Loading [MathJax]/jax/output/SVG/jax.js
Research article Special Issues

A dissipation model for concrete based on an enhanced Timoshenko beam

  • A novel Timoshenko beam model enriched to account for dissipation in cement-based materials was presented in this paper. The model introduced a new variable representing the relative sliding inside microcracks within the material. In the paper, the microcrack density was not supposed to increase, assuming a small deformation regime that implied no damage growth. The model utilized an expanded version of the principle of virtual work whose contributions came from external forces, internal elastic forces, and dissipation due to the microcrack's microstructure. The elastic energy included terms related to microcrack sliding and micro-macro interactions, accounting for nonlinearity in the material behavior. Numerical simulations, conducted using the finite element method, evaluated the mechanical properties of cement-based materials under three-point flexural tests and compression tests. These tests enabled the assessment of the material dissipative behavior under cyclic loading. Results showed dissipated energy cycles and mechanical responses influenced by the microcrack mechanics. Additionally, a parametric study, varying the friction force amplitude, revealed its impact on dissipated energy. The study highlighted a non-monotonic relationship between friction force amplitude and dissipated energy, with an optimal value maximizing dissipation. Overall, the model provided insights into the mechanics of cement-based materials, particularly regarding dissipation, which was essential for understanding their behavior in structural applications.

    Citation: Giuliano Aretusi, Christian Cardillo, Larry Murcia Terranova, Ewa Bednarczyk. A dissipation model for concrete based on an enhanced Timoshenko beam[J]. Networks and Heterogeneous Media, 2024, 19(2): 700-723. doi: 10.3934/nhm.2024031

    Related Papers:

    [1] Jiafan Zhang . On the distribution of primitive roots and Lehmer numbers. Electronic Research Archive, 2023, 31(11): 6913-6927. doi: 10.3934/era.2023350
    [2] Yang Gao, Qingzhong Ji . On the inverse stability of zn+c. Electronic Research Archive, 2025, 33(3): 1414-1428. doi: 10.3934/era.2025066
    [3] J. Bravo-Olivares, E. Fernández-Cara, E. Notte-Cuello, M.A. Rojas-Medar . Regularity criteria for 3D MHD flows in terms of spectral components. Electronic Research Archive, 2022, 30(9): 3238-3248. doi: 10.3934/era.2022164
    [4] Zhefeng Xu, Xiaoying Liu, Luyao Chen . Hybrid mean value involving some two-term exponential sums and fourth Gauss sums. Electronic Research Archive, 2025, 33(3): 1510-1522. doi: 10.3934/era.2025071
    [5] Jorge Garcia Villeda . A computable formula for the class number of the imaginary quadratic field Q(p), p=4n1. Electronic Research Archive, 2021, 29(6): 3853-3865. doi: 10.3934/era.2021065
    [6] Li Wang, Yuanyuan Meng . Generalized polynomial exponential sums and their fourth power mean. Electronic Research Archive, 2023, 31(7): 4313-4323. doi: 10.3934/era.2023220
    [7] Qingjie Chai, Hanyu Wei . The binomial sums for four types of polynomials involving floor and ceiling functions. Electronic Research Archive, 2025, 33(3): 1384-1397. doi: 10.3934/era.2025064
    [8] Hai-Liang Wu, Li-Yuan Wang . Permutations involving squares in finite fields. Electronic Research Archive, 2022, 30(6): 2109-2120. doi: 10.3934/era.2022106
    [9] Li Rui, Nilanjan Bag . Fourth power mean values of one kind special Kloosterman's sum. Electronic Research Archive, 2023, 31(10): 6445-6453. doi: 10.3934/era.2023326
    [10] Hongliang Chang, Yin Chen, Runxuan Zhang . A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29(3): 2457-2473. doi: 10.3934/era.2020124
  • A novel Timoshenko beam model enriched to account for dissipation in cement-based materials was presented in this paper. The model introduced a new variable representing the relative sliding inside microcracks within the material. In the paper, the microcrack density was not supposed to increase, assuming a small deformation regime that implied no damage growth. The model utilized an expanded version of the principle of virtual work whose contributions came from external forces, internal elastic forces, and dissipation due to the microcrack's microstructure. The elastic energy included terms related to microcrack sliding and micro-macro interactions, accounting for nonlinearity in the material behavior. Numerical simulations, conducted using the finite element method, evaluated the mechanical properties of cement-based materials under three-point flexural tests and compression tests. These tests enabled the assessment of the material dissipative behavior under cyclic loading. Results showed dissipated energy cycles and mechanical responses influenced by the microcrack mechanics. Additionally, a parametric study, varying the friction force amplitude, revealed its impact on dissipated energy. The study highlighted a non-monotonic relationship between friction force amplitude and dissipated energy, with an optimal value maximizing dissipation. Overall, the model provided insights into the mechanics of cement-based materials, particularly regarding dissipation, which was essential for understanding their behavior in structural applications.



    Let Fq be the finite field of q elements with characteristic p, where q=pr, p is a prime number. Let Fq=Fq{0} and Z+ denote the set of positive integers. Let sZ+ and bFq. Let f(x1,,xs) be a diagonal polynomial over Fq of the following form

    f(x1,,xs)=a1xm11+a2xm22++asxmss,

    where aiFq, miZ+, i=1,,s. Denote by Nq(f=b) the number of Fq-rational points on the affine hypersurface f=b, namely,

    Nq(f=b)=#{(x1,,xs)As(Fq)f(x1,,xs)=b}.

    In 1949, Hua and Vandiver [1] and Weil [2] independently obtained the formula of Nq(f=b) in terms of character sum as follows

    Nq(f=b)=qs1+ψ1(a11)ψs(ass)J0q(ψ1,,ψs), (1.1)

    where the sum is taken over all s multiplicative characters of Fq that satisfy ψmii=ε, ψiε, i=1,,s and ψ1ψs=ε. Here ε is the trivial multiplicative character of Fq, and J0q(ψ1,,ψs) is the Jacobi sum over Fq defined by

    J0q(ψ1,,ψs)=c1++cs=0,ciFqψ1(c1)ψs(cs).

    Though the explicit formula for Nq(f=b) are difficult to obtain in general, it has been studied extensively because of their theoretical importance as well as their applications in cryptology and coding theory; see[3,4,5,6,7,8,9]. In this paper, we use the Jacobi sums, Gauss sums and the results of quadratic form to deduce the formula of the number of Fq2-rational points on a class of hypersurfaces over Fq2 under certain conditions. The main result of this paper can be stated as

    Theorem 1.1. Let q=2r with rZ+ and Fq2 be the finite field of q2 elements. Let f(X)=a1xm11+a2xm22++asxmss, g(Y)=y1y2+y3y4++yn1yn+y2n2t1+ +y2n3+y2n1+bty2n2t++b1y2n2+b0y2n, and l(X,Y)=f(X)+g(Y), where ai,bjFq2, mi1, (mi,mk)=1, ik, mi|(q+1), miZ+, 2|n, n>2, 0tn22, TrFq2/F2(bj)=1 for i,k=1,,s and j=0,1,,t. For hFq2, we have

    (1) If h=0, then

    Nq2(l(X,Y)=0)=q2(s+n1)+γFq2(si=1((γai)mimi1)(qs+2n3+(1)tqs+n3)).

    (2) If hFq2, then

    Nq2(l(X,Y)=h)=q2(s+n1)+(qs+2n3+(1)t+1(q21)qs+n3)si=1((hai)mimi1)+γFq2{h}[si=1((γai)mimi1)(q2n+s3+(1)tqn+s3)].

    Here,

    (γai)mi={1,ifγaiisaresidueofordermi,0,otherwise.

    To prove Theorem 1.1, we need the lemmas and theorems below which are related to the Jacobi sums and Gauss sums.

    Definition 2.1. Let χ be an additive character and ψ a multiplicative character of Fq. The Gauss sum Gq(ψ,χ) in Fq is defined by

    Gq(ψ,χ)=xFqψ(x)χ(x).

    In particular, if χ is the canonical additive character, i.e., χ(x)=e2πiTrFq/Fp(x)/p where TrFq/Fp(y)=y+yp++ypr1 is the absolute trace of y from Fq to Fp, we simply write Gq(ψ):=Gq(ψ,χ).

    Let ψ be a multiplicative character of Fq which is defined for all nonzero elements of Fq. We extend the definition of ψ by setting ψ(0)=0 if ψε and ε(0)=1.

    Definition 2.2. Let ψ1,,ψs be s multiplicative characters of Fq. Then, Jq(ψ1,,ψs) is the Jacobi sum over Fq defined by

    Jq(ψ1,,ψs)=c1++cs=1,ciFqψ1(c1)ψs(cs).

    The Jacobi sums Jq(ψ1,,ψs) as well as the sums J0q(ψ1,,ψs) can be evaluated easily in case some of the multiplicative characters ψi are trivial.

    Lemma 2.3. ([10,Theorem 5.19,p. 206]) If the multiplicative characters ψ1,,ψs of Fq are trivial, then

    Jq(ψ1,,ψs)=J0q(ψ1,,ψs)=qs1.

    If some, but not all, of the ψi are trivial, then

    Jq(ψ1,,ψs)=J0q(ψ1,,ψs)=0.

    Lemma 2.4. ([10,Theorem 5.20,p. 206]) If ψ1,,ψs are multiplicative characters of Fq with ψs nontrivial, then

    J0q(ψ1,,ψs)=0

    if ψ1ψs is nontrivial and

    J0q(ψ1,,ψs)=ψs(1)(q1)Jq(ψ1,,ψs1)

    if ψ1ψs is trivial.

    If all ψi are nontrivial, there exists an important connection between Jacobi sums and Gauss sums.

    Lemma 2.5. ([10,Theorem 5.21,p. 207]) If ψ1,,ψs are nontrivial multiplicative characters of Fq and χ is a nontrivial additive character of Fq, then

    Jq(ψ1,,ψs)=Gq(ψ1,χ)Gq(ψs,χ)Gq(ψ1ψs,χ)

    if ψ1ψs is nontrivial and

    Jq(ψ1,,ψs)=ψs(1)Jq(ψ1,,ψs1)=1qGq(ψ1,χ)Gq(ψs,χ)

    if ψ1ψs is trivial.

    We turn to another special formula for Gauss sums which applies to a wider range of multiplicative characters but needs a restriction on the underlying field.

    Lemma 2.6. ([10,Theorem 5.16,p. 202]) Let q be a prime power, let ψ be a nontrivial multiplicative character of Fq2 of order m dividing q+1. Then

    Gq2(ψ)={q,ifmoddorq+1meven,q,ifmevenandq+1modd.

    For hFq2, define v(h)=1 if hFq2 and v(0)=q21. The property of the function v(h) will be used in the later proofs.

    Lemma 2.7. ([10,Lemma 6.23,p. 281]) For any finite field Fq, we have

    cFqv(c)=0,

    for any bFq,

    c1++cm=bv(c1)v(ck)={0,1k<m,v(b)qm1,k=m,

    where the sum is over all c1,,cmFq with c1++cm=b.

    The quadratic forms have been studied intensively. A quadratic form f in n indeterminates is called nondegenerate if f is not equivalent to a quadratic form in fewer than n indeterminates. For any finite field Fq, two quadratic forms f and g over Fq are called equivalent if f can be transformed into g by means of a nonsingular linear substitution of indeterminates.

    Lemma 2.8. ([10,Theorem 6.30,p. 287]) Let fFq[x1,,xn], q even, be a nondegenerate quadratic form. If n is even, then f is either equivalent to

    x1x2+x3x4++xn1xn

    or to a quadratic form of the type

    x1x2+x3x4++xn1xn+x2n1+ax2n,

    where aFq satisfies TrFq/Fp(a)=1.

    Lemma 2.9. ([10,Corollary 3.79,p. 127]) Let aFq and let p be the characteristic of Fq, the trinomial xpxa is irreducible in Fq if and only if TrFq/Fp(a)0.

    Lemma 2.10. ([10,Lemma 6.31,p. 288]) For even q, let aFq with TrFq/Fp(a)=1 and bFq. Then

    Nq(x21+x1x2+ax22=b)=qv(b).

    Lemma 2.11. ([10,Theorem 6.32,p. 288]) Let Fq be a finite field with q even and let bFq. Then for even n, the number of solutions of the equation

    x1x2+x3x4++xn1xn=b

    in Fnq is qn1+v(b)q(n2)/2. For even n and aFq with TrFq/Fp(a)=1, the number of solutions of the equation

    x1x2+x3x4++xn1xn+x2n1+ax2n=b

    in Fnq is qn1v(b)q(n2)/2.

    Lemma 2.12. Let q=2r and hFq2. Let g(Y)Fq2[y1,y2,,yn] be a polynomial of the form

    g(Y)=y1y2+y3y4++yn1yn+y2n2t1++y2n3+y2n1+bty2n2t++b1y2n2+b0y2n,

    where bjFq2, 2|n, n>2, 0tn22, TrFq2/F2(bj)=1, j=0,1,,t. Then

    Nq2(g(Y)=h)=q2(n1)+(1)t+1qn2v(h). (2.1)

    Proof. We provide two proofs here. The first proof is as follows. Let q1=q2. Then by Lemmas 2.7 and 2.10, the number of solutions of g(Y)=h in Fq2 can be deduced as

    Nq2(g(Y)=h)=c1+c2++ct+2=hNq2(y1y2+y3y4++yn2t3yn2t2=c1)Nq2(yn2t1yn2t+y2n2t1+bty2n2t=c2)Nq2(yn1yn+y2n1+b0y2n=ct+2)=c1+c2++ct+2=h(qn2t31+v(c1)q(n2t4)/21)(q1v(c2))(q1v(ct+2))=c1+c2++ct+2=h(qn2t21+v(c1)q(n2t2)/21v(c2)qn2t31v(c1)v(c2)q(n2t4)/21)(q1v(c3))(q1v(ct+2))=c1+c2++ct+2=h(qnt21+v(c1)q(n2)/21v(c2)qnt31++(1)t+1v(c1)v(c2)v(ct+2)q(n2t4)/21)=qn11+q(n2)/21c1Fq2v(c1)++(1)t+1c1+c2++ct+2=hv(c1)v(c2)v(ct+2)q(n2t4)/21. (2.2)

    By Lamma 2.7 and (2.2), we have

    Nq2(g(Y)=h)=qn11+(1)t+1v(h)q(n2)/21=q2(n1)+(1)t+1v(h)qn2.

    Next we give the second proof. Note that if f and g are equivalent, then for any bFq2 the equation f(x1,,xn)=b and g(x1,,xn)=b have the same number of solutions in Fq2. So we can get the number of solutions of g(Y)=h for hFq2 by means of a nonsingular linear substitution of indeterminates.

    Let k(X)Fq2[x1,x2,x3,x4] and k(X)=x1x2+x21+Ax22+x3x4+x23+Bx24, where TrFq2/F2(A)=TrFq2/F2(B)=1. We first show that k(x) is equivalent to x1x2+x3x4.

    Let x3=y1+y3 and xi=yi for i3, then k(X) is equivalent to y1y2+y1y4+y3y4+Ay22+y23+By24.

    Let y2=z2+z4 and yi=zi for i2, then k(X) is equivalent to z1z2+z3z4+Az22+z23+Az24+Bz24.

    Let z1=α1+Aα2 and zi=αi for i1, then k(X) is equivalent to α1α2+α23+α3α4+(A+B)α24.

    Since TrFq2/F2(A+B)=0, we have α23+α3α4+(A+B)α24 is reducible by Lemma 2.9. Then k(X) is equivalent to x1x2+x3x4. It follows that if t is odd, then g(Y) is equivalent to x1x2+x3x4++xn1xn, and if t is even, then g(Y) is equivalent to x1x2+x3x4++xn1xn+x2n1+ax2n with TrFq2/F2(a)=1. By Lemma 2.11, we get the desired result.

    From (1.1), we know that the formula for the number of solutions of f(X)=0 over Fq2 is

    Nq2(f(X)=0)=q2(s1)+d11j1=1ds1js=1¯ψj11(a1)¯ψjss(as)J0q2(ψj11,,ψjss),

    where di=(mi,q21) and ψi is a multiplicative character of Fq2 of order di. Since mi|q+1, we have di=mi. Let H={(j1,,js)1ji<mi, 1is}. It follows that ψj11ψjss is nontrivial for any (j1,,js)H as (mi,mj)=1. By Lemma 2, we have J0q2(ψj11,,ψjss)=0 and hence Nq2(f(X)=0)=q2(s1).

    Let Nq2(f(X)=c) denote the number of solutions of the equation f(X)=c over Fq2 with cFq2. Let V={(j1,,js)|0ji<mi,1is}. Then

    Nq2(f(X)=c)=γ1++γs=cNq2(a1xm11=γ1)Nq2(asxmss=γs)=γ1++γs=cm11j1=0ψj11(γ1a1)ms1js=0ψjss(γsas).

    Since ψi is a multiplicative character of Fq2 of order mi, we have

    Nq2(f(X)=c)=γ1c++γsc=1(j1,,js)Vψj11(γ1c)ψj11(ca1)ψjss(γsc)ψjss(cas)=(j1,,js)Vψj11(ca1)ψjss(cas)γ1c++γsc=1ψj11(γ1c)ψjss(γsc)=(j1,,js)Vψj11(ca1)ψjss(cas)Jq2(ψj11,,ψjss).

    By Lemma 2.3,

    Nq2(f(X)=c)=q2(s1)+(j1,,js)Hψj11(ca1)ψjss(cas)Jq2(ψj11,,ψjss).

    By Lemma 2.5,

    Jq2(ψj11,,ψjss)=Gq2(ψj11)Gq2(ψjss)Gq2(ψj11ψjss).

    Since mi|q+1 and 2mi, by Lemma 2.6, we have

    Gq2(ψj11)==Gq2(ψjss)=Gq2(ψj11ψjss)=q.

    Then

    Nq2(f(X)=c)=q2(s1)+qs1m11j1=1ψj11(ca1)ms1js=1ψjss(cas)=q2(s1)+qs1(m11j1=0ψj11(ca1)1)(ms1js=0ψjss(cas)1).

    It follows that

    Nq2(f(X)=c)=q2(s1)+qs1si=1((cai)mimi1), (3.1)

    where

    (cai)mi={1,ifcai is a residue of ordermi,0,otherwise.

    For a given hFq2. We discuss the two cases according to whether h is zero or not.

    Case 1: h=0. If f(X)=0, then g(Y)=0; if f(X)0, then g(Y)0. Then

    Nq2(l(X,Y)=0)=c1+c2=0Nq2(f(X)=c1)Nq2(g(Y)=c2)=q2(s1)(q2(n1)+(1)t+1(q21)qn2)+c1+c2=0c1,c2Fq2Nq2(f(X)=c1)Nq2(g(Y)=c2). (3.2)

    By Lemma 2.12, (3.1) and (3.2), we have

    Nq2(l(X,Y)=0)=q2(s+n2)+(1)t+1q2(s1)+hn(1)t+1q2(s2)+n+c1Fq2[q2(s+n2)(1)t+1q2(s2)+n+si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)]=q2(s+n2)+(1)t+1q2(s1)+n(1)t+1q2(s2)+n+q2(s+n1)(1)t+1q2(s1)+nq2(s+n2)+(1)t+1q2(s2)+n+c1Fq2[si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)]=q2(s+n1)+c1Fq2[si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)]. (3.3)

    Case 2: hFq2. If f(X)=h, then g(Y)=0; if f(X)=0, then g(Y)=h; if f(X){0,h}, then g(Y){0,h}. So we have

    Nq2(l(X,Y))=h)=c1+c2=hNq2(f(X)=c1)Nq2(g(Y)=c2)=Nq2(f(X)=0)Nq2(g(Y)=h)+Nq2(f(X)=h)Nq2(g(Y)=0)+c1+c2=hc1,c2Fq2{h}Nq2(f(X)=c1)Nq2(g(Y)=c2). (3.4)

    By Lemma 2.12, (3.1) and (3.4),

    Nq2(l(X,Y)=h)=2q2(s+n2)+(1)t+1q2s+n2(1)t+12q2s+n4+(qs+2n3+(1)t+1(q21)qs+n3)si=1((hai)mimi1)+c1Fq2{h}[q2(s+n2)(1)t+1q2s+n4+si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)].

    It follows that

    Nq2(l(X,Y)=h)=2q2(s+n2)+(1)t+1q2s+n2(1)t+12q2s+n4+(qs+2n3+(1)t+1(q21)qs+n3)si=1((hai)mimi1)+c1Fq2{h}[q2(s+n2)(1)t+1q2s+n4+si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)]=q2(s+n1)+(qs+2n3+(1)t+1(q21)qs+n3)si=1((hai)mimi1)+c1Fq2{h}[si=1((c1ai)mimi1)(q2n+s3+(1)tqn+s3)]. (3.5)

    By (3.3) and (3.5), we get the desired result. The proof of Theorem 1.1 is complete.

    There is a direct corollary of Theorem 1.1 and we omit its proof.

    Corollary 4.1. Under the conditions of Theorem 1.1, if a1==as=hFq2, then we have

    Nq2(l(X,Y)=h)=q2(s+n1)+(qs+2n3+(1)t+1(q21)qs+n3)si=1(mi1)+γFq2{h}[si=1((γh)mimi1)(q2n+s3+(1)tqn+s3)],

    where

    (γh)mi={1,ifγhisaresidueofordermi,0,otherwise.

    Finally, we give two examples to conclude the paper.

    Example 4.2. Let F210=α=F2[x]/(x10+x3+1) where α is a root of x10+x3+1. Suppose l(X,Y)=α33x31+x112+y23+α10y24+y1y2+y3y4. Clearly, TrF210/F2(α10)=1, m1=3, m2=11, s=2, n=4, t=0, a2=1. By Theorem 1.1, we have

    N210(l(X,Y)=0)=10245+(327+323)×20=1126587102265344.

    Example 4.3. Let F212=β=F2[x]/(x12+x6+x4+x+1) where β is a root of x12+x6+x4+x+1. Suppose l(X,Y)=x51+x132+y23+β10y24+y1y2+y3y4. Clearly, TrF212/F2(β10)=1, m1=5, m2=13, s=2, n=4, t=0, a1=a2=1. By Corollary 1, we have

    N212(l(X,Y)=1)=25×12+(647643×4095)×48=1153132559312355328.

    This work was jointly supported by the Natural Science Foundation of Fujian Province, China under Grant No. 2022J02046, Fujian Key Laboratory of Granular Computing and Applications (Minnan Normal University), Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics.

    The authors declare there is no conflicts of interest.



    [1] A. Misra, Stabilization characteristics of clays using class C fly ash, Transp. Res. Rec., 1611 (1998), 46–54. https://doi.org/10.3141/1611-06 doi: 10.3141/1611-06
    [2] D. A. S. Sclofani, L. Contrafatto, Experimental behaviour of polyvinyl-alcohol modified concrete, Adv. Mater. Res., 687 (2013), 155–160. https://doi.org/10.4028/www.scientific.net/AMR.687.155 doi: 10.4028/www.scientific.net/AMR.687.155
    [3] L. Contrafatto, 8-Volcanic ash, in Sustainable Concrete Made with Ashes and Dust from Different Sources, (eds. R. Siddique and R. Belarbi), Elsevier, (2022), 331–418. https://doi.org/10.1016/B978-0-12-824050-2.00011-5
    [4] L. Contrafatto, R. Cosenza, R. Barbagallo, S. Ognibene, Use of recycled aggregates in road sub-base construction and concrete manufacturing, Ann. Geophys., 61 (2018), SE223. https://doi.org/10.4401/ag-7785 doi: 10.4401/ag-7785
    [5] G. Goel, P. Sachdeva, A. K. Chaudhary, Y. Singh, The use of nanomaterials in concrete: A review, Mater. Today Proc., 69 (2022), 365–371. https://doi.org/10.1016/j.matpr.2022.09.051 doi: 10.1016/j.matpr.2022.09.051
    [6] P. Franciosi, M. Spagnuolo, O. U. Salman, Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates, Continuum Mech. Thermodyn., 31 (2019), 101–132. https://doi.org/10.1007/s00161-018-0668-0 doi: 10.1007/s00161-018-0668-0
    [7] M. Spagnuolo, Symmetrization of mechanical response in fibrous metamaterials through micro-shear deformability, Symmetry, 14 (2022), 2660. https://doi.org/10.3390/sym14122660 doi: 10.3390/sym14122660
    [8] M. Spagnuolo, P. Franciosi, F. Dell'Isola, A Green operator-based elastic modeling for two-phase pantographic-inspired bi-continuous materials, Int. J. Solids Struct., 188 (2020), 282–308. https://doi.org/10.1016/j.ijsolstr.2019.10.018 doi: 10.1016/j.ijsolstr.2019.10.018
    [9] I. Giorgio, N. L. Rizzi, E. Turco, Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis, Proc. R. Soc. A, 473 (2017), 1–21. https://doi.org/10.1098/rspa.2017.0636 doi: 10.1098/rspa.2017.0636
    [10] E. Turco, M. Golaszewski, I. Giorgio, F. D'Annibale, Pantographic lattices with non-orthogonal fibres: Experiments and their numerical simulations, Composites, Part B, 118 (2017), 1–14. https://doi.org/10.1016/j.compositesb.2017.02.039 doi: 10.1016/j.compositesb.2017.02.039
    [11] E. Turco, I. Giorgio, A. Misra, F. Dell'Isola, King post truss as a motif for internal structure of (meta) material with controlled elastic properties, R. Soc. Open Sci., 4 (2017), 171153. https://doi.org/10.1098/rsos.171153 doi: 10.1098/rsos.171153
    [12] P. P. Abhilash, D. K. Nayak, B. Sangoju, R. Kumar, V. Kumar, Effect of nano-silica in concrete; a review, Constr. Build. Mater., 278 (2021), 122347. https://doi.org/10.1016/j.conbuildmat.2021.122347 doi: 10.1016/j.conbuildmat.2021.122347
    [13] N. De Belie, E. Gruyaert, A. Al‐Tabbaa, P. Antonaci, C. Baera, D. Bajare, et al., A review of self-healing concrete for damage management of structures, Adv. Mater. Interfaces, 5 (2018), 1800074. https://doi.org/10.1002/admi.201800074 doi: 10.1002/admi.201800074
    [14] S. Sangadji, E. Schlangen, Mimicking bone healing process to self repair concrete structure novel approach using porous network concrete, Procedia Eng., 54 (2013), 315–326. https://doi.org/10.1016/j.proeng.2013.03.029 doi: 10.1016/j.proeng.2013.03.029
    [15] A. Casalotti, F. D'annibale, G. Rosi, Multi-scale design of an architected composite structure with optimized graded properties, Composite Structures, 252 (2020), 112608. https://doi.org/10.1016/j.compstruct.2020.112608 doi: 10.1016/j.compstruct.2020.112608
    [16] T. Lekszycki, F. Dell'Isola, A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials, ZAMM Z. Angew. Math. Mech., 92 (2012), 426–444. https://doi.org/10.1002/zamm.201100082 doi: 10.1002/zamm.201100082
    [17] E. I. Bednarczyk, T. Lekszycki, W. Glinkowski, Effect of micro-cracks on the angiogenesis and osteophyte development during degenerative joint disease, Comput. Assisted Methods Eng. Sci., 24 (2018), 149–156. http://doi.org/10.24423/cames.191 doi: 10.24423/cames.191
    [18] I. Giorgio, F. Dell'Isola, U. Andreaus, F. Alzahrani, T. Hayat, T. Lekszycki, On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon, Biomech. Model. Mechanobiol., 18 (2019), 1639–1663. https://doi.org/10.1007/s10237-019-01166-w doi: 10.1007/s10237-019-01166-w
    [19] I. Giorgio, F. Dell'Isola, U. Andreaus, A. Misra, An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff's law, Biomech. Model. Mechanobiol., 22 (2023), 2135–2152. https://doi.org/10.1007/s10237-023-01755-w doi: 10.1007/s10237-023-01755-w
    [20] M. Rajczakowska, K. Habermehl-Cwirzen, H. Hedlund, A. Cwirzen, Autogenous self-healing: A better solution for concrete, J. Mater. Civ. Eng., 31 (2019), 03119001. https://doi.org/10.1061/(ASCE)MT.1943-5533.0002764 doi: 10.1061/(ASCE)MT.1943-5533.0002764
    [21] M. E. Espitia-Nery, D. E. Corredor-Pulido, P. A. Castaño-Oliveros, J. A. Rodríguez-Medina, Q. Y. Ordoñez-Bello, M. S. Pérez-Fuentes, Mechanisms of encapsulation of bacteria in self-healing concrete, Dyna, 86 (2019), 17–22. https://doi.org/10.15446/dyna.v86n210.75343 doi: 10.15446/dyna.v86n210.75343
    [22] J. Xue, B. Briseghella, F. Huang, C. Nuti, H. Tabatabai, B. Chen, Review of ultra-high performance concrete and its application in bridge engineering, Constr. Build. Mater., 260 (2020), 119844. https://doi.org/10.1016/j.conbuildmat.2020.119844 doi: 10.1016/j.conbuildmat.2020.119844
    [23] L. Placidi, F. Dell'Isola, A. Kandalaft, R. Luciano, C. Majorana, A. Misra, A granular micromechanic-based model for Ultra High Performance Fiber-Reinforced Concrete (UHP FRC), Int. J. Solids Struct., 297 (2024), 112844. https://doi.org/10.1016/j.ijsolstr.2024.112844 doi: 10.1016/j.ijsolstr.2024.112844
    [24] V. Nguyen-Van, B. Panda, G. Zhang, H. Nguyen-Xuan, P. Tran, Digital design computing and modelling for 3-D concrete printing, Autom. Constr., 123 (2021), 103529. https://doi.org/10.1016/j.autcon.2020.103529 doi: 10.1016/j.autcon.2020.103529
    [25] A. Kezmane, B. Chiaia, O. Kumpyak, V. Maksimov, L. Placidi, 3D modelling of reinforced concrete slab with yielding supports subject to impact load, Eur. J. Environ. Civ. Eng., 21 (2017), 988–1025. https://doi.org/10.1080/19648189.2016.1194330 doi: 10.1080/19648189.2016.1194330
    [26] A. Casalotti, F. D'Annibale, G. Rosi, Optimization of an architected composite with tailored graded properties, Z. Angew. Math. Phys., 75 (2014), 126. https://doi.org/10.1007/s00033-024-02255-2 doi: 10.1007/s00033-024-02255-2
    [27] I. Giorgio, A. Ciallella, D. Scerrato, A study about the impact of the topological arrangement of fibers on fiber-reinforced composites: Some guidelines aiming at the development of new ultra-stiff and ultra-soft metamaterials, Int. J. Solids Struct., 203 (2020), 73–83. https://doi.org/10.1016/j.ijsolstr.2020.07.016 doi: 10.1016/j.ijsolstr.2020.07.016
    [28] A. Ciallella, F. D'Annibale, D. Del Vescovo, I. Giorgio, Deformation patterns in a second-gradient lattice annular plate composed of "spira mirabilis" fibers, Continuum Mech. Thermodyn., 35 (2023), 1561–1580. https://doi.org/10.1007/s00161-022-01169-6 doi: 10.1007/s00161-022-01169-6
    [29] T. Wangler, N. Roussel, F. P. Bos, T. A. M. Salet, R. J. Flatt, Digital concrete: A review, Cem. Concr. Res., 123 (2019), 105780. https://doi.org/10.1016/j.cemconres.2019.105780 doi: 10.1016/j.cemconres.2019.105780
    [30] N. Rezaei, E. Barchiesi, D. Timofeev, C. A. Tran, A. Misra, L. Placidi, Solution of a paradox related to the rigid bar pull-out problem in standard elasticity, Mech. Res. Commun., 126 (2022), 104015. https://doi.org/10.1016/j.mechrescom.2022.104015 doi: 10.1016/j.mechrescom.2022.104015
    [31] M. F. Funari, S. Spadea, F. Fabbrocino, R. Luciano, A moving interface finite element formulation to predict dynamic edge debonding in FRP-strengthened concrete beams in service conditions, Fibers, 8 (2020), 42. https://doi.org/10.3390/fib8060042 doi: 10.3390/fib8060042
    [32] W. Pietraszkiewicz, V. A. Eremeyev, On vectorially parameterized natural strain measures of the non-linear Cosserat continuum, Int. J. Solids Struct., 46 (2009), 2477–2480. https://doi.org/10.1016/j.ijsolstr.2009.01.030 doi: 10.1016/j.ijsolstr.2009.01.030
    [33] G. La Valle, A new deformation measure for the nonlinear micropolar continuum, Z. Angew. Math. Phys., 73 (2022), 78. https://doi.org/10.1007/s00033-022-01715-x doi: 10.1007/s00033-022-01715-x
    [34] I. Giorgio, A. Misra, L. Placidi, Geometrically nonlinear Cosserat elasticity with chiral effects based upon granular micromechanics, in Sixty Shades of Generalized Continua, (eds. H. Altenbach, A. Berezovski, F. Dell'Isola and A. Porubov), Springer, 170 (2023), 273–292. https://doi.org/10.1007/978-3-031-26186-2_17
    [35] I. Giorgio, M. De Angelo, E. Turco, A. Misra, A Biot–Cosserat two-dimensional elastic nonlinear model for a micromorphic medium, Continuum Mech. Thermodyn., 32 (2020), 1357–1369. https://doi.org/10.1007/s00161-019-00848-1 doi: 10.1007/s00161-019-00848-1
    [36] E. Turco, F. Dell'Isola, A. Misra, A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations, Int. J. Numer. Anal. Methods Geomech., 43 (2019), 1051–1079. https://doi.org/10.1002/nag.2915 doi: 10.1002/nag.2915
    [37] E. Turco, Forecasting nonlinear vibrations of patches of granular materials by elastic interactions between spheres, Mech. Res. Commun., 122 (2022), 103879. https://doi.org/10.1016/j.mechrescom.2022.103879 doi: 10.1016/j.mechrescom.2022.103879
    [38] D. Scerrato, I. Giorgio, A. Della Corte, A. Madeo, N. E. Dowling, F. Darve, Towards the design of an enriched concrete with enhanced dissipation performances, Cem. Concr. Res., 84 (2016), 48–61. https://doi.org/10.1016/j.cemconres.2016.03.002 doi: 10.1016/j.cemconres.2016.03.002
    [39] A. Scrofani, E. Barchiesi, B. Chiaia, A. Misra, L. Placidi, Fluid diffusion related aging effect in a concrete dam modeled as a Timoshenko beam, Math. Mech. Complex Syst., 11 (2023), 313–334. https://doi.org/10.2140/memocs.2023.11.313 doi: 10.2140/memocs.2023.11.313
    [40] I. Giorgio, D. Scerrato, Multi-scale concrete model with rate-dependent internal friction, Eur. J. Environ. Civ. Eng., 21 (2017), 821–839. https://doi.org/10.1080/19648189.2016.1144539 doi: 10.1080/19648189.2016.1144539
    [41] G. Jouan, P. Kotronis, F. Collin, Using a second gradient model to simulate the behaviour of concrete structural elements, Finite Elem. Anal. Des., 90 (2014), 50–60. https://doi.org/10.1016/j.finel.2014.06.002 doi: 10.1016/j.finel.2014.06.002
    [42] P. Germain, The method of virtual power in the mechanics of continuous media, Ⅰ: Second-gradient theory, Math. Mech. Complex Syst., 8 (2020), 153–190. https://doi.org/10.2140/memocs.2020.8.153 doi: 10.2140/memocs.2020.8.153
    [43] G. La Valle, B. E. Abali, G. Falsone, C. Soize, Sensitivity of a homogeneous and isotropic second-gradient continuum model for particle-based materials with respect to uncertainties, ZAMM Z. Angew. Math. Mech., 103 (2023), e202300068. https://doi.org/10.1002/zamm.202300068 doi: 10.1002/zamm.202300068
    [44] F. Dell'Isola, S. R. Eugster, R. Fedele, P. Seppecher, Second-gradient continua: From Lagrangian to Eulerian and back, Math. Mech. Solids, 27 (2022), 2715–2750. https://doi.org/10.1177/10812865221078822 doi: 10.1177/10812865221078822
    [45] F. Dell'Isola, U. Andreaus, L. Placidi, At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola, Math. Mech. Solids, 20 (2015), 887–928. https://doi.org/10.1177/1081286513509811 doi: 10.1177/1081286513509811
    [46] A. Berezovski, I. Giorgio, A. D. Corte, Interfaces in micromorphic materials: Wave transmission and reflection with numerical simulations, Math. Mech. Solids, 21 (2016), 37–51. https://doi.org/10.1177/1081286515572244 doi: 10.1177/1081286515572244
    [47] M. Golaszewski, R. Grygoruk, I. Giorgio, M. Laudato, F. D. Cosmo, Metamaterials with relative displacements in their microstructure: Technological challenges in 3D printing, experiments and numerical predictions, Continuum Mech. Thermodyn., 31 (2019), 1015–1034. https://doi.org/10.1007/s00161-018-0692-0 doi: 10.1007/s00161-018-0692-0
    [48] A. Ciallella, I. Giorgio, S. R. Eugster, N. L. Rizzi, F. Dell'Isola, Generalized beam model for the analysis of wave propagation with a symmetric pattern of deformation in planar pantographic sheets, Wave Motion, 113 (2022), 102986. https://doi.org/10.1016/j.wavemoti.2022.102986 doi: 10.1016/j.wavemoti.2022.102986
    [49] O. Szlachetka, I. Giorgio, Heat conduction in multi-component step-wise FGMs, Continuum Mech. Thermodyn., (2024), 1–19. https://doi.org/10.1007/s00161-024-01296-2 doi: 10.1007/s00161-024-01296-2
    [50] V. A. Eremeyev, W. Pietraszkiewicz, Material symmetry group and constitutive equations of micropolar anisotropic elastic solids, Math. Mech. Solids, 21 (2016), 210–221. https://doi.org/10.1177/1081286515582862 doi: 10.1177/1081286515582862
    [51] G. La Valle, S. Massoumi, A new deformation measure for micropolar plates subjected to in-plane loads, Continuum Mech. Thermodyn., 34 (2022), 243–257. https://doi.org/10.1007/s00161-021-01055-7 doi: 10.1007/s00161-021-01055-7
    [52] I. Giorgio, F. Dell'Isola, A. Misrav, Chirality in 2D Cosserat media related to stretch-micro-rotation coupling with links to granular micromechanics, Int. J. Solids Struct., 202 (2020), 28–38. https://doi.org/10.1016/j.ijsolstr.2020.06.005 doi: 10.1016/j.ijsolstr.2020.06.005
    [53] I. Giorgio, F. Hild, E. Gerami, F. Dell'Isola, A. Misra, Experimental verification of 2D Cosserat chirality with stretch-micro-rotation coupling in orthotropic metamaterials with granular motif, Mech. Res. Commun., 126 (2022), 104020. https://doi.org/10.1016/j.mechrescom.2022.104020 doi: 10.1016/j.mechrescom.2022.104020
    [54] G. La Valle, C. Soize, A higher-order nonlocal elasticity continuum model for deterministic and stochastic particle-based materials, Z. Angew. Math. Phys., 75 (2024). https://doi.org/10.1007/s00033-024-02196-w doi: 10.1007/s00033-024-02196-w
    [55] A. Madeo, F. Dell'Isola, F. Darve, A continuum model for deformable, second gradient porous media partially saturated with compressible fluids, J. Mech. Phys. Solids, 61 (2013), 2196–2211. https://doi.org/10.1016/j.jmps.2013.06.009 doi: 10.1016/j.jmps.2013.06.009
    [56] F. Dell'Isola, A. Madeo, P. Seppecher, Boundary conditions at fluid-permeable interfaces in porous media: A variational approach, Int. J. Solids Struct., 46 (2009), 3150–3164. https://doi.org/10.1016/j.ijsolstr.2009.04.008 doi: 10.1016/j.ijsolstr.2009.04.008
    [57] I. Giorgio, A variational formulation for one-dimensional linear thermoviscoelasticity, Math. Mech. Complex Syst., 9 (2022), 397–412. https://doi.org/10.2140/memocs.2021.9.397 doi: 10.2140/memocs.2021.9.397
    [58] E. Barchiesi, N. Hamila, Maximum mechano-damage power release-based phase-field modeling of mass diffusion in damaging deformable solids, Z. Angew. Math. Phys., 73 (2022). https://doi.org/10.1007/s00033-021-01668-7 doi: 10.1007/s00033-021-01668-7
    [59] A. Ramírez-Torres, R. Penta, A. Grillo, Effective properties of fractional viscoelastic composites via two-scale asymptotic homogenization, Math. Methods Appl. Sci., 46 (2023), 16500–16520. https://doi.org/10.1002/mma.9457 doi: 10.1002/mma.9457
    [60] L. Placidi, E. Barchiesi, Energy approach to brittle fracture in strain-gradient modelling, Proc. R. Soc. A, 474 (2018), 20170878. https://doi.org/10.1098/rspa.2017.0878 doi: 10.1098/rspa.2017.0878
    [61] F. Fabbrocino, M. F. Funari, F. Greco, P. Lonetti, R. Luciano, R. Penna, Dynamic crack growth based on moving mesh method, Composites, Part B, 174 (2019), 107053. https://doi.org/10.1016/j.compositesb.2019.107053 doi: 10.1016/j.compositesb.2019.107053
    [62] C. Comi, R. Fedele, U. Perego, A chemo-thermo-damage model for the analysis of concrete dams affected by alkali-silica reaction, Mech. Mater., 41 (2009), 210–230. https://doi.org/10.1016/j.mechmat.2008.10.010 doi: 10.1016/j.mechmat.2008.10.010
    [63] A. Bilotta, A. Morassi, E. Turco, Simple convolutional neural networks for the damage identification in composite steel-concrete beams, in International Conference on Experimental Vibration Analysis for Civil Engineering Structures, Springer, 433 (2023), 422–431. https://doi.org/10.1007/978-3-031-39117-0_43
    [64] L. Placidi, A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model, Continuum Mech. Thermodyn., 28 (2016), 119–137. https://doi.org/10.1007/s00161-014-0405-2 doi: 10.1007/s00161-014-0405-2
    [65] L. Placidi, E. Barchiesi, A. Misra, A strain gradient variational approach to damage: A comparison with damage gradient models and numerical results, Math. Mech. Complex Syst., 6 (2018), 77–100. https://doi.org/10.2140/memocs.2018.6.77 doi: 10.2140/memocs.2018.6.77
    [66] L. Placidi, A. Misra, E. Barchiesi, Simulation results for damage with evolving microstructure and growing strain gradient moduli, Continuum Mech. Thermodyn., 31 (2019), 1143–1163. https://doi.org/10.1007/s00161-018-0693-z doi: 10.1007/s00161-018-0693-z
    [67] R. Luciano, A. Caporale, H. Darban, C. Bartolomeo, Variational approaches for bending and buckling of non-local stress-driven Timoshenko nano-beams for smart materials, Mech. Res. Commun., 103 (2020), 103470. https://doi.org/10.1016/j.mechrescom.2019.103470 doi: 10.1016/j.mechrescom.2019.103470
    [68] Ciallella, A and Pasquali, D and Gołaszewski, M and D'Annibale, F and Giorgio, I, A rate-independent internal friction to describe the hysteretic behavior of pantographic structures under cyclic loads, Mech. Res. Commun., 116 (2021), 103761.
    [69] A. Ciallella, D. Pasquali, M. Gołaszewski, F. D'Annibale, I. Giorgio, Shear rupture mechanism and dissipation phenomena in bias-extension test of pantographic sheets: Numerical modeling and experiments, Math. Mech. Solids, 27 (2022), 2170–2188. https://doi.org/10.1016/j.mechrescom.2021.103761 doi: 10.1016/j.mechrescom.2021.103761
    [70] I. Giorgio, L. Placidi, A variational formulation for three-dimensional linear thermoelasticity with 'thermal inertia', Meccanica, (2024). https://doi.org/10.1007/s11012-024-01796-0 doi: 10.1007/s11012-024-01796-0
    [71] A. Battista, L. Rosa, R. Dell'Erba, L. Greco, Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena, Math. Mech. Solids, 22 (2017), 2120–2134. https://doi.org/10.1177/1081286516657889 doi: 10.1177/1081286516657889
    [72] D. Scerrato, I. Giorgio, A. Madeo, A. Limam, F. Darve, A simple non-linear model for internal friction in modified concrete, Int. J. Eng. Sci., 80 (2014), 136–152. https://doi.org/10.1016/j.ijengsci.2014.02.021 doi: 10.1016/j.ijengsci.2014.02.021
    [73] D. Scerrato, I. Giorgio, A. Della Corte, A. Madeo, A. Limam, A micro-structural model for dissipation phenomena in the concrete, Int. J. Numer. Anal. Methods Geomech., 39 (2015), 2037–2052. https://doi.org/10.1002/nag.2394 doi: 10.1002/nag.2394
    [74] I. Giorgio, M. Spagnuolo, L. Greco, F. D'Annibale, A. Cazzani, A variational approach to address the problem of planar nonlinear beams, in Comprehensive Mechanics of Materials, (eds. V. Silberschmidt), Elsevier, 1 (2024), 67–97. https://doi.org/10.1016/B978-0-323-90646-3.00027-7
    [75] V. Pensée, D. Kondo, L. Dormieux, Micromechanical analysis of anisotropic damage in brittle materials, J. Eng. Mech., 128 (2002), 889–897. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:8(889) doi: 10.1061/(ASCE)0733-9399(2002)128:8(889)
    [76] B. R. Raveendra, G. S. Benipal, A. K. Singh, Constitutive modelling of concrete: An overview, Asian J. Civ. Eng., 6 (2005), 211–214.
    [77] A. M. Bersani, P. Caressa, F. Dell'Isola, Approximation of dissipative systems by elastic chains: Numerical evidence, Math. Mech. Solids, 28 (2023), 501–520. https://doi.org/10.1177/10812865221081851 doi: 10.1177/10812865221081851
    [78] A. M. Bersani, P. Caressa, A. Ciallella, Numerical evidence for the approximation of dissipative systems by gyroscopically coupled oscillator chains, Math. Mech. Complex Syst., 10 (2022), 265–278. https://doi.org/10.2140/memocs.2022.10.265 doi: 10.2140/memocs.2022.10.265
    [79] A. Ciallella, D. Scerrato, M. Spagnuolo, I. Giorgio, A continuum model based on Rayleigh dissipation functions to describe a Coulomb-type constitutive law for internal friction in woven fabrics, Z. Angew. Math. Phys., 73 (2022). https://doi.org/10.1007/s00033-022-01845-2 doi: 10.1007/s00033-022-01845-2
    [80] L. Greco, M. Cuomo, L. Contrafatto, Two new triangular G1-conforming finite elements with cubic edge rotation for the analysis of Kirchhoff plates, Comput. Methods Appl. Mech. Eng., 356 (2019), 354–386. https://doi.org/10.1016/j.cma.2019.07.026 doi: 10.1016/j.cma.2019.07.026
    [81] L. Greco, D. Castello, M. Cuomo, An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods, Math. Mech. Solids, 29 (2024), 645–685. https://doi.org/10.1177/10812865231204972 doi: 10.1177/10812865231204972
    [82] I. Giorgio, A discrete formulation of Kirchhoff rods in large-motion dynamics, Math. Mech. Solids, 25 (2020), 1081–1100. https://doi.org/10.1177/1081286519900902 doi: 10.1177/1081286519900902
    [83] I. Giorgio, D. Del Vescovo, Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators, Math. Mecha. Complex Syst., 7 (2019), 159–174. https://doi.org/10.2140/memocs.2019.7.159 doi: 10.2140/memocs.2019.7.159
    [84] D. Baroudi, I. Giorgio, A. Battista, E. Turco, L. A. Igumnov, Nonlinear dynamics of uniformly loaded elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations, ZAMM Z. Angew. Math. Mech., 99 (2019), e201800121. https://doi.org/10.1002/zamm.201800121 doi: 10.1002/zamm.201800121
    [85] E. Turco, Discrete is it enough? The revival of Piola–Hencky keynotes to analyze three-dimensional Elastica, Continuum Mech. Thermodyn., 30 (2018), 1039–1057. https://doi.org/10.1007/s00161-018-0656-4 doi: 10.1007/s00161-018-0656-4
    [86] R. Fedele, G. Maier, B. Miller, Identification of elastic stiffness and local stresses in concrete dams by in situ tests and neural networks, Struct. Infrastruct. Eng., 1 (2005), 165–180. https://doi.org/10.1080/15732470500030513 doi: 10.1080/15732470500030513
    [87] R. Fedele, G. Maier, B. Miller, Image correlation-based identification of fracture parameters for structural adhesives, Tech. Mech. Eur. J. Eng. Mech., 32 (2012), 195–204.
    [88] B. E. Abali, C. C. Wu, W. H. Müller, An energy-based method to determine material constants in nonlinear rheology with applications, Continuum Mech. Thermodyn., 28 (2016), 1221–1246. https://doi.org/10.1007/s00161-015-0472-z doi: 10.1007/s00161-015-0472-z
    [89] M. De Angelo, L. Placidi, N. Nejadsadeghi, A. Misra, Non-standard Timoshenko beam model for chiral metamaterial: Identification of stiffness parameters, Mech. Res. Commun., 103 (2020), 103462. https://doi.org/10.1016/j.mechrescom.2019.103462 doi: 10.1016/j.mechrescom.2019.103462
    [90] A. Ciallella, G. La Valle, A. Vintache, B. Smaniotto, F. Hild, Deformation mode in 3-point flexure on pantographic block, Int. J. Solids Struct., 265 (2023), 112129. https://doi.org/10.1016/j.ijsolstr.2023.112129 doi: 10.1016/j.ijsolstr.2023.112129
    [91] M. De Angelo, E. Barchiesi, I. Giorgio, B. E. Abali, Numerical identification of constitutive parameters in reduced-order bi-dimensional models for pantographic structures: Application to out-of-plane buckling, Arch. Appl. Mech., 89 (2019), 1333–1358. https://doi.org/10.1007/s00419-018-01506-9 doi: 10.1007/s00419-018-01506-9
    [92] N. Shekarchizadeh, B. E. Abali, E. Barchiesi, A. M. Bersani, Inverse analysis of metamaterials and parameter determination by means of an automatized optimization problem, ZAMM Z. Angew. Math. Mech., 101 (2021), e202000277. https://doi.org/10.1002/zamm.202000277 doi: 10.1002/zamm.202000277
    [93] I. Giorgio, P. Harrison, F. Dell'Isola, J. Alsayednoor, E. Turco, Wrinkling in engineering fabrics: A comparison between two different comprehensive modelling approaches, Proc. R. Soc. A, 474 (2018), 20180063. https://doi.org/10.1098/rspa.2018.0063 doi: 10.1098/rspa.2018.0063
    [94] R. Fedele, A. Ciani, L. Galantucci, V. Casalegno, A. Ventrella, M. Ferraris, Characterization of innovative CFC/Cu joints by full-field measurements and finite elements, Mater. Sci. Eng. A, 595 (2014), 306–317. https://doi.org/10.1016/j.msea.2013.12.015 doi: 10.1016/j.msea.2013.12.015
    [95] N. Cefis, R. Fedele, M. G. Beghi, An integrated methodology to estimate the effective elastic parameters of amorphous TiO2 nanostructured films, combining SEM images, finite element simulations and homogenization techniques, Mech. Res. Commun., 131 (2023), 104153. https://doi.org/10.1016/j.mechrescom.2023.104153 doi: 10.1016/j.mechrescom.2023.104153
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1156) PDF downloads(67) Cited by(1)

Figures and Tables

Figures(10)  /  Tables(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog