quantitative information on the error caused by using the solution of the linear problem to describe the response of the elastic material on a corner

Quantitative Justication of Linearization in Nonlinear Hencky Material Problems 1 Weimin Han and Hong-ci Huang 3 Abstract. The classical linear elasticity theory is based on the assumption that the size of the displacement gradient tensor is small. In many engineering applications, although the basic assumption is violated due to, e.g., corner singularities, the linear elasticity system is still used to approximate the reality. In this paper, we give a- posteriori error analysis for the eect of the linearization on solutions of nonlinear Hencky materials. A-posteriori error estimates for the eect of idealizations on solutions of more realistic materials will be given in forthcoming papers. 1 Introduction The classical linear elasticity theory is based on the assumption that the size of the displacement gradient tensor is small. The linear Hooke's law in the linear elasticity theory characterizes a large class of elastic materials occupying smooth domains subject to moderate size body forces and boundary tractions. For engineering applications, however, the ideal assumption is not always satised. For instance, consider an elasticity problem on a domain with reentrant corners. The solution of the linear elasticity system exhibits singularities; in a neighborhood of the corners, the strain and stress tensors computed from the solution become arbitrarily large in sizes. Thus, the basic assumption of the linear elasticity theory is grossly violated around the corners. On the other hand, it is a usual practice among engineers to adopt the linear elasticity model, as long as it can be reasonably expected that the violation of the basic assumption is limited to a small region of the domain. Qualitatively, when the input data are small, the expectation is guaranteed by the continuous dependence of the deformation of an elastic material on input data. Obviously, it is more useful to have 1 The work was supported by Research Grants Council of Hong Kong UGC. Department of Mathematics, University of Iowa, Iowa City, IA 54, U.S.A. 3 Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong. 1

quantitative information on the error caused by using the solution of the linear problem to describe the response of the elastic material on a corner domain. The purpose of the paper is to provide quantitative error analysis for the eect of linearization on solutions of nonlinear Hencky materials. We notice that in practice there are material models more realistic than the nonlinear Hencky materials considered here. However, we may view the analysis given in this paper as a preliminary step towards a general a-posteriori error analysis for linearization of more realistic materials. A typical result has the feature that we solve the linear elasticity system and use its solution u to bound the error u? u, with u the solution of a nonlinear Hencky material problem. We emphasize here that to obtain an error bound, all we need to know is the solution of the linear elasticity problem. We give a brief description of nonlinear Hencky materials in the next section. We indicate a procedure to derive a-posteriori error estimates in Section 3. The derivation of a-posteriori error estimates is based on the duality theory in convex analysis. For the sake of completeness, we include some standard results of the duality theory. To have problems to make specic computations, we introduce in Section 4 a family of nonlinear Hencky materials, and apply the theory presented earlier to give an error bound in terms of the solution of the linear elasticity problem for the error caused by the linearization of a nonlinear Hencky material. Finally, we present some numerical examples. We will extend the results of the paper to more realistic materials and their approximations and idealizations in forthcoming papers. The idea of applying dual variational principles has been used to derive quantitative error estimates for the eects of idealizations on solutions in linear torsion problems ([5]), in modeling a Bingham laminar stationary ow uid ([6]) and in linearization of a nonlinear elliptic boundary value problem ([8]). It is also proved useful to obtain computable error estimates for some regularization procedures in numerical analysis ([4], [7]). In [9], quantitative error estimates are given for coecient idealization in linear elliptic problems. Nonlinear Hencky materials For a detailed description of nonlinear Hencky materials, one is referred to [1] or [1]. Let R N be a bounded domain. For a displacement eld v :! R N, we use the notation v = (v 1 ; : : : ; v N ) and introduce the innitesimal strain tensor "(v) :! R N N with components " ij (v) = 1 @v i + @v! j ; 1 i; j N: @x j @x i

Denote a stress tensor with components ij, 1 i; j N. Deviators of the stress and strain tensors are dened by ij = ij? 1 N tr ij; " ij = " ij? 1 N tr" ij; 1 i; j N where tr = ii, tr" = " ii. Throughout the paper, we adopt the summation convention over repeated indices. For the dot product of two tensors, we use the notation " = " ij ij and " = " ". For a linear isotropic, homogeneous material, we have the stored energy function M (") = 1 (tr") + " = 1 k (tr") + " (.1) where k = + =N, and are the Lame coecients. A nonlinear Hencky material is characterized by a stored energy function of the form M(") = 1 k (tr") + (" ) (.) for a non-negative, continuously dierentiable function satisfying The constitutive law = M (") from (.) is then () = ; () > : (.3) = k tr" I + (" ) " or, equivalently = (" ) ": For the special case when () =, we obtain a linear material, and the constitutive law becomes = k tr" I + " or, equivalently = ": Now we assume a nonlinear Hencky material, initially occupying the domain, is subject to the action of a body force of density f, a traction g on part of the boundary @. We assume on another part of the boundary? 1 @, the displacement eld g 1 is given. We may specify other boundary conditions, e.g., normal or tangential components of tractions or displacements. One can easily modify the proofs of the results presented in this paper to derive a-posteriori error estimates for the cases of other boundary conditions. 3

Let us assume f (L ()) N ; g 1 (H 1 ()) N ; g (L ( )) N and We denote the Sobolev spaces meas(? 1 ) > : (.4) V = (H 1 ()) N ; V?1 = fv V j v = on? 1 g Then the displacement eld u :! R N total potential energy: of the nonlinear Hencky material minimizes the where the total potential energy u? g 1 V?1 ; E(u) = inffe(v) j v? g 1 V?1 g (.5) E(v) = [M("(v))? fv] dx? We now study the minimization problem (.6). We need g v ds: (.6) Lemma.1 If is a Lipschitz-continuous domain or a crack domain, then E(v)! 1 as kvk V! 1, v? g 1 V?1. Proof. We will use Korn's inequality " (v) dx c kvk V ; 8 v V?1 : (.7) Usually, Korn's inequality is proved for a Lipschitz-continuous domain ([1]). For a domain with cracks, we decompose into 1 and in such a way that both 1 and are Lipschitz-continuous, meas(@ 1 \? 1 ) > and meas(@ \? 1 ) >. Then for a v V?1, Korn's inequality holds for both vj 1 and vj. Thus, Korn's inequality also holds for v. For v V, v? g 1 V?1, using the assumption (.3) and Korn's inequality (.7) on v? g 1, we have E(v) = " k " k (tr"(v)) + (" (v))? fv (tr"(v)) + " (v)? fv min(; 1) = min(; 1) " (v) dx? # # dx? dx? fv dx? g v ds g v ds g v ds [" (v? g 1 ) + "(v) "(g 1 )? " (g 1 )] dx? fv dx? g v ds min(; 1) c kv? g 1 k V? c (kvk V + 1): 4

So, E(v)! 1 as kvk V! 1, v? g 1 V?1. Theorem. Let be a Lipschitz-continuous domain or a crack domain. Assume (.3) and (.4). If we further assume () is second-order continuously dierentiable and satises, for some >, () + () ; 8 R (.8) then the minimization problem (.5) has a unique solution u which solves the nonlinear elasticity system [k tr"(u) tr"(v) + (" (u)) "(u) "(v)] dx = 8 v V?1 : fv dx + g v ds (.9) Proof. First we prove the stored energy function M(") dened in (.) is strictly convex. We have M (") = k (tr) + 4 (" ) (" ) + (" ) : (.1) If (" ), then M (") k (tr) + (" ) k (tr) + min(; 1) min(; 1) : If (" ) <, we use the assumption (.8) to obtain M (") k (tr) + 4 (" ) " + (" ) = k (tr) + ( (" ) + (" ) " ) k (tr) + min(; 1) min(; 1) : Thus, M(") is strictly convex. Hence the total potential energy function E(v) is strictly convex. By the assumptions on the function, E(v) is continuous. By Lemma.1, E(v) is coercive. Therefore, the minimization problem (.5) has a unique solution u V with u? g 1 V?1 ([3]). It is easy to verify the solution u satises the equation (.1). In practice, because of its simplicity, linear elasticity theory is preferred as long as the solution of the linear elasticity system can provide suciently accurate information of physical interest. As mentioned in the Introduction, if the material occupies a corner domain, and if the region where the basic assumption of the linear elasticity theory is violated is limited to small neighborhoods of corners, then for a large part of the domain, the governing function 5

is close to the identity function (assuming there the material is isotropic and homogeneous). So it is intuitively reasonable to use the linear elasticity theory. The solution u of the linear elasticity theory is the minimizer of the following potential energy function E (v) = [M ("(v))? fv] dx? The solution can also be equivalently characterized by the relation [k tr"(u ) tr"(v) + "(u ) "(v)] dx = 8 v V?1 : g v ds; v? g 1 V?1 : (.11) fv dx + g v ds (.1) The main purpose of the paper is to derive computable bounds in terms of u error u? u. on the 3 A-posteriori error estimates for the linearization First, we recall a standard result from the duality theory in convex analysis. One is referred to [3] for a complete exposition of the duality theory. Let V, Q be two normed spaces, V, Q their dual spaces. Assume there exists a linear continuous operator L(V; Q), with transpose L(Q ; V ). Let J be a function mapping V Q into R the extended real line. Consider the minimization problem: Dene the conjugate function of J by: J (v ; q ) = We have the following standard result. inf J(v; v): (3.1) vv sup [< v; v > + < q; q >?J(v; q)] : (3.) vv;qq Theorem 3.1 Assume: (1) V is a reexive Banach space, Q a normed space. () J : V Q! R is a proper, lower semi-continuous, strictly convex function. (3) 9 u V, such that J(u ; u ) < 1 and q 7! J(u ; q) is continuous at u. (4) J(v; v)! +1, as kvk! 1; v V. Then problem (3.1) has a unique solution u V. And, for any v V with J(v; v) < 1, J(v; v)? J(u; u) J(v; v) + J ( q ;?q ); 8q Q : (3.3) 6

In [8], we discussed the question of how to derive a-posteriori error estimates from the above theorem. For the paper to be self-contained, we give a similar discussion below. In our applications later, the problem (3.1) will be the boundary value problem of a nonlinear elliptic system for a nonlinear Hencky material, while v = u in (3.3) will be the solution of a related linear elasticity problem. An application of Theorem 3.1 then allows us to get a computable estimate for the error u? u. To do this, we need: Step 1. Find a suitable lower bound for J(u ; u )?J(u; u), which measures the dierence between u and u. Usually, this lower bound will be a quantity depending on ku? uk in some norm. Step. Take an appropriate q so that the estimate (3.3) is as accurate as possible. If q is chosen to be a solution of the dual variational problem, (3.3) becomes an equality. But to get a solution of the dual variational problem is even more dicult than to solve the problem (3.1). Therefore, it is desirable to have a strategy on choosing a q, which can be obtained easily, and which produces a reasonable error bound. Often, we need to calculate the conjugate function for a function dened by an integral of the form: G(v) = g (x; v(x)) dx: Before stating a theorem on how to calculate its conjugate function, we introduce the following notion: Denition 3. Let be an open set of R n, g : R l! R. g is said to be a Caratheodory function, if: 8 R l ; x 7! g(x; ) is a measurable function; for a.e. x ; 7! g(x; ) is a continuous function. Let m i (1; 1), i = 1; ; l. For each i, denote L m i () = v measurable jvj m i dx < 1 : We have the following theorem, a proof of which can be found in [3]. Theorem 3.3 Let g : R l! R be a Caratheodory function. For any v V = l i=1l m i (), dene: G(v) = g (x; v(x)) dx: Then for the conjugate function of G, G (v ) = g (x; v (x)) dx; 8 v V 7

where g (x; y) = sup R l [ y? g(x; ) ] : To apply Theorem 3.1, let us take V = (H 1 ()) N ; V = (H 1 () ) N Q = Q = L () (L ()) N N L ( ) v = (tr"(v); "(v); vj? ) F (v) = 8 < : G(v) =? fv dx if v? g 1 V?1 +1 otherwise M("(v)) dx? J(v; v) = F (v) + G(v) g v ds We notice that when v? g 1 V?1, J(v; v) = E(v), the total potential energy function. The minimization problem (.5) can be equivalently written as u V; J(u; u) = inffj(v; v) j v V g: (3.4) From the corresponding properties of the potential energy function E(v) proved in the last section, it is easy to verify that J(v; v) satises all the assumptions of Theorem 3.1. Therefore, we may apply the inequality (3.3) with v = u, the solution of the linear elasticity problem (.1), to obtain an a-posteriori bound on the error u? u. Let us compute the conjugate functions. For q Q, we use the notation q = (q 1 ; q ; q 3 ) with q 1 L (), q (L ()) N N and q 3 L ( ). We use a similar notation for q Q. We have J ( q ;?q ) = F ( q ) + G (?q ) F ( q ) = supf< v; q >?F (v)g vv = supf< v; q >?F (v)g vv = sup = v?g 1 V?1 8 >< >: [q 1tr"(v) + q "(v) + fv] dx + q 3v ds [q1tr"(g 1 ) + q "(g 1 ) + fg 1 ] dx + q3g 1 ds if [q1tr"(v) + q "(v) + fv] dx + q3v ds = ; 8 v V?1 ; +1; otherwise: 8

To compute the conjugate function of G, we will use Theorem 3.3. G (?q ) = supf< q;?q >?G(q)g qq = sup qq ( = sup q 1 L () = = " + sup q (L ()) NN sup t 1 R?q 1q 1? q q? k jq 1j? (q )?q 1q 1? k jq 1j!?q 1t 1? k t 1 dx # dx + (g? q 3)q 3 ds h?q q? (q) i dx + sup (g? q3)q 3 ds q 3 L ( )! dx + + sup(g? q3)t 3 ds t 3 R " jq 1j k + q (q)? ( (q) ) +1; otherwise 8 >< >: where t =? (q ) solves h sup?q t? (t ) i dx t R NN # dx; if q 3 = g on (t )t =? 1 q : (3.5) Before giving an estimate for the error u?u, we analyze the system (3.5) in detail. Taking inner products of both sides of the system (3.5) with themselves, we obtain an equation for the unknown scale = t : ( ()) = 1 4 q (3.6) Let f() = ( ()). We have f () = () () + ( ()) = ()( () + () ) > by assumptions (.3) and (.8). Thus, f is a strictly increasing continuous function. Since f() =, f()! 1 as! 1, we conclude that f : R +! R + is a one-to-one mapping. Therefore, (3.6) is uniquely solvable. Having the solution of the nonlinear scalar equation (3.6), we obtain the solution of the system (3.5) from t =? 1 q () : ) 9

Now we apply Theorem 3.1. For any q 1 L (), q (L ()) N N such that we have the estimate [q 1tr"(v) + q "(v) + fv] dx + J(u ; u )? J(u; u) + " k Taking v = u? g 1 in (.1), we obtain = + (tr"(u )) + ("(u ) )? fu g v ds = ; 8 v V?1 (3.7) # dx? g u ds [q1tr"(g 1 ) + q "(g 1 ) + fg 1 ] dx + g g 1 ds 1 k jq 1j + q (q)? ( (q) ) dx (3.8) [ktr"(u )tr"(u? g 1 ) + "(u ) "(u? g 1 )] dx f(u? g 1 ) dx + Using the above relation in (3.8), we then have g (u? g 1 ) ds: J(u ; u )? J(u; u) + "? k (tr"(u )) + jq 1j k + (q 1 + ktr"(u ))tr"(g 1 ) + (q + "(u )) "(g 1 ) dx h ("(u ) ) "(u ) + q (q)? ( (q) ) i dx (3.9) # The inequality (3.9) holds for any auxiliary functions q 1 L (), q (L ()) N N satisfying the constraint (3.7). In particular, from (.1), q 1 =?k tr"(u ); q =? "(u ) (3.1) is an admissible pair of auxiliary functions. With the choice (3.1), we derive the following inequality from (3.9), J(u ; u )? J(u; u) h ("(u ) )? ( (? "(u )) ) i dx ("(u ) + (? "(u ))) "(u ) dx (3.11) Next, we derive a lower bound for the dierence of the potential energies. We have J(u ; u )? J(u; u) 1

= J(u ; u )? J(u; u)? < J (u; u); (u? u; u? u) > = " k (tr"(u ))? k (tr"(u))? k tr"(u )tr"(u? u) dx h + ("(u ) )? ("(u) ) ("(u ) ) "(u) "(u? u) i dx = k ktr"(u? u)k L () + ( 1 d dt (("(u) + t"(u? u)) ) dt ("(u ) )"(u) "(u? u) # ) dx where = k ktr"(u? u)k L () 1 h + (("(u) + t"(u? u)) ) ("(u) + t"(u? u)) "(u? u)? ("(u) ) "(u) "(u? u) i dt dx = k ktr"(u? u)k L () + 1 1 h(; t; u; u ) d dt dx h(; t; u; u ) = @ h ("(u) + t "(u? u)) ("(u) + t "(u? u)) "(u? u) i @ = t ("(u) + t "(u? u)) [("(u) + t "(u? u)) "(u? u)] + t ("(u) + t "(u? u)) "(u? u) If (("(u) + t "(u? u)) ), then h(; t; u; u ) t ("(u) + t "(u? u)) "(u? u) t "(u? u) by using the assumption (.3). If (("(u) + t "(u? u)) ) <, then h(; t; u; u ) t ("(u) + t "(u? u)) ("(u) + t "(u? u)) "(u? u) + t ("(u) + t "(u? u)) "(u? u) t "(u? u) by using the assumption (.8). So h(; t; u; u ) t min(; ) "(u? u) 11

and J(u ; u )? J(u; u) k ktr"(u? u)k L () + min(; ) k"(u? u)k L () (3.1) In conclusion, we have proved a-posteriori bounds for the error between the displacement eld of a nonlinear Hencky material and that of a linear elastic material. Theorem 3.4 Under the assumptions (.3), (.4) and (.8), we have, for the error in the total potential energy E(u )? E(u) + "? k (tr"(u )) + jq 1j k + (q 1 + ktr"(u ))tr"(g 1 ) + (q + "(u )) "(g 1 ) dx h ("(u ) ) "(u ) + q (q)? ( (q) ) i dx (3.13) and for the error in energy norm k ktr"(u? u)k L () + min(; ) k"(u? u)k L () (3.14) + "? k (tr"(u )) + jq 1j k + (q 1 + ktr"(u ))tr"(g 1 ) + (q + "(u )) "(g 1 ) dx h ("(u ) ) "(u ) + q (q)? ( (q) ) i dx (3.15) # # for any auxiliary functions q 1 L (), q (L ()) N N particular, with the choice (3.1), we have subject to the constraint (3.7). In and E(u )? E(u) h ("(u ) )? ( (? "(u )) ) i dx ("(u ) + (? "(u ))) "(u ) dx (3.16) k ktr"(u? u)k L () + min(; ) k"(u? u)k L () (3.17) h ("(u ) )? ( (? "(u )) ) i dx ("(u ) + (? "(u ))) "(u ) dx (3.18) 1

4 Numerical examples To make specic computations, we introduce a family of nonlinear Hencky materials whose governing function in the stored energy function M is of the form () = 8 >< >: ; if + 1? 1?? (1? )(1? ) ; if > (4.1) with parameters (; 1), 6= ((:5? )=(1? ); 1). A material with the governing function (4.1) is linear on the region where the size of the strain deviator tensor is small (bounded by p ). On the region where the size of the strain deviator tensor is large, the slope of the governing function is close to (; 1). When approaches, the material on the nonlinear elasticity region becomes softer. The parameter is introduced to ensure C 1. We have () = 8 < : 1; if + (1? ) 1??1 ; if > (4.) So the assumption (.3) is satised. With the choice (4.1), () is only piecewisely twice dierentiable, so it does not satisfy the assumption (.8). Nevertheless, we will show all the results presented in Sections and 3 still hold. First, let us verify that the corresponding stored energy function is strictly convex, which implies the conclusions of Theorem.. If " <, then from (.1) and (4.1), M (") = k (tr) + = (tr) + : If " >, again from (.1) and (4.1), M (") = k (tr)? 4 (1? ) (1? ) 1? (" )? (" ) + h + (1? ) 1? (" )?1i k (tr)? 4 (1? ) (1? ) 1? (" )?1 + h + (1? ) 1? (" )?1 i = k (tr) + h + (1? ) (? 1) 1? (" )?1i k (tr) + [ + (1? ) (? 1)] = k (tr) + 4 (1? )? :5? 1? = + 4 N (1? ) (1? ) (tr) + 4 (1? ) 4 (1? )? :5?! 1?!? :5?! 1? 13

! where the coecient 4 (1? )? :5? is a positive constant by the restrictions on 1? and. Therefore, M(") is strictly convex. Next, we show that the equation (3.6) is uniquely solvable. Obviously, we only need to prove the function f() = ( ()) is strictly increasing when >. In that case, and If 1=, then If < 1=, then f() = h + (1? ) 1??1 i f () = h + (1? ) 1??1 i h + (1? ) (? 1) 1??1 i : f () h + (1? ) 1??1 i > : f () h + (1? ) 1??1 i [ + (1? ) (? 1)] = h + (1? ) 1? i :5??1 (1? )? 1? > : Hence, f() is a strictly increasing function. Finally, we modify the inequality (3.1), which is inadequate when the parameter is close to zero. As in the last section, we have where J(u ; u )? J(u; u) = k ktr"(u? u)k L () + 1 1! h(; t; u; u ) d dt dx h(; t; u; u ) = @ h ("(u) + t "(u? u)) ("(u) + t "(u? u)) "(u? u) i @ Denote = "(u) + t "(u? u). When <, we have When, we have h(; t; u; u ) h(; t; u; u ) = t "(u? u) : = ( ) t ( "(u? u)) + ( ) t "(u? u) = (1? ) (? 1) 1? ( )? t ( "(u? u)) + t h + (1? ) 1? ( )?1i "(u? u) t (1? ) (? 1) 1? ( )?1 "(u? u) + t h + (1? ) 1? ( )?1i "(u? u) t h + (1? ) (? 1) 1? ( )?1 i "(u? u) : 14

If 1=, then h(; t; u; u ) t "(u? u) : If < 1=, then h(; t; u; u ) t [ + (1? ) (? 1)] "(u? u) = t (1? )? :5? 1?! "(u? u) : Now, we dene l (u; u ) = fx j "(u(x)) ; "(u (x)) g the intersection of the linear ealsticity region of u and that of u, and c = 8 >< >: ; if 1 ; (1? )? :5? 1?! ; if < 1 : Since on l (u; u ), = ("(u) + t "(u? u)), we have the following relation J(u ; u )? J(u; u ) k ktr"(u? u)k L () + k"(u? u)k L ( l (u;u )) + c k"(u? u)k L (n l (u;u )): (4.3) In conclusion, we have Theorem 4.1 For a nonlinear Hencky material dened by the governing function (4.1), the following a-posteriori error estimates for the linearization of the material hold. For the error in the total potential energy E(u )? E(u) h ("(u ) )? ( (? "(u )) ) i dx and for the error in energy norm ("(u ) + (? "(u ))) "(u ) dx (4.4) k ktr"(u? u)k L () + k"(u? u)k L ( l (u;u )) + c k"(u? u)k L (n l (u;u )) h ("(u ) )? ( (? "(u )) ) i dx ("(u ) + (? "(u ))) "(u ) dx (4.5) 15

As for numerical experiments, we consider elasticity problems on a plane corner domain (N = ). Near the reentrant corner O, the domain coincides with a cone: C! = f(r; ) j r > ;?!= < <!=g: Assume the plane elastic body is loaded by boundary tractions only and, in the neighborhood of the corner, has stress-free boundaries. We have the following singularity expansion for the solution u of the linear elasticity problem ([11]): u = K I p r I I() + K II p r II II() + ~u (4.6) G G in which, G is the modulus of rigidity, K I and K II are Mode 1 and Mode stress-intensity factors, I() = II() = @ (? Q I( I + 1)) cos I? I cos( I ) ( + Q I ( I + 1)) sin I + I sin( I ) @ (? Q II( II + 1)) sin II? II sin( II )?( + Q II ( II + 1)) cos II? II cos( II ) correspond to the most singular Mode 1 and Mode displacement components, where Q I =? cos(( I? 1)!=) cos(( I + 1)!=) depends only on Poisson's ratio, if! < ; Q I =? 1? I sin(( I? 1)!=) 1 + I sin(( I + 1)!=) Q II =? sin(( II? 1)!=) sin(( II + 1)!=) ; = 3? 4 for plane strain; = 3? for plane stress; 1 + if! = ; and ~u is a smooth remainder such that "(~u ) is bounded when x! O. Following is a table for values of singular exponents I and II for some typical angles (the angles are in degrees). 1 A 1 A! I II 36.5.5 7.544484.9859 4.615731 1.148913 5.673583 1.386 1.751975 1.48581 16

For the convenience of later use, let us denote: It is easy to verify following expressions: u I = r I I() u II = r II II() " 11 (u I ) = I r I?1 f(? Q I ( I + 1)? 1) cos( I? 1)? ( I? 1) cos( I? 3)g " (u I ) = I r I?1 f( + Q I ( I + 1)? 1) cos( I? 1) + ( I? 1) cos( I? 3)g " 1 (u I ) = I r I?1 fq I ( I + 1) sin( I? 1) + ( I? 1) sin( I? 3)g " 11 (u II ) = II r II?1 f(? Q II ( II + 1)? 1) sin( II? 1)? ( II? 1) sin( II? 3)g " (u II ) = II r II?1 f( + Q II ( II + 1)? 1) sin( II? 1) + ( II? 1) sin( II? 3)g " 1 (u II ) =? II r II?1 fq II ( II + 1) cos( II? 1)? ( II? 1) cos( II? 3)g Note that when the angle is 36 or 7 degree, both "(u I ) and "(u II ) are unbounded around the corner, whereas when the angle is 4, 5 or 1 degree, only the most singular Mode 1 strain components are unbounded around the corner. Experiment 4. We consider a nonlinear Hencky elasticity problem as shown in Figure 1, where homogeneous natural boundary conditions are not written explicitly. The part of the boundary represented by thick line segments is xed. Assume the stored energy function is given by (.) and (4.1). Owing to the symmetry of the problem, we consider only a quarter problem, shown in Figure, where again homogeneous natural boundary conditions are not written explicitly. We are going to use estimates (4.4) and (4.5) to examine how much error we are making if we take the material as a linear one. Let a = 5, b = 4, c = 4, h = :5. By using MSC/PROBE, a software package for solving elliptic problems by h-p-version nite element method, we obtain that for the L-shape singularity ( = j n j), K I = 1:51 (); K II = 59:9 (): The strain energy is: where, E is the Young's modulus. :541 1 4 () E 17

6 n?????????? 6? h c a - a - b -? Figure 1. A nonlinear elasticity problem u; v =????? u = u = Figure. The working problem u; v = Around the L-shape corner, "(u ) is mainly determined by the rst two terms in the expansion (4.6), since strain components for ~u are negligible. Hence, we use the rst two terms to approxiamtely represent the linear solution u for producing estimates for error in the (nonlinear) energy between the linear solution u and the nonlinear solution u, the corresponding estimates from (4.4) and/or (4.5) being denoted by Ested. For typical materials, the Young's modulus is of order 1 6 1 7 psi, and the strain yield value is of the order 1?4. In the following, we take E = 1 6 psi; = :3; = 1?4 then the linear energy of the linear solution is :541 1? (). estimates for various and. We compute the 18

Linear Energy Ested 5 :541(+).9.8 :388(?5).5 :961(?5). :171(?4).5.8 :86(?4).5 :433(?3). :97(?3).3.8 :194(?3).5 :1414(?) :1(?).51 :8618(+).51 :864(+).51 :864(+) Experiment 4.3 We consider the Edge-cracked Panel Problem from the paper [11]. By using a mesh ner than that presented in the paper, we obtain that the linear energy of the linear solution is: 1:754 E and the Mode 1 and Mode stress intensity factors are: K I = :599778 ; K II =?:9515 : Again, we take E = 1 6 psi; = :3; = 1?4 : Linear Energy Ested 1 1.754.9.8 :491(?7).5 :1795(?6). :33(?6).5.8 :1696(?5).5 :877(?5). :155(?4).3.8 :4169(?5).5 :638(?4) :1(?3).51 :1614(+).51 :1615(+) 19

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