Computational plasticity and viscoplasticity for composite materials and structures

Transcription

1 Coputational plasticity and viscoplasticity for coposite aterials and structures Jacob Fish and Kalun Shek Departents of Civil, Mechanical and Aerospace Engineering Rensselaer Polytechnic Institute Troy, NY 280 The paper extends a recently developed theory of atheatical hoogenization with eigenstrains to account for viscous effects. A unified atheatical and nuerical odel of plasticity and viscoplasticity is eployed within the fraework of the power-law and the Bodner-Parto odels..0 Introduction A typical aerospace coposite coponent can be odeled on (at least) three different scales: (i) acroscale (structural level), (ii) esoscale (lainate level), and (iii) icroscale (the level of icroconstituents). Additional scales can be introduced into the odel. For exaple, the scale of aterial heterogeneity in each icrophase, such as dislocations and grain boundaries, could be considered as another scale, not to ention the atoic and electronic scales as the sallest spatial scales. Coputational odels can be either deterinistic or stochastic. Various deterinistic approaches can be classified into the following three categories (i) the classical uncoupled approach [2], [2], [4], [5], [], [3], (ii) the coupled approach assuing icro/esostructure periodicity [4], [5], and (iii) the coupled approach free of periodicity assuptions [7], [6], [8]. The uncoupled approach eploys representative volue eleents (RVE) at one level to produce averaged paraeters for the use at the next level. The atheatical hoogenization theory [2], [5] provides a theoretical fraework for the uncoupled approach. For three or ore spatial scales the atheatical hoogenization theory has been eployed in [3]. The coputational coplexity of solving nonlinear heterogeneous systes is uch greater. While for linear probles the RVE proble has to be solved only once, for nonlinear history dependent systes it has to be solved at every increent and for each integration point. Moreover, history data has to be updated at a nuber of integration points equal to the product of integration points at all odeling scales considered. The priary objective of the present anuscript is to extend the recent forulation of the atheatical hoogenization theory with eigenstrains developed by the authors in [4] to account for viscous effects. We focus on the uncoupled approach assuing solution peri-

2 odicity. In Section 2 we derive a closed for expression relating arbitrary transforation fields to echanical fields in the phases. In Section 3 the 2-point approxiation schee (for two phase aterials), each point represents an average response within a phase is derived. The local response within each phase is recovered by eans of post-processing. A unified atheatical and nuerical odel of plasticity and viscoplasticity is eployed within the fraework of the power-law [7], [8] and the Bodner-Parto [], [6], [0] odels. Nuerical experients are conducted in Section 4. A discussion on future work conclude the anuscript. 2.0 Matheatical hoogenization with eigenstrains In this section we suarize our recent results on atheatical hoogenization with eigenstrains [4]. We regard all inelastic strains, phase transforation as eigenstrains in an otherwise elastic body. Attention is restricted to sall deforations. For extensions to large deforation theory we refer to []. The icrostructure of a coposite aterial is assued to be locally periodic (Y-periodic) with a periodic arrangeent represented by a Unit Cell or a Representative Volue Eleent, denoted by Θ. Let x be a acroscopic coordinate vector in acro-doain Ω and y x ς be a icroscopic position vector in Θ. For any Y-periodic function f, we have f y) f y + kŷ) in which vector ŷ is the basic period of the icrostructure and k is a 3 by 3 diagonal atrix with integer coponents. Adopting the classical noenclature, any Y-periodic function f can be represented as f ς ( x) f y( x) ) () superscript ς denotes a Y-periodic function f. The indirect acroscopic spatial derivatives of f ς can be calculated by the chain rule as f ς,xi ( x) f ; xi y) f,xi y) + -- f ς,yi y) (2) a coa followed by a subscript variable or denotes a partial derivative with respect to the subscript variable (i.e. f,xi f x i or f,yi f y i ). A sei-colon followed by a subscript variable x i denotes a total partial derivative with respect to x i y depends on x, as defined in (2). Suation convention for repeated right hand side subscripts is eployed, except for subscripts x and y. We assue that icro-constituents possess hoogeneous properties and satisfy equilibriu, constitutive and kineatical conditions. The corresponding boundary value proble is governed by the following equations: x i y i 2

3 σς ; xj + b i 0 in Ω (3) σ ς µ kl L ( ες ς kl ) in Ω (4) ε ς uς ( ix ) ; j -- ( u ς 2 ix ; j + uς jx ; i ) in Ω (5) and appropriate set of boundary and interface conditions. In the above equations, ε ς are coponents of stress and strain tensors; L and µ ς are coponents of elastic stiffness and eigenstrain tensors, respectively; is a body force assued to be independent of y ; and u ς i denotes the coponents of the displaceent vector. Subscript pairs with parenthesizes denote syetric gradients. In the following, displaceents u ς i ( x) u i y) and eigenstrains µ ς ( x) µ y) are approxiated in ters of double scale asyptotic expansions on Ω Θ: b i σ ς and u i y) u 0 i y) + ςu i y) + µ y) µ 0 y) + ςµ y) + (6) (7) The corresponding expansions of strain and stress tensors can be obtained by anipulating (6), (7), (4) and (5) with consideration of the indirect differentiation rule (2) ε y) σ y) ς --ε ς --σ y) + ε 0 y) + ςε y) + y) + σ 0 y) + ςσ y) + (8) (9) strain coponents for various orders of ς are given as ε ε y ( u 0 ), ε s ε x ( u s ) + ε y ( u s + ), s 0,, (0) and ε x ( u s ) us i,xj, ε y ( u s ) us i,yj ( ) ( ) () are related by the following constitutive equa- Stresses and strains for different orders of ς tions σ L ε kl, σs L ( ε s kl µ s kl ), s 0,, (2) 3

4 Inserting the stress expansion (9) into equilibriu equation (3) and aking the use of (2) yields a set of equilibriu equations for various orders [4]. Fro the lowest order O( ς 2 ) equilibriu equation we get u0 ( i,yj ) 0 which iplies u i 0 u 0 i ( x) and σ y) ε y) 0 (3) In order to solve the O( ς ) equilibriu equation for up to a constant we introduce the following separation of variables u i y) H ikl ( y) { ε xkl ( u 0 ) + d µ kl ( x) } (4) H ikl d kl is a Y-periodic function, and µ is a acroscopic portion of the solution resulting fro eigenstrains, i.e. if µ 0 kl y) 0 then d µ kl ( x) 0. It should be noted that both and µ are syetric with respect to indices k and l. Based on (4) the H ikl O( ς ) d kl equilibriu equation takes the following for: { L (( I kln + H k,yl )ε xn ( u 0 ) + H µ k,yl ( x) µ 0 kl )},yj 0 on Θ ( )n ( )n d n (5) I kln -- ( δ 2 k δ nl + δ nk δ l ) (6) δ k and is the Kronecker delta. Since equation (5) is valid for arbitrary cobination of acroscopic strain field εxn ( u0 ) and eigenstrain field µ 0 kl, we first consider µ 0 kl 0, ε xn ( u 0 ) 0 and then ε xn ( u 0 ) 0, µ 0 kl 0 which yields the following two governing equations on Θ : { L ( I kln + H k,yl )},yj 0 ( )n { L ( H µ k,yl µ 0 kl )},yj 0 ( )n d n (7) (8) Equation (7) together with Y-periodic boundary conditions coprise a standard linear boundary value proble on Θ. For coplex icrostructures a finite eleent ethod is often eployed for discretization of H ikl ( y), which yields a set of linear algebraic syste with six right hand side vectors [3]. After solving (7) for, the elastic hoogenized stiffness follows fro the O( ς 0 ) H in equilibriu equation [3]: L 4

5 Θ L L Θ n A nkl dθ A Θ n L nst A stkl dθ Θ (9) A kln of a unit cell. A kln I kln + H k,yl ( )n is often referred to as an elastic strain concentration function and d kl A close for expression for µ follows fro (8) and (9): Θ (20) is the volue dµ ( L L Θ ) H,yn Θ ( )kl L nst µ st d 0 Θ (2) L L Θ dθ Θ (22) The superscript denotes a reciprocal tensor. In the case of piecewise constant eigenstrains [2], the average strain (8) over a subdoain ρ is given as: Θ ρ Θ ρ ε ρ ε dθ A n ρ ε kl + F ρη µ η kl + η O( ς) (23) Fρη are the eigenstrains influence factors: Fρη c η ( Aρ n I n )( L npq L npq ) ( Aη rspq I rspq )Lη rskl (24) and Aρ Θ ρ Θ ρ ( I + H ) d Θ ( i, y j )kl (25) The superscripts ρ (or η) denote the ρ-th subdoain, Θ ρ, and n is the total nuber of subdoains in the unit cell doain. The subdoain could be ade of a single finite eleent or a group of eleents; c η is the η -th subdoain volue fraction given by c η Θ η n Θ and satisfying c η. η We refer to the piecewise constant odel defined by (23) as the n-point schee odel. As a special case we consider a coposite ediu consisting of two phases, atrix and reinforceent, with respective volue fractions c and c f such that c + c f. Super- 5

6 scripts and f represent atrix and reinforceent phases, respectively. Θ and Θ f denote the atrix and reinforceent doains such that Θ Θ Θ f. We assue that eigenstrains and elastic strain concentration factors are constant within each phase. This yields the siplest variant of (23) n 2. The corresponding approxiation schee is tered as the 2-point odel. The overall elastic properties are given as: L c L n A nkl + c f L f n Af nkl (26) The influence factors Fρη reduce to F L nst ρ ( I A ρ )( L f n n nst ) L stkl ρf ( I A ρ )( Lf n n nst L nst ) L f stkl ρ f, F (27) and the overall stress is given as: σ c σ + c f σf (28) 3.0 The 2-point schee We consider a two-phase coposite the atrix phase behaves as a rate-dependent or rate independent elastoplastic aterial, as the reinforceent is elastic, i.e. f 0. A unified atheatical and nuerical odel of plasticity and viscoplasticity for µ the atrix phase is subsequently described. 3. The 2-phase ediu Cobining equations (4) and (23) phase stresses can be expressed as: σ ρ R ρ µ kl ρ ε Q kl (29) µ kl is the atrix plastic strain and Rρ Q ρη Lρ pq Aρ pqkl L ρ δ I pq ( F ρη ) ρη pqkl pqkl ρη,, f (30) A unified rate-dependent / rate-independent flow rule is given as µ ℵ λ (3) 6

7 ℵ ξ , ξ P ( σ kl α kl ) ξ n ξ n (32) and µ represents the plastic strain rate in the atrix phase; P I δ δ kl 3 is a projection operator which transfors an arbitrary syetric second order tensor to the deviatoric space; and α is a back stress. A unified rate-dependent / rate independent consistency condition can be expressed as: Φ ξ g( λ, λ ) 0 (33) ξ is an effective stress defined as ξ 3 --ξ 2 n ξ n (34) Attention is restricted to two widely used classes of flow functions: The power-law odel [7], [8]: λ ξ a b -- Y g Y λ a b (35) The Bodner-Parto odel [], [6], [0]: λ a Y b 2 ξ λ exp g Y log a b (36) a and b are aterial constants; Y is a drag or yield stress which is a function of λ α. The back stress is assued to be governed by the following evolution equation α 2 --H ( λ, λ )µ 3 (37) in which H is a kineatic hardening odulus, which in general depends on the effective plastic strain λ and its rate of change λ. For siplicity, in the present anuscript H is assued to be constant. The rate-independent plasticity odel can be obtained by taking b Iplicit Integration of Constitutive Equations The constitutive equations are integrated over a finite tie step t schee, which yields: using a backward Euler t + t µ t µ + λ t + t ℵ (38) 7

8 t + t α t α 2 t + t + -- λ H ℵ 3 (39) λ t + t λ λ. t for siplicity. Sub- In the following we oit the left superscript for the current step t + t stituting (38) into (29) yields: σ σ tr λ Q ℵ kl (40) σ tr is a trial stress in the atrix phase defined as σ tr t σ + R ε kl (4) Subtracting (39) fro (40), we arrive at the following result: σ α ( I + λ ) ( σ kl α kl ) tr t (42) Q st 2 P stkl + --HP 3 (43) Integrating λ using backward Euler schee, λ λ + λ t, or λ λ t, and then cobining equations (35) and (36) with the generalized consistency condition in the atrix phase (33) yields: t Power law: Φ ξ λ Y a t b Bodner-Parto: Φ ξ λ Y 2 log a t 0 b (44) The value of λ is obtained by satisfying the generalized consistency condition (44). For calculation of λ, equation (42) is substituted into (44), which produces a nonlinear equation for λ. The Newton ethod is applied to solve for λ : λ Φ k λ λ k + Φ λ k (45) 8

9 k is an iteration count. It can be shown that the derivative Φ λ required in (45) has the following for: Φ λ dξ dg d λ d λ Power law: Bodner-Parto: dg d λ λ b dy a t d λ by λ dg λ d λ log a t b dy d λ b λ log λ a t (46) dξ d λ ξ 2ξ ( I + λ ) kln ξ n (47) and dy d λ is a rate-independent isotropic hardening odulus. The converged value of λ is used to copute the phase stresses fro equations (42), (38), (29) and the back stress fro (39). The overall stress is coputed fro (28). 3.3 Consistent Linearization While integration of the constitutive equations affects the accuracy of the solution, the foration of a tangent stiffness atrix consistent with the integration procedure is essential to aintain the quadratic rate of convergence if one is to adopt the Newton ethod for the solution of nonlinear syste of equations on the acro level [9]. The starting point is the increental for of the phase constitutive equations (29): σρ R ρ ε Q kl ρ µ kl (48) Taking the aterial tie derivative of (48) and keeping in ind that all quantities are known at the previous step yields σ ρ R ρ ε kl Q ρ { λ ℵ kl + λ ℵ kl } (49) ℵ Ξ ( σ kl α kl), Ξ ξ 3 -- ( P 2 n ℵ ℵ n ) (50) 9

10 To calculate λ we first linearize equation (42) σ α Λ λ, Λ ( I + λ ) kln ξ n (5) and then substitute (5) into the linearized for of (39), which yields α 2 --H ( λ Ξ 3 Λ kl + ℵ )λ (52) Subsequently, we linearize the generalized consistency condition (44) Φ ℵ kl ( σ kl α kl) g 0 λ Power law: g λ b dy by a t dλ λ Bodner-Parto: g λ 2 log λ a t b dy b d λ λ λ log a t and then cobine (52) and (53) to get the expression for λ : (53) λ γℵ kl σ kl Power law: Bodner-Parto: γ 2 2 λ --H -- b dy by a t dλ λ γ 2 2 λ --H log a t b dy b d λ λ log λ a t (54) The rate of phase stress can now be expressed as σ ρ R ρ ε kl γq ρ ( λ Ξ kln Λ n + ℵ kl )ℵ st σ st (55) Evaluating (55) for the atrix phase, ρ, gives σ D ε kl (56) 0

11 D { I st + γq λξ Λ ℵ pq ( + )ℵ pqn n pq st } R stkl (57) Starting fro (49) and proceeding in an analogous anner for the rate of the reinforceent stress yields σ f Df ε kl (58) Df Rf γq pq f ( λξ Λ + ℵ )ℵ pqn n pq st D stkl (59) D The overall consistent instantaneous stiffness is obtained fro the rate for of (28), (56) and (58) σ D ε kl (60) D c D + c f D f (6) 3.4 Consistent tangent stiffness atrix We start fro the syste of nonlinear equations arising fro the finite eleent discretization r A ext ( q A ) f A 0 f int A N ia, xj σ dω f A int Ω (62) N ka is set of C 0 continuous shape functions, such that v k N ka q A ; the upper case subscripts denote the degree-of-freedo and the suation convention over repeated indexes is eployed for the degrees-of-freedo and for the spatial diensions; q A and int ext q A are coponents of nodal displaceent and velocity vectors; f A and f A are coponents of the internal and external force vectors, respectively. The consistent tangent stiffness atrix, K AB, is obtained via consistent linearization of the discrete equilibriu equations (62). It is convenient to forulate such a linearization procedure as:

12 K AB q B d ra dt (63) For siplicity, assuing that the external force vector is not a function of the solution, the consistent linearization procedure for sall deforation theory yields: K AB N ia, xj D N kb, xl dω Ω (64) D is defined in (6); Reark: In the 2-point schee described in Section 3 the eigenstrains have been approxiated by a constant within each phase. A better resolution of local fields can be obtained by approxiating eigenstrains as piecewise constant within each phase. In [4], the eigenstrains have been assued to be constant within each eleent in the unit cell, giving rise to the so called n-point schee. Typically, the coputational cost resulting fro the n-point schee is orders of agnitude higher then that of the 2-point schee since the nuber of nonlinear equations to be solved at each acro-gauss point is equal to the nuber of Gauss points in a unit cell experiencing inelastic deforation. Therefore, it is advantageous to erge the two approaches. In such a cobined schee the overall structural response is coputed using the 2-point schee, while local fields of interest in the unit cell corresponding to the critical regions in the acro-doain are recovered by eans of the n-point schee [4]. An alternative procedure has been presented in []. 4.0 Nuerical experients We consider two nuerical exaples to test the accuracy and efficiency of the coputational schees. For both exaples, continuous fibre icrostructure shown in Figure is considered. The unit cell is discretized with 98 tetrahedral eleents in the reinforceent doain and 253 in the atrix doain, totaling 330 degrees of freedo. 4. The nozzle flap proble The finite eleent esh for one-half of the nozzle flap (due to syetry) is shown in Figure 2. The esh contains 788 linear tetrahedral eleents (993 degrees of freedo). The flap is subjected to an aerodynaic force (siulated by a unifor pressure) on the back of the flap and a syetric boundary condition is applied on the syetric face. We assue that the pin-eyes are rigid and a rotation is not allowed so that all the degrees of freedo on the pin-eye surfaces are fixed. There is about 5 percent of plastic zone in the odel due to the loading. The nozzle flap proble is solved using the 2-point averaging schee with icro-history recovery and the n -point procedures for the reference solution. The objective of this nuerical study is to investigate the 2-point averaging schee with icro-history recovery in ters of accuracy (in both acro- and icro-scales), coputational efficiency and eory consuption. 2

13 The phase properties are given as follows: Titaniu Matrix: Young s odulus 68.9 GPa, Poisson s ratio 0.33, initial yield stress is 24 MPa, isotropic hardening odulus is 4 GPa kineatic hardening odulus is 0 GPa SiC Fiber: Young s odulus GPa, Poisson s ratio 0.2, fiber volue fraction is 0.27 For this proble, the power law is adopted with a and b 0.. The CPU tie is 30 seconds for the acroscopic analysis using the 2-point schee, plus approxiately 70 seconds for each Gaussian point a icro-history recovery procedure is applied. On the other hand, the CPU tie for the n-point schee is over 7 hours. Moreover, eory requireent ratio for these two approaches is roughly :250. Effective icro-stresses obtained by both approaches are copared using the relative error easure defined as Error and are the effective icro-stresses obtained via the -point and the 2-point schees, respectively. The axiu effective stress appears at the pineye of the iddle flap (Gaussian point A). Micro-history is recovered for the unit cell corresponding to the acro Gauss point A. The relative error at this point is approxiately 5%, as shown in Figure Thick-walled cylinder proble point ξn ξ 2 point ξ n point ξ n point ξ 2 point n For our second test proble we consider a coposite thick-walled cylinder with an inner radius r inner, and outer radius r outer. The ratio of r outer to r inner is taken to be 2. The finite eleent discretization is shown in Figure 4. Boundary conditions are assigned in such a way that plain strain and axisyetric conditions are preserved. A constant displaceent rate of is iposed on the inner wall of the cylinder. A power law is eployed with both the kineatic and isotropic hardening taken to be zero, i.e., Y( λ ) Y 0. Other aterial constants eployed are: Matrix: Youngs odulus 500Y 0, Poisson ratio 0.499, a 0.002, b 0.2 Fiber: Youngs odulus 5000Y 0, Poisson ratio 0.2 An exact solution for the steady state proble for the hoogeneous aterial ( c f 0 ) is available in [8]. The inner wall pressures vs. tie as obtained with various fiber volue fractions are depicted in Figure 5. The relative error for the steady state proble with c f 0 is less than %. (65) 3

14 5.0 Future work An uncoupled two-scale approach assuing solution periodicity in inelastic heterogeneous edia has been presented. The two issues that have not been addressed in the present anuscript are: i. Stochastic nature of data So far only the deterinistic aspect has been considered. The deterinistic approach utilizes ean values associated with icrostructural characteristics. The uncertainties in input data pose a treendous challenge not only because we have to deal with nuerical ethods for stochastic differential equations [9], but ore iportantly, because the probability fields, especially the correlations, are usually unknown. ii. Multiple tie scales The difficulty of estiating the average behavior at the larger length scales fro the essential physics at saller scales is copounded in tie dependent probles. Multiple length and tie scales can be accounted for by incorporating the into appropriate asyptotic expansions. Acknowledgent This work was supported in part by the Air Force Office of Scientific Research under grant nuber F , the National Science Foundation under grant nuber CMS , and Sandia National Laboratories under grant nuber AX-856. References S. R. Bodner and Y. Parto, Constitutive Equations for Elastic-Viscoplastic Strain- Hardening Materials, Journal of Applied Mechanics, 42, (975). 2 G. J. Dvorak, Transforation Field Analysis of Inelastic Coposite Materials, Proceedings Royal Society of London, A437, (992). 3 J. Fish, P. Nayak and M. H. Holes, Microscale Reduction Error Indicators and Estiators for a Periodic Heterogeneous Mediu, Coputational Mechanics, 4, (994). 4 J. Fish, K. Shek, M. Pandheeradi and M. S. Shephard, Coputational Plasticity for Coposite Structures Based on Matheatical Hoogenization: Theory and Practice, Coputer Methods in Applied Mechanics and Engineering, 48, (997). 5 J. M. Guedes and N. Kikuchi, Preprocessing and Postprocessing for Materials Based on the Hoogenization Method with Adaptive Finite Eleent Methods, Coputer Methods in Applied Mechanics and Engineering, 83, (990). 6 A. M. Merzer, Modeling of Adiabatic Shear Band Developent fro Sall Iperfections, Journal of the Mechanics and Physics of Solids, 30, (982). 4

15 7 F. K. G. Odqvist, Matheatical Theory of Creep and Creep Rupture, The Clarendon Press, Oxford, (974). 8 D. Pierce, C. F. Shih and A. Needlean, A Tangent Modulus Method for Rate Dependent Solid, Coputers and Structures, 8, (984). 9 J. C. Sio and R. L. Taylor, Consistent Tangent Operators for Rate-Independent Elastoplasticity, Coputer Methods in Applied Mechanics and Engineering, 48, (985). 0 D. C. Stouffer and S. R. Bodner, A Constitutive Model for the Deforation Induced Anisotropic Plastic Flow of Metals, International Journal of Engineering Science, 7, (979). J. Fish and K. L. Shek, Finite deforation plasticity for coposite structures: coputational odels and adaptive strategies, to appear in Coputational Advances in Modeling Coposites and Heterogeneous Materials, edited by J. Fish. Special Issue of Coputer Methods in Applied Mechanics and Engineering, (998). 2 A. Benssousan, J. L. Lions and G. Papanicoulau, Asyptotic Analysis for Periodic Structure, North-Holland, (978). 3 P. Tong and C. C. Mey, Mechanics of coposites of ultiple scales, Coputational Mechanics, Vol. 9, pp , (992) 4 J.Fish and V.Belsky, Multigrid ethod for a periodic heterogeneous ediu. Part I: Convergence studies for one-diensional case, Cop. Meth. Appl. Mech. Engng., Vol. 26, pp. -6, (995). 5 J.Fish and V.Belsky, Multigrid ethod for a periodic heterogeneous ediu. Part 2: Multiscale odeling and quality control in utidiensional case, Cop. Meth. Appl. Mech. Engng., Vol. 26, 7-38, (995). 6 J. Fish and V. Belsky, Generalized Aggregation Multilevel Solver, accepted in International Journal for Nuerical Methods in Engineering, (997). 7 J. Fish and A. Suvorov, Autoated Adaptive Multilevel Solver, accepted in Cop. Meth. Appl. Mech. Engng., (997) 8 J. T. Oden and T. I. Zohdi, Analysis and adaptive odeling of highly heterogeneous elastic structures, TICAM report, Deceber (996). 9 W. K. Liu, T. Belytschko and A. Mani, Rando fields finite eleents, International Journal for Nuerical Methods in Engineering, Vol. 23, pp , (986). 5

16 Figure : Unit cell discretization A Figure 2: Finite eleent esh for the nozzle flap proble 6

17 5% 2.5% 0% -2.5% -5% Figure 3: Relative error (%) in effective stress at acro Gauss point A Figure 4: Finite eleent esh for the thick-walled cylinder proble 7

18 Pressure / Matrix Yield Stress vol. fraction 0.00 vol. fraction 0.08 vol. fraction 0.27 vol. fraction Tie Figure 5: Inner wall pressure vs. tie 8