = (J1, J2, J3) 로 (I1, I2, I3) 와 다름. J1 = 항상 0.

Constitutive Equations (구성방정식) : s_ij = C_ijkl * e_kl

• C_ijkl = 4th order tensor : 3^4 = 81 elements
• s = C e, where C : stiffness tensor
• e = S s, where S : compliance tensor
• Symmetrization : 81 - 3*9 - 3*6 = 36 elements
• Thus, anisotrophy constitutive equation : [s] (6x1) = [C] (6x6) * [e] (6x1)
• Isotrophiy constitutive equation : s_ij = lamda * e_kk * d_ij + 2 * mu * e_ij
• s_11 = lamda*(e_11+e_22+e_22) + 2mu*e_11
• Thus, [C] (6x6) = [ [lamda+2mu, lamda, lamda, 0, 0, 0] ...
• These can be transformed with ( K, E, v, G (=mu) ) in engineering.
• 추가 : Compatibility conditions = No gaps or overlaps during deformation.
• 즉, 적합방정식은 복잡해보여도 결국은 3차원 Strain 들의 관계식.

한마디로 결론은?

수치해석에서 등방성 탄성체를 가정했다면, Hooke's Law 는 딱 두 문자로 표현된다.
바로 Lame's parameters : lamda 와 mu .
복잡하게 Young's modulus (E), Poisson's ratio (v), bulk modulus (K), shear modulus (G) 로도 표현되곤 하는데, 결국은 똑같은거다. 더 배우고 싶다면 구조방에서...... ㅡ.ㅡㅋ

Failure Criteria (4 classic models)

Brittle material : Elasticity

• 1st. Mohr-Coulomb theory : t = c + s * tan(phi) 
• t : shear strength
• c : cohesion
• s : normal stress
• phi : angle of internal friction.

Ductile material : Plasticity

Beyond the elastic range yield occurs, Elastic and Plastic deformations are assumed to happen concurrently (Total strain e = eElastic + ePlastic) . Plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces.

소성변형 들어가기에 앞서서..

1. EPP model : Elastic Perfectly Plastic model = no work hardening.
2. 그간 써오던 total accumulated strain 대신, increments in strain 을 사용한다. (loading, unloading 으로 인한 stress-strain 관계가 더이상 1:1 대응이 안되기 때문)
3. 
4. Principal plastic strain increments (d ePlastic) 와 principal deviator stress (s') 는 비례
5. Flow rule : plastic stress-strain law (between d ePlastic & Cauchy stress (s))
6. Prandtl-Reuss equations : full elastic-plastic stress-strain relations (w/ Hooke's law)

• 2nd. Tresca yield criterion
• Yield occurs when s1 - s3 reaches k (= direct shear strength).
• This assumes s2 has no effect.
• Tresca's criterion is more conservative than von Mises's.

• 3rd. von Mises yield criterion ( called J2-plasticity or J2 flow theory )
• Yield occurs : 2nd invariant of deviator stress tensor (J2) reaches k^2.
• von Mises 응력 : 각 지점에서의 비틀림에너지 (max. distortion E) 나타냄.
• J2 = 1/6 * [(s_xx-s_yy)^2 +...] + s_xy^2 +... = 1/2 * s'_ij * s'_ij
• material yield strength (k) = direct shear strength
• 식으로 쓰면 f = J2 - k^2 = 1/2 * (s'_ij)^2 - k^2
• f = 0 : yield
• f < 0 : elastic
• f > 0 : not possible
• All components of shear stress contribute to yield.

• 4th. Drucker-Prager yield criterion
• Pressure-dependent model (for plastic deformation of soils)
• sqrt(J2) = A + B * I1
• I1 : 1st invariant of Cauchy stress = 3*mean stress(s_m)
• J2 : 2nd deviator stress invariant
• A & B from experiments

추가 : Hyper-elastic = Green elastic

s_ij = B_ij + C_ijkl * e_kl

where, B_ij : components of initial stress tensor = storing of past Energy
아쉽게도 모든 지반 탄성체는 B_ij 가 당연히 존재함.

그리하여 

지반의 불균질 & 비등방성으로 인해, 복잡하지만 탄소성변형을 알아야 함.