On the indentification of nonlinear constitutive laws from indentation tests
On the indentification of nonlinear constitutive laws from indentation tests
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论压痕试验非线性本构律的辨识
DOI：
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发表时间：
1999
期刊：
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通讯作者：
N. Tardieu
中科院分区：
文献类型：
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作者：
A. Constantinescu;N. Tardieu
本文讨论了由压痕试验识别非线性本构关系参数的问题。用伴随态方法广泛地处理了没有加工硬化的标准广义材料的情况。这提供了一个通用的框架来执行涉及接触条件和非线性材料行为的优化。给出了一个数值辨识，证明了该方法的准确性和鲁棒性。压痕试验包括在材料样品上压上冲头。它最初用于评估金属的硬度，现在被认为是确定材料机械特性的一种有效的非破坏性方法(Taljat等人，1998年)。本构定律应该从压痕曲线的知识中确定，该曲线表示施加在凸模上的载荷与穿透深度之间的关系。压痕曲线的力学解释并不像经典的牵引力曲线那样直接。这意味着压痕试验用于材料表征取决于鉴定的可靠性。大多数的识别策略都是基于半经验公式来描述特定的本构行为：弹性、完美塑性(Johnson，1985)、幂定律(Jayaraman et al.，1998)、：：只有少数研究从一般的角度提出了这个问题，即将识别定义为成本泛函的最小化(BUI，1994)。识别方法通常基于简单的试错法(Hasanov&Seyidmamedov，1995)。这在一定程度上是由于接触描述的数学复杂性，看起来与材料的本构行为无关。在线弹性的情况下，首次尝试从一般的观点来解决这个问题(Constantinescu&Tardieu，1995)。接触条件通过惩罚被正则化，因此问题用变分方程来描述，而不是像以前那样用变分不等式来描述。这使得经典的最优控制技术(Lions，1968)的应用成为可能，特别是伴随状态方法。然后利用梯度下降法对代价泛函进行极小化，解决了辨识问题。本文的目的是将该方法推广到无加工硬化的标准广义本构关系的参数识别中。本文提出的方法不像Constantinescu&Tardieu，1995那样基于接触条件的正则化，而是使用拉格朗日乘子。代价泛函的梯度是由一个直接问题和一个伴随问题的解计算出来的。通过对Maxwell粘弹性本构方程的数值算例，说明了该方法的准确性和鲁棒性。1版权所有1999年由ASME直接问题(P)让我们考虑一个轴对称体，其截面在其参考构形中占据具有光滑边界Ω的开放子集ΓR2(见图1)。边界被划分为三部分Γ=ΓD[ΓF[ΓC：施加位移的部分ΓD、自由表面ΓF和可能发生接触的表面ΓC。N和t表示边界Γ的法线和切线向量。为了简化表示和计算负担，采用了轴对称假设，并且不限制方法的通用性。这个问题将在小应变和旋转理论的范围内处理。这一假设的正确性将在后面讨论。让我们用u、ε和σ分别表示位移的矢量场和小应变和应力的张量场。在续集中考虑的问题是通过刚性冲头对车身Ω进行压痕，其轮廓以缺口为特征。接触被认为是无摩擦的。压痕实验是由垂直位移U或施加于冲头的力F驱动的。力F可以表示为接触压力的积分：
This paper addresses the identification of the parameters of a nonlinear constitutive law from indentation tests. The case of a standard generalized material without work hardening is extensively treated using the adjoint state method. This provides a general framework to perform optimization involving contact conditions and nonlinear material behaviour. A numerical identification is presented and proves the accuracy and the robustness of the method. INTRODUCTION The indentation test consists in pressing a punch on a material sample. It was initialy used to evaluate the hardness of metals and is now being considered as an efficient non destructive method for determining material mechanical characteristics (Taljat et al.,1998). The constitutive law should be identified from the knowledge of the indentation curve, representing the load applied on the punch versus the penetration depth. The mechanical interpretation of the indentation curve is not as straightforward as for the classical traction curve. This implies that the use of the indentation test for material characterization depends on the reliability of the identification. Most of the identification strategies are based on semiempirical formulas dedicated to a given constitutive behaviour : elasticity, perfect plasticity (Johnson,1985), power laws (Jayaraman et al., 1998), : : : Only a few studies present this problem from a general point of view, i.e. defining the identification as the minimization of a cost functional (Bui, 1994). The identification methods are generally based on simple trial & error techniques (Hasanov & Seyidmamedov , 1995). This is partly due to the mathematical complexity of the contact description, appearing independently of the constitutive behaviour of the material. A first attempt to solve the problem from a general point of view has been presented in the case of linear elasticity (Constantinescu & Tardieu, 1995). The contact conditions have been regularized by penalization, the problem was therefore described by variational equalities, and not variational inequalities as before. This enables the application of classical optimal control (Lions,1968) techniques, in particular the adjoint state method. The identification problem has been solved afterwards by the minimization of cost functional using a gradient descent method. The goal of this paper is to extend this method to the identification of the parameters of a standard generalized constitutive law without work hardening. The method presented in this paper is not based on the regularization of the contact conditions as in (Constantinescu & Tardieu, 1995), instead Lagrange multipliers are used. The gradient of the cost functional is computed from the solution of a direct and an adjoint problem. The accuracy and robustness of the method are illustrated through a numerical example for a Maxwell viscoelastic constitutive law. 1 Copyright 1999 by ASME THE DIRECT PROBLEM (P ) Let us consider an axisymmetric body, with its section occupying in its reference configuration an open subset Ω R2 with smooth boundary Γ (see Figure 1). The boundary is partioned in three partsΓ = ΓD[ΓF [ΓC : the partΓD where displacements are imposed, the free surface ΓF , and the surface ΓC where contact might occur. n andt denote the normal and tangent vector to the boundaryΓ. The axisymmetric hypothesis is taken in order to simplify the presentation and the computational burden and does not restrict the generality of the method. The problem will be treated within the theory of small strains and rotations. The validity of this hypothesis will be discussed later. Let us denote respectively by u, ε andσ the vector field of displacements and the tensor fields of small strains and stresses. The problem considered in the sequel is the indentation of the bodyΩ by a rigid punch whose profile is characterized by the gapg. The contact is considered without friction. An indentation experiment is driven either by the vertical displacement U or by the forceF applied to the punch. The forceF can be expressed as integral of the contact pressure: